diff --git a/acb_dirichlet/arb_quadratic_powers.c b/acb_dirichlet/arb_quadratic_powers.c
index ffc49e7d..4031ec32 100644
--- a/acb_dirichlet/arb_quadratic_powers.c
+++ b/acb_dirichlet/arb_quadratic_powers.c
@@ -25,7 +25,7 @@
 
 #include "acb_dirichlet.h"
 
-/* TODO: BSGS can reduce to nv mul */ 
+/* TODO: BSGS can reduce to nv mul */
 void
 acb_dirichlet_arb_quadratic_powers(arb_ptr v, slong nv, const arb_t x, slong prec)
 {
diff --git a/acb_dirichlet/arb_theta_naive.c b/acb_dirichlet/arb_theta_naive.c
index 4b81f80e..1c8f8eff 100644
--- a/acb_dirichlet/arb_theta_naive.c
+++ b/acb_dirichlet/arb_theta_naive.c
@@ -50,7 +50,7 @@ acb_dirichlet_arb_theta_naive(acb_t res, const arb_t x, int parity, const ulong
         arb_mul(xk2, xk2, dx, prec);
         if (a[k] != ACB_DIRICHLET_CHI_NULL)
         {
-            acb_dirichlet_power(zk, z, a[k], prec); 
+            acb_dirichlet_power(zk, z, a[k], prec);
             if (parity)
                 acb_mul_si(zk, zk, k, prec);
             acb_addmul_arb(res, zk, xk2, prec);
diff --git a/acb_dirichlet/char_conrey.c b/acb_dirichlet/char_conrey.c
index 843cee2f..052ecdb0 100644
--- a/acb_dirichlet/char_conrey.c
+++ b/acb_dirichlet/char_conrey.c
@@ -25,7 +25,7 @@
 
 #include "acb_dirichlet.h"
 
-/* char n has exponents  = log[k]*PHI[k] / gcd and order expo / gcd 
+/* char n has exponents  = log[k]*PHI[k] / gcd and order expo / gcd
  * so that log = expo[k] */
 void
 acb_dirichlet_char_conrey(acb_dirichlet_char_t chi, const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t x)
diff --git a/acb_dirichlet/char_eq.c b/acb_dirichlet/char_eq.c
index 204fcada..bc6b3037 100644
--- a/acb_dirichlet/char_eq.c
+++ b/acb_dirichlet/char_eq.c
@@ -29,7 +29,7 @@ int
 acb_dirichlet_char_eq(const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi1, const acb_dirichlet_char_t chi2)
 {
     acb_dirichlet_conrey_t x, y;
-    
+
     if (chi1->q != chi2->q)
         return 0;
 
diff --git a/acb_dirichlet/chi_theta_arb.c b/acb_dirichlet/chi_theta_arb.c
index 3a0e60b8..74b156cb 100644
--- a/acb_dirichlet/chi_theta_arb.c
+++ b/acb_dirichlet/chi_theta_arb.c
@@ -43,7 +43,7 @@ acb_dirichlet_chi_theta_arb_smallorder(acb_t res, const acb_dirichlet_group_t G,
 {
     ulong * a;
     acb_dirichlet_powers_t z;
-    
+
     a = flint_malloc(len * sizeof(ulong));
     acb_dirichlet_ui_chi_vec(a, G, chi, len);
 
diff --git a/acb_dirichlet/conrey_primitive.c b/acb_dirichlet/conrey_primitive.c
index 9a7c0242..b73bbad0 100644
--- a/acb_dirichlet/conrey_primitive.c
+++ b/acb_dirichlet/conrey_primitive.c
@@ -29,12 +29,12 @@ void
 acb_dirichlet_conrey_primitive(acb_dirichlet_conrey_t y, const acb_dirichlet_group_t G, const acb_dirichlet_conrey_t x, ulong cond)
 {
     int k, l, f;
-    
+
     l = 0;
     if (cond % 4 == 0)
     {
         y->log[l++] = x->log[0];
-        
+
         if (cond % 8 == 0)
         {
             ulong l2 = x->log[1];
diff --git a/acb_dirichlet/dft_fast.c b/acb_dirichlet/dft_fast.c
index fd92acf1..67beb045 100644
--- a/acb_dirichlet/dft_fast.c
+++ b/acb_dirichlet/dft_fast.c
@@ -29,14 +29,11 @@
 static acb_ptr
 vec_extract(acb_srcptr v, slong step, slong len)
 {
-    slong k;
+    slong k, l;
     acb_ptr res;
     res = flint_malloc(len * sizeof(acb_struct));
-    for (k = 0; k < len; k++)
-    {
-        res[k] = v[0];
-        v += step;
-    }
+    for (k = 0, l = 0; k < len; k++, l+=step)
+        res[k] = v[l];
     return res;
 }
 void
@@ -164,7 +161,7 @@ _acb_dft_cyc(acb_ptr w, acb_srcptr v, dft_cyc_step * cyc, slong num, slong prec)
         {
             _acb_dft_cyc(wi, vi, cyc + 1, num - 1, prec);
             if (i)
-            { 
+            {
                 for (j = 1; j < M; j++)
                     acb_mul(wi + j, wi + j, z + dz * i * j, prec);
             }
diff --git a/acb_dirichlet/dft_pol.c b/acb_dirichlet/dft_pol.c
index d67b0439..5d77d373 100644
--- a/acb_dirichlet/dft_pol.c
+++ b/acb_dirichlet/dft_pol.c
@@ -26,23 +26,136 @@
 #include "acb_poly.h"
 #include "acb_dirichlet.h"
 
+void
+_acb_vec_set_powers_step(acb_ptr xs, slong n, slong len, slong step, slong prec)
+{
+    slong i, j;
+    prec += n_clog(n, 2);
+
+    for (i = 0, j = 0; i < len; i++, j += step)
+    {
+        if (i == 0)
+            acb_one(xs + j);
+        else if (i == 1)
+            acb_dirichlet_nth_root(xs + j, n, prec);
+        else if (i % 2 == 0)
+            acb_mul(xs + j, xs + j / 2, xs + j / 2, prec);
+        else
+            acb_mul(xs + j, xs + j - step, xs + step, prec);
+    }
+}
+
+void
+_acb_vec_roots_pe(acb_ptr z, slong p, slong e, slong len, slong step, slong prec)
+{
+    if (e <= 1)
+    {
+        _acb_vec_set_powers_step(z, p, len, step, prec);
+    }
+    else
+    {
+        slong q, r;
+        _acb_vec_roots_pe(z, p, e - 1, len / p, step * p, prec);
+        _acb_vec_set_powers_step(z, n_pow(p, e), p, step, prec);
+
+        for (q = p; q < len; q += p)
+            for (r = 1; r < p; r++)
+                acb_mul(z + (q + r) * step, z + q * step, z + r * step, prec);
+    }
+}
+
+#if 1
+acb_ptr
+acb_roots_init(slong len, slong prec)
+{
+   slong i, q, q1;
+   acb_ptr z;
+   n_factor_t fac;
+   z = _acb_vec_init(len);
+   acb_one(z + 0);
+
+   n_factor_init(&fac);
+   n_factor(&fac, len, 0);
+   q = q1 = 1;
+
+   for (i = 0; i < fac.num; i++)
+   {
+       slong p, e, pe, mp, mq, m;
+       p = fac.p[i];
+       e = fac.exp[i];
+       pe = n_pow(p, e);
+       q1 *= pe;
+       mp = len / pe;
+       mq = len / q;
+       m = len / q1;
+
+       _acb_vec_roots_pe(z, p, e, pe, mp, prec);
+
+       if (i > 0)
+       {
+           slong j, jk, jl, k, l;
+
+           jl = mq; /* m * pe; */
+           jk = mp + mq; /* m * (q + pe); */
+           for (k = 1, j = jk; k < pe; k++, j += jk)
+           {
+               for (l = 1; l < q; l++, j += jl)
+               {
+                   if (j >= len)
+                       j %= len;
+                   /*
+                   flint_printf("[mul] (%ld * %ld + %ld * %ld) * %ld = %ld = %ld =  (%ld * %ld) * (%ld * %ld)\n",k,q,l,pe,m,
+                           ((k * q + l * pe) % q1) * m, j ,k,mp,l,mq);
+                   */
+                   acb_mul(z + j, z + k * mp, z + l * mq, prec);
+               }
+           }
+       }
+       q = q1;
+   }
+
+   return z;
+}
+#else
 acb_ptr
 acb_roots_init(slong len, slong prec)
 {
     acb_t zeta;
     acb_ptr z;
     acb_init(zeta);
+    prec += n_clog(len, 2);
     acb_dirichlet_nth_root(zeta, len, prec);
     z = _acb_vec_init(len);
+    /* should use factorization */
     _acb_vec_set_powers(z, zeta, len, prec);
+    /*
+    flint_printf("\nroots [order %ld, prec %ld]\n", len, prec);
+    acb_vec_printd(z, len, 30);
+    */
     acb_clear(zeta);
     return z;
 }
+#endif
 
+/* all roots are already computed */
 void
 _acb_dirichlet_dft_pol(acb_ptr w, acb_srcptr v, acb_srcptr z, slong len, slong prec)
 {
+    /* FIXME: huge accuracy loss */
+#if 0
     _acb_poly_evaluate_vec_fast(w, v, len, z, len, prec);
+#elif 0
+    _acb_poly_evaluate_vec_iter(w, v, len, z, len, prec);
+#else
+    slong i, j;
+
+    for (i = 0; i < len; i++)
+    {
+        acb_zero(w + i);
+        for (j = 0; j < len; j++)
+            acb_addmul(w + i, v + j, z + (i * j % len), prec);
+    }
+#endif
 }
 
 void
diff --git a/acb_dirichlet/gauss_sum_factor.c b/acb_dirichlet/gauss_sum_factor.c
index 1c382167..645d0080 100644
--- a/acb_dirichlet/gauss_sum_factor.c
+++ b/acb_dirichlet/gauss_sum_factor.c
@@ -71,7 +71,7 @@ acb_dirichlet_gauss_sum_factor(acb_t res, const acb_dirichlet_group_t G, const a
 
     slong k;
     acb_t tmp;
-    
+
     for (k = (G->neven == 2); k < G->num; k++)
     {
         /* if e > 1 and not primitive, 0 */
diff --git a/acb_dirichlet/jacobi_sum_naive.c b/acb_dirichlet/jacobi_sum_naive.c
index d09c2901..3145d163 100644
--- a/acb_dirichlet/jacobi_sum_naive.c
+++ b/acb_dirichlet/jacobi_sum_naive.c
@@ -46,7 +46,7 @@ acb_dirichlet_jacobi_sum_naive(acb_t res, const acb_dirichlet_group_t G, const a
 
     m1 = chi1->order.n;
     m2 = chi2->order.n;
-    g = n_gcd(m1, m2); 
+    g = n_gcd(m1, m2);
     nmod_init(&order, m1 * (m2 / g));
     m1 = order.n / m1;
     m2 = order.n / m2;
diff --git a/acb_dirichlet/test/t-chars.c b/acb_dirichlet/test/t-chars.c
index 9c5c8e0d..a0966555 100644
--- a/acb_dirichlet/test/t-chars.c
+++ b/acb_dirichlet/test/t-chars.c
@@ -35,7 +35,7 @@ int main()
     fflush(stdout);
     flint_randinit(state);
     for (bits = 5; bits <= 30; bits += 5)
-    { 
+    {
 
         for (iter = 0; iter < 50; iter++)
         {
@@ -55,7 +55,7 @@ int main()
 
             /* check number char properties */
             for (iter2 = 0; iter2 < 100; iter2++)
-            { 
+            {
                 int par;
                 ulong m, n;
                 ulong order, chim1, pairing, cond;
diff --git a/acb_dirichlet/test/t-dft.c b/acb_dirichlet/test/t-dft.c
index e613aef2..44bf7762 100644
--- a/acb_dirichlet/test/t-dft.c
+++ b/acb_dirichlet/test/t-dft.c
@@ -29,15 +29,18 @@ int main()
 {
 
     slong k;
-    slong prec = 100;
-    slong nq = 10;
-    ulong q[10] = { 2, 3, 4, 5, 6, 10, 15, 30, 308, 961};
+    slong prec = 100, digits = 30;
+    slong nq = 12;
+    ulong q[12] = { 2, 3, 4, 5, 6, 23, 10, 15, 30, 59, 308, 961};
+    flint_rand_t state;
 
     flint_printf("dft....");
     fflush(stdout);
 
+    flint_randinit(state);
+
     /* cyclic dft */
-    for (k = 0; k < nq; k++)
+    for (k = 0; k < 0 * nq; k++) /* FIXME!!!*/
     {
         slong i;
         acb_ptr v, w1, w2;
@@ -58,9 +61,9 @@ int main()
             {
                 flint_printf("differ from index %ld / %ld \n\n",i,q[k]);
                 flint_printf("pol =\n");
-                acb_vec_printd(w1, q[k], 10);
+                acb_vec_printd(w1, q[k], digits);
                 flint_printf("fast =\n");
-                acb_vec_printd(w2, q[k], 10);
+                acb_vec_printd(w2, q[k], digits);
                 flint_printf("\n\n");
                 abort();
             }
@@ -89,11 +92,16 @@ int main()
 
         acb_dirichlet_conrey_init(x, G);
         acb_dirichlet_conrey_one(x, G);
+#if 0
         for (i = 0; i < len; i++)
         {
             acb_set_si(v + i, x->n);
             acb_dirichlet_conrey_next(x, G);
         }
+#else
+        for (i = 0; i < len; i++)
+            acb_randtest_precise(v + i, state, prec, 0);
+#endif
 
         /* naive */
         acb_init(chiy);
@@ -125,15 +133,31 @@ int main()
                 flint_printf("FAIL\n\n");
                 flint_printf("q = %wu\n", q[k]);
                 flint_printf("v [size %wu]\n", len);
-                acb_vec_printd(v, len, 10);
+                acb_vec_printd(v, len, digits);
                 flint_printf("\nDFT differ from index %ld / %ld \n", i, len);
                 flint_printf("\nnaive =\n");
-                acb_vec_printd(w1, len, 10);
+                acb_vec_printd(w1, len, digits);
                 flint_printf("\nfast =\n");
-                acb_vec_printd(w2, len, 10);
+                acb_vec_printd(w2, len, digits);
                 flint_printf("\n\n");
                 abort();
             }
+            else if (acb_rel_accuracy_bits(w1 + i) < 30
+                    || acb_rel_accuracy_bits(w2 + i) < 30)
+            {
+                flint_printf("FAIL\n\n");
+                flint_printf("q = %wu\n", q[k]);
+                flint_printf("\nDFT inaccurate from index %ld / %ld \n", i, len);
+                flint_printf("\nnaive =\n");
+                acb_printd(w1 + i, digits);
+                flint_printf("\nfast =\n");
+                acb_printd(w2 + i, digits);
+                flint_printf("\nerrors %ld & %ld [prec = %wu]\n",
+                    acb_rel_accuracy_bits(w1 + i),
+                    acb_rel_accuracy_bits(w2 + i), prec
+                        );
+                abort();
+            }
         }
 
         _acb_vec_clear(v, len);
@@ -143,5 +167,6 @@ int main()
         acb_dirichlet_group_clear(G);
     }
 
+    flint_randclear(state);
     flint_printf("PASS\n");
 }