LMFDB - Embedded newform 256.4.b.f.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,Mod(129,256)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.1044889615$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.4.b.f.129.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +6.00000i q^{5} +20.0000 q^{7} +23.0000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +6.00000i q^{5} +20.0000 q^{7} +23.0000 q^{9} -14.0000i q^{11} -54.0000i q^{13} -12.0000 q^{15} -66.0000 q^{17} +162.000i q^{19} +40.0000i q^{21} +172.000 q^{23} +89.0000 q^{25} +100.000i q^{27} +2.00000i q^{29} +128.000 q^{31} +28.0000 q^{33} +120.000i q^{35} +158.000i q^{37} +108.000 q^{39} -202.000 q^{41} +298.000i q^{43} +138.000i q^{45} +408.000 q^{47} +57.0000 q^{49} -132.000i q^{51} -690.000i q^{53} +84.0000 q^{55} -324.000 q^{57} +322.000i q^{59} +298.000i q^{61} +460.000 q^{63} +324.000 q^{65} +202.000i q^{67} +344.000i q^{69} -700.000 q^{71} +418.000 q^{73} +178.000i q^{75} -280.000i q^{77} -744.000 q^{79} +421.000 q^{81} -678.000i q^{83} -396.000i q^{85} -4.00000 q^{87} +82.0000 q^{89} -1080.00i q^{91} +256.000i q^{93} -972.000 q^{95} -1122.00 q^{97} -322.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 40 q^{7} + 46 q^{9}+O(q^{10}) $ 2 * q + 40 * q^7 + 46 * q^9	$ 2 q + 40 q^{7} + 46 q^{9} - 24 q^{15} - 132 q^{17} + 344 q^{23} + 178 q^{25} + 256 q^{31} + 56 q^{33} + 216 q^{39} - 404 q^{41} + 816 q^{47} + 114 q^{49} + 168 q^{55} - 648 q^{57} + 920 q^{63} + 648 q^{65} - 1400 q^{71} + 836 q^{73} - 1488 q^{79} + 842 q^{81} - 8 q^{87} + 164 q^{89} - 1944 q^{95} - 2244 q^{97}+O(q^{100}) $ 2 * q + 40 * q^7 + 46 * q^9 - 24 * q^15 - 132 * q^17 + 344 * q^23 + 178 * q^25 + 256 * q^31 + 56 * q^33 + 216 * q^39 - 404 * q^41 + 816 * q^47 + 114 * q^49 + 168 * q^55 - 648 * q^57 + 920 * q^63 + 648 * q^65 - 1400 * q^71 + 836 * q^73 - 1488 * q^79 + 842 * q^81 - 8 * q^87 + 164 * q^89 - 1944 * q^95 - 2244 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.00000i	0.384900i	0.981307	+	0.192450i	$0.0616434\pi$
$3$	2.00000i	0.384900i	−0.981307	+	0.192450i	$0.938357\pi$
$4$	0	0
$5$	6.00000i	0.536656i	0.963328	+	0.268328i	$0.0864711\pi$
$5$	6.00000i	0.536656i	−0.963328	+	0.268328i	$0.913529\pi$
$6$	0	0
$7$	20.0000	1.07990	0.539949	−	0.841698i	$-0.318443\pi$
$7$	20.0000	1.07990	0.539949	+	0.841698i	$0.318443\pi$
$8$	0	0
$9$	23.0000	0.851852
$10$	0	0
$11$	− 14.0000i	− 0.383742i	−0.981420	−	0.191871i	$-0.938545\pi$
$11$	− 14.0000i	− 0.383742i	0.981420	−	0.191871i	$-0.0614555\pi$
$12$	0	0
$13$	− 54.0000i	− 1.15207i	−0.817425	−	0.576035i	$-0.804599\pi$
$13$	− 54.0000i	− 1.15207i	0.817425	−	0.576035i	$-0.195401\pi$
$14$	0	0
$15$	−12.0000	−0.206559
$16$	0	0
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	0	0
$19$	162.000i	1.95607i	0.208438	+	0.978035i	$0.433162\pi$
$19$	162.000i	1.95607i	−0.208438	+	0.978035i	$0.566838\pi$
$20$	0	0
$21$	40.0000i	0.415653i
$22$	0	0
$23$	172.000	1.55933	0.779663	−	0.626200i	$-0.215391\pi$
$23$	172.000	1.55933	0.779663	+	0.626200i	$0.215391\pi$
$24$	0	0
$25$	89.0000	0.712000
$26$	0	0
$27$	100.000i	0.712778i
$28$	0	0
$29$	2.00000i	0.0128066i	0.999979	+	0.00640329i	$0.00203824\pi$
$29$	2.00000i	0.0128066i	−0.999979	+	0.00640329i	$0.997962\pi$
$30$	0	0
$31$	128.000	0.741596	0.370798	−	0.928714i	$-0.379084\pi$
$31$	128.000	0.741596	0.370798	+	0.928714i	$0.379084\pi$
$32$	0	0
$33$	28.0000	0.147702
$34$	0	0
$35$	120.000i	0.579534i
$36$	0	0
$37$	158.000i	0.702028i	0.936370	+	0.351014i	$0.114163\pi$
$37$	158.000i	0.702028i	−0.936370	+	0.351014i	$0.885837\pi$
$38$	0	0
$39$	108.000	0.443432
$40$	0	0
$41$	−202.000	−0.769441	−0.384721	−	0.923033i	$-0.625702\pi$
$41$	−202.000	−0.769441	−0.384721	+	0.923033i	$0.625702\pi$
$42$	0	0
$43$	298.000i	1.05685i	0.848980	+	0.528425i	$0.177217\pi$
$43$	298.000i	1.05685i	−0.848980	+	0.528425i	$0.822783\pi$
$44$	0	0
$45$	138.000i	0.457152i
$46$	0	0
$47$	408.000	1.26623	0.633116	−	0.774057i	$-0.281776\pi$
$47$	408.000	1.26623	0.633116	+	0.774057i	$0.281776\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	− 132.000i	− 0.362425i
$52$	0	0
$53$	− 690.000i	− 1.78828i	−0.447788	−	0.894140i	$-0.647788\pi$
$53$	− 690.000i	− 1.78828i	0.447788	−	0.894140i	$-0.352212\pi$
$54$	0	0
$55$	84.0000	0.205937
$56$	0	0
$57$	−324.000	−0.752892
$58$	0	0
$59$	322.000i	0.710523i	0.934767	+	0.355261i	$0.115608\pi$
$59$	322.000i	0.710523i	−0.934767	+	0.355261i	$0.884392\pi$
$60$	0	0
$61$	298.000i	0.625492i	0.949837	+	0.312746i	$0.101249\pi$
$61$	298.000i	0.625492i	−0.949837	+	0.312746i	$0.898751\pi$
$62$	0	0
$63$	460.000	0.919914
$64$	0	0
$65$	324.000	0.618265
$66$	0	0
$67$	202.000i	0.368332i	0.982895	+	0.184166i	$0.0589584\pi$
$67$	202.000i	0.368332i	−0.982895	+	0.184166i	$0.941042\pi$
$68$	0	0
$69$	344.000i	0.600185i
$70$	0	0
$71$	−700.000	−1.17007	−0.585033	−	0.811009i	$-0.698919\pi$
$71$	−700.000	−1.17007	−0.585033	+	0.811009i	$0.698919\pi$
$72$	0	0
$73$	418.000	0.670181	0.335090	−	0.942186i	$-0.391233\pi$
$73$	418.000	0.670181	0.335090	+	0.942186i	$0.391233\pi$
$74$	0	0
$75$	178.000i	0.274049i
$76$	0	0
$77$	− 280.000i	− 0.414402i
$78$	0	0
$79$	−744.000	−1.05958	−0.529788	−	0.848130i	$-0.677729\pi$
$79$	−744.000	−1.05958	−0.529788	+	0.848130i	$0.677729\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	− 678.000i	− 0.896629i	−0.893876	−	0.448314i	$-0.852025\pi$
$83$	− 678.000i	− 0.896629i	0.893876	−	0.448314i	$-0.147975\pi$
$84$	0	0
$85$	− 396.000i	− 0.505320i
$86$	0	0
$87$	−4.00000	−0.00492925
$88$	0	0
$89$	82.0000	0.0976627	0.0488314	−	0.998807i	$-0.484450\pi$
$89$	82.0000	0.0976627	0.0488314	+	0.998807i	$0.484450\pi$
$90$	0	0
$91$	− 1080.00i	− 1.24412i
$92$	0	0
$93$	256.000i	0.285440i
$94$	0	0
$95$	−972.000	−1.04974
$96$	0	0
$97$	−1122.00	−1.17445	−0.587226	−	0.809423i	$-0.699780\pi$
$97$	−1122.00	−1.17445	−0.587226	+	0.809423i	$0.699780\pi$
$98$	0	0
$99$	− 322.000i	− 0.326891i
$100$	0	0
$101$	1390.00i	1.36941i	0.728821	+	0.684704i	$0.240068\pi$
$101$	1390.00i	1.36941i	−0.728821	+	0.684704i	$0.759932\pi$
$102$	0	0
$103$	788.000	0.753825	0.376912	−	0.926249i	$-0.376986\pi$
$103$	788.000	0.753825	0.376912	+	0.926249i	$0.376986\pi$
$104$	0	0
$105$	−240.000	−0.223063
$106$	0	0
$107$	− 1614.00i	− 1.45824i	−0.684388	−	0.729118i	$-0.739930\pi$
$107$	− 1614.00i	− 1.45824i	0.684388	−	0.729118i	$-0.260070\pi$
$108$	0	0
$109$	− 2014.00i	− 1.76978i	−0.465798	−	0.884891i	$-0.654233\pi$
$109$	− 2014.00i	− 1.76978i	0.465798	−	0.884891i	$-0.345767\pi$
$110$	0	0
$111$	−316.000	−0.270211
$112$	0	0
$113$	−542.000	−0.451213	−0.225607	−	0.974219i	$-0.572436\pi$
$113$	−542.000	−0.451213	−0.225607	+	0.974219i	$0.572436\pi$
$114$	0	0
$115$	1032.00i	0.836822i
$116$	0	0
$117$	− 1242.00i	− 0.981393i
$118$	0	0
$119$	−1320.00	−1.01684
$120$	0	0
$121$	1135.00	0.852742
$122$	0	0
$123$	− 404.000i	− 0.296158i
$124$	0	0
$125$	1284.00i	0.918756i
$126$	0	0
$127$	−1712.00	−1.19618	−0.598092	−	0.801427i	$-0.704074\pi$
$127$	−1712.00	−1.19618	−0.598092	+	0.801427i	$0.704074\pi$
$128$	0	0
$129$	−596.000	−0.406782
$130$	0	0
$131$	− 2118.00i	− 1.41260i	−0.707913	−	0.706300i	$-0.750363\pi$
$131$	− 2118.00i	− 1.41260i	0.707913	−	0.706300i	$-0.249637\pi$
$132$	0	0
$133$	3240.00i	2.11236i
$134$	0	0
$135$	−600.000	−0.382517
$136$	0	0
$137$	486.000	0.303079	0.151539	−	0.988451i	$-0.451577\pi$
$137$	486.000	0.303079	0.151539	+	0.988451i	$0.451577\pi$
$138$	0	0
$139$	− 1286.00i	− 0.784727i	−0.919810	−	0.392364i	$-0.871657\pi$
$139$	− 1286.00i	− 0.784727i	0.919810	−	0.392364i	$-0.128343\pi$
$140$	0	0
$141$	816.000i	0.487373i
$142$	0	0
$143$	−756.000	−0.442097
$144$	0	0
$145$	−12.0000	−0.00687273
$146$	0	0
$147$	114.000i	0.0639630i
$148$	0	0
$149$	− 2666.00i	− 1.46582i	−0.680325	−	0.732910i	$-0.738162\pi$
$149$	− 2666.00i	− 1.46582i	0.680325	−	0.732910i	$-0.261838\pi$
$150$	0	0
$151$	172.000	0.0926964	0.0463482	−	0.998925i	$-0.485242\pi$
$151$	172.000	0.0926964	0.0463482	+	0.998925i	$0.485242\pi$
$152$	0	0
$153$	−1518.00	−0.802111
$154$	0	0
$155$	768.000i	0.397982i
$156$	0	0
$157$	− 838.000i	− 0.425985i	−0.977054	−	0.212993i	$-0.931679\pi$
$157$	− 838.000i	− 0.425985i	0.977054	−	0.212993i	$-0.0683210\pi$
$158$	0	0
$159$	1380.00	0.688309
$160$	0	0
$161$	3440.00	1.68391
$162$	0	0
$163$	1346.00i	0.646791i	0.946264	+	0.323395i	$0.104824\pi$
$163$	1346.00i	0.646791i	−0.946264	+	0.323395i	$0.895176\pi$
$164$	0	0
$165$	168.000i	0.0792653i
$166$	0	0
$167$	−1052.00	−0.487462	−0.243731	−	0.969843i	$-0.578371\pi$
$167$	−1052.00	−0.487462	−0.243731	+	0.969843i	$0.578371\pi$
$168$	0	0
$169$	−719.000	−0.327264
$170$	0	0
$171$	3726.00i	1.66628i
$172$	0	0
$173$	− 38.0000i	− 0.0166999i	−0.999965	−	0.00834996i	$-0.997342\pi$
$173$	− 38.0000i	− 0.0166999i	0.999965	−	0.00834996i	$-0.00265791\pi$
$174$	0	0
$175$	1780.00	0.768888
$176$	0	0
$177$	−644.000	−0.273480
$178$	0	0
$179$	− 2790.00i	− 1.16500i	−0.812832	−	0.582498i	$-0.802075\pi$
$179$	− 2790.00i	− 1.16500i	0.812832	−	0.582498i	$-0.197925\pi$
$180$	0	0
$181$	− 3418.00i	− 1.40364i	−0.712357	−	0.701818i	$-0.752372\pi$
$181$	− 3418.00i	− 1.40364i	0.712357	−	0.701818i	$-0.247628\pi$
$182$	0	0
$183$	−596.000	−0.240752
$184$	0	0
$185$	−948.000	−0.376748
$186$	0	0
$187$	924.000i	0.361335i
$188$	0	0
$189$	2000.00i	0.769728i
$190$	0	0
$191$	1968.00	0.745547	0.372774	−	0.927922i	$-0.378407\pi$
$191$	1968.00	0.745547	0.372774	+	0.927922i	$0.378407\pi$
$192$	0	0
$193$	−1058.00	−0.394593	−0.197297	−	0.980344i	$-0.563216\pi$
$193$	−1058.00	−0.394593	−0.197297	+	0.980344i	$0.563216\pi$
$194$	0	0
$195$	648.000i	0.237970i
$196$	0	0
$197$	726.000i	0.262565i	0.991345	+	0.131283i	$0.0419095\pi$
$197$	726.000i	0.262565i	−0.991345	+	0.131283i	$0.958090\pi$
$198$	0	0
$199$	4116.00	1.46621	0.733104	−	0.680116i	$-0.238071\pi$
$199$	4116.00	1.46621	0.733104	+	0.680116i	$0.238071\pi$
$200$	0	0
$201$	−404.000	−0.141771
$202$	0	0
$203$	40.0000i	0.0138298i
$204$	0	0
$205$	− 1212.00i	− 0.412926i
$206$	0	0
$207$	3956.00	1.32831
$208$	0	0
$209$	2268.00	0.750626
$210$	0	0
$211$	1482.00i	0.483531i	0.970335	+	0.241766i	$0.0777265\pi$
$211$	1482.00i	0.483531i	−0.970335	+	0.241766i	$0.922273\pi$
$212$	0	0
$213$	− 1400.00i	− 0.450359i
$214$	0	0
$215$	−1788.00	−0.567166
$216$	0	0
$217$	2560.00	0.800848
$218$	0	0
$219$	836.000i	0.257953i
$220$	0	0
$221$	3564.00i	1.08480i
$222$	0	0
$223$	896.000	0.269061	0.134530	−	0.990909i	$-0.457047\pi$
$223$	896.000	0.269061	0.134530	+	0.990909i	$0.457047\pi$
$224$	0	0
$225$	2047.00	0.606519
$226$	0	0
$227$	3410.00i	0.997047i	0.866876	+	0.498523i	$0.166124\pi$
$227$	3410.00i	0.997047i	−0.866876	+	0.498523i	$0.833876\pi$
$228$	0	0
$229$	4502.00i	1.29913i	0.760307	+	0.649564i	$0.225049\pi$
$229$	4502.00i	1.29913i	−0.760307	+	0.649564i	$0.774951\pi$
$230$	0	0
$231$	560.000	0.159503
$232$	0	0
$233$	−2302.00	−0.647249	−0.323625	−	0.946186i	$-0.604901\pi$
$233$	−2302.00	−0.647249	−0.323625	+	0.946186i	$0.604901\pi$
$234$	0	0
$235$	2448.00i	0.679532i
$236$	0	0
$237$	− 1488.00i	− 0.407831i
$238$	0	0
$239$	−4024.00	−1.08908	−0.544542	−	0.838734i	$-0.683296\pi$
$239$	−4024.00	−1.08908	−0.544542	+	0.838734i	$0.683296\pi$
$240$	0	0
$241$	−3586.00	−0.958484	−0.479242	−	0.877683i	$-0.659088\pi$
$241$	−3586.00	−0.958484	−0.479242	+	0.877683i	$0.659088\pi$
$242$	0	0
$243$	3542.00i	0.935059i
$244$	0	0
$245$	342.000i	0.0891820i
$246$	0	0
$247$	8748.00	2.25353
$248$	0	0
$249$	1356.00	0.345112
$250$	0	0
$251$	1250.00i	0.314340i	0.987572	+	0.157170i	$0.0502370\pi$
$251$	1250.00i	0.314340i	−0.987572	+	0.157170i	$0.949763\pi$
$252$	0	0
$253$	− 2408.00i	− 0.598378i
$254$	0	0
$255$	792.000	0.194498
$256$	0	0
$257$	−6638.00	−1.61116	−0.805578	−	0.592490i	$-0.798145\pi$
$257$	−6638.00	−1.61116	−0.805578	+	0.592490i	$0.798145\pi$
$258$	0	0
$259$	3160.00i	0.758119i
$260$	0	0
$261$	46.0000i	0.0109093i
$262$	0	0
$263$	−1724.00	−0.404207	−0.202103	−	0.979364i	$-0.564778\pi$
$263$	−1724.00	−0.404207	−0.202103	+	0.979364i	$0.564778\pi$
$264$	0	0
$265$	4140.00	0.959691
$266$	0	0
$267$	164.000i	0.0375904i
$268$	0	0
$269$	− 4814.00i	− 1.09113i	−0.838068	−	0.545566i	$-0.816315\pi$
$269$	− 4814.00i	− 1.09113i	0.838068	−	0.545566i	$-0.183685\pi$
$270$	0	0
$271$	1640.00	0.367612	0.183806	−	0.982963i	$-0.441158\pi$
$271$	1640.00	0.367612	0.183806	+	0.982963i	$0.441158\pi$
$272$	0	0
$273$	2160.00	0.478861
$274$	0	0
$275$	− 1246.00i	− 0.273224i
$276$	0	0
$277$	3982.00i	0.863737i	0.901937	+	0.431869i	$0.142146\pi$
$277$	3982.00i	0.863737i	−0.901937	+	0.431869i	$0.857854\pi$
$278$	0	0
$279$	2944.00	0.631730
$280$	0	0
$281$	−4126.00	−0.875931	−0.437965	−	0.898992i	$-0.644301\pi$
$281$	−4126.00	−0.875931	−0.437965	+	0.898992i	$0.644301\pi$
$282$	0	0
$283$	− 3446.00i	− 0.723828i	−0.932211	−	0.361914i	$-0.882123\pi$
$283$	− 3446.00i	− 0.723828i	0.932211	−	0.361914i	$-0.117877\pi$
$284$	0	0
$285$	− 1944.00i	− 0.404044i
$286$	0	0
$287$	−4040.00	−0.830919
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	− 2244.00i	− 0.452047i
$292$	0	0
$293$	− 1514.00i	− 0.301873i	−0.988543	−	0.150937i	$-0.951771\pi$
$293$	− 1514.00i	− 0.301873i	0.988543	−	0.150937i	$-0.0482289\pi$
$294$	0	0
$295$	−1932.00	−0.381306
$296$	0	0
$297$	1400.00	0.273523
$298$	0	0
$299$	− 9288.00i	− 1.79645i
$300$	0	0
$301$	5960.00i	1.14129i
$302$	0	0
$303$	−2780.00	−0.527085
$304$	0	0
$305$	−1788.00	−0.335674
$306$	0	0
$307$	5490.00i	1.02062i	0.859990	+	0.510311i	$0.170470\pi$
$307$	5490.00i	1.02062i	−0.859990	+	0.510311i	$0.829530\pi$
$308$	0	0
$309$	1576.00i	0.290147i
$310$	0	0
$311$	−5556.00	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$311$	−5556.00	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$312$	0	0
$313$	2054.00	0.370923	0.185462	−	0.982652i	$-0.440622\pi$
$313$	2054.00	0.370923	0.185462	+	0.982652i	$0.440622\pi$
$314$	0	0
$315$	2760.00i	0.493677i
$316$	0	0
$317$	− 2494.00i	− 0.441883i	−0.975287	−	0.220942i	$-0.929087\pi$
$317$	− 2494.00i	− 0.441883i	0.975287	−	0.220942i	$-0.0709130\pi$
$318$	0	0
$319$	28.0000	0.00491442
$320$	0	0
$321$	3228.00	0.561275
$322$	0	0
$323$	− 10692.0i	− 1.84185i
$324$	0	0
$325$	− 4806.00i	− 0.820274i
$326$	0	0
$327$	4028.00	0.681189
$328$	0	0
$329$	8160.00	1.36740
$330$	0	0
$331$	2914.00i	0.483891i	0.970290	+	0.241946i	$0.0777855\pi$
$331$	2914.00i	0.483891i	−0.970290	+	0.241946i	$0.922214\pi$
$332$	0	0
$333$	3634.00i	0.598024i
$334$	0	0
$335$	−1212.00	−0.197668
$336$	0	0
$337$	7186.00	1.16156	0.580781	−	0.814060i	$-0.302747\pi$
$337$	7186.00	1.16156	0.580781	+	0.814060i	$0.302747\pi$
$338$	0	0
$339$	− 1084.00i	− 0.173672i
$340$	0	0
$341$	− 1792.00i	− 0.284581i
$342$	0	0
$343$	−5720.00	−0.900440
$344$	0	0
$345$	−2064.00	−0.322093
$346$	0	0
$347$	− 4446.00i	− 0.687821i	−0.939002	−	0.343910i	$-0.888248\pi$
$347$	− 4446.00i	− 0.687821i	0.939002	−	0.343910i	$-0.111752\pi$
$348$	0	0
$349$	− 430.000i	− 0.0659524i	−0.999456	−	0.0329762i	$-0.989501\pi$
$349$	− 430.000i	− 0.0659524i	0.999456	−	0.0329762i	$-0.0104985\pi$
$350$	0	0
$351$	5400.00	0.821170
$352$	0	0
$353$	850.000	0.128161	0.0640806	−	0.997945i	$-0.479589\pi$
$353$	850.000	0.128161	0.0640806	+	0.997945i	$0.479589\pi$
$354$	0	0
$355$	− 4200.00i	− 0.627924i
$356$	0	0
$357$	− 2640.00i	− 0.391383i
$358$	0	0
$359$	−10988.0	−1.61539	−0.807694	−	0.589602i	$-0.799285\pi$
$359$	−10988.0	−1.61539	−0.807694	+	0.589602i	$0.799285\pi$
$360$	0	0
$361$	−19385.0	−2.82621
$362$	0	0
$363$	2270.00i	0.328221i
$364$	0	0
$365$	2508.00i	0.359657i
$366$	0	0
$367$	−872.000	−0.124027	−0.0620137	−	0.998075i	$-0.519752\pi$
$367$	−872.000	−0.124027	−0.0620137	+	0.998075i	$0.519752\pi$
$368$	0	0
$369$	−4646.00	−0.655450
$370$	0	0
$371$	− 13800.0i	− 1.93116i
$372$	0	0
$373$	3454.00i	0.479467i	0.970839	+	0.239734i	$0.0770601\pi$
$373$	3454.00i	0.479467i	−0.970839	+	0.239734i	$0.922940\pi$
$374$	0	0
$375$	−2568.00	−0.353629
$376$	0	0
$377$	108.000	0.0147541
$378$	0	0
$379$	1490.00i	0.201942i	0.994889	+	0.100971i	$0.0321950\pi$
$379$	1490.00i	0.201942i	−0.994889	+	0.100971i	$0.967805\pi$
$380$	0	0
$381$	− 3424.00i	− 0.460412i
$382$	0	0
$383$	−10240.0	−1.36616	−0.683080	−	0.730343i	$-0.739360\pi$
$383$	−10240.0	−1.36616	−0.683080	+	0.730343i	$0.739360\pi$
$384$	0	0
$385$	1680.00	0.222392
$386$	0	0
$387$	6854.00i	0.900280i
$388$	0	0
$389$	− 7458.00i	− 0.972071i	−0.873939	−	0.486035i	$-0.838443\pi$
$389$	− 7458.00i	− 0.972071i	0.873939	−	0.486035i	$-0.161557\pi$
$390$	0	0
$391$	−11352.0	−1.46827
$392$	0	0
$393$	4236.00	0.543710
$394$	0	0
$395$	− 4464.00i	− 0.568628i
$396$	0	0
$397$	4706.00i	0.594930i	0.954733	+	0.297465i	$0.0961412\pi$
$397$	4706.00i	0.594930i	−0.954733	+	0.297465i	$0.903859\pi$
$398$	0	0
$399$	−6480.00	−0.813047
$400$	0	0
$401$	5598.00	0.697134	0.348567	−	0.937284i	$-0.386668\pi$
$401$	5598.00	0.697134	0.348567	+	0.937284i	$0.386668\pi$
$402$	0	0
$403$	− 6912.00i	− 0.854370i
$404$	0	0
$405$	2526.00i	0.309921i
$406$	0	0
$407$	2212.00	0.269397
$408$	0	0
$409$	838.000	0.101312	0.0506558	−	0.998716i	$-0.483869\pi$
$409$	838.000	0.101312	0.0506558	+	0.998716i	$0.483869\pi$
$410$	0	0
$411$	972.000i	0.116655i
$412$	0	0
$413$	6440.00i	0.767292i
$414$	0	0
$415$	4068.00	0.481181
$416$	0	0
$417$	2572.00	0.302042
$418$	0	0
$419$	26.0000i	0.00303146i	0.999999	+	0.00151573i	$0.000482473\pi$
$419$	26.0000i	0.00303146i	−0.999999	+	0.00151573i	$0.999518\pi$
$420$	0	0
$421$	− 15578.0i	− 1.80339i	−0.432377	−	0.901693i	$-0.642325\pi$
$421$	− 15578.0i	− 1.80339i	0.432377	−	0.901693i	$-0.357675\pi$
$422$	0	0
$423$	9384.00	1.07864
$424$	0	0
$425$	−5874.00	−0.670426
$426$	0	0
$427$	5960.00i	0.675467i
$428$	0	0
$429$	− 1512.00i	− 0.170163i
$430$	0	0
$431$	6792.00	0.759070	0.379535	−	0.925177i	$-0.376084\pi$
$431$	6792.00	0.759070	0.379535	+	0.925177i	$0.376084\pi$
$432$	0	0
$433$	−9314.00	−1.03372	−0.516862	−	0.856069i	$-0.672900\pi$
$433$	−9314.00	−1.03372	−0.516862	+	0.856069i	$0.672900\pi$
$434$	0	0
$435$	− 24.0000i	− 0.00264531i
$436$	0	0
$437$	27864.0i	3.05015i
$438$	0	0
$439$	−3828.00	−0.416174	−0.208087	−	0.978110i	$-0.566724\pi$
$439$	−3828.00	−0.416174	−0.208087	+	0.978110i	$0.566724\pi$
$440$	0	0
$441$	1311.00	0.141561
$442$	0	0
$443$	− 15414.0i	− 1.65314i	−0.562834	−	0.826570i	$-0.690289\pi$
$443$	− 15414.0i	− 1.65314i	0.562834	−	0.826570i	$-0.309711\pi$
$444$	0	0
$445$	492.000i	0.0524113i
$446$	0	0
$447$	5332.00	0.564195
$448$	0	0
$449$	−3650.00	−0.383640	−0.191820	−	0.981430i	$-0.561439\pi$
$449$	−3650.00	−0.383640	−0.191820	+	0.981430i	$0.561439\pi$
$450$	0	0
$451$	2828.00i	0.295267i
$452$	0	0
$453$	344.000i	0.0356789i
$454$	0	0
$455$	6480.00	0.667664
$456$	0	0
$457$	15862.0	1.62362	0.811809	−	0.583924i	$-0.198483\pi$
$457$	15862.0	1.62362	0.811809	+	0.583924i	$0.198483\pi$
$458$	0	0
$459$	− 6600.00i	− 0.671158i
$460$	0	0
$461$	− 78.0000i	− 0.00788031i	−0.999992	−	0.00394015i	$-0.998746\pi$
$461$	− 78.0000i	− 0.00788031i	0.999992	−	0.00394015i	$-0.00125419\pi$
$462$	0	0
$463$	4376.00	0.439244	0.219622	−	0.975585i	$-0.429518\pi$
$463$	4376.00	0.439244	0.219622	+	0.975585i	$0.429518\pi$
$464$	0	0
$465$	−1536.00	−0.153183
$466$	0	0
$467$	13714.0i	1.35890i	0.733720	+	0.679452i	$0.237782\pi$
$467$	13714.0i	1.35890i	−0.733720	+	0.679452i	$0.762218\pi$
$468$	0	0
$469$	4040.00i	0.397761i
$470$	0	0
$471$	1676.00	0.163962
$472$	0	0
$473$	4172.00	0.405558
$474$	0	0
$475$	14418.0i	1.39272i
$476$	0	0
$477$	− 15870.0i	− 1.52335i
$478$	0	0
$479$	−19072.0	−1.81925	−0.909626	−	0.415428i	$-0.863632\pi$
$479$	−19072.0	−1.81925	−0.909626	+	0.415428i	$0.863632\pi$
$480$	0	0
$481$	8532.00	0.808785
$482$	0	0
$483$	6880.00i	0.648138i
$484$	0	0
$485$	− 6732.00i	− 0.630277i
$486$	0	0
$487$	19428.0	1.80773	0.903867	−	0.427813i	$-0.140716\pi$
$487$	19428.0	1.80773	0.903867	+	0.427813i	$0.140716\pi$
$488$	0	0
$489$	−2692.00	−0.248950
$490$	0	0
$491$	2490.00i	0.228864i	0.993431	+	0.114432i	$0.0365048\pi$
$491$	2490.00i	0.228864i	−0.993431	+	0.114432i	$0.963495\pi$
$492$	0	0
$493$	− 132.000i	− 0.0120588i
$494$	0	0
$495$	1932.00	0.175428
$496$	0	0
$497$	−14000.0	−1.26355
$498$	0	0
$499$	2826.00i	0.253525i	0.991933	+	0.126763i	$0.0404587\pi$
$499$	2826.00i	0.253525i	−0.991933	+	0.126763i	$0.959541\pi$
$500$	0	0
$501$	− 2104.00i	− 0.187624i
$502$	0	0
$503$	2268.00	0.201044	0.100522	−	0.994935i	$-0.467949\pi$
$503$	2268.00	0.201044	0.100522	+	0.994935i	$0.467949\pi$
$504$	0	0
$505$	−8340.00	−0.734901
$506$	0	0
$507$	− 1438.00i	− 0.125964i
$508$	0	0
$509$	− 10534.0i	− 0.917311i	−0.888614	−	0.458656i	$-0.848331\pi$
$509$	− 10534.0i	− 0.917311i	0.888614	−	0.458656i	$-0.151669\pi$
$510$	0	0
$511$	8360.00	0.723727
$512$	0	0
$513$	−16200.0	−1.39424
$514$	0	0
$515$	4728.00i	0.404545i
$516$	0	0
$517$	− 5712.00i	− 0.485906i
$518$	0	0
$519$	76.0000	0.00642780
$520$	0	0
$521$	9478.00	0.797003	0.398502	−	0.917168i	$-0.369530\pi$
$521$	9478.00	0.797003	0.398502	+	0.917168i	$0.369530\pi$
$522$	0	0
$523$	5858.00i	0.489775i	0.969551	+	0.244888i	$0.0787511\pi$
$523$	5858.00i	0.489775i	−0.969551	+	0.244888i	$0.921249\pi$
$524$	0	0
$525$	3560.00i	0.295945i
$526$	0	0
$527$	−8448.00	−0.698293
$528$	0	0
$529$	17417.0	1.43150
$530$	0	0
$531$	7406.00i	0.605260i
$532$	0	0
$533$	10908.0i	0.886450i
$534$	0	0
$535$	9684.00	0.782572
$536$	0	0
$537$	5580.00	0.448407
$538$	0	0
$539$	− 798.000i	− 0.0637705i
$540$	0	0
$541$	− 1910.00i	− 0.151788i	−0.997116	−	0.0758940i	$-0.975819\pi$
$541$	− 1910.00i	− 0.151788i	0.997116	−	0.0758940i	$-0.0241811\pi$
$542$	0	0
$543$	6836.00	0.540259
$544$	0	0
$545$	12084.0	0.949765
$546$	0	0
$547$	1754.00i	0.137104i	0.997648	+	0.0685518i	$0.0218378\pi$
$547$	1754.00i	0.137104i	−0.997648	+	0.0685518i	$0.978162\pi$
$548$	0	0
$549$	6854.00i	0.532826i
$550$	0	0
$551$	−324.000	−0.0250506
$552$	0	0
$553$	−14880.0	−1.14424
$554$	0	0
$555$	− 1896.00i	− 0.145010i
$556$	0	0
$557$	− 2294.00i	− 0.174506i	−0.996186	−	0.0872531i	$-0.972191\pi$
$557$	− 2294.00i	− 0.174506i	0.996186	−	0.0872531i	$-0.0278089\pi$
$558$	0	0
$559$	16092.0	1.21757
$560$	0	0
$561$	−1848.00	−0.139078
$562$	0	0
$563$	16242.0i	1.21584i	0.793998	+	0.607921i	$0.207996\pi$
$563$	16242.0i	1.21584i	−0.793998	+	0.607921i	$0.792004\pi$
$564$	0	0
$565$	− 3252.00i	− 0.242146i
$566$	0	0
$567$	8420.00	0.623645
$568$	0	0
$569$	15990.0	1.17809	0.589047	−	0.808099i	$-0.299503\pi$
$569$	15990.0	1.17809	0.589047	+	0.808099i	$0.299503\pi$
$570$	0	0
$571$	21674.0i	1.58849i	0.607597	+	0.794246i	$0.292134\pi$
$571$	21674.0i	1.58849i	−0.607597	+	0.794246i	$0.707866\pi$
$572$	0	0
$573$	3936.00i	0.286961i
$574$	0	0
$575$	15308.0	1.11024
$576$	0	0
$577$	−12542.0	−0.904905	−0.452453	−	0.891788i	$-0.649451\pi$
$577$	−12542.0	−0.904905	−0.452453	+	0.891788i	$0.649451\pi$
$578$	0	0
$579$	− 2116.00i	− 0.151879i
$580$	0	0
$581$	− 13560.0i	− 0.968268i
$582$	0	0
$583$	−9660.00	−0.686237
$584$	0	0
$585$	7452.00	0.526671
$586$	0	0
$587$	18578.0i	1.30630i	0.757230	+	0.653148i	$0.226552\pi$
$587$	18578.0i	1.30630i	−0.757230	+	0.653148i	$0.773448\pi$
$588$	0	0
$589$	20736.0i	1.45061i
$590$	0	0
$591$	−1452.00	−0.101061
$592$	0	0
$593$	14514.0	1.00509	0.502545	−	0.864551i	$-0.332397\pi$
$593$	14514.0	1.00509	0.502545	+	0.864551i	$0.332397\pi$
$594$	0	0
$595$	− 7920.00i	− 0.545695i
$596$	0	0
$597$	8232.00i	0.564344i
$598$	0	0
$599$	−3460.00	−0.236013	−0.118006	−	0.993013i	$-0.537650\pi$
$599$	−3460.00	−0.236013	−0.118006	+	0.993013i	$0.537650\pi$
$600$	0	0
$601$	−20686.0	−1.40399	−0.701996	−	0.712181i	$-0.747708\pi$
$601$	−20686.0	−1.40399	−0.701996	+	0.712181i	$0.747708\pi$
$602$	0	0
$603$	4646.00i	0.313764i
$604$	0	0
$605$	6810.00i	0.457630i
$606$	0	0
$607$	9776.00	0.653700	0.326850	−	0.945076i	$-0.394013\pi$
$607$	9776.00	0.653700	0.326850	+	0.945076i	$0.394013\pi$
$608$	0	0
$609$	−80.0000	−0.00532309
$610$	0	0
$611$	− 22032.0i	− 1.45879i
$612$	0	0
$613$	− 8794.00i	− 0.579423i	−0.957114	−	0.289712i	$-0.906441\pi$
$613$	− 8794.00i	− 0.579423i	0.957114	−	0.289712i	$-0.0935594\pi$
$614$	0	0
$615$	2424.00	0.158935
$616$	0	0
$617$	−16398.0	−1.06995	−0.534975	−	0.844868i	$-0.679679\pi$
$617$	−16398.0	−1.06995	−0.534975	+	0.844868i	$0.679679\pi$
$618$	0	0
$619$	− 21374.0i	− 1.38787i	−0.720036	−	0.693937i	$-0.755875\pi$
$619$	− 21374.0i	− 1.38787i	0.720036	−	0.693937i	$-0.244125\pi$
$620$	0	0
$621$	17200.0i	1.11145i
$622$	0	0
$623$	1640.00	0.105466
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	4536.00i	0.288916i
$628$	0	0
$629$	− 10428.0i	− 0.661036i
$630$	0	0
$631$	−18916.0	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$631$	−18916.0	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$632$	0	0
$633$	−2964.00	−0.186111
$634$	0	0
$635$	− 10272.0i	− 0.641940i
$636$	0	0
$637$	− 3078.00i	− 0.191452i
$638$	0	0
$639$	−16100.0	−0.996723
$640$	0	0
$641$	−5250.00	−0.323498	−0.161749	−	0.986832i	$-0.551714\pi$
$641$	−5250.00	−0.323498	−0.161749	+	0.986832i	$0.551714\pi$
$642$	0	0
$643$	− 4502.00i	− 0.276114i	−0.990424	−	0.138057i	$-0.955914\pi$
$643$	− 4502.00i	− 0.276114i	0.990424	−	0.138057i	$-0.0440858\pi$
$644$	0	0
$645$	− 3576.00i	− 0.218302i
$646$	0	0
$647$	11076.0	0.673018	0.336509	−	0.941680i	$-0.390754\pi$
$647$	11076.0	0.673018	0.336509	+	0.941680i	$0.390754\pi$
$648$	0	0
$649$	4508.00	0.272657
$650$	0	0
$651$	5120.00i	0.308247i
$652$	0	0
$653$	− 30766.0i	− 1.84375i	−0.387491	−	0.921873i	$-0.626658\pi$
$653$	− 30766.0i	− 1.84375i	0.387491	−	0.921873i	$-0.373342\pi$
$654$	0	0
$655$	12708.0	0.758080
$656$	0	0
$657$	9614.00	0.570895
$658$	0	0
$659$	− 20518.0i	− 1.21285i	−0.795141	−	0.606425i	$-0.792603\pi$
$659$	− 20518.0i	− 1.21285i	0.795141	−	0.606425i	$-0.207397\pi$
$660$	0	0
$661$	70.0000i	0.00411904i	0.999998	+	0.00205952i	$0.000655566\pi$
$661$	70.0000i	0.00411904i	−0.999998	+	0.00205952i	$0.999344\pi$
$662$	0	0
$663$	−7128.00	−0.417539
$664$	0	0
$665$	−19440.0	−1.13361
$666$	0	0
$667$	344.000i	0.0199696i
$668$	0	0
$669$	1792.00i	0.103562i
$670$	0	0
$671$	4172.00	0.240027
$672$	0	0
$673$	23070.0	1.32137	0.660686	−	0.750663i	$-0.270266\pi$
$673$	23070.0	1.32137	0.660686	+	0.750663i	$0.270266\pi$
$674$	0	0
$675$	8900.00i	0.507498i
$676$	0	0
$677$	2622.00i	0.148850i	0.997227	+	0.0744251i	$0.0237122\pi$
$677$	2622.00i	0.148850i	−0.997227	+	0.0744251i	$0.976288\pi$
$678$	0	0
$679$	−22440.0	−1.26829
$680$	0	0
$681$	−6820.00	−0.383764
$682$	0	0
$683$	14682.0i	0.822535i	0.911515	+	0.411267i	$0.134914\pi$
$683$	14682.0i	0.822535i	−0.911515	+	0.411267i	$0.865086\pi$
$684$	0	0
$685$	2916.00i	0.162649i
$686$	0	0
$687$	−9004.00	−0.500035
$688$	0	0
$689$	−37260.0	−2.06022
$690$	0	0
$691$	− 23270.0i	− 1.28109i	−0.767921	−	0.640545i	$-0.778709\pi$
$691$	− 23270.0i	− 1.28109i	0.767921	−	0.640545i	$-0.221291\pi$
$692$	0	0
$693$	− 6440.00i	− 0.353009i
$694$	0	0
$695$	7716.00	0.421129
$696$	0	0
$697$	13332.0	0.724513
$698$	0	0
$699$	− 4604.00i	− 0.249126i
$700$	0	0
$701$	31530.0i	1.69882i	0.527735	+	0.849409i	$0.323041\pi$
$701$	31530.0i	1.69882i	−0.527735	+	0.849409i	$0.676959\pi$
$702$	0	0
$703$	−25596.0	−1.37322
$704$	0	0
$705$	−4896.00	−0.261552
$706$	0	0
$707$	27800.0i	1.47882i
$708$	0	0
$709$	22014.0i	1.16608i	0.812442	+	0.583042i	$0.198138\pi$
$709$	22014.0i	1.16608i	−0.812442	+	0.583042i	$0.801862\pi$
$710$	0	0
$711$	−17112.0	−0.902602
$712$	0	0
$713$	22016.0	1.15639
$714$	0	0
$715$	− 4536.00i	− 0.237254i
$716$	0	0
$717$	− 8048.00i	− 0.419188i
$718$	0	0
$719$	−19016.0	−0.986338	−0.493169	−	0.869933i	$-0.664162\pi$
$719$	−19016.0	−0.986338	−0.493169	+	0.869933i	$0.664162\pi$
$720$	0	0
$721$	15760.0	0.814054
$722$	0	0
$723$	− 7172.00i	− 0.368921i
$724$	0	0
$725$	178.000i	0.00911828i
$726$	0	0
$727$	15996.0	0.816037	0.408018	−	0.912974i	$-0.366220\pi$
$727$	15996.0	0.816037	0.408018	+	0.912974i	$0.366220\pi$
$728$	0	0
$729$	4283.00	0.217599
$730$	0	0
$731$	− 19668.0i	− 0.995140i
$732$	0	0
$733$	− 10622.0i	− 0.535242i	−0.963524	−	0.267621i	$-0.913762\pi$
$733$	− 10622.0i	− 0.535242i	0.963524	−	0.267621i	$-0.0862375\pi$
$734$	0	0
$735$	−684.000	−0.0343261
$736$	0	0
$737$	2828.00	0.141344
$738$	0	0
$739$	− 8878.00i	− 0.441925i	−0.975282	−	0.220962i	$-0.929080\pi$
$739$	− 8878.00i	− 0.441925i	0.975282	−	0.220962i	$-0.0709198\pi$
$740$	0	0
$741$	17496.0i	0.867384i
$742$	0	0
$743$	−21852.0	−1.07897	−0.539483	−	0.841996i	$-0.681380\pi$
$743$	−21852.0	−1.07897	−0.539483	+	0.841996i	$0.681380\pi$
$744$	0	0
$745$	15996.0	0.786642
$746$	0	0
$747$	− 15594.0i	− 0.763795i
$748$	0	0
$749$	− 32280.0i	− 1.57475i
$750$	0	0
$751$	−32024.0	−1.55602	−0.778011	−	0.628251i	$-0.783771\pi$
$751$	−32024.0	−1.55602	−0.778011	+	0.628251i	$0.783771\pi$
$752$	0	0
$753$	−2500.00	−0.120989
$754$	0	0
$755$	1032.00i	0.0497461i
$756$	0	0
$757$	− 26602.0i	− 1.27723i	−0.769525	−	0.638617i	$-0.779507\pi$
$757$	− 26602.0i	− 1.27723i	0.769525	−	0.638617i	$-0.220493\pi$
$758$	0	0
$759$	4816.00	0.230316
$760$	0	0
$761$	13958.0	0.664885	0.332442	−	0.943124i	$-0.392127\pi$
$761$	13958.0	0.664885	0.332442	+	0.943124i	$0.392127\pi$
$762$	0	0
$763$	− 40280.0i	− 1.91118i
$764$	0	0
$765$	− 9108.00i	− 0.430458i
$766$	0	0
$767$	17388.0	0.818571
$768$	0	0
$769$	−11970.0	−0.561312	−0.280656	−	0.959808i	$-0.590552\pi$
$769$	−11970.0	−0.561312	−0.280656	+	0.959808i	$0.590552\pi$
$770$	0	0
$771$	− 13276.0i	− 0.620134i
$772$	0	0
$773$	12318.0i	0.573154i	0.958057	+	0.286577i	$0.0925173\pi$
$773$	12318.0i	0.573154i	−0.958057	+	0.286577i	$0.907483\pi$
$774$	0	0
$775$	11392.0	0.528016
$776$	0	0
$777$	−6320.00	−0.291800
$778$	0	0
$779$	− 32724.0i	− 1.50508i
$780$	0	0
$781$	9800.00i	0.449003i
$782$	0	0
$783$	−200.000	−0.00912825
$784$	0	0
$785$	5028.00	0.228608
$786$	0	0
$787$	26698.0i	1.20925i	0.796510	+	0.604626i	$0.206677\pi$
$787$	26698.0i	1.20925i	−0.796510	+	0.604626i	$0.793323\pi$
$788$	0	0
$789$	− 3448.00i	− 0.155579i
$790$	0	0
$791$	−10840.0	−0.487264
$792$	0	0
$793$	16092.0	0.720610
$794$	0	0
$795$	8280.00i	0.369385i
$796$	0	0
$797$	4794.00i	0.213064i	0.994309	+	0.106532i	$0.0339747\pi$
$797$	4794.00i	0.213064i	−0.994309	+	0.106532i	$0.966025\pi$
$798$	0	0
$799$	−26928.0	−1.19230
$800$	0	0
$801$	1886.00	0.0831942
$802$	0	0
$803$	− 5852.00i	− 0.257176i
$804$	0	0
$805$	20640.0i	0.903683i
$806$	0	0
$807$	9628.00	0.419977
$808$	0	0
$809$	−39114.0	−1.69985	−0.849923	−	0.526907i	$-0.823351\pi$
$809$	−39114.0	−1.69985	−0.849923	+	0.526907i	$0.823351\pi$
$810$	0	0
$811$	1090.00i	0.0471949i	0.999722	+	0.0235975i	$0.00751200\pi$
$811$	1090.00i	0.0471949i	−0.999722	+	0.0235975i	$0.992488\pi$
$812$	0	0
$813$	3280.00i	0.141494i
$814$	0	0
$815$	−8076.00	−0.347104
$816$	0	0
$817$	−48276.0	−2.06727
$818$	0	0
$819$	− 24840.0i	− 1.05980i
$820$	0	0
$821$	− 9730.00i	− 0.413617i	−0.978381	−	0.206808i	$-0.933692\pi$
$821$	− 9730.00i	− 0.413617i	0.978381	−	0.206808i	$-0.0663077\pi$
$822$	0	0
$823$	1100.00	0.0465900	0.0232950	−	0.999729i	$-0.492584\pi$
$823$	1100.00	0.0465900	0.0232950	+	0.999729i	$0.492584\pi$
$824$	0	0
$825$	2492.00	0.105164
$826$	0	0
$827$	38074.0i	1.60092i	0.599385	+	0.800461i	$0.295412\pi$
$827$	38074.0i	1.60092i	−0.599385	+	0.800461i	$0.704588\pi$
$828$	0	0
$829$	− 24230.0i	− 1.01513i	−0.861613	−	0.507565i	$-0.830546\pi$
$829$	− 24230.0i	− 1.01513i	0.861613	−	0.507565i	$-0.169454\pi$
$830$	0	0
$831$	−7964.00	−0.332453
$832$	0	0
$833$	−3762.00	−0.156477
$834$	0	0
$835$	− 6312.00i	− 0.261600i
$836$	0	0
$837$	12800.0i	0.528593i
$838$	0	0
$839$	16820.0	0.692123	0.346061	−	0.938212i	$-0.387519\pi$
$839$	16820.0	0.692123	0.346061	+	0.938212i	$0.387519\pi$
$840$	0	0
$841$	24385.0	0.999836
$842$	0	0
$843$	− 8252.00i	− 0.337146i
$844$	0	0
$845$	− 4314.00i	− 0.175629i
$846$	0	0
$847$	22700.0	0.920875
$848$	0	0
$849$	6892.00	0.278602
$850$	0	0
$851$	27176.0i	1.09469i
$852$	0	0
$853$	− 22162.0i	− 0.889581i	−0.895635	−	0.444790i	$-0.853278\pi$
$853$	− 22162.0i	− 0.889581i	0.895635	−	0.444790i	$-0.146722\pi$
$854$	0	0
$855$	−22356.0	−0.894221
$856$	0	0
$857$	8790.00	0.350363	0.175181	−	0.984536i	$-0.443949\pi$
$857$	8790.00	0.350363	0.175181	+	0.984536i	$0.443949\pi$
$858$	0	0
$859$	− 10558.0i	− 0.419365i	−0.977770	−	0.209682i	$-0.932757\pi$
$859$	− 10558.0i	− 0.419365i	0.977770	−	0.209682i	$-0.0672430\pi$
$860$	0	0
$861$	− 8080.00i	− 0.319821i
$862$	0	0
$863$	7392.00	0.291572	0.145786	−	0.989316i	$-0.453429\pi$
$863$	7392.00	0.291572	0.145786	+	0.989316i	$0.453429\pi$
$864$	0	0
$865$	228.000	0.00896212
$866$	0	0
$867$	− 1114.00i	− 0.0436372i
$868$	0	0
$869$	10416.0i	0.406604i
$870$	0	0
$871$	10908.0	0.424344
$872$	0	0
$873$	−25806.0	−1.00046
$874$	0	0
$875$	25680.0i	0.992163i
$876$	0	0
$877$	− 6574.00i	− 0.253122i	−0.991959	−	0.126561i	$-0.959606\pi$
$877$	− 6574.00i	− 0.253122i	0.991959	−	0.126561i	$-0.0403940\pi$
$878$	0	0
$879$	3028.00	0.116191
$880$	0	0
$881$	47154.0	1.80324	0.901622	−	0.432524i	$-0.142377\pi$
$881$	47154.0	1.80324	0.901622	+	0.432524i	$0.142377\pi$
$882$	0	0
$883$	642.000i	0.0244677i	0.999925	+	0.0122339i	$0.00389426\pi$
$883$	642.000i	0.0244677i	−0.999925	+	0.0122339i	$0.996106\pi$
$884$	0	0
$885$	− 3864.00i	− 0.146765i
$886$	0	0
$887$	23308.0	0.882307	0.441153	−	0.897432i	$-0.354569\pi$
$887$	23308.0	0.882307	0.441153	+	0.897432i	$0.354569\pi$
$888$	0	0
$889$	−34240.0	−1.29176
$890$	0	0
$891$	− 5894.00i	− 0.221612i
$892$	0	0
$893$	66096.0i	2.47684i
$894$	0	0
$895$	16740.0	0.625203
$896$	0	0
$897$	18576.0	0.691454
$898$	0	0
$899$	256.000i	0.00949731i
$900$	0	0
$901$	45540.0i	1.68386i
$902$	0	0
$903$	−11920.0	−0.439283
$904$	0	0
$905$	20508.0	0.753270
$906$	0	0
$907$	21450.0i	0.785265i	0.919695	+	0.392633i	$0.128436\pi$
$907$	21450.0i	0.785265i	−0.919695	+	0.392633i	$0.871564\pi$
$908$	0	0
$909$	31970.0i	1.16653i
$910$	0	0
$911$	40904.0	1.48761	0.743804	−	0.668398i	$-0.233020\pi$
$911$	40904.0	1.48761	0.743804	+	0.668398i	$0.233020\pi$
$912$	0	0
$913$	−9492.00	−0.344074
$914$	0	0
$915$	− 3576.00i	− 0.129201i
$916$	0	0
$917$	− 42360.0i	− 1.52546i
$918$	0	0
$919$	−27380.0	−0.982789	−0.491394	−	0.870937i	$-0.663513\pi$
$919$	−27380.0	−0.982789	−0.491394	+	0.870937i	$0.663513\pi$
$920$	0	0
$921$	−10980.0	−0.392837
$922$	0	0
$923$	37800.0i	1.34800i
$924$	0	0
$925$	14062.0i	0.499844i
$926$	0	0
$927$	18124.0	0.642147
$928$	0	0
$929$	10302.0	0.363830	0.181915	−	0.983314i	$-0.441770\pi$
$929$	10302.0	0.363830	0.181915	+	0.983314i	$0.441770\pi$
$930$	0	0
$931$	9234.00i	0.325061i
$932$	0	0
$933$	− 11112.0i	− 0.389915i
$934$	0	0
$935$	−5544.00	−0.193912
$936$	0	0
$937$	−5054.00	−0.176208	−0.0881040	−	0.996111i	$-0.528081\pi$
$937$	−5054.00	−0.176208	−0.0881040	+	0.996111i	$0.528081\pi$
$938$	0	0
$939$	4108.00i	0.142768i
$940$	0	0
$941$	− 30462.0i	− 1.05530i	−0.849463	−	0.527648i	$-0.823074\pi$
$941$	− 30462.0i	− 1.05530i	0.849463	−	0.527648i	$-0.176926\pi$
$942$	0	0
$943$	−34744.0	−1.19981
$944$	0	0
$945$	−12000.0	−0.413079
$946$	0	0
$947$	32082.0i	1.10087i	0.834878	+	0.550436i	$0.185538\pi$
$947$	32082.0i	1.10087i	−0.834878	+	0.550436i	$0.814462\pi$
$948$	0	0
$949$	− 22572.0i	− 0.772095i
$950$	0	0
$951$	4988.00	0.170081
$952$	0	0
$953$	−12970.0	−0.440860	−0.220430	−	0.975403i	$-0.570746\pi$
$953$	−12970.0	−0.440860	−0.220430	+	0.975403i	$0.570746\pi$
$954$	0	0
$955$	11808.0i	0.400103i
$956$	0	0
$957$	56.0000i	0.00189156i
$958$	0	0
$959$	9720.00	0.327294
$960$	0	0
$961$	−13407.0	−0.450035
$962$	0	0
$963$	− 37122.0i	− 1.24220i
$964$	0	0
$965$	− 6348.00i	− 0.211761i
$966$	0	0
$967$	1652.00	0.0549377	0.0274688	−	0.999623i	$-0.491255\pi$
$967$	1652.00	0.0549377	0.0274688	+	0.999623i	$0.491255\pi$
$968$	0	0
$969$	21384.0	0.708930
$970$	0	0
$971$	24650.0i	0.814682i	0.913276	+	0.407341i	$0.133544\pi$
$971$	24650.0i	0.814682i	−0.913276	+	0.407341i	$0.866456\pi$
$972$	0	0
$973$	− 25720.0i	− 0.847426i
$974$	0	0
$975$	9612.00	0.315723
$976$	0	0
$977$	23646.0	0.774312	0.387156	−	0.922014i	$-0.373458\pi$
$977$	23646.0	0.774312	0.387156	+	0.922014i	$0.373458\pi$
$978$	0	0
$979$	− 1148.00i	− 0.0374773i
$980$	0	0
$981$	− 46322.0i	− 1.50759i
$982$	0	0
$983$	38108.0	1.23648	0.618238	−	0.785991i	$-0.287847\pi$
$983$	38108.0	1.23648	0.618238	+	0.785991i	$0.287847\pi$
$984$	0	0
$985$	−4356.00	−0.140907
$986$	0	0
$987$	16320.0i	0.526313i
$988$	0	0
$989$	51256.0i	1.64797i
$990$	0	0
$991$	18640.0	0.597497	0.298748	−	0.954332i	$-0.403431\pi$
$991$	18640.0	0.597497	0.298748	+	0.954332i	$0.403431\pi$
$992$	0	0
$993$	−5828.00	−0.186250
$994$	0	0
$995$	24696.0i	0.786850i
$996$	0	0
$997$	29022.0i	0.921902i	0.887426	+	0.460951i	$0.152492\pi$
$997$	29022.0i	0.921902i	−0.887426	+	0.460951i	$0.847508\pi$
$998$	0	0
$999$	−15800.0	−0.500390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.4.b.f.129.2		2
4.3	odd	2		256.4.b.b.129.1		2
8.3	odd	2		256.4.b.b.129.2		2
8.5	even	2	inner	256.4.b.f.129.1		2
16.3	odd	4		128.4.a.d.1.1	yes	1
16.5	even	4		128.4.a.c.1.1	yes	1
16.11	odd	4		128.4.a.a.1.1	✓	1
16.13	even	4		128.4.a.b.1.1	yes	1
48.5	odd	4		1152.4.a.i.1.1		1
48.11	even	4		1152.4.a.j.1.1		1
48.29	odd	4		1152.4.a.c.1.1		1
48.35	even	4		1152.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.a.a.1.1	✓	1	16.11	odd	4
128.4.a.b.1.1	yes	1	16.13	even	4
128.4.a.c.1.1	yes	1	16.5	even	4
128.4.a.d.1.1	yes	1	16.3	odd	4
256.4.b.b.129.1		2	4.3	odd	2
256.4.b.b.129.2		2	8.3	odd	2
256.4.b.f.129.1		2	8.5	even	2	inner
256.4.b.f.129.2		2	1.1	even	1	trivial
1152.4.a.c.1.1		1	48.29	odd	4
1152.4.a.d.1.1		1	48.35	even	4
1152.4.a.i.1.1		1	48.5	odd	4
1152.4.a.j.1.1		1	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +6.00000i q^{5} +20.0000 q^{7} +23.0000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +6.00000i q^{5} +20.0000 q^{7} +23.0000 q^{9} -14.0000i q^{11} -54.0000i q^{13} -12.0000 q^{15} -66.0000 q^{17} +162.000i q^{19} +40.0000i q^{21} +172.000 q^{23} +89.0000 q^{25} +100.000i q^{27} +2.00000i q^{29} +128.000 q^{31} +28.0000 q^{33} +120.000i q^{35} +158.000i q^{37} +108.000 q^{39} -202.000 q^{41} +298.000i q^{43} +138.000i q^{45} +408.000 q^{47} +57.0000 q^{49} -132.000i q^{51} -690.000i q^{53} +84.0000 q^{55} -324.000 q^{57} +322.000i q^{59} +298.000i q^{61} +460.000 q^{63} +324.000 q^{65} +202.000i q^{67} +344.000i q^{69} -700.000 q^{71} +418.000 q^{73} +178.000i q^{75} -280.000i q^{77} -744.000 q^{79} +421.000 q^{81} -678.000i q^{83} -396.000i q^{85} -4.00000 q^{87} +82.0000 q^{89} -1080.00i q^{91} +256.000i q^{93} -972.000 q^{95} -1122.00 q^{97} -322.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 40 q^{7} + 46 q^{9}+O(q^{10}) \) 2 * q + 40 * q^7 + 46 * q^9	\( 2 q + 40 q^{7} + 46 q^{9} - 24 q^{15} - 132 q^{17} + 344 q^{23} + 178 q^{25} + 256 q^{31} + 56 q^{33} + 216 q^{39} - 404 q^{41} + 816 q^{47} + 114 q^{49} + 168 q^{55} - 648 q^{57} + 920 q^{63} + 648 q^{65} - 1400 q^{71} + 836 q^{73} - 1488 q^{79} + 842 q^{81} - 8 q^{87} + 164 q^{89} - 1944 q^{95} - 2244 q^{97}+O(q^{100}) \) 2 * q + 40 * q^7 + 46 * q^9 - 24 * q^15 - 132 * q^17 + 344 * q^23 + 178 * q^25 + 256 * q^31 + 56 * q^33 + 216 * q^39 - 404 * q^41 + 816 * q^47 + 114 * q^49 + 168 * q^55 - 648 * q^57 + 920 * q^63 + 648 * q^65 - 1400 * q^71 + 836 * q^73 - 1488 * q^79 + 842 * q^81 - 8 * q^87 + 164 * q^89 - 1944 * q^95 - 2244 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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