LMFDB - Embedded newform 3380.2.f.i.3041.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(3041,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3380.3041");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.f (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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3041.3
Root			$0.665665 + 1.24775i$ of defining polynomial
Character	$\chi$	$=$	3380.3041
Dual form			3380.2.f.i.3041.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0947876 q^{3} -1.00000i q^{5} -0.826838i q^{7} -2.99102 q^{9} +O(q^{10})$	$q-0.0947876 q^{3} -1.00000i q^{5} -0.826838i q^{7} -2.99102 q^{9} +1.73205i q^{11} +0.0947876i q^{15} -1.43213 q^{17} +1.06939i q^{19} +0.0783740i q^{21} -3.08580 q^{23} -1.00000 q^{25} +0.567874 q^{27} +7.45512 q^{29} +5.84325i q^{31} -0.164177i q^{33} -0.826838 q^{35} +0.983586i q^{37} -4.26795i q^{41} +9.54092 q^{43} +2.99102i q^{45} +3.46410i q^{47} +6.31634 q^{49} +0.135748 q^{51} +0.334308 q^{53} +1.73205 q^{55} -0.101365i q^{57} +11.5245i q^{59} +2.71649 q^{61} +2.47309i q^{63} -13.7550i q^{67} +0.292496 q^{69} -9.77689i q^{71} +11.1806i q^{73} +0.0947876 q^{75} +1.43213 q^{77} -0.252387 q^{79} +8.91922 q^{81} +5.67165i q^{83} +1.43213i q^{85} -0.706653 q^{87} +4.59630i q^{89} -0.553868i q^{93} +1.06939 q^{95} -9.53434i q^{97} -5.18059i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 8 * q^9	$ 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) $ 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−0.0947876	−0.0547256	−0.0273628	−	0.999626i	$-0.508711\pi$
$3$	−0.0947876	−0.0547256	−0.0273628	+	0.999626i	$0.508711\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 0.826838i	− 0.312516i	−0.987716	−	0.156258i	$-0.950057\pi$
$7$	− 0.826838i	− 0.312516i	0.987716	−	0.156258i	$-0.0499431\pi$
$8$	0	0
$9$	−2.99102	−0.997005
$10$	0	0
$11$	1.73205i	0.522233i	0.965307	+	0.261116i	$0.0840907\pi$
$11$	1.73205i	0.522233i	−0.965307	+	0.261116i	$0.915909\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.0947876i	0.0244740i
$16$	0	0
$17$	−1.43213	−0.347342	−0.173671	−	0.984804i	$-0.555563\pi$
$17$	−1.43213	−0.347342	−0.173671	+	0.984804i	$0.555563\pi$
$18$	0	0
$19$	1.06939i	0.245335i	0.992448	+	0.122667i	$0.0391448\pi$
$19$	1.06939i	0.245335i	−0.992448	+	0.122667i	$0.960855\pi$
$20$	0	0
$21$	0.0783740i	0.0171026i
$22$	0	0
$23$	−3.08580	−0.643434	−0.321717	−	0.946836i	$-0.604260\pi$
$23$	−3.08580	−0.643434	−0.321717	+	0.946836i	$0.604260\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0.567874	0.109287
$28$	0	0
$29$	7.45512	1.38438	0.692190	−	0.721715i	$-0.256646\pi$
$29$	7.45512	1.38438	0.692190	+	0.721715i	$0.256646\pi$
$30$	0	0
$31$	5.84325i	1.04948i	0.851263	+	0.524740i	$0.175837\pi$
$31$	5.84325i	1.04948i	−0.851263	+	0.524740i	$0.824163\pi$
$32$	0	0
$33$	− 0.164177i	− 0.0285795i
$34$	0	0
$35$	−0.826838	−0.139761
$36$	0	0
$37$	0.983586i	0.161701i	0.996726	+	0.0808503i	$0.0257636\pi$
$37$	0.983586i	0.161701i	−0.996726	+	0.0808503i	$0.974236\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.26795i	− 0.666542i	−0.942831	−	0.333271i	$-0.891848\pi$
$41$	− 4.26795i	− 0.666542i	0.942831	−	0.333271i	$-0.108152\pi$
$42$	0	0
$43$	9.54092	1.45498	0.727488	−	0.686120i	$-0.240688\pi$
$43$	9.54092	1.45498	0.727488	+	0.686120i	$0.240688\pi$
$44$	0	0
$45$	2.99102i	0.445874i
$46$	0	0
$47$	3.46410i	0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$	3.46410i	0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$	0	0
$49$	6.31634	0.902334
$50$	0	0
$51$	0.135748	0.0190085
$52$	0	0
$53$	0.334308	0.0459207	0.0229603	−	0.999736i	$-0.492691\pi$
$53$	0.334308	0.0459207	0.0229603	+	0.999736i	$0.492691\pi$
$54$	0	0
$55$	1.73205	0.233550
$56$	0	0
$57$	− 0.101365i	− 0.0134261i
$58$	0	0
$59$	11.5245i	1.50036i	0.661232	+	0.750181i	$0.270034\pi$
$59$	11.5245i	1.50036i	−0.661232	+	0.750181i	$0.729966\pi$
$60$	0	0
$61$	2.71649	0.347811	0.173905	−	0.984762i	$-0.444361\pi$
$61$	2.71649	0.347811	0.173905	+	0.984762i	$0.444361\pi$
$62$	0	0
$63$	2.47309i	0.311580i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.7550i	− 1.68045i	−0.542241	−	0.840223i	$-0.682424\pi$
$67$	− 13.7550i	− 1.68045i	0.542241	−	0.840223i	$-0.317576\pi$
$68$	0	0
$69$	0.292496	0.0352124
$70$	0	0
$71$	− 9.77689i	− 1.16030i	−0.814508	−	0.580152i	$-0.802993\pi$
$71$	− 9.77689i	− 1.16030i	0.814508	−	0.580152i	$-0.197007\pi$
$72$	0	0
$73$	11.1806i	1.30859i	0.756240	+	0.654295i	$0.227034\pi$
$73$	11.1806i	1.30859i	−0.756240	+	0.654295i	$0.772966\pi$
$74$	0	0
$75$	0.0947876	0.0109451
$76$	0	0
$77$	1.43213	0.163206
$78$	0	0
$79$	−0.252387	−0.0283958	−0.0141979	−	0.999899i	$-0.504519\pi$
$79$	−0.252387	−0.0283958	−0.0141979	+	0.999899i	$0.504519\pi$
$80$	0	0
$81$	8.91922	0.991024
$82$	0	0
$83$	5.67165i	0.622544i	0.950321	+	0.311272i	$0.100755\pi$
$83$	5.67165i	0.622544i	−0.950321	+	0.311272i	$0.899245\pi$
$84$	0	0
$85$	1.43213i	0.155336i
$86$	0	0
$87$	−0.706653	−0.0757611
$88$	0	0
$89$	4.59630i	0.487207i	0.969875	+	0.243604i	$0.0783296\pi$
$89$	4.59630i	0.487207i	−0.969875	+	0.243604i	$0.921670\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 0.553868i	− 0.0574334i
$94$	0	0
$95$	1.06939	0.109717
$96$	0	0
$97$	− 9.53434i	− 0.968066i	−0.875050	−	0.484033i	$-0.839171\pi$
$97$	− 9.53434i	− 0.968066i	0.875050	−	0.484033i	$-0.160829\pi$
$98$	0	0
$99$	− 5.18059i	− 0.520669i
$100$	0	0
$101$	5.80144	0.577265	0.288632	−	0.957440i	$-0.406799\pi$
$101$	5.80144	0.577265	0.288632	+	0.957440i	$0.406799\pi$
$102$	0	0
$103$	10.0760	0.992814	0.496407	−	0.868090i	$-0.334652\pi$
$103$	10.0760	0.992814	0.496407	+	0.868090i	$0.334652\pi$
$104$	0	0
$105$	0.0783740	0.00764852
$106$	0	0
$107$	16.2795	1.57380	0.786902	−	0.617078i	$-0.211684\pi$
$107$	16.2795	1.57380	0.786902	+	0.617078i	$0.211684\pi$
$108$	0	0
$109$	3.12979i	0.299780i	0.988703	+	0.149890i	$0.0478919\pi$
$109$	3.12979i	0.299780i	−0.988703	+	0.149890i	$0.952108\pi$
$110$	0	0
$111$	− 0.0932318i	− 0.00884917i
$112$	0	0
$113$	−10.1708	−0.956784	−0.478392	−	0.878146i	$-0.658780\pi$
$113$	−10.1708	−0.956784	−0.478392	+	0.878146i	$0.658780\pi$
$114$	0	0
$115$	3.08580i	0.287753i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.18414i	0.108550i
$120$	0	0
$121$	8.00000	0.727273
$122$	0	0
$123$	0.404549i	0.0364769i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−5.96802	−0.529577	−0.264788	−	0.964307i	$-0.585302\pi$
$127$	−5.96802	−0.529577	−0.264788	+	0.964307i	$0.585302\pi$
$128$	0	0
$129$	−0.904361	−0.0796245
$130$	0	0
$131$	16.6267	1.45268	0.726342	−	0.687334i	$-0.241219\pi$
$131$	16.6267	1.45268	0.726342	+	0.687334i	$0.241219\pi$
$132$	0	0
$133$	0.884212	0.0766709
$134$	0	0
$135$	− 0.567874i	− 0.0488748i
$136$	0	0
$137$	0.404549i	0.0345629i	0.999851	+	0.0172815i	$0.00550113\pi$
$137$	0.404549i	0.0345629i	−0.999851	+	0.0172815i	$0.994499\pi$
$138$	0	0
$139$	−9.31634	−0.790201	−0.395101	−	0.918638i	$-0.629290\pi$
$139$	−9.31634	−0.790201	−0.395101	+	0.918638i	$0.629290\pi$
$140$	0	0
$141$	− 0.328354i	− 0.0276524i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 7.45512i	− 0.619114i
$146$	0	0
$147$	−0.598710	−0.0493808
$148$	0	0
$149$	− 10.8678i	− 0.890325i	−0.895450	−	0.445162i	$-0.853146\pi$
$149$	− 10.8678i	− 0.890325i	0.895450	−	0.445162i	$-0.146854\pi$
$150$	0	0
$151$	− 0.991015i	− 0.0806477i	−0.999187	−	0.0403238i	$-0.987161\pi$
$151$	− 0.991015i	− 0.0806477i	0.999187	−	0.0403238i	$-0.0128390\pi$
$152$	0	0
$153$	4.28351	0.346301
$154$	0	0
$155$	5.84325	0.469341
$156$	0	0
$157$	17.5729	1.40247	0.701235	−	0.712930i	$-0.252632\pi$
$157$	17.5729	1.40247	0.701235	+	0.712930i	$0.252632\pi$
$158$	0	0
$159$	−0.0316882	−0.00251304
$160$	0	0
$161$	2.55146i	0.201083i
$162$	0	0
$163$	17.2820i	1.35363i	0.736154	+	0.676814i	$0.236640\pi$
$163$	17.2820i	1.35363i	−0.736154	+	0.676814i	$0.763360\pi$
$164$	0	0
$165$	−0.164177	−0.0127812
$166$	0	0
$167$	− 3.05955i	− 0.236755i	−0.992969	−	0.118378i	$-0.962231\pi$
$167$	− 3.05955i	− 0.236755i	0.992969	−	0.118378i	$-0.0377693\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 3.19856i	− 0.244600i
$172$	0	0
$173$	3.42011	0.260026	0.130013	−	0.991512i	$-0.458498\pi$
$173$	3.42011	0.260026	0.130013	+	0.991512i	$0.458498\pi$
$174$	0	0
$175$	0.826838i	0.0625031i
$176$	0	0
$177$	− 1.09238i	− 0.0821083i
$178$	0	0
$179$	10.3822	0.776001	0.388000	−	0.921659i	$-0.373166\pi$
$179$	10.3822	0.776001	0.388000	+	0.921659i	$0.373166\pi$
$180$	0	0
$181$	−10.3492	−0.769247	−0.384624	−	0.923073i	$-0.625669\pi$
$181$	−10.3492	−0.769247	−0.384624	+	0.923073i	$0.625669\pi$
$182$	0	0
$183$	−0.257489	−0.0190342
$184$	0	0
$185$	0.983586	0.0723147
$186$	0	0
$187$	− 2.48052i	− 0.181393i
$188$	0	0
$189$	− 0.469540i	− 0.0341540i
$190$	0	0
$191$	−15.5059	−1.12197	−0.560984	−	0.827826i	$-0.689577\pi$
$191$	−15.5059	−1.12197	−0.560984	+	0.827826i	$0.689577\pi$
$192$	0	0
$193$	5.57028i	0.400958i	0.979698	+	0.200479i	$0.0642498\pi$
$193$	5.57028i	0.400958i	−0.979698	+	0.200479i	$0.935750\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 24.8728i	− 1.77212i	−0.463575	−	0.886058i	$-0.653434\pi$
$197$	− 24.8728i	− 1.77212i	0.463575	−	0.886058i	$-0.346566\pi$
$198$	0	0
$199$	18.6489	1.32198	0.660991	−	0.750393i	$-0.270136\pi$
$199$	18.6489	1.32198	0.660991	+	0.750393i	$0.270136\pi$
$200$	0	0
$201$	1.30381i	0.0919635i
$202$	0	0
$203$	− 6.16418i	− 0.432640i
$204$	0	0
$205$	−4.26795	−0.298087
$206$	0	0
$207$	9.22968	0.641507
$208$	0	0
$209$	−1.85224	−0.128122
$210$	0	0
$211$	9.64469	0.663968	0.331984	−	0.943285i	$-0.392282\pi$
$211$	9.64469	0.663968	0.331984	+	0.943285i	$0.392282\pi$
$212$	0	0
$213$	0.926728i	0.0634984i
$214$	0	0
$215$	− 9.54092i	− 0.650685i
$216$	0	0
$217$	4.83143	0.327979
$218$	0	0
$219$	− 1.05978i	− 0.0716134i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 1.16115i	− 0.0777561i	−0.999244	−	0.0388780i	$-0.987622\pi$
$223$	− 1.16115i	− 0.0777561i	0.999244	−	0.0388780i	$-0.0123784\pi$
$224$	0	0
$225$	2.99102	0.199401
$226$	0	0
$227$	27.2460i	1.80838i	0.427129	+	0.904191i	$0.359525\pi$
$227$	27.2460i	1.80838i	−0.427129	+	0.904191i	$0.640475\pi$
$228$	0	0
$229$	24.3432i	1.60864i	0.594193	+	0.804322i	$0.297471\pi$
$229$	24.3432i	1.60864i	−0.594193	+	0.804322i	$0.702529\pi$
$230$	0	0
$231$	−0.135748	−0.00893155
$232$	0	0
$233$	−23.0238	−1.50834	−0.754171	−	0.656678i	$-0.771961\pi$
$233$	−23.0238	−1.50834	−0.754171	+	0.656678i	$0.771961\pi$
$234$	0	0
$235$	3.46410	0.225973
$236$	0	0
$237$	0.0239232	0.00155398
$238$	0	0
$239$	23.7057i	1.53340i	0.642008	+	0.766698i	$0.278102\pi$
$239$	23.7057i	1.53340i	−0.642008	+	0.766698i	$0.721898\pi$
$240$	0	0
$241$	10.8307i	0.697668i	0.937185	+	0.348834i	$0.113422\pi$
$241$	10.8307i	0.697668i	−0.937185	+	0.348834i	$0.886578\pi$
$242$	0	0
$243$	−2.54905	−0.163522
$244$	0	0
$245$	− 6.31634i	− 0.403536i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 0.537602i	− 0.0340691i
$250$	0	0
$251$	1.12081	0.0707449	0.0353724	−	0.999374i	$-0.488738\pi$
$251$	1.12081	0.0707449	0.0353724	+	0.999374i	$0.488738\pi$
$252$	0	0
$253$	− 5.34477i	− 0.336023i
$254$	0	0
$255$	− 0.135748i	− 0.00850086i
$256$	0	0
$257$	−16.6307	−1.03739	−0.518697	−	0.854958i	$-0.673583\pi$
$257$	−16.6307	−1.03739	−0.518697	+	0.854958i	$0.673583\pi$
$258$	0	0
$259$	0.813267	0.0505340
$260$	0	0
$261$	−22.2984	−1.38023
$262$	0	0
$263$	24.5020	1.51085	0.755427	−	0.655232i	$-0.227429\pi$
$263$	24.5020	1.51085	0.755427	+	0.655232i	$0.227429\pi$
$264$	0	0
$265$	− 0.334308i	− 0.0205364i
$266$	0	0
$267$	− 0.435672i	− 0.0266627i
$268$	0	0
$269$	−6.53287	−0.398316	−0.199158	−	0.979967i	$-0.563821\pi$
$269$	−6.53287	−0.398316	−0.199158	+	0.979967i	$0.563821\pi$
$270$	0	0
$271$	5.65608i	0.343583i	0.985133	+	0.171791i	$0.0549555\pi$
$271$	5.65608i	0.343583i	−0.985133	+	0.171791i	$0.945045\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.73205i	− 0.104447i
$276$	0	0
$277$	−3.71564	−0.223251	−0.111626	−	0.993750i	$-0.535606\pi$
$277$	−3.71564	−0.223251	−0.111626	+	0.993750i	$0.535606\pi$
$278$	0	0
$279$	− 17.4773i	− 1.04634i
$280$	0	0
$281$	9.70447i	0.578920i	0.957190	+	0.289460i	$0.0934758\pi$
$281$	9.70447i	0.578920i	−0.957190	+	0.289460i	$0.906524\pi$
$282$	0	0
$283$	24.1976	1.43840	0.719200	−	0.694803i	$-0.244509\pi$
$283$	24.1976	1.43840	0.719200	+	0.694803i	$0.244509\pi$
$284$	0	0
$285$	−0.101365	−0.00600433
$286$	0	0
$287$	−3.52890	−0.208305
$288$	0	0
$289$	−14.9490	−0.879354
$290$	0	0
$291$	0.903737i	0.0529780i
$292$	0	0
$293$	3.62828i	0.211966i	0.994368	+	0.105983i	$0.0337990\pi$
$293$	3.62828i	0.211966i	−0.994368	+	0.105983i	$0.966201\pi$
$294$	0	0
$295$	11.5245	0.670983
$296$	0	0
$297$	0.983586i	0.0570735i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 7.88880i	− 0.454703i
$302$	0	0
$303$	−0.549905	−0.0315912
$304$	0	0
$305$	− 2.71649i	− 0.155546i
$306$	0	0
$307$	− 9.40129i	− 0.536560i	−0.963341	−	0.268280i	$-0.913545\pi$
$307$	− 9.40129i	− 0.536560i	0.963341	−	0.268280i	$-0.0864552\pi$
$308$	0	0
$309$	−0.955077	−0.0543324
$310$	0	0
$311$	25.5370	1.44807	0.724034	−	0.689764i	$-0.242286\pi$
$311$	25.5370	1.44807	0.724034	+	0.689764i	$0.242286\pi$
$312$	0	0
$313$	5.25656	0.297118	0.148559	−	0.988904i	$-0.452536\pi$
$313$	5.25656	0.297118	0.148559	+	0.988904i	$0.452536\pi$
$314$	0	0
$315$	2.47309	0.139343
$316$	0	0
$317$	− 14.1536i	− 0.794947i	−0.917614	−	0.397474i	$-0.869887\pi$
$317$	− 14.1536i	− 0.794947i	0.917614	−	0.397474i	$-0.130113\pi$
$318$	0	0
$319$	12.9126i	0.722969i
$320$	0	0
$321$	−1.54310	−0.0861274
$322$	0	0
$323$	− 1.53150i	− 0.0852150i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 0.296666i	− 0.0164056i
$328$	0	0
$329$	2.86425	0.157911
$330$	0	0
$331$	18.5843i	1.02148i	0.859734	+	0.510742i	$0.170629\pi$
$331$	18.5843i	1.02148i	−0.859734	+	0.510742i	$0.829371\pi$
$332$	0	0
$333$	− 2.94192i	− 0.161216i
$334$	0	0
$335$	−13.7550	−0.751518
$336$	0	0
$337$	22.4060	1.22053	0.610267	−	0.792196i	$-0.291062\pi$
$337$	22.4060	1.22053	0.610267	+	0.792196i	$0.291062\pi$
$338$	0	0
$339$	0.964061	0.0523606
$340$	0	0
$341$	−10.1208	−0.548073
$342$	0	0
$343$	− 11.0105i	− 0.594509i
$344$	0	0
$345$	− 0.292496i	− 0.0157474i
$346$	0	0
$347$	20.1725	1.08291	0.541457	−	0.840728i	$-0.317873\pi$
$347$	20.1725	1.08291	0.541457	+	0.840728i	$0.317873\pi$
$348$	0	0
$349$	28.5943i	1.53062i	0.643662	+	0.765310i	$0.277414\pi$
$349$	28.5943i	1.53062i	−0.643662	+	0.765310i	$0.722586\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.1563i	0.593792i	0.954910	+	0.296896i	$0.0959514\pi$
$353$	11.1563i	0.593792i	−0.954910	+	0.296896i	$0.904049\pi$
$354$	0	0
$355$	−9.77689	−0.518904
$356$	0	0
$357$	− 0.112241i	− 0.00594045i
$358$	0	0
$359$	11.0490i	0.583145i	0.956549	+	0.291572i	$0.0941784\pi$
$359$	11.0490i	0.583145i	−0.956549	+	0.291572i	$0.905822\pi$
$360$	0	0
$361$	17.8564	0.939811
$362$	0	0
$363$	−0.758301	−0.0398005
$364$	0	0
$365$	11.1806	0.585219
$366$	0	0
$367$	24.4052	1.27394	0.636970	−	0.770889i	$-0.280188\pi$
$367$	24.4052	1.27394	0.636970	+	0.770889i	$0.280188\pi$
$368$	0	0
$369$	12.7655i	0.664545i
$370$	0	0
$371$	− 0.276418i	− 0.0143509i
$372$	0	0
$373$	−5.31132	−0.275010	−0.137505	−	0.990501i	$-0.543908\pi$
$373$	−5.31132	−0.275010	−0.137505	+	0.990501i	$0.543908\pi$
$374$	0	0
$375$	− 0.0947876i	− 0.00489481i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	33.5405i	1.72286i	0.507876	+	0.861430i	$0.330431\pi$
$379$	33.5405i	1.72286i	−0.507876	+	0.861430i	$0.669569\pi$
$380$	0	0
$381$	0.565695	0.0289814
$382$	0	0
$383$	− 23.8027i	− 1.21626i	−0.793836	−	0.608131i	$-0.791919\pi$
$383$	− 23.8027i	− 1.21626i	0.793836	−	0.608131i	$-0.208081\pi$
$384$	0	0
$385$	− 1.43213i	− 0.0729879i
$386$	0	0
$387$	−28.5370	−1.45062
$388$	0	0
$389$	−26.2787	−1.33238	−0.666191	−	0.745781i	$-0.732077\pi$
$389$	−26.2787	−1.33238	−0.666191	+	0.745781i	$0.732077\pi$
$390$	0	0
$391$	4.41926	0.223492
$392$	0	0
$393$	−1.57601	−0.0794990
$394$	0	0
$395$	0.252387i	0.0126990i
$396$	0	0
$397$	− 33.2920i	− 1.67088i	−0.549584	−	0.835439i	$-0.685214\pi$
$397$	− 33.2920i	− 1.67088i	0.549584	−	0.835439i	$-0.314786\pi$
$398$	0	0
$399$	−0.0838123	−0.00419586
$400$	0	0
$401$	14.1692i	0.707576i	0.935326	+	0.353788i	$0.115107\pi$
$401$	14.1692i	0.707576i	−0.935326	+	0.353788i	$0.884893\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 8.91922i	− 0.443200i
$406$	0	0
$407$	−1.70362	−0.0844454
$408$	0	0
$409$	− 6.84983i	− 0.338702i	−0.985556	−	0.169351i	$-0.945833\pi$
$409$	− 6.84983i	− 0.338702i	0.985556	−	0.169351i	$-0.0541672\pi$
$410$	0	0
$411$	− 0.0383462i	− 0.00189148i
$412$	0	0
$413$	9.52890	0.468887
$414$	0	0
$415$	5.67165	0.278410
$416$	0	0
$417$	0.883073	0.0432443
$418$	0	0
$419$	−16.3822	−0.800322	−0.400161	−	0.916445i	$-0.631046\pi$
$419$	−16.3822	−0.800322	−0.400161	+	0.916445i	$0.631046\pi$
$420$	0	0
$421$	− 21.7045i	− 1.05781i	−0.848681	−	0.528906i	$-0.822603\pi$
$421$	− 21.7045i	− 1.05781i	0.848681	−	0.528906i	$-0.177397\pi$
$422$	0	0
$423$	− 10.3612i	− 0.503778i
$424$	0	0
$425$	1.43213	0.0694683
$426$	0	0
$427$	− 2.24610i	− 0.108696i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.3888i	1.56011i	0.625710	+	0.780056i	$0.284809\pi$
$431$	32.3888i	1.56011i	−0.625710	+	0.780056i	$0.715191\pi$
$432$	0	0
$433$	−28.7975	−1.38392	−0.691959	−	0.721937i	$-0.743252\pi$
$433$	−28.7975	−1.38392	−0.691959	+	0.721937i	$0.743252\pi$
$434$	0	0
$435$	0.706653i	0.0338814i
$436$	0	0
$437$	− 3.29992i	− 0.157857i
$438$	0	0
$439$	17.5998	0.839995	0.419997	−	0.907525i	$-0.362031\pi$
$439$	17.5998	0.839995	0.419997	+	0.907525i	$0.362031\pi$
$440$	0	0
$441$	−18.8923	−0.899632
$442$	0	0
$443$	−14.4043	−0.684370	−0.342185	−	0.939633i	$-0.611167\pi$
$443$	−14.4043	−0.684370	−0.342185	+	0.939633i	$0.611167\pi$
$444$	0	0
$445$	4.59630	0.217886
$446$	0	0
$447$	1.03013i	0.0487236i
$448$	0	0
$449$	2.98861i	0.141041i	0.997510	+	0.0705206i	$0.0224660\pi$
$449$	2.98861i	0.141041i	−0.997510	+	0.0705206i	$0.977534\pi$
$450$	0	0
$451$	7.39230	0.348090
$452$	0	0
$453$	0.0939360i	0.00441350i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 26.6204i	− 1.24525i	−0.782520	−	0.622626i	$-0.786066\pi$
$457$	− 26.6204i	− 1.24525i	0.782520	−	0.622626i	$-0.213934\pi$
$458$	0	0
$459$	−0.813267	−0.0379601
$460$	0	0
$461$	− 8.38876i	− 0.390703i	−0.980733	−	0.195352i	$-0.937415\pi$
$461$	− 8.38876i	− 0.390703i	0.980733	−	0.195352i	$-0.0625848\pi$
$462$	0	0
$463$	− 21.3014i	− 0.989960i	−0.868904	−	0.494980i	$-0.835175\pi$
$463$	− 21.3014i	− 0.989960i	0.868904	−	0.494980i	$-0.164825\pi$
$464$	0	0
$465$	−0.553868	−0.0256850
$466$	0	0
$467$	2.12392	0.0982833	0.0491417	−	0.998792i	$-0.484351\pi$
$467$	2.12392	0.0982833	0.0491417	+	0.998792i	$0.484351\pi$
$468$	0	0
$469$	−11.3732	−0.525165
$470$	0	0
$471$	−1.66569	−0.0767511
$472$	0	0
$473$	16.5254i	0.759837i
$474$	0	0
$475$	− 1.06939i	− 0.0490669i
$476$	0	0
$477$	−0.999919	−0.0457832
$478$	0	0
$479$	− 29.0543i	− 1.32752i	−0.747944	−	0.663762i	$-0.768959\pi$
$479$	− 29.0543i	− 1.32752i	0.747944	−	0.663762i	$-0.231041\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 0.241847i	− 0.0110044i
$484$	0	0
$485$	−9.53434	−0.432932
$486$	0	0
$487$	30.3190i	1.37388i	0.726713	+	0.686941i	$0.241047\pi$
$487$	30.3190i	1.37388i	−0.726713	+	0.686941i	$0.758953\pi$
$488$	0	0
$489$	− 1.63811i	− 0.0740781i
$490$	0	0
$491$	−38.1519	−1.72177	−0.860884	−	0.508800i	$-0.830089\pi$
$491$	−38.1519	−1.72177	−0.860884	+	0.508800i	$0.830089\pi$
$492$	0	0
$493$	−10.6767	−0.480853
$494$	0	0
$495$	−5.18059	−0.232850
$496$	0	0
$497$	−8.08391	−0.362613
$498$	0	0
$499$	− 16.5179i	− 0.739444i	−0.929142	−	0.369722i	$-0.879453\pi$
$499$	− 16.5179i	− 0.739444i	0.929142	−	0.369722i	$-0.120547\pi$
$500$	0	0
$501$	0.290008i	0.0129566i
$502$	0	0
$503$	11.7616	0.524425	0.262212	−	0.965010i	$-0.415548\pi$
$503$	11.7616	0.524425	0.262212	+	0.965010i	$0.415548\pi$
$504$	0	0
$505$	− 5.80144i	− 0.258161i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.8617i	1.41224i	0.708091	+	0.706122i	$0.249557\pi$
$509$	31.8617i	1.41224i	−0.708091	+	0.706122i	$0.750443\pi$
$510$	0	0
$511$	9.24454	0.408954
$512$	0	0
$513$	0.607278i	0.0268120i
$514$	0	0
$515$	− 10.0760i	− 0.444000i
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−0.324184	−0.0142301
$520$	0	0
$521$	−19.5013	−0.854367	−0.427183	−	0.904165i	$-0.640494\pi$
$521$	−19.5013	−0.854367	−0.427183	+	0.904165i	$0.640494\pi$
$522$	0	0
$523$	−44.5659	−1.94873	−0.974365	−	0.224971i	$-0.927771\pi$
$523$	−44.5659	−1.94873	−0.974365	+	0.224971i	$0.927771\pi$
$524$	0	0
$525$	− 0.0783740i	− 0.00342052i
$526$	0	0
$527$	− 8.36827i	− 0.364528i
$528$	0	0
$529$	−13.4778	−0.585992
$530$	0	0
$531$	− 34.4700i	− 1.49587i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 16.2795i	− 0.703826i
$536$	0	0
$537$	−0.984102	−0.0424671
$538$	0	0
$539$	10.9402i	0.471229i
$540$	0	0
$541$	− 3.74450i	− 0.160989i	−0.996755	−	0.0804943i	$-0.974350\pi$
$541$	− 3.74450i	− 0.160989i	0.996755	−	0.0804943i	$-0.0256499\pi$
$542$	0	0
$543$	0.980972	0.0420976
$544$	0	0
$545$	3.12979	0.134066
$546$	0	0
$547$	38.3803	1.64102	0.820511	−	0.571630i	$-0.193689\pi$
$547$	38.3803	1.64102	0.820511	+	0.571630i	$0.193689\pi$
$548$	0	0
$549$	−8.12506	−0.346769
$550$	0	0
$551$	7.97242i	0.339637i
$552$	0	0
$553$	0.208683i	0.00887412i
$554$	0	0
$555$	−0.0932318	−0.00395747
$556$	0	0
$557$	− 26.6462i	− 1.12903i	−0.825421	−	0.564517i	$-0.809062\pi$
$557$	− 26.6462i	− 1.12903i	0.825421	−	0.564517i	$-0.190938\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.235122i	0.00992686i
$562$	0	0
$563$	16.6935	0.703547	0.351774	−	0.936085i	$-0.385579\pi$
$563$	16.6935	0.703547	0.351774	+	0.936085i	$0.385579\pi$
$564$	0	0
$565$	10.1708i	0.427887i
$566$	0	0
$567$	− 7.37475i	− 0.309710i
$568$	0	0
$569$	43.9240	1.84139	0.920694	−	0.390285i	$-0.127624\pi$
$569$	43.9240	1.84139	0.920694	+	0.390285i	$0.127624\pi$
$570$	0	0
$571$	−11.4641	−0.479758	−0.239879	−	0.970803i	$-0.577108\pi$
$571$	−11.4641	−0.479758	−0.239879	+	0.970803i	$0.577108\pi$
$572$	0	0
$573$	1.46977	0.0614004
$574$	0	0
$575$	3.08580	0.128687
$576$	0	0
$577$	− 44.9354i	− 1.87069i	−0.353743	−	0.935343i	$-0.615091\pi$
$577$	− 44.9354i	− 1.87069i	0.353743	−	0.935343i	$-0.384909\pi$
$578$	0	0
$579$	− 0.527994i	− 0.0219427i
$580$	0	0
$581$	4.68953	0.194555
$582$	0	0
$583$	0.579038i	0.0239813i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 24.5217i	− 1.01212i	−0.862499	−	0.506059i	$-0.831102\pi$
$587$	− 24.5217i	− 1.01212i	0.862499	−	0.506059i	$-0.168898\pi$
$588$	0	0
$589$	−6.24871	−0.257474
$590$	0	0
$591$	2.35763i	0.0969801i
$592$	0	0
$593$	18.7655i	0.770607i	0.922790	+	0.385303i	$0.125903\pi$
$593$	18.7655i	0.770607i	−0.922790	+	0.385303i	$0.874097\pi$
$594$	0	0
$595$	1.18414	0.0485449
$596$	0	0
$597$	−1.76768	−0.0723464
$598$	0	0
$599$	−35.9293	−1.46803	−0.734015	−	0.679133i	$-0.762356\pi$
$599$	−35.9293	−1.46803	−0.734015	+	0.679133i	$0.762356\pi$
$600$	0	0
$601$	−39.7726	−1.62236	−0.811179	−	0.584798i	$-0.801174\pi$
$601$	−39.7726	−1.62236	−0.811179	+	0.584798i	$0.801174\pi$
$602$	0	0
$603$	41.1415i	1.67541i
$604$	0	0
$605$	− 8.00000i	− 0.325246i
$606$	0	0
$607$	33.4613	1.35815	0.679076	−	0.734068i	$-0.262381\pi$
$607$	33.4613	1.35815	0.679076	+	0.734068i	$0.262381\pi$
$608$	0	0
$609$	0.584287i	0.0236765i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	19.7699i	0.798499i	0.916842	+	0.399249i	$0.130729\pi$
$613$	19.7699i	0.798499i	−0.916842	+	0.399249i	$0.869271\pi$
$614$	0	0
$615$	0.404549	0.0163130
$616$	0	0
$617$	19.5087i	0.785391i	0.919668	+	0.392696i	$0.128457\pi$
$617$	19.5087i	0.785391i	−0.919668	+	0.392696i	$0.871543\pi$
$618$	0	0
$619$	− 40.4640i	− 1.62639i	−0.581994	−	0.813193i	$-0.697727\pi$
$619$	− 40.4640i	− 1.62639i	0.581994	−	0.813193i	$-0.302273\pi$
$620$	0	0
$621$	−1.75235	−0.0703193
$622$	0	0
$623$	3.80040	0.152260
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.175569	0.00701155
$628$	0	0
$629$	− 1.40862i	− 0.0561653i
$630$	0	0
$631$	19.5961i	0.780109i	0.920792	+	0.390054i	$0.127544\pi$
$631$	19.5961i	0.780109i	−0.920792	+	0.390054i	$0.872456\pi$
$632$	0	0
$633$	−0.914197	−0.0363361
$634$	0	0
$635$	5.96802i	0.236834i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	29.2428i	1.15683i
$640$	0	0
$641$	−31.0477	−1.22631	−0.613155	−	0.789963i	$-0.710100\pi$
$641$	−31.0477	−1.22631	−0.613155	+	0.789963i	$0.710100\pi$
$642$	0	0
$643$	5.94659i	0.234511i	0.993102	+	0.117255i	$0.0374096\pi$
$643$	5.94659i	0.234511i	−0.993102	+	0.117255i	$0.962590\pi$
$644$	0	0
$645$	0.904361i	0.0356092i
$646$	0	0
$647$	−42.5020	−1.67092	−0.835462	−	0.549548i	$-0.814800\pi$
$647$	−42.5020	−1.67092	−0.835462	+	0.549548i	$0.814800\pi$
$648$	0	0
$649$	−19.9610	−0.783539
$650$	0	0
$651$	−0.457959	−0.0179488
$652$	0	0
$653$	−30.6107	−1.19789	−0.598945	−	0.800790i	$-0.704413\pi$
$653$	−30.6107	−1.19789	−0.598945	+	0.800790i	$0.704413\pi$
$654$	0	0
$655$	− 16.6267i	− 0.649660i
$656$	0	0
$657$	− 33.4413i	− 1.30467i
$658$	0	0
$659$	−25.4383	−0.990935	−0.495467	−	0.868626i	$-0.665003\pi$
$659$	−25.4383	−0.990935	−0.495467	+	0.868626i	$0.665003\pi$
$660$	0	0
$661$	− 0.333603i	− 0.0129757i	−0.999979	−	0.00648784i	$-0.997935\pi$
$661$	− 0.333603i	− 0.0129757i	0.999979	−	0.00648784i	$-0.00206516\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 0.884212i	− 0.0342883i
$666$	0	0
$667$	−23.0050	−0.890758
$668$	0	0
$669$	0.110062i	0.00425525i
$670$	0	0
$671$	4.70510i	0.181638i
$672$	0	0
$673$	−9.81412	−0.378306	−0.189153	−	0.981948i	$-0.560574\pi$
$673$	−9.81412	−0.378306	−0.189153	+	0.981948i	$0.560574\pi$
$674$	0	0
$675$	−0.567874	−0.0218575
$676$	0	0
$677$	−23.2414	−0.893241	−0.446620	−	0.894724i	$-0.647373\pi$
$677$	−23.2414	−0.893241	−0.446620	+	0.894724i	$0.647373\pi$
$678$	0	0
$679$	−7.88336	−0.302536
$680$	0	0
$681$	− 2.58258i	− 0.0989648i
$682$	0	0
$683$	− 5.26710i	− 0.201540i	−0.994910	−	0.100770i	$-0.967869\pi$
$683$	− 5.26710i	− 0.201540i	0.994910	−	0.100770i	$-0.0321306\pi$
$684$	0	0
$685$	0.404549	0.0154570
$686$	0	0
$687$	− 2.30743i	− 0.0880341i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	19.7840i	0.752618i	0.926494	+	0.376309i	$0.122807\pi$
$691$	19.7840i	0.752618i	−0.926494	+	0.376309i	$0.877193\pi$
$692$	0	0
$693$	−4.28351	−0.162717
$694$	0	0
$695$	9.31634i	0.353389i
$696$	0	0
$697$	6.11224i	0.231518i
$698$	0	0
$699$	2.18237	0.0825450
$700$	0	0
$701$	18.1256	0.684595	0.342298	−	0.939592i	$-0.388795\pi$
$701$	18.1256	0.684595	0.342298	+	0.939592i	$0.388795\pi$
$702$	0	0
$703$	−1.05184	−0.0396708
$704$	0	0
$705$	−0.328354	−0.0123665
$706$	0	0
$707$	− 4.79685i	− 0.180404i
$708$	0	0
$709$	25.8092i	0.969283i	0.874713	+	0.484642i	$0.161050\pi$
$709$	25.8092i	0.969283i	−0.874713	+	0.484642i	$0.838950\pi$
$710$	0	0
$711$	0.754894	0.0283107
$712$	0	0
$713$	− 18.0311i	− 0.675271i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.24701i	− 0.0839161i
$718$	0	0
$719$	9.02677	0.336642	0.168321	−	0.985732i	$-0.446166\pi$
$719$	9.02677	0.336642	0.168321	+	0.985732i	$0.446166\pi$
$720$	0	0
$721$	− 8.33120i	− 0.310270i
$722$	0	0
$723$	− 1.02662i	− 0.0381803i
$724$	0	0
$725$	−7.45512	−0.276876
$726$	0	0
$727$	24.8934	0.923245	0.461623	−	0.887076i	$-0.347267\pi$
$727$	24.8934	0.923245	0.461623	+	0.887076i	$0.347267\pi$
$728$	0	0
$729$	−26.5160	−0.982075
$730$	0	0
$731$	−13.6638	−0.505374
$732$	0	0
$733$	− 13.2793i	− 0.490484i	−0.969462	−	0.245242i	$-0.921133\pi$
$733$	− 13.2793i	− 0.490484i	0.969462	−	0.245242i	$-0.0788674\pi$
$734$	0	0
$735$	0.598710i	0.0220838i
$736$	0	0
$737$	23.8244	0.877584
$738$	0	0
$739$	19.5902i	0.720636i	0.932830	+	0.360318i	$0.117332\pi$
$739$	19.5902i	0.720636i	−0.932830	+	0.360318i	$0.882668\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	51.2460i	1.88003i	0.341128	+	0.940017i	$0.389191\pi$
$743$	51.2460i	1.88003i	−0.341128	+	0.940017i	$0.610809\pi$
$744$	0	0
$745$	−10.8678	−0.398165
$746$	0	0
$747$	− 16.9640i	− 0.620680i
$748$	0	0
$749$	− 13.4606i	− 0.491838i
$750$	0	0
$751$	21.1985	0.773543	0.386772	−	0.922175i	$-0.373590\pi$
$751$	21.1985	0.773543	0.386772	+	0.922175i	$0.373590\pi$
$752$	0	0
$753$	−0.106239	−0.00387156
$754$	0	0
$755$	−0.991015	−0.0360667
$756$	0	0
$757$	−32.9494	−1.19757	−0.598783	−	0.800911i	$-0.704349\pi$
$757$	−32.9494	−1.19757	−0.598783	+	0.800911i	$0.704349\pi$
$758$	0	0
$759$	0.506618i	0.0183891i
$760$	0	0
$761$	− 52.2349i	− 1.89351i	−0.321952	−	0.946756i	$-0.604339\pi$
$761$	− 52.2349i	− 1.89351i	0.321952	−	0.946756i	$-0.395661\pi$
$762$	0	0
$763$	2.58783	0.0936859
$764$	0	0
$765$	− 4.28351i	− 0.154871i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	26.8719i	0.969025i	0.874784	+	0.484513i	$0.161003\pi$
$769$	26.8719i	0.969025i	−0.874784	+	0.484513i	$0.838997\pi$
$770$	0	0
$771$	1.57638	0.0567720
$772$	0	0
$773$	− 11.9549i	− 0.429989i	−0.976615	−	0.214994i	$-0.931027\pi$
$773$	− 11.9549i	− 0.429989i	0.976615	−	0.214994i	$-0.0689733\pi$
$774$	0	0
$775$	− 5.84325i	− 0.209896i
$776$	0	0
$777$	−0.0770876	−0.00276550
$778$	0	0
$779$	4.56410	0.163526
$780$	0	0
$781$	16.9341	0.605949
$782$	0	0
$783$	4.23357	0.151295
$784$	0	0
$785$	− 17.5729i	− 0.627204i
$786$	0	0
$787$	− 34.4937i	− 1.22957i	−0.788696	−	0.614783i	$-0.789243\pi$
$787$	− 34.4937i	− 1.22957i	0.788696	−	0.614783i	$-0.210757\pi$
$788$	0	0
$789$	−2.32248	−0.0826825
$790$	0	0
$791$	8.40957i	0.299010i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.0316882i	0.00112387i
$796$	0	0
$797$	−33.2049	−1.17618	−0.588089	−	0.808796i	$-0.700120\pi$
$797$	−33.2049	−1.17618	−0.588089	+	0.808796i	$0.700120\pi$
$798$	0	0
$799$	− 4.96103i	− 0.175509i
$800$	0	0
$801$	− 13.7476i	− 0.485748i
$802$	0	0
$803$	−19.3654	−0.683388
$804$	0	0
$805$	2.55146	0.0899272
$806$	0	0
$807$	0.619235	0.0217981
$808$	0	0
$809$	−8.70277	−0.305973	−0.152987	−	0.988228i	$-0.548889\pi$
$809$	−8.70277	−0.305973	−0.152987	+	0.988228i	$0.548889\pi$
$810$	0	0
$811$	− 7.69132i	− 0.270079i	−0.990840	−	0.135039i	$-0.956884\pi$
$811$	− 7.69132i	− 0.270079i	0.990840	−	0.135039i	$-0.0431161\pi$
$812$	0	0
$813$	− 0.536127i	− 0.0188028i
$814$	0	0
$815$	17.2820	0.605360
$816$	0	0
$817$	10.2030i	0.356956i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 0.615411i	− 0.0214780i	−0.999942	−	0.0107390i	$-0.996582\pi$
$821$	− 0.615411i	− 0.0214780i	0.999942	−	0.0107390i	$-0.00341840\pi$
$822$	0	0
$823$	22.3375	0.778638	0.389319	−	0.921103i	$-0.372710\pi$
$823$	22.3375	0.778638	0.389319	+	0.921103i	$0.372710\pi$
$824$	0	0
$825$	0.164177i	0.00571591i
$826$	0	0
$827$	− 33.8701i	− 1.17778i	−0.808213	−	0.588890i	$-0.799565\pi$
$827$	− 33.8701i	− 1.17778i	0.808213	−	0.588890i	$-0.200435\pi$
$828$	0	0
$829$	34.5293	1.19925	0.599626	−	0.800280i	$-0.295316\pi$
$829$	34.5293	1.19925	0.599626	+	0.800280i	$0.295316\pi$
$830$	0	0
$831$	0.352196	0.0122176
$832$	0	0
$833$	−9.04579	−0.313418
$834$	0	0
$835$	−3.05955	−0.105880
$836$	0	0
$837$	3.31823i	0.114695i
$838$	0	0
$839$	29.9513i	1.03404i	0.855975	+	0.517018i	$0.172958\pi$
$839$	29.9513i	1.03404i	−0.855975	+	0.517018i	$0.827042\pi$
$840$	0	0
$841$	26.5788	0.916509
$842$	0	0
$843$	− 0.919864i	− 0.0316818i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6.61471i	− 0.227284i
$848$	0	0
$849$	−2.29363	−0.0787173
$850$	0	0
$851$	− 3.03515i	− 0.104044i
$852$	0	0
$853$	54.1009i	1.85238i	0.377059	+	0.926189i	$0.376935\pi$
$853$	54.1009i	1.85238i	−0.377059	+	0.926189i	$0.623065\pi$
$854$	0	0
$855$	−3.19856	−0.109388
$856$	0	0
$857$	−16.4383	−0.561521	−0.280761	−	0.959778i	$-0.590587\pi$
$857$	−16.4383	−0.561521	−0.280761	+	0.959778i	$0.590587\pi$
$858$	0	0
$859$	32.7187	1.11635	0.558174	−	0.829724i	$-0.311502\pi$
$859$	32.7187	1.11635	0.558174	+	0.829724i	$0.311502\pi$
$860$	0	0
$861$	0.334496	0.0113996
$862$	0	0
$863$	− 55.7922i	− 1.89919i	−0.313482	−	0.949594i	$-0.601496\pi$
$863$	− 55.7922i	− 1.89919i	0.313482	−	0.949594i	$-0.398504\pi$
$864$	0	0
$865$	− 3.42011i	− 0.116287i
$866$	0	0
$867$	1.41698	0.0481232
$868$	0	0
$869$	− 0.437148i	− 0.0148292i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	28.5174i	0.965167i
$874$	0	0
$875$	0.826838	0.0279522
$876$	0	0
$877$	38.6921i	1.30654i	0.757125	+	0.653271i	$0.226604\pi$
$877$	38.6921i	1.30654i	−0.757125	+	0.653271i	$0.773396\pi$
$878$	0	0
$879$	− 0.343916i	− 0.0116000i
$880$	0	0
$881$	5.42115	0.182643	0.0913216	−	0.995821i	$-0.470891\pi$
$881$	5.42115	0.182643	0.0913216	+	0.995821i	$0.470891\pi$
$882$	0	0
$883$	21.2583	0.715397	0.357699	−	0.933837i	$-0.383561\pi$
$883$	21.2583	0.715397	0.357699	+	0.933837i	$0.383561\pi$
$884$	0	0
$885$	−1.09238	−0.0367200
$886$	0	0
$887$	0.566171	0.0190101	0.00950507	−	0.999955i	$-0.496974\pi$
$887$	0.566171	0.0190101	0.00950507	+	0.999955i	$0.496974\pi$
$888$	0	0
$889$	4.93459i	0.165501i
$890$	0	0
$891$	15.4485i	0.517546i
$892$	0	0
$893$	−3.70447	−0.123965
$894$	0	0
$895$	− 10.3822i	− 0.347038i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43.5621i	1.45288i
$900$	0	0
$901$	−0.478771	−0.0159502
$902$	0	0
$903$	0.747760i	0.0248839i
$904$	0	0
$905$	10.3492i	0.344018i
$906$	0	0
$907$	−37.0516	−1.23028	−0.615139	−	0.788419i	$-0.710900\pi$
$907$	−37.0516	−1.23028	−0.615139	+	0.788419i	$0.710900\pi$
$908$	0	0
$909$	−17.3522	−0.575536
$910$	0	0
$911$	−27.9952	−0.927522	−0.463761	−	0.885960i	$-0.653500\pi$
$911$	−27.9952	−0.927522	−0.463761	+	0.885960i	$0.653500\pi$
$912$	0	0
$913$	−9.82358	−0.325113
$914$	0	0
$915$	0.257489i	0.00851234i
$916$	0	0
$917$	− 13.7476i	− 0.453986i
$918$	0	0
$919$	36.9441	1.21867	0.609337	−	0.792911i	$-0.291436\pi$
$919$	36.9441	1.21867	0.609337	+	0.792911i	$0.291436\pi$
$920$	0	0
$921$	0.891126i	0.0293636i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 0.983586i	− 0.0323401i
$926$	0	0
$927$	−30.1374	−0.989841
$928$	0	0
$929$	− 13.3290i	− 0.437310i	−0.975802	−	0.218655i	$-0.929833\pi$
$929$	− 13.3290i	− 0.437310i	0.975802	−	0.218655i	$-0.0701669\pi$
$930$	0	0
$931$	6.75462i	0.221374i
$932$	0	0
$933$	−2.42059	−0.0792464
$934$	0	0
$935$	−2.48052	−0.0811215
$936$	0	0
$937$	13.0922	0.427702	0.213851	−	0.976866i	$-0.431399\pi$
$937$	13.0922	0.427702	0.213851	+	0.976866i	$0.431399\pi$
$938$	0	0
$939$	−0.498256	−0.0162600
$940$	0	0
$941$	43.0399i	1.40306i	0.712639	+	0.701531i	$0.247500\pi$
$941$	43.0399i	1.40306i	−0.712639	+	0.701531i	$0.752500\pi$
$942$	0	0
$943$	13.1700i	0.428876i
$944$	0	0
$945$	−0.469540	−0.0152741
$946$	0	0
$947$	− 30.4045i	− 0.988015i	−0.869458	−	0.494008i	$-0.835532\pi$
$947$	− 30.4045i	− 0.988015i	0.869458	−	0.494008i	$-0.164468\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1.34159i	0.0435040i
$952$	0	0
$953$	21.4022	0.693287	0.346643	−	0.937997i	$-0.387321\pi$
$953$	21.4022	0.693287	0.346643	+	0.937997i	$0.387321\pi$
$954$	0	0
$955$	15.5059i	0.501760i
$956$	0	0
$957$	− 1.22396i	− 0.0395649i
$958$	0	0
$959$	0.334496	0.0108014
$960$	0	0
$961$	−3.14359	−0.101406
$962$	0	0
$963$	−48.6924	−1.56909
$964$	0	0
$965$	5.57028	0.179314
$966$	0	0
$967$	− 41.8892i	− 1.34707i	−0.739157	−	0.673533i	$-0.764776\pi$
$967$	− 41.8892i	− 1.34707i	0.739157	−	0.673533i	$-0.235224\pi$
$968$	0	0
$969$	0.145167i	0.00466344i
$970$	0	0
$971$	−28.8253	−0.925047	−0.462524	−	0.886607i	$-0.653056\pi$
$971$	−28.8253	−0.925047	−0.462524	+	0.886607i	$0.653056\pi$
$972$	0	0
$973$	7.70311i	0.246950i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.4678i	− 1.74258i	−0.490770	−	0.871289i	$-0.663284\pi$
$977$	− 54.4678i	− 1.74258i	0.490770	−	0.871289i	$-0.336716\pi$
$978$	0	0
$979$	−7.96103	−0.254436
$980$	0	0
$981$	− 9.36126i	− 0.298882i
$982$	0	0
$983$	− 56.5991i	− 1.80523i	−0.430447	−	0.902616i	$-0.641644\pi$
$983$	− 56.5991i	− 1.80523i	0.430447	−	0.902616i	$-0.358356\pi$
$984$	0	0
$985$	−24.8728	−0.792514
$986$	0	0
$987$	−0.271496	−0.00864180
$988$	0	0
$989$	−29.4414	−0.936182
$990$	0	0
$991$	22.6925	0.720850	0.360425	−	0.932788i	$-0.382632\pi$
$991$	22.6925	0.720850	0.360425	+	0.932788i	$0.382632\pi$
$992$	0	0
$993$	− 1.76156i	− 0.0559014i
$994$	0	0
$995$	− 18.6489i	− 0.591209i
$996$	0	0
$997$	46.7228	1.47972	0.739862	−	0.672758i	$-0.234891\pi$
$997$	46.7228	1.47972	0.739862	+	0.672758i	$0.234891\pi$
$998$	0	0
$999$	0.558553i	0.0176718i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.f.i.3041.3		8
13.5	odd	4		3380.2.a.p.1.2		4
13.8	odd	4		3380.2.a.q.1.2		4
13.9	even	3		260.2.x.a.101.3	✓	8
13.10	even	6		260.2.x.a.121.3	yes	8
13.12	even	2	inner	3380.2.f.i.3041.4		8
39.23	odd	6		2340.2.dj.d.901.2		8
39.35	odd	6		2340.2.dj.d.361.4		8
52.23	odd	6		1040.2.da.c.641.2		8
52.35	odd	6		1040.2.da.c.881.2		8
65.9	even	6		1300.2.y.b.101.2		8
65.22	odd	12		1300.2.ba.b.49.3		8
65.23	odd	12		1300.2.ba.b.849.3		8
65.48	odd	12		1300.2.ba.c.49.2		8
65.49	even	6		1300.2.y.b.901.2		8
65.62	odd	12		1300.2.ba.c.849.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.x.a.101.3	✓	8	13.9	even	3
260.2.x.a.121.3	yes	8	13.10	even	6
1040.2.da.c.641.2		8	52.23	odd	6
1040.2.da.c.881.2		8	52.35	odd	6
1300.2.y.b.101.2		8	65.9	even	6
1300.2.y.b.901.2		8	65.49	even	6
1300.2.ba.b.49.3		8	65.22	odd	12
1300.2.ba.b.849.3		8	65.23	odd	12
1300.2.ba.c.49.2		8	65.48	odd	12
1300.2.ba.c.849.2		8	65.62	odd	12
2340.2.dj.d.361.4		8	39.35	odd	6
2340.2.dj.d.901.2		8	39.23	odd	6
3380.2.a.p.1.2		4	13.5	odd	4
3380.2.a.q.1.2		4	13.8	odd	4
3380.2.f.i.3041.3		8	1.1	even	1	trivial
3380.2.f.i.3041.4		8	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.0947876 q^{3} -1.00000i q^{5} -0.826838i q^{7} -2.99102 q^{9} +O(q^{10})\)	\(q-0.0947876 q^{3} -1.00000i q^{5} -0.826838i q^{7} -2.99102 q^{9} +1.73205i q^{11} +0.0947876i q^{15} -1.43213 q^{17} +1.06939i q^{19} +0.0783740i q^{21} -3.08580 q^{23} -1.00000 q^{25} +0.567874 q^{27} +7.45512 q^{29} +5.84325i q^{31} -0.164177i q^{33} -0.826838 q^{35} +0.983586i q^{37} -4.26795i q^{41} +9.54092 q^{43} +2.99102i q^{45} +3.46410i q^{47} +6.31634 q^{49} +0.135748 q^{51} +0.334308 q^{53} +1.73205 q^{55} -0.101365i q^{57} +11.5245i q^{59} +2.71649 q^{61} +2.47309i q^{63} -13.7550i q^{67} +0.292496 q^{69} -9.77689i q^{71} +11.1806i q^{73} +0.0947876 q^{75} +1.43213 q^{77} -0.252387 q^{79} +8.91922 q^{81} +5.67165i q^{83} +1.43213i q^{85} -0.706653 q^{87} +4.59630i q^{89} -0.553868i q^{93} +1.06939 q^{95} -9.53434i q^{97} -5.18059i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 8 * q^9	\( 8 q + 4 q^{3} + 8 q^{9} - 12 q^{17} + 12 q^{23} - 8 q^{25} + 4 q^{27} + 12 q^{35} - 20 q^{43} + 8 q^{49} + 24 q^{53} + 8 q^{61} + 48 q^{69} - 4 q^{75} + 12 q^{77} - 16 q^{79} - 16 q^{81} + 12 q^{87}+O(q^{100}) \) 8 * q + 4 * q^3 + 8 * q^9 - 12 * q^17 + 12 * q^23 - 8 * q^25 + 4 * q^27 + 12 * q^35 - 20 * q^43 + 8 * q^49 + 24 * q^53 + 8 * q^61 + 48 * q^69 - 4 * q^75 + 12 * q^77 - 16 * q^79 - 16 * q^81 + 12 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more