LMFDB - Embedded newform 3380.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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, 0]))

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

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

chi := DirichletCharacter("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.a (trivial)

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:	yes
Analytic conductor:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} -1.00000 q^{15} -3.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +4.00000 q^{31} -3.00000 q^{33} -1.00000 q^{35} +7.00000 q^{37} +9.00000 q^{41} +11.0000 q^{43} +2.00000 q^{45} -6.00000 q^{49} -3.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} +7.00000 q^{57} +3.00000 q^{59} +11.0000 q^{61} -2.00000 q^{63} +7.00000 q^{67} -3.00000 q^{69} +3.00000 q^{71} -2.00000 q^{73} +1.00000 q^{75} -3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +3.00000 q^{85} +3.00000 q^{87} -15.0000 q^{89} +4.00000 q^{93} -7.00000 q^{95} +7.00000 q^{97} +6.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	7.00000	0.927173
$58$	0	0
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−7.00000	−0.718185
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	9.00000	0.811503
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.0000	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$128$	0	0
$129$	11.0000	0.968496
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	7.00000	0.606977
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.00000	−0.249136
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−14.0000	−1.07061
$172$	0	0
$173$	−3.00000	−0.228086	−0.114043	−	0.993476i	$-0.536380\pi$
$173$	−3.00000	−0.228086	−0.114043	+	0.993476i	$0.536380\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	3.00000	0.225494
$178$	0	0
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	11.0000	0.813143
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	9.00000	0.658145
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	7.00000	0.493742
$202$	0	0
$203$	3.00000	0.210559
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−21.0000	−1.45260
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	0	0
$213$	3.00000	0.205557
$214$	0	0
$215$	−11.0000	−0.750194
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	−27.0000	−1.79205	−0.896026	−	0.444001i	$-0.853559\pi$
$227$	−27.0000	−1.79205	−0.896026	+	0.444001i	$0.853559\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−15.0000	−0.917985
$268$	0	0
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	0	0
$271$	−23.0000	−1.39715	−0.698575	−	0.715537i	$-0.746182\pi$
$271$	−23.0000	−1.39715	−0.698575	+	0.715537i	$0.746182\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	0	0
$285$	−7.00000	−0.414644
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	7.00000	0.410347
$292$	0	0
$293$	27.0000	1.57736	0.788678	−	0.614806i	$-0.210766\pi$
$293$	27.0000	1.57736	0.788678	+	0.614806i	$0.210766\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	0	0
$297$	15.0000	0.870388
$298$	0	0
$299$	0	0
$300$	0	0
$301$	11.0000	0.634029
$302$	0	0
$303$	−9.00000	−0.517036
$304$	0	0
$305$	−11.0000	−0.629858
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	−21.0000	−1.16847
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.0000	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$332$	0	0
$333$	−14.0000	−0.767195
$334$	0	0
$335$	−7.00000	−0.382451
$336$	0	0
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	0	0
$339$	9.00000	0.488813
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−33.0000	−1.77153	−0.885766	−	0.464131i	$-0.846367\pi$
$347$	−33.0000	−1.77153	−0.885766	+	0.464131i	$0.846367\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	0	0
$355$	−3.00000	−0.159223
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	5.00000	0.260998	0.130499	−	0.991448i	$-0.458342\pi$
$367$	5.00000	0.260998	0.130499	+	0.991448i	$0.458342\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−31.0000	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1.00000	0.0513665	0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000	0.0513665	0.0256833	+	0.999670i	$0.491824\pi$
$380$	0	0
$381$	−19.0000	−0.973399
$382$	0	0
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	0	0
$387$	−22.0000	−1.11832
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−5.00000	−0.250943	−0.125471	−	0.992097i	$-0.540044\pi$
$397$	−5.00000	−0.250943	−0.125471	+	0.992097i	$0.540044\pi$
$398$	0	0
$399$	7.00000	0.350438
$400$	0	0
$401$	33.0000	1.64794	0.823971	−	0.566632i	$-0.191754\pi$
$401$	33.0000	1.64794	0.823971	+	0.566632i	$0.191754\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−21.0000	−1.04093
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	15.0000	0.739895
$412$	0	0
$413$	3.00000	0.147620
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	5.00000	0.244851
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	11.0000	0.532327
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.00000	0.433515	0.216757	−	0.976226i	$-0.430452\pi$
$431$	9.00000	0.433515	0.216757	+	0.976226i	$0.430452\pi$
$432$	0	0
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	0	0
$437$	−21.0000	−1.00457
$438$	0	0
$439$	11.0000	0.525001	0.262501	−	0.964932i	$-0.415453\pi$
$439$	11.0000	0.525001	0.262501	+	0.964932i	$0.415453\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	0	0
$447$	21.0000	0.993266
$448$	0	0
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	−33.0000	−1.51734
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−7.00000	−0.317854
$486$	0	0
$487$	−11.0000	−0.498458	−0.249229	−	0.968445i	$-0.580177\pi$
$487$	−11.0000	−0.498458	−0.249229	+	0.968445i	$0.580177\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	3.00000	0.134568
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	0	0
$503$	15.0000	0.668817	0.334408	−	0.942428i	$-0.391463\pi$
$503$	15.0000	0.668817	0.334408	+	0.942428i	$0.391463\pi$
$504$	0	0
$505$	9.00000	0.400495
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	−35.0000	−1.54529
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−9.00000	−0.389104
$536$	0	0
$537$	−21.0000	−0.906217
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	0	0
$551$	21.0000	0.894630
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	−7.00000	−0.297133
$556$	0	0
$557$	39.0000	1.65248	0.826242	−	0.563316i	$-0.190475\pi$
$557$	39.0000	1.65248	0.826242	+	0.563316i	$0.190475\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	9.00000	0.379980
$562$	0	0
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	−9.00000	−0.378633
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	−3.00000	−0.125327
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	−5.00000	−0.207793
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	18.0000	0.745484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.0000	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$588$	0	0
$589$	28.0000	1.15372
$590$	0	0
$591$	−21.0000	−0.863825
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	17.0000	0.695764
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	−14.0000	−0.570124
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	0	0
$612$	0	0
$613$	43.0000	1.73675	0.868377	−	0.495905i	$-0.165164\pi$
$613$	43.0000	1.73675	0.868377	+	0.495905i	$0.165164\pi$
$614$	0	0
$615$	−9.00000	−0.362915
$616$	0	0
$617$	−33.0000	−1.32853	−0.664265	−	0.747497i	$-0.731255\pi$
$617$	−33.0000	−1.32853	−0.664265	+	0.747497i	$0.731255\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	15.0000	0.601929
$622$	0	0
$623$	−15.0000	−0.600962
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−21.0000	−0.838659
$628$	0	0
$629$	−21.0000	−0.837325
$630$	0	0
$631$	−17.0000	−0.676759	−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000	−0.676759	−0.338380	+	0.941010i	$0.609879\pi$
$632$	0	0
$633$	11.0000	0.437211
$634$	0	0
$635$	19.0000	0.753992
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	0	0
$643$	−11.0000	−0.433798	−0.216899	−	0.976194i	$-0.569594\pi$
$643$	−11.0000	−0.433798	−0.216899	+	0.976194i	$0.569594\pi$
$644$	0	0
$645$	−11.0000	−0.433125
$646$	0	0
$647$	45.0000	1.76913	0.884566	−	0.466415i	$-0.154454\pi$
$647$	45.0000	1.76913	0.884566	+	0.466415i	$0.154454\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	−39.0000	−1.52619	−0.763094	−	0.646288i	$-0.776321\pi$
$653$	−39.0000	−1.52619	−0.763094	+	0.646288i	$0.776321\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	−39.0000	−1.51922	−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000	−1.51922	−0.759612	+	0.650376i	$0.774611\pi$
$660$	0	0
$661$	1.00000	0.0388955	0.0194477	−	0.999811i	$-0.493809\pi$
$661$	1.00000	0.0388955	0.0194477	+	0.999811i	$0.493809\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.00000	−0.271448
$666$	0	0
$667$	−9.00000	−0.348481
$668$	0	0
$669$	19.0000	0.734582
$670$	0	0
$671$	−33.0000	−1.27395
$672$	0	0
$673$	17.0000	0.655302	0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000	0.655302	0.327651	+	0.944799i	$0.393743\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	7.00000	0.268635
$680$	0	0
$681$	−27.0000	−1.03464
$682$	0	0
$683$	−51.0000	−1.95146	−0.975730	−	0.218975i	$-0.929729\pi$
$683$	−51.0000	−1.95146	−0.975730	+	0.218975i	$0.929729\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	0	0
$687$	22.0000	0.839352
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−23.0000	−0.874961	−0.437481	−	0.899228i	$-0.644129\pi$
$691$	−23.0000	−0.874961	−0.437481	+	0.899228i	$0.644129\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	−5.00000	−0.189661
$696$	0	0
$697$	−27.0000	−1.02270
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	49.0000	1.84807
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.00000	−0.338480
$708$	0	0
$709$	37.0000	1.38956	0.694782	−	0.719220i	$-0.255501\pi$
$709$	37.0000	1.38956	0.694782	+	0.719220i	$0.255501\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	−12.0000	−0.449404
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.00000	0.335643	0.167822	−	0.985817i	$-0.446327\pi$
$719$	9.00000	0.335643	0.167822	+	0.985817i	$0.446327\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	1.00000	0.0371904
$724$	0	0
$725$	3.00000	0.111417
$726$	0	0
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−33.0000	−1.22055
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	−21.0000	−0.773545
$738$	0	0
$739$	−47.0000	−1.72892	−0.864461	−	0.502699i	$-0.832340\pi$
$739$	−47.0000	−1.72892	−0.864461	+	0.502699i	$0.832340\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	0	0
$745$	−21.0000	−0.769380
$746$	0	0
$747$	−24.0000	−0.878114
$748$	0	0
$749$	9.00000	0.328853
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	0	0
$753$	21.0000	0.765283
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	29.0000	1.05402	0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000	1.05402	0.527011	+	0.849858i	$0.323312\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	0	0
$768$	0	0
$769$	13.0000	0.468792	0.234396	−	0.972141i	$-0.424689\pi$
$769$	13.0000	0.468792	0.234396	+	0.972141i	$0.424689\pi$
$770$	0	0
$771$	9.00000	0.324127
$772$	0	0
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	7.00000	0.251124
$778$	0	0
$779$	63.0000	2.25721
$780$	0	0
$781$	−9.00000	−0.322045
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	37.0000	1.31891	0.659454	−	0.751745i	$-0.270788\pi$
$787$	37.0000	1.31891	0.659454	+	0.751745i	$0.270788\pi$
$788$	0	0
$789$	−3.00000	−0.106803
$790$	0	0
$791$	9.00000	0.320003
$792$	0	0
$793$	0	0
$794$	0	0
$795$	6.00000	0.212798
$796$	0	0
$797$	−51.0000	−1.80651	−0.903256	−	0.429101i	$-0.858830\pi$
$797$	−51.0000	−1.80651	−0.903256	+	0.429101i	$0.858830\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	30.0000	1.06000
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	3.00000	0.105736
$806$	0	0
$807$	27.0000	0.950445
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	−32.0000	−1.12367	−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000	−1.12367	−0.561836	+	0.827249i	$0.689905\pi$
$812$	0	0
$813$	−23.0000	−0.806645
$814$	0	0
$815$	−1.00000	−0.0350285
$816$	0	0
$817$	77.0000	2.69389
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	−13.0000	−0.453152	−0.226576	−	0.973994i	$-0.572753\pi$
$823$	−13.0000	−0.453152	−0.226576	+	0.973994i	$0.572753\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	−3.00000	−0.103819
$836$	0	0
$837$	−20.0000	−0.691301
$838$	0	0
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	5.00000	0.171600
$850$	0	0
$851$	−21.0000	−0.719871
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	14.0000	0.478790
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	3.00000	0.102003
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−41.0000	−1.38447	−0.692236	−	0.721671i	$-0.743374\pi$
$877$	−41.0000	−1.38447	−0.692236	+	0.721671i	$0.743374\pi$
$878$	0	0
$879$	27.0000	0.910687
$880$	0	0
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	−3.00000	−0.100844
$886$	0	0
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	0	0
$889$	−19.0000	−0.637240
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	0	0
$894$	0	0
$895$	21.0000	0.701953
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.0000	0.400222
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	11.0000	0.366057
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−19.0000	−0.630885	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	−19.0000	−0.630885	−0.315442	+	0.948945i	$0.602153\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	−11.0000	−0.363649
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	0	0
$924$	0	0
$925$	7.00000	0.230159
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−51.0000	−1.67326	−0.836628	−	0.547772i	$-0.815476\pi$
$929$	−51.0000	−1.67326	−0.836628	+	0.547772i	$0.815476\pi$
$930$	0	0
$931$	−42.0000	−1.37649
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−9.00000	−0.294331
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−27.0000	−0.879241
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	−21.0000	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$947$	−21.0000	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−51.0000	−1.65205	−0.826026	−	0.563632i	$-0.809404\pi$
$953$	−51.0000	−1.65205	−0.826026	+	0.563632i	$0.809404\pi$
$954$	0	0
$955$	3.00000	0.0970777
$956$	0	0
$957$	−9.00000	−0.290929
$958$	0	0
$959$	15.0000	0.484375
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	5.00000	0.160956
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	−21.0000	−0.674617
$970$	0	0
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	0	0
$973$	5.00000	0.160293
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	0	0
$979$	45.0000	1.43821
$980$	0	0
$981$	4.00000	0.127710
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	21.0000	0.669116
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−33.0000	−1.04934
$990$	0	0
$991$	−61.0000	−1.93773	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−61.0000	−1.93773	−0.968864	+	0.247592i	$0.920361\pi$
$992$	0	0
$993$	19.0000	0.602947
$994$	0	0
$995$	−17.0000	−0.538936
$996$	0	0
$997$	−31.0000	−0.981780	−0.490890	−	0.871222i	$-0.663328\pi$
$997$	−31.0000	−0.981780	−0.490890	+	0.871222i	$0.663328\pi$
$998$	0	0
$999$	−35.0000	−1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.g.1.1		1
13.4	even	6		260.2.i.b.81.1	yes	2
13.5	odd	4		3380.2.f.e.3041.2		2
13.8	odd	4		3380.2.f.e.3041.1		2
13.10	even	6		260.2.i.b.61.1	✓	2
13.12	even	2		3380.2.a.h.1.1		1
39.17	odd	6		2340.2.q.b.2161.1		2
39.23	odd	6		2340.2.q.b.1621.1		2
52.23	odd	6		1040.2.q.j.321.1		2
52.43	odd	6		1040.2.q.j.81.1		2
65.4	even	6		1300.2.i.e.601.1		2
65.17	odd	12		1300.2.bb.a.549.1		4
65.23	odd	12		1300.2.bb.a.1049.1		4
65.43	odd	12		1300.2.bb.a.549.2		4
65.49	even	6		1300.2.i.e.1101.1		2
65.62	odd	12		1300.2.bb.a.1049.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.i.b.61.1	✓	2	13.10	even	6
260.2.i.b.81.1	yes	2	13.4	even	6
1040.2.q.j.81.1		2	52.43	odd	6
1040.2.q.j.321.1		2	52.23	odd	6
1300.2.i.e.601.1		2	65.4	even	6
1300.2.i.e.1101.1		2	65.49	even	6
1300.2.bb.a.549.1		4	65.17	odd	12
1300.2.bb.a.549.2		4	65.43	odd	12
1300.2.bb.a.1049.1		4	65.23	odd	12
1300.2.bb.a.1049.2		4	65.62	odd	12
2340.2.q.b.1621.1		2	39.23	odd	6
2340.2.q.b.2161.1		2	39.17	odd	6
3380.2.a.g.1.1		1	1.1	even	1	trivial
3380.2.a.h.1.1		1	13.12	even	2
3380.2.f.e.3041.1		2	13.8	odd	4
3380.2.f.e.3041.2		2	13.5	odd	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

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