LMFDB - Embedded newform 3680.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{3} -1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+2.00000 q^{3} -1.00000 q^{5} +5.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +4.00000 q^{13} -2.00000 q^{15} +3.00000 q^{17} +10.0000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -4.00000 q^{27} -1.00000 q^{29} -1.00000 q^{31} +4.00000 q^{33} -5.00000 q^{35} +7.00000 q^{37} +8.00000 q^{39} -3.00000 q^{41} +4.00000 q^{43} -1.00000 q^{45} -12.0000 q^{47} +18.0000 q^{49} +6.00000 q^{51} -1.00000 q^{53} -2.00000 q^{55} +3.00000 q^{59} -14.0000 q^{61} +5.00000 q^{63} -4.00000 q^{65} +5.00000 q^{67} -2.00000 q^{69} -15.0000 q^{71} +8.00000 q^{73} +2.00000 q^{75} +10.0000 q^{77} -4.00000 q^{79} -11.0000 q^{81} +17.0000 q^{83} -3.00000 q^{85} -2.00000 q^{87} -14.0000 q^{89} +20.0000 q^{91} -2.00000 q^{93} -6.00000 q^{97} +2.00000 q^{99} +O(q^{100})$

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	5.00000	1.88982	0.944911	−	0.327327i	$-0.106148\pi$
$7$	5.00000	1.88982	0.944911	+	0.327327i	$0.106148\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	10.0000	2.18218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	−5.00000	−0.845154
$36$	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
$39$	8.00000	1.28103
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	18.0000	2.57143
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	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
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	5.00000	0.629941
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	2.00000	0.230940
$76$	0	0
$77$	10.0000	1.13961
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	17.0000	1.86599	0.932996	−	0.359886i	$-0.117184\pi$
$83$	17.0000	1.86599	0.932996	+	0.359886i	$0.117184\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	20.0000	2.09657
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	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
$105$	−10.0000	−0.975900
$106$	0	0
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	14.0000	1.32882
$112$	0	0
$113$	19.0000	1.78737	0.893685	−	0.448695i	$-0.148111\pi$
$113$	19.0000	1.78737	0.893685	+	0.448695i	$0.148111\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	15.0000	1.37505
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	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
$141$	−24.0000	−2.02116
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	36.0000	2.96923
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	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
$153$	3.00000	0.242536
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−5.00000	−0.394055
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	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
$183$	−28.0000	−2.06982
$184$	0	0
$185$	−7.00000	−0.514650
$186$	0	0
$187$	6.00000	0.438763
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	−8.00000	−0.572892
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	3.00000	0.209529
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	0	0
$213$	−30.0000	−2.05557
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−5.00000	−0.339422
$218$	0	0
$219$	16.0000	1.08118
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	20.0000	1.31590
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	19.0000	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	−18.0000	−1.14998
$246$	0	0
$247$	0	0
$248$	0	0
$249$	34.0000	2.15466
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	0	0
$259$	35.0000	2.17479
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	1.00000	0.0614295
$266$	0	0
$267$	−28.0000	−1.71357
$268$	0	0
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	0	0
$273$	40.0000	2.42091
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	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
$283$	15.0000	0.891657	0.445829	−	0.895118i	$-0.352909\pi$
$283$	15.0000	0.891657	0.445829	+	0.895118i	$0.352909\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.0000	−0.885422
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	20.0000	1.15278
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	14.0000	0.801638
$306$	0	0
$307$	−6.00000	−0.342438	−0.171219	−	0.985233i	$-0.554771\pi$
$307$	−6.00000	−0.342438	−0.171219	+	0.985233i	$0.554771\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−27.0000	−1.52613	−0.763065	−	0.646322i	$-0.776306\pi$
$313$	−27.0000	−1.52613	−0.763065	+	0.646322i	$0.776306\pi$
$314$	0	0
$315$	−5.00000	−0.281718
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−8.00000	−0.442401
$328$	0	0
$329$	−60.0000	−3.30791
$330$	0	0
$331$	15.0000	0.824475	0.412237	−	0.911077i	$-0.364747\pi$
$331$	15.0000	0.824475	0.412237	+	0.911077i	$0.364747\pi$
$332$	0	0
$333$	7.00000	0.383598
$334$	0	0
$335$	−5.00000	−0.273179
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	38.0000	2.06388
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	55.0000	2.96972
$344$	0	0
$345$	2.00000	0.107676
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	−21.0000	−1.12410	−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000	−1.12410	−0.562052	+	0.827102i	$0.689988\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	15.0000	0.796117
$356$	0	0
$357$	30.0000	1.58777
$358$	0	0
$359$	−34.0000	−1.79445	−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000	−1.79445	−0.897226	+	0.441572i	$0.854421\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−8.00000	−0.418739
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	0	0
$369$	−3.00000	−0.156174
$370$	0	0
$371$	−5.00000	−0.259587
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−17.0000	−0.868659	−0.434330	−	0.900754i	$-0.643015\pi$
$383$	−17.0000	−0.868659	−0.434330	+	0.900754i	$0.643015\pi$
$384$	0	0
$385$	−10.0000	−0.509647
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	11.0000	0.546594
$406$	0	0
$407$	14.0000	0.693954
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	15.0000	0.738102
$414$	0	0
$415$	−17.0000	−0.834497
$416$	0	0
$417$	10.0000	0.489702
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	−70.0000	−3.38754
$428$	0	0
$429$	16.0000	0.772487
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	14.0000	0.663664
$446$	0	0
$447$	−36.0000	−1.70274
$448$	0	0
$449$	−13.0000	−0.613508	−0.306754	−	0.951789i	$-0.599243\pi$
$449$	−13.0000	−0.613508	−0.306754	+	0.951789i	$0.599243\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	−20.0000	−0.937614
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	−12.0000	−0.560112
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	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
$465$	2.00000	0.0927478
$466$	0	0
$467$	−15.0000	−0.694117	−0.347059	−	0.937843i	$-0.612820\pi$
$467$	−15.0000	−0.694117	−0.347059	+	0.937843i	$0.612820\pi$
$468$	0	0
$469$	25.0000	1.15439
$470$	0	0
$471$	26.0000	1.19802
$472$	0	0
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.00000	−0.0457869
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	28.0000	1.27669
$482$	0	0
$483$	−10.0000	−0.455016
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	29.0000	1.30875	0.654376	−	0.756169i	$-0.272931\pi$
$491$	29.0000	1.30875	0.654376	+	0.756169i	$0.272931\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	−75.0000	−3.36421
$498$	0	0
$499$	21.0000	0.940089	0.470045	−	0.882643i	$-0.344238\pi$
$499$	21.0000	0.940089	0.470045	+	0.882643i	$0.344238\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.0000	−0.579641	−0.289821	−	0.957081i	$-0.593596\pi$
$503$	−13.0000	−0.579641	−0.289821	+	0.957081i	$0.593596\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	0	0
$507$	6.00000	0.266469
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	10.0000	0.436436
$526$	0	0
$527$	−3.00000	−0.130682
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.00000	0.130189
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	3.00000	0.129701
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	4.00000	0.171656
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	−14.0000	−0.597505
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	−14.0000	−0.594267
$556$	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
$559$	16.0000	0.676728
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−5.00000	−0.210725	−0.105362	−	0.994434i	$-0.533600\pi$
$563$	−5.00000	−0.210725	−0.105362	+	0.994434i	$0.533600\pi$
$564$	0	0
$565$	−19.0000	−0.799336
$566$	0	0
$567$	−55.0000	−2.30978
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−38.0000	−1.59025	−0.795125	−	0.606445i	$-0.792595\pi$
$571$	−38.0000	−1.59025	−0.795125	+	0.606445i	$0.792595\pi$
$572$	0	0
$573$	40.0000	1.67102
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	0	0
$579$	20.0000	0.831172
$580$	0	0
$581$	85.0000	3.52639
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	36.0000	1.48084
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	−15.0000	−0.614940
$596$	0	0
$597$	−40.0000	−1.63709
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	0	0
$603$	5.00000	0.203616
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−6.00000	−0.243532	−0.121766	−	0.992559i	$-0.538856\pi$
$607$	−6.00000	−0.243532	−0.121766	+	0.992559i	$0.538856\pi$
$608$	0	0
$609$	−10.0000	−0.405220
$610$	0	0
$611$	−48.0000	−1.94187
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−23.0000	−0.925945	−0.462973	−	0.886373i	$-0.653217\pi$
$617$	−23.0000	−0.925945	−0.462973	+	0.886373i	$0.653217\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	−70.0000	−2.80449
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	21.0000	0.837325
$630$	0	0
$631$	−26.0000	−1.03504	−0.517522	−	0.855670i	$-0.673145\pi$
$631$	−26.0000	−1.03504	−0.517522	+	0.855670i	$0.673145\pi$
$632$	0	0
$633$	46.0000	1.82834
$634$	0	0
$635$	2.00000	0.0793676
$636$	0	0
$637$	72.0000	2.85274
$638$	0	0
$639$	−15.0000	−0.593391
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	−10.0000	−0.391931
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	24.0000	0.927894
$670$	0	0
$671$	−28.0000	−1.08093
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	−30.0000	−1.15129
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	−36.0000	−1.37349
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	10.0000	0.379869
$694$	0	0
$695$	−5.00000	−0.189661
$696$	0	0
$697$	−9.00000	−0.340899
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	15.0000	0.564133
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	1.00000	0.0374503
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	38.0000	1.41914
$718$	0	0
$719$	−33.0000	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$719$	−33.0000	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$720$	0	0
$721$	40.0000	1.48968
$722$	0	0
$723$	56.0000	2.08266
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	47.0000	1.74313	0.871567	−	0.490277i	$-0.163104\pi$
$727$	47.0000	1.74313	0.871567	+	0.490277i	$0.163104\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	7.00000	0.258551	0.129275	−	0.991609i	$-0.458735\pi$
$733$	7.00000	0.258551	0.129275	+	0.991609i	$0.458735\pi$
$734$	0	0
$735$	−36.0000	−1.32788
$736$	0	0
$737$	10.0000	0.368355
$738$	0	0
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	17.0000	0.621997
$748$	0	0
$749$	−15.0000	−0.548088
$750$	0	0
$751$	−52.0000	−1.89751	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−52.0000	−1.89751	−0.948753	+	0.316017i	$0.897654\pi$
$752$	0	0
$753$	−32.0000	−1.16614
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	−51.0000	−1.85363	−0.926813	−	0.375523i	$-0.877463\pi$
$757$	−51.0000	−1.85363	−0.926813	+	0.375523i	$0.877463\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	49.0000	1.77625	0.888124	−	0.459603i	$-0.152008\pi$
$761$	49.0000	1.77625	0.888124	+	0.459603i	$0.152008\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	0	0
$765$	−3.00000	−0.108465
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−20.0000	−0.721218	−0.360609	−	0.932717i	$-0.617431\pi$
$769$	−20.0000	−0.721218	−0.360609	+	0.932717i	$0.617431\pi$
$770$	0	0
$771$	−56.0000	−2.01679
$772$	0	0
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	70.0000	2.51124
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−30.0000	−1.07348
$782$	0	0
$783$	4.00000	0.142948
$784$	0	0
$785$	−13.0000	−0.463990
$786$	0	0
$787$	−15.0000	−0.534692	−0.267346	−	0.963601i	$-0.586147\pi$
$787$	−15.0000	−0.534692	−0.267346	+	0.963601i	$0.586147\pi$
$788$	0	0
$789$	38.0000	1.35284
$790$	0	0
$791$	95.0000	3.37781
$792$	0	0
$793$	−56.0000	−1.98862
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	16.0000	0.564628
$804$	0	0
$805$	5.00000	0.176227
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	0	0
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	0	0
$813$	−38.0000	−1.33272
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	0	0
$818$	0	0
$819$	20.0000	0.698857
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	7.00000	0.243414	0.121707	−	0.992566i	$-0.461163\pi$
$827$	7.00000	0.243414	0.121707	+	0.992566i	$0.461163\pi$
$828$	0	0
$829$	41.0000	1.42399	0.711994	−	0.702185i	$-0.247792\pi$
$829$	41.0000	1.42399	0.711994	+	0.702185i	$0.247792\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	0	0
$833$	54.0000	1.87099
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−12.0000	−0.413302
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−35.0000	−1.20261
$848$	0	0
$849$	30.0000	1.02960
$850$	0	0
$851$	−7.00000	−0.239957
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−45.0000	−1.53538	−0.767690	−	0.640821i	$-0.778594\pi$
$859$	−45.0000	−1.53538	−0.767690	+	0.640821i	$0.778594\pi$
$860$	0	0
$861$	−30.0000	−1.02240
$862$	0	0
$863$	10.0000	0.340404	0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000	0.340404	0.170202	+	0.985409i	$0.445558\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	−5.00000	−0.169031
$876$	0	0
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	0	0
$879$	42.0000	1.41662
$880$	0	0
$881$	22.0000	0.741199	0.370599	−	0.928793i	$-0.379152\pi$
$881$	22.0000	0.741199	0.370599	+	0.928793i	$0.379152\pi$
$882$	0	0
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	−22.0000	−0.737028
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	1.00000	0.0333519
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	40.0000	1.33112
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	1.00000	0.0332045	0.0166022	−	0.999862i	$-0.494715\pi$
$907$	1.00000	0.0332045	0.0166022	+	0.999862i	$0.494715\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	−50.0000	−1.65657	−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000	−1.65657	−0.828287	+	0.560304i	$0.810684\pi$
$912$	0	0
$913$	34.0000	1.12524
$914$	0	0
$915$	28.0000	0.925651
$916$	0	0
$917$	0	0
$918$	0	0
$919$	42.0000	1.38545	0.692726	−	0.721201i	$-0.256409\pi$
$919$	42.0000	1.38545	0.692726	+	0.721201i	$0.256409\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	−60.0000	−1.97492
$924$	0	0
$925$	7.00000	0.230159
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−17.0000	−0.557752	−0.278876	−	0.960327i	$-0.589962\pi$
$929$	−17.0000	−0.557752	−0.278876	+	0.960327i	$0.589962\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	−6.00000	−0.196221
$936$	0	0
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	0	0
$939$	−54.0000	−1.76222
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	3.00000	0.0976934
$944$	0	0
$945$	20.0000	0.650600
$946$	0	0
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	0	0
$949$	32.0000	1.03876
$950$	0	0
$951$	4.00000	0.129709
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−20.0000	−0.647185
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−3.00000	−0.0966736
$964$	0	0
$965$	−10.0000	−0.321911
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	25.0000	0.801463
$974$	0	0
$975$	8.00000	0.256205
$976$	0	0
$977$	−47.0000	−1.50366	−0.751832	−	0.659355i	$-0.770829\pi$
$977$	−47.0000	−1.50366	−0.751832	+	0.659355i	$0.770829\pi$
$978$	0	0
$979$	−28.0000	−0.894884
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−27.0000	−0.861166	−0.430583	−	0.902551i	$-0.641692\pi$
$983$	−27.0000	−0.861166	−0.430583	+	0.902551i	$0.641692\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	−120.000	−3.81964
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	30.0000	0.952021
$994$	0	0
$995$	20.0000	0.634043
$996$	0	0
$997$	−16.0000	−0.506725	−0.253363	−	0.967371i	$-0.581537\pi$
$997$	−16.0000	−0.506725	−0.253363	+	0.967371i	$0.581537\pi$
$998$	0	0
$999$	−28.0000	−0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.i.1.1	yes	1
4.3	odd	2		3680.2.a.a.1.1	✓	1
8.3	odd	2		7360.2.a.x.1.1		1
8.5	even	2		7360.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.a.1.1	✓	1	4.3	odd	2
3680.2.a.i.1.1	yes	1	1.1	even	1	trivial
7360.2.a.f.1.1		1	8.5	even	2
7360.2.a.x.1.1		1	8.3	odd	2

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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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