LMFDB - Embedded newform 2368.2.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2368,2,Mod(1,2368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2368.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,2,0,1,0,-2,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -1.00000 q^{11} +6.00000 q^{13} +2.00000 q^{15} -4.00000 q^{17} +8.00000 q^{19} +1.00000 q^{21} +6.00000 q^{23} -1.00000 q^{25} -5.00000 q^{27} -2.00000 q^{29} -4.00000 q^{31} -1.00000 q^{33} +2.00000 q^{35} +1.00000 q^{37} +6.00000 q^{39} +7.00000 q^{41} -2.00000 q^{43} -4.00000 q^{45} +9.00000 q^{47} -6.00000 q^{49} -4.00000 q^{51} +3.00000 q^{53} -2.00000 q^{55} +8.00000 q^{57} +12.0000 q^{59} -4.00000 q^{61} -2.00000 q^{63} +12.0000 q^{65} +6.00000 q^{69} +7.00000 q^{71} +7.00000 q^{73} -1.00000 q^{75} -1.00000 q^{77} +1.00000 q^{81} -3.00000 q^{83} -8.00000 q^{85} -2.00000 q^{87} -12.0000 q^{89} +6.00000 q^{91} -4.00000 q^{93} +16.0000 q^{95} -8.00000 q^{97} +2.00000 q^{99} +O(q^{100})$

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	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
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−4.00000	−0.596285
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	16.0000	1.64157
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	−12.0000	−1.10940
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	7.00000	0.631169
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	0	0
$135$	−10.0000	−0.860663
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	0	0
$143$	−6.00000	−0.501745
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−19.0000	−1.51637	−0.758183	−	0.652042i	$-0.773912\pi$
$157$	−19.0000	−1.51637	−0.758183	+	0.652042i	$0.773912\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	22.0000	1.72317	0.861586	−	0.507611i	$-0.169471\pi$
$163$	22.0000	1.72317	0.861586	+	0.507611i	$0.169471\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−16.0000	−1.22355
$172$	0	0
$173$	3.00000	0.228086	0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000	0.228086	0.114043	+	0.993476i	$0.463620\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	22.0000	1.64436	0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000	1.64436	0.822179	+	0.569230i	$0.192758\pi$
$180$	0	0
$181$	23.0000	1.70958	0.854788	−	0.518977i	$-0.173687\pi$
$181$	23.0000	1.70958	0.854788	+	0.518977i	$0.173687\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	12.0000	0.859338
$196$	0	0
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	14.0000	0.977802
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−9.00000	−0.619586	−0.309793	−	0.950804i	$-0.600260\pi$
$211$	−9.00000	−0.619586	−0.309793	+	0.950804i	$0.600260\pi$
$212$	0	0
$213$	7.00000	0.479632
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	25.0000	1.67412	0.837062	−	0.547108i	$-0.184271\pi$
$223$	25.0000	1.67412	0.837062	+	0.547108i	$0.184271\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	48.0000	3.05417
$248$	0	0
$249$	−3.00000	−0.190117
$250$	0	0
$251$	−26.0000	−1.64111	−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000	−1.64111	−0.820553	+	0.571571i	$0.806334\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	13.0000	0.801614	0.400807	−	0.916162i	$-0.368730\pi$
$263$	13.0000	0.801614	0.400807	+	0.916162i	$0.368730\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	−12.0000	−0.734388
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−9.00000	−0.546711	−0.273356	−	0.961913i	$-0.588134\pi$
$271$	−9.00000	−0.546711	−0.273356	+	0.961913i	$0.588134\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	0	0
$285$	16.0000	0.947758
$286$	0	0
$287$	7.00000	0.413197
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.00000	−0.468968
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	36.0000	2.08193
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	17.0000	0.976624
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−29.0000	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$307$	−29.0000	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$308$	0	0
$309$	2.00000	0.113776
$310$	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
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	−32.0000	−1.78053
$324$	0	0
$325$	−6.00000	−0.332820
$326$	0	0
$327$	−16.0000	−0.884802
$328$	0	0
$329$	9.00000	0.496186
$330$	0	0
$331$	22.0000	1.20923	0.604615	−	0.796518i	$-0.293327\pi$
$331$	22.0000	1.20923	0.604615	+	0.796518i	$0.293327\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−30.0000	−1.60128
$352$	0	0
$353$	−36.0000	−1.91609	−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000	−1.91609	−0.958043	+	0.286623i	$0.907467\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−14.0000	−0.728811
$370$	0	0
$371$	3.00000	0.155752
$372$	0	0
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−37.0000	−1.90056	−0.950281	−	0.311393i	$-0.899204\pi$
$379$	−37.0000	−1.90056	−0.950281	+	0.311393i	$0.899204\pi$
$380$	0	0
$381$	−17.0000	−0.870936
$382$	0	0
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.00000	0.0501886	0.0250943	−	0.999685i	$-0.492011\pi$
$397$	1.00000	0.0501886	0.0250943	+	0.999685i	$0.492011\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	−1.00000	−0.0495682
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	0	0
$423$	−18.0000	−0.875190
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	−39.0000	−1.87422	−0.937110	−	0.349034i	$-0.886510\pi$
$433$	−39.0000	−1.87422	−0.937110	+	0.349034i	$0.886510\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	0	0
$437$	48.0000	2.29615
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	13.0000	0.617649	0.308824	−	0.951119i	$-0.400064\pi$
$443$	13.0000	0.617649	0.308824	+	0.951119i	$0.400064\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	0	0
$447$	9.00000	0.425685
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−7.00000	−0.329617
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	20.0000	0.933520
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−19.0000	−0.875474
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	−16.0000	−0.726523
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	22.0000	0.994874
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	7.00000	0.313993
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	0	0
$505$	34.0000	1.51298
$506$	0	0
$507$	23.0000	1.02147
$508$	0	0
$509$	−45.0000	−1.99459	−0.997295	−	0.0735034i	$-0.976582\pi$
$509$	−45.0000	−1.99459	−0.997295	+	0.0735034i	$0.976582\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	−40.0000	−1.76604
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	−9.00000	−0.395820
$518$	0	0
$519$	3.00000	0.131685
$520$	0	0
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	0	0
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−24.0000	−1.04151
$532$	0	0
$533$	42.0000	1.81922
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	22.0000	0.949370
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	36.0000	1.54776	0.773880	−	0.633332i	$-0.218313\pi$
$541$	36.0000	1.54776	0.773880	+	0.633332i	$0.218313\pi$
$542$	0	0
$543$	23.0000	0.987024
$544$	0	0
$545$	−32.0000	−1.37073
$546$	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
$549$	8.00000	0.341432
$550$	0	0
$551$	−16.0000	−0.681623
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	−21.0000	−0.878823	−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000	−0.878823	−0.439411	+	0.898286i	$0.644813\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	0	0
$579$	−10.0000	−0.415586
$580$	0	0
$581$	−3.00000	−0.124461
$582$	0	0
$583$	−3.00000	−0.124247
$584$	0	0
$585$	−24.0000	−0.992278
$586$	0	0
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	−7.00000	−0.287942
$592$	0	0
$593$	3.00000	0.123195	0.0615976	−	0.998101i	$-0.480380\pi$
$593$	3.00000	0.123195	0.0615976	+	0.998101i	$0.480380\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	0	0
$597$	−14.0000	−0.572982
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20.0000	−0.813116
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	54.0000	2.18461
$612$	0	0
$613$	−35.0000	−1.41364	−0.706818	−	0.707395i	$-0.749870\pi$
$613$	−35.0000	−1.41364	−0.706818	+	0.707395i	$0.749870\pi$
$614$	0	0
$615$	14.0000	0.564534
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	0	0
$619$	−5.00000	−0.200967	−0.100483	−	0.994939i	$-0.532039\pi$
$619$	−5.00000	−0.200967	−0.100483	+	0.994939i	$0.532039\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	−9.00000	−0.357718
$634$	0	0
$635$	−34.0000	−1.34925
$636$	0	0
$637$	−36.0000	−1.42637
$638$	0	0
$639$	−14.0000	−0.553831
$640$	0	0
$641$	31.0000	1.22443	0.612213	−	0.790693i	$-0.290279\pi$
$641$	31.0000	1.22443	0.612213	+	0.790693i	$0.290279\pi$
$642$	0	0
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	0	0
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	8.00000	0.313064	0.156532	−	0.987673i	$-0.449969\pi$
$653$	8.00000	0.313064	0.156532	+	0.987673i	$0.449969\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	36.0000	1.40024	0.700119	−	0.714026i	$-0.253130\pi$
$661$	36.0000	1.40024	0.700119	+	0.714026i	$0.253130\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	25.0000	0.966556
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	−42.0000	−1.60709	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000	−1.60709	−0.803543	+	0.595247i	$0.797054\pi$
$684$	0	0
$685$	−28.0000	−1.06983
$686$	0	0
$687$	5.00000	0.190762
$688$	0	0
$689$	18.0000	0.685745
$690$	0	0
$691$	−52.0000	−1.97817	−0.989087	−	0.147335i	$-0.952930\pi$
$691$	−52.0000	−1.97817	−0.989087	+	0.147335i	$0.952930\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	−28.0000	−1.06058
$698$	0	0
$699$	−2.00000	−0.0756469
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	18.0000	0.677919
$706$	0	0
$707$	17.0000	0.639351
$708$	0	0
$709$	−12.0000	−0.450669	−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000	−0.450669	−0.225335	+	0.974281i	$0.572348\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	10.0000	0.373457
$718$	0	0
$719$	1.00000	0.0372937	0.0186469	−	0.999826i	$-0.494064\pi$
$719$	1.00000	0.0372937	0.0186469	+	0.999826i	$0.494064\pi$
$720$	0	0
$721$	2.00000	0.0744839
$722$	0	0
$723$	−30.0000	−1.11571
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−19.0000	−0.701781	−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000	−0.701781	−0.350891	+	0.936416i	$0.614121\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−21.0000	−0.772497	−0.386249	−	0.922395i	$-0.626229\pi$
$739$	−21.0000	−0.772497	−0.386249	+	0.922395i	$0.626229\pi$
$740$	0	0
$741$	48.0000	1.76332
$742$	0	0
$743$	43.0000	1.57752	0.788759	−	0.614703i	$-0.210724\pi$
$743$	43.0000	1.57752	0.788759	+	0.614703i	$0.210724\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−33.0000	−1.20419	−0.602094	−	0.798426i	$-0.705667\pi$
$751$	−33.0000	−1.20419	−0.602094	+	0.798426i	$0.705667\pi$
$752$	0	0
$753$	−26.0000	−0.947493
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	16.0000	0.578481
$766$	0	0
$767$	72.0000	2.59977
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	−19.0000	−0.683383	−0.341691	−	0.939812i	$-0.611000\pi$
$773$	−19.0000	−0.683383	−0.341691	+	0.939812i	$0.611000\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	1.00000	0.0358748
$778$	0	0
$779$	56.0000	2.00641
$780$	0	0
$781$	−7.00000	−0.250480
$782$	0	0
$783$	10.0000	0.357371
$784$	0	0
$785$	−38.0000	−1.35628
$786$	0	0
$787$	39.0000	1.39020	0.695100	−	0.718913i	$-0.255360\pi$
$787$	39.0000	1.39020	0.695100	+	0.718913i	$0.255360\pi$
$788$	0	0
$789$	13.0000	0.462812
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	6.00000	0.212798
$796$	0	0
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	24.0000	0.847998
$802$	0	0
$803$	−7.00000	−0.247025
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	−45.0000	−1.58016	−0.790082	−	0.613001i	$-0.789962\pi$
$811$	−45.0000	−1.58016	−0.790082	+	0.613001i	$0.789962\pi$
$812$	0	0
$813$	−9.00000	−0.315644
$814$	0	0
$815$	44.0000	1.54125
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	−29.0000	−1.01211	−0.506053	−	0.862502i	$-0.668896\pi$
$821$	−29.0000	−1.01211	−0.506053	+	0.862502i	$0.668896\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	54.0000	1.87776	0.938882	−	0.344239i	$-0.111863\pi$
$827$	54.0000	1.87776	0.938882	+	0.344239i	$0.111863\pi$
$828$	0	0
$829$	40.0000	1.38926	0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000	1.38926	0.694629	+	0.719368i	$0.255569\pi$
$830$	0	0
$831$	−12.0000	−0.416275
$832$	0	0
$833$	24.0000	0.831551
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−20.0000	−0.688837
$844$	0	0
$845$	46.0000	1.58245
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	−32.0000	−1.09438
$856$	0	0
$857$	−52.0000	−1.77629	−0.888143	−	0.459567i	$-0.848005\pi$
$857$	−52.0000	−1.77629	−0.888143	+	0.459567i	$0.848005\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	7.00000	0.238559
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.0000	0.541518
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	0	0
$887$	47.0000	1.57811	0.789053	−	0.614325i	$-0.210572\pi$
$887$	47.0000	1.57811	0.789053	+	0.614325i	$0.210572\pi$
$888$	0	0
$889$	−17.0000	−0.570162
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	72.0000	2.40939
$894$	0	0
$895$	44.0000	1.47076
$896$	0	0
$897$	36.0000	1.20201
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	−2.00000	−0.0665558
$904$	0	0
$905$	46.0000	1.52909
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	0	0
$909$	−34.0000	−1.12771
$910$	0	0
$911$	−54.0000	−1.78910	−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000	−1.78910	−0.894550	+	0.446968i	$0.852504\pi$
$912$	0	0
$913$	3.00000	0.0992855
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	0	0
$921$	−29.0000	−0.955582
$922$	0	0
$923$	42.0000	1.38245
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−48.0000	−1.57314
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−19.0000	−0.620703	−0.310351	−	0.950622i	$-0.600447\pi$
$937$	−19.0000	−0.620703	−0.310351	+	0.950622i	$0.600447\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	42.0000	1.36771
$944$	0	0
$945$	−10.0000	−0.325300
$946$	0	0
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	42.0000	1.36338
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	−19.0000	−0.615470	−0.307735	−	0.951472i	$-0.599571\pi$
$953$	−19.0000	−0.615470	−0.307735	+	0.951472i	$0.599571\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.00000	0.0646508
$958$	0	0
$959$	−14.0000	−0.452084
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−8.00000	−0.257796
$964$	0	0
$965$	−20.0000	−0.643823
$966$	0	0
$967$	34.0000	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$967$	34.0000	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$968$	0	0
$969$	−32.0000	−1.02799
$970$	0	0
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	0	0
$973$	4.00000	0.128234
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	4.00000	0.127971	0.0639857	−	0.997951i	$-0.479619\pi$
$977$	4.00000	0.127971	0.0639857	+	0.997951i	$0.479619\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	32.0000	1.02168
$982$	0	0
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	0	0
$987$	9.00000	0.286473
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−22.0000	−0.698853	−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000	−0.698853	−0.349427	+	0.936964i	$0.613624\pi$
$992$	0	0
$993$	22.0000	0.698149
$994$	0	0
$995$	−28.0000	−0.887660
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	0	0
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.n.1.1		1
4.3	odd	2		2368.2.a.g.1.1		1
8.3	odd	2		592.2.a.c.1.1		1
8.5	even	2		296.2.a.a.1.1	✓	1
24.5	odd	2		2664.2.a.f.1.1		1
24.11	even	2		5328.2.a.o.1.1		1
40.29	even	2		7400.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.a.1.1	✓	1	8.5	even	2
592.2.a.c.1.1		1	8.3	odd	2
2368.2.a.g.1.1		1	4.3	odd	2
2368.2.a.n.1.1		1	1.1	even	1	trivial
2664.2.a.f.1.1		1	24.5	odd	2
5328.2.a.o.1.1		1	24.11	even	2
7400.2.a.f.1.1		1	40.29	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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