LMFDB - Embedded newform 4477.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4477,2,Mod(1,4477)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4477, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4477.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4477 = 11^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4477.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:	$35.7490249849$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 37)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4477.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{12} +4.00000 q^{13} +4.00000 q^{16} -6.00000 q^{17} -2.00000 q^{19} +1.00000 q^{21} +6.00000 q^{23} -5.00000 q^{25} -5.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} +4.00000 q^{36} +1.00000 q^{37} +4.00000 q^{39} +9.00000 q^{41} -8.00000 q^{43} +3.00000 q^{47} +4.00000 q^{48} -6.00000 q^{49} -6.00000 q^{51} -8.00000 q^{52} -3.00000 q^{53} -2.00000 q^{57} +12.0000 q^{59} -8.00000 q^{61} -2.00000 q^{63} -8.00000 q^{64} -4.00000 q^{67} +12.0000 q^{68} +6.00000 q^{69} -15.0000 q^{71} -11.0000 q^{73} -5.00000 q^{75} +4.00000 q^{76} +10.0000 q^{79} +1.00000 q^{81} -9.00000 q^{83} -2.00000 q^{84} +6.00000 q^{87} +6.00000 q^{89} +4.00000 q^{91} -12.0000 q^{92} -4.00000 q^{93} +8.00000 q^{97} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−2.00000	−1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	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
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.577350
$13$	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
$15$	0	0
$16$	4.00000	1.00000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	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
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	−2.00000	−0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$33$	0	0
$34$	0	0
$35$	0	0
$36$	4.00000	0.666667
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	4.00000	0.640513
$40$	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
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	4.00000	0.577350
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	−8.00000	−1.10940
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	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
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	12.0000	1.45521
$69$	6.00000	0.722315
$70$	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
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	−5.00000	−0.577350
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	−12.0000	−1.25109
$93$	−4.00000	−0.414781
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	10.0000	0.962250
$109$	−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
$111$	1.00000	0.0949158
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	−12.0000	−1.11417
$117$	−8.00000	−0.739600
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	0	0
$122$	0	0
$123$	9.00000	0.811503
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	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
$141$	3.00000	0.252646
$142$	0	0
$143$	0	0
$144$	−8.00000	−0.666667
$145$	0	0
$146$	0	0
$147$	−6.00000	−0.494872
$148$	−2.00000	−0.164399
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	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
$153$	12.0000	0.970143
$154$	0	0
$155$	0	0
$156$	−8.00000	−0.640513
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−18.0000	−1.40556
$165$	0	0
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	4.00000	0.305888
$172$	16.0000	1.21999
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−6.00000	−0.437595
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−8.00000	−0.577350
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	12.0000	0.857143
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	6.00000	0.421117
$204$	12.0000	0.840168
$205$	0	0
$206$	0	0
$207$	−12.0000	−0.834058
$208$	16.0000	1.10940
$209$	0	0
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	6.00000	0.412082
$213$	−15.0000	−1.02778
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−11.0000	−0.743311
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	0	0
$225$	10.0000	0.666667
$226$	0	0
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	4.00000	0.264906
$229$	23.0000	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$230$	0	0
$231$	0	0
$232$	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
$236$	−24.0000	−1.56227
$237$	10.0000	0.649570
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	16.0000	1.02430
$245$	0	0
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−9.00000	−0.570352
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	−12.0000	−0.742781
$262$	0	0
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	8.00000	0.488678
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	−24.0000	−1.45521
$273$	4.00000	0.242091
$274$	0	0
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	30.0000	1.78017
$285$	0	0
$286$	0	0
$287$	9.00000	0.531253
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	8.00000	0.468968
$292$	22.0000	1.28745
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.0000	1.38796
$300$	10.0000	0.577350
$301$	−8.00000	−0.461112
$302$	0	0
$303$	−3.00000	−0.172345
$304$	−8.00000	−0.458831
$305$	0	0
$306$	0	0
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	0	0
$315$	0	0
$316$	−20.0000	−1.12509
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	12.0000	0.667698
$324$	−2.00000	−0.111111
$325$	−20.0000	−1.10940
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	18.0000	0.987878
$333$	−2.00000	−0.109599
$334$	0	0
$335$	0	0
$336$	4.00000	0.218218
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	−12.0000	−0.643268
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	−12.0000	−0.635999
$357$	−6.00000	−0.317554
$358$	0	0
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	−8.00000	−0.419314
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	24.0000	1.25109
$369$	−18.0000	−0.937043
$370$	0	0
$371$	−3.00000	−0.155752
$372$	8.00000	0.414781
$373$	−5.00000	−0.258890	−0.129445	−	0.991587i	$-0.541320\pi$
$373$	−5.00000	−0.258890	−0.129445	+	0.991587i	$0.541320\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0000	0.813326
$388$	−16.0000	−0.812277
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11.0000	0.552074	0.276037	−	0.961147i	$-0.410979\pi$
$397$	11.0000	0.552074	0.276037	+	0.961147i	$0.410979\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	−20.0000	−1.00000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	8.00000	0.394132
$413$	12.0000	0.590481
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	30.0000	1.45521
$426$	0	0
$427$	−8.00000	−0.387147
$428$	24.0000	1.16008
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−20.0000	−0.962250
$433$	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
$435$	0	0
$436$	4.00000	0.191565
$437$	−12.0000	−0.574038
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	0	0
$447$	−15.0000	−0.709476
$448$	−8.00000	−0.377964
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	12.0000	0.564433
$453$	−8.00000	−0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	0	0
$459$	30.0000	1.40028
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	16.0000	0.739600
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	0	0
$474$	0	0
$475$	10.0000	0.458831
$476$	12.0000	0.550019
$477$	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
$481$	4.00000	0.182384
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−18.0000	−0.811503
$493$	−36.0000	−1.62136
$494$	0	0
$495$	0	0
$496$	−16.0000	−0.718421
$497$	−15.0000	−0.672842
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000	0.133235
$508$	−14.0000	−0.621150
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	0	0
$513$	10.0000	0.441511
$514$	0	0
$515$	0	0
$516$	16.0000	0.704361
$517$	0	0
$518$	0	0
$519$	−9.00000	−0.395056
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−12.0000	−0.524222
$525$	−5.00000	−0.218218
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−24.0000	−1.04151
$532$	4.00000	0.173422
$533$	36.0000	1.55933
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.0000	0.776757
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	−7.00000	−0.300399
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	12.0000	0.512615
$549$	16.0000	0.682863
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	0	0
$556$	−8.00000	−0.339276
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	−30.0000	−1.25109
$576$	16.0000	0.666667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	0	0
$579$	4.00000	0.166234
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	12.0000	0.494872
$589$	8.00000	0.329634
$590$	0	0
$591$	−15.0000	−0.617018
$592$	4.00000	0.164399
$593$	−27.0000	−1.10876	−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000	−1.10876	−0.554379	+	0.832265i	$0.687044\pi$
$594$	0	0
$595$	0	0
$596$	30.0000	1.22885
$597$	2.00000	0.0818546
$598$	0	0
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	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
$603$	8.00000	0.325785
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	12.0000	0.485468
$612$	−24.0000	−0.970143
$613$	25.0000	1.00974	0.504870	−	0.863195i	$-0.331540\pi$
$613$	25.0000	1.00974	0.504870	+	0.863195i	$0.331540\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	0	0
$619$	−37.0000	−1.48716	−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000	−1.48716	−0.743578	+	0.668649i	$0.766873\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	0	0
$623$	6.00000	0.240385
$624$	16.0000	0.640513
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	26.0000	1.03751
$629$	−6.00000	−0.239236
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	0	0
$633$	13.0000	0.516704
$634$	0	0
$635$	0	0
$636$	6.00000	0.237915
$637$	−24.0000	−0.950915
$638$	0	0
$639$	30.0000	1.18678
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−4.00000	−0.156772
$652$	32.0000	1.25322
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	0	0
$656$	36.0000	1.40556
$657$	22.0000	0.858302
$658$	0	0
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.0000	1.39393
$668$	36.0000	1.39288
$669$	−1.00000	−0.0386622
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	25.0000	0.962250
$676$	−6.00000	−0.230769
$677$	−21.0000	−0.807096	−0.403548	−	0.914959i	$-0.632223\pi$
$677$	−21.0000	−0.807096	−0.403548	+	0.914959i	$0.632223\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	0	0
$687$	23.0000	0.877505
$688$	−32.0000	−1.21999
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−54.0000	−2.04540
$698$	0	0
$699$	18.0000	0.680823
$700$	10.0000	0.377964
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.00000	−0.112827
$708$	−24.0000	−0.901975
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	−36.0000	−1.34538
$717$	6.00000	0.224074
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	10.0000	0.371904
$724$	14.0000	0.520306
$725$	−30.0000	−1.11417
$726$	0	0
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	48.0000	1.77534
$732$	16.0000	0.591377
$733$	13.0000	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$733$	13.0000	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	27.0000	0.990534	0.495267	−	0.868741i	$-0.335070\pi$
$743$	27.0000	0.990534	0.495267	+	0.868741i	$0.335070\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.0000	0.658586
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	0	0
$755$	0	0
$756$	10.0000	0.363696
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	48.0000	1.73658
$765$	0	0
$766$	0	0
$767$	48.0000	1.73318
$768$	16.0000	0.577350
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	−8.00000	−0.287926
$773$	3.00000	0.107903	0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000	0.107903	0.0539513	+	0.998544i	$0.482818\pi$
$774$	0	0
$775$	20.0000	0.718421
$776$	0	0
$777$	1.00000	0.0358748
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−30.0000	−1.07211
$784$	−24.0000	−0.857143
$785$	0	0
$786$	0	0
$787$	−23.0000	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$788$	30.0000	1.06871
$789$	−15.0000	−0.534014
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−32.0000	−1.13635
$794$	0	0
$795$	0	0
$796$	−4.00000	−0.141776
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	0	0
$804$	8.00000	0.282138
$805$	0	0
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−11.0000	−0.386262	−0.193131	−	0.981173i	$-0.561864\pi$
$811$	−11.0000	−0.386262	−0.193131	+	0.981173i	$0.561864\pi$
$812$	−12.0000	−0.421117
$813$	−29.0000	−1.01707
$814$	0	0
$815$	0	0
$816$	−24.0000	−0.840168
$817$	16.0000	0.559769
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	3.00000	0.104701	0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000	0.104701	0.0523504	+	0.998629i	$0.483329\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	24.0000	0.834058
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	−32.0000	−1.10940
$833$	36.0000	1.24733
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	−26.0000	−0.894957
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−12.0000	−0.412082
$849$	4.00000	0.137280
$850$	0	0
$851$	6.00000	0.205677
$852$	30.0000	1.02778
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	0	0
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000	0.645274
$868$	8.00000	0.271538
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−16.0000	−0.541518
$874$	0	0
$875$	0	0
$876$	22.0000	0.743311
$877$	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
$879$	18.0000	0.607125
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	48.0000	1.61441
$885$	0	0
$886$	0	0
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	0	0
$891$	0	0
$892$	2.00000	0.0669650
$893$	−6.00000	−0.200782
$894$	0	0
$895$	0	0
$896$	0	0
$897$	24.0000	0.801337
$898$	0	0
$899$	−24.0000	−0.800445
$900$	−20.0000	−0.666667
$901$	18.0000	0.599667
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	48.0000	1.59294
$909$	6.00000	0.199007
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	−8.00000	−0.264906
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−46.0000	−1.51988
$917$	6.00000	0.198137
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	25.0000	0.823778
$922$	0	0
$923$	−60.0000	−1.97492
$924$	0	0
$925$	−5.00000	−0.164399
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	−36.0000	−1.17922
$933$	−6.00000	−0.196431
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	−28.0000	−0.913745
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	54.0000	1.75848
$944$	48.0000	1.56227
$945$	0	0
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	−20.0000	−0.649570
$949$	−44.0000	−1.42830
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	3.00000	0.0971795	0.0485898	−	0.998819i	$-0.484527\pi$
$953$	3.00000	0.0971795	0.0485898	+	0.998819i	$0.484527\pi$
$954$	0	0
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	24.0000	0.773389
$964$	−20.0000	−0.644157
$965$	0	0
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−32.0000	−1.02640
$973$	4.00000	0.128234
$974$	0	0
$975$	−20.0000	−0.640513
$976$	−32.0000	−1.02430
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	0	0
$979$	0	0
$980$	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
$985$	0	0
$986$	0	0
$987$	3.00000	0.0954911
$988$	16.0000	0.509028
$989$	−48.0000	−1.52631
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	−10.0000	−0.317340
$994$	0	0
$995$	0	0
$996$	18.0000	0.570352
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	0	0
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4477.2.a.a.1.1		1
11.10	odd	2		37.2.a.b.1.1	✓	1
33.32	even	2		333.2.a.b.1.1		1
44.43	even	2		592.2.a.a.1.1		1
55.32	even	4		925.2.b.e.149.1		2
55.43	even	4		925.2.b.e.149.2		2
55.54	odd	2		925.2.a.b.1.1		1
77.76	even	2		1813.2.a.b.1.1		1
88.21	odd	2		2368.2.a.d.1.1		1
88.43	even	2		2368.2.a.m.1.1		1
132.131	odd	2		5328.2.a.k.1.1		1
143.142	odd	2		6253.2.a.b.1.1		1
165.164	even	2		8325.2.a.p.1.1		1
407.43	even	4		1369.2.b.a.1368.1		2
407.142	even	4		1369.2.b.a.1368.2		2
407.406	odd	2		1369.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.a.b.1.1	✓	1	11.10	odd	2
333.2.a.b.1.1		1	33.32	even	2
592.2.a.a.1.1		1	44.43	even	2
925.2.a.b.1.1		1	55.54	odd	2
925.2.b.e.149.1		2	55.32	even	4
925.2.b.e.149.2		2	55.43	even	4
1369.2.a.c.1.1		1	407.406	odd	2
1369.2.b.a.1368.1		2	407.43	even	4
1369.2.b.a.1368.2		2	407.142	even	4
1813.2.a.b.1.1		1	77.76	even	2
2368.2.a.d.1.1		1	88.21	odd	2
2368.2.a.m.1.1		1	88.43	even	2
4477.2.a.a.1.1		1	1.1	even	1	trivial
5328.2.a.k.1.1		1	132.131	odd	2
6253.2.a.b.1.1		1	143.142	odd	2
8325.2.a.p.1.1		1	165.164	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4477 = 11^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4477.a (trivial)

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