LMFDB - Embedded newform 464.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,2,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	464.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:	$3.70505865379$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} +3.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} +8.00000 q^{17} +2.00000 q^{21} -4.00000 q^{23} -4.00000 q^{25} -5.00000 q^{27} -1.00000 q^{29} +3.00000 q^{31} +3.00000 q^{33} +2.00000 q^{35} +8.00000 q^{37} -1.00000 q^{39} +2.00000 q^{41} +11.0000 q^{43} -2.00000 q^{45} -13.0000 q^{47} -3.00000 q^{49} +8.00000 q^{51} -11.0000 q^{53} +3.00000 q^{55} -8.00000 q^{61} -4.00000 q^{63} -1.00000 q^{65} +12.0000 q^{67} -4.00000 q^{69} -2.00000 q^{71} +4.00000 q^{73} -4.00000 q^{75} +6.00000 q^{77} -15.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +8.00000 q^{85} -1.00000 q^{87} -10.0000 q^{89} -2.00000 q^{91} +3.00000 q^{93} -2.00000 q^{97} -6.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	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
$63$	−4.00000	−0.503953
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	−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
$115$	−4.00000	−0.373002
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	16.0000	1.46672
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	11.0000	0.968496
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−13.0000	−1.09480
$142$	0	0
$143$	−3.00000	−0.250873
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	−16.0000	−1.29352
$154$	0	0
$155$	3.00000	0.240966
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−11.0000	−0.872357
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	24.0000	1.75505
$188$	0	0
$189$	−10.0000	−0.727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	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
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	3.00000	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$212$	0	0
$213$	−2.00000	−0.137038
$214$	0	0
$215$	11.0000	0.750194
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	0	0
$235$	−13.0000	−0.848026
$236$	0	0
$237$	−15.0000	−0.974355
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	8.00000	0.500979
$256$	0	0
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	−11.0000	−0.675725
$266$	0	0
$267$	−10.0000	−0.611990
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	−12.0000	−0.723627
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	47.0000	2.76471
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−15.0000	−0.870388
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	22.0000	1.26806
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	2.00000	0.111629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	5.00000	0.276501
$328$	0	0
$329$	−26.0000	−1.43343
$330$	0	0
$331$	23.0000	1.26419	0.632097	−	0.774889i	$-0.282194\pi$
$331$	23.0000	1.26419	0.632097	+	0.774889i	$0.282194\pi$
$332$	0	0
$333$	−16.0000	−0.876795
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	9.00000	0.487377
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	0	0
$357$	16.0000	0.846810
$358$	0	0
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−22.0000	−1.14218
$372$	0	0
$373$	−21.0000	−1.08734	−0.543669	−	0.839299i	$-0.682965\pi$
$373$	−21.0000	−1.08734	−0.543669	+	0.839299i	$0.682965\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	−22.0000	−1.11832
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−15.0000	−0.754732
$396$	0	0
$397$	−17.0000	−0.853206	−0.426603	−	0.904439i	$-0.640290\pi$
$397$	−17.0000	−0.853206	−0.426603	+	0.904439i	$0.640290\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	−3.00000	−0.149441
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	0	0
$423$	26.0000	1.26416
$424$	0	0
$425$	−32.0000	−1.55223
$426$	0	0
$427$	−16.0000	−0.774294
$428$	0	0
$429$	−3.00000	−0.144841
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	−1.00000	−0.0479463
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	15.0000	0.709476
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	−40.0000	−1.86704
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	3.00000	0.139122
$466$	0	0
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	33.0000	1.51734
$474$	0	0
$475$	0	0
$476$	0	0
$477$	22.0000	1.00731
$478$	0	0
$479$	5.00000	0.228456	0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000	0.228456	0.114228	+	0.993455i	$0.463561\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	−8.00000	−0.364013
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	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
$489$	−9.00000	−0.406994
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	0	0
$503$	−19.0000	−0.847168	−0.423584	−	0.905857i	$-0.639228\pi$
$503$	−19.0000	−0.847168	−0.423584	+	0.905857i	$0.639228\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.0000	−0.616914
$516$	0	0
$517$	−39.0000	−1.71522
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−13.0000	−0.569540	−0.284770	−	0.958596i	$-0.591917\pi$
$521$	−13.0000	−0.569540	−0.284770	+	0.958596i	$0.591917\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	10.0000	0.431532
$538$	0	0
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	0	0
$543$	7.00000	0.300399
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	−38.0000	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$548$	0	0
$549$	16.0000	0.682863
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−30.0000	−1.27573
$554$	0	0
$555$	8.00000	0.339581
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	16.0000	0.667246
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−33.0000	−1.36672
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	39.0000	1.60154	0.800769	−	0.598973i	$-0.204424\pi$
$593$	39.0000	1.60154	0.800769	+	0.598973i	$0.204424\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	10.0000	0.409273
$598$	0	0
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−3.00000	−0.121766	−0.0608831	−	0.998145i	$-0.519392\pi$
$607$	−3.00000	−0.121766	−0.0608831	+	0.998145i	$0.519392\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	13.0000	0.525924
$612$	0	0
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	0	0
$621$	20.0000	0.802572
$622$	0	0
$623$	−20.0000	−0.801283
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	64.0000	2.55185
$630$	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
$633$	3.00000	0.119239
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	0	0
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	0	0
$645$	11.0000	0.433125
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.00000	0.235159
$652$	0	0
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	−8.00000	−0.312110
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.00000	0.154881
$668$	0	0
$669$	26.0000	1.00522
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	9.00000	0.346925	0.173462	−	0.984841i	$-0.444505\pi$
$673$	9.00000	0.346925	0.173462	+	0.984841i	$0.444505\pi$
$674$	0	0
$675$	20.0000	0.769800
$676$	0	0
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	10.0000	0.381524
$688$	0	0
$689$	11.0000	0.419067
$690$	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
$693$	−12.0000	−0.455842
$694$	0	0
$695$	20.0000	0.758643
$696$	0	0
$697$	16.0000	0.606043
$698$	0	0
$699$	−1.00000	−0.0378235
$700$	0	0
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−13.0000	−0.489608
$706$	0	0
$707$	−16.0000	−0.601742
$708$	0	0
$709$	15.0000	0.563337	0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000	0.563337	0.281668	+	0.959512i	$0.409112\pi$
$710$	0	0
$711$	30.0000	1.12509
$712$	0	0
$713$	−12.0000	−0.449404
$714$	0	0
$715$	−3.00000	−0.112194
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.0000	1.86469	0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000	1.86469	0.932343	+	0.361576i	$0.117761\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	17.0000	0.632237
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	88.0000	3.25480
$732$	0	0
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	36.0000	1.32608
$738$	0	0
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	−27.0000	−0.983935
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	−12.0000	−0.435572
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	−16.0000	−0.578481
$766$	0	0
$767$	0	0
$768$	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
$771$	13.0000	0.468184
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−12.0000	−0.431053
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	5.00000	0.178685
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	0	0
$789$	−9.00000	−0.320408
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	−11.0000	−0.390130
$796$	0	0
$797$	−32.0000	−1.13350	−0.566749	−	0.823890i	$-0.691799\pi$
$797$	−32.0000	−1.13350	−0.566749	+	0.823890i	$0.691799\pi$
$798$	0	0
$799$	−104.000	−3.67926
$800$	0	0
$801$	20.0000	0.706665
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	18.0000	0.632065	0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000	0.632065	0.316033	+	0.948748i	$0.397649\pi$
$812$	0	0
$813$	13.0000	0.455930
$814$	0	0
$815$	−9.00000	−0.315256
$816$	0	0
$817$	0	0
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−13.0000	−0.452054	−0.226027	−	0.974121i	$-0.572574\pi$
$827$	−13.0000	−0.452054	−0.226027	+	0.974121i	$0.572574\pi$
$828$	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
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	−24.0000	−0.831551
$834$	0	0
$835$	2.00000	0.0692129
$836$	0	0
$837$	−15.0000	−0.518476
$838$	0	0
$839$	−45.0000	−1.55357	−0.776786	−	0.629764i	$-0.783151\pi$
$839$	−45.0000	−1.55357	−0.776786	+	0.629764i	$0.783151\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	27.0000	0.929929
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	−4.00000	−0.137442
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.0000	−0.922302	−0.461151	−	0.887322i	$-0.652563\pi$
$857$	−27.0000	−0.922302	−0.461151	+	0.887322i	$0.652563\pi$
$858$	0	0
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	46.0000	1.56586	0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000	1.56586	0.782929	+	0.622111i	$0.213725\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	47.0000	1.59620
$868$	0	0
$869$	−45.0000	−1.52652
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	−18.0000	−0.608511
$876$	0	0
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33.0000	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$887$	−33.0000	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	0	0
$894$	0	0
$895$	10.0000	0.334263
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	−3.00000	−0.100056
$900$	0	0
$901$	−88.0000	−2.93171
$902$	0	0
$903$	22.0000	0.732114
$904$	0	0
$905$	7.00000	0.232688
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	16.0000	0.530687
$910$	0	0
$911$	13.0000	0.430709	0.215355	−	0.976536i	$-0.430909\pi$
$911$	13.0000	0.430709	0.215355	+	0.976536i	$0.430909\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	−24.0000	−0.792550
$918$	0	0
$919$	−30.0000	−0.989609	−0.494804	−	0.869004i	$-0.664760\pi$
$919$	−30.0000	−0.989609	−0.494804	+	0.869004i	$0.664760\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	2.00000	0.0658308
$924$	0	0
$925$	−32.0000	−1.05215
$926$	0	0
$927$	28.0000	0.919641
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	9.00000	0.293704
$940$	0	0
$941$	37.0000	1.20617	0.603083	−	0.797679i	$-0.293939\pi$
$941$	37.0000	1.20617	0.603083	+	0.797679i	$0.293939\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	−10.0000	−0.325300
$946$	0	0
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−1.00000	−0.0323932	−0.0161966	−	0.999869i	$-0.505156\pi$
$953$	−1.00000	−0.0323932	−0.0161966	+	0.999869i	$0.505156\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	−3.00000	−0.0969762
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	40.0000	1.28234
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	13.0000	0.415907	0.207953	−	0.978139i	$-0.433320\pi$
$977$	13.0000	0.415907	0.207953	+	0.978139i	$0.433320\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−49.0000	−1.56286	−0.781429	−	0.623995i	$-0.785509\pi$
$983$	−49.0000	−1.56286	−0.781429	+	0.623995i	$0.785509\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	−26.0000	−0.827589
$988$	0	0
$989$	−44.0000	−1.39912
$990$	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
$993$	23.0000	0.729883
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	0	0
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.2.a.e.1.1		1
3.2	odd	2		4176.2.a.n.1.1		1
4.3	odd	2		58.2.a.b.1.1	✓	1
8.3	odd	2		1856.2.a.k.1.1		1
8.5	even	2		1856.2.a.f.1.1		1
12.11	even	2		522.2.a.b.1.1		1
20.3	even	4		1450.2.b.b.349.1		2
20.7	even	4		1450.2.b.b.349.2		2
20.19	odd	2		1450.2.a.c.1.1		1
28.27	even	2		2842.2.a.e.1.1		1
44.43	even	2		7018.2.a.a.1.1		1
52.51	odd	2		9802.2.a.a.1.1		1
116.75	even	4		1682.2.b.a.1681.1		2
116.99	even	4		1682.2.b.a.1681.2		2
116.115	odd	2		1682.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.a.b.1.1	✓	1	4.3	odd	2
464.2.a.e.1.1		1	1.1	even	1	trivial
522.2.a.b.1.1		1	12.11	even	2
1450.2.a.c.1.1		1	20.19	odd	2
1450.2.b.b.349.1		2	20.3	even	4
1450.2.b.b.349.2		2	20.7	even	4
1682.2.a.d.1.1		1	116.115	odd	2
1682.2.b.a.1681.1		2	116.75	even	4
1682.2.b.a.1681.2		2	116.99	even	4
1856.2.a.f.1.1		1	8.5	even	2
1856.2.a.k.1.1		1	8.3	odd	2
2842.2.a.e.1.1		1	28.27	even	2
4176.2.a.n.1.1		1	3.2	odd	2
7018.2.a.a.1.1		1	44.43	even	2
9802.2.a.a.1.1		1	52.51	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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