LMFDB - Embedded newform 464.2.a.b.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 232)
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)

$f(q)$

$=$

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

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

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	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	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	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−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
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	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	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	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	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	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
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.00000	0.107211
$88$	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
$91$	2.00000	0.209657
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	16.0000	1.59206	0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000	1.59206	0.796030	+	0.605257i	$0.206930\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	6.00000	0.541002
$124$	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
$129$	−5.00000	−0.440225
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	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	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	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
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.00000	−0.240966
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−5.00000	−0.396526
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	0	0
$188$	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
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	−6.00000	−0.419058
$206$	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	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	5.00000	0.340997
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	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
$233$	−17.0000	−1.11371	−0.556854	−	0.830611i	$-0.687992\pi$
$233$	−17.0000	−1.11371	−0.556854	+	0.830611i	$0.687992\pi$
$234$	0	0
$235$	−3.00000	−0.195698
$236$	0	0
$237$	1.00000	0.0649570
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−31.0000	−1.99689	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−31.0000	−1.99689	−0.998443	+	0.0557856i	$0.982234\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	25.0000	1.54157	0.770783	−	0.637098i	$-0.219865\pi$
$263$	25.0000	1.54157	0.770783	+	0.637098i	$0.219865\pi$
$264$	0	0
$265$	5.00000	0.307148
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−15.0000	−0.870388
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	−16.0000	−0.919176
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−23.0000	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	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
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	11.0000	0.608301
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	16.0000	0.876795
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	24.0000	1.30736	0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000	1.30736	0.653682	+	0.756770i	$0.273224\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	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
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	0	0
$357$	0	0
$358$	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
$361$	−19.0000	−1.00000
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	27.0000	1.39801	0.699004	−	0.715118i	$-0.253627\pi$
$373$	27.0000	1.39801	0.699004	+	0.715118i	$0.253627\pi$
$374$	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	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	−10.0000	−0.508329
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	−1.00000	−0.0503155
$396$	0	0
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	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
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	−16.0000	−0.787309
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	22.0000	1.07477	0.537385	−	0.843337i	$-0.319412\pi$
$419$	22.0000	1.07477	0.537385	+	0.843337i	$0.319412\pi$
$420$	0	0
$421$	16.0000	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.00000	−0.144841
$430$	0	0
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	0	0
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	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
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	1.00000	0.0472984
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	0	0
$453$	22.0000	1.03365
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	0	0
$465$	3.00000	0.139122
$466$	0	0
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−15.0000	−0.689701
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	−8.00000	−0.364013
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	0	0
$489$	−9.00000	−0.406994
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	12.0000	0.538274
$498$	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
$501$	2.00000	0.0893534
$502$	0	0
$503$	19.0000	0.847168	0.423584	−	0.905857i	$-0.360772\pi$
$503$	19.0000	0.847168	0.423584	+	0.905857i	$0.360772\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	1.00000	0.0443242	0.0221621	−	0.999754i	$-0.492945\pi$
$509$	1.00000	0.0443242	0.0221621	+	0.999754i	$0.492945\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.0000	−0.440653
$516$	0	0
$517$	9.00000	0.395820
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−45.0000	−1.97149	−0.985743	−	0.168259i	$-0.946186\pi$
$521$	−45.0000	−1.97149	−0.985743	+	0.168259i	$0.946186\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−16.0000	−0.694341
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	−18.0000	−0.778208
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	9.00000	0.387657
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	25.0000	1.07285
$544$	0	0
$545$	−11.0000	−0.471188
$546$	0	0
$547$	14.0000	0.598597	0.299298	−	0.954160i	$-0.403247\pi$
$547$	14.0000	0.598597	0.299298	+	0.954160i	$0.403247\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	8.00000	0.339581
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.0000	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$563$	−27.0000	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	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	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	−15.0000	−0.621237
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	7.00000	0.287456	0.143728	−	0.989617i	$-0.454091\pi$
$593$	7.00000	0.287456	0.143728	+	0.989617i	$0.454091\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−14.0000	−0.572982
$598$	0	0
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−45.0000	−1.82649	−0.913247	−	0.407407i	$-0.866433\pi$
$607$	−45.0000	−1.82649	−0.913247	+	0.407407i	$0.866433\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	0	0
$621$	−20.0000	−0.802572
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−6.00000	−0.238856	−0.119428	−	0.992843i	$-0.538106\pi$
$631$	−6.00000	−0.238856	−0.119428	+	0.992843i	$0.538106\pi$
$632$	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
$639$	12.0000	0.474713
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	0	0
$645$	−5.00000	−0.196875
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	−1.00000	−0.0389545	−0.0194772	−	0.999810i	$-0.506200\pi$
$659$	−1.00000	−0.0389545	−0.0194772	+	0.999810i	$0.506200\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.00000	0.154881
$668$	0	0
$669$	26.0000	1.00522
$670$	0	0
$671$	0	0
$672$	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
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−28.0000	−1.07454
$680$	0	0
$681$	6.00000	0.229920
$682$	0	0
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−5.00000	−0.190485
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−12.0000	−0.455842
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	0	0
$698$	0	0
$699$	17.0000	0.642999
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	3.00000	0.112987
$706$	0	0
$707$	−32.0000	−1.20348
$708$	0	0
$709$	−33.0000	−1.23934	−0.619671	−	0.784862i	$-0.712734\pi$
$709$	−33.0000	−1.23934	−0.619671	+	0.784862i	$0.712734\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	3.00000	0.112194
$716$	0	0
$717$	−16.0000	−0.597531
$718$	0	0
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	31.0000	1.15290
$724$	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
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	−36.0000	−1.32608
$738$	0	0
$739$	43.0000	1.58178	0.790890	−	0.611958i	$-0.209618\pi$
$739$	43.0000	1.58178	0.790890	+	0.611958i	$0.209618\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	−1.00000	−0.0366372
$746$	0	0
$747$	−24.0000	−0.878114
$748$	0	0
$749$	36.0000	1.31541
$750$	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
$753$	21.0000	0.765283
$754$	0	0
$755$	−22.0000	−0.800662
$756$	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
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	22.0000	0.796453
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	0	0
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	12.0000	0.431053
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	0	0
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	0	0
$789$	−25.0000	−0.890024
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−5.00000	−0.177332
$796$	0	0
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	8.00000	0.281963
$806$	0	0
$807$	24.0000	0.844840
$808$	0	0
$809$	−32.0000	−1.12506	−0.562530	−	0.826777i	$-0.690172\pi$
$809$	−32.0000	−1.12506	−0.562530	+	0.826777i	$0.690172\pi$
$810$	0	0
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	0	0
$813$	29.0000	1.01707
$814$	0	0
$815$	9.00000	0.315256
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	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
$827$	−35.0000	−1.21707	−0.608535	−	0.793527i	$-0.708242\pi$
$827$	−35.0000	−1.21707	−0.608535	+	0.793527i	$0.708242\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	−14.0000	−0.485655
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	−15.0000	−0.518476
$838$	0	0
$839$	−35.0000	−1.20833	−0.604167	−	0.796858i	$-0.706494\pi$
$839$	−35.0000	−1.20833	−0.604167	+	0.796858i	$0.706494\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	5.00000	0.172209
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	4.00000	0.137442
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.0000	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$858$	0	0
$859$	55.0000	1.87658	0.938288	−	0.345855i	$-0.112411\pi$
$859$	55.0000	1.87658	0.938288	+	0.345855i	$0.112411\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	3.00000	0.101768
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	−28.0000	−0.947656
$874$	0	0
$875$	18.0000	0.608511
$876$	0	0
$877$	29.0000	0.979260	0.489630	−	0.871930i	$-0.337132\pi$
$877$	29.0000	0.979260	0.489630	+	0.871930i	$0.337132\pi$
$878$	0	0
$879$	−22.0000	−0.742042
$880$	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
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	0	0
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	0	0
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	3.00000	0.100056
$900$	0	0
$901$	0	0
$902$	0	0
$903$	10.0000	0.332779
$904$	0	0
$905$	−25.0000	−0.831028
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	0	0
$909$	−32.0000	−1.06137
$910$	0	0
$911$	−13.0000	−0.430709	−0.215355	−	0.976536i	$-0.569091\pi$
$911$	−13.0000	−0.430709	−0.215355	+	0.976536i	$0.569091\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	6.00000	0.197922	0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000	0.197922	0.0989609	+	0.995091i	$0.468448\pi$
$920$	0	0
$921$	23.0000	0.757876
$922$	0	0
$923$	6.00000	0.197492
$924$	0	0
$925$	32.0000	1.05215
$926$	0	0
$927$	20.0000	0.656886
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	−9.00000	−0.293704
$940$	0	0
$941$	5.00000	0.162995	0.0814977	−	0.996674i	$-0.474030\pi$
$941$	5.00000	0.162995	0.0814977	+	0.996674i	$0.474030\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	−10.0000	−0.325300
$946$	0	0
$947$	−15.0000	−0.487435	−0.243717	−	0.969846i	$-0.578367\pi$
$947$	−15.0000	−0.487435	−0.243717	+	0.969846i	$0.578367\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	28.0000	0.907962
$952$	0	0
$953$	47.0000	1.52248	0.761240	−	0.648471i	$-0.224591\pi$
$953$	47.0000	1.52248	0.761240	+	0.648471i	$0.224591\pi$
$954$	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
$963$	36.0000	1.16008
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	29.0000	0.932577	0.466289	−	0.884633i	$-0.345591\pi$
$967$	29.0000	0.932577	0.466289	+	0.884633i	$0.345591\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−35.0000	−1.11975	−0.559875	−	0.828577i	$-0.689151\pi$
$977$	−35.0000	−1.11975	−0.559875	+	0.828577i	$0.689151\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	22.0000	0.702406
$982$	0	0
$983$	1.00000	0.0318950	0.0159475	−	0.999873i	$-0.494924\pi$
$983$	1.00000	0.0318950	0.0159475	+	0.999873i	$0.494924\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	−20.0000	−0.635963
$990$	0	0
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	0	0
$993$	7.00000	0.222138
$994$	0	0
$995$	14.0000	0.443830
$996$	0	0
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	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.b.1.1		1
3.2	odd	2		4176.2.a.l.1.1		1
4.3	odd	2		232.2.a.b.1.1	✓	1
8.3	odd	2		1856.2.a.e.1.1		1
8.5	even	2		1856.2.a.j.1.1		1
12.11	even	2		2088.2.a.d.1.1		1
20.19	odd	2		5800.2.a.e.1.1		1
116.115	odd	2		6728.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.a.b.1.1	✓	1	4.3	odd	2
464.2.a.b.1.1		1	1.1	even	1	trivial
1856.2.a.e.1.1		1	8.3	odd	2
1856.2.a.j.1.1		1	8.5	even	2
2088.2.a.d.1.1		1	12.11	even	2
4176.2.a.l.1.1		1	3.2	odd	2
5800.2.a.e.1.1		1	20.19	odd	2
6728.2.a.b.1.1		1	116.115	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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