LMFDB - Embedded newform 3136.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(1,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, 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("3136.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.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:	$25.0410860739$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3136.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} -3.00000 q^{11} -6.00000 q^{13} +1.00000 q^{15} +5.00000 q^{17} +1.00000 q^{19} -7.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} -2.00000 q^{29} +5.00000 q^{31} +3.00000 q^{33} -3.00000 q^{37} +6.00000 q^{39} +2.00000 q^{41} +4.00000 q^{43} +2.00000 q^{45} -5.00000 q^{47} -5.00000 q^{51} +1.00000 q^{53} +3.00000 q^{55} -1.00000 q^{57} +15.0000 q^{59} -5.00000 q^{61} +6.00000 q^{65} +9.00000 q^{67} +7.00000 q^{69} -7.00000 q^{73} +4.00000 q^{75} +1.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -5.00000 q^{85} +2.00000 q^{87} -7.00000 q^{89} -5.00000 q^{93} -1.00000 q^{95} +2.00000 q^{97} +6.00000 q^{99} +O(q^{100})$

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	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	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
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	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
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	0	0
$69$	7.00000	0.842701
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	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
$85$	−5.00000	−0.542326
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.00000	−0.518476
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	15.0000	1.47799	0.738997	−	0.673709i	$-0.235300\pi$
$103$	15.0000	1.47799	0.738997	+	0.673709i	$0.235300\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	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
$111$	3.00000	0.284747
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	7.00000	0.652753
$116$	0	0
$117$	12.0000	1.10940
$118$	0	0
$119$	0	0
$120$	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	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	5.00000	0.436852	0.218426	−	0.975854i	$-0.429908\pi$
$131$	5.00000	0.436852	0.218426	+	0.975854i	$0.429908\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	11.0000	0.939793	0.469897	−	0.882721i	$-0.344291\pi$
$137$	11.0000	0.939793	0.469897	+	0.882721i	$0.344291\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	0	0
$143$	18.0000	1.50524
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	−10.0000	−0.808452
$154$	0	0
$155$	−5.00000	−0.401610
$156$	0	0
$157$	−9.00000	−0.718278	−0.359139	−	0.933284i	$-0.616930\pi$
$157$	−9.00000	−0.718278	−0.359139	+	0.933284i	$0.616930\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.0000	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−15.0000	−1.12747
$178$	0	0
$179$	13.0000	0.971666	0.485833	−	0.874052i	$-0.338516\pi$
$179$	13.0000	0.971666	0.485833	+	0.874052i	$0.338516\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	−15.0000	−1.09691
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	0	0
$193$	3.00000	0.215945	0.107972	−	0.994154i	$-0.465564\pi$
$193$	3.00000	0.215945	0.107972	+	0.994154i	$0.465564\pi$
$194$	0	0
$195$	−6.00000	−0.429669
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	13.0000	0.921546	0.460773	−	0.887518i	$-0.347572\pi$
$199$	13.0000	0.921546	0.460773	+	0.887518i	$0.347572\pi$
$200$	0	0
$201$	−9.00000	−0.634811
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	14.0000	0.973067
$208$	0	0
$209$	−3.00000	−0.207514
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	−30.0000	−2.01802
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	0	0
$229$	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
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	0	0
$235$	5.00000	0.326164
$236$	0	0
$237$	−1.00000	−0.0649570
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	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
$245$	0	0
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	21.0000	1.32026
$254$	0	0
$255$	5.00000	0.313112
$256$	0	0
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	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
$265$	−1.00000	−0.0614295
$266$	0	0
$267$	7.00000	0.428393
$268$	0	0
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−15.0000	−0.873334
$296$	0	0
$297$	−15.0000	−0.870388
$298$	0	0
$299$	42.0000	2.42892
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.00000	−0.172345
$304$	0	0
$305$	5.00000	0.286299
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−15.0000	−0.853320
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	5.00000	0.278207
$324$	0	0
$325$	24.0000	1.33128
$326$	0	0
$327$	−5.00000	−0.276501
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−29.0000	−1.59398	−0.796992	−	0.603990i	$-0.793577\pi$
$331$	−29.0000	−1.59398	−0.796992	+	0.603990i	$0.793577\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−9.00000	−0.491723
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	−15.0000	−0.812296
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.00000	−0.376867
$346$	0	0
$347$	−15.0000	−0.805242	−0.402621	−	0.915367i	$-0.631901\pi$
$347$	−15.0000	−0.805242	−0.402621	+	0.915367i	$0.631901\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−30.0000	−1.60128
$352$	0	0
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.0000	1.10834	0.554169	−	0.832404i	$-0.313036\pi$
$359$	21.0000	1.10834	0.554169	+	0.832404i	$0.313036\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	7.00000	0.366397
$366$	0	0
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	3.00000	0.153293	0.0766464	−	0.997058i	$-0.475579\pi$
$383$	3.00000	0.153293	0.0766464	+	0.997058i	$0.475579\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	29.0000	1.47036	0.735179	−	0.677873i	$-0.237098\pi$
$389$	29.0000	1.47036	0.735179	+	0.677873i	$0.237098\pi$
$390$	0	0
$391$	−35.0000	−1.77003
$392$	0	0
$393$	−5.00000	−0.252217
$394$	0	0
$395$	−1.00000	−0.0503155
$396$	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
$403$	−30.0000	−1.49441
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	9.00000	0.446113
$408$	0	0
$409$	−27.0000	−1.33506	−0.667532	−	0.744581i	$-0.732649\pi$
$409$	−27.0000	−1.33506	−0.667532	+	0.744581i	$0.732649\pi$
$410$	0	0
$411$	−11.0000	−0.542590
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	0	0
$423$	10.0000	0.486217
$424$	0	0
$425$	−20.0000	−0.970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−18.0000	−0.869048
$430$	0	0
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	−7.00000	−0.334855
$438$	0	0
$439$	−21.0000	−1.00228	−0.501138	−	0.865368i	$-0.667085\pi$
$439$	−21.0000	−1.00228	−0.501138	+	0.865368i	$0.667085\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.0000	1.47285	0.736427	−	0.676517i	$-0.236511\pi$
$443$	31.0000	1.47285	0.736427	+	0.676517i	$0.236511\pi$
$444$	0	0
$445$	7.00000	0.331832
$446$	0	0
$447$	−17.0000	−0.804072
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	5.00000	0.234920
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	25.0000	1.16690
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	5.00000	0.231869
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	9.00000	0.414698
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−31.0000	−1.41643	−0.708213	−	0.705999i	$-0.750498\pi$
$479$	−31.0000	−1.41643	−0.708213	+	0.705999i	$0.750498\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	7.00000	0.317200	0.158600	−	0.987343i	$-0.449302\pi$
$487$	7.00000	0.317200	0.158600	+	0.987343i	$0.449302\pi$
$488$	0	0
$489$	13.0000	0.587880
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	−10.0000	−0.450377
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	0	0
$498$	0	0
$499$	35.0000	1.56682	0.783408	−	0.621508i	$-0.213480\pi$
$499$	35.0000	1.56682	0.783408	+	0.621508i	$0.213480\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	−9.00000	−0.398918	−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000	−0.398918	−0.199459	+	0.979906i	$0.563918\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.00000	0.220755
$514$	0	0
$515$	−15.0000	−0.660979
$516$	0	0
$517$	15.0000	0.659699
$518$	0	0
$519$	13.0000	0.570637
$520$	0	0
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	0	0
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.0000	1.08902
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	−30.0000	−1.30189
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	9.00000	0.389104
$536$	0	0
$537$	−13.0000	−0.560991
$538$	0	0
$539$	0	0
$540$	0	0
$541$	41.0000	1.76273	0.881364	−	0.472438i	$-0.156626\pi$
$541$	41.0000	1.76273	0.881364	+	0.472438i	$0.156626\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	−5.00000	−0.214176
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.00000	−0.127343
$556$	0	0
$557$	5.00000	0.211857	0.105928	−	0.994374i	$-0.466219\pi$
$557$	5.00000	0.211857	0.105928	+	0.994374i	$0.466219\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	−41.0000	−1.72794	−0.863972	−	0.503540i	$-0.832031\pi$
$563$	−41.0000	−1.72794	−0.863972	+	0.503540i	$0.832031\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−33.0000	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$570$	0	0
$571$	13.0000	0.544033	0.272017	−	0.962293i	$-0.412309\pi$
$571$	13.0000	0.544033	0.272017	+	0.962293i	$0.412309\pi$
$572$	0	0
$573$	11.0000	0.459532
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	33.0000	1.37381	0.686904	−	0.726748i	$-0.258969\pi$
$577$	33.0000	1.37381	0.686904	+	0.726748i	$0.258969\pi$
$578$	0	0
$579$	−3.00000	−0.124676
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.00000	−0.124247
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	5.00000	0.206021
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	45.0000	1.84793	0.923964	−	0.382479i	$-0.124930\pi$
$593$	45.0000	1.84793	0.923964	+	0.382479i	$0.124930\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.0000	−0.532055
$598$	0	0
$599$	−17.0000	−0.694601	−0.347301	−	0.937754i	$-0.612902\pi$
$599$	−17.0000	−0.694601	−0.347301	+	0.937754i	$0.612902\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−18.0000	−0.733017
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.0000	1.21367
$612$	0	0
$613$	−43.0000	−1.73675	−0.868377	−	0.495905i	$-0.834836\pi$
$613$	−43.0000	−1.73675	−0.868377	+	0.495905i	$0.834836\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−9.00000	−0.361741	−0.180870	−	0.983507i	$-0.557891\pi$
$619$	−9.00000	−0.361741	−0.180870	+	0.983507i	$0.557891\pi$
$620$	0	0
$621$	−35.0000	−1.40450
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	−15.0000	−0.598089
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.0000	−0.671460	−0.335730	−	0.941958i	$-0.608983\pi$
$641$	−17.0000	−0.671460	−0.335730	+	0.941958i	$0.608983\pi$
$642$	0	0
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	−15.0000	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$647$	−15.0000	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$648$	0	0
$649$	−45.0000	−1.76640
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.0000	−1.36966	−0.684828	−	0.728705i	$-0.740123\pi$
$653$	−35.0000	−1.36966	−0.684828	+	0.728705i	$0.740123\pi$
$654$	0	0
$655$	−5.00000	−0.195366
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	0	0
$663$	30.0000	1.16510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.0000	0.542082
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	15.0000	0.579069
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	39.0000	1.49889	0.749446	−	0.662066i	$-0.230320\pi$
$677$	39.0000	1.49889	0.749446	+	0.662066i	$0.230320\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11.0000	−0.421521
$682$	0	0
$683$	−27.0000	−1.03313	−0.516563	−	0.856249i	$-0.672789\pi$
$683$	−27.0000	−1.03313	−0.516563	+	0.856249i	$0.672789\pi$
$684$	0	0
$685$	−11.0000	−0.420288
$686$	0	0
$687$	−23.0000	−0.877505
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	10.0000	0.378777
$698$	0	0
$699$	−11.0000	−0.416058
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	−3.00000	−0.113147
$704$	0	0
$705$	−5.00000	−0.188311
$706$	0	0
$707$	0	0
$708$	0	0
$709$	29.0000	1.08912	0.544559	−	0.838723i	$-0.316697\pi$
$709$	29.0000	1.08912	0.544559	+	0.838723i	$0.316697\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	−35.0000	−1.31076
$714$	0	0
$715$	−18.0000	−0.673162
$716$	0	0
$717$	−20.0000	−0.746914
$718$	0	0
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−17.0000	−0.632237
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	19.0000	0.701781	0.350891	−	0.936416i	$-0.385879\pi$
$733$	19.0000	0.701781	0.350891	+	0.936416i	$0.385879\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27.0000	−0.994558
$738$	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
$741$	6.00000	0.220416
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−17.0000	−0.622832
$746$	0	0
$747$	−24.0000	−0.878114
$748$	0	0
$749$	0	0
$750$	0	0
$751$	53.0000	1.93400	0.966999	−	0.254781i	$-0.0820034\pi$
$751$	53.0000	1.93400	0.966999	+	0.254781i	$0.0820034\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	5.00000	0.181969
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	−21.0000	−0.762252
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	10.0000	0.361551
$766$	0	0
$767$	−90.0000	−3.24971
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−21.0000	−0.756297
$772$	0	0
$773$	19.0000	0.683383	0.341691	−	0.939812i	$-0.389000\pi$
$773$	19.0000	0.683383	0.341691	+	0.939812i	$0.389000\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−10.0000	−0.357371
$784$	0	0
$785$	9.00000	0.321224
$786$	0	0
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	0	0
$789$	9.00000	0.320408
$790$	0	0
$791$	0	0
$792$	0	0
$793$	30.0000	1.06533
$794$	0	0
$795$	1.00000	0.0354663
$796$	0	0
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	−25.0000	−0.884436
$800$	0	0
$801$	14.0000	0.494666
$802$	0	0
$803$	21.0000	0.741074
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.00000	0.316815
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	−7.00000	−0.245501
$814$	0	0
$815$	13.0000	0.455370
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.00000	0.314102	0.157051	−	0.987590i	$-0.449801\pi$
$821$	9.00000	0.314102	0.157051	+	0.987590i	$0.449801\pi$
$822$	0	0
$823$	47.0000	1.63832	0.819159	−	0.573567i	$-0.194441\pi$
$823$	47.0000	1.63832	0.819159	+	0.573567i	$0.194441\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	−17.0000	−0.589723
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	25.0000	0.864126
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	−23.0000	−0.791224
$846$	0	0
$847$	0	0
$848$	0	0
$849$	13.0000	0.446159
$850$	0	0
$851$	21.0000	0.719871
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−39.0000	−1.33221	−0.666107	−	0.745856i	$-0.732041\pi$
$857$	−39.0000	−1.33221	−0.666107	+	0.745856i	$0.732041\pi$
$858$	0	0
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.0000	0.578687	0.289343	−	0.957225i	$-0.406563\pi$
$863$	17.0000	0.578687	0.289343	+	0.957225i	$0.406563\pi$
$864$	0	0
$865$	13.0000	0.442013
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−3.00000	−0.101768
$870$	0	0
$871$	−54.0000	−1.82972
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	15.0000	0.504219
$886$	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
$889$	0	0
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−5.00000	−0.167319
$894$	0	0
$895$	−13.0000	−0.434542
$896$	0	0
$897$	−42.0000	−1.40234
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	5.00000	0.166574
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	1.00000	0.0332045	0.0166022	−	0.999862i	$-0.494715\pi$
$907$	1.00000	0.0332045	0.0166022	+	0.999862i	$0.494715\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	−5.00000	−0.165295
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	12.0000	0.394558
$926$	0	0
$927$	−30.0000	−0.985329
$928$	0	0
$929$	−7.00000	−0.229663	−0.114831	−	0.993385i	$-0.536633\pi$
$929$	−7.00000	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15.0000	0.491078
$934$	0	0
$935$	15.0000	0.490552
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	−1.00000	−0.0326338
$940$	0	0
$941$	−33.0000	−1.07577	−0.537885	−	0.843018i	$-0.680776\pi$
$941$	−33.0000	−1.07577	−0.537885	+	0.843018i	$0.680776\pi$
$942$	0	0
$943$	−14.0000	−0.455903
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37.0000	−1.20234	−0.601169	−	0.799122i	$-0.705298\pi$
$947$	−37.0000	−1.20234	−0.601169	+	0.799122i	$0.705298\pi$
$948$	0	0
$949$	42.0000	1.36338
$950$	0	0
$951$	3.00000	0.0972817
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	11.0000	0.355952
$956$	0	0
$957$	−6.00000	−0.193952
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	18.0000	0.580042
$964$	0	0
$965$	−3.00000	−0.0965734
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	−5.00000	−0.160623
$970$	0	0
$971$	−35.0000	−1.12320	−0.561602	−	0.827408i	$-0.689815\pi$
$971$	−35.0000	−1.12320	−0.561602	+	0.827408i	$0.689815\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−24.0000	−0.768615
$976$	0	0
$977$	3.00000	0.0959785	0.0479893	−	0.998848i	$-0.484719\pi$
$977$	3.00000	0.0959785	0.0479893	+	0.998848i	$0.484719\pi$
$978$	0	0
$979$	21.0000	0.671163
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	21.0000	0.669796	0.334898	−	0.942254i	$-0.391298\pi$
$983$	21.0000	0.669796	0.334898	+	0.942254i	$0.391298\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−28.0000	−0.890348
$990$	0	0
$991$	15.0000	0.476491	0.238245	−	0.971205i	$-0.423428\pi$
$991$	15.0000	0.476491	0.238245	+	0.971205i	$0.423428\pi$
$992$	0	0
$993$	29.0000	0.920287
$994$	0	0
$995$	−13.0000	−0.412128
$996$	0	0
$997$	−41.0000	−1.29848	−0.649242	−	0.760582i	$-0.724914\pi$
$997$	−41.0000	−1.29848	−0.649242	+	0.760582i	$0.724914\pi$
$998$	0	0
$999$	−15.0000	−0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.a.i.1.1		1
4.3	odd	2		3136.2.a.t.1.1		1
7.3	odd	6		448.2.i.b.65.1		2
7.5	odd	6		448.2.i.b.193.1		2
7.6	odd	2		3136.2.a.u.1.1		1
8.3	odd	2		784.2.a.c.1.1		1
8.5	even	2		392.2.a.e.1.1		1
24.5	odd	2		3528.2.a.j.1.1		1
24.11	even	2		7056.2.a.u.1.1		1
28.3	even	6		448.2.i.d.65.1		2
28.19	even	6		448.2.i.d.193.1		2
28.27	even	2		3136.2.a.j.1.1		1
40.29	even	2		9800.2.a.s.1.1		1
56.3	even	6		112.2.i.a.65.1		2
56.5	odd	6		56.2.i.b.25.1	yes	2
56.11	odd	6		784.2.i.h.177.1		2
56.13	odd	2		392.2.a.c.1.1		1
56.19	even	6		112.2.i.a.81.1		2
56.27	even	2		784.2.a.h.1.1		1
56.37	even	6		392.2.i.b.361.1		2
56.45	odd	6		56.2.i.b.9.1	✓	2
56.51	odd	6		784.2.i.h.753.1		2
56.53	even	6		392.2.i.b.177.1		2
168.5	even	6		504.2.s.c.361.1		2
168.53	odd	6		3528.2.s.q.3313.1		2
168.59	odd	6		1008.2.s.g.289.1		2
168.83	odd	2		7056.2.a.bj.1.1		1
168.101	even	6		504.2.s.c.289.1		2
168.125	even	2		3528.2.a.p.1.1		1
168.131	odd	6		1008.2.s.g.865.1		2
168.149	odd	6		3528.2.s.q.361.1		2
280.69	odd	2		9800.2.a.be.1.1		1
280.117	even	12		1400.2.bh.a.249.1		4
280.157	even	12		1400.2.bh.a.849.2		4
280.173	even	12		1400.2.bh.a.249.2		4
280.213	even	12		1400.2.bh.a.849.1		4
280.229	odd	6		1400.2.q.d.1201.1		2
280.269	odd	6		1400.2.q.d.401.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.i.b.9.1	✓	2	56.45	odd	6
56.2.i.b.25.1	yes	2	56.5	odd	6
112.2.i.a.65.1		2	56.3	even	6
112.2.i.a.81.1		2	56.19	even	6
392.2.a.c.1.1		1	56.13	odd	2
392.2.a.e.1.1		1	8.5	even	2
392.2.i.b.177.1		2	56.53	even	6
392.2.i.b.361.1		2	56.37	even	6
448.2.i.b.65.1		2	7.3	odd	6
448.2.i.b.193.1		2	7.5	odd	6
448.2.i.d.65.1		2	28.3	even	6
448.2.i.d.193.1		2	28.19	even	6
504.2.s.c.289.1		2	168.101	even	6
504.2.s.c.361.1		2	168.5	even	6
784.2.a.c.1.1		1	8.3	odd	2
784.2.a.h.1.1		1	56.27	even	2
784.2.i.h.177.1		2	56.11	odd	6
784.2.i.h.753.1		2	56.51	odd	6
1008.2.s.g.289.1		2	168.59	odd	6
1008.2.s.g.865.1		2	168.131	odd	6
1400.2.q.d.401.1		2	280.269	odd	6
1400.2.q.d.1201.1		2	280.229	odd	6
1400.2.bh.a.249.1		4	280.117	even	12
1400.2.bh.a.249.2		4	280.173	even	12
1400.2.bh.a.849.1		4	280.213	even	12
1400.2.bh.a.849.2		4	280.157	even	12
3136.2.a.i.1.1		1	1.1	even	1	trivial
3136.2.a.j.1.1		1	28.27	even	2
3136.2.a.t.1.1		1	4.3	odd	2
3136.2.a.u.1.1		1	7.6	odd	2
3528.2.a.j.1.1		1	24.5	odd	2
3528.2.a.p.1.1		1	168.125	even	2
3528.2.s.q.361.1		2	168.149	odd	6
3528.2.s.q.3313.1		2	168.53	odd	6
7056.2.a.u.1.1		1	24.11	even	2
7056.2.a.bj.1.1		1	168.83	odd	2
9800.2.a.s.1.1		1	40.29	even	2
9800.2.a.be.1.1		1	280.69	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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.a (trivial)

Introduction

L-functions

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