LMFDB - Embedded newform 8.13.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,13,Mod(3,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	8.d (of order $2$, degree $1$, minimal)

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:	$7.31195053821$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			3.1
Character	$\chi$	$=$	8.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+64.0000 q^{2} +658.000 q^{3} +4096.00 q^{4} +42112.0 q^{6} +262144. q^{8} -98477.0 q^{9} +O(q^{10})$

$q+64.0000 q^{2} +658.000 q^{3} +4096.00 q^{4} +42112.0 q^{6} +262144. q^{8} -98477.0 q^{9} +1.92312e6 q^{11} +2.69517e6 q^{12} +1.67772e7 q^{16} -4.52961e7 q^{17} -6.30253e6 q^{18} -8.79314e7 q^{19} +1.23080e8 q^{22} +1.72491e8 q^{24} +2.44141e8 q^{25} -4.14486e8 q^{27} +1.07374e9 q^{32} +1.26541e9 q^{33} -2.89895e9 q^{34} -4.03362e8 q^{36} -5.62761e9 q^{38} +8.62826e9 q^{41} -7.03062e9 q^{43} +7.87711e9 q^{44} +1.10394e10 q^{48} +1.38413e10 q^{49} +1.56250e10 q^{50} -2.98048e10 q^{51} -2.65271e10 q^{54} -5.78589e10 q^{57} +8.63831e9 q^{59} +6.87195e10 q^{64} +8.09865e10 q^{66} +1.75046e11 q^{67} -1.85533e11 q^{68} -2.58152e10 q^{72} +4.91395e10 q^{73} +1.60645e11 q^{75} -3.60167e11 q^{76} -2.20397e11 q^{81} +5.52209e11 q^{82} -1.92940e11 q^{83} -4.49960e11 q^{86} +5.04135e11 q^{88} -8.66326e11 q^{89} +7.06522e11 q^{96} +1.65649e12 q^{97} +8.85842e11 q^{98} -1.89383e11 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$.

$n$	$5$	$7$
$\chi(n)$	$-1$	$-1$

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 $

$2$	64.0000	1.00000
$2$	64.0000	1.00000
$3$	658.000	0.902606	0.451303	−	0.892371i	$-0.350959\pi$
$3$	658.000	0.902606	0.451303	+	0.892371i	$0.350959\pi$
$4$	4096.00	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	42112.0	0.902606
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	262144.	1.00000
$9$	−98477.0	−0.185302
$10$	0	0
$11$	1.92312e6	1.08555	0.542776	−	0.839877i	$-0.317373\pi$
$11$	1.92312e6	1.08555	0.542776	+	0.839877i	$0.317373\pi$
$12$	2.69517e6	0.902606
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.67772e7	1.00000
$17$	−4.52961e7	−1.87658	−0.938290	−	0.345850i	$-0.887590\pi$
$17$	−4.52961e7	−1.87658	−0.938290	+	0.345850i	$0.887590\pi$
$18$	−6.30253e6	−0.185302
$19$	−8.79314e7	−1.86906	−0.934529	−	0.355888i	$-0.884178\pi$
$19$	−8.79314e7	−1.86906	−0.934529	+	0.355888i	$0.884178\pi$
$20$	0	0
$21$	0	0
$22$	1.23080e8	1.08555
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.72491e8	0.902606
$25$	2.44141e8	1.00000
$26$	0	0
$27$	−4.14486e8	−1.06986
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.07374e9	1.00000
$33$	1.26541e9	0.979826
$34$	−2.89895e9	−1.87658
$35$	0	0
$36$	−4.03362e8	−0.185302
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−5.62761e9	−1.86906
$39$	0	0
$40$	0	0
$41$	8.62826e9	1.81644	0.908218	−	0.418498i	$-0.137443\pi$
$41$	8.62826e9	1.81644	0.908218	+	0.418498i	$0.137443\pi$
$42$	0	0
$43$	−7.03062e9	−1.11220	−0.556100	−	0.831115i	$-0.687703\pi$
$43$	−7.03062e9	−1.11220	−0.556100	+	0.831115i	$0.687703\pi$
$44$	7.87711e9	1.08555
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.10394e10	0.902606
$49$	1.38413e10	1.00000
$50$	1.56250e10	1.00000
$51$	−2.98048e10	−1.69381
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−2.65271e10	−1.06986
$55$	0	0
$56$	0	0
$57$	−5.78589e10	−1.68702
$58$	0	0
$59$	8.63831e9	0.204794	0.102397	−	0.994744i	$-0.467349\pi$
$59$	8.63831e9	0.204794	0.102397	+	0.994744i	$0.467349\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	6.87195e10	1.00000
$65$	0	0
$66$	8.09865e10	0.979826
$67$	1.75046e11	1.93510	0.967549	−	0.252684i	$-0.0813132\pi$
$67$	1.75046e11	1.93510	0.967549	+	0.252684i	$0.0813132\pi$
$68$	−1.85533e11	−1.87658
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−2.58152e10	−0.185302
$73$	4.91395e10	0.324708	0.162354	−	0.986733i	$-0.448091\pi$
$73$	4.91395e10	0.324708	0.162354	+	0.986733i	$0.448091\pi$
$74$	0	0
$75$	1.60645e11	0.902606
$76$	−3.60167e11	−1.86906
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−2.20397e11	−0.780361
$82$	5.52209e11	1.81644
$83$	−1.92940e11	−0.590139	−0.295069	−	0.955476i	$-0.595343\pi$
$83$	−1.92940e11	−0.590139	−0.295069	+	0.955476i	$0.595343\pi$
$84$	0	0
$85$	0	0
$86$	−4.49960e11	−1.11220
$87$	0	0
$88$	5.04135e11	1.08555
$89$	−8.66326e11	−1.74318	−0.871589	−	0.490238i	$-0.836910\pi$
$89$	−8.66326e11	−1.74318	−0.871589	+	0.490238i	$0.836910\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	7.06522e11	0.902606
$97$	1.65649e12	1.98865	0.994324	−	0.106394i	$-0.0339306\pi$
$97$	1.65649e12	1.98865	0.994324	+	0.106394i	$0.0339306\pi$
$98$	8.85842e11	1.00000
$99$	−1.89383e11	−0.201155
$100$	1.00000e12	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−1.90751e12	−1.69381
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.77677e12	−1.85028	−0.925140	−	0.379626i	$-0.876053\pi$
$107$	−2.77677e12	−1.85028	−0.925140	+	0.379626i	$0.876053\pi$
$108$	−1.69773e12	−1.06986
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.74847e12	1.80046	0.900230	−	0.435416i	$-0.143399\pi$
$113$	3.74847e12	1.80046	0.900230	+	0.435416i	$0.143399\pi$
$114$	−3.70297e12	−1.68702
$115$	0	0
$116$	0	0
$117$	0	0
$118$	5.52852e11	0.204794
$119$	0	0
$120$	0	0
$121$	5.59970e11	0.178424
$122$	0	0
$123$	5.67739e12	1.63953
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	4.39805e12	1.00000
$129$	−4.62615e12	−1.00388
$130$	0	0
$131$	−1.38397e12	−0.273841	−0.136920	−	0.990582i	$-0.543720\pi$
$131$	−1.38397e12	−0.273841	−0.136920	+	0.990582i	$0.543720\pi$
$132$	5.18314e12	0.979826
$133$	0	0
$134$	1.12029e13	1.93510
$135$	0	0
$136$	−1.18741e13	−1.87658
$137$	−1.32173e13	−1.99903	−0.999514	−	0.0311883i	$-0.990071\pi$
$137$	−1.32173e13	−1.99903	−0.999514	+	0.0311883i	$0.990071\pi$
$138$	0	0
$139$	−4.19430e12	−0.581528	−0.290764	−	0.956795i	$-0.593909\pi$
$139$	−4.19430e12	−0.581528	−0.290764	+	0.956795i	$0.593909\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−1.65217e12	−0.185302
$145$	0	0
$146$	3.14493e12	0.324708
$147$	9.10757e12	0.902606
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.02812e13	0.902606
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−2.30507e13	−1.86906
$153$	4.46062e12	0.347734
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−1.41054e13	−0.780361
$163$	2.20968e13	1.17816	0.589080	−	0.808075i	$-0.299490\pi$
$163$	2.20968e13	1.17816	0.589080	+	0.808075i	$0.299490\pi$
$164$	3.53414e13	1.81644
$165$	0	0
$166$	−1.23482e13	−0.590139
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	2.32981e13	1.00000
$170$	0	0
$171$	8.65922e12	0.346340
$172$	−2.87974e13	−1.11220
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	3.22646e13	1.08555
$177$	5.68401e12	0.184848
$178$	−5.54449e13	−1.74318
$179$	−5.53626e13	−1.68305	−0.841527	−	0.540215i	$-0.818343\pi$
$179$	−5.53626e13	−1.68305	−0.841527	+	0.540215i	$0.818343\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−8.71099e13	−2.03712
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	4.52174e13	0.902606
$193$	−3.16470e12	−0.0612335	−0.0306167	−	0.999531i	$-0.509747\pi$
$193$	−3.16470e12	−0.0612335	−0.0306167	+	0.999531i	$0.509747\pi$
$194$	1.06015e14	1.98865
$195$	0	0
$196$	5.66939e13	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.21205e13	−0.201155
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	6.40000e13	1.00000
$201$	1.15180e14	1.74663
$202$	0	0
$203$	0	0
$204$	−1.22080e14	−1.69381
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.69103e14	−2.02896
$210$	0	0
$211$	1.71035e14	1.93816	0.969079	−	0.246750i	$-0.0793627\pi$
$211$	1.71035e14	1.93816	0.969079	+	0.246750i	$0.0793627\pi$
$212$	0	0
$213$	0	0
$214$	−1.77713e14	−1.85028
$215$	0	0
$216$	−1.08655e14	−1.06986
$217$	0	0
$218$	0	0
$219$	3.23338e13	0.293084
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−2.40422e13	−0.185302
$226$	2.39902e14	1.80046
$227$	1.17238e14	0.856870	0.428435	−	0.903573i	$-0.359065\pi$
$227$	1.17238e14	0.856870	0.428435	+	0.903573i	$0.359065\pi$
$228$	−2.36990e14	−1.68702
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.97639e14	−1.23520	−0.617599	−	0.786494i	$-0.711894\pi$
$233$	−1.97639e14	−1.23520	−0.617599	+	0.786494i	$0.711894\pi$
$234$	0	0
$235$	0	0
$236$	3.53825e13	0.204794
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	3.10482e14	1.58465	0.792326	−	0.610097i	$-0.208870\pi$
$241$	3.10482e14	1.58465	0.792326	+	0.610097i	$0.208870\pi$
$242$	3.58381e13	0.178424
$243$	7.52536e13	0.365502
$244$	0	0
$245$	0	0
$246$	3.63353e14	1.63953
$247$	0	0
$248$	0	0
$249$	−1.26955e14	−0.532663
$250$	0	0
$251$	−3.28002e14	−1.31170	−0.655849	−	0.754892i	$-0.727689\pi$
$251$	−3.28002e14	−1.31170	−0.655849	+	0.754892i	$0.727689\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	2.81475e14	1.00000
$257$	−2.16326e14	−0.750775	−0.375388	−	0.926868i	$-0.622490\pi$
$257$	−2.16326e14	−0.750775	−0.375388	+	0.926868i	$0.622490\pi$
$258$	−2.96073e14	−1.00388
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−8.85739e13	−0.273841
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	3.31721e14	0.979826
$265$	0	0
$266$	0	0
$267$	−5.70043e14	−1.57340
$268$	7.16988e14	1.93510
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−7.59942e14	−1.87658
$273$	0	0
$274$	−8.45906e14	−1.99903
$275$	4.69512e14	1.08555
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−2.68435e14	−0.581528
$279$	0	0
$280$	0	0
$281$	8.55524e14	1.73778	0.868889	−	0.495007i	$-0.164834\pi$
$281$	8.55524e14	1.73778	0.868889	+	0.495007i	$0.164834\pi$
$282$	0	0
$283$	−6.60679e14	−1.28609	−0.643046	−	0.765828i	$-0.722329\pi$
$283$	−6.60679e14	−1.28609	−0.643046	+	0.765828i	$0.722329\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.05739e14	−0.185302
$289$	1.46911e15	2.52155
$290$	0	0
$291$	1.08997e15	1.79497
$292$	2.01275e14	0.324708
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	5.82884e14	0.902606
$295$	0	0
$296$	0	0
$297$	−7.97107e14	−1.16139
$298$	0	0
$299$	0	0
$300$	6.58000e14	0.902606
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−1.47524e15	−1.86906
$305$	0	0
$306$	2.85480e14	0.347734
$307$	−1.63875e15	−1.95741	−0.978706	−	0.205269i	$-0.934193\pi$
$307$	−1.63875e15	−1.95741	−0.978706	+	0.205269i	$0.934193\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.79845e15	−1.91264	−0.956319	−	0.292326i	$-0.905571\pi$
$313$	−1.79845e15	−1.91264	−0.956319	+	0.292326i	$0.905571\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.82712e15	−1.67007
$322$	0	0
$323$	3.98295e15	3.50743
$324$	−9.02747e14	−0.780361
$325$	0	0
$326$	1.41420e15	1.17816
$327$	0	0
$328$	2.26185e15	1.81644
$329$	0	0
$330$	0	0
$331$	−2.60911e15	−1.98392	−0.991959	−	0.126557i	$-0.959607\pi$
$331$	−2.60911e15	−1.98392	−0.991959	+	0.126557i	$0.959607\pi$
$332$	−7.90283e14	−0.590139
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5.20794e13	−0.0355538	−0.0177769	−	0.999842i	$-0.505659\pi$
$337$	−5.20794e13	−0.0355538	−0.0177769	+	0.999842i	$0.505659\pi$
$338$	1.49108e15	1.00000
$339$	2.46649e15	1.62511
$340$	0	0
$341$	0	0
$342$	5.54190e14	0.346340
$343$	0	0
$344$	−1.84303e15	−1.11220
$345$	0	0
$346$	0	0
$347$	−1.26334e15	−0.723675	−0.361837	−	0.932241i	$-0.617850\pi$
$347$	−1.26334e15	−0.723675	−0.361837	+	0.932241i	$0.617850\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	2.06494e15	1.08555
$353$	3.84078e14	0.198505	0.0992523	−	0.995062i	$-0.468355\pi$
$353$	3.84078e14	0.198505	0.0992523	+	0.995062i	$0.468355\pi$
$354$	3.63777e14	0.184848
$355$	0	0
$356$	−3.54847e15	−1.74318
$357$	0	0
$358$	−3.54320e15	−1.68305
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	5.51862e15	2.49337
$362$	0	0
$363$	3.68460e14	0.161046
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−8.49685e14	−0.336589
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−5.57503e15	−2.03712
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−5.18672e15	−1.75008	−0.875039	−	0.484052i	$-0.839165\pi$
$379$	−5.18672e15	−1.75008	−0.875039	+	0.484052i	$0.839165\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	2.89391e15	0.902606
$385$	0	0
$386$	−2.02541e14	−0.0612335
$387$	6.92354e14	0.206093
$388$	6.78498e15	1.98865
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	3.62841e15	1.00000
$393$	−9.10650e14	−0.247170
$394$	0	0
$395$	0	0
$396$	−7.75714e14	−0.201155
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	4.09600e15	1.00000
$401$	−1.92614e15	−0.463257	−0.231629	−	0.972804i	$-0.574405\pi$
$401$	−1.92614e15	−0.463257	−0.231629	+	0.972804i	$0.574405\pi$
$402$	7.37153e15	1.74663
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−7.81315e15	−1.69381
$409$	7.63062e15	1.63012	0.815061	−	0.579375i	$-0.196703\pi$
$409$	7.63062e15	1.63012	0.815061	+	0.579375i	$0.196703\pi$
$410$	0	0
$411$	−8.69697e15	−1.80433
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.75985e15	−0.524891
$418$	−1.08226e16	−2.02896
$419$	7.38079e15	1.36401	0.682007	−	0.731346i	$-0.261108\pi$
$419$	7.38079e15	1.36401	0.682007	+	0.731346i	$0.261108\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1.09462e16	1.93816
$423$	0	0
$424$	0	0
$425$	−1.10586e16	−1.87658
$426$	0	0
$427$	0	0
$428$	−1.13737e16	−1.85028
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−6.95392e15	−1.06986
$433$	4.24415e15	0.643966	0.321983	−	0.946745i	$-0.395651\pi$
$433$	4.24415e15	0.643966	0.321983	+	0.946745i	$0.395651\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	2.06936e15	0.293084
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−1.36305e15	−0.185302
$442$	0	0
$443$	1.04567e16	1.38348	0.691738	−	0.722149i	$-0.256845\pi$
$443$	1.04567e16	1.38348	0.691738	+	0.722149i	$0.256845\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.86206e14	−0.0715438	−0.0357719	−	0.999360i	$-0.511389\pi$
$449$	−5.86206e14	−0.0715438	−0.0357719	+	0.999360i	$0.511389\pi$
$450$	−1.53870e15	−0.185302
$451$	1.65932e16	1.97184
$452$	1.53537e16	1.80046
$453$	0	0
$454$	7.50326e15	0.856870
$455$	0	0
$456$	−1.51674e16	−1.68702
$457$	1.78261e14	0.0195686	0.00978428	−	0.999952i	$-0.496886\pi$
$457$	1.78261e14	0.0195686	0.00978428	+	0.999952i	$0.496886\pi$
$458$	0	0
$459$	1.87746e16	2.00768
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−1.26489e16	−1.23520
$467$	−2.02528e16	−1.95246	−0.976232	−	0.216730i	$-0.930461\pi$
$467$	−2.02528e16	−1.95246	−0.976232	+	0.216730i	$0.930461\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	2.26448e15	0.204794
$473$	−1.35207e16	−1.20735
$474$	0	0
$475$	−2.14676e16	−1.86906
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	1.98708e16	1.58465
$483$	0	0
$484$	2.29364e15	0.178424
$485$	0	0
$486$	4.81623e15	0.365502
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	1.45397e16	1.06341
$490$	0	0
$491$	−2.03908e16	−1.45527	−0.727637	−	0.685963i	$-0.759381\pi$
$491$	−2.03908e16	−1.45527	−0.727637	+	0.685963i	$0.759381\pi$
$492$	2.32546e16	1.63953
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−8.12510e15	−0.532663
$499$	−2.39468e16	−1.55112	−0.775559	−	0.631275i	$-0.782532\pi$
$499$	−2.39468e16	−1.55112	−0.775559	+	0.631275i	$0.782532\pi$
$500$	0	0
$501$	0	0
$502$	−2.09921e16	−1.31170
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.53301e16	0.902606
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.80144e16	1.00000
$513$	3.64464e16	1.99963
$514$	−1.38449e16	−0.750775
$515$	0	0
$516$	−1.89487e16	−1.00388
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.38740e14	−0.0219372	−0.0109686	−	0.999940i	$-0.503491\pi$
$521$	−4.38740e14	−0.0219372	−0.0109686	+	0.999940i	$0.503491\pi$
$522$	0	0
$523$	2.85281e16	1.39400	0.697001	−	0.717070i	$-0.254517\pi$
$523$	2.85281e16	1.39400	0.697001	+	0.717070i	$0.254517\pi$
$524$	−5.66873e15	−0.273841
$525$	0	0
$526$	0	0
$527$	0	0
$528$	2.12301e16	0.979826
$529$	2.19146e16	1.00000
$530$	0	0
$531$	−8.50675e14	−0.0379487
$532$	0	0
$533$	0	0
$534$	−3.64827e16	−1.57340
$535$	0	0
$536$	4.58872e16	1.93510
$537$	−3.64286e16	−1.51914
$538$	0	0
$539$	2.66185e16	1.08555
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−4.86363e16	−1.87658
$545$	0	0
$546$	0	0
$547$	−3.10476e16	−1.15906	−0.579528	−	0.814952i	$-0.696763\pi$
$547$	−3.10476e16	−1.15906	−0.579528	+	0.814952i	$0.696763\pi$
$548$	−5.41380e16	−1.99903
$549$	0	0
$550$	3.00488e16	1.08555
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−1.71799e16	−0.581528
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−5.73183e16	−1.83872
$562$	5.47535e16	1.73778
$563$	−2.23353e16	−0.701361	−0.350681	−	0.936495i	$-0.614050\pi$
$563$	−2.23353e16	−0.701361	−0.350681	+	0.936495i	$0.614050\pi$
$564$	0	0
$565$	0	0
$566$	−4.22834e16	−1.28609
$567$	0	0
$568$	0	0
$569$	6.36836e16	1.87652	0.938262	−	0.345925i	$-0.112435\pi$
$569$	6.36836e16	1.87652	0.938262	+	0.345925i	$0.112435\pi$
$570$	0	0
$571$	5.93581e16	1.71263	0.856314	−	0.516455i	$-0.172749\pi$
$571$	5.93581e16	1.71263	0.856314	+	0.516455i	$0.172749\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−6.76729e15	−0.185302
$577$	−7.38009e16	−1.99989	−0.999946	−	0.0103984i	$-0.996690\pi$
$577$	−7.38009e16	−1.99989	−0.999946	+	0.0103984i	$0.996690\pi$
$578$	9.40231e16	2.52155
$579$	−2.08237e15	−0.0552697
$580$	0	0
$581$	0	0
$582$	6.97580e16	1.79497
$583$	0	0
$584$	1.28816e16	0.324708
$585$	0	0
$586$	0	0
$587$	6.62782e15	0.162010	0.0810051	−	0.996714i	$-0.474187\pi$
$587$	6.62782e15	0.162010	0.0810051	+	0.996714i	$0.474187\pi$
$588$	3.73046e16	0.902606
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.46818e16	−0.337638	−0.168819	−	0.985647i	$-0.553995\pi$
$593$	−1.46818e16	−0.337638	−0.168819	+	0.985647i	$0.553995\pi$
$594$	−5.10149e16	−1.16139
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	4.21120e16	0.902606
$601$	−4.27830e16	−0.907872	−0.453936	−	0.891034i	$-0.649980\pi$
$601$	−4.27830e16	−0.907872	−0.453936	+	0.891034i	$0.649980\pi$
$602$	0	0
$603$	−1.72380e16	−0.358577
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−9.44157e16	−1.86906
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.82707e16	0.347734
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−1.04880e17	−1.95741
$615$	0	0
$616$	0	0
$617$	8.12519e15	0.147273	0.0736364	−	0.997285i	$-0.476540\pi$
$617$	8.12519e15	0.147273	0.0736364	+	0.997285i	$0.476540\pi$
$618$	0	0
$619$	1.06992e17	1.90198	0.950989	−	0.309225i	$-0.100070\pi$
$619$	1.06992e17	1.90198	0.950989	+	0.309225i	$0.100070\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	5.96046e16	1.00000
$626$	−1.15101e17	−1.91264
$627$	−1.11270e17	−1.83135
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	1.12541e17	1.74939
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.37398e16	1.35137	0.675687	−	0.737188i	$-0.263847\pi$
$641$	9.37398e16	1.35137	0.675687	+	0.737188i	$0.263847\pi$
$642$	−1.16935e17	−1.67007
$643$	−6.10304e16	−0.863536	−0.431768	−	0.901985i	$-0.642110\pi$
$643$	−6.10304e16	−0.863536	−0.431768	+	0.901985i	$0.642110\pi$
$644$	0	0
$645$	0	0
$646$	2.54909e17	3.50743
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−5.77758e16	−0.780361
$649$	1.66125e16	0.222314
$650$	0	0
$651$	0	0
$652$	9.05086e16	1.17816
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	1.44758e17	1.81644
$657$	−4.83911e15	−0.0601691
$658$	0	0
$659$	−6.58930e16	−0.804501	−0.402251	−	0.915530i	$-0.631772\pi$
$659$	−6.58930e16	−0.804501	−0.402251	+	0.915530i	$0.631772\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−1.66983e17	−1.98392
$663$	0	0
$664$	−5.05781e16	−0.590139
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.27565e17	−1.37290	−0.686451	−	0.727176i	$-0.740833\pi$
$673$	−1.27565e17	−1.37290	−0.686451	+	0.727176i	$0.740833\pi$
$674$	−3.33308e15	−0.0355538
$675$	−1.01193e17	−1.06986
$676$	9.54290e16	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	1.57856e17	1.62511
$679$	0	0
$680$	0	0
$681$	7.71429e16	0.773416
$682$	0	0
$683$	4.09526e16	0.403419	0.201710	−	0.979445i	$-0.435350\pi$
$683$	4.09526e16	0.403419	0.201710	+	0.979445i	$0.435350\pi$
$684$	3.54682e16	0.346340
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−1.17954e17	−1.11220
$689$	0	0
$690$	0	0
$691$	2.12572e17	1.95271	0.976354	−	0.216178i	$-0.0693592\pi$
$691$	2.12572e17	1.95271	0.976354	+	0.216178i	$0.0693592\pi$
$692$	0	0
$693$	0	0
$694$	−8.08538e16	−0.723675
$695$	0	0
$696$	0	0
$697$	−3.90826e17	−3.40869
$698$	0	0
$699$	−1.30046e17	−1.11490
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.32156e17	1.08555
$705$	0	0
$706$	2.45810e16	0.198505
$707$	0	0
$708$	2.32817e16	0.184848
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−2.27102e17	−1.74318
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−2.26765e17	−1.68305
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	3.53192e17	2.49337
$723$	2.04297e17	1.43032
$724$	0	0
$725$	0	0
$726$	2.35815e16	0.161046
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.66645e17	1.11027
$730$	0	0
$731$	3.18459e17	2.08713
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.36634e17	2.10065
$738$	−5.43798e16	−0.336589
$739$	−8.15585e16	−0.500729	−0.250364	−	0.968152i	$-0.580550\pi$
$739$	−8.15585e16	−0.500729	−0.250364	+	0.968152i	$0.580550\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.90002e16	0.109354
$748$	−3.56802e17	−2.03712
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−2.15825e17	−1.18395
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−3.31950e17	−1.75008
$759$	0	0
$760$	0	0
$761$	−3.06091e17	−1.57595	−0.787975	−	0.615707i	$-0.788871\pi$
$761$	−3.06091e17	−1.57595	−0.787975	+	0.615707i	$0.788871\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.85211e17	0.902606
$769$	7.49631e16	0.362485	0.181242	−	0.983438i	$-0.441988\pi$
$769$	7.49631e16	0.362485	0.181242	+	0.983438i	$0.441988\pi$
$770$	0	0
$771$	−1.42342e17	−0.677654
$772$	−1.29626e16	−0.0612335
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	4.43107e16	0.206093
$775$	0	0
$776$	4.34238e17	1.98865
$777$	0	0
$778$	0	0
$779$	−7.58695e17	−3.39502
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	2.32218e17	1.00000
$785$	0	0
$786$	−5.82816e16	−0.247170
$787$	3.83356e17	1.61345	0.806723	−	0.590930i	$-0.201239\pi$
$787$	3.83356e17	1.61345	0.806723	+	0.590930i	$0.201239\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−4.96457e16	−0.201155
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	2.62144e17	1.00000
$801$	8.53132e16	0.323014
$802$	−1.23273e17	−0.463257
$803$	9.45012e16	0.352488
$804$	4.71778e17	1.74663
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.67164e17	0.596283	0.298142	−	0.954522i	$-0.403633\pi$
$809$	1.67164e17	0.596283	0.298142	+	0.954522i	$0.403633\pi$
$810$	0	0
$811$	−4.49628e17	−1.58026	−0.790130	−	0.612939i	$-0.789987\pi$
$811$	−4.49628e17	−1.58026	−0.790130	+	0.612939i	$0.789987\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−5.00042e17	−1.69381
$817$	6.18212e17	2.07876
$818$	4.88360e17	1.63012
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−5.56606e17	−1.80433
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	3.08939e17	0.979826
$826$	0	0
$827$	−3.04079e17	−0.950502	−0.475251	−	0.879850i	$-0.657643\pi$
$827$	−3.04079e17	−0.950502	−0.475251	+	0.879850i	$0.657643\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.26956e17	−1.87658
$834$	−1.76630e17	−0.524891
$835$	0	0
$836$	−6.92645e17	−2.02896
$837$	0	0
$838$	4.72370e17	1.36401
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	3.53815e17	1.00000
$842$	0	0
$843$	5.62935e17	1.56853
$844$	7.00558e17	1.93816
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.34727e17	−1.16083
$850$	−7.07751e17	−1.87658
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−7.27914e17	−1.85028
$857$	3.89760e16	0.0983813	0.0491907	−	0.998789i	$-0.484336\pi$
$857$	3.89760e16	0.0983813	0.0491907	+	0.998789i	$0.484336\pi$
$858$	0	0
$859$	−1.37849e17	−0.343119	−0.171560	−	0.985174i	$-0.554881\pi$
$859$	−1.37849e17	−0.343119	−0.171560	+	0.985174i	$0.554881\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−4.45051e17	−1.06986
$865$	0	0
$866$	2.71625e17	0.643966
$867$	9.66675e17	2.27597
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.63126e17	−0.368500
$874$	0	0
$875$	0	0
$876$	1.32439e17	0.293084
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.94685e17	−1.69957	−0.849786	−	0.527128i	$-0.823269\pi$
$881$	−7.94685e17	−1.69957	−0.849786	+	0.527128i	$0.823269\pi$
$882$	−8.72351e16	−0.185302
$883$	8.71688e17	1.83906	0.919532	−	0.393015i	$-0.128568\pi$
$883$	8.71688e17	1.83906	0.919532	+	0.393015i	$0.128568\pi$
$884$	0	0
$885$	0	0
$886$	6.69228e17	1.38348
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.23851e17	−0.847123
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−3.75172e16	−0.0715438
$899$	0	0
$900$	−9.84770e16	−0.185302
$901$	0	0
$902$	1.06196e18	1.97184
$903$	0	0
$904$	9.82639e17	1.80046
$905$	0	0
$906$	0	0
$907$	−4.64915e17	−0.835084	−0.417542	−	0.908658i	$-0.637108\pi$
$907$	−4.64915e17	−0.835084	−0.417542	+	0.908658i	$0.637108\pi$
$908$	4.80209e17	0.856870
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−9.70711e17	−1.68702
$913$	−3.71048e17	−0.640627
$914$	1.14087e16	0.0195686
$915$	0	0
$916$	0	0
$917$	0	0
$918$	1.20157e18	2.00768
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1.07830e18	−1.76677
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.13239e18	1.76157	0.880786	−	0.473515i	$-0.157015\pi$
$929$	1.13239e18	1.76157	0.880786	+	0.473515i	$0.157015\pi$
$930$	0	0
$931$	−1.21708e18	−1.86906
$932$	−8.09528e17	−1.23520
$933$	0	0
$934$	−1.29618e18	−1.95246
$935$	0	0
$936$	0	0
$937$	9.59909e17	1.41838	0.709190	−	0.705018i	$-0.249061\pi$
$937$	9.59909e17	1.41838	0.709190	+	0.705018i	$0.249061\pi$
$938$	0	0
$939$	−1.18338e18	−1.72636
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1.44927e17	0.204794
$945$	0	0
$946$	−8.65327e17	−1.20735
$947$	1.42092e18	1.97001	0.985005	−	0.172528i	$-0.0551935\pi$
$947$	1.42092e18	1.97001	0.985005	+	0.172528i	$0.0551935\pi$
$948$	0	0
$949$	0	0
$950$	−1.37393e18	−1.86906
$951$	0	0
$952$	0	0
$953$	−1.35078e18	−1.80313	−0.901565	−	0.432645i	$-0.857581\pi$
$953$	−1.35078e18	−1.80313	−0.901565	+	0.432645i	$0.857581\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.87663e17	1.00000
$962$	0	0
$963$	2.73448e17	0.342860
$964$	1.27173e18	1.58465
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.46793e17	0.178424
$969$	2.62078e18	3.16583
$970$	0	0
$971$	1.67618e18	1.99989	0.999943	−	0.0107080i	$-0.00340853\pi$
$971$	1.67618e18	1.99989	0.999943	+	0.0107080i	$0.00340853\pi$
$972$	3.08239e17	0.365502
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.04930e17	0.810548	0.405274	−	0.914195i	$-0.367176\pi$
$977$	7.04930e17	0.810548	0.405274	+	0.914195i	$0.367176\pi$
$978$	9.30542e17	1.06341
$979$	−1.66605e18	−1.89231
$980$	0	0
$981$	0	0
$982$	−1.30501e18	−1.45527
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	1.48830e18	1.63953
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	−1.71679e18	−1.79070
$994$	0	0
$995$	0	0
$996$	−5.20006e17	−0.532663
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.53260e18	−1.55112
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.13.d.a.3.1	✓	1
3.2	odd	2		72.13.b.a.19.1		1
4.3	odd	2		32.13.d.a.15.1		1
8.3	odd	2	CM	8.13.d.a.3.1	✓	1
8.5	even	2		32.13.d.a.15.1		1
24.11	even	2		72.13.b.a.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.13.d.a.3.1	✓	1	1.1	even	1	trivial
8.13.d.a.3.1	✓	1	8.3	odd	2	CM
32.13.d.a.15.1		1	4.3	odd	2
32.13.d.a.15.1		1	8.5	even	2
72.13.b.a.19.1		1	3.2	odd	2
72.13.b.a.19.1		1	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	8.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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