LMFDB - Embedded newform 8.31.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,31,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, 31, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 31);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 31 $
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:	$45.6114095221$
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)

$f(q)$

$=$

$q-32768.0 q^{2} -2.65953e7 q^{3} +1.07374e9 q^{4} +8.71475e11 q^{6} -3.51844e13 q^{8} +5.01420e14 q^{9} +O(q^{10})$

$q-32768.0 q^{2} -2.65953e7 q^{3} +1.07374e9 q^{4} +8.71475e11 q^{6} -3.51844e13 q^{8} +5.01420e14 q^{9} +6.65619e15 q^{11} -2.85565e16 q^{12} +1.15292e18 q^{16} -4.42278e18 q^{17} -1.64305e19 q^{18} -2.39648e19 q^{19} -2.18110e20 q^{22} +9.35739e20 q^{24} +9.31323e20 q^{25} -7.85967e21 q^{27} -3.77789e22 q^{32} -1.77023e23 q^{33} +1.44926e23 q^{34} +5.38395e23 q^{36} +7.85278e23 q^{38} +1.46736e24 q^{41} -4.03884e24 q^{43} +7.14703e24 q^{44} -3.06623e25 q^{48} +2.25393e25 q^{49} -3.05176e25 q^{50} +1.17625e26 q^{51} +2.57546e26 q^{54} +6.37351e26 q^{57} -6.30942e26 q^{59} +1.23794e27 q^{64} +5.80070e27 q^{66} -3.95203e27 q^{67} -4.74893e27 q^{68} -1.76421e28 q^{72} +1.65626e28 q^{73} -2.47688e28 q^{75} -2.57320e28 q^{76} +1.05793e29 q^{81} -4.80825e28 q^{82} -4.47013e27 q^{83} +1.32345e29 q^{86} -2.34194e29 q^{88} -3.33711e29 q^{89} +1.00474e30 q^{96} +1.22180e30 q^{97} -7.38569e29 q^{98} +3.33754e30 q^{99} +O(q^{100})$

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 $

$2$	−32768.0	−1.00000
$2$	−32768.0	−1.00000
$3$	−2.65953e7	−1.85347	−0.926737	−	0.375712i	$-0.877398\pi$
$3$	−2.65953e7	−1.85347	−0.926737	+	0.375712i	$0.877398\pi$
$4$	1.07374e9	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	8.71475e11	1.85347
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−3.51844e13	−1.00000
$9$	5.01420e14	2.43536
$10$	0	0
$11$	6.65619e15	1.59344	0.796719	−	0.604350i	$-0.206567\pi$
$11$	6.65619e15	1.59344	0.796719	+	0.604350i	$0.206567\pi$
$12$	−2.85565e16	−1.85347
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.15292e18	1.00000
$17$	−4.42278e18	−1.54512	−0.772560	−	0.634942i	$-0.781024\pi$
$17$	−4.42278e18	−1.54512	−0.772560	+	0.634942i	$0.781024\pi$
$18$	−1.64305e19	−2.43536
$19$	−2.39648e19	−1.57859	−0.789295	−	0.614014i	$-0.789554\pi$
$19$	−2.39648e19	−1.57859	−0.789295	+	0.614014i	$0.789554\pi$
$20$	0	0
$21$	0	0
$22$	−2.18110e20	−1.59344
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	9.35739e20	1.85347
$25$	9.31323e20	1.00000
$26$	0	0
$27$	−7.85967e21	−2.66041
$28$	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			$\pi$
$32$	−3.77789e22	−1.00000
$33$	−1.77023e23	−2.95340
$34$	1.44926e23	1.54512
$35$	0	0
$36$	5.38395e23	2.43536
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	7.85278e23	1.57859
$39$	0	0
$40$	0	0
$41$	1.46736e24	0.943581	0.471790	−	0.881711i	$-0.343608\pi$
$41$	1.46736e24	0.943581	0.471790	+	0.881711i	$0.343608\pi$
$42$	0	0
$43$	−4.03884e24	−1.27125	−0.635624	−	0.771999i	$-0.719257\pi$
$43$	−4.03884e24	−1.27125	−0.635624	+	0.771999i	$0.719257\pi$
$44$	7.14703e24	1.59344
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−3.06623e25	−1.85347
$49$	2.25393e25	1.00000
$50$	−3.05176e25	−1.00000
$51$	1.17625e26	2.86384
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	2.57546e26	2.66041
$55$	0	0
$56$	0	0
$57$	6.37351e26	2.92588
$58$	0	0
$59$	−6.30942e26	−1.72667	−0.863334	−	0.504633i	$-0.831628\pi$
$59$	−6.30942e26	−1.72667	−0.863334	+	0.504633i	$0.831628\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.23794e27	1.00000
$65$	0	0
$66$	5.80070e27	2.95340
$67$	−3.95203e27	−1.60582	−0.802912	−	0.596098i	$-0.796717\pi$
$67$	−3.95203e27	−1.60582	−0.802912	+	0.596098i	$0.796717\pi$
$68$	−4.74893e27	−1.54512
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.76421e28	−2.43536
$73$	1.65626e28	1.85902	0.929512	−	0.368791i	$-0.120228\pi$
$73$	1.65626e28	1.85902	0.929512	+	0.368791i	$0.120228\pi$
$74$	0	0
$75$	−2.47688e28	−1.85347
$76$	−2.57320e28	−1.57859
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.05793e29	2.49563
$82$	−4.80825e28	−0.943581
$83$	−4.47013e27	−0.0731390	−0.0365695	−	0.999331i	$-0.511643\pi$
$83$	−4.47013e27	−0.0731390	−0.0365695	+	0.999331i	$0.511643\pi$
$84$	0	0
$85$	0	0
$86$	1.32345e29	1.27125
$87$	0	0
$88$	−2.34194e29	−1.59344
$89$	−3.33711e29	−1.91655	−0.958276	−	0.285846i	$-0.907725\pi$
$89$	−3.33711e29	−1.91655	−0.958276	+	0.285846i	$0.907725\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.00474e30	1.85347
$97$	1.22180e30	1.92940	0.964701	−	0.263347i	$-0.0848265\pi$
$97$	1.22180e30	1.92940	0.964701	+	0.263347i	$0.0848265\pi$
$98$	−7.38569e29	−1.00000
$99$	3.33754e30	3.88060
$100$	1.00000e30	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−3.85435e30	−2.86384
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.56260e30	1.65370	0.826848	−	0.562425i	$-0.190132\pi$
$107$	4.56260e30	1.65370	0.826848	+	0.562425i	$0.190132\pi$
$108$	−8.43926e30	−2.66041
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.37928e30	−0.860098	−0.430049	−	0.902806i	$-0.641504\pi$
$113$	−5.37928e30	−0.860098	−0.430049	+	0.902806i	$0.641504\pi$
$114$	−2.08847e31	−2.92588
$115$	0	0
$116$	0	0
$117$	0	0
$118$	2.06747e31	1.72667
$119$	0	0
$120$	0	0
$121$	2.68554e31	1.53905
$122$	0	0
$123$	−3.90249e31	−1.74890
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−4.05648e31	−1.00000
$129$	1.07414e32	2.35622
$130$	0	0
$131$	4.91252e31	0.855530	0.427765	−	0.903890i	$-0.359301\pi$
$131$	4.91252e31	0.855530	0.427765	+	0.903890i	$0.359301\pi$
$132$	−1.90077e32	−2.95340
$133$	0	0
$134$	1.29500e32	1.60582
$135$	0	0
$136$	1.55613e32	1.54512
$137$	−1.75146e31	−0.155809	−0.0779045	−	0.996961i	$-0.524823\pi$
$137$	−1.75146e31	−0.155809	−0.0779045	+	0.996961i	$0.524823\pi$
$138$	0	0
$139$	1.33608e31	0.0956335	0.0478167	−	0.998856i	$-0.484774\pi$
$139$	1.33608e31	0.0956335	0.0478167	+	0.998856i	$0.484774\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	5.78097e32	2.43536
$145$	0	0
$146$	−5.42723e32	−1.85902
$147$	−5.99441e32	−1.85347
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	8.11625e32	1.85347
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	8.43186e32	1.57859
$153$	−2.21767e33	−3.76293
$154$	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
$162$	−3.46661e33	−2.49563
$163$	2.14577e33	1.40854	0.704270	−	0.709932i	$-0.251274\pi$
$163$	2.14577e33	1.40854	0.704270	+	0.709932i	$0.251274\pi$
$164$	1.57557e33	0.943581
$165$	0	0
$166$	1.46477e32	0.0731390
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	2.62000e33	1.00000
$170$	0	0
$171$	−1.20164e34	−3.84444
$172$	−4.33667e33	−1.27125
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	7.67406e33	1.59344
$177$	1.67801e34	3.20033
$178$	1.09350e34	1.91655
$179$	1.22830e34	1.97928	0.989641	−	0.143567i	$-0.0458573\pi$
$179$	1.22830e34	1.97928	0.989641	+	0.143567i	$0.0458573\pi$
$180$	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
$187$	−2.94389e34	−2.46205
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−3.29234e34	−1.85347
$193$	2.50001e34	1.30191	0.650957	−	0.759115i	$-0.274368\pi$
$193$	2.50001e34	1.30191	0.650957	+	0.759115i	$0.274368\pi$
$194$	−4.00358e34	−1.92940
$195$	0	0
$196$	2.42014e34	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.09365e35	−3.88060
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−3.27680e34	−1.00000
$201$	1.05105e35	2.97635
$202$	0	0
$203$	0	0
$204$	1.26299e35	2.86384
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.59514e35	−2.51539
$210$	0	0
$211$	1.18794e35	1.62390	0.811949	−	0.583729i	$-0.198407\pi$
$211$	1.18794e35	1.62390	0.811949	+	0.583729i	$0.198407\pi$
$212$	0	0
$213$	0	0
$214$	−1.49507e35	−1.65370
$215$	0	0
$216$	2.76538e35	2.66041
$217$	0	0
$218$	0	0
$219$	−4.40487e35	−3.44565
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	4.66983e35	2.43536
$226$	1.76268e35	0.860098
$227$	−4.15504e35	−1.89752	−0.948762	−	0.315992i	$-0.897663\pi$
$227$	−4.15504e35	−1.89752	−0.948762	+	0.315992i	$0.897663\pi$
$228$	6.84351e35	2.92588
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.98713e35	1.53997	0.769984	−	0.638063i	$-0.220264\pi$
$233$	4.98713e35	1.53997	0.769984	+	0.638063i	$0.220264\pi$
$234$	0	0
$235$	0	0
$236$	−6.77468e35	−1.72667
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	7.48053e34	0.139212	0.0696060	−	0.997575i	$-0.477826\pi$
$241$	7.48053e34	0.139212	0.0696060	+	0.997575i	$0.477826\pi$
$242$	−8.79999e35	−1.53905
$243$	−1.19535e36	−1.96518
$244$	0	0
$245$	0	0
$246$	1.27877e36	1.74890
$247$	0	0
$248$	0	0
$249$	1.18885e35	0.135561
$250$	0	0
$251$	−1.66714e36	−1.68604	−0.843018	−	0.537886i	$-0.819223\pi$
$251$	−1.66714e36	−1.68604	−0.843018	+	0.537886i	$0.819223\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.32923e36	1.00000
$257$	−4.95283e35	−0.351444	−0.175722	−	0.984440i	$-0.556226\pi$
$257$	−4.95283e35	−0.351444	−0.175722	+	0.984440i	$0.556226\pi$
$258$	−3.51975e36	−2.35622
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−1.60974e36	−0.855530
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	6.22846e36	2.95340
$265$	0	0
$266$	0	0
$267$	8.87515e36	3.55228
$268$	−4.24346e36	−1.60582
$269$	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			$\pi$
$272$	−5.09912e36	−1.54512
$273$	0	0
$274$	5.73919e35	0.155809
$275$	6.19906e36	1.59344
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−4.37806e35	−0.0956335
$279$	0	0
$280$	0	0
$281$	2.93290e36	0.545384	0.272692	−	0.962101i	$-0.412086\pi$
$281$	2.93290e36	0.545384	0.272692	+	0.962101i	$0.412086\pi$
$282$	0	0
$283$	9.80498e36	1.63927	0.819635	−	0.572887i	$-0.194176\pi$
$283$	9.80498e36	1.63927	0.819635	+	0.572887i	$0.194176\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.89431e37	−2.43536
$289$	1.13676e37	1.38739
$290$	0	0
$291$	−3.24941e37	−3.57610
$292$	1.77839e37	1.85902
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	1.96425e37	1.85347
$295$	0	0
$296$	0	0
$297$	−5.23155e37	−4.23919
$298$	0	0
$299$	0	0
$300$	−2.65953e37	−1.85347
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−2.76295e37	−1.57859
$305$	0	0
$306$	7.26686e37	3.76293
$307$	2.00427e37	0.988281	0.494141	−	0.869382i	$-0.335483\pi$
$307$	2.00427e37	0.988281	0.494141	+	0.869382i	$0.335483\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	3.66288e37	1.35101	0.675503	−	0.737357i	$-0.263927\pi$
$313$	3.66288e37	1.35101	0.675503	+	0.737357i	$0.263927\pi$
$314$	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
$321$	−1.21344e38	−3.06508
$322$	0	0
$323$	1.05991e38	2.43911
$324$	1.13594e38	2.49563
$325$	0	0
$326$	−7.03126e37	−1.40854
$327$	0	0
$328$	−5.16282e37	−0.943581
$329$	0	0
$330$	0	0
$331$	−3.91322e37	−0.623899	−0.311949	−	0.950099i	$-0.600982\pi$
$331$	−3.91322e37	−0.623899	−0.311949	+	0.950099i	$0.600982\pi$
$332$	−4.79977e36	−0.0731390
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.10861e38	−1.34998	−0.674992	−	0.737825i	$-0.735853\pi$
$337$	−1.10861e38	−1.34998	−0.674992	+	0.737825i	$0.735853\pi$
$338$	−8.58520e37	−1.00000
$339$	1.43064e38	1.59417
$340$	0	0
$341$	0	0
$342$	3.93754e38	3.84444
$343$	0	0
$344$	1.42104e38	1.27125
$345$	0	0
$346$	0	0
$347$	−3.55867e37	−0.279476	−0.139738	−	0.990188i	$-0.544626\pi$
$347$	−3.55867e37	−0.279476	−0.139738	+	0.990188i	$0.544626\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−2.51464e38	−1.59344
$353$	2.83012e38	1.71864	0.859320	−	0.511439i	$-0.170887\pi$
$353$	2.83012e38	1.71864	0.859320	+	0.511439i	$0.170887\pi$
$354$	−5.49850e38	−3.20033
$355$	0	0
$356$	−3.58319e38	−1.91655
$357$	0	0
$358$	−4.02488e38	−1.97928
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	3.43845e38	1.49195
$362$	0	0
$363$	−7.14229e38	−2.85258
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	7.35764e38	2.29796
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	9.64653e38	2.46205
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	9.11465e38	1.90613	0.953063	−	0.302771i	$-0.0979117\pi$
$379$	9.11465e38	1.90613	0.953063	+	0.302771i	$0.0979117\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.07883e39	1.85347
$385$	0	0
$386$	−8.19204e38	−1.30191
$387$	−2.02516e39	−3.09595
$388$	1.31189e39	1.92940
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−7.93033e38	−1.00000
$393$	−1.30650e39	−1.58570
$394$	0	0
$395$	0	0
$396$	3.58366e39	3.88060
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.07374e39	1.00000
$401$	4.45148e38	0.399337	0.199668	−	0.979864i	$-0.436014\pi$
$401$	4.45148e38	0.399337	0.199668	+	0.979864i	$0.436014\pi$
$402$	−3.44409e39	−2.97635
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−4.13857e39	−2.86384
$409$	7.76251e37	0.0517789	0.0258894	−	0.999665i	$-0.491758\pi$
$409$	7.76251e37	0.0517789	0.0258894	+	0.999665i	$0.491758\pi$
$410$	0	0
$411$	4.65807e38	0.288788
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.55334e38	−0.177254
$418$	5.22696e39	2.51539
$419$	−1.98894e39	−0.923447	−0.461724	−	0.887024i	$-0.652769\pi$
$419$	−1.98894e39	−0.923447	−0.461724	+	0.887024i	$0.652769\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−3.89265e39	−1.62390
$423$	0	0
$424$	0	0
$425$	−4.11904e39	−1.54512
$426$	0	0
$427$	0	0
$428$	4.89905e39	1.65370
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−9.06159e39	−2.66041
$433$	7.04847e39	1.99883	0.999417	−	0.0341532i	$-0.0108734\pi$
$433$	7.04847e39	1.99883	0.999417	+	0.0341532i	$0.0108734\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.44339e40	3.44565
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.13017e40	2.43536
$442$	0	0
$443$	−4.28890e39	−0.863556	−0.431778	−	0.901980i	$-0.642114\pi$
$443$	−4.28890e39	−0.863556	−0.431778	+	0.901980i	$0.642114\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.79221e39	−1.28223	−0.641114	−	0.767445i	$-0.721528\pi$
$449$	−7.79221e39	−1.28223	−0.641114	+	0.767445i	$0.721528\pi$
$450$	−1.53021e40	−2.43536
$451$	9.76703e39	1.50354
$452$	−5.77596e39	−0.860098
$453$	0	0
$454$	1.36152e40	1.89752
$455$	0	0
$456$	−2.24248e40	−2.92588
$457$	−1.14716e40	−1.44838	−0.724190	−	0.689600i	$-0.757786\pi$
$457$	−1.14716e40	−1.44838	−0.724190	+	0.689600i	$0.757786\pi$
$458$	0	0
$459$	3.47616e40	4.11065
$460$	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
$466$	−1.63418e40	−1.53997
$467$	1.13839e40	1.03882	0.519409	−	0.854526i	$-0.326152\pi$
$467$	1.13839e40	1.03882	0.519409	+	0.854526i	$0.326152\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	2.21993e40	1.72667
$473$	−2.68833e40	−2.02565
$474$	0	0
$475$	−2.23189e40	−1.57859
$476$	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
$482$	−2.45122e39	−0.139212
$483$	0	0
$484$	2.88358e40	1.53905
$485$	0	0
$486$	3.91693e40	1.96518
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−5.70674e40	−2.61069
$490$	0	0
$491$	−4.41334e40	−1.89909	−0.949544	−	0.313635i	$-0.898453\pi$
$491$	−4.41334e40	−1.89909	−0.949544	+	0.313635i	$0.898453\pi$
$492$	−4.19027e40	−1.74890
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−3.89561e39	−0.135561
$499$	5.86730e40	1.98121	0.990604	−	0.136763i	$-0.0436700\pi$
$499$	5.86730e40	1.98121	0.990604	+	0.136763i	$0.0436700\pi$
$500$	0	0
$501$	0	0
$502$	5.46289e40	1.68604
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.96796e40	−1.85347
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−4.35561e40	−1.00000
$513$	1.88355e41	4.19969
$514$	1.62294e40	0.351444
$515$	0	0
$516$	1.15335e41	2.35622
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.77748e40	1.37491	0.687453	−	0.726229i	$-0.258729\pi$
$521$	7.77748e40	1.37491	0.687453	+	0.726229i	$0.258729\pi$
$522$	0	0
$523$	4.97539e40	0.830428	0.415214	−	0.909724i	$-0.363707\pi$
$523$	4.97539e40	0.830428	0.415214	+	0.909724i	$0.363707\pi$
$524$	5.27478e40	0.855530
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−2.04094e41	−2.95340
$529$	7.10943e40	1.00000
$530$	0	0
$531$	−3.16366e41	−4.20506
$532$	0	0
$533$	0	0
$534$	−2.90821e41	−3.55228
$535$	0	0
$536$	1.39050e41	1.60582
$537$	−3.26669e41	−3.66854
$538$	0	0
$539$	1.50026e41	1.59344
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.67088e41	1.54512
$545$	0	0
$546$	0	0
$547$	1.61807e41	1.37781	0.688903	−	0.724853i	$-0.258092\pi$
$547$	1.61807e41	1.37781	0.688903	+	0.724853i	$0.258092\pi$
$548$	−1.88062e40	−0.155809
$549$	0	0
$550$	−2.03131e41	−1.59344
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	1.43460e40	0.0956335
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	7.82936e41	4.56335
$562$	−9.61052e40	−0.545384
$563$	3.98594e40	0.220245	0.110122	−	0.993918i	$-0.464876\pi$
$563$	3.98594e40	0.220245	0.110122	+	0.993918i	$0.464876\pi$
$564$	0	0
$565$	0	0
$566$	−3.21290e41	−1.63927
$567$	0	0
$568$	0	0
$569$	−2.69367e41	−1.26958	−0.634790	−	0.772685i	$-0.718913\pi$
$569$	−2.69367e41	−1.26958	−0.634790	+	0.772685i	$0.718913\pi$
$570$	0	0
$571$	9.50010e40	0.424802	0.212401	−	0.977183i	$-0.431872\pi$
$571$	9.50010e40	0.424802	0.212401	+	0.977183i	$0.431872\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	6.20727e41	2.43536
$577$	−1.35998e40	−0.0519872	−0.0259936	−	0.999662i	$-0.508275\pi$
$577$	−1.35998e40	−0.0519872	−0.0259936	+	0.999662i	$0.508275\pi$
$578$	−3.72492e41	−1.38739
$579$	−6.64886e41	−2.41306
$580$	0	0
$581$	0	0
$582$	1.06477e42	3.57610
$583$	0	0
$584$	−5.82744e41	−1.85902
$585$	0	0
$586$	0	0
$587$	5.65312e41	1.67000	0.835000	−	0.550250i	$-0.185468\pi$
$587$	5.65312e41	1.67000	0.835000	+	0.550250i	$0.185468\pi$
$588$	−6.43645e41	−1.85347
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.78773e41	−0.707017	−0.353509	−	0.935431i	$-0.615011\pi$
$593$	−2.78773e41	−0.707017	−0.353509	+	0.935431i	$0.615011\pi$
$594$	1.71427e42	4.23919
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	8.71475e41	1.85347
$601$	−3.68830e41	−0.765084	−0.382542	−	0.923938i	$-0.624951\pi$
$601$	−3.68830e41	−0.765084	−0.382542	+	0.923938i	$0.624951\pi$
$602$	0	0
$603$	−1.98162e42	−3.91076
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	9.05364e41	1.57859
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−2.38121e42	−3.76293
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−6.56759e41	−0.988281
$615$	0	0
$616$	0	0
$617$	−1.17923e42	−1.64938	−0.824691	−	0.565583i	$-0.808651\pi$
$617$	−1.17923e42	−1.64938	−0.824691	+	0.565583i	$0.808651\pi$
$618$	0	0
$619$	1.06081e42	1.41344	0.706720	−	0.707494i	$-0.250174\pi$
$619$	1.06081e42	1.41344	0.706720	+	0.707494i	$0.250174\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.67362e41	1.00000
$626$	−1.20025e42	−1.35101
$627$	4.24233e42	4.66220
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−3.15937e42	−3.00985
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.21836e42	0.961401	0.480701	−	0.876885i	$-0.340382\pi$
$641$	1.21836e42	0.961401	0.480701	+	0.876885i	$0.340382\pi$
$642$	3.97619e42	3.06508
$643$	8.62431e41	0.649471	0.324736	−	0.945805i	$-0.394725\pi$
$643$	8.62431e41	0.649471	0.324736	+	0.945805i	$0.394725\pi$
$644$	0	0
$645$	0	0
$646$	−3.47312e42	−2.43911
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−3.72225e42	−2.49563
$649$	−4.19967e42	−2.75134
$650$	0	0
$651$	0	0
$652$	2.30400e42	1.40854
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	1.69175e42	0.943581
$657$	8.30480e42	4.52740
$658$	0	0
$659$	9.48263e41	0.493910	0.246955	−	0.969027i	$-0.420570\pi$
$659$	9.48263e41	0.493910	0.246955	+	0.969027i	$0.420570\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	1.28228e42	0.623899
$663$	0	0
$664$	1.57279e41	0.0731390
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.70513e42	−1.78791	−0.893956	−	0.448155i	$-0.852081\pi$
$673$	−4.70513e42	−1.78791	−0.893956	+	0.448155i	$0.852081\pi$
$674$	3.63268e42	1.34998
$675$	−7.31989e42	−2.66041
$676$	2.81320e42	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−4.68791e42	−1.59417
$679$	0	0
$680$	0	0
$681$	1.10505e43	3.51701
$682$	0	0
$683$	6.31513e42	1.92341	0.961705	−	0.274087i	$-0.0883756\pi$
$683$	6.31513e42	1.92341	0.961705	+	0.274087i	$0.0883756\pi$
$684$	−1.29025e43	−3.84444
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−4.65647e42	−1.27125
$689$	0	0
$690$	0	0
$691$	−6.68803e42	−1.71052	−0.855259	−	0.518201i	$-0.826602\pi$
$691$	−6.68803e42	−1.71052	−0.855259	+	0.518201i	$0.826602\pi$
$692$	0	0
$693$	0	0
$694$	1.16610e42	0.279476
$695$	0	0
$696$	0	0
$697$	−6.48982e42	−1.45795
$698$	0	0
$699$	−1.32634e43	−2.85429
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	8.23996e42	1.59344
$705$	0	0
$706$	−9.27375e42	−1.71864
$707$	0	0
$708$	1.80175e43	3.20033
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	1.17414e43	1.91655
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.31887e43	1.97928
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−1.12671e43	−1.49195
$723$	−1.98947e42	−0.258026
$724$	0	0
$725$	0	0
$726$	2.34038e43	2.85258
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.00090e43	1.14677
$730$	0	0
$731$	1.78629e43	1.96423
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.63054e43	−2.55878
$738$	−2.41095e43	−2.29796
$739$	−3.25842e42	−0.304327	−0.152163	−	0.988355i	$-0.548624\pi$
$739$	−3.25842e42	−0.304327	−0.152163	+	0.988355i	$0.548624\pi$
$740$	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
$747$	−2.24141e42	−0.178120
$748$	−3.16098e43	−2.46205
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	4.43382e43	3.12502
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−2.98669e43	−1.90613
$759$	0	0
$760$	0	0
$761$	3.31237e43	1.99236	0.996182	−	0.0872990i	$-0.0278236\pi$
$761$	3.31237e43	1.99236	0.996182	+	0.0872990i	$0.0278236\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−3.53512e43	−1.85347
$769$	−3.68018e43	−1.89223	−0.946115	−	0.323829i	$-0.895030\pi$
$769$	−3.68018e43	−1.89223	−0.946115	+	0.323829i	$0.895030\pi$
$770$	0	0
$771$	1.31722e43	0.651392
$772$	2.68437e43	1.30191
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	6.63603e43	3.09595
$775$	0	0
$776$	−4.29881e43	−1.92940
$777$	0	0
$778$	0	0
$779$	−3.51650e43	−1.48953
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	2.59861e43	1.00000
$785$	0	0
$786$	4.28114e43	1.58570
$787$	5.35627e41	0.0194644	0.00973221	−	0.999953i	$-0.496902\pi$
$787$	5.35627e41	0.0194644	0.00973221	+	0.999953i	$0.496902\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.17429e44	−3.88060
$793$	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
$800$	−3.51844e43	−1.00000
$801$	−1.67329e44	−4.66750
$802$	−1.45866e43	−0.399337
$803$	1.10244e44	2.96224
$804$	1.12856e44	2.97635
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.31919e43	1.99919	0.999593	−	0.0285315i	$-0.00908308\pi$
$809$	8.31919e43	1.99919	0.999593	+	0.0285315i	$0.00908308\pi$
$810$	0	0
$811$	8.60991e43	1.99382	0.996909	−	0.0785606i	$-0.0250324\pi$
$811$	8.60991e43	1.99382	0.996909	+	0.0785606i	$0.0250324\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	1.35613e44	2.86384
$817$	9.67900e43	2.00678
$818$	−2.54362e42	−0.0517789
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−1.52636e43	−0.288788
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−1.64866e44	−2.95340
$826$	0	0
$827$	−5.06418e43	−0.874836	−0.437418	−	0.899258i	$-0.644107\pi$
$827$	−5.06418e43	−0.874836	−0.437418	+	0.899258i	$0.644107\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.96867e43	−1.54512
$834$	1.16436e43	0.177254
$835$	0	0
$836$	−1.71277e44	−2.51539
$837$	0	0
$838$	6.51737e43	0.923447
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	7.44629e43	1.00000
$842$	0	0
$843$	−7.80013e43	−1.01085
$844$	1.27554e44	1.62390
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.60767e44	−3.03834
$850$	1.34973e44	1.54512
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−1.60532e44	−1.65370
$857$	1.55799e44	1.57707	0.788536	−	0.614989i	$-0.210840\pi$
$857$	1.55799e44	1.57707	0.788536	+	0.614989i	$0.210840\pi$
$858$	0	0
$859$	−7.09990e43	−0.693993	−0.346996	−	0.937866i	$-0.612798\pi$
$859$	−7.09990e43	−0.693993	−0.346996	+	0.937866i	$0.612798\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	2.96930e44	2.66041
$865$	0	0
$866$	−2.30964e44	−1.99883
$867$	−3.02324e44	−2.57150
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.12633e44	4.69879
$874$	0	0
$875$	0	0
$876$	−4.72969e44	−3.44565
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.94019e44	1.96669	0.983347	−	0.181737i	$-0.0581719\pi$
$881$	2.94019e44	1.96669	0.983347	+	0.181737i	$0.0581719\pi$
$882$	−3.70333e44	−2.43536
$883$	−1.64587e44	−1.06411	−0.532055	−	0.846710i	$-0.678580\pi$
$883$	−1.64587e44	−1.06411	−0.532055	+	0.846710i	$0.678580\pi$
$884$	0	0
$885$	0	0
$886$	1.40539e44	0.863556
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.04175e44	3.97663
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	2.55335e44	1.28223
$899$	0	0
$900$	5.01420e44	2.43536
$901$	0	0
$902$	−3.20046e44	−1.50354
$903$	0	0
$904$	1.89267e44	0.860098
$905$	0	0
$906$	0	0
$907$	−1.32903e44	−0.574680	−0.287340	−	0.957829i	$-0.592771\pi$
$907$	−1.32903e44	−0.574680	−0.287340	+	0.957829i	$0.592771\pi$
$908$	−4.46144e44	−1.89752
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	7.34816e44	2.92588
$913$	−2.97540e43	−0.116542
$914$	3.75903e44	1.44838
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.13907e45	−4.11065
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−5.33042e44	−1.83175
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.19478e44	−0.662452	−0.331226	−	0.943551i	$-0.607462\pi$
$929$	−2.19478e44	−0.662452	−0.331226	+	0.943551i	$0.607462\pi$
$930$	0	0
$931$	−5.40151e44	−1.57859
$932$	5.35489e44	1.53997
$933$	0	0
$934$	−3.73029e44	−1.03882
$935$	0	0
$936$	0	0
$937$	−2.83236e44	−0.751718	−0.375859	−	0.926677i	$-0.622652\pi$
$937$	−2.83236e44	−0.751718	−0.375859	+	0.926677i	$0.622652\pi$
$938$	0	0
$939$	−9.74153e44	−2.50405
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−7.27426e44	−1.72667
$945$	0	0
$946$	8.80912e44	2.02565
$947$	8.01917e44	1.81501	0.907506	−	0.420039i	$-0.137984\pi$
$947$	8.01917e44	1.81501	0.907506	+	0.420039i	$0.137984\pi$
$948$	0	0
$949$	0	0
$950$	7.31347e44	1.57859
$951$	0	0
$952$	0	0
$953$	8.73798e44	1.79894	0.899472	−	0.436978i	$-0.143951\pi$
$953$	8.73798e44	1.79894	0.899472	+	0.436978i	$0.143951\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.50619e44	1.00000
$962$	0	0
$963$	2.28778e45	4.02735
$964$	8.03215e43	0.139212
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−9.44892e44	−1.53905
$969$	−2.81887e45	−4.52083
$970$	0	0
$971$	−1.28577e45	−1.99928	−0.999642	−	0.0267673i	$-0.991479\pi$
$971$	−1.28577e45	−1.99928	−0.999642	+	0.0267673i	$0.991479\pi$
$972$	−1.28350e45	−1.96518
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.36412e45	1.93391	0.966954	−	0.254952i	$-0.0820598\pi$
$977$	1.36412e45	1.93391	0.966954	+	0.254952i	$0.0820598\pi$
$978$	1.86999e45	2.61069
$979$	−2.22124e45	−3.05391
$980$	0	0
$981$	0	0
$982$	1.44616e45	1.89909
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	1.37307e45	1.74890
$985$	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
$993$	1.04073e45	1.15638
$994$	0	0
$995$	0	0
$996$	1.27651e44	0.135561
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.92260e45	−1.98121
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.31.d.a.3.1	✓	1
4.3	odd	2		32.31.d.a.15.1		1
8.3	odd	2	CM	8.31.d.a.3.1	✓	1
8.5	even	2		32.31.d.a.15.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.31.d.a.3.1	✓	1	1.1	even	1	trivial
8.31.d.a.3.1	✓	1	8.3	odd	2	CM
32.31.d.a.15.1		1	4.3	odd	2
32.31.d.a.15.1		1	8.5	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:	\( k \)	\(=\)	\( 31 \)
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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