LMFDB - Embedded newform 8.45.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,45,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, 45, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 45);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 45 $
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:	$98.0957879834$
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)

$f(q)$

$=$

$q+4.19430e6 q^{2} -2.31380e10 q^{3} +1.75922e13 q^{4} -9.70479e16 q^{6} +7.37870e19 q^{8} -4.49402e20 q^{9} +O(q^{10})$

$q+4.19430e6 q^{2} -2.31380e10 q^{3} +1.75922e13 q^{4} -9.70479e16 q^{6} +7.37870e19 q^{8} -4.49402e20 q^{9} +1.40229e23 q^{11} -4.07049e23 q^{12} +3.09485e26 q^{16} -6.40003e26 q^{17} -1.88493e27 q^{18} -1.96678e28 q^{19} +5.88165e29 q^{22} -1.70729e30 q^{24} +5.68434e30 q^{25} +3.31839e31 q^{27} +1.29807e33 q^{32} -3.24463e33 q^{33} -2.68437e33 q^{34} -7.90597e33 q^{36} -8.24927e34 q^{38} -5.20766e35 q^{41} +1.55084e36 q^{43} +2.46694e36 q^{44} -7.16087e36 q^{48} +1.52867e37 q^{49} +2.38419e37 q^{50} +1.48084e37 q^{51} +1.39184e38 q^{54} +4.55074e38 q^{57} +6.68046e38 q^{59} +5.44452e39 q^{64} -1.36090e40 q^{66} +1.19749e40 q^{67} -1.12591e40 q^{68} -3.31600e40 q^{72} +8.54899e40 q^{73} -1.31524e41 q^{75} -3.45999e41 q^{76} -3.25253e41 q^{81} -2.18425e42 q^{82} +2.78421e42 q^{83} +6.50468e42 q^{86} +1.03471e43 q^{88} +1.03786e43 q^{89} -3.00349e43 q^{96} -1.35442e43 q^{97} +6.41171e43 q^{98} -6.30194e43 q^{99} +O(q^{100})$

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 $

$2$	4.19430e6	1.00000
$2$	4.19430e6	1.00000
$3$	−2.31380e10	−0.737325	−0.368662	−	0.929563i	$-0.620184\pi$
$3$	−2.31380e10	−0.737325	−0.368662	+	0.929563i	$0.620184\pi$
$4$	1.75922e13	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−9.70479e16	−0.737325
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	7.37870e19	1.00000
$9$	−4.49402e20	−0.456352
$10$	0	0
$11$	1.40229e23	1.72266	0.861331	−	0.508045i	$-0.169632\pi$
$11$	1.40229e23	1.72266	0.861331	+	0.508045i	$0.169632\pi$
$12$	−4.07049e23	−0.737325
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	3.09485e26	1.00000
$17$	−6.40003e26	−0.544886	−0.272443	−	0.962172i	$-0.587832\pi$
$17$	−6.40003e26	−0.544886	−0.272443	+	0.962172i	$0.587832\pi$
$18$	−1.88493e27	−0.456352
$19$	−1.96678e28	−1.44936	−0.724680	−	0.689085i	$-0.758013\pi$
$19$	−1.96678e28	−1.44936	−0.724680	+	0.689085i	$0.758013\pi$
$20$	0	0
$21$	0	0
$22$	5.88165e29	1.72266
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.70729e30	−0.737325
$25$	5.68434e30	1.00000
$26$	0	0
$27$	3.31839e31	1.07380
$28$	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			$\pi$
$32$	1.29807e33	1.00000
$33$	−3.24463e33	−1.27016
$34$	−2.68437e33	−0.544886
$35$	0	0
$36$	−7.90597e33	−0.456352
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−8.24927e34	−1.44936
$39$	0	0
$40$	0	0
$41$	−5.20766e35	−1.71948	−0.859741	−	0.510730i	$-0.829375\pi$
$41$	−5.20766e35	−1.71948	−0.859741	+	0.510730i	$0.829375\pi$
$42$	0	0
$43$	1.55084e36	1.79581	0.897904	−	0.440192i	$-0.145090\pi$
$43$	1.55084e36	1.79581	0.897904	+	0.440192i	$0.145090\pi$
$44$	2.46694e36	1.72266
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−7.16087e36	−0.737325
$49$	1.52867e37	1.00000
$50$	2.38419e37	1.00000
$51$	1.48084e37	0.401758
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	1.39184e38	1.07380
$55$	0	0
$56$	0	0
$57$	4.55074e38	1.06865
$58$	0	0
$59$	6.68046e38	0.734619	0.367309	−	0.930099i	$-0.380279\pi$
$59$	6.68046e38	0.734619	0.367309	+	0.930099i	$0.380279\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	5.44452e39	1.00000
$65$	0	0
$66$	−1.36090e40	−1.27016
$67$	1.19749e40	0.802832	0.401416	−	0.915896i	$-0.368518\pi$
$67$	1.19749e40	0.802832	0.401416	+	0.915896i	$0.368518\pi$
$68$	−1.12591e40	−0.544886
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−3.31600e40	−0.456352
$73$	8.54899e40	0.868584	0.434292	−	0.900772i	$-0.356999\pi$
$73$	8.54899e40	0.868584	0.434292	+	0.900772i	$0.356999\pi$
$74$	0	0
$75$	−1.31524e41	−0.737325
$76$	−3.45999e41	−1.44936
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−3.25253e41	−0.335390
$82$	−2.18425e42	−1.71948
$83$	2.78421e42	1.67874	0.839371	−	0.543559i	$-0.182924\pi$
$83$	2.78421e42	1.67874	0.839371	+	0.543559i	$0.182924\pi$
$84$	0	0
$85$	0	0
$86$	6.50468e42	1.79581
$87$	0	0
$88$	1.03471e43	1.72266
$89$	1.03786e43	1.34759	0.673793	−	0.738920i	$-0.264664\pi$
$89$	1.03786e43	1.34759	0.673793	+	0.738920i	$0.264664\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−3.00349e43	−0.737325
$97$	−1.35442e43	−0.264714	−0.132357	−	0.991202i	$-0.542254\pi$
$97$	−1.35442e43	−0.264714	−0.132357	+	0.991202i	$0.542254\pi$
$98$	6.41171e43	1.00000
$99$	−6.30194e43	−0.786140
$100$	1.00000e44	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	6.21110e43	0.401758
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.22683e44	1.85690	0.928452	−	0.371453i	$-0.121140\pi$
$107$	8.22683e44	1.85690	0.928452	+	0.371453i	$0.121140\pi$
$108$	5.83778e44	1.07380
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.65925e45	1.80731	0.903657	−	0.428256i	$-0.140872\pi$
$113$	2.65925e45	1.80731	0.903657	+	0.428256i	$0.140872\pi$
$114$	1.90872e45	1.06865
$115$	0	0
$116$	0	0
$117$	0	0
$118$	2.80199e45	0.734619
$119$	0	0
$120$	0	0
$121$	1.30379e46	1.96756
$122$	0	0
$123$	1.20495e46	1.26782
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	2.28360e46	1.00000
$129$	−3.58833e46	−1.32409
$130$	0	0
$131$	−3.93238e46	−1.03440	−0.517198	−	0.855866i	$-0.673025\pi$
$131$	−3.93238e46	−1.03440	−0.517198	+	0.855866i	$0.673025\pi$
$132$	−5.70802e46	−1.27016
$133$	0	0
$134$	5.02262e46	0.802832
$135$	0	0
$136$	−4.72239e46	−0.544886
$137$	8.10313e46	0.795793	0.397897	−	0.917430i	$-0.369740\pi$
$137$	8.10313e46	0.795793	0.397897	+	0.917430i	$0.369740\pi$
$138$	0	0
$139$	−2.47292e47	−1.76557	−0.882786	−	0.469776i	$-0.844335\pi$
$139$	−2.47292e47	−1.76557	−0.882786	+	0.469776i	$0.844335\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−1.39083e47	−0.456352
$145$	0	0
$146$	3.58571e47	0.868584
$147$	−3.53704e47	−0.737325
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−5.51654e47	−0.737325
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−1.45123e48	−1.44936
$153$	2.87619e47	0.248660
$154$	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
$162$	−1.36421e48	−0.335390
$163$	−8.87532e48	−1.90571	−0.952855	−	0.303427i	$-0.901869\pi$
$163$	−8.87532e48	−1.90571	−0.952855	+	0.303427i	$0.901869\pi$
$164$	−9.16141e48	−1.71948
$165$	0	0
$166$	1.16778e49	1.67874
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.03159e49	1.00000
$170$	0	0
$171$	8.83875e48	0.661419
$172$	2.72826e49	1.79581
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	4.33989e49	1.72266
$177$	−1.54573e49	−0.541653
$178$	4.35308e49	1.34759
$179$	3.64223e49	0.996784	0.498392	−	0.866952i	$-0.333924\pi$
$179$	3.64223e49	0.996784	0.498392	+	0.866952i	$0.333924\pi$
$180$	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
$187$	−8.97472e49	−0.938654
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.25975e50	−0.737325
$193$	−3.08208e50	−1.60910	−0.804551	−	0.593884i	$-0.797594\pi$
$193$	−3.08208e50	−1.60910	−0.804551	+	0.593884i	$0.797594\pi$
$194$	−5.68087e49	−0.264714
$195$	0	0
$196$	2.68926e50	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−2.64323e50	−0.786140
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	4.19430e50	1.00000
$201$	−2.77075e50	−0.591948
$202$	0	0
$203$	0	0
$204$	2.60512e50	0.401758
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.75800e51	−2.49676
$210$	0	0
$211$	−2.70017e51	−1.98234	−0.991168	−	0.132613i	$-0.957663\pi$
$211$	−2.70017e51	−1.98234	−0.991168	+	0.132613i	$0.957663\pi$
$212$	0	0
$213$	0	0
$214$	3.45058e51	1.85690
$215$	0	0
$216$	2.44854e51	1.07380
$217$	0	0
$218$	0	0
$219$	−1.97807e51	−0.640428
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−2.55456e51	−0.456352
$226$	1.11537e52	1.80731
$227$	−7.41150e51	−1.08978	−0.544888	−	0.838509i	$-0.683428\pi$
$227$	−7.41150e51	−1.08978	−0.544888	+	0.838509i	$0.683428\pi$
$228$	8.00575e51	1.06865
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.19937e51	−0.679126	−0.339563	−	0.940583i	$-0.610279\pi$
$233$	−8.19937e51	−0.679126	−0.339563	+	0.940583i	$0.610279\pi$
$234$	0	0
$235$	0	0
$236$	1.17524e52	0.734619
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	4.83025e52	1.90368	0.951841	−	0.306592i	$-0.0991888\pi$
$241$	4.83025e52	1.90368	0.951841	+	0.306592i	$0.0991888\pi$
$242$	5.46848e52	1.96756
$243$	−2.51529e52	−0.826513
$244$	0	0
$245$	0	0
$246$	5.05393e52	1.26782
$247$	0	0
$248$	0	0
$249$	−6.44211e52	−1.23778
$250$	0	0
$251$	−6.25765e52	−1.00830	−0.504150	−	0.863616i	$-0.668194\pi$
$251$	−6.25765e52	−1.00830	−0.504150	+	0.863616i	$0.668194\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	9.57810e52	1.00000
$257$	−2.06058e53	−1.97452	−0.987258	−	0.159129i	$-0.949131\pi$
$257$	−2.06058e53	−1.97452	−0.987258	+	0.159129i	$0.949131\pi$
$258$	−1.50505e53	−1.32409
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−1.64936e53	−1.03440
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−2.39412e53	−1.27016
$265$	0	0
$266$	0	0
$267$	−2.40139e53	−0.993609
$268$	2.10664e53	0.802832
$269$	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			$\pi$
$272$	−1.98071e53	−0.544886
$273$	0	0
$274$	3.39870e53	0.795793
$275$	7.97112e53	1.72266
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.03722e54	−1.76557
$279$	0	0
$280$	0	0
$281$	1.45950e54	1.96185	0.980924	−	0.194393i	$-0.0622738\pi$
$281$	1.45950e54	1.96185	0.980924	+	0.194393i	$0.0622738\pi$
$282$	0	0
$283$	1.73631e54	1.99676	0.998379	−	0.0569121i	$-0.0181255\pi$
$283$	1.73631e54	1.99676	0.998379	+	0.0569121i	$0.0181255\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−5.83358e53	−0.456352
$289$	−9.69994e53	−0.703099
$290$	0	0
$291$	3.13387e53	0.195180
$292$	1.50395e54	0.868584
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.48354e54	−0.737325
$295$	0	0
$296$	0	0
$297$	4.65336e54	1.84980
$298$	0	0
$299$	0	0
$300$	−2.31380e54	−0.737325
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−6.08689e54	−1.44936
$305$	0	0
$306$	1.20636e54	0.248660
$307$	−2.42172e54	−0.464601	−0.232300	−	0.972644i	$-0.574625\pi$
$307$	−2.42172e54	−0.464601	−0.232300	+	0.972644i	$0.574625\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.59459e55	1.99836	0.999178	−	0.0405395i	$-0.0129076\pi$
$313$	1.59459e55	1.99836	0.999178	+	0.0405395i	$0.0129076\pi$
$314$	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
$321$	−1.90353e55	−1.36914
$322$	0	0
$323$	1.25874e55	0.789736
$324$	−5.72191e54	−0.335390
$325$	0	0
$326$	−3.72258e55	−1.90571
$327$	0	0
$328$	−3.84257e55	−1.71948
$329$	0	0
$330$	0	0
$331$	−4.88015e55	−1.78738	−0.893691	−	0.448683i	$-0.851893\pi$
$331$	−4.88015e55	−1.78738	−0.893691	+	0.448683i	$0.851893\pi$
$332$	4.89803e55	1.67874
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5.28118e54	−0.130279	−0.0651393	−	0.997876i	$-0.520749\pi$
$337$	−5.28118e54	−0.130279	−0.0651393	+	0.997876i	$0.520749\pi$
$338$	4.32681e55	1.00000
$339$	−6.15299e55	−1.33258
$340$	0	0
$341$	0	0
$342$	3.70724e55	0.661419
$343$	0	0
$344$	1.14431e56	1.79581
$345$	0	0
$346$	0	0
$347$	1.03663e56	1.34392	0.671962	−	0.740585i	$-0.265452\pi$
$347$	1.03663e56	1.34392	0.671962	+	0.740585i	$0.265452\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	1.82028e56	1.72266
$353$	−2.22104e56	−1.97475	−0.987374	−	0.158403i	$-0.949365\pi$
$353$	−2.22104e56	−1.97475	−0.987374	+	0.158403i	$0.949365\pi$
$354$	−6.48325e55	−0.541653
$355$	0	0
$356$	1.82581e56	1.34759
$357$	0	0
$358$	1.52766e56	0.996784
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	2.02678e56	1.10065
$362$	0	0
$363$	−3.01671e56	−1.45073
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	2.34033e56	0.784689
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−3.76427e56	−0.938654
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−1.04299e57	−1.94186	−0.970931	−	0.239358i	$-0.923063\pi$
$379$	−1.04299e57	−1.94186	−0.970931	+	0.239358i	$0.923063\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−5.28379e56	−0.737325
$385$	0	0
$386$	−1.29272e57	−1.60910
$387$	−6.96949e56	−0.819521
$388$	−2.38273e56	−0.264714
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.12796e57	1.00000
$393$	9.09876e56	0.762686
$394$	0	0
$395$	0	0
$396$	−1.10865e57	−0.786140
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.75922e57	1.00000
$401$	3.51232e57	1.88981	0.944906	−	0.327343i	$-0.106153\pi$
$401$	3.51232e57	1.88981	0.944906	+	0.327343i	$0.106153\pi$
$402$	−1.16214e57	−0.591948
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	1.09267e57	0.401758
$409$	5.65623e57	1.97067	0.985336	−	0.170625i	$-0.0545787\pi$
$409$	5.65623e57	1.97067	0.985336	+	0.170625i	$0.0545787\pi$
$410$	0	0
$411$	−1.87490e57	−0.586758
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.72186e57	1.30180
$418$	−1.15679e58	−2.49676
$419$	3.71084e57	0.759912	0.379956	−	0.925005i	$-0.375939\pi$
$419$	3.71084e57	0.759912	0.379956	+	0.925005i	$0.375939\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−1.13253e58	−1.98234
$423$	0	0
$424$	0	0
$425$	−3.63800e57	−0.544886
$426$	0	0
$427$	0	0
$428$	1.44728e58	1.85690
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.02699e58	1.07380
$433$	1.87732e58	1.86554	0.932769	−	0.360476i	$-0.117386\pi$
$433$	1.87732e58	1.86554	0.932769	+	0.360476i	$0.117386\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−8.29662e57	−0.640428
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−6.86988e57	−0.456352
$442$	0	0
$443$	−3.27042e58	−1.96662	−0.983312	−	0.181927i	$-0.941767\pi$
$443$	−3.27042e58	−1.96662	−0.983312	+	0.181927i	$0.941767\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.13159e58	1.84798	0.923991	−	0.382414i	$-0.124907\pi$
$449$	4.13159e58	1.84798	0.923991	+	0.382414i	$0.124907\pi$
$450$	−1.07146e58	−0.456352
$451$	−7.30267e58	−2.96209
$452$	4.67821e58	1.80731
$453$	0	0
$454$	−3.10861e58	−1.08978
$455$	0	0
$456$	3.35785e58	1.06865
$457$	−5.82566e58	−1.76680	−0.883402	−	0.468615i	$-0.844753\pi$
$457$	−5.82566e58	−1.76680	−0.883402	+	0.468615i	$0.844753\pi$
$458$	0	0
$459$	−2.12378e58	−0.585101
$460$	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
$466$	−3.43907e58	−0.679126
$467$	−7.38905e58	−1.39193	−0.695965	−	0.718076i	$-0.745023\pi$
$467$	−7.38905e58	−1.39193	−0.695965	+	0.718076i	$0.745023\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	4.92931e58	0.734619
$473$	2.17473e59	3.09357
$474$	0	0
$475$	−1.11798e59	−1.44936
$476$	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
$482$	2.02595e59	1.90368
$483$	0	0
$484$	2.29365e59	1.96756
$485$	0	0
$486$	−1.05499e59	−0.826513
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	2.05357e59	1.40513
$490$	0	0
$491$	−4.89647e58	−0.306260	−0.153130	−	0.988206i	$-0.548935\pi$
$491$	−4.89647e58	−0.306260	−0.153130	+	0.988206i	$0.548935\pi$
$492$	2.11977e59	1.26782
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−2.70202e59	−1.23778
$499$	5.14251e58	0.225405	0.112702	−	0.993629i	$-0.464049\pi$
$499$	5.14251e58	0.225405	0.112702	+	0.993629i	$0.464049\pi$
$500$	0	0
$501$	0	0
$502$	−2.62465e59	−1.00830
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.38690e59	−0.737325
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	4.01735e59	1.00000
$513$	−6.52655e59	−1.55633
$514$	−8.64270e59	−1.97452
$515$	0	0
$516$	−6.31265e59	−1.32409
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.04379e60	1.77086	0.885429	−	0.464774i	$-0.153865\pi$
$521$	1.04379e60	1.77086	0.885429	+	0.464774i	$0.153865\pi$
$522$	0	0
$523$	8.58488e59	1.33874	0.669371	−	0.742928i	$-0.266564\pi$
$523$	8.58488e59	1.33874	0.669371	+	0.742928i	$0.266564\pi$
$524$	−6.91792e59	−1.03440
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−1.00416e60	−1.27016
$529$	8.24185e59	1.00000
$530$	0	0
$531$	−3.00221e59	−0.335245
$532$	0	0
$533$	0	0
$534$	−1.00722e60	−0.993609
$535$	0	0
$536$	8.83589e59	0.802832
$537$	−8.42741e59	−0.734953
$538$	0	0
$539$	2.14364e60	1.72266
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−8.30771e59	−0.544886
$545$	0	0
$546$	0	0
$547$	−5.89545e59	−0.342607	−0.171303	−	0.985218i	$-0.554798\pi$
$547$	−5.89545e59	−0.342607	−0.171303	+	0.985218i	$0.554798\pi$
$548$	1.42552e60	0.795793
$549$	0	0
$550$	3.34333e60	1.72266
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−4.35041e60	−1.76557
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.07657e60	0.692093
$562$	6.12159e60	1.96185
$563$	1.70945e60	0.526830	0.263415	−	0.964683i	$-0.415151\pi$
$563$	1.70945e60	0.526830	0.263415	+	0.964683i	$0.415151\pi$
$564$	0	0
$565$	0	0
$566$	7.28261e60	1.99676
$567$	0	0
$568$	0	0
$569$	−7.94346e60	−1.93883	−0.969416	−	0.245424i	$-0.921073\pi$
$569$	−7.94346e60	−1.93883	−0.969416	+	0.245424i	$0.921073\pi$
$570$	0	0
$571$	−5.20111e60	−1.17517	−0.587586	−	0.809162i	$-0.699922\pi$
$571$	−5.20111e60	−1.17517	−0.587586	+	0.809162i	$0.699922\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−2.44678e60	−0.456352
$577$	−1.11322e61	−1.99855	−0.999273	−	0.0381190i	$-0.987863\pi$
$577$	−1.11322e61	−1.99855	−0.999273	+	0.0381190i	$0.987863\pi$
$578$	−4.06845e60	−0.703099
$579$	7.13133e60	1.18643
$580$	0	0
$581$	0	0
$582$	1.31444e60	0.195180
$583$	0	0
$584$	6.30804e60	0.868584
$585$	0	0
$586$	0	0
$587$	1.10804e61	1.36306	0.681531	−	0.731789i	$-0.261314\pi$
$587$	1.10804e61	1.36306	0.681531	+	0.731789i	$0.261314\pi$
$588$	−6.22243e60	−0.737325
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.18478e61	−1.16530	−0.582648	−	0.812725i	$-0.697983\pi$
$593$	−1.18478e61	−1.16530	−0.582648	+	0.812725i	$0.697983\pi$
$594$	1.95176e61	1.84980
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−9.70479e60	−0.737325
$601$	−2.69714e61	−1.97545	−0.987723	−	0.156213i	$-0.950072\pi$
$601$	−2.69714e61	−1.97545	−0.987723	+	0.156213i	$0.950072\pi$
$602$	0	0
$603$	−5.38153e60	−0.366374
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−2.55303e61	−1.44936
$609$	0	0
$610$	0	0
$611$	0	0
$612$	5.05984e60	0.248660
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−1.01574e61	−0.464601
$615$	0	0
$616$	0	0
$617$	−4.71209e61	−1.93615	−0.968077	−	0.250652i	$-0.919355\pi$
$617$	−4.71209e61	−1.93615	−0.968077	+	0.250652i	$0.919355\pi$
$618$	0	0
$619$	3.07555e61	1.17687	0.588434	−	0.808545i	$-0.299745\pi$
$619$	3.07555e61	1.17687	0.588434	+	0.808545i	$0.299745\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.23117e61	1.00000
$626$	6.68818e61	1.99836
$627$	6.38147e61	1.84092
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	6.24766e61	1.46163
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.60193e61	1.17164	0.585820	−	0.810441i	$-0.300772\pi$
$641$	6.60193e61	1.17164	0.585820	+	0.810441i	$0.300772\pi$
$642$	−7.98397e61	−1.36914
$643$	5.32992e61	0.883241	0.441620	−	0.897202i	$-0.354404\pi$
$643$	5.32992e61	0.883241	0.441620	+	0.897202i	$0.354404\pi$
$644$	0	0
$645$	0	0
$646$	5.27956e61	0.789736
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−2.39994e61	−0.335390
$649$	9.36797e61	1.26550
$650$	0	0
$651$	0	0
$652$	−1.56136e62	−1.90571
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−1.61169e62	−1.71948
$657$	−3.84194e61	−0.396380
$658$	0	0
$659$	−2.06967e62	−1.99720	−0.998602	−	0.0528673i	$-0.983164\pi$
$659$	−2.06967e62	−1.99720	−0.998602	+	0.0528673i	$0.983164\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−2.04688e62	−1.78738
$663$	0	0
$664$	2.05438e62	1.67874
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.06900e62	−1.25727	−0.628635	−	0.777701i	$-0.716386\pi$
$673$	−2.06900e62	−1.25727	−0.628635	+	0.777701i	$0.716386\pi$
$674$	−2.21509e61	−0.130279
$675$	1.88629e62	1.07380
$676$	1.81479e62	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−2.58075e62	−1.33258
$679$	0	0
$680$	0	0
$681$	1.71488e62	0.803519
$682$	0	0
$683$	−4.44203e62	−1.95131	−0.975654	−	0.219316i	$-0.929618\pi$
$683$	−4.44203e62	−1.95131	−0.975654	+	0.219316i	$0.929618\pi$
$684$	1.55493e62	0.661419
$685$	0	0
$686$	0	0
$687$	0	0
$688$	4.79960e62	1.79581
$689$	0	0
$690$	0	0
$691$	−5.70199e62	−1.93869	−0.969347	−	0.245695i	$-0.920984\pi$
$691$	−5.70199e62	−1.93869	−0.969347	+	0.245695i	$0.920984\pi$
$692$	0	0
$693$	0	0
$694$	4.34794e62	1.34392
$695$	0	0
$696$	0	0
$697$	3.33292e62	0.936922
$698$	0	0
$699$	1.89717e62	0.500737
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	7.63481e62	1.72266
$705$	0	0
$706$	−9.31570e62	−1.97475
$707$	0	0
$708$	−2.71927e62	−0.541653
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	7.65802e62	1.34759
$713$	0	0
$714$	0	0
$715$	0	0
$716$	6.40748e62	0.996784
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	8.50092e62	1.10065
$723$	−1.11762e63	−1.40363
$724$	0	0
$725$	0	0
$726$	−1.26530e63	−1.45073
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	9.02288e62	0.944799
$730$	0	0
$731$	−9.92539e62	−0.978510
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.67923e63	1.38301
$738$	9.81607e62	0.784689
$739$	−3.06833e62	−0.238081	−0.119040	−	0.992889i	$-0.537982\pi$
$739$	−3.06833e62	−0.238081	−0.119040	+	0.992889i	$0.537982\pi$
$740$	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
$747$	−1.25123e63	−0.766098
$748$	−1.57885e63	−0.938654
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	1.44790e63	0.743444
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−4.37463e63	−1.94186
$759$	0	0
$760$	0	0
$761$	−4.63716e63	−1.88708	−0.943539	−	0.331262i	$-0.892525\pi$
$761$	−4.63716e63	−1.88708	−0.943539	+	0.331262i	$0.892525\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−2.21618e63	−0.737325
$769$	3.83292e63	1.23922	0.619612	−	0.784908i	$-0.287290\pi$
$769$	3.83292e63	1.23922	0.619612	+	0.784908i	$0.287290\pi$
$770$	0	0
$771$	4.76777e63	1.45586
$772$	−5.42206e63	−1.60910
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−2.92322e63	−0.819521
$775$	0	0
$776$	−9.99389e62	−0.264714
$777$	0	0
$778$	0	0
$779$	1.02423e64	2.49215
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	4.73100e63	1.00000
$785$	0	0
$786$	3.81630e63	0.762686
$787$	−3.04018e63	−0.590821	−0.295410	−	0.955370i	$-0.595456\pi$
$787$	−3.04018e63	−0.590821	−0.295410	+	0.955370i	$0.595456\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−4.65001e63	−0.786140
$793$	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
$800$	7.37870e63	1.00000
$801$	−4.66415e63	−0.614974
$802$	1.47318e64	1.88981
$803$	1.19882e64	1.49628
$804$	−4.87435e63	−0.591948
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.18588e63	−0.125653	−0.0628265	−	0.998024i	$-0.520011\pi$
$809$	−1.18588e63	−0.125653	−0.0628265	+	0.998024i	$0.520011\pi$
$810$	0	0
$811$	1.49510e64	1.50042	0.750210	−	0.661200i	$-0.229952\pi$
$811$	1.49510e64	1.50042	0.750210	+	0.661200i	$0.229952\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	4.58298e63	0.401758
$817$	−3.05015e64	−2.60277
$818$	2.37240e64	1.97067
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−7.86392e63	−0.586758
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−1.84436e64	−1.27016
$826$	0	0
$827$	−2.97147e64	−1.94022	−0.970109	−	0.242671i	$-0.921977\pi$
$827$	−2.97147e64	−1.94022	−0.970109	+	0.242671i	$0.921977\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.78353e63	−0.544886
$834$	2.39992e64	1.30180
$835$	0	0
$836$	−4.85193e64	−2.49676
$837$	0	0
$838$	1.55644e64	0.759912
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.21570e64	1.00000
$842$	0	0
$843$	−3.37700e64	−1.44652
$844$	−4.75019e64	−1.98234
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.01748e64	−1.47226
$850$	−1.52589e64	−0.544886
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	6.07033e64	1.85690
$857$	1.20387e64	0.358922	0.179461	−	0.983765i	$-0.442565\pi$
$857$	1.20387e64	0.358922	0.179461	+	0.983765i	$0.442565\pi$
$858$	0	0
$859$	7.01963e64	1.98822	0.994111	−	0.108368i	$-0.0345624\pi$
$859$	7.01963e64	1.98822	0.994111	+	0.108368i	$0.0345624\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	4.30752e64	1.07380
$865$	0	0
$866$	7.87404e64	1.86554
$867$	2.24438e64	0.518412
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.08682e63	0.120803
$874$	0	0
$875$	0	0
$876$	−3.47985e64	−0.640428
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.77254e64	1.09972	0.549859	−	0.835257i	$-0.314681\pi$
$881$	6.77254e64	1.09972	0.549859	+	0.835257i	$0.314681\pi$
$882$	−2.88144e64	−0.456352
$883$	1.16105e64	0.179356	0.0896782	−	0.995971i	$-0.471416\pi$
$883$	1.16105e64	0.179356	0.0896782	+	0.995971i	$0.471416\pi$
$884$	0	0
$885$	0	0
$886$	−1.37171e65	−1.96662
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.56100e64	−0.577764
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	1.73291e65	1.84798
$899$	0	0
$900$	−4.49402e64	−0.456352
$901$	0	0
$902$	−3.06296e65	−2.96209
$903$	0	0
$904$	1.96218e65	1.80731
$905$	0	0
$906$	0	0
$907$	1.15038e65	0.985104	0.492552	−	0.870283i	$-0.336064\pi$
$907$	1.15038e65	0.985104	0.492552	+	0.870283i	$0.336064\pi$
$908$	−1.30384e65	−1.08978
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	1.40839e65	1.06865
$913$	3.90428e65	2.89190
$914$	−2.44346e65	−1.76680
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−8.90779e64	−0.585101
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	5.60339e64	0.342562
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.86418e65	−1.44762	−0.723811	−	0.689998i	$-0.757611\pi$
$929$	−2.86418e65	−1.44762	−0.723811	+	0.689998i	$0.757611\pi$
$930$	0	0
$931$	−3.00656e65	−1.44936
$932$	−1.44245e65	−0.679126
$933$	0	0
$934$	−3.09919e65	−1.39193
$935$	0	0
$936$	0	0
$937$	3.41534e65	1.42943	0.714715	−	0.699416i	$-0.246557\pi$
$937$	3.41534e65	1.42943	0.714715	+	0.699416i	$0.246557\pi$
$938$	0	0
$939$	−3.68956e65	−1.47344
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	2.06750e65	0.734619
$945$	0	0
$946$	9.12147e65	3.09357
$947$	−5.53208e65	−1.83311	−0.916556	−	0.399906i	$-0.869043\pi$
$947$	−5.53208e65	−1.83311	−0.916556	+	0.399906i	$0.869043\pi$
$948$	0	0
$949$	0	0
$950$	−4.68917e65	−1.44936
$951$	0	0
$952$	0	0
$953$	4.83412e64	0.139404	0.0697018	−	0.997568i	$-0.477795\pi$
$953$	4.83412e64	0.139404	0.0697018	+	0.997568i	$0.477795\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.16787e65	1.00000
$962$	0	0
$963$	−3.69716e65	−0.847402
$964$	8.49746e65	1.90368
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	9.62025e65	1.96756
$969$	−2.91249e65	−0.582292
$970$	0	0
$971$	−5.58568e65	−1.06722	−0.533609	−	0.845731i	$-0.679165\pi$
$971$	−5.58568e65	−1.06722	−0.533609	+	0.845731i	$0.679165\pi$
$972$	−4.42494e65	−0.826513
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.56535e65	−0.928565	−0.464282	−	0.885687i	$-0.653688\pi$
$977$	−5.56535e65	−0.928565	−0.464282	+	0.885687i	$0.653688\pi$
$978$	8.61331e65	1.40513
$979$	1.45538e66	2.32143
$980$	0	0
$981$	0	0
$982$	−2.05373e65	−0.306260
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	8.89096e65	1.26782
$985$	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
$993$	1.12917e66	1.31788
$994$	0	0
$995$	0	0
$996$	−1.13331e66	−1.23778
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	2.15692e65	0.225405
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.45.d.a.3.1	✓	1
8.3	odd	2	CM	8.45.d.a.3.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.45.d.a.3.1	✓	1	1.1	even	1	trivial
8.45.d.a.3.1	✓	1	8.3	odd	2	CM

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:	\( k \)	\(=\)	\( 45 \)
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

Downloads

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