LMFDB - Embedded newform 8.47.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,47,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, 47, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 47);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 47 $
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:	$107.214694906$
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)

$f(q)$

$=$

$q-8.38861e6 q^{2} +1.88152e11 q^{3} +7.03687e13 q^{4} -1.57834e18 q^{6} -5.90296e20 q^{8} +2.65384e22 q^{9} +O(q^{10})$

$q-8.38861e6 q^{2} +1.88152e11 q^{3} +7.03687e13 q^{4} -1.57834e18 q^{6} -5.90296e20 q^{8} +2.65384e22 q^{9} +2.79767e23 q^{11} +1.32400e25 q^{12} +4.95176e27 q^{16} -3.89979e28 q^{17} -2.22620e29 q^{18} +1.75663e29 q^{19} -2.34685e30 q^{22} -1.11066e32 q^{24} +1.42109e32 q^{25} +3.32567e33 q^{27} -4.15384e34 q^{32} +5.26388e34 q^{33} +3.27138e35 q^{34} +1.86747e36 q^{36} -1.47357e36 q^{38} +2.24777e37 q^{41} +4.31122e37 q^{43} +1.96868e37 q^{44} +9.31685e38 q^{48} +7.49048e38 q^{49} -1.19209e39 q^{50} -7.33755e39 q^{51} -2.78978e40 q^{54} +3.30514e40 q^{57} -9.91593e40 q^{59} +3.48449e41 q^{64} -4.41566e41 q^{66} +1.99410e42 q^{67} -2.74424e42 q^{68} -1.56655e43 q^{72} -9.07893e42 q^{73} +2.67381e43 q^{75} +1.23612e43 q^{76} +3.90525e44 q^{81} -1.88557e44 q^{82} +1.74056e44 q^{83} -3.61651e44 q^{86} -1.65145e44 q^{88} +1.78172e44 q^{89} -7.81554e45 q^{96} -7.97012e45 q^{97} -6.28347e45 q^{98} +7.42456e45 q^{99} +O(q^{100})$

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 $

$2$	−8.38861e6	−1.00000
$2$	−8.38861e6	−1.00000
$3$	1.88152e11	1.99858	0.999288	−	0.0377238i	$-0.0120107\pi$
$3$	1.88152e11	1.99858	0.999288	+	0.0377238i	$0.0120107\pi$
$4$	7.03687e13	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−1.57834e18	−1.99858
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−5.90296e20	−1.00000
$9$	2.65384e22	2.99431
$10$	0	0
$11$	2.79767e23	0.312439	0.156219	−	0.987722i	$-0.450069\pi$
$11$	2.79767e23	0.312439	0.156219	+	0.987722i	$0.450069\pi$
$12$	1.32400e25	1.99858
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	4.95176e27	1.00000
$17$	−3.89979e28	−1.95306	−0.976532	−	0.215373i	$-0.930903\pi$
$17$	−3.89979e28	−1.95306	−0.976532	+	0.215373i	$0.930903\pi$
$18$	−2.22620e29	−2.99431
$19$	1.75663e29	0.681314	0.340657	−	0.940188i	$-0.389351\pi$
$19$	1.75663e29	0.681314	0.340657	+	0.940188i	$0.389351\pi$
$20$	0	0
$21$	0	0
$22$	−2.34685e30	−0.312439
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.11066e32	−1.99858
$25$	1.42109e32	1.00000
$26$	0	0
$27$	3.32567e33	3.98578
$28$	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			$\pi$
$32$	−4.15384e34	−1.00000
$33$	5.26388e34	0.624432
$34$	3.27138e35	1.95306
$35$	0	0
$36$	1.86747e36	2.99431
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.47357e36	−0.681314
$39$	0	0
$40$	0	0
$41$	2.24777e37	1.81019	0.905093	−	0.425214i	$-0.139801\pi$
$41$	2.24777e37	1.81019	0.905093	+	0.425214i	$0.139801\pi$
$42$	0	0
$43$	4.31122e37	1.16098	0.580491	−	0.814267i	$-0.302861\pi$
$43$	4.31122e37	1.16098	0.580491	+	0.814267i	$0.302861\pi$
$44$	1.96868e37	0.312439
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	9.31685e38	1.99858
$49$	7.49048e38	1.00000
$50$	−1.19209e39	−1.00000
$51$	−7.33755e39	−3.90335
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−2.78978e40	−3.98578
$55$	0	0
$56$	0	0
$57$	3.30514e40	1.36166
$58$	0	0
$59$	−9.91593e40	−1.84815	−0.924075	−	0.382211i	$-0.875163\pi$
$59$	−9.91593e40	−1.84815	−0.924075	+	0.382211i	$0.875163\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	3.48449e41	1.00000
$65$	0	0
$66$	−4.41566e41	−0.624432
$67$	1.99410e42	1.99538	0.997692	−	0.0679086i	$-0.0216326\pi$
$67$	1.99410e42	1.99538	0.997692	+	0.0679086i	$0.0216326\pi$
$68$	−2.74424e42	−1.95306
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1.56655e43	−2.99431
$73$	−9.07893e42	−1.26360	−0.631799	−	0.775132i	$-0.717683\pi$
$73$	−9.07893e42	−1.26360	−0.631799	+	0.775132i	$0.717683\pi$
$74$	0	0
$75$	2.67381e43	1.99858
$76$	1.23612e43	0.681314
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	3.90525e44	4.97157
$82$	−1.88557e44	−1.81019
$83$	1.74056e44	1.26442	0.632212	−	0.774796i	$-0.282147\pi$
$83$	1.74056e44	1.26442	0.632212	+	0.774796i	$0.282147\pi$
$84$	0	0
$85$	0	0
$86$	−3.61651e44	−1.16098
$87$	0	0
$88$	−1.65145e44	−0.312439
$89$	1.78172e44	0.259938	0.129969	−	0.991518i	$-0.458512\pi$
$89$	1.78172e44	0.259938	0.129969	+	0.991518i	$0.458512\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−7.81554e45	−1.99858
$97$	−7.97012e45	−1.60589	−0.802943	−	0.596055i	$-0.796734\pi$
$97$	−7.97012e45	−1.60589	−0.802943	+	0.596055i	$0.796734\pi$
$98$	−6.28347e45	−1.00000
$99$	7.42456e45	0.935537
$100$	1.00000e46	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	6.15519e46	3.90335
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.36690e46	−1.13213	−0.566066	−	0.824360i	$-0.691535\pi$
$107$	−5.36690e46	−1.13213	−0.566066	+	0.824360i	$0.691535\pi$
$108$	2.34023e47	3.98578
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.58627e47	1.55550	0.777750	−	0.628574i	$-0.216361\pi$
$113$	2.58627e47	1.55550	0.777750	+	0.628574i	$0.216361\pi$
$114$	−2.77255e47	−1.36166
$115$	0	0
$116$	0	0
$117$	0	0
$118$	8.31809e47	1.84815
$119$	0	0
$120$	0	0
$121$	−7.23526e47	−0.902382
$122$	0	0
$123$	4.22923e48	3.61779
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−2.92300e48	−1.00000
$129$	8.11165e48	2.32031
$130$	0	0
$131$	7.06346e48	1.41833	0.709166	−	0.705042i	$-0.249072\pi$
$131$	7.06346e48	1.41833	0.709166	+	0.705042i	$0.249072\pi$
$132$	3.70413e48	0.624432
$133$	0	0
$134$	−1.67277e49	−1.99538
$135$	0	0
$136$	2.30203e49	1.95306
$137$	−2.23254e49	−1.60039	−0.800197	−	0.599738i	$-0.795272\pi$
$137$	−2.23254e49	−1.60039	−0.800197	+	0.599738i	$0.795272\pi$
$138$	0	0
$139$	−1.31881e49	−0.677395	−0.338697	−	0.940895i	$-0.609986\pi$
$139$	−1.31881e49	−0.677395	−0.338697	+	0.940895i	$0.609986\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.31412e50	2.99431
$145$	0	0
$146$	7.61596e49	1.26360
$147$	1.40935e50	1.99858
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.24295e50	−1.99858
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−1.03693e50	−0.681314
$153$	−1.03494e51	−5.84807
$154$	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
$162$	−3.27596e51	−4.97157
$163$	1.35698e51	1.78755	0.893777	−	0.448512i	$-0.148046\pi$
$163$	1.35698e51	1.78755	0.893777	+	0.448512i	$0.148046\pi$
$164$	1.58173e51	1.81019
$165$	0	0
$166$	−1.46009e51	−1.26442
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.74339e51	1.00000
$170$	0	0
$171$	4.66181e51	2.04006
$172$	3.03375e51	1.16098
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	1.38534e51	0.312439
$177$	−1.86571e52	−3.69367
$178$	−1.49462e51	−0.259938
$179$	1.06704e52	1.63140	0.815700	−	0.578475i	$-0.196352\pi$
$179$	1.06704e52	1.63140	0.815700	+	0.578475i	$0.196352\pi$
$180$	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
$187$	−1.09103e52	−0.610212
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	6.55615e52	1.99858
$193$	2.73677e52	0.740321	0.370161	−	0.928968i	$-0.379303\pi$
$193$	2.73677e52	0.740321	0.370161	+	0.928968i	$0.379303\pi$
$194$	6.68582e52	1.60589
$195$	0	0
$196$	5.27096e52	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−6.22817e52	−0.935537
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−8.38861e52	−1.00000
$201$	3.75194e53	3.98793
$202$	0	0
$203$	0	0
$204$	−5.16334e53	−3.90335
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.91447e52	0.212869
$210$	0	0
$211$	2.40745e53	0.837645	0.418822	−	0.908068i	$-0.362443\pi$
$211$	2.40745e53	0.837645	0.418822	+	0.908068i	$0.362443\pi$
$212$	0	0
$213$	0	0
$214$	4.50209e53	1.13213
$215$	0	0
$216$	−1.96313e54	−3.98578
$217$	0	0
$218$	0	0
$219$	−1.70822e54	−2.52540
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	3.77133e54	2.99431
$226$	−2.16952e54	−1.55550
$227$	−1.16888e54	−0.757137	−0.378569	−	0.925573i	$-0.623584\pi$
$227$	−1.16888e54	−0.757137	−0.378569	+	0.925573i	$0.623584\pi$
$228$	2.32578e54	1.36166
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.70645e54	−1.31757	−0.658783	−	0.752333i	$-0.728928\pi$
$233$	−3.70645e54	−1.31757	−0.658783	+	0.752333i	$0.728928\pi$
$234$	0	0
$235$	0	0
$236$	−6.97772e54	−1.84815
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	1.25288e54	0.204888	0.102444	−	0.994739i	$-0.467334\pi$
$241$	1.25288e54	0.204888	0.102444	+	0.994739i	$0.467334\pi$
$242$	6.06937e54	0.902382
$243$	4.40030e55	5.95029
$244$	0	0
$245$	0	0
$246$	−3.54774e55	−3.61779
$247$	0	0
$248$	0	0
$249$	3.27490e55	2.52705
$250$	0	0
$251$	2.45857e55	1.57829	0.789144	−	0.614208i	$-0.210524\pi$
$251$	2.45857e55	1.57829	0.789144	+	0.614208i	$0.210524\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	2.45199e55	1.00000
$257$	−3.41327e55	−1.27265	−0.636326	−	0.771421i	$-0.719547\pi$
$257$	−3.41327e55	−1.27265	−0.636326	+	0.771421i	$0.719547\pi$
$258$	−6.80455e55	−2.32031
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−5.92526e55	−1.41833
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−3.10725e55	−0.624432
$265$	0	0
$266$	0	0
$267$	3.35235e55	0.519505
$268$	1.40322e56	1.99538
$269$	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			$\pi$
$272$	−1.93108e56	−1.95306
$273$	0	0
$274$	1.87279e56	1.60039
$275$	3.97573e55	0.312439
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	1.10630e56	0.677395
$279$	0	0
$280$	0	0
$281$	−2.47308e56	−1.18302	−0.591510	−	0.806298i	$-0.701468\pi$
$281$	−2.47308e56	−1.18302	−0.591510	+	0.806298i	$0.701468\pi$
$282$	0	0
$283$	−9.89038e55	−0.401906	−0.200953	−	0.979601i	$-0.564404\pi$
$283$	−9.89038e55	−0.401906	−0.200953	+	0.979601i	$0.564404\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.10236e57	−2.99431
$289$	1.12214e57	2.81446
$290$	0	0
$291$	−1.49960e57	−3.20949
$292$	−6.38873e56	−1.26360
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.18225e57	−1.99858
$295$	0	0
$296$	0	0
$297$	9.30413e56	1.24531
$298$	0	0
$299$	0	0
$300$	1.88152e57	1.99858
$301$	0	0
$302$	0	0
$303$	0	0
$304$	8.69841e56	0.681314
$305$	0	0
$306$	8.68172e57	5.84807
$307$	8.06373e56	0.503911	0.251955	−	0.967739i	$-0.418926\pi$
$307$	8.06373e56	0.503911	0.251955	+	0.967739i	$0.418926\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−4.08397e57	−1.63517	−0.817585	−	0.575808i	$-0.804687\pi$
$313$	−4.08397e57	−1.63517	−0.817585	+	0.575808i	$0.804687\pi$
$314$	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
$321$	−1.00980e58	−2.26265
$322$	0	0
$323$	−6.85049e57	−1.33065
$324$	2.74808e58	4.97157
$325$	0	0
$326$	−1.13832e58	−1.78755
$327$	0	0
$328$	−1.32685e58	−1.81019
$329$	0	0
$330$	0	0
$331$	−8.44966e57	−0.934966	−0.467483	−	0.884002i	$-0.654839\pi$
$331$	−8.44966e57	−0.934966	−0.467483	+	0.884002i	$0.654839\pi$
$332$	1.22481e58	1.26442
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.79594e58	−1.31463	−0.657315	−	0.753616i	$-0.728308\pi$
$337$	−1.79594e58	−1.31463	−0.657315	+	0.753616i	$0.728308\pi$
$338$	−1.46246e58	−1.00000
$339$	4.86613e58	3.10878
$340$	0	0
$341$	0	0
$342$	−3.91061e58	−2.04006
$343$	0	0
$344$	−2.54489e58	−1.16098
$345$	0	0
$346$	0	0
$347$	−2.15024e58	−0.803359	−0.401679	−	0.915780i	$-0.631573\pi$
$347$	−2.15024e58	−0.803359	−0.401679	+	0.915780i	$0.631573\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−1.16211e58	−0.312439
$353$	−9.45020e57	−0.238025	−0.119013	−	0.992893i	$-0.537973\pi$
$353$	−9.45020e57	−0.238025	−0.119013	+	0.992893i	$0.537973\pi$
$354$	1.56507e59	3.69367
$355$	0	0
$356$	1.25377e58	0.259938
$357$	0	0
$358$	−8.95097e58	−1.63140
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−3.56187e58	−0.535811
$362$	0	0
$363$	−1.36133e59	−1.80348
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	5.96521e59	5.42025
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	9.15225e58	0.610212
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.16293e59	−1.55377	−0.776886	−	0.629641i	$-0.783202\pi$
$379$	−3.16293e59	−1.55377	−0.776886	+	0.629641i	$0.783202\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−5.49970e59	−1.99858
$385$	0	0
$386$	−2.29577e59	−0.740321
$387$	1.14413e60	3.47634
$388$	−5.60847e59	−1.60589
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−4.42160e59	−1.00000
$393$	1.32901e60	2.83464
$394$	0	0
$395$	0	0
$396$	5.22457e59	0.935537
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	7.03687e59	1.00000
$401$	−1.20068e60	−1.61104	−0.805518	−	0.592572i	$-0.798113\pi$
$401$	−1.20068e60	−1.61104	−0.805518	+	0.592572i	$0.798113\pi$
$402$	−3.14736e60	−3.98793
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	4.33133e60	3.90335
$409$	−5.78908e59	−0.493143	−0.246572	−	0.969125i	$-0.579304\pi$
$409$	−5.78908e59	−0.493143	−0.246572	+	0.969125i	$0.579304\pi$
$410$	0	0
$411$	−4.20058e60	−3.19851
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.48137e60	−1.35383
$418$	−4.12255e59	−0.212869
$419$	2.03592e60	0.995033	0.497516	−	0.867455i	$-0.334245\pi$
$419$	2.03592e60	0.995033	0.497516	+	0.867455i	$0.334245\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−2.01951e60	−0.837645
$423$	0	0
$424$	0	0
$425$	−5.54194e60	−1.95306
$426$	0	0
$427$	0	0
$428$	−3.77662e60	−1.13213
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.64679e61	3.98578
$433$	7.76477e60	1.78200	0.890998	−	0.454007i	$-0.150006\pi$
$433$	7.76477e60	1.78200	0.890998	+	0.454007i	$0.150006\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.43296e61	2.52540
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.98785e61	2.99431
$442$	0	0
$443$	−1.47165e61	−1.99765	−0.998826	−	0.0484428i	$-0.984574\pi$
$443$	−1.47165e61	−1.99765	−0.998826	+	0.0484428i	$0.984574\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.99211e61	1.98448	0.992242	−	0.124322i	$-0.0396756\pi$
$449$	1.99211e61	1.98448	0.992242	+	0.124322i	$0.0396756\pi$
$450$	−3.16362e61	−2.99431
$451$	6.28852e60	0.565572
$452$	1.81993e61	1.55550
$453$	0	0
$454$	9.80526e60	0.757137
$455$	0	0
$456$	−1.95101e61	−1.36166
$457$	−6.70301e60	−0.444833	−0.222417	−	0.974952i	$-0.571394\pi$
$457$	−6.70301e60	−0.444833	−0.222417	+	0.974952i	$0.571394\pi$
$458$	0	0
$459$	−1.29694e62	−7.78448
$460$	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
$466$	3.10920e61	1.31757
$467$	3.68357e61	1.48587	0.742935	−	0.669364i	$-0.233433\pi$
$467$	3.68357e61	1.48587	0.742935	+	0.669364i	$0.233433\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	5.85333e61	1.84815
$473$	1.20614e61	0.362735
$474$	0	0
$475$	2.49632e61	0.681314
$476$	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
$482$	−1.05099e61	−0.204888
$483$	0	0
$484$	−5.09136e61	−0.902382
$485$	0	0
$486$	−3.69124e62	−5.95029
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	2.55319e62	3.57256
$490$	0	0
$491$	−1.12989e62	−1.43933	−0.719666	−	0.694320i	$-0.755705\pi$
$491$	−1.12989e62	−1.43933	−0.719666	+	0.694320i	$0.755705\pi$
$492$	2.97606e62	3.61779
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−2.74719e62	−2.52705
$499$	1.14027e62	1.00160	0.500800	−	0.865563i	$-0.333039\pi$
$499$	1.14027e62	1.00160	0.500800	+	0.865563i	$0.333039\pi$
$500$	0	0
$501$	0	0
$502$	−2.06239e62	−1.57829
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.28023e62	1.99858
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−2.05688e62	−1.00000
$513$	5.84197e62	2.71556
$514$	2.86326e62	1.27265
$515$	0	0
$516$	5.70807e62	2.32031
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.99413e62	−1.95190	−0.975951	−	0.217988i	$-0.930051\pi$
$521$	−5.99413e62	−1.95190	−0.975951	+	0.217988i	$0.930051\pi$
$522$	0	0
$523$	−2.90006e62	−0.864706	−0.432353	−	0.901705i	$-0.642317\pi$
$523$	−2.90006e62	−0.864706	−0.432353	+	0.901705i	$0.642317\pi$
$524$	4.97047e62	1.41833
$525$	0	0
$526$	0	0
$527$	0	0
$528$	2.60655e62	0.624432
$529$	4.35994e62	1.00000
$530$	0	0
$531$	−2.63153e63	−5.53393
$532$	0	0
$533$	0	0
$534$	−2.81215e62	−0.519505
$535$	0	0
$536$	−1.17711e63	−1.99538
$537$	2.00766e63	3.26048
$538$	0	0
$539$	2.09559e62	0.312439
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.61991e63	1.95306
$545$	0	0
$546$	0	0
$547$	−9.62983e62	−1.02308	−0.511541	−	0.859259i	$-0.670925\pi$
$547$	−9.62983e62	−1.02308	−0.511541	+	0.859259i	$0.670925\pi$
$548$	−1.57101e63	−1.60039
$549$	0	0
$550$	−3.33508e62	−0.312439
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−9.28030e62	−0.677395
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.05280e63	−1.21956
$562$	2.07457e63	1.18302
$563$	−3.64603e63	−1.99584	−0.997922	−	0.0644329i	$-0.979476\pi$
$563$	−3.64603e63	−1.99584	−0.997922	+	0.0644329i	$0.979476\pi$
$564$	0	0
$565$	0	0
$566$	8.29665e62	0.401906
$567$	0	0
$568$	0	0
$569$	−3.44189e63	−1.47644	−0.738219	−	0.674561i	$-0.764333\pi$
$569$	−3.44189e63	−1.47644	−0.738219	+	0.674561i	$0.764333\pi$
$570$	0	0
$571$	5.05380e63	1.99980	0.999902	−	0.0140218i	$-0.00446344\pi$
$571$	5.05380e63	1.99980	0.999902	+	0.0140218i	$0.00446344\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	9.24727e63	2.99431
$577$	−2.56160e62	−0.0797016	−0.0398508	−	0.999206i	$-0.512688\pi$
$577$	−2.56160e62	−0.0797016	−0.0398508	+	0.999206i	$0.512688\pi$
$578$	−9.41315e63	−2.81446
$579$	5.14930e63	1.47959
$580$	0	0
$581$	0	0
$582$	1.25795e64	3.20949
$583$	0	0
$584$	5.35926e63	1.26360
$585$	0	0
$586$	0	0
$587$	−8.02093e63	−1.68093	−0.840463	−	0.541869i	$-0.817717\pi$
$587$	−8.02093e63	−1.68093	−0.840463	+	0.541869i	$0.817717\pi$
$588$	9.91743e63	1.99858
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.62465e63	−0.269465	−0.134733	−	0.990882i	$-0.543018\pi$
$593$	−1.62465e63	−0.269465	−0.134733	+	0.990882i	$0.543018\pi$
$594$	−7.80487e63	−1.24531
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−1.57834e64	−1.99858
$601$	−1.39909e64	−1.70503	−0.852517	−	0.522699i	$-0.824925\pi$
$601$	−1.39909e64	−1.70503	−0.852517	+	0.522699i	$0.824925\pi$
$602$	0	0
$603$	5.29201e64	5.97479
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−7.29675e63	−0.681314
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−7.28275e64	−5.84807
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−6.76435e63	−0.503911
$615$	0	0
$616$	0	0
$617$	6.22498e62	0.0414552	0.0207276	−	0.999785i	$-0.493402\pi$
$617$	6.22498e62	0.0414552	0.0207276	+	0.999785i	$0.493402\pi$
$618$	0	0
$619$	−3.19505e64	−1.97511	−0.987557	−	0.157263i	$-0.949733\pi$
$619$	−3.19505e64	−1.97511	−0.987557	+	0.157263i	$0.949733\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.01948e64	1.00000
$626$	3.42588e64	1.63517
$627$	9.24668e63	0.425434
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	4.52966e64	1.67410
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.83384e64	−1.06145	−0.530725	−	0.847544i	$-0.678080\pi$
$641$	−3.83384e64	−1.06145	−0.530725	+	0.847544i	$0.678080\pi$
$642$	8.47078e64	2.26265
$643$	9.38224e63	0.241799	0.120899	−	0.992665i	$-0.461422\pi$
$643$	9.38224e63	0.241799	0.120899	+	0.992665i	$0.461422\pi$
$644$	0	0
$645$	0	0
$646$	5.74661e64	1.33065
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−2.30525e65	−4.97157
$649$	−2.77415e64	−0.577433
$650$	0	0
$651$	0	0
$652$	9.54891e64	1.78755
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	1.11304e65	1.81019
$657$	−2.40940e65	−3.78360
$658$	0	0
$659$	9.81210e64	1.43680	0.718402	−	0.695629i	$-0.244874\pi$
$659$	9.81210e64	1.43680	0.718402	+	0.695629i	$0.244874\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	7.08809e64	0.934966
$663$	0	0
$664$	−1.02744e65	−1.26442
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.94056e65	1.75218	0.876092	−	0.482145i	$-0.160142\pi$
$673$	1.94056e65	1.75218	0.876092	+	0.482145i	$0.160142\pi$
$674$	1.50654e65	1.31463
$675$	4.72607e65	3.98578
$676$	1.22680e65	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−4.08201e65	−3.10878
$679$	0	0
$680$	0	0
$681$	−2.19927e65	−1.51320
$682$	0	0
$683$	−1.53636e65	−0.988137	−0.494068	−	0.869423i	$-0.664491\pi$
$683$	−1.53636e65	−0.988137	−0.494068	+	0.869423i	$0.664491\pi$
$684$	3.28045e65	2.04006
$685$	0	0
$686$	0	0
$687$	0	0
$688$	2.13481e65	1.16098
$689$	0	0
$690$	0	0
$691$	−1.24646e65	−0.613313	−0.306657	−	0.951820i	$-0.599210\pi$
$691$	−1.24646e65	−0.613313	−0.306657	+	0.951820i	$0.599210\pi$
$692$	0	0
$693$	0	0
$694$	1.80375e65	0.803359
$695$	0	0
$696$	0	0
$697$	−8.76584e65	−3.53541
$698$	0	0
$699$	−6.97377e65	−2.63325
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	9.74845e64	0.312439
$705$	0	0
$706$	7.92741e64	0.238025
$707$	0	0
$708$	−1.31287e66	−3.69367
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−1.05174e65	−0.259938
$713$	0	0
$714$	0	0
$715$	0	0
$716$	7.50862e65	1.63140
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	2.98791e65	0.535811
$723$	2.35732e65	0.409484
$724$	0	0
$725$	0	0
$726$	1.14197e66	1.80348
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	4.81807e66	6.92053
$730$	0	0
$731$	−1.68128e66	−2.26747
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.57883e65	0.623435
$738$	−5.00398e66	−5.42025
$739$	−1.79644e66	−1.88621	−0.943105	−	0.332494i	$-0.892110\pi$
$739$	−1.79644e66	−1.88621	−0.943105	+	0.332494i	$0.892110\pi$
$740$	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
$747$	4.61916e66	3.78607
$748$	−7.67746e65	−0.610212
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	4.62585e66	3.15433
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	2.65326e66	1.55377
$759$	0	0
$760$	0	0
$761$	−3.72942e66	−1.99432	−0.997158	−	0.0753389i	$-0.975996\pi$
$761$	−3.72942e66	−1.99432	−0.997158	+	0.0753389i	$0.975996\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	4.61348e66	1.99858
$769$	−4.73913e66	−1.99247	−0.996237	−	0.0866660i	$-0.972379\pi$
$769$	−4.73913e66	−1.99247	−0.996237	+	0.0866660i	$0.972379\pi$
$770$	0	0
$771$	−6.42216e66	−2.54349
$772$	1.92583e66	0.740321
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−9.59762e66	−3.47634
$775$	0	0
$776$	4.70473e66	1.60589
$777$	0	0
$778$	0	0
$779$	3.94850e66	1.23330
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.70911e66	1.00000
$785$	0	0
$786$	−1.11485e67	−2.83464
$787$	−7.66471e66	−1.89268	−0.946339	−	0.323175i	$-0.895250\pi$
$787$	−7.66471e66	−1.89268	−0.946339	+	0.323175i	$0.895250\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−4.38268e66	−0.935537
$793$	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
$800$	−5.90296e66	−1.00000
$801$	4.72839e66	0.778333
$802$	1.00720e67	1.61104
$803$	−2.53999e66	−0.394797
$804$	2.64020e67	3.98793
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.25891e66	−0.295858	−0.147929	−	0.988998i	$-0.547261\pi$
$809$	−2.25891e66	−0.295858	−0.147929	+	0.988998i	$0.547261\pi$
$810$	0	0
$811$	9.29148e66	1.14975	0.574877	−	0.818240i	$-0.305050\pi$
$811$	9.29148e66	1.14975	0.574877	+	0.818240i	$0.305050\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−3.63338e67	−3.90335
$817$	7.57321e66	0.790993
$818$	4.85623e66	0.493143
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	3.52370e67	3.19851
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	7.48042e66	0.624432
$826$	0	0
$827$	−2.27235e67	−1.79411	−0.897054	−	0.441921i	$-0.854297\pi$
$827$	−2.27235e67	−1.79411	−0.897054	+	0.441921i	$0.854297\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.92113e67	−1.95306
$834$	2.08153e67	1.35383
$835$	0	0
$836$	3.45825e66	0.212869
$837$	0	0
$838$	−1.70785e67	−0.995033
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.86341e67	1.00000
$842$	0	0
$843$	−4.65315e67	−2.36436
$844$	1.69409e67	0.837645
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.86090e67	−0.803240
$850$	4.64892e67	1.95306
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	3.16806e67	1.13213
$857$	4.75532e67	1.65432	0.827162	−	0.561964i	$-0.189954\pi$
$857$	4.75532e67	1.65432	0.827162	+	0.561964i	$0.189954\pi$
$858$	0	0
$859$	5.58886e67	1.84281	0.921405	−	0.388604i	$-0.127042\pi$
$859$	5.58886e67	1.84281	0.921405	+	0.388604i	$0.127042\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.38143e68	−3.98578
$865$	0	0
$866$	−6.51356e67	−1.78200
$867$	2.11132e68	5.62491
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.11514e68	−4.80852
$874$	0	0
$875$	0	0
$876$	−1.20205e68	−2.52540
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.68207e66	0.0678651	0.0339326	−	0.999424i	$-0.489197\pi$
$881$	3.68207e66	0.0678651	0.0339326	+	0.999424i	$0.489197\pi$
$882$	−1.66753e68	−2.99431
$883$	−2.56980e66	−0.0449577	−0.0224788	−	0.999747i	$-0.507156\pi$
$883$	−2.56980e66	−0.0449577	−0.0224788	+	0.999747i	$0.507156\pi$
$884$	0	0
$885$	0	0
$886$	1.23451e68	1.99765
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.09256e68	1.55331
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−1.67110e68	−1.98448
$899$	0	0
$900$	2.65384e68	2.99431
$901$	0	0
$902$	−5.27519e67	−0.565572
$903$	0	0
$904$	−1.52667e68	−1.55550
$905$	0	0
$906$	0	0
$907$	−3.82229e67	−0.360874	−0.180437	−	0.983587i	$-0.557751\pi$
$907$	−3.82229e67	−0.360874	−0.180437	+	0.983587i	$0.557751\pi$
$908$	−8.22525e67	−0.757137
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	1.63663e68	1.36166
$913$	4.86951e67	0.395054
$914$	5.62289e67	0.444833
$915$	0	0
$916$	0	0
$917$	0	0
$918$	1.08796e69	7.78448
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	1.51721e68	1.00710
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.60028e68	−1.95873	−0.979366	−	0.202097i	$-0.935224\pi$
$929$	−3.60028e68	−1.95873	−0.979366	+	0.202097i	$0.935224\pi$
$930$	0	0
$931$	1.31580e68	0.681314
$932$	−2.60818e68	−1.31757
$933$	0	0
$934$	−3.09000e68	−1.48587
$935$	0	0
$936$	0	0
$937$	1.66659e68	0.744421	0.372211	−	0.928148i	$-0.378600\pi$
$937$	1.66659e68	0.744421	0.372211	+	0.928148i	$0.378600\pi$
$938$	0	0
$939$	−7.68408e68	−3.26801
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−4.91013e68	−1.84815
$945$	0	0
$946$	−1.01178e68	−0.362735
$947$	8.03737e67	0.281232	0.140616	−	0.990064i	$-0.455092\pi$
$947$	8.03737e67	0.281232	0.140616	+	0.990064i	$0.455092\pi$
$948$	0	0
$949$	0	0
$950$	−2.09407e68	−0.681314
$951$	0	0
$952$	0	0
$953$	6.54075e68	1.97921	0.989603	−	0.143828i	$-0.0459413\pi$
$953$	6.54075e68	1.97921	0.989603	+	0.143828i	$0.0459413\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.00533e68	1.00000
$962$	0	0
$963$	−1.42429e69	−3.38995
$964$	8.81633e67	0.204888
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	4.27094e68	0.902382
$969$	−1.28894e69	−2.65940
$970$	0	0
$971$	4.71659e68	0.928080	0.464040	−	0.885814i	$-0.346399\pi$
$971$	4.71659e68	0.928080	0.464040	+	0.885814i	$0.346399\pi$
$972$	3.09644e69	5.95029
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.35528e68	0.572999	0.286499	−	0.958080i	$-0.407508\pi$
$977$	3.35528e68	0.572999	0.286499	+	0.958080i	$0.407508\pi$
$978$	−2.14177e69	−3.57256
$979$	4.98466e67	0.0812146
$980$	0	0
$981$	0	0
$982$	9.47819e68	1.43933
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−2.49650e69	−3.61779
$985$	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
$993$	−1.58982e69	−1.86860
$994$	0	0
$995$	0	0
$996$	2.30451e69	2.52705
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−9.56525e68	−1.00160
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.47.d.a.3.1	✓	1
8.3	odd	2	CM	8.47.d.a.3.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.47.d.a.3.1	✓	1	1.1	even	1	trivial
8.47.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:	\( k \)	\(=\)	\( 47 \)
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

Learn more