LMFDB - Embedded newform 8.69.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,69,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, 69, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 69);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 69 $
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:	$234.271461498$
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)

$f(q)$

$=$

$q+1.71799e10 q^{2} -1.76857e16 q^{3} +2.95148e20 q^{4} -3.03839e26 q^{6} +5.07060e30 q^{8} +3.46568e31 q^{9} +O(q^{10})$

$q+1.71799e10 q^{2} -1.76857e16 q^{3} +2.95148e20 q^{4} -3.03839e26 q^{6} +5.07060e30 q^{8} +3.46568e31 q^{9} +5.58858e34 q^{11} -5.21991e36 q^{12} +8.71123e40 q^{16} +5.70998e41 q^{17} +5.95399e41 q^{18} -6.00376e43 q^{19} +9.60111e44 q^{22} -8.96773e46 q^{24} +3.38813e47 q^{25} +4.30597e48 q^{27} +1.49658e51 q^{32} -9.88382e50 q^{33} +9.80967e51 q^{34} +1.02289e52 q^{36} -1.03144e54 q^{38} -2.32817e54 q^{41} -5.17253e55 q^{43} +1.64946e55 q^{44} -1.54064e57 q^{48} +2.92864e57 q^{49} +5.82077e57 q^{50} -1.00985e58 q^{51} +7.39761e58 q^{54} +1.06181e60 q^{57} -1.77680e60 q^{59} +2.57110e61 q^{64} -1.69803e61 q^{66} +1.94794e62 q^{67} +1.68529e62 q^{68} +1.75731e62 q^{72} -5.53564e62 q^{73} -5.99216e63 q^{75} -1.77200e64 q^{76} -8.57933e64 q^{81} -3.99977e64 q^{82} -1.37113e65 q^{83} -8.88633e65 q^{86} +2.83375e65 q^{88} -1.07109e66 q^{89} -2.64681e67 q^{96} +5.70563e66 q^{97} +5.03137e67 q^{98} +1.93682e66 q^{99} +O(q^{100})$

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 $

$2$	1.71799e10	1.00000
$2$	1.71799e10	1.00000
$3$	−1.76857e16	−1.06047	−0.530237	−	0.847849i	$-0.677897\pi$
$3$	−1.76857e16	−1.06047	−0.530237	+	0.847849i	$0.677897\pi$
$4$	2.95148e20	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−3.03839e26	−1.06047
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	5.07060e30	1.00000
$9$	3.46568e31	0.124607
$10$	0	0
$11$	5.58858e34	0.218751	0.109376	−	0.994000i	$-0.465115\pi$
$11$	5.58858e34	0.218751	0.109376	+	0.994000i	$0.465115\pi$
$12$	−5.21991e36	−1.06047
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	8.71123e40	1.00000
$17$	5.70998e41	0.834394	0.417197	−	0.908816i	$-0.363013\pi$
$17$	5.70998e41	0.834394	0.417197	+	0.908816i	$0.363013\pi$
$18$	5.95399e41	0.124607
$19$	−6.00376e43	−1.99894	−0.999472	−	0.0324918i	$-0.989656\pi$
$19$	−6.00376e43	−1.99894	−0.999472	+	0.0324918i	$0.989656\pi$
$20$	0	0
$21$	0	0
$22$	9.60111e44	0.218751
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−8.96773e46	−1.06047
$25$	3.38813e47	1.00000
$26$	0	0
$27$	4.30597e48	0.928332
$28$	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			$\pi$
$32$	1.49658e51	1.00000
$33$	−9.88382e50	−0.231980
$34$	9.80967e51	0.834394
$35$	0	0
$36$	1.02289e52	0.124607
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.03144e54	−1.99894
$39$	0	0
$40$	0	0
$41$	−2.32817e54	−0.340694	−0.170347	−	0.985384i	$-0.554489\pi$
$41$	−2.32817e54	−0.340694	−0.170347	+	0.985384i	$0.554489\pi$
$42$	0	0
$43$	−5.17253e55	−1.49891	−0.749454	−	0.662056i	$-0.769684\pi$
$43$	−5.17253e55	−1.49891	−0.749454	+	0.662056i	$0.769684\pi$
$44$	1.64946e55	0.218751
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.54064e57	−1.06047
$49$	2.92864e57	1.00000
$50$	5.82077e57	1.00000
$51$	−1.00985e58	−0.884854
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	7.39761e58	0.928332
$55$	0	0
$56$	0	0
$57$	1.06181e60	2.11983
$58$	0	0
$59$	−1.77680e60	−1.09817	−0.549084	−	0.835767i	$-0.685023\pi$
$59$	−1.77680e60	−1.09817	−0.549084	+	0.835767i	$0.685023\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	2.57110e61	1.00000
$65$	0	0
$66$	−1.69803e61	−0.231980
$67$	1.94794e62	1.59600	0.798000	−	0.602658i	$-0.205892\pi$
$67$	1.94794e62	1.59600	0.798000	+	0.602658i	$0.205892\pi$
$68$	1.68529e62	0.834394
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.75731e62	0.124607
$73$	−5.53564e62	−0.245579	−0.122789	−	0.992433i	$-0.539184\pi$
$73$	−5.53564e62	−0.245579	−0.122789	+	0.992433i	$0.539184\pi$
$74$	0	0
$75$	−5.99216e63	−1.06047
$76$	−1.77200e64	−1.99894
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−8.57933e64	−1.10908
$82$	−3.99977e64	−0.340694
$83$	−1.37113e65	−0.773433	−0.386717	−	0.922199i	$-0.626391\pi$
$83$	−1.37113e65	−0.773433	−0.386717	+	0.922199i	$0.626391\pi$
$84$	0	0
$85$	0	0
$86$	−8.88633e65	−1.49891
$87$	0	0
$88$	2.83375e65	0.218751
$89$	−1.07109e66	−0.563074	−0.281537	−	0.959550i	$-0.590844\pi$
$89$	−1.07109e66	−0.563074	−0.281537	+	0.959550i	$0.590844\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−2.64681e67	−1.06047
$97$	5.70563e66	0.160718	0.0803590	−	0.996766i	$-0.474393\pi$
$97$	5.70563e66	0.160718	0.0803590	+	0.996766i	$0.474393\pi$
$98$	5.03137e67	1.00000
$99$	1.93682e66	0.0272580
$100$	1.00000e68	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−1.73491e68	−0.884854
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.98107e68	0.799857	0.399929	−	0.916546i	$-0.369035\pi$
$107$	7.98107e68	0.799857	0.399929	+	0.916546i	$0.369035\pi$
$108$	1.27090e69	0.928332
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.14384e70	1.79349	0.896745	−	0.442547i	$-0.145925\pi$
$113$	1.14384e70	1.79349	0.896745	+	0.442547i	$0.145925\pi$
$114$	1.82417e70	2.11983
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−3.05251e70	−1.09817
$119$	0	0
$120$	0	0
$121$	−6.21451e70	−0.952148
$122$	0	0
$123$	4.11754e70	0.361297
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	4.41712e71	1.00000
$129$	9.14799e71	1.58956
$130$	0	0
$131$	5.15884e71	0.531286	0.265643	−	0.964072i	$-0.414416\pi$
$131$	5.15884e71	0.531286	0.265643	+	0.964072i	$0.414416\pi$
$132$	−2.91719e71	−0.231980
$133$	0	0
$134$	3.34654e72	1.59600
$135$	0	0
$136$	2.89530e72	0.834394
$137$	3.02629e72	0.679848	0.339924	−	0.940453i	$-0.389599\pi$
$137$	3.02629e72	0.679848	0.339924	+	0.940453i	$0.389599\pi$
$138$	0	0
$139$	1.44982e73	1.98980	0.994902	−	0.100842i	$-0.0321536\pi$
$139$	1.44982e73	1.98980	0.994902	+	0.100842i	$0.0321536\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	3.01903e72	0.124607
$145$	0	0
$146$	−9.51016e72	−0.245579
$147$	−5.17952e73	−1.06047
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−1.02945e74	−1.06047
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−3.04427e74	−1.99894
$153$	1.97889e73	0.103971
$154$	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
$162$	−1.47392e75	−1.10908
$163$	1.90174e75	1.16084	0.580419	−	0.814318i	$-0.302889\pi$
$163$	1.90174e75	1.16084	0.580419	+	0.814318i	$0.302889\pi$
$164$	−6.87155e74	−0.340694
$165$	0	0
$166$	−2.35558e75	−0.773433
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	5.59948e75	1.00000
$170$	0	0
$171$	−2.08071e75	−0.249083
$172$	−1.52666e76	−1.49891
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	4.86834e75	0.218751
$177$	3.14239e76	1.16458
$178$	−1.84012e76	−0.563074
$179$	7.87371e76	1.99148	0.995740	−	0.0922005i	$-0.0293901\pi$
$179$	7.87371e76	1.99148	0.995740	+	0.0922005i	$0.0293901\pi$
$180$	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
$187$	3.19107e76	0.182525
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−4.54718e77	−1.06047
$193$	7.84514e77	1.53339	0.766694	−	0.642013i	$-0.221900\pi$
$193$	7.84514e77	1.53339	0.766694	+	0.642013i	$0.221900\pi$
$194$	9.80220e76	0.160718
$195$	0	0
$196$	8.64383e77	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	3.32744e76	0.0272580
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.71799e78	1.00000
$201$	−3.44508e78	−1.69252
$202$	0	0
$203$	0	0
$204$	−2.98056e78	−0.884854
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.35525e78	−0.437271
$210$	0	0
$211$	−1.98121e79	−1.86779	−0.933893	−	0.357552i	$-0.883611\pi$
$211$	−1.98121e79	−1.86779	−0.933893	+	0.357552i	$0.883611\pi$
$212$	0	0
$213$	0	0
$214$	1.37114e79	0.799857
$215$	0	0
$216$	2.18339e79	0.928332
$217$	0	0
$218$	0	0
$219$	9.79019e78	0.260430
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	1.17422e79	0.124607
$226$	1.96511e80	1.79349
$227$	2.53118e80	1.98813	0.994063	−	0.108805i	$-0.0347025\pi$
$227$	2.53118e80	1.98813	0.994063	+	0.108805i	$0.0347025\pi$
$228$	3.13391e80	2.11983
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.14666e80	1.98856	0.994278	−	0.106823i	$-0.0340679\pi$
$233$	6.14666e80	1.98856	0.994278	+	0.106823i	$0.0340679\pi$
$234$	0	0
$235$	0	0
$236$	−5.24417e80	−1.09817
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−1.03554e80	−0.106313	−0.0531564	−	0.998586i	$-0.516928\pi$
$241$	−1.03554e80	−0.106313	−0.0531564	+	0.998586i	$0.516928\pi$
$242$	−1.06764e81	−0.952148
$243$	3.19705e80	0.247820
$244$	0	0
$245$	0	0
$246$	7.07389e80	0.361297
$247$	0	0
$248$	0	0
$249$	2.42494e81	0.820207
$250$	0	0
$251$	7.18379e81	1.85117	0.925586	−	0.378536i	$-0.123572\pi$
$251$	7.18379e81	1.85117	0.925586	+	0.378536i	$0.123572\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	7.58855e81	1.00000
$257$	1.42049e82	1.63950	0.819750	−	0.572721i	$-0.194112\pi$
$257$	1.42049e82	1.63950	0.819750	+	0.572721i	$0.194112\pi$
$258$	1.57161e82	1.58956
$259$	0	0
$260$	0	0
$261$	0	0
$262$	8.86282e81	0.531286
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−5.01169e81	−0.231980
$265$	0	0
$266$	0	0
$267$	1.89430e82	0.597126
$268$	5.74931e82	1.59600
$269$	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			$\pi$
$272$	4.97409e82	0.834394
$273$	0	0
$274$	5.19913e82	0.679848
$275$	1.89349e82	0.218751
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	2.49077e83	1.98980
$279$	0	0
$280$	0	0
$281$	2.40685e83	1.33486	0.667428	−	0.744674i	$-0.267395\pi$
$281$	2.40685e83	1.33486	0.667428	+	0.744674i	$0.267395\pi$
$282$	0	0
$283$	−1.04990e83	−0.457517	−0.228759	−	0.973483i	$-0.573467\pi$
$283$	−1.04990e83	−0.457517	−0.228759	+	0.973483i	$0.573467\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	5.18666e82	0.124607
$289$	−1.42264e83	−0.303787
$290$	0	0
$291$	−1.00908e83	−0.170437
$292$	−1.63383e83	−0.245579
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−8.89835e83	−1.06047
$295$	0	0
$296$	0	0
$297$	2.40643e83	0.203074
$298$	0	0
$299$	0	0
$300$	−1.76857e84	−1.06047
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−5.23001e84	−1.99894
$305$	0	0
$306$	3.39972e83	0.103971
$307$	−4.40994e84	−1.20706	−0.603528	−	0.797341i	$-0.706239\pi$
$307$	−4.40994e84	−1.20706	−0.603528	+	0.797341i	$0.706239\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.13693e85	1.61149	0.805743	−	0.592266i	$-0.201766\pi$
$313$	1.13693e85	1.61149	0.805743	+	0.592266i	$0.201766\pi$
$314$	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
$321$	−1.41151e85	−0.848229
$322$	0	0
$323$	−3.42813e85	−1.66791
$324$	−2.53217e85	−1.10908
$325$	0	0
$326$	3.26716e85	1.16084
$327$	0	0
$328$	−1.18052e85	−0.340694
$329$	0	0
$330$	0	0
$331$	−7.10616e85	−1.50482	−0.752408	−	0.658698i	$-0.771108\pi$
$331$	−7.10616e85	−1.50482	−0.752408	+	0.658698i	$0.771108\pi$
$332$	−4.04686e85	−0.773433
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.74952e85	0.201142	0.100571	−	0.994930i	$-0.467933\pi$
$337$	1.74952e85	0.201142	0.100571	+	0.994930i	$0.467933\pi$
$338$	9.61984e85	1.00000
$339$	−2.02297e86	−1.90195
$340$	0	0
$341$	0	0
$342$	−3.57463e85	−0.249083
$343$	0	0
$344$	−2.62278e86	−1.49891
$345$	0	0
$346$	0	0
$347$	1.70221e84	0.00724119	0.00362059	−	0.999993i	$-0.498848\pi$
$347$	1.70221e84	0.00724119	0.00362059	+	0.999993i	$0.498848\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	8.36375e85	0.218751
$353$	8.41447e86	1.99842	0.999210	−	0.0397482i	$-0.0126556\pi$
$353$	8.41447e86	1.99842	0.999210	+	0.0397482i	$0.0126556\pi$
$354$	5.39859e86	1.16458
$355$	0	0
$356$	−3.16130e86	−0.563074
$357$	0	0
$358$	1.35269e87	1.99148
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	2.70243e87	2.99578
$362$	0	0
$363$	1.09908e87	1.00973
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−8.06870e85	−0.0424529
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	5.48222e86	0.182525
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−6.78135e87	−1.43742	−0.718709	−	0.695311i	$-0.755266\pi$
$379$	−6.78135e87	−1.43742	−0.718709	+	0.695311i	$0.755266\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−7.81200e87	−1.06047
$385$	0	0
$386$	1.34778e88	1.53339
$387$	−1.79263e87	−0.186775
$388$	1.68400e87	0.160718
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.48500e88	1.00000
$393$	−9.12379e87	−0.563415
$394$	0	0
$395$	0	0
$396$	5.71649e86	0.0272580
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	2.95148e88	1.00000
$401$	−4.47244e88	−1.39199	−0.695994	−	0.718048i	$-0.745036\pi$
$401$	−4.47244e88	−1.39199	−0.695994	+	0.718048i	$0.745036\pi$
$402$	−5.91860e88	−1.69252
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−5.12056e88	−0.884854
$409$	2.04628e88	0.325366	0.162683	−	0.986678i	$-0.447985\pi$
$409$	2.04628e88	0.325366	0.162683	+	0.986678i	$0.447985\pi$
$410$	0	0
$411$	−5.35222e88	−0.720962
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.56411e89	−2.11014
$418$	−5.76428e88	−0.437271
$419$	2.56295e89	1.79251	0.896257	−	0.443535i	$-0.146276\pi$
$419$	2.56295e89	1.79251	0.896257	+	0.443535i	$0.146276\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−3.40369e89	−1.86779
$423$	0	0
$424$	0	0
$425$	1.93462e89	0.834394
$426$	0	0
$427$	0	0
$428$	2.35559e89	0.799857
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	3.75103e89	0.928332
$433$	−8.38490e89	−1.91827	−0.959135	−	0.282950i	$-0.908687\pi$
$433$	−8.38490e89	−1.91827	−0.959135	+	0.282950i	$0.908687\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.68194e89	0.260430
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.01497e89	0.124607
$442$	0	0
$443$	−2.65011e89	−0.278957	−0.139478	−	0.990225i	$-0.544543\pi$
$443$	−2.65011e89	−0.278957	−0.139478	+	0.990225i	$0.544543\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.84877e90	−1.89794	−0.948968	−	0.315371i	$-0.897871\pi$
$449$	−2.84877e90	−1.89794	−0.948968	+	0.315371i	$0.897871\pi$
$450$	2.01729e89	0.124607
$451$	−1.30112e89	−0.0745272
$452$	3.37603e90	1.79349
$453$	0	0
$454$	4.34853e90	1.98813
$455$	0	0
$456$	5.38401e90	2.11983
$457$	4.88362e90	1.78481	0.892403	−	0.451239i	$-0.149018\pi$
$457$	4.88362e90	1.78481	0.892403	+	0.451239i	$0.149018\pi$
$458$	0	0
$459$	2.45870e90	0.774595
$460$	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
$466$	1.05599e91	1.98856
$467$	−3.73237e90	−0.653448	−0.326724	−	0.945120i	$-0.605945\pi$
$467$	−3.73237e90	−0.653448	−0.326724	+	0.945120i	$0.605945\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−9.00942e90	−1.09817
$473$	−2.89071e90	−0.327888
$474$	0	0
$475$	−2.03415e91	−1.99894
$476$	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
$482$	−1.77904e90	−0.106313
$483$	0	0
$484$	−1.83420e91	−0.952148
$485$	0	0
$486$	5.49248e90	0.247820
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−3.36336e91	−1.23104
$490$	0	0
$491$	−6.24952e91	−1.99102	−0.995511	−	0.0946484i	$-0.969827\pi$
$491$	−6.24952e91	−1.99102	−0.995511	+	0.0946484i	$0.969827\pi$
$492$	1.21528e91	0.361297
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	4.16602e91	0.820207
$499$	−1.03324e92	−1.90013	−0.950067	−	0.312047i	$-0.898985\pi$
$499$	−1.03324e92	−1.90013	−0.950067	+	0.312047i	$0.898985\pi$
$500$	0	0
$501$	0	0
$502$	1.23417e92	1.85117
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.90310e91	−1.06047
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.30370e92	1.00000
$513$	−2.58520e92	−1.85568
$514$	2.44038e92	1.63950
$515$	0	0
$516$	2.70001e92	1.58956
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.22216e92	−1.79082	−0.895411	−	0.445240i	$-0.853118\pi$
$521$	−4.22216e92	−1.79082	−0.895411	+	0.445240i	$0.853118\pi$
$522$	0	0
$523$	−4.09099e92	−1.52325	−0.761625	−	0.648018i	$-0.775598\pi$
$523$	−4.09099e92	−1.52325	−0.761625	+	0.648018i	$0.775598\pi$
$524$	1.52262e92	0.531286
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−8.61002e91	−0.231980
$529$	3.95816e92	1.00000
$530$	0	0
$531$	−6.15780e91	−0.136840
$532$	0	0
$533$	0	0
$534$	3.25438e92	0.597126
$535$	0	0
$536$	9.87724e92	1.59600
$537$	−1.39252e93	−2.11192
$538$	0	0
$539$	1.63670e92	0.218751
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	8.54543e92	0.834394
$545$	0	0
$546$	0	0
$547$	2.43697e93	1.97370	0.986851	−	0.161633i	$-0.0516761\pi$
$547$	2.43697e93	1.97370	0.986851	+	0.161633i	$0.0516761\pi$
$548$	8.93204e92	0.679848
$549$	0	0
$550$	3.25298e92	0.218751
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	4.27911e93	1.98980
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−5.64364e92	−0.193563
$562$	4.13494e93	1.33486
$563$	−5.47734e93	−1.66450	−0.832251	−	0.554399i	$-0.812948\pi$
$563$	−5.47734e93	−1.66450	−0.832251	+	0.554399i	$0.812948\pi$
$564$	0	0
$565$	0	0
$566$	−1.80371e93	−0.457517
$567$	0	0
$568$	0	0
$569$	−5.45564e93	−1.15619	−0.578094	−	0.815970i	$-0.696203\pi$
$569$	−5.45564e93	−1.15619	−0.578094	+	0.815970i	$0.696203\pi$
$570$	0	0
$571$	4.69516e93	0.883126	0.441563	−	0.897230i	$-0.354424\pi$
$571$	4.69516e93	0.883126	0.441563	+	0.897230i	$0.354424\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	8.91061e92	0.124607
$577$	−1.51444e94	−1.99653	−0.998264	−	0.0588914i	$-0.981243\pi$
$577$	−1.51444e94	−1.99653	−0.998264	+	0.0588914i	$0.981243\pi$
$578$	−2.44408e93	−0.303787
$579$	−1.38747e94	−1.62612
$580$	0	0
$581$	0	0
$582$	−1.73359e93	−0.170437
$583$	0	0
$584$	−2.80691e93	−0.245579
$585$	0	0
$586$	0	0
$587$	−1.50856e94	−1.10884	−0.554420	−	0.832237i	$-0.687060\pi$
$587$	−1.50856e94	−1.10884	−0.554420	+	0.832237i	$0.687060\pi$
$588$	−1.52873e94	−1.06047
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.15249e94	1.63981	0.819905	−	0.572499i	$-0.194026\pi$
$593$	3.15249e94	1.63981	0.819905	+	0.572499i	$0.194026\pi$
$594$	4.13421e93	0.203074
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−3.03839e94	−1.06047
$601$	2.75489e94	0.908600	0.454300	−	0.890849i	$-0.349889\pi$
$601$	2.75489e94	0.908600	0.454300	+	0.890849i	$0.349889\pi$
$602$	0	0
$603$	6.75094e93	0.198873
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−8.98509e94	−1.99894
$609$	0	0
$610$	0	0
$611$	0	0
$612$	5.84067e93	0.103971
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−7.57622e94	−1.20706
$615$	0	0
$616$	0	0
$617$	1.47327e95	1.98879	0.994393	−	0.105750i	$-0.0337243\pi$
$617$	1.47327e95	1.98879	0.994393	+	0.105750i	$0.0337243\pi$
$618$	0	0
$619$	1.57360e95	1.90288	0.951439	−	0.307837i	$-0.0996052\pi$
$619$	1.57360e95	1.90288	0.951439	+	0.307837i	$0.0996052\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.14794e95	1.00000
$626$	1.95323e95	1.61149
$627$	5.93401e94	0.463715
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	3.50392e95	1.98074
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.65406e95	−1.71656	−0.858280	−	0.513182i	$-0.828467\pi$
$641$	−4.65406e95	−1.71656	−0.858280	+	0.513182i	$0.828467\pi$
$642$	−2.42496e95	−0.848229
$643$	2.54317e95	0.843728	0.421864	−	0.906659i	$-0.361376\pi$
$643$	2.54317e95	0.843728	0.421864	+	0.906659i	$0.361376\pi$
$644$	0	0
$645$	0	0
$646$	−5.88949e95	−1.66791
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−4.35024e95	−1.10908
$649$	−9.92977e94	−0.240226
$650$	0	0
$651$	0	0
$652$	5.61293e95	1.16084
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−2.02812e95	−0.340694
$657$	−1.91848e94	−0.0306009
$658$	0	0
$659$	9.93264e95	1.42876	0.714382	−	0.699756i	$-0.246708\pi$
$659$	9.93264e95	1.42876	0.714382	+	0.699756i	$0.246708\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−1.22083e96	−1.50482
$663$	0	0
$664$	−6.95245e95	−0.773433
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	2.30898e96	1.62520	0.812601	−	0.582820i	$-0.198051\pi$
$673$	2.30898e96	1.62520	0.812601	+	0.582820i	$0.198051\pi$
$674$	3.00565e95	0.201142
$675$	1.45892e96	0.928332
$676$	1.65268e96	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−3.47543e96	−1.90195
$679$	0	0
$680$	0	0
$681$	−4.47658e96	−2.10836
$682$	0	0
$683$	3.79845e96	1.61921	0.809604	−	0.586977i	$-0.199682\pi$
$683$	3.79845e96	1.61921	0.809604	+	0.586977i	$0.199682\pi$
$684$	−6.14117e95	−0.249083
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−4.50591e96	−1.49891
$689$	0	0
$690$	0	0
$691$	−6.84856e96	−1.96492	−0.982462	−	0.186462i	$-0.940298\pi$
$691$	−6.84856e96	−1.96492	−0.982462	+	0.186462i	$0.940298\pi$
$692$	0	0
$693$	0	0
$694$	2.92437e94	0.00724119
$695$	0	0
$696$	0	0
$697$	−1.32938e96	−0.284273
$698$	0	0
$699$	−1.08708e97	−2.10881
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.43688e96	0.218751
$705$	0	0
$706$	1.44559e97	1.99842
$707$	0	0
$708$	9.27471e96	1.16458
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−5.43107e96	−0.563074
$713$	0	0
$714$	0	0
$715$	0	0
$716$	2.32391e97	1.99148
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	4.64274e97	2.99578
$723$	1.83142e96	0.112742
$724$	0	0
$725$	0	0
$726$	1.88821e97	1.00973
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.82074e97	0.846274
$730$	0	0
$731$	−2.95350e97	−1.25068
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.08862e97	0.349127
$738$	−1.38619e96	−0.0424529
$739$	−2.57948e97	−0.754435	−0.377217	−	0.926125i	$-0.623119\pi$
$739$	−2.57948e97	−0.754435	−0.377217	+	0.926125i	$0.623119\pi$
$740$	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
$747$	−4.75189e96	−0.0963753
$748$	9.41838e96	0.182525
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−1.27051e98	−1.96312
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−1.16503e98	−1.43742
$759$	0	0
$760$	0	0
$761$	1.72930e97	0.186549	0.0932745	−	0.995640i	$-0.470267\pi$
$761$	1.72930e97	0.186549	0.0932745	+	0.995640i	$0.470267\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−1.34209e98	−1.06047
$769$	−2.27181e98	−1.71742	−0.858710	−	0.512461i	$-0.828734\pi$
$769$	−2.27181e98	−1.71742	−0.858710	+	0.512461i	$0.828734\pi$
$770$	0	0
$771$	−2.51224e98	−1.73865
$772$	2.31548e98	1.53339
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−3.07972e97	−0.186775
$775$	0	0
$776$	2.89310e97	0.160718
$777$	0	0
$778$	0	0
$779$	1.39778e98	0.681028
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	2.55121e98	1.00000
$785$	0	0
$786$	−1.56746e98	−0.563415
$787$	4.77452e98	1.64357	0.821786	−	0.569797i	$-0.192978\pi$
$787$	4.77452e98	1.64357	0.821786	+	0.569797i	$0.192978\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	9.82086e96	0.0272580
$793$	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
$800$	5.07060e98	1.00000
$801$	−3.71205e97	−0.0701630
$802$	−7.68359e98	−1.39199
$803$	−3.09364e97	−0.0537207
$804$	−1.01681e99	−1.69252
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.19306e98	1.23940	0.619698	−	0.784840i	$-0.287255\pi$
$809$	9.19306e98	1.23940	0.619698	+	0.784840i	$0.287255\pi$
$810$	0	0
$811$	1.33419e99	1.65390	0.826951	−	0.562274i	$-0.190073\pi$
$811$	1.33419e99	1.65390	0.826951	+	0.562274i	$0.190073\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−8.79705e98	−0.884854
$817$	3.10546e99	2.99623
$818$	3.51547e98	0.325366
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−9.19505e98	−0.720962
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−3.34877e98	−0.231980
$826$	0	0
$827$	1.03118e99	0.657878	0.328939	−	0.944351i	$-0.393309\pi$
$827$	1.03118e99	0.657878	0.328939	+	0.944351i	$0.393309\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.67225e99	0.834394
$834$	−4.40511e99	−2.11014
$835$	0	0
$836$	−9.90295e98	−0.437271
$837$	0	0
$838$	4.40311e99	1.79251
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.77373e99	1.00000
$842$	0	0
$843$	−4.25670e99	−1.41558
$844$	−5.84750e99	−1.86779
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.85682e99	0.485186
$850$	3.32364e99	0.834394
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	4.04688e99	0.799857
$857$	−2.89815e99	−0.550519	−0.275260	−	0.961370i	$-0.588764\pi$
$857$	−2.89815e99	−0.550519	−0.275260	+	0.961370i	$0.588764\pi$
$858$	0	0
$859$	−1.02456e100	−1.79792	−0.898960	−	0.438030i	$-0.855676\pi$
$859$	−1.02456e100	−1.79792	−0.898960	+	0.438030i	$0.855676\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	6.44422e99	0.928332
$865$	0	0
$866$	−1.44052e100	−1.91827
$867$	2.51605e99	0.322158
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.97739e98	0.0200266
$874$	0	0
$875$	0	0
$876$	2.88955e99	0.260430
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.35723e100	−1.00803	−0.504013	−	0.863696i	$-0.668144\pi$
$881$	−1.35723e100	−1.00803	−0.504013	+	0.863696i	$0.668144\pi$
$882$	1.74371e99	0.124607
$883$	−1.91359e100	−1.31579	−0.657893	−	0.753111i	$-0.728552\pi$
$883$	−1.91359e100	−1.31579	−0.657893	+	0.753111i	$0.728552\pi$
$884$	0	0
$885$	0	0
$886$	−4.55285e99	−0.278957
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.79463e99	−0.242613
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−4.89415e100	−1.89794
$899$	0	0
$900$	3.46568e99	0.124607
$901$	0	0
$902$	−2.23531e99	−0.0745272
$903$	0	0
$904$	5.79997e100	1.79349
$905$	0	0
$906$	0	0
$907$	2.45813e100	0.679136	0.339568	−	0.940582i	$-0.389719\pi$
$907$	2.45813e100	0.679136	0.339568	+	0.940582i	$0.389719\pi$
$908$	7.47073e100	1.98813
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	9.24966e100	2.11983
$913$	−7.66267e99	−0.169189
$914$	8.38999e100	1.78481
$915$	0	0
$916$	0	0
$917$	0	0
$918$	4.22402e100	0.774595
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	7.79930e100	1.28005
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.88311e100	0.352636	0.176318	−	0.984333i	$-0.443581\pi$
$929$	2.88311e100	0.352636	0.176318	+	0.984333i	$0.443581\pi$
$930$	0	0
$931$	−1.75829e101	−1.99894
$932$	1.81417e101	1.98856
$933$	0	0
$934$	−6.41217e100	−0.653448
$935$	0	0
$936$	0	0
$937$	−1.52152e101	−1.39037	−0.695187	−	0.718829i	$-0.744678\pi$
$937$	−1.52152e101	−1.39037	−0.695187	+	0.718829i	$0.744678\pi$
$938$	0	0
$939$	−2.01074e101	−1.70894
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−1.54781e101	−1.09817
$945$	0	0
$946$	−4.96620e100	−0.327888
$947$	−3.13344e101	−1.99583	−0.997913	−	0.0645788i	$-0.979430\pi$
$947$	−3.13344e101	−1.99583	−0.997913	+	0.0645788i	$0.979430\pi$
$948$	0	0
$949$	0	0
$950$	−3.49465e101	−1.99894
$951$	0	0
$952$	0	0
$953$	3.19866e101	1.64365	0.821825	−	0.569740i	$-0.192956\pi$
$953$	3.19866e101	1.64365	0.821825	+	0.569740i	$0.192956\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.58580e101	1.00000
$962$	0	0
$963$	2.76598e100	0.0996679
$964$	−3.05636e100	−0.106313
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−3.15113e101	−0.952148
$969$	6.06290e101	1.76877
$970$	0	0
$971$	−4.05608e101	−1.10320	−0.551598	−	0.834110i	$-0.685982\pi$
$971$	−4.05608e101	−1.10320	−0.551598	+	0.834110i	$0.685982\pi$
$972$	9.43601e100	0.247820
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.77704e101	1.93612	0.968062	−	0.250711i	$-0.0806645\pi$
$977$	8.77704e101	1.93612	0.968062	+	0.250711i	$0.0806645\pi$
$978$	−5.77821e101	−1.23104
$979$	−5.98587e100	−0.123173
$980$	0	0
$981$	0	0
$982$	−1.07366e102	−1.99102
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	2.08784e101	0.361297
$985$	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
$993$	1.25678e102	1.59582
$994$	0	0
$995$	0	0
$996$	7.15716e101	0.820207
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.77510e102	−1.90013
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.69.d.a.3.1	✓	1
8.3	odd	2	CM	8.69.d.a.3.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.69.d.a.3.1	✓	1	1.1	even	1	trivial
8.69.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:	\( k \)	\(=\)	\( 69 \)
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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