LMFDB - Embedded newform 8.77.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,77,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, 77, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 77);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 77 $
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:	$292.632463664$
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)

$f(q)$

$=$

$q+2.74878e11 q^{2} -2.54028e18 q^{3} +7.55579e22 q^{4} -6.98266e29 q^{6} +2.07692e34 q^{8} +4.62821e36 q^{9} +O(q^{10})$

$q+2.74878e11 q^{2} -2.54028e18 q^{3} +7.55579e22 q^{4} -6.98266e29 q^{6} +2.07692e34 q^{8} +4.62821e36 q^{9} -3.53446e39 q^{11} -1.91938e41 q^{12} +5.70899e45 q^{16} +7.06077e46 q^{17} +1.27219e48 q^{18} +2.41431e48 q^{19} -9.71545e50 q^{22} -5.27595e52 q^{24} +1.32349e53 q^{25} -7.12143e54 q^{27} +1.56928e57 q^{32} +8.97851e57 q^{33} +1.94085e58 q^{34} +3.49697e59 q^{36} +6.63642e59 q^{38} +3.09708e61 q^{41} -2.35387e62 q^{43} -2.67056e62 q^{44} -1.45024e64 q^{48} +1.68831e64 q^{49} +3.63798e64 q^{50} -1.79363e65 q^{51} -1.95752e66 q^{54} -6.13303e66 q^{57} +2.37190e67 q^{59} +4.31359e68 q^{64} +2.46799e69 q^{66} -4.13949e69 q^{67} +5.33497e69 q^{68} +9.61241e70 q^{72} -1.11783e71 q^{73} -3.36203e71 q^{75} +1.82420e71 q^{76} +9.64486e72 q^{81} +8.51320e72 q^{82} +1.26395e73 q^{83} -6.47026e73 q^{86} -7.34079e73 q^{88} -9.73814e73 q^{89} -3.98639e75 q^{96} -5.85560e75 q^{97} +4.64078e75 q^{98} -1.63582e76 q^{99} +O(q^{100})$

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 $

$2$	2.74878e11	1.00000
$2$	2.74878e11	1.00000
$3$	−2.54028e18	−1.88050	−0.940250	−	0.340484i	$-0.889409\pi$
$3$	−2.54028e18	−1.88050	−0.940250	+	0.340484i	$0.889409\pi$
$4$	7.55579e22	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−6.98266e29	−1.88050
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	2.07692e34	1.00000
$9$	4.62821e36	2.53628
$10$	0	0
$11$	−3.53446e39	−0.944934	−0.472467	−	0.881348i	$-0.656636\pi$
$11$	−3.53446e39	−0.944934	−0.472467	+	0.881348i	$0.656636\pi$
$12$	−1.91938e41	−1.88050
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	5.70899e45	1.00000
$17$	7.06077e46	1.23536	0.617680	−	0.786430i	$-0.288073\pi$
$17$	7.06077e46	1.23536	0.617680	+	0.786430i	$0.288073\pi$
$18$	1.27219e48	2.53628
$19$	2.41431e48	0.616818	0.308409	−	0.951254i	$-0.400203\pi$
$19$	2.41431e48	0.616818	0.308409	+	0.951254i	$0.400203\pi$
$20$	0	0
$21$	0	0
$22$	−9.71545e50	−0.944934
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−5.27595e52	−1.88050
$25$	1.32349e53	1.00000
$26$	0	0
$27$	−7.12143e54	−2.88898
$28$	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			$\pi$
$32$	1.56928e57	1.00000
$33$	8.97851e57	1.77695
$34$	1.94085e58	1.23536
$35$	0	0
$36$	3.49697e59	2.53628
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	6.63642e59	0.616818
$39$	0	0
$40$	0	0
$41$	3.09708e61	1.60386	0.801930	−	0.597418i	$-0.203807\pi$
$41$	3.09708e61	1.60386	0.801930	+	0.597418i	$0.203807\pi$
$42$	0	0
$43$	−2.35387e62	−1.99517	−0.997587	−	0.0694293i	$-0.977882\pi$
$43$	−2.35387e62	−1.99517	−0.997587	+	0.0694293i	$0.977882\pi$
$44$	−2.67056e62	−0.944934
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.45024e64	−1.88050
$49$	1.68831e64	1.00000
$50$	3.63798e64	1.00000
$51$	−1.79363e65	−2.32309
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.95752e66	−2.88898
$55$	0	0
$56$	0	0
$57$	−6.13303e66	−1.15993
$58$	0	0
$59$	2.37190e67	1.20982	0.604910	−	0.796294i	$-0.293209\pi$
$59$	2.37190e67	1.20982	0.604910	+	0.796294i	$0.293209\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	4.31359e68	1.00000
$65$	0	0
$66$	2.46799e69	1.77695
$67$	−4.13949e69	−1.68308	−0.841538	−	0.540197i	$-0.818350\pi$
$67$	−4.13949e69	−1.68308	−0.841538	+	0.540197i	$0.818350\pi$
$68$	5.33497e69	1.23536
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	9.61241e70	2.53628
$73$	−1.11783e71	−1.74625	−0.873127	−	0.487493i	$-0.837911\pi$
$73$	−1.11783e71	−1.74625	−0.873127	+	0.487493i	$0.837911\pi$
$74$	0	0
$75$	−3.36203e71	−1.88050
$76$	1.82420e71	0.616818
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	9.64486e72	2.89644
$82$	8.51320e72	1.60386
$83$	1.26395e73	1.50232	0.751158	−	0.660123i	$-0.229496\pi$
$83$	1.26395e73	1.50232	0.751158	+	0.660123i	$0.229496\pi$
$84$	0	0
$85$	0	0
$86$	−6.47026e73	−1.99517
$87$	0	0
$88$	−7.34079e73	−0.944934
$89$	−9.73814e73	−0.815935	−0.407968	−	0.912996i	$-0.633762\pi$
$89$	−9.73814e73	−0.815935	−0.407968	+	0.912996i	$0.633762\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−3.98639e75	−1.88050
$97$	−5.85560e75	−1.86314	−0.931570	−	0.363562i	$-0.881560\pi$
$97$	−5.85560e75	−1.86314	−0.931570	+	0.363562i	$0.881560\pi$
$98$	4.64078e75	1.00000
$99$	−1.63582e76	−2.39662
$100$	1.00000e76	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−4.93030e76	−2.32309
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.43717e77	−1.86338	−0.931691	−	0.363251i	$-0.881667\pi$
$107$	−2.43717e77	−1.86338	−0.931691	+	0.363251i	$0.881667\pi$
$108$	−5.38080e77	−2.88898
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.84602e77	0.466020	0.233010	−	0.972474i	$-0.425143\pi$
$113$	4.84602e77	0.466020	0.233010	+	0.972474i	$0.425143\pi$
$114$	−1.68583e78	−1.15993
$115$	0	0
$116$	0	0
$117$	0	0
$118$	6.51984e78	1.20982
$119$	0	0
$120$	0	0
$121$	−1.49843e78	−0.107101
$122$	0	0
$123$	−7.86745e79	−3.01606
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.18571e80	1.00000
$129$	5.97948e80	3.75193
$130$	0	0
$131$	5.37974e80	1.88127	0.940636	−	0.339416i	$-0.110229\pi$
$131$	5.37974e80	1.88127	0.940636	+	0.339416i	$0.110229\pi$
$132$	6.78397e80	1.77695
$133$	0	0
$134$	−1.13785e81	−1.68308
$135$	0	0
$136$	1.46647e81	1.23536
$137$	2.07073e81	1.32051	0.660254	−	0.751042i	$-0.270449\pi$
$137$	2.07073e81	1.32051	0.660254	+	0.751042i	$0.270449\pi$
$138$	0	0
$139$	−5.20064e81	−1.91203	−0.956015	−	0.293319i	$-0.905240\pi$
$139$	−5.20064e81	−1.91203	−0.956015	+	0.293319i	$0.905240\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	2.64224e82	2.53628
$145$	0	0
$146$	−3.07267e82	−1.74625
$147$	−4.28876e82	−1.88050
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−9.24147e82	−1.88050
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	5.01434e82	0.616818
$153$	3.26787e83	3.13322
$154$	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
$162$	2.65116e84	2.89644
$163$	2.19235e84	1.89575	0.947873	−	0.318648i	$-0.103229\pi$
$163$	2.19235e84	1.89575	0.947873	+	0.318648i	$0.103229\pi$
$164$	2.34009e84	1.60386
$165$	0	0
$166$	3.47431e84	1.50232
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	4.56767e84	1.00000
$170$	0	0
$171$	1.11740e85	1.56442
$172$	−1.77853e85	−1.99517
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−2.01782e85	−0.944934
$177$	−6.02529e85	−2.27507
$178$	−2.67680e85	−0.815935
$179$	7.22739e85	1.78060	0.890300	−	0.455375i	$-0.150495\pi$
$179$	7.22739e85	1.78060	0.890300	+	0.455375i	$0.150495\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.49560e86	−1.16733
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.09577e87	−1.88050
$193$	1.34329e87	1.89230	0.946152	−	0.323724i	$-0.104935\pi$
$193$	1.34329e87	1.89230	0.946152	+	0.323724i	$0.104935\pi$
$194$	−1.60957e87	−1.86314
$195$	0	0
$196$	1.27565e87	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−4.49651e87	−2.39662
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	2.74878e87	1.00000
$201$	1.05154e88	3.16503
$202$	0	0
$203$	0	0
$204$	−1.35523e88	−2.32309
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.53330e87	−0.582852
$210$	0	0
$211$	3.65886e88	1.74026	0.870128	−	0.492826i	$-0.164036\pi$
$211$	3.65886e88	1.74026	0.870128	+	0.492826i	$0.164036\pi$
$212$	0	0
$213$	0	0
$214$	−6.69924e88	−1.86338
$215$	0	0
$216$	−1.47906e89	−2.88898
$217$	0	0
$218$	0	0
$219$	2.83960e89	3.28383
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	6.12538e89	2.53628
$226$	1.33206e89	0.466020
$227$	4.40579e89	1.30329	0.651645	−	0.758524i	$-0.274079\pi$
$227$	4.40579e89	1.30329	0.651645	+	0.758524i	$0.274079\pi$
$228$	−4.63399e89	−1.15993
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.95877e88	−0.0544313	−0.0272156	−	0.999630i	$-0.508664\pi$
$233$	−4.95877e88	−0.0544313	−0.0272156	+	0.999630i	$0.508664\pi$
$234$	0	0
$235$	0	0
$236$	1.79216e90	1.20982
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	6.56813e90	1.99891	0.999457	−	0.0329557i	$-0.0104920\pi$
$241$	6.56813e90	1.99891	0.999457	+	0.0329557i	$0.0104920\pi$
$242$	−4.11885e89	−0.107101
$243$	−1.15054e91	−2.55778
$244$	0	0
$245$	0	0
$246$	−2.16259e91	−3.01606
$247$	0	0
$248$	0	0
$249$	−3.21077e91	−2.82511
$250$	0	0
$251$	−1.03165e91	−0.669777	−0.334888	−	0.942258i	$-0.608699\pi$
$251$	−1.03165e91	−0.669777	−0.334888	+	0.942258i	$0.608699\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	3.25926e91	1.00000
$257$	−4.89549e91	−1.29520	−0.647601	−	0.761980i	$-0.724227\pi$
$257$	−4.89549e91	−1.29520	−0.647601	+	0.761980i	$0.724227\pi$
$258$	1.64363e92	3.75193
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.47877e92	1.88127
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	1.86476e92	1.77695
$265$	0	0
$266$	0	0
$267$	2.47376e92	1.53437
$268$	−3.12771e92	−1.68308
$269$	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			$\pi$
$272$	4.03099e92	1.23536
$273$	0	0
$274$	5.69197e92	1.32051
$275$	−4.67782e92	−0.944934
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−1.42954e93	−1.91203
$279$	0	0
$280$	0	0
$281$	1.38150e93	1.22888	0.614440	−	0.788964i	$-0.289382\pi$
$281$	1.38150e93	1.22888	0.614440	+	0.788964i	$0.289382\pi$
$282$	0	0
$283$	−2.13696e93	−1.45182	−0.725908	−	0.687791i	$-0.758580\pi$
$283$	−2.13696e93	−1.45182	−0.725908	+	0.687791i	$0.758580\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	7.26293e93	2.53628
$289$	1.71869e93	0.526113
$290$	0	0
$291$	1.48748e94	3.50364
$292$	−8.44609e93	−1.74625
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.17889e94	−1.88050
$295$	0	0
$296$	0	0
$297$	2.51704e94	2.72989
$298$	0	0
$299$	0	0
$300$	−2.54028e94	−1.88050
$301$	0	0
$302$	0	0
$303$	0	0
$304$	1.37833e94	0.616818
$305$	0	0
$306$	8.98266e94	3.13322
$307$	6.26891e94	1.93167	0.965836	−	0.259154i	$-0.0834436\pi$
$307$	6.26891e94	1.93167	0.965836	+	0.259154i	$0.0834436\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.32296e95	−1.95371	−0.976857	−	0.213892i	$-0.931386\pi$
$313$	−1.32296e95	−1.95371	−0.976857	+	0.213892i	$0.931386\pi$
$314$	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
$321$	6.19108e95	3.50409
$322$	0	0
$323$	1.70469e95	0.761992
$324$	7.28745e95	2.89644
$325$	0	0
$326$	6.02628e95	1.89575
$327$	0	0
$328$	6.43239e95	1.60386
$329$	0	0
$330$	0	0
$331$	−7.86846e95	−1.38812	−0.694058	−	0.719919i	$-0.744179\pi$
$331$	−7.86846e95	−1.38812	−0.694058	+	0.719919i	$0.744179\pi$
$332$	9.55011e95	1.50232
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.52092e95	−0.224711	−0.112355	−	0.993668i	$-0.535839\pi$
$337$	−2.52092e95	−0.224711	−0.112355	+	0.993668i	$0.535839\pi$
$338$	1.25555e96	1.00000
$339$	−1.23102e96	−0.876350
$340$	0	0
$341$	0	0
$342$	3.07147e96	1.56442
$343$	0	0
$344$	−4.88879e96	−1.99517
$345$	0	0
$346$	0	0
$347$	6.56361e96	1.92585	0.962925	−	0.269769i	$-0.0869472\pi$
$347$	6.56361e96	1.92585	0.962925	+	0.269769i	$0.0869472\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−5.54654e96	−0.944934
$353$	5.30269e96	0.811068	0.405534	−	0.914080i	$-0.367086\pi$
$353$	5.30269e96	0.811068	0.405534	+	0.914080i	$0.367086\pi$
$354$	−1.65622e97	−2.27507
$355$	0	0
$356$	−7.35793e96	−0.815935
$357$	0	0
$358$	1.98665e97	1.78060
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−9.49161e96	−0.619536
$362$	0	0
$363$	3.80643e96	0.201403
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	1.43340e98	4.06784
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−6.85986e97	−1.16733
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−1.07032e98	−1.09957	−0.549785	−	0.835306i	$-0.685290\pi$
$379$	−1.07032e98	−1.09957	−0.549785	+	0.835306i	$0.685290\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−3.01203e98	−1.88050
$385$	0	0
$386$	3.69240e98	1.89230
$387$	−1.08942e99	−5.06032
$388$	−4.42437e98	−1.86314
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	3.50647e98	1.00000
$393$	−1.36660e99	−3.53773
$394$	0	0
$395$	0	0
$396$	−1.23599e99	−2.39662
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	7.55579e98	1.00000
$401$	6.97529e98	0.839607	0.419803	−	0.907615i	$-0.362099\pi$
$401$	6.97529e98	0.839607	0.419803	+	0.907615i	$0.362099\pi$
$402$	2.89046e99	3.16503
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−3.72523e99	−2.32309
$409$	3.38749e99	1.92483	0.962416	−	0.271579i	$-0.0875459\pi$
$409$	3.38749e99	1.92483	0.962416	+	0.271579i	$0.0875459\pi$
$410$	0	0
$411$	−5.26022e99	−2.48322
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.32111e100	3.59557
$418$	−2.34562e99	−0.582852
$419$	−8.80646e99	−1.99834	−0.999168	−	0.0407846i	$-0.987014\pi$
$419$	−8.80646e99	−1.99834	−0.999168	+	0.0407846i	$0.987014\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1.00574e100	1.74026
$423$	0	0
$424$	0	0
$425$	9.34486e99	1.23536
$426$	0	0
$427$	0	0
$428$	−1.84147e100	−1.86338
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−4.06562e100	−2.88898
$433$	2.68884e100	1.74994	0.874972	−	0.484173i	$-0.160879\pi$
$433$	2.68884e100	1.74994	0.874972	+	0.484173i	$0.160879\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	7.80543e100	3.28383
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	7.81383e100	2.53628
$442$	0	0
$443$	2.83615e100	0.775153	0.387576	−	0.921838i	$-0.373312\pi$
$443$	2.83615e100	0.775153	0.387576	+	0.921838i	$0.373312\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.92556e100	−1.46310	−0.731551	−	0.681787i	$-0.761203\pi$
$449$	−8.92556e100	−1.46310	−0.731551	+	0.681787i	$0.761203\pi$
$450$	1.68373e101	2.53628
$451$	−1.09465e101	−1.51554
$452$	3.66155e100	0.466020
$453$	0	0
$454$	1.21105e101	1.30329
$455$	0	0
$456$	−1.27378e101	−1.15993
$457$	1.98929e101	1.66680	0.833399	−	0.552672i	$-0.186392\pi$
$457$	1.98929e101	1.66680	0.833399	+	0.552672i	$0.186392\pi$
$458$	0	0
$459$	−5.02828e101	−3.56893
$460$	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
$466$	−1.36306e100	−0.0544313
$467$	−1.01116e101	−0.372203	−0.186102	−	0.982530i	$-0.559585\pi$
$467$	−1.01116e101	−0.372203	−0.186102	+	0.982530i	$0.559585\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	4.92625e101	1.20982
$473$	8.31966e101	1.88531
$474$	0	0
$475$	3.19532e101	0.616818
$476$	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
$482$	1.80543e102	1.99891
$483$	0	0
$484$	−1.13218e101	−0.107101
$485$	0	0
$486$	−3.16258e102	−2.55778
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−5.56917e102	−3.56495
$490$	0	0
$491$	3.28782e102	1.80223	0.901117	−	0.433575i	$-0.142748\pi$
$491$	3.28782e102	1.80223	0.901117	+	0.433575i	$0.142748\pi$
$492$	−5.94448e102	−3.01606
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−8.82571e102	−2.82511
$499$	4.14336e102	1.22894	0.614471	−	0.788939i	$-0.289369\pi$
$499$	4.14336e102	1.22894	0.614471	+	0.788939i	$0.289369\pi$
$500$	0	0
$501$	0	0
$502$	−2.83577e102	−0.669777
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.16031e103	−1.88050
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	8.95898e102	1.00000
$513$	−1.71934e103	−1.78197
$514$	−1.34566e103	−1.29520
$515$	0	0
$516$	4.51796e103	3.75193
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.88097e103	−1.65846	−0.829230	−	0.558908i	$-0.811221\pi$
$521$	−2.88097e103	−1.65846	−0.829230	+	0.558908i	$0.811221\pi$
$522$	0	0
$523$	2.57517e103	1.28157	0.640783	−	0.767722i	$-0.278610\pi$
$523$	2.57517e103	1.28157	0.640783	+	0.767722i	$0.278610\pi$
$524$	4.06481e103	1.88127
$525$	0	0
$526$	0	0
$527$	0	0
$528$	5.12582e103	1.77695
$529$	3.09967e103	1.00000
$530$	0	0
$531$	1.09777e104	3.06844
$532$	0	0
$533$	0	0
$534$	6.79981e103	1.53437
$535$	0	0
$536$	−8.59738e103	−1.68308
$537$	−1.83596e104	−3.34842
$538$	0	0
$539$	−5.96725e103	−0.944934
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.10803e104	1.23536
$545$	0	0
$546$	0	0
$547$	5.97876e103	0.540869	0.270435	−	0.962738i	$-0.412833\pi$
$547$	5.97876e103	0.540869	0.270435	+	0.962738i	$0.412833\pi$
$548$	1.56460e104	1.32051
$549$	0	0
$550$	−1.28583e104	−0.944934
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−3.92949e104	−1.91203
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	6.33953e104	2.19517
$562$	3.79743e104	1.22888
$563$	−1.15010e104	−0.347869	−0.173934	−	0.984757i	$-0.555648\pi$
$563$	−1.15010e104	−0.347869	−0.173934	+	0.984757i	$0.555648\pi$
$564$	0	0
$565$	0	0
$566$	−5.87403e104	−1.45182
$567$	0	0
$568$	0	0
$569$	9.79174e104	1.97967	0.989837	−	0.142207i	$-0.0454199\pi$
$569$	9.79174e104	1.97967	0.989837	+	0.142207i	$0.0454199\pi$
$570$	0	0
$571$	2.55546e104	0.452164	0.226082	−	0.974108i	$-0.427408\pi$
$571$	2.55546e104	0.452164	0.226082	+	0.974108i	$0.427408\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.99642e105	2.53628
$577$	−1.67791e105	−1.99566	−0.997832	−	0.0658103i	$-0.979037\pi$
$577$	−1.67791e105	−1.99566	−0.997832	+	0.0658103i	$0.979037\pi$
$578$	4.72429e104	0.526113
$579$	−3.41232e105	−3.55848
$580$	0	0
$581$	0	0
$582$	4.08877e105	3.50364
$583$	0	0
$584$	−2.32164e105	−1.74625
$585$	0	0
$586$	0	0
$587$	−3.23040e105	−1.99990	−0.999950	−	0.0100039i	$-0.996816\pi$
$587$	−3.23040e105	−1.99990	−0.999950	+	0.0100039i	$0.996816\pi$
$588$	−3.24050e105	−1.88050
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.18054e105	−1.75855	−0.879273	−	0.476318i	$-0.841971\pi$
$593$	−4.18054e105	−1.75855	−0.879273	+	0.476318i	$0.841971\pi$
$594$	6.91879e105	2.72989
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−6.98266e105	−1.88050
$601$	−1.24067e105	−0.313635	−0.156818	−	0.987628i	$-0.550124\pi$
$601$	−1.24067e105	−0.313635	−0.156818	+	0.987628i	$0.550124\pi$
$602$	0	0
$603$	−1.91584e106	−4.26876
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	3.78872e105	0.616818
$609$	0	0
$610$	0	0
$611$	0	0
$612$	2.46914e106	3.13322
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.72318e106	1.93167
$615$	0	0
$616$	0	0
$617$	1.17744e106	1.09674	0.548368	−	0.836237i	$-0.315249\pi$
$617$	1.17744e106	1.09674	0.548368	+	0.836237i	$0.315249\pi$
$618$	0	0
$619$	−1.42450e106	−1.17333	−0.586663	−	0.809831i	$-0.699559\pi$
$619$	−1.42450e106	−1.17333	−0.586663	+	0.809831i	$0.699559\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.75162e106	1.00000
$626$	−3.63652e106	−1.95371
$627$	2.16770e106	1.09605
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−9.29453e106	−3.27255
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.46812e106	0.976158	0.488079	−	0.872800i	$-0.337698\pi$
$641$	4.46812e106	0.976158	0.488079	+	0.872800i	$0.337698\pi$
$642$	1.70179e107	3.50409
$643$	1.00796e107	1.95625	0.978127	−	0.208006i	$-0.0666975\pi$
$643$	1.00796e107	1.95625	0.978127	+	0.208006i	$0.0666975\pi$
$644$	0	0
$645$	0	0
$646$	4.68583e106	0.761992
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	2.00316e107	2.89644
$649$	−8.38340e106	−1.14320
$650$	0	0
$651$	0	0
$652$	1.65649e107	1.89575
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	1.76812e107	1.60386
$657$	−5.17355e107	−4.42899
$658$	0	0
$659$	−1.30240e107	−0.993340	−0.496670	−	0.867940i	$-0.665444\pi$
$659$	−1.30240e107	−0.993340	−0.496670	+	0.867940i	$0.665444\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−2.16287e107	−1.38812
$663$	0	0
$664$	2.62511e107	1.50232
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−3.30576e107	−1.13422	−0.567112	−	0.823641i	$-0.691939\pi$
$673$	−3.30576e107	−1.13422	−0.567112	+	0.823641i	$0.691939\pi$
$674$	−6.92946e106	−0.224711
$675$	−9.42514e107	−2.88898
$676$	3.45123e107	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−3.38381e107	−0.876350
$679$	0	0
$680$	0	0
$681$	−1.11919e108	−2.45084
$682$	0	0
$683$	−2.41819e107	−0.473701	−0.236851	−	0.971546i	$-0.576115\pi$
$683$	−2.41819e107	−0.473701	−0.236851	+	0.971546i	$0.576115\pi$
$684$	8.44280e107	1.56442
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−1.34382e108	−1.99517
$689$	0	0
$690$	0	0
$691$	1.20070e108	1.51101	0.755505	−	0.655143i	$-0.227392\pi$
$691$	1.20070e108	1.51101	0.755505	+	0.655143i	$0.227392\pi$
$692$	0	0
$693$	0	0
$694$	1.80419e108	1.92585
$695$	0	0
$696$	0	0
$697$	2.18678e108	1.98134
$698$	0	0
$699$	1.25966e107	0.102358
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−1.52462e108	−0.944934
$705$	0	0
$706$	1.45759e108	0.811068
$707$	0	0
$708$	−4.55258e108	−2.27507
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−2.02253e108	−0.815935
$713$	0	0
$714$	0	0
$715$	0	0
$716$	5.46086e108	1.78060
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−2.60903e108	−0.619536
$723$	−1.66849e109	−3.75896
$724$	0	0
$725$	0	0
$726$	1.04630e108	0.201403
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.16270e109	1.91347
$730$	0	0
$731$	−1.66201e109	−2.46476
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.46309e109	1.59040
$738$	3.94009e109	4.06784
$739$	9.62894e108	0.944256	0.472128	−	0.881530i	$-0.343486\pi$
$739$	9.62894e108	0.944256	0.472128	+	0.881530i	$0.343486\pi$
$740$	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
$747$	5.84981e109	3.81030
$748$	−1.88562e109	−1.16733
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	2.62067e109	1.25952
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−2.94207e109	−1.09957
$759$	0	0
$760$	0	0
$761$	−6.21750e109	−1.99986	−0.999928	−	0.0119996i	$-0.996180\pi$
$761$	−6.21750e109	−1.99986	−0.999928	+	0.0119996i	$0.996180\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−8.27942e109	−1.88050
$769$	8.46078e109	1.82898	0.914491	−	0.404607i	$-0.132592\pi$
$769$	8.46078e109	1.82898	0.914491	+	0.404607i	$0.132592\pi$
$770$	0	0
$771$	1.24359e110	2.43563
$772$	1.01496e110	1.89230
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−2.99457e110	−5.06032
$775$	0	0
$776$	−1.21616e110	−1.86314
$777$	0	0
$778$	0	0
$779$	7.47734e109	0.989290
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	9.63852e109	1.00000
$785$	0	0
$786$	−3.75649e110	−3.53773
$787$	−7.40783e109	−0.664739	−0.332369	−	0.943149i	$-0.607848\pi$
$787$	−7.40783e109	−0.664739	−0.332369	+	0.943149i	$0.607848\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−3.39747e110	−2.39662
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	2.07692e110	1.00000
$801$	−4.50701e110	−2.06944
$802$	1.91735e110	0.839607
$803$	3.95093e110	1.65009
$804$	7.94524e110	3.16503
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.11892e110	−0.352173	−0.176086	−	0.984375i	$-0.556344\pi$
$809$	−1.11892e110	−0.352173	−0.176086	+	0.984375i	$0.556344\pi$
$810$	0	0
$811$	3.55129e110	1.01764	0.508819	−	0.860873i	$-0.330082\pi$
$811$	3.55129e110	1.01764	0.508819	+	0.860873i	$0.330082\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−1.02398e111	−2.32309
$817$	−5.68298e110	−1.23066
$818$	9.31147e110	1.92483
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−1.44592e111	−2.48322
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	1.18830e111	1.77695
$826$	0	0
$827$	−7.35170e108	−0.0100272	−0.00501358	−	0.999987i	$-0.501596\pi$
$827$	−7.35170e108	−0.0100272	−0.00501358	+	0.999987i	$0.501596\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.19207e111	1.23536
$834$	3.63143e111	3.59557
$835$	0	0
$836$	−6.44758e110	−0.582852
$837$	0	0
$838$	−2.42070e111	−1.99834
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.38755e111	1.00000
$842$	0	0
$843$	−3.50938e111	−2.31091
$844$	2.76456e111	1.74026
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5.42847e111	2.73014
$850$	2.56869e111	1.23536
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−5.06180e111	−1.86338
$857$	1.74156e111	0.613291	0.306645	−	0.951824i	$-0.400793\pi$
$857$	1.74156e111	0.613291	0.306645	+	0.951824i	$0.400793\pi$
$858$	0	0
$859$	2.77594e110	0.0894685	0.0447342	−	0.998999i	$-0.485756\pi$
$859$	2.77594e110	0.0894685	0.0447342	+	0.998999i	$0.485756\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.11755e112	−2.88898
$865$	0	0
$866$	7.39102e111	1.74994
$867$	−4.36594e111	−0.989356
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.71009e112	−4.72545
$874$	0	0
$875$	0	0
$876$	2.14554e112	3.28383
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.40520e111	−0.296529	−0.148265	−	0.988948i	$-0.547369\pi$
$881$	−2.40520e111	−0.296529	−0.148265	+	0.988948i	$0.547369\pi$
$882$	2.14785e112	2.53628
$883$	−1.47564e112	−1.66907	−0.834536	−	0.550953i	$-0.814264\pi$
$883$	−1.47564e112	−1.66907	−0.834536	+	0.550953i	$0.814264\pi$
$884$	0	0
$885$	0	0
$886$	7.79595e111	0.775153
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.40894e112	−2.73695
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−2.45344e112	−1.46310
$899$	0	0
$900$	4.62821e112	2.53628
$901$	0	0
$902$	−3.00896e112	−1.51554
$903$	0	0
$904$	1.00648e112	0.466020
$905$	0	0
$906$	0	0
$907$	4.86938e112	1.98791	0.993954	−	0.109796i	$-0.0350197\pi$
$907$	4.86938e112	1.98791	0.993954	+	0.109796i	$0.0350197\pi$
$908$	3.32892e112	1.30329
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−3.50134e112	−1.15993
$913$	−4.46737e112	−1.41959
$914$	5.46812e112	1.66680
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−1.38216e113	−3.56893
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1.59248e113	−3.63251
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.27339e112	1.03016	0.515080	−	0.857142i	$-0.327762\pi$
$929$	6.27339e112	1.03016	0.515080	+	0.857142i	$0.327762\pi$
$930$	0	0
$931$	4.07610e112	0.616818
$932$	−3.74674e111	−0.0544313
$933$	0	0
$934$	−2.77947e112	−0.372203
$935$	0	0
$936$	0	0
$937$	1.21502e113	1.44039	0.720195	−	0.693772i	$-0.244052\pi$
$937$	1.21502e113	1.44039	0.720195	+	0.693772i	$0.244052\pi$
$938$	0	0
$939$	3.36068e113	3.67396
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	1.35412e113	1.20982
$945$	0	0
$946$	2.28689e113	1.88531
$947$	1.37249e113	1.08695	0.543475	−	0.839425i	$-0.317108\pi$
$947$	1.37249e113	1.08695	0.543475	+	0.839425i	$0.317108\pi$
$948$	0	0
$949$	0	0
$950$	8.78323e112	0.616818
$951$	0	0
$952$	0	0
$953$	3.05942e113	1.90594	0.952970	−	0.303065i	$-0.0980099\pi$
$953$	3.05942e113	1.90594	0.952970	+	0.303065i	$0.0980099\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.20541e113	1.00000
$962$	0	0
$963$	−1.12797e114	−4.72606
$964$	4.96274e113	1.99891
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−3.11212e112	−0.107101
$969$	−4.33039e113	−1.43293
$970$	0	0
$971$	−2.87723e113	−0.880326	−0.440163	−	0.897918i	$-0.645079\pi$
$971$	−2.87723e113	−0.880326	−0.440163	+	0.897918i	$0.645079\pi$
$972$	−8.69325e113	−2.55778
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.30634e113	1.04259	0.521297	−	0.853376i	$-0.325449\pi$
$977$	4.30634e113	1.04259	0.521297	+	0.853376i	$0.325449\pi$
$978$	−1.53084e114	−3.56495
$979$	3.44191e113	0.771004
$980$	0	0
$981$	0	0
$982$	9.03749e113	1.80223
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−1.63401e114	−3.01606
$985$	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
$993$	1.99881e114	2.61035
$994$	0	0
$995$	0	0
$996$	−2.42599e114	−2.82511
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.13892e114	1.22894
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.77.d.a.3.1	✓	1
8.3	odd	2	CM	8.77.d.a.3.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.77.d.a.3.1	✓	1	1.1	even	1	trivial
8.77.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:	\( k \)	\(=\)	\( 77 \)
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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