LMFDB - Embedded newform 8.41.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,41,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, 41, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 41);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 41 $
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:	$81.0738281572$
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)

$f(q)$

$=$

$q+1.04858e6 q^{2} +6.07401e9 q^{3} +1.09951e12 q^{4} +6.36906e15 q^{6} +1.15292e18 q^{8} +2.47359e19 q^{9} +O(q^{10})$

$q+1.04858e6 q^{2} +6.07401e9 q^{3} +1.09951e12 q^{4} +6.36906e15 q^{6} +1.15292e18 q^{8} +2.47359e19 q^{9} +4.50821e20 q^{11} +6.67844e21 q^{12} +1.20893e24 q^{16} +3.11774e24 q^{17} +2.59375e25 q^{18} -7.41504e25 q^{19} +4.72720e26 q^{22} +7.00286e27 q^{24} +9.09495e27 q^{25} +7.64005e28 q^{27} +1.26765e30 q^{32} +2.73829e30 q^{33} +3.26919e30 q^{34} +2.71974e31 q^{36} -7.77524e31 q^{38} +2.85743e32 q^{41} -9.26378e32 q^{43} +4.95683e32 q^{44} +7.34303e33 q^{48} +6.36681e33 q^{49} +9.53674e33 q^{50} +1.89372e34 q^{51} +8.01118e34 q^{54} -4.50390e35 q^{57} -4.92235e35 q^{59} +1.32923e36 q^{64} +2.87130e36 q^{66} -6.51864e36 q^{67} +3.42799e36 q^{68} +2.85186e37 q^{72} -7.38290e35 q^{73} +5.52428e37 q^{75} -8.15293e37 q^{76} +1.63326e38 q^{81} +2.99623e38 q^{82} +4.80923e38 q^{83} -9.71377e38 q^{86} +5.19761e38 q^{88} +1.80114e39 q^{89} +7.69972e39 q^{96} -8.37497e39 q^{97} +6.67608e39 q^{98} +1.11515e40 q^{99} +O(q^{100})$

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 $

$2$	1.04858e6	1.00000
$2$	1.04858e6	1.00000
$3$	6.07401e9	1.74201	0.871005	−	0.491275i	$-0.163469\pi$
$3$	6.07401e9	1.74201	0.871005	+	0.491275i	$0.163469\pi$
$4$	1.09951e12	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	6.36906e15	1.74201
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.15292e18	1.00000
$9$	2.47359e19	2.03460
$10$	0	0
$11$	4.50821e20	0.670116	0.335058	−	0.942197i	$-0.391244\pi$
$11$	4.50821e20	0.670116	0.335058	+	0.942197i	$0.391244\pi$
$12$	6.67844e21	1.74201
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.20893e24	1.00000
$17$	3.11774e24	0.767117	0.383558	−	0.923517i	$-0.374699\pi$
$17$	3.11774e24	0.767117	0.383558	+	0.923517i	$0.374699\pi$
$18$	2.59375e25	2.03460
$19$	−7.41504e25	−1.97261	−0.986306	−	0.164926i	$-0.947261\pi$
$19$	−7.41504e25	−1.97261	−0.986306	+	0.164926i	$0.947261\pi$
$20$	0	0
$21$	0	0
$22$	4.72720e26	0.670116
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	7.00286e27	1.74201
$25$	9.09495e27	1.00000
$26$	0	0
$27$	7.64005e28	1.80228
$28$	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			$\pi$
$32$	1.26765e30	1.00000
$33$	2.73829e30	1.16735
$34$	3.26919e30	0.767117
$35$	0	0
$36$	2.71974e31	2.03460
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−7.77524e31	−1.97261
$39$	0	0
$40$	0	0
$41$	2.85743e32	1.58598	0.792991	−	0.609234i	$-0.208523\pi$
$41$	2.85743e32	1.58598	0.792991	+	0.609234i	$0.208523\pi$
$42$	0	0
$43$	−9.26378e32	−1.98344	−0.991720	−	0.128420i	$-0.959009\pi$
$43$	−9.26378e32	−1.98344	−0.991720	+	0.128420i	$0.959009\pi$
$44$	4.95683e32	0.670116
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	7.34303e33	1.74201
$49$	6.36681e33	1.00000
$50$	9.53674e33	1.00000
$51$	1.89372e34	1.33632
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	8.01118e34	1.80228
$55$	0	0
$56$	0	0
$57$	−4.50390e35	−3.43631
$58$	0	0
$59$	−4.92235e35	−1.88422	−0.942112	−	0.335299i	$-0.891163\pi$
$59$	−4.92235e35	−1.88422	−0.942112	+	0.335299i	$0.891163\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.32923e36	1.00000
$65$	0	0
$66$	2.87130e36	1.16735
$67$	−6.51864e36	−1.96183	−0.980914	−	0.194441i	$-0.937711\pi$
$67$	−6.51864e36	−1.96183	−0.980914	+	0.194441i	$0.937711\pi$
$68$	3.42799e36	0.767117
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	2.85186e37	2.03460
$73$	−7.38290e35	−0.0399733	−0.0199866	−	0.999800i	$-0.506362\pi$
$73$	−7.38290e35	−0.0399733	−0.0199866	+	0.999800i	$0.506362\pi$
$74$	0	0
$75$	5.52428e37	1.74201
$76$	−8.15293e37	−1.97261
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.63326e38	1.10498
$82$	2.99623e38	1.58598
$83$	4.80923e38	1.99762	0.998811	−	0.0487509i	$-0.0155240\pi$
$83$	4.80923e38	1.99762	0.998811	+	0.0487509i	$0.0155240\pi$
$84$	0	0
$85$	0	0
$86$	−9.71377e38	−1.98344
$87$	0	0
$88$	5.19761e38	0.670116
$89$	1.80114e39	1.85245	0.926227	−	0.376966i	$-0.123033\pi$
$89$	1.80114e39	1.85245	0.926227	+	0.376966i	$0.123033\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	7.69972e39	1.74201
$97$	−8.37497e39	−1.54010	−0.770050	−	0.637984i	$-0.779769\pi$
$97$	−8.37497e39	−1.54010	−0.770050	+	0.637984i	$0.779769\pi$
$98$	6.67608e39	1.00000
$99$	1.11515e40	1.36342
$100$	1.00000e40	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	1.98571e40	1.33632
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.41190e40	1.39854	0.699269	−	0.714858i	$-0.253509\pi$
$107$	5.41190e40	1.39854	0.699269	+	0.714858i	$0.253509\pi$
$108$	8.40033e40	1.80228
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.07069e41	−1.79699	−0.898494	−	0.438985i	$-0.855338\pi$
$113$	−2.07069e41	−1.79699	−0.898494	+	0.438985i	$0.855338\pi$
$114$	−4.72269e41	−3.43631
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−5.16146e41	−1.88422
$119$	0	0
$120$	0	0
$121$	−2.49353e41	−0.550944
$122$	0	0
$123$	1.73560e42	2.76279
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.39380e42	1.00000
$129$	−5.62683e42	−3.45517
$130$	0	0
$131$	2.91139e41	0.131424	0.0657121	−	0.997839i	$-0.479068\pi$
$131$	2.91139e41	0.131424	0.0657121	+	0.997839i	$0.479068\pi$
$132$	3.01078e42	1.16735
$133$	0	0
$134$	−6.83529e42	−1.96183
$135$	0	0
$136$	3.59451e42	0.767117
$137$	−6.37113e42	−1.17437	−0.587185	−	0.809453i	$-0.699764\pi$
$137$	−6.37113e42	−1.17437	−0.587185	+	0.809453i	$0.699764\pi$
$138$	0	0
$139$	−8.03485e42	−1.10836	−0.554181	−	0.832396i	$-0.686968\pi$
$139$	−8.03485e42	−1.10836	−0.554181	+	0.832396i	$0.686968\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	2.99039e43	2.03460
$145$	0	0
$146$	−7.74153e41	−0.0399733
$147$	3.86720e43	1.74201
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	5.79263e43	1.74201
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−8.54896e43	−1.97261
$153$	7.71202e43	1.56077
$154$	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
$162$	1.71260e44	1.10498
$163$	−3.50571e44	−1.99997	−0.999986	−	0.00533843i	$-0.998301\pi$
$163$	−3.50571e44	−1.99997	−0.999986	+	0.00533843i	$0.998301\pi$
$164$	3.14178e44	1.58598
$165$	0	0
$166$	5.04284e44	1.99762
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	3.61189e44	1.00000
$170$	0	0
$171$	−1.83418e45	−4.01347
$172$	−1.01856e45	−1.98344
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	5.45009e44	0.670116
$177$	−2.98984e45	−3.28233
$178$	1.88863e45	1.85245
$179$	−7.42345e44	−0.650946	−0.325473	−	0.945551i	$-0.605524\pi$
$179$	−7.42345e44	−0.650946	−0.325473	+	0.945551i	$0.605524\pi$
$180$	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
$187$	1.40554e45	0.514057
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	8.07374e45	1.74201
$193$	4.20788e45	0.818309	0.409154	−	0.912465i	$-0.365824\pi$
$193$	4.20788e45	0.818309	0.409154	+	0.912465i	$0.365824\pi$
$194$	−8.78180e45	−1.54010
$195$	0	0
$196$	7.00038e45	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	1.16932e46	1.36342
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.04858e46	1.00000
$201$	−3.95943e46	−3.41752
$202$	0	0
$203$	0	0
$204$	2.08217e46	1.33632
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.34285e46	−1.32188
$210$	0	0
$211$	1.85199e46	0.605326	0.302663	−	0.953098i	$-0.402124\pi$
$211$	1.85199e46	0.605326	0.302663	+	0.953098i	$0.402124\pi$
$212$	0	0
$213$	0	0
$214$	5.67479e46	1.39854
$215$	0	0
$216$	8.80838e46	1.80228
$217$	0	0
$218$	0	0
$219$	−4.48438e45	−0.0696338
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	2.24972e47	2.03460
$226$	−2.17127e47	−1.79699
$227$	−2.15061e47	−1.62946	−0.814732	−	0.579837i	$-0.803116\pi$
$227$	−2.15061e47	−1.62946	−0.814732	+	0.579837i	$0.803116\pi$
$228$	−4.95210e47	−3.43631
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.72814e47	0.777071	0.388535	−	0.921434i	$-0.372981\pi$
$233$	1.72814e47	0.777071	0.388535	+	0.921434i	$0.372981\pi$
$234$	0	0
$235$	0	0
$236$	−5.41218e47	−1.88422
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−3.64795e47	−0.835042	−0.417521	−	0.908667i	$-0.637101\pi$
$241$	−3.64795e47	−0.835042	−0.417521	+	0.908667i	$0.637101\pi$
$242$	−2.61466e47	−0.550944
$243$	6.31943e46	0.122617
$244$	0	0
$245$	0	0
$246$	1.81991e48	2.76279
$247$	0	0
$248$	0	0
$249$	2.92113e48	3.47988
$250$	0	0
$251$	4.56130e47	0.463035	0.231518	−	0.972831i	$-0.425631\pi$
$251$	4.56130e47	0.463035	0.231518	+	0.972831i	$0.425631\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.46150e48	1.00000
$257$	−8.97804e47	−0.568223	−0.284112	−	0.958791i	$-0.591699\pi$
$257$	−8.97804e47	−0.568223	−0.284112	+	0.958791i	$0.591699\pi$
$258$	−5.90016e48	−3.45517
$259$	0	0
$260$	0	0
$261$	0	0
$262$	3.05282e47	0.131424
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	3.15703e48	1.16735
$265$	0	0
$266$	0	0
$267$	1.09401e49	3.22699
$268$	−7.16732e48	−1.96183
$269$	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			$\pi$
$272$	3.76912e48	0.767117
$273$	0	0
$274$	−6.68061e48	−1.17437
$275$	4.10019e48	0.670116
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−8.42515e48	−1.10836
$279$	0	0
$280$	0	0
$281$	−1.46645e49	−1.55647	−0.778236	−	0.627972i	$-0.783885\pi$
$281$	−1.46645e49	−1.55647	−0.778236	+	0.627972i	$0.783885\pi$
$282$	0	0
$283$	−2.11240e49	−1.94557	−0.972785	−	0.231708i	$-0.925569\pi$
$283$	−2.11240e49	−1.94557	−0.972785	+	0.231708i	$0.925569\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	3.13565e49	2.03460
$289$	−6.79768e48	−0.411532
$290$	0	0
$291$	−5.08697e49	−2.68287
$292$	−8.11758e47	−0.0399733
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	4.05506e49	1.74201
$295$	0	0
$296$	0	0
$297$	3.44429e49	1.20773
$298$	0	0
$299$	0	0
$300$	6.07401e49	1.74201
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−8.96424e49	−1.97261
$305$	0	0
$306$	8.08664e49	1.56077
$307$	−1.03595e50	−1.87315	−0.936575	−	0.350467i	$-0.886023\pi$
$307$	−1.03595e50	−1.87315	−0.936575	+	0.350467i	$0.886023\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	7.30857e49	0.897316	0.448658	−	0.893704i	$-0.351902\pi$
$313$	7.30857e49	0.897316	0.448658	+	0.893704i	$0.351902\pi$
$314$	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
$321$	3.28720e50	2.43627
$322$	0	0
$323$	−2.31182e50	−1.51322
$324$	1.79579e50	1.10498
$325$	0	0
$326$	−3.67600e50	−1.99997
$327$	0	0
$328$	3.29439e50	1.58598
$329$	0	0
$330$	0	0
$331$	4.54486e50	1.82373	0.911865	−	0.410491i	$-0.134643\pi$
$331$	4.54486e50	1.82373	0.911865	+	0.410491i	$0.134643\pi$
$332$	5.28780e50	1.99762
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.12631e50	−1.99649	−0.998245	−	0.0592248i	$-0.981137\pi$
$337$	−7.12631e50	−1.99649	−0.998245	+	0.0592248i	$0.981137\pi$
$338$	3.78734e50	1.00000
$339$	−1.25774e51	−3.13037
$340$	0	0
$341$	0	0
$342$	−1.92328e51	−4.01347
$343$	0	0
$344$	−1.06804e51	−1.98344
$345$	0	0
$346$	0	0
$347$	1.25889e51	1.96516	0.982580	−	0.185843i	$-0.0595015\pi$
$347$	1.25889e51	1.96516	0.982580	+	0.185843i	$0.0595015\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	5.71483e50	0.670116
$353$	1.36213e51	1.50912	0.754561	−	0.656229i	$-0.227850\pi$
$353$	1.36213e51	1.50912	0.754561	+	0.656229i	$0.227850\pi$
$354$	−3.13508e51	−3.28233
$355$	0	0
$356$	1.98037e51	1.85245
$357$	0	0
$358$	−7.78405e50	−0.650946
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	4.08528e51	2.89120
$362$	0	0
$363$	−1.51457e51	−0.959750
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	7.06812e51	3.22683
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	1.47382e51	0.514057
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−6.01155e51	−1.60769	−0.803844	−	0.594840i	$-0.797216\pi$
$379$	−6.01155e51	−1.60769	−0.803844	+	0.594840i	$0.797216\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	8.46593e51	1.74201
$385$	0	0
$386$	4.41228e51	0.818309
$387$	−2.29148e52	−4.03550
$388$	−9.20838e51	−1.54010
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	7.34043e51	1.00000
$393$	1.76838e51	0.228942
$394$	0	0
$395$	0	0
$396$	1.22612e52	1.36342
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.09951e52	1.00000
$401$	2.22909e52	1.92859	0.964296	−	0.264828i	$-0.0853153\pi$
$401$	2.22909e52	1.92859	0.964296	+	0.264828i	$0.0853153\pi$
$402$	−4.15176e52	−3.41752
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	2.18331e52	1.33632
$409$	−1.81735e52	−1.05919	−0.529594	−	0.848251i	$-0.677656\pi$
$409$	−1.81735e52	−1.05919	−0.529594	+	0.848251i	$0.677656\pi$
$410$	0	0
$411$	−3.86983e52	−2.04576
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.88037e52	−1.93078
$418$	−3.50524e52	−1.32188
$419$	4.46235e52	1.60429	0.802146	−	0.597128i	$-0.203692\pi$
$419$	4.46235e52	1.60429	0.802146	+	0.597128i	$0.203692\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1.94195e52	0.605326
$423$	0	0
$424$	0	0
$425$	2.83557e52	0.767117
$426$	0	0
$427$	0	0
$428$	5.95045e52	1.39854
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	9.23626e52	1.80228
$433$	−4.93848e52	−0.920101	−0.460051	−	0.887893i	$-0.652169\pi$
$433$	−4.93848e52	−0.920101	−0.460051	+	0.887893i	$0.652169\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−4.70221e51	−0.0696338
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	1.57489e53	2.03460
$442$	0	0
$443$	−1.52462e53	−1.79923	−0.899613	−	0.436688i	$-0.856151\pi$
$443$	−1.52462e53	−1.79923	−0.899613	+	0.436688i	$0.856151\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.32965e53	1.19898	0.599490	−	0.800382i	$-0.295370\pi$
$449$	1.32965e53	1.19898	0.599490	+	0.800382i	$0.295370\pi$
$450$	2.35900e53	2.03460
$451$	1.28819e53	1.06279
$452$	−2.27674e53	−1.79699
$453$	0	0
$454$	−2.25508e53	−1.62946
$455$	0	0
$456$	−5.19265e53	−3.43631
$457$	1.66712e53	1.05595	0.527974	−	0.849260i	$-0.322952\pi$
$457$	1.66712e53	1.05595	0.527974	+	0.849260i	$0.322952\pi$
$458$	0	0
$459$	2.38197e53	1.38256
$460$	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
$466$	1.81209e53	0.777071
$467$	3.63345e53	1.49273	0.746365	−	0.665536i	$-0.231797\pi$
$467$	3.63345e53	1.49273	0.746365	+	0.665536i	$0.231797\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−5.67509e53	−1.88422
$473$	−4.17630e53	−1.32914
$474$	0	0
$475$	−6.74394e53	−1.97261
$476$	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
$482$	−3.82516e53	−0.835042
$483$	0	0
$484$	−2.74167e53	−0.550944
$485$	0	0
$486$	6.62640e52	0.122617
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−2.12937e54	−3.48397
$490$	0	0
$491$	1.20816e54	1.82178	0.910891	−	0.412647i	$-0.135396\pi$
$491$	1.20816e54	1.82178	0.910891	+	0.412647i	$0.135396\pi$
$492$	1.90832e54	2.76279
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	3.06303e54	3.47988
$499$	1.80191e54	1.96663	0.983316	−	0.181906i	$-0.0582267\pi$
$499$	1.80191e54	1.96663	0.983316	+	0.181906i	$0.0582267\pi$
$500$	0	0
$501$	0	0
$502$	4.78287e53	0.463035
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.19386e54	1.74201
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.53250e54	1.00000
$513$	−5.66513e54	−3.55519
$514$	−9.41416e53	−0.568223
$515$	0	0
$516$	−6.18676e54	−3.45517
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.30751e54	1.06265	0.531323	−	0.847169i	$-0.321695\pi$
$521$	2.30751e54	1.06265	0.531323	+	0.847169i	$0.321695\pi$
$522$	0	0
$523$	2.21725e53	0.0945758	0.0472879	−	0.998881i	$-0.484942\pi$
$523$	2.21725e53	0.0945758	0.0472879	+	0.998881i	$0.484942\pi$
$524$	3.20111e53	0.131424
$525$	0	0
$526$	0	0
$527$	0	0
$528$	3.31039e54	1.16735
$529$	2.94519e54	1.00000
$530$	0	0
$531$	−1.21759e55	−3.83363
$532$	0	0
$533$	0	0
$534$	1.14716e55	3.22699
$535$	0	0
$536$	−7.51548e54	−1.96183
$537$	−4.50901e54	−1.13395
$538$	0	0
$539$	2.87029e54	0.670116
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	3.95220e54	0.767117
$545$	0	0
$546$	0	0
$547$	6.08533e54	1.05813	0.529063	−	0.848582i	$-0.322543\pi$
$547$	6.08533e54	1.05813	0.529063	+	0.848582i	$0.322543\pi$
$548$	−7.00513e54	−1.17437
$549$	0	0
$550$	4.29936e54	0.670116
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−8.83441e54	−1.10836
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.53727e54	0.895493
$562$	−1.53769e55	−1.55647
$563$	−1.27256e55	−1.24311	−0.621556	−	0.783370i	$-0.713499\pi$
$563$	−1.27256e55	−1.24311	−0.621556	+	0.783370i	$0.713499\pi$
$564$	0	0
$565$	0	0
$566$	−2.21501e55	−1.94557
$567$	0	0
$568$	0	0
$569$	1.53934e55	1.21644	0.608219	−	0.793769i	$-0.291884\pi$
$569$	1.53934e55	1.21644	0.608219	+	0.793769i	$0.291884\pi$
$570$	0	0
$571$	2.60509e55	1.91911	0.959556	−	0.281518i	$-0.0908378\pi$
$571$	2.60509e55	1.91911	0.959556	+	0.281518i	$0.0908378\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	3.28797e55	2.03460
$577$	3.34415e55	1.99880	0.999399	−	0.0346551i	$-0.0110333\pi$
$577$	3.34415e55	1.99880	0.999399	+	0.0346551i	$0.0110333\pi$
$578$	−7.12788e54	−0.411532
$579$	2.55587e55	1.42550
$580$	0	0
$581$	0	0
$582$	−5.33407e55	−2.68287
$583$	0	0
$584$	−8.51190e53	−0.0399733
$585$	0	0
$586$	0	0
$587$	1.18236e55	0.501172	0.250586	−	0.968094i	$-0.419377\pi$
$587$	1.18236e55	0.501172	0.250586	+	0.968094i	$0.419377\pi$
$588$	4.25204e55	1.74201
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.88257e55	−1.68871	−0.844354	−	0.535786i	$-0.820015\pi$
$593$	−4.88257e55	−1.68871	−0.844354	+	0.535786i	$0.820015\pi$
$594$	3.61160e55	1.20773
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	6.36906e55	1.74201
$601$	−1.54380e52	−0.000408416 0	−0.000204208	−	1.00000i	$-0.500065\pi$
$601$	−1.54380e52	−0.000408416 0	−0.000204208		1.00000i	$0.500065\pi$
$602$	0	0
$603$	−1.61245e56	−3.99153
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−9.39968e55	−1.97261
$609$	0	0
$610$	0	0
$611$	0	0
$612$	8.47946e55	1.56077
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−1.08628e56	−1.87315
$615$	0	0
$616$	0	0
$617$	8.89411e55	1.39123	0.695615	−	0.718415i	$-0.255132\pi$
$617$	8.89411e55	1.39123	0.695615	+	0.718415i	$0.255132\pi$
$618$	0	0
$619$	−1.36409e56	−2.00000	−1.00000		0.000729572i	$-0.999768\pi$
$619$	−1.36409e56	−2.00000	−1.00000		0.000729572i	$0.999768\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.27181e55	1.00000
$626$	7.66359e55	0.897316
$627$	−2.03045e56	−2.30273
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	1.12490e56	1.05448
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.96326e55	0.288996	0.144498	−	0.989505i	$-0.453843\pi$
$641$	3.96326e55	0.288996	0.144498	+	0.989505i	$0.453843\pi$
$642$	3.44687e56	2.43627
$643$	2.63985e56	1.80867	0.904335	−	0.426823i	$-0.140367\pi$
$643$	2.63985e56	1.80867	0.904335	+	0.426823i	$0.140367\pi$
$644$	0	0
$645$	0	0
$646$	−2.42412e56	−1.51322
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1.88303e56	1.10498
$649$	−2.21910e56	−1.26265
$650$	0	0
$651$	0	0
$652$	−3.85457e56	−1.99997
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	3.45442e56	1.58598
$657$	−1.82623e55	−0.0813295
$658$	0	0
$659$	−9.05452e55	−0.379452	−0.189726	−	0.981837i	$-0.560760\pi$
$659$	−9.05452e55	−0.379452	−0.189726	+	0.981837i	$0.560760\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	4.76563e56	1.82373
$663$	0	0
$664$	5.54466e56	1.99762
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6.96535e55	0.191709	0.0958543	−	0.995395i	$-0.469442\pi$
$673$	6.96535e55	0.191709	0.0958543	+	0.995395i	$0.469442\pi$
$674$	−7.47248e56	−1.99649
$675$	6.94859e56	1.80228
$676$	3.97131e56	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	−1.31883e57	−3.13037
$679$	0	0
$680$	0	0
$681$	−1.30628e57	−2.83854
$682$	0	0
$683$	9.09874e56	1.86452	0.932260	−	0.361790i	$-0.117834\pi$
$683$	9.09874e56	1.86452	0.932260	+	0.361790i	$0.117834\pi$
$684$	−2.01670e57	−4.01347
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−1.11992e57	−1.98344
$689$	0	0
$690$	0	0
$691$	2.48044e56	0.402687	0.201343	−	0.979521i	$-0.435469\pi$
$691$	2.48044e56	0.402687	0.201343	+	0.979521i	$0.435469\pi$
$692$	0	0
$693$	0	0
$694$	1.32004e57	1.96516
$695$	0	0
$696$	0	0
$697$	8.90872e56	1.21663
$698$	0	0
$699$	1.04967e57	1.35366
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	5.99244e56	0.670116
$705$	0	0
$706$	1.42830e57	1.50912
$707$	0	0
$708$	−3.28737e57	−3.28233
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	2.07657e57	1.85245
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−8.16217e56	−0.650946
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	4.28373e57	2.89120
$723$	−2.21577e57	−1.45465
$724$	0	0
$725$	0	0
$726$	−1.58815e57	−0.959750
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.60183e57	−0.891384
$730$	0	0
$731$	−2.88820e57	−1.52153
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.93874e57	−1.31465
$738$	7.41146e57	3.22683
$739$	−1.48433e57	−0.628987	−0.314494	−	0.949260i	$-0.601835\pi$
$739$	−1.48433e57	−0.628987	−0.314494	+	0.949260i	$0.601835\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.18961e58	4.06435
$748$	1.54541e57	0.514057
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	2.77054e57	0.806612
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−6.30356e57	−1.60769
$759$	0	0
$760$	0	0
$761$	−3.35979e57	−0.791807	−0.395904	−	0.918292i	$-0.629569\pi$
$761$	−3.35979e57	−0.791807	−0.395904	+	0.918292i	$0.629569\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	8.87718e57	1.74201
$769$	−8.58898e57	−1.64216	−0.821078	−	0.570815i	$-0.806627\pi$
$769$	−8.58898e57	−1.64216	−0.821078	+	0.570815i	$0.806627\pi$
$770$	0	0
$771$	−5.45327e57	−0.989850
$772$	4.62661e57	0.818309
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−2.40279e58	−4.03550
$775$	0	0
$776$	−9.65569e57	−1.54010
$777$	0	0
$778$	0	0
$779$	−2.11879e58	−3.12853
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	7.69700e57	1.00000
$785$	0	0
$786$	1.85428e57	0.228942
$787$	1.66145e58	1.99983	0.999916	−	0.0129761i	$-0.00413054\pi$
$787$	1.66145e58	1.99983	0.999916	+	0.0129761i	$0.00413054\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	1.28568e58	1.36342
$793$	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
$800$	1.15292e58	1.00000
$801$	4.45529e58	3.76899
$802$	2.33737e58	1.92859
$803$	−3.32836e56	−0.0267867
$804$	−4.35344e58	−3.41752
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.34597e58	−0.933393	−0.466696	−	0.884418i	$-0.654556\pi$
$809$	−1.34597e58	−0.933393	−0.466696	+	0.884418i	$0.654556\pi$
$810$	0	0
$811$	−1.78134e58	−1.17579	−0.587895	−	0.808937i	$-0.700043\pi$
$811$	−1.78134e58	−1.17579	−0.587895	+	0.808937i	$0.700043\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	2.28936e58	1.33632
$817$	6.86913e58	3.91256
$818$	−1.90563e58	−1.05919
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−4.05781e58	−2.04576
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	2.49046e58	1.16735
$826$	0	0
$827$	3.58059e57	0.159899	0.0799493	−	0.996799i	$-0.474524\pi$
$827$	3.58059e57	0.159899	0.0799493	+	0.996799i	$0.474524\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.98500e58	0.767117
$834$	−5.11744e58	−1.93078
$835$	0	0
$836$	−3.67551e58	−1.32188
$837$	0	0
$838$	4.67911e58	1.60429
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	3.13271e58	1.00000
$842$	0	0
$843$	−8.90724e58	−2.71139
$844$	2.03628e58	0.605326
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.28307e59	−3.38920
$850$	2.97331e58	0.767117
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	6.23950e58	1.39854
$857$	−9.01109e58	−1.97315	−0.986576	−	0.163300i	$-0.947786\pi$
$857$	−9.01109e58	−1.97315	−0.986576	+	0.163300i	$0.947786\pi$
$858$	0	0
$859$	8.52113e58	1.78088	0.890438	−	0.455104i	$-0.150398\pi$
$859$	8.52113e58	1.78088	0.890438	+	0.455104i	$0.150398\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	9.68492e58	1.80228
$865$	0	0
$866$	−5.17837e58	−0.920101
$867$	−4.12892e58	−0.716893
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.07163e59	−3.13348
$874$	0	0
$875$	0	0
$876$	−4.93063e57	−0.0696338
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.54002e59	1.94092	0.970459	−	0.241266i	$-0.0775625\pi$
$881$	1.54002e59	1.94092	0.970459	+	0.241266i	$0.0775625\pi$
$882$	1.65139e59	2.03460
$883$	3.69558e58	0.445111	0.222556	−	0.974920i	$-0.428560\pi$
$883$	3.69558e58	0.445111	0.222556	+	0.974920i	$0.428560\pi$
$884$	0	0
$885$	0	0
$886$	−1.59868e59	−1.79923
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.36309e58	0.740468
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	1.39424e59	1.19898
$899$	0	0
$900$	2.47359e59	2.03460
$901$	0	0
$902$	1.35076e59	1.06279
$903$	0	0
$904$	−2.38734e59	−1.79699
$905$	0	0
$906$	0	0
$907$	2.62739e59	1.85088	0.925441	−	0.378891i	$-0.123695\pi$
$907$	2.62739e59	1.85088	0.925441	+	0.378891i	$0.123695\pi$
$908$	−2.36462e59	−1.62946
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−5.44489e59	−3.43631
$913$	2.16810e59	1.33864
$914$	1.74810e59	1.05595
$915$	0	0
$916$	0	0
$917$	0	0
$918$	2.49768e59	1.38256
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−6.29239e59	−3.26304
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.12832e59	1.80077	0.900387	−	0.435089i	$-0.143283\pi$
$929$	4.12832e59	1.80077	0.900387	+	0.435089i	$0.143283\pi$
$930$	0	0
$931$	−4.72101e59	−1.97261
$932$	1.90011e59	0.777071
$933$	0	0
$934$	3.80995e59	1.49273
$935$	0	0
$936$	0	0
$937$	−5.35254e57	−0.0196684	−0.00983418	−	0.999952i	$-0.503130\pi$
$937$	−5.35254e57	−0.0196684	−0.00983418	+	0.999952i	$0.503130\pi$
$938$	0	0
$939$	4.43923e59	1.56313
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−5.95076e59	−1.88422
$945$	0	0
$946$	−4.37917e59	−1.32914
$947$	3.65848e58	0.108718	0.0543590	−	0.998521i	$-0.482688\pi$
$947$	3.65848e58	0.108718	0.0543590	+	0.998521i	$0.482688\pi$
$948$	0	0
$949$	0	0
$950$	−7.07154e59	−1.97261
$951$	0	0
$952$	0	0
$953$	6.05563e58	0.158599	0.0792996	−	0.996851i	$-0.474732\pi$
$953$	6.05563e58	0.158599	0.0792996	+	0.996851i	$0.474732\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4.51303e59	1.00000
$962$	0	0
$963$	1.33868e60	2.84546
$964$	−4.01097e59	−0.835042
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−2.87485e59	−0.550944
$969$	−1.40420e60	−2.63605
$970$	0	0
$971$	−5.20451e59	−0.937552	−0.468776	−	0.883317i	$-0.655305\pi$
$971$	−5.20451e59	−0.937552	−0.468776	+	0.883317i	$0.655305\pi$
$972$	6.94828e58	0.122617
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.57681e59	−1.52521	−0.762605	−	0.646865i	$-0.776080\pi$
$977$	−9.57681e59	−1.52521	−0.762605	+	0.646865i	$0.776080\pi$
$978$	−2.23281e60	−3.48397
$979$	8.11991e59	1.24136
$980$	0	0
$981$	0	0
$982$	1.26685e60	1.82178
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	2.00102e60	2.76279
$985$	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
$993$	2.76055e60	3.17695
$994$	0	0
$995$	0	0
$996$	3.21181e60	3.47988
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.88944e60	1.96663
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.41.d.a.3.1	✓	1
8.3	odd	2	CM	8.41.d.a.3.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.41.d.a.3.1	✓	1	1.1	even	1	trivial
8.41.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:	\( k \)	\(=\)	\( 41 \)
Character orbit:	\([\chi]\)	\(=\)	8.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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