LMFDB - Embedded newform 8.21.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,21,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, 21, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 21);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 21 $
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:	$20.2811012082$
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)

$f(q)$

$=$

$q+1024.00 q^{2} +114226. q^{3} +1.04858e6 q^{4} +1.16967e8 q^{6} +1.07374e9 q^{8} +9.56079e9 q^{9} +O(q^{10})$

$q+1024.00 q^{2} +114226. q^{3} +1.04858e6 q^{4} +1.16967e8 q^{6} +1.07374e9 q^{8} +9.56079e9 q^{9} -4.23830e10 q^{11} +1.19775e11 q^{12} +1.09951e12 q^{16} -3.35354e12 q^{17} +9.79025e12 q^{18} -1.01465e12 q^{19} -4.34002e13 q^{22} +1.22649e14 q^{24} +9.53674e13 q^{25} +6.93810e14 q^{27} +1.12590e15 q^{32} -4.84124e15 q^{33} -3.43402e15 q^{34} +1.00252e16 q^{36} -1.03901e15 q^{38} -2.54181e16 q^{41} +2.78111e15 q^{43} -4.44418e16 q^{44} +1.25593e17 q^{48} +7.97923e16 q^{49} +9.76562e16 q^{50} -3.83061e17 q^{51} +7.10461e17 q^{54} -1.15900e17 q^{57} -1.73912e17 q^{59} +1.15292e18 q^{64} -4.95743e18 q^{66} -3.56138e17 q^{67} -3.51644e18 q^{68} +1.02658e19 q^{72} -6.01672e18 q^{73} +1.08934e19 q^{75} -1.06394e18 q^{76} +4.59147e19 q^{81} -2.60281e19 q^{82} -3.10229e19 q^{83} +2.84786e18 q^{86} -4.55084e19 q^{88} +6.12024e19 q^{89} +1.28607e20 q^{96} -5.00091e19 q^{97} +8.17073e19 q^{98} -4.05215e20 q^{99} +O(q^{100})$

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 $

$2$	1024.00	1.00000
$2$	1024.00	1.00000
$3$	114226.	1.93443	0.967214	−	0.253964i	$-0.0817345\pi$
$3$	114226.	1.93443	0.967214	+	0.253964i	$0.0817345\pi$
$4$	1.04858e6	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	1.16967e8	1.93443
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.07374e9	1.00000
$9$	9.56079e9	2.74201
$10$	0	0
$11$	−4.23830e10	−1.63405	−0.817025	−	0.576603i	$-0.804378\pi$
$11$	−4.23830e10	−1.63405	−0.817025	+	0.576603i	$0.804378\pi$
$12$	1.19775e11	1.93443
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	1.09951e12	1.00000
$17$	−3.35354e12	−1.66347	−0.831733	−	0.555176i	$-0.812651\pi$
$17$	−3.35354e12	−1.66347	−0.831733	+	0.555176i	$0.812651\pi$
$18$	9.79025e12	2.74201
$19$	−1.01465e12	−0.165494	−0.0827470	−	0.996571i	$-0.526369\pi$
$19$	−1.01465e12	−0.165494	−0.0827470	+	0.996571i	$0.526369\pi$
$20$	0	0
$21$	0	0
$22$	−4.34002e13	−1.63405
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.22649e14	1.93443
$25$	9.53674e13	1.00000
$26$	0	0
$27$	6.93810e14	3.36979
$28$	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			$\pi$
$32$	1.12590e15	1.00000
$33$	−4.84124e15	−3.16095
$34$	−3.43402e15	−1.66347
$35$	0	0
$36$	1.00252e16	2.74201
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	−1.03901e15	−0.165494
$39$	0	0
$40$	0	0
$41$	−2.54181e16	−1.89367	−0.946834	−	0.321721i	$-0.895739\pi$
$41$	−2.54181e16	−1.89367	−0.946834	+	0.321721i	$0.895739\pi$
$42$	0	0
$43$	2.78111e15	0.128687	0.0643434	−	0.997928i	$-0.479505\pi$
$43$	2.78111e15	0.128687	0.0643434	+	0.997928i	$0.479505\pi$
$44$	−4.44418e16	−1.63405
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.25593e17	1.93443
$49$	7.97923e16	1.00000
$50$	9.76562e16	1.00000
$51$	−3.83061e17	−3.21785
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	7.10461e17	3.36979
$55$	0	0
$56$	0	0
$57$	−1.15900e17	−0.320136
$58$	0	0
$59$	−1.73912e17	−0.340259	−0.170130	−	0.985422i	$-0.554419\pi$
$59$	−1.73912e17	−0.340259	−0.170130	+	0.985422i	$0.554419\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.15292e18	1.00000
$65$	0	0
$66$	−4.95743e18	−3.16095
$67$	−3.56138e17	−0.195375	−0.0976877	−	0.995217i	$-0.531145\pi$
$67$	−3.56138e17	−0.195375	−0.0976877	+	0.995217i	$0.531145\pi$
$68$	−3.51644e18	−1.66347
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.02658e19	2.74201
$73$	−6.01672e18	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$73$	−6.01672e18	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$74$	0	0
$75$	1.08934e19	1.93443
$76$	−1.06394e18	−0.165494
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	4.59147e19	3.77660
$82$	−2.60281e19	−1.89367
$83$	−3.10229e19	−1.99941	−0.999703	−	0.0243827i	$-0.992238\pi$
$83$	−3.10229e19	−1.99941	−0.999703	+	0.0243827i	$0.992238\pi$
$84$	0	0
$85$	0	0
$86$	2.84786e18	0.128687
$87$	0	0
$88$	−4.55084e19	−1.63405
$89$	6.12024e19	1.96277	0.981383	−	0.192059i	$-0.0615164\pi$
$89$	6.12024e19	1.96277	0.981383	+	0.192059i	$0.0615164\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	1.28607e20	1.93443
$97$	−5.00091e19	−0.678160	−0.339080	−	0.940758i	$-0.610116\pi$
$97$	−5.00091e19	−0.678160	−0.339080	+	0.940758i	$0.610116\pi$
$98$	8.17073e19	1.00000
$99$	−4.05215e20	−4.48058
$100$	1.00000e20	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−3.92254e20	−3.21785
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.62647e20	1.84351	0.921756	−	0.387770i	$-0.126754\pi$
$107$	3.62647e20	1.84351	0.921756	+	0.387770i	$0.126754\pi$
$108$	7.27512e20	3.36979
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.52948e20	0.450567	0.225284	−	0.974293i	$-0.427669\pi$
$113$	1.52948e20	0.450567	0.225284	+	0.974293i	$0.427669\pi$
$114$	−1.18682e20	−0.320136
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−1.78086e20	−0.340259
$119$	0	0
$120$	0	0
$121$	1.12357e21	1.67012
$122$	0	0
$123$	−2.90340e21	−3.66316
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	1.18059e21	1.00000
$129$	3.17676e20	0.248935
$130$	0	0
$131$	2.17294e21	1.45994	0.729970	−	0.683479i	$-0.239534\pi$
$131$	2.17294e21	1.45994	0.729970	+	0.683479i	$0.239534\pi$
$132$	−5.07641e21	−3.16095
$133$	0	0
$134$	−3.64685e20	−0.195375
$135$	0	0
$136$	−3.60083e21	−1.66347
$137$	2.11640e21	0.908642	0.454321	−	0.890838i	$-0.349882\pi$
$137$	2.11640e21	0.908642	0.454321	+	0.890838i	$0.349882\pi$
$138$	0	0
$139$	2.54239e21	0.944266	0.472133	−	0.881527i	$-0.343484\pi$
$139$	2.54239e21	0.944266	0.472133	+	0.881527i	$0.343484\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.05122e22	2.74201
$145$	0	0
$146$	−6.16112e21	−1.40001
$147$	9.11435e21	1.93443
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.11549e22	1.93443
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	−1.08948e21	−0.165494
$153$	−3.20625e22	−4.56124
$154$	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
$162$	4.70167e22	3.77660
$163$	7.06792e19	0.00533845	0.00266923	−	0.999996i	$-0.499150\pi$
$163$	7.06792e19	0.00533845	0.00266923	+	0.999996i	$0.499150\pi$
$164$	−2.66528e22	−1.89367
$165$	0	0
$166$	−3.17674e22	−1.99941
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.90050e22	1.00000
$170$	0	0
$171$	−9.70090e21	−0.453786
$172$	2.91621e21	0.128687
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−4.66006e22	−1.63405
$177$	−1.98653e22	−0.658207
$178$	6.26713e22	1.96277
$179$	−3.92234e22	−1.16149	−0.580744	−	0.814086i	$-0.697238\pi$
$179$	−3.92234e22	−1.16149	−0.580744	+	0.814086i	$0.697238\pi$
$180$	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
$187$	1.42133e23	2.71818
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.31694e23	1.93443
$193$	1.20384e23	1.67878	0.839391	−	0.543528i	$-0.182912\pi$
$193$	1.20384e23	1.67878	0.839391	+	0.543528i	$0.182912\pi$
$194$	−5.12093e22	−0.678160
$195$	0	0
$196$	8.36683e22	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−4.14941e23	−4.48058
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.02400e23	1.00000
$201$	−4.06802e22	−0.377939
$202$	0	0
$203$	0	0
$204$	−4.01669e23	−3.21785
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.30041e22	0.270425
$210$	0	0
$211$	−2.82329e23	−1.61410	−0.807051	−	0.590482i	$-0.798938\pi$
$211$	−2.82329e23	−1.61410	−0.807051	+	0.590482i	$0.798938\pi$
$212$	0	0
$213$	0	0
$214$	3.71350e23	1.84351
$215$	0	0
$216$	7.44973e23	3.36979
$217$	0	0
$218$	0	0
$219$	−6.87266e23	−2.70822
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	9.11788e23	2.74201
$226$	1.56619e23	0.450567
$227$	−2.21143e23	−0.608716	−0.304358	−	0.952558i	$-0.598442\pi$
$227$	−2.21143e23	−0.608716	−0.304358	+	0.952558i	$0.598442\pi$
$228$	−1.21530e23	−0.320136
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.85874e23	−1.66645	−0.833227	−	0.552931i	$-0.813509\pi$
$233$	−7.85874e23	−1.66645	−0.833227	+	0.552931i	$0.813509\pi$
$234$	0	0
$235$	0	0
$236$	−1.82360e23	−0.340259
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	7.13387e23	1.07933	0.539666	−	0.841879i	$-0.318551\pi$
$241$	7.13387e23	1.07933	0.539666	+	0.841879i	$0.318551\pi$
$242$	1.15054e24	1.67012
$243$	2.82549e24	3.93578
$244$	0	0
$245$	0	0
$246$	−2.97309e24	−3.66316
$247$	0	0
$248$	0	0
$249$	−3.54362e24	−3.86770
$250$	0	0
$251$	−1.55766e24	−1.56941	−0.784703	−	0.619872i	$-0.787185\pi$
$251$	−1.55766e24	−1.56941	−0.784703	+	0.619872i	$0.787185\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.20893e24	1.00000
$257$	1.50407e24	1.19657	0.598284	−	0.801284i	$-0.295849\pi$
$257$	1.50407e24	1.19657	0.598284	+	0.801284i	$0.295849\pi$
$258$	3.25300e23	0.248935
$259$	0	0
$260$	0	0
$261$	0	0
$262$	2.22509e24	1.45994
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−5.19825e24	−3.16095
$265$	0	0
$266$	0	0
$267$	6.99091e24	3.79683
$268$	−3.73437e23	−0.195375
$269$	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			$\pi$
$272$	−3.68725e24	−1.66347
$273$	0	0
$274$	2.16720e24	0.908642
$275$	−4.04196e24	−1.63405
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	2.60341e24	0.944266
$279$	0	0
$280$	0	0
$281$	2.04420e24	0.665979	0.332989	−	0.942931i	$-0.391943\pi$
$281$	2.04420e24	0.665979	0.332989	+	0.942931i	$0.391943\pi$
$282$	0	0
$283$	−7.68742e23	−0.233301	−0.116650	−	0.993173i	$-0.537216\pi$
$283$	−7.68742e23	−0.233301	−0.116650	+	0.993173i	$0.537216\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.07645e25	2.74201
$289$	7.18197e24	1.76712
$290$	0	0
$291$	−5.71234e24	−1.31185
$292$	−6.30899e24	−1.40001
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	9.33310e24	1.93443
$295$	0	0
$296$	0	0
$297$	−2.94058e25	−5.50640
$298$	0	0
$299$	0	0
$300$	1.14226e25	1.93443
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−1.11562e24	−0.165494
$305$	0	0
$306$	−3.28320e25	−4.56124
$307$	2.64867e24	0.356160	0.178080	−	0.984016i	$-0.443011\pi$
$307$	2.64867e24	0.356160	0.178080	+	0.984016i	$0.443011\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.53618e25	1.70215	0.851075	−	0.525044i	$-0.175951\pi$
$313$	1.53618e25	1.70215	0.851075	+	0.525044i	$0.175951\pi$
$314$	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
$321$	4.14237e25	3.56614
$322$	0	0
$323$	3.40268e24	0.275293
$324$	4.81451e25	3.77660
$325$	0	0
$326$	7.23755e22	0.00533845
$327$	0	0
$328$	−2.72924e25	−1.89367
$329$	0	0
$330$	0	0
$331$	−3.08691e25	−1.95544	−0.977718	−	0.209923i	$-0.932679\pi$
$331$	−3.08691e25	−1.95544	−0.977718	+	0.209923i	$0.932679\pi$
$332$	−3.25298e25	−1.99941
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.11942e24	0.0592508	0.0296254	−	0.999561i	$-0.490569\pi$
$337$	1.11942e24	0.0592508	0.0296254	+	0.999561i	$0.490569\pi$
$338$	1.94611e25	1.00000
$339$	1.74707e25	0.871590
$340$	0	0
$341$	0	0
$342$	−9.93372e24	−0.453786
$343$	0	0
$344$	2.98620e24	0.128687
$345$	0	0
$346$	0	0
$347$	−5.03994e25	−1.99127	−0.995635	−	0.0933286i	$-0.970249\pi$
$347$	−5.03994e25	−1.99127	−0.995635	+	0.0933286i	$0.970249\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	−4.77190e25	−1.63405
$353$	5.62790e25	1.87327	0.936633	−	0.350313i	$-0.113925\pi$
$353$	5.62790e25	1.87327	0.936633	+	0.350313i	$0.113925\pi$
$354$	−2.03421e25	−0.658207
$355$	0	0
$356$	6.41754e25	1.96277
$357$	0	0
$358$	−4.01647e25	−1.16149
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−3.65604e25	−0.972612
$362$	0	0
$363$	1.28341e26	3.23072
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−2.43017e26	−5.19246
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	1.45544e26	2.71818
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.83008e25	−0.626347	−0.313174	−	0.949696i	$-0.601392\pi$
$379$	−3.83008e25	−0.626347	−0.313174	+	0.949696i	$0.601392\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.34854e26	1.93443
$385$	0	0
$386$	1.23273e26	1.67878
$387$	2.65897e25	0.352861
$388$	−5.24384e25	−0.678160
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	8.56763e25	1.00000
$393$	2.48206e26	2.82415
$394$	0	0
$395$	0	0
$396$	−4.24899e26	−4.48058
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	1.04858e26	1.00000
$401$	−2.13090e26	−1.98207	−0.991034	−	0.133612i	$-0.957342\pi$
$401$	−2.13090e26	−1.98207	−0.991034	+	0.133612i	$0.957342\pi$
$402$	−4.16565e25	−0.377939
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−4.11309e26	−3.21785
$409$	1.27053e26	0.969955	0.484977	−	0.874527i	$-0.338828\pi$
$409$	1.27053e26	0.969955	0.484977	+	0.874527i	$0.338828\pi$
$410$	0	0
$411$	2.41748e26	1.75770
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.90407e26	1.82661
$418$	4.40362e25	0.270425
$419$	−3.16629e26	−1.89850	−0.949249	−	0.314527i	$-0.898154\pi$
$419$	−3.16629e26	−1.89850	−0.949249	+	0.314527i	$0.898154\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	−2.89105e26	−1.61410
$423$	0	0
$424$	0	0
$425$	−3.19818e26	−1.66347
$426$	0	0
$427$	0	0
$428$	3.80263e26	1.84351
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	7.62852e26	3.36979
$433$	−2.40752e26	−1.03918	−0.519591	−	0.854415i	$-0.673916\pi$
$433$	−2.40752e26	−1.03918	−0.519591	+	0.854415i	$0.673916\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−7.03760e26	−2.70822
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	7.62877e26	2.74201
$442$	0	0
$443$	1.30434e26	0.448078	0.224039	−	0.974580i	$-0.428076\pi$
$443$	1.30434e26	0.448078	0.224039	+	0.974580i	$0.428076\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.95620e26	−1.78857	−0.894285	−	0.447499i	$-0.852315\pi$
$449$	−5.95620e26	−1.78857	−0.894285	+	0.447499i	$0.852315\pi$
$450$	9.33671e26	2.74201
$451$	1.07729e27	3.09435
$452$	1.60378e26	0.450567
$453$	0	0
$454$	−2.26450e26	−0.608716
$455$	0	0
$456$	−1.24447e26	−0.320136
$457$	6.94600e26	1.74813	0.874064	−	0.485812i	$-0.161476\pi$
$457$	6.94600e26	1.74813	0.874064	+	0.485812i	$0.161476\pi$
$458$	0	0
$459$	−2.32672e27	−5.60553
$460$	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
$466$	−8.04735e26	−1.66645
$467$	−9.22043e26	−1.86888	−0.934442	−	0.356114i	$-0.884101\pi$
$467$	−9.22043e26	−1.86888	−0.934442	+	0.356114i	$0.884101\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.86737e26	−0.340259
$473$	−1.17872e26	−0.210281
$474$	0	0
$475$	−9.67650e25	−0.165494
$476$	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
$482$	7.30509e26	1.07933
$483$	0	0
$484$	1.17815e27	1.67012
$485$	0	0
$486$	2.89330e27	3.93578
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	8.07340e24	0.0103268
$490$	0	0
$491$	1.59202e27	1.95494	0.977469	−	0.211079i	$-0.0676979\pi$
$491$	1.59202e27	1.95494	0.977469	+	0.211079i	$0.0676979\pi$
$492$	−3.04444e27	−3.66316
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−3.62866e27	−3.86770
$499$	1.90641e27	1.99164	0.995820	−	0.0913348i	$-0.0291133\pi$
$499$	1.90641e27	1.99164	0.995820	+	0.0913348i	$0.0291133\pi$
$500$	0	0
$501$	0	0
$502$	−1.59504e27	−1.56941
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.17086e27	1.93443
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.23794e27	1.00000
$513$	−7.03977e26	−0.557680
$514$	1.54017e27	1.19657
$515$	0	0
$516$	3.33107e26	0.248935
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.57885e27	−1.75004	−0.875021	−	0.484085i	$-0.839152\pi$
$521$	−2.57885e27	−1.75004	−0.875021	+	0.484085i	$0.839152\pi$
$522$	0	0
$523$	2.21598e27	1.44726	0.723632	−	0.690186i	$-0.242471\pi$
$523$	2.21598e27	1.44726	0.723632	+	0.690186i	$0.242471\pi$
$524$	2.27849e27	1.45994
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−5.32300e27	−3.16095
$529$	1.71616e27	1.00000
$530$	0	0
$531$	−1.66274e27	−0.932994
$532$	0	0
$533$	0	0
$534$	7.15869e27	3.79683
$535$	0	0
$536$	−3.82400e26	−0.195375
$537$	−4.48033e27	−2.24681
$538$	0	0
$539$	−3.38184e27	−1.63405
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−3.77575e27	−1.66347
$545$	0	0
$546$	0	0
$547$	−4.19373e27	−1.74875	−0.874375	−	0.485251i	$-0.838728\pi$
$547$	−4.19373e27	−1.74875	−0.874375	+	0.485251i	$0.838728\pi$
$548$	2.21921e27	0.908642
$549$	0	0
$550$	−4.13897e27	−1.63405
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	2.66589e27	0.944266
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.62353e28	5.25813
$562$	2.09326e27	0.665979
$563$	2.78356e27	0.869994	0.434997	−	0.900432i	$-0.356749\pi$
$563$	2.78356e27	0.869994	0.434997	+	0.900432i	$0.356749\pi$
$564$	0	0
$565$	0	0
$566$	−7.87192e26	−0.233301
$567$	0	0
$568$	0	0
$569$	−6.37985e27	−1.79344	−0.896722	−	0.442595i	$-0.854058\pi$
$569$	−6.37985e27	−1.79344	−0.896722	+	0.442595i	$0.854058\pi$
$570$	0	0
$571$	−7.29382e27	−1.97967	−0.989837	−	0.142204i	$-0.954581\pi$
$571$	−7.29382e27	−1.97967	−0.989837	+	0.142204i	$0.954581\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.10228e28	2.74201
$577$	−8.17943e27	−1.99970	−0.999850	−	0.0173302i	$-0.994483\pi$
$577$	−8.17943e27	−1.99970	−0.999850	+	0.0173302i	$0.994483\pi$
$578$	7.35434e27	1.76712
$579$	1.37509e28	3.24748
$580$	0	0
$581$	0	0
$582$	−5.84944e27	−1.31185
$583$	0	0
$584$	−6.46040e27	−1.40001
$585$	0	0
$586$	0	0
$587$	−7.68162e27	−1.58151	−0.790755	−	0.612133i	$-0.790312\pi$
$587$	−7.68162e27	−1.58151	−0.790755	+	0.612133i	$0.790312\pi$
$588$	9.55709e27	1.93443
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.00007e27	0.557936	0.278968	−	0.960300i	$-0.410008\pi$
$593$	3.00007e27	0.557936	0.278968	+	0.960300i	$0.410008\pi$
$594$	−3.01115e28	−5.50640
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	1.16967e28	1.93443
$601$	8.69391e27	1.41407	0.707035	−	0.707179i	$-0.250033\pi$
$601$	8.69391e27	1.41407	0.707035	+	0.707179i	$0.250033\pi$
$602$	0	0
$603$	−3.40496e27	−0.535721
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−1.14240e27	−0.165494
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−3.36199e28	−4.56124
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	2.71224e27	0.356160
$615$	0	0
$616$	0	0
$617$	1.47242e28	1.84153	0.920765	−	0.390119i	$-0.127566\pi$
$617$	1.47242e28	1.84153	0.920765	+	0.390119i	$0.127566\pi$
$618$	0	0
$619$	6.02524e24	0.000729572 0	0.000364786	−	1.00000i	$-0.499884\pi$
$619$	6.02524e24	0.000729572 0	0.000364786		1.00000i	$0.499884\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	9.09495e27	1.00000
$626$	1.57305e28	1.70215
$627$	4.91219e27	0.523118
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−3.22493e28	−3.12236
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.77175e28	1.51294	0.756471	−	0.654027i	$-0.226922\pi$
$641$	1.77175e28	1.51294	0.756471	+	0.654027i	$0.226922\pi$
$642$	4.24179e28	3.56614
$643$	−2.35774e28	−1.95158	−0.975791	−	0.218706i	$-0.929816\pi$
$643$	−2.35774e28	−1.95158	−0.975791	+	0.218706i	$0.929816\pi$
$644$	0	0
$645$	0	0
$646$	3.48434e27	0.275293
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	4.93005e28	3.77660
$649$	7.37092e27	0.556000
$650$	0	0
$651$	0	0
$652$	7.41125e25	0.00533845
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−2.79475e28	−1.89367
$657$	−5.75246e28	−3.83884
$658$	0	0
$659$	1.96646e28	1.27301	0.636504	−	0.771274i	$-0.280380\pi$
$659$	1.96646e28	1.27301	0.636504	+	0.771274i	$0.280380\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−3.16099e28	−1.95544
$663$	0	0
$664$	−3.33105e28	−1.99941
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.82190e28	−1.48044	−0.740221	−	0.672364i	$-0.765279\pi$
$673$	−2.82190e28	−1.48044	−0.740221	+	0.672364i	$0.765279\pi$
$674$	1.14629e27	0.0592508
$675$	6.61669e28	3.36979
$676$	1.99281e28	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	1.78900e28	0.871590
$679$	0	0
$680$	0	0
$681$	−2.52603e28	−1.17752
$682$	0	0
$683$	4.34265e28	1.96584	0.982919	−	0.184038i	$-0.0589171\pi$
$683$	4.34265e28	1.96584	0.982919	+	0.184038i	$0.0589171\pi$
$684$	−1.01721e28	−0.453786
$685$	0	0
$686$	0	0
$687$	0	0
$688$	3.05787e27	0.128687
$689$	0	0
$690$	0	0
$691$	−3.84706e28	−1.55006	−0.775030	−	0.631924i	$-0.782265\pi$
$691$	−3.84706e28	−1.55006	−0.775030	+	0.631924i	$0.782265\pi$
$692$	0	0
$693$	0	0
$694$	−5.16089e28	−1.99127
$695$	0	0
$696$	0	0
$697$	8.52404e28	3.15005
$698$	0	0
$699$	−8.97672e28	−3.22364
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−4.88643e28	−1.63405
$705$	0	0
$706$	5.76297e28	1.87327
$707$	0	0
$708$	−2.08303e28	−0.658207
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	6.57156e28	1.96277
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.11287e28	−1.16149
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	−3.74379e28	−0.972612
$723$	8.14874e28	2.08789
$724$	0	0
$725$	0	0
$726$	1.31421e29	3.23072
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.62649e29	3.83687
$730$	0	0
$731$	−9.32657e27	−0.214066
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.50942e28	0.319253
$738$	−2.48849e29	−5.19246
$739$	5.68807e28	1.17090	0.585451	−	0.810708i	$-0.300917\pi$
$739$	5.68807e28	1.17090	0.585451	+	0.810708i	$0.300917\pi$
$740$	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
$747$	−2.96603e29	−5.48239
$748$	1.49037e29	2.71818
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−1.77925e29	−3.03590
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−3.92200e28	−0.626347
$759$	0	0
$760$	0	0
$761$	−7.16003e28	−1.09918	−0.549589	−	0.835435i	$-0.685216\pi$
$761$	−7.16003e28	−1.09918	−0.549589	+	0.835435i	$0.685216\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.38091e29	1.93443
$769$	−4.32623e28	−0.598200	−0.299100	−	0.954222i	$-0.596686\pi$
$769$	−4.32623e28	−0.598200	−0.299100	+	0.954222i	$0.596686\pi$
$770$	0	0
$771$	1.71804e29	2.31468
$772$	1.26231e29	1.67878
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	2.72278e28	0.352861
$775$	0	0
$776$	−5.36969e28	−0.678160
$777$	0	0
$778$	0	0
$779$	2.57906e28	0.313391
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	8.77325e28	1.00000
$785$	0	0
$786$	2.54163e29	2.82415
$787$	−1.82292e29	−1.99996	−0.999979	−	0.00648819i	$-0.997935\pi$
$787$	−1.82292e29	−1.99996	−0.999979	+	0.00648819i	$0.997935\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−4.35097e29	−4.48058
$793$	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
$800$	1.07374e29	1.00000
$801$	5.85144e29	5.38192
$802$	−2.18204e29	−1.98207
$803$	2.55007e29	2.28768
$804$	−4.26562e28	−0.377939
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.24019e29	−1.03277	−0.516383	−	0.856357i	$-0.672722\pi$
$809$	−1.24019e29	−1.03277	−0.516383	+	0.856357i	$0.672722\pi$
$810$	0	0
$811$	−1.11745e29	−0.907860	−0.453930	−	0.891037i	$-0.649978\pi$
$811$	−1.11745e29	−0.907860	−0.453930	+	0.891037i	$0.649978\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−4.21180e29	−3.21785
$817$	−2.82187e27	−0.0212969
$818$	1.30102e29	0.969955
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.47550e29	1.75770
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−4.61697e29	−3.16095
$826$	0	0
$827$	2.19924e29	1.46966	0.734830	−	0.678252i	$-0.237262\pi$
$827$	2.19924e29	1.46966	0.734830	+	0.678252i	$0.237262\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.67586e29	−1.66347
$834$	2.97377e29	1.82661
$835$	0	0
$836$	4.50931e28	0.270425
$837$	0	0
$838$	−3.24228e29	−1.89850
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.76995e29	1.00000
$842$	0	0
$843$	2.33501e29	1.28829
$844$	−2.96043e29	−1.61410
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.78103e28	−0.451303
$850$	−3.27494e29	−1.66347
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	3.89389e29	1.84351
$857$	−3.50153e28	−0.163851	−0.0819256	−	0.996638i	$-0.526107\pi$
$857$	−3.50153e28	−0.163851	−0.0819256	+	0.996638i	$0.526107\pi$
$858$	0	0
$859$	−4.25332e29	−1.94445	−0.972224	−	0.234053i	$-0.924801\pi$
$859$	−4.25332e29	−1.94445	−0.972224	+	0.234053i	$0.924801\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	7.81160e29	3.36979
$865$	0	0
$866$	−2.46530e29	−1.03918
$867$	8.20368e29	3.41836
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−4.78127e29	−1.85952
$874$	0	0
$875$	0	0
$876$	−7.20650e29	−2.70822
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.59188e29	1.98517	0.992587	−	0.121534i	$-0.0387812\pi$
$881$	5.59188e29	1.98517	0.992587	+	0.121534i	$0.0387812\pi$
$882$	7.81187e29	2.74201
$883$	4.50564e29	1.56369	0.781843	−	0.623476i	$-0.214280\pi$
$883$	4.50564e29	1.56369	0.781843	+	0.623476i	$0.214280\pi$
$884$	0	0
$885$	0	0
$886$	1.33564e29	0.448078
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.94600e30	−6.17116
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−6.09915e29	−1.78857
$899$	0	0
$900$	9.56079e29	2.74201
$901$	0	0
$902$	1.10315e30	3.09435
$903$	0	0
$904$	1.64227e29	0.450567
$905$	0	0
$906$	0	0
$907$	−7.39355e29	−1.96237	−0.981183	−	0.193079i	$-0.938153\pi$
$907$	−7.39355e29	−1.96237	−0.981183	+	0.193079i	$0.938153\pi$
$908$	−2.31885e29	−0.608716
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−1.27433e29	−0.320136
$913$	1.31484e30	3.26713
$914$	7.11271e29	1.74813
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−2.38256e30	−5.60553
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	3.02547e29	0.688965
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−9.33455e29	−1.94956	−0.974779	−	0.223173i	$-0.928358\pi$
$929$	−9.33455e29	−1.94956	−0.974779	+	0.223173i	$0.928358\pi$
$930$	0	0
$931$	−8.09616e28	−0.165494
$932$	−8.24048e29	−1.66645
$933$	0	0
$934$	−9.44172e29	−1.86888
$935$	0	0
$936$	0	0
$937$	7.34116e29	1.40724	0.703621	−	0.710575i	$-0.251565\pi$
$937$	7.34116e29	1.40724	0.703621	+	0.710575i	$0.251565\pi$
$938$	0	0
$939$	1.75471e30	3.29269
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	−1.91218e29	−0.340259
$945$	0	0
$946$	−1.20701e29	−0.210281
$947$	−8.42382e29	−1.45214	−0.726071	−	0.687619i	$-0.758656\pi$
$947$	−8.42382e29	−1.45214	−0.726071	+	0.687619i	$0.758656\pi$
$948$	0	0
$949$	0	0
$950$	−9.90873e28	−0.165494
$951$	0	0
$952$	0	0
$953$	−9.07852e29	−1.46922	−0.734609	−	0.678491i	$-0.762634\pi$
$953$	−9.07852e29	−1.46922	−0.734609	+	0.678491i	$0.762634\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	6.71791e29	1.00000
$962$	0	0
$963$	3.46719e30	5.05493
$964$	7.48041e29	1.07933
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.20642e30	1.67012
$969$	3.88675e29	0.532535
$970$	0	0
$971$	−7.67973e29	−1.03075	−0.515376	−	0.856964i	$-0.672348\pi$
$971$	−7.67973e29	−1.03075	−0.515376	+	0.856964i	$0.672348\pi$
$972$	2.96274e30	3.93578
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.46005e29	−0.689051	−0.344525	−	0.938777i	$-0.611960\pi$
$977$	−5.46005e29	−0.689051	−0.344525	+	0.938777i	$0.611960\pi$
$978$	8.26716e27	0.0103268
$979$	−2.59394e30	−3.20726
$980$	0	0
$981$	0	0
$982$	1.63023e30	1.95494
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−3.11751e30	−3.66316
$985$	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
$993$	−3.52605e30	−3.78265
$994$	0	0
$995$	0	0
$996$	−3.71575e30	−3.86770
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.95216e30	1.99164
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.21.d.a.3.1	✓	1
3.2	odd	2		72.21.b.a.19.1		1
4.3	odd	2		32.21.d.a.15.1		1
8.3	odd	2	CM	8.21.d.a.3.1	✓	1
8.5	even	2		32.21.d.a.15.1		1
24.11	even	2		72.21.b.a.19.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.21.d.a.3.1	✓	1	1.1	even	1	trivial
8.21.d.a.3.1	✓	1	8.3	odd	2	CM
32.21.d.a.15.1		1	4.3	odd	2
32.21.d.a.15.1		1	8.5	even	2
72.21.b.a.19.1		1	3.2	odd	2
72.21.b.a.19.1		1	24.11	even	2

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:	\( k \)	\(=\)	\( 21 \)
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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