LMFDB - Embedded newform 8.29.d.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,29,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, 29, names="a")

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

chi := DirichletCharacter("8.3");

S:= CuspForms(chi, 29);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 29 $
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:	$39.7347793985$
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)

$f(q)$

$=$

$q+16384.0 q^{2} -433454. q^{3} +2.68435e8 q^{4} -7.10171e9 q^{6} +4.39805e12 q^{8} -2.26889e13 q^{9} +O(q^{10})$

$q+16384.0 q^{2} -433454. q^{3} +2.68435e8 q^{4} -7.10171e9 q^{6} +4.39805e12 q^{8} -2.26889e13 q^{9} +7.39139e14 q^{11} -1.16354e14 q^{12} +7.20576e16 q^{16} -2.28754e17 q^{17} -3.71735e17 q^{18} +1.56673e18 q^{19} +1.21101e19 q^{22} -1.90635e18 q^{24} +3.72529e19 q^{25} +1.97506e19 q^{27} +1.18059e21 q^{32} -3.20383e20 q^{33} -3.74790e21 q^{34} -6.09051e21 q^{36} +2.56694e22 q^{38} +3.52928e22 q^{41} +9.72667e22 q^{43} +1.98411e23 q^{44} -3.12337e22 q^{48} +4.59987e23 q^{49} +6.10352e23 q^{50} +9.91543e22 q^{51} +3.23594e23 q^{54} -6.79107e23 q^{57} +2.93650e24 q^{59} +1.93428e25 q^{64} -5.24915e24 q^{66} -6.61144e25 q^{67} -6.14057e25 q^{68} -9.97869e25 q^{72} -2.41596e26 q^{73} -1.61474e25 q^{75} +4.20567e26 q^{76} +5.10488e26 q^{81} +5.78238e26 q^{82} -5.02641e26 q^{83} +1.59362e27 q^{86} +3.25077e27 q^{88} -2.43567e27 q^{89} -5.11732e26 q^{96} -9.11100e27 q^{97} +7.53642e27 q^{98} -1.67703e28 q^{99} +O(q^{100})$

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 $

$2$	16384.0	1.00000
$2$	16384.0	1.00000
$3$	−433454.	−0.0906245	−0.0453122	−	0.998973i	$-0.514428\pi$
$3$	−433454.	−0.0906245	−0.0453122	+	0.998973i	$0.514428\pi$
$4$	2.68435e8	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−7.10171e9	−0.0906245
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	4.39805e12	1.00000
$9$	−2.26889e13	−0.991787
$10$	0	0
$11$	7.39139e14	1.94638	0.973192	−	0.229995i	$-0.0738708\pi$
$11$	7.39139e14	1.94638	0.973192	+	0.229995i	$0.0738708\pi$
$12$	−1.16354e14	−0.0906245
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	7.20576e16	1.00000
$17$	−2.28754e17	−1.35857	−0.679287	−	0.733872i	$-0.737711\pi$
$17$	−2.28754e17	−1.35857	−0.679287	+	0.733872i	$0.737711\pi$
$18$	−3.71735e17	−0.991787
$19$	1.56673e18	1.96085	0.980426	−	0.196886i	$-0.0630829\pi$
$19$	1.56673e18	1.96085	0.980426	+	0.196886i	$0.0630829\pi$
$20$	0	0
$21$	0	0
$22$	1.21101e19	1.94638
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−1.90635e18	−0.0906245
$25$	3.72529e19	1.00000
$26$	0	0
$27$	1.97506e19	0.180505
$28$	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			$\pi$
$32$	1.18059e21	1.00000
$33$	−3.20383e20	−0.176390
$34$	−3.74790e21	−1.35857
$35$	0	0
$36$	−6.09051e21	−0.991787
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	2.56694e22	1.96085
$39$	0	0
$40$	0	0
$41$	3.52928e22	0.930491	0.465246	−	0.885182i	$-0.345966\pi$
$41$	3.52928e22	0.930491	0.465246	+	0.885182i	$0.345966\pi$
$42$	0	0
$43$	9.72667e22	1.31645	0.658227	−	0.752819i	$-0.271307\pi$
$43$	9.72667e22	1.31645	0.658227	+	0.752819i	$0.271307\pi$
$44$	1.98411e23	1.94638
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−3.12337e22	−0.0906245
$49$	4.59987e23	1.00000
$50$	6.10352e23	1.00000
$51$	9.91543e22	0.123120
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	3.23594e23	0.180505
$55$	0	0
$56$	0	0
$57$	−6.79107e23	−0.177701
$58$	0	0
$59$	2.93650e24	0.474134	0.237067	−	0.971493i	$-0.423814\pi$
$59$	2.93650e24	0.474134	0.237067	+	0.971493i	$0.423814\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	1.93428e25	1.00000
$65$	0	0
$66$	−5.24915e24	−0.176390
$67$	−6.61144e25	−1.79990	−0.899950	−	0.435992i	$-0.856398\pi$
$67$	−6.61144e25	−1.79990	−0.899950	+	0.435992i	$0.856398\pi$
$68$	−6.14057e25	−1.35857
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−9.97869e25	−0.991787
$73$	−2.41596e26	−1.97956	−0.989780	−	0.142600i	$-0.954454\pi$
$73$	−2.41596e26	−1.97956	−0.989780	+	0.142600i	$0.954454\pi$
$74$	0	0
$75$	−1.61474e25	−0.0906245
$76$	4.20567e26	1.96085
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	5.10488e26	0.975429
$82$	5.78238e26	0.930491
$83$	−5.02641e26	−0.682598	−0.341299	−	0.939955i	$-0.610867\pi$
$83$	−5.02641e26	−0.682598	−0.341299	+	0.939955i	$0.610867\pi$
$84$	0	0
$85$	0	0
$86$	1.59362e27	1.31645
$87$	0	0
$88$	3.25077e27	1.94638
$89$	−2.43567e27	−1.24497	−0.622485	−	0.782631i	$-0.713877\pi$
$89$	−2.43567e27	−1.24497	−0.622485	+	0.782631i	$0.713877\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−5.11732e26	−0.0906245
$97$	−9.11100e27	−1.39560	−0.697801	−	0.716292i	$-0.745838\pi$
$97$	−9.11100e27	−1.39560	−0.697801	+	0.716292i	$0.745838\pi$
$98$	7.53642e27	1.00000
$99$	−1.67703e28	−1.93040
$100$	1.00000e28	1.00000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	1.62454e27	0.123120
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.93670e28	−0.751086	−0.375543	−	0.926805i	$-0.622544\pi$
$107$	−1.93670e28	−0.751086	−0.375543	+	0.926805i	$0.622544\pi$
$108$	5.30177e27	0.180505
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.10694e29	−1.99998	−0.999992	−	0.00396865i	$-0.998737\pi$
$113$	−1.10694e29	−1.99998	−0.999992	+	0.00396865i	$0.998737\pi$
$114$	−1.11265e28	−0.177701
$115$	0	0
$116$	0	0
$117$	0	0
$118$	4.81115e28	0.474134
$119$	0	0
$120$	0	0
$121$	4.02116e29	2.78841
$122$	0	0
$123$	−1.52978e28	−0.0843253
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	3.16913e29	1.00000
$129$	−4.21606e28	−0.119303
$130$	0	0
$131$	8.58633e29	1.95889	0.979444	−	0.201718i	$-0.0646523\pi$
$131$	8.58633e29	1.95889	0.979444	+	0.201718i	$0.0646523\pi$
$132$	−8.60021e28	−0.176390
$133$	0	0
$134$	−1.08322e30	−1.79990
$135$	0	0
$136$	−1.00607e30	−1.35857
$137$	9.21694e29	1.12331	0.561654	−	0.827372i	$-0.310165\pi$
$137$	9.21694e29	1.12331	0.561654	+	0.827372i	$0.310165\pi$
$138$	0	0
$139$	−1.27707e30	−1.27059	−0.635296	−	0.772268i	$-0.719122\pi$
$139$	−1.27707e30	−1.27059	−0.635296	+	0.772268i	$0.719122\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−1.63491e30	−0.991787
$145$	0	0
$146$	−3.95830e30	−1.97956
$147$	−1.99383e29	−0.0906245
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.64559e29	−0.0906245
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	6.89057e30	1.96085
$153$	5.19018e30	1.34742
$154$	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
$162$	8.36384e30	0.975429
$163$	−1.09303e31	−1.16952	−0.584758	−	0.811208i	$-0.698811\pi$
$163$	−1.09303e31	−1.16952	−0.584758	+	0.811208i	$0.698811\pi$
$164$	9.47384e30	0.930491
$165$	0	0
$166$	−8.23527e30	−0.682598
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	1.55029e31	1.00000
$170$	0	0
$171$	−3.55475e31	−1.94475
$172$	2.61098e31	1.31645
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	5.32606e31	1.94638
$177$	−1.27284e30	−0.0429681
$178$	−3.99061e31	−1.24497
$179$	−1.64267e31	−0.473813	−0.236906	−	0.971532i	$-0.576134\pi$
$179$	−1.64267e31	−0.473813	−0.236906	+	0.971532i	$0.576134\pi$
$180$	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
$187$	−1.69081e32	−2.64431
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−8.38422e30	−0.0906245
$193$	1.78994e32	1.79902	0.899510	−	0.436899i	$-0.143923\pi$
$193$	1.78994e32	1.79902	0.899510	+	0.436899i	$0.143923\pi$
$194$	−1.49275e32	−1.39560
$195$	0	0
$196$	1.23477e32	1.00000
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−2.74764e32	−1.93040
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.63840e32	1.00000
$201$	2.86575e31	0.163115
$202$	0	0
$203$	0	0
$204$	2.66165e31	0.123120
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.15803e33	3.81657
$210$	0	0
$211$	4.03056e31	0.116255	0.0581274	−	0.998309i	$-0.481487\pi$
$211$	4.03056e31	0.116255	0.0581274	+	0.998309i	$0.481487\pi$
$212$	0	0
$213$	0	0
$214$	−3.17309e32	−0.751086
$215$	0	0
$216$	8.68642e31	0.180505
$217$	0	0
$218$	0	0
$219$	1.04721e32	0.179397
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−8.45228e32	−0.991787
$226$	−1.81361e33	−1.99998
$227$	−1.68422e33	−1.74597	−0.872985	−	0.487748i	$-0.837819\pi$
$227$	−1.68422e33	−1.74597	−0.872985	+	0.487748i	$0.837819\pi$
$228$	−1.82296e32	−0.177701
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.34749e33	0.969485	0.484743	−	0.874657i	$-0.338913\pi$
$233$	1.34749e33	0.969485	0.484743	+	0.874657i	$0.338913\pi$
$234$	0	0
$235$	0	0
$236$	7.88259e32	0.474134
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−3.94734e33	−1.77038	−0.885189	−	0.465232i	$-0.845971\pi$
$241$	−3.94734e33	−1.77038	−0.885189	+	0.465232i	$0.845971\pi$
$242$	6.58828e33	2.78841
$243$	−6.73104e32	−0.268902
$244$	0	0
$245$	0	0
$246$	−2.50639e32	−0.0843253
$247$	0	0
$248$	0	0
$249$	2.17872e32	0.0618601
$250$	0	0
$251$	4.58975e33	1.16508	0.582541	−	0.812801i	$-0.302058\pi$
$251$	4.58975e33	1.16508	0.582541	+	0.812801i	$0.302058\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	5.19230e33	1.00000
$257$	−8.57654e33	−1.56404	−0.782021	−	0.623252i	$-0.785811\pi$
$257$	−8.57654e33	−1.56404	−0.782021	+	0.623252i	$0.785811\pi$
$258$	−6.90760e32	−0.119303
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.40678e34	1.95889
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−1.40906e33	−0.176390
$265$	0	0
$266$	0	0
$267$	1.05575e33	0.112825
$268$	−1.77474e34	−1.79990
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−1.64835e34	−1.35857
$273$	0	0
$274$	1.51010e34	1.12331
$275$	2.75351e34	1.94638
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−2.09235e34	−1.27059
$279$	0	0
$280$	0	0
$281$	−3.77797e34	−1.97410	−0.987051	−	0.160406i	$-0.948720\pi$
$281$	−3.77797e34	−1.97410	−0.987051	+	0.160406i	$0.948720\pi$
$282$	0	0
$283$	1.89414e34	0.896194	0.448097	−	0.893985i	$-0.352102\pi$
$283$	1.89414e34	0.896194	0.448097	+	0.893985i	$0.352102\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−2.67863e34	−0.991787
$289$	2.39772e34	0.845726
$290$	0	0
$291$	3.94920e33	0.126476
$292$	−6.48528e34	−1.97956
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−3.26669e33	−0.0906245
$295$	0	0
$296$	0	0
$297$	1.45985e34	0.351331
$298$	0	0
$299$	0	0
$300$	−4.33454e33	−0.0906245
$301$	0	0
$302$	0	0
$303$	0	0
$304$	1.12895e35	1.96085
$305$	0	0
$306$	8.50358e34	1.34742
$307$	1.11600e35	1.68938	0.844691	−	0.535254i	$-0.179784\pi$
$307$	1.11600e35	1.68938	0.844691	+	0.535254i	$0.179784\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−2.90710e34	−0.335614	−0.167807	−	0.985820i	$-0.553669\pi$
$313$	−2.90710e34	−0.335614	−0.167807	+	0.985820i	$0.553669\pi$
$314$	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
$321$	8.39471e33	0.0680668
$322$	0	0
$323$	−3.58397e35	−2.66397
$324$	1.37033e35	0.975429
$325$	0	0
$326$	−1.79082e35	−1.16952
$327$	0	0
$328$	1.55219e35	0.930491
$329$	0	0
$330$	0	0
$331$	−3.62493e35	−1.91297	−0.956483	−	0.291787i	$-0.905750\pi$
$331$	−3.62493e35	−1.91297	−0.956483	+	0.291787i	$0.905750\pi$
$332$	−1.34927e35	−0.682598
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.02106e34	−0.0829395	−0.0414697	−	0.999140i	$-0.513204\pi$
$337$	−2.02106e34	−0.0829395	−0.0414697	+	0.999140i	$0.513204\pi$
$338$	2.54000e35	1.00000
$339$	4.79808e34	0.181247
$340$	0	0
$341$	0	0
$342$	−5.82410e35	−1.94475
$343$	0	0
$344$	4.27784e35	1.31645
$345$	0	0
$346$	0	0
$347$	−1.33780e35	−0.364568	−0.182284	−	0.983246i	$-0.558349\pi$
$347$	−1.33780e35	−0.364568	−0.182284	+	0.983246i	$0.558349\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	8.72621e35	1.94638
$353$	6.79104e35	1.45576	0.727881	−	0.685703i	$-0.240505\pi$
$353$	6.79104e35	1.45576	0.727881	+	0.685703i	$0.240505\pi$
$354$	−2.08541e34	−0.0429681
$355$	0	0
$356$	−6.53821e35	−1.24497
$357$	0	0
$358$	−2.69134e35	−0.473813
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.81625e36	2.84494
$362$	0	0
$363$	−1.74299e35	−0.252698
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−8.00756e35	−0.922849
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	−2.77022e36	−2.64431
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.50148e36	1.98266	0.991329	−	0.131406i	$-0.0419491\pi$
$379$	2.50148e36	1.98266	0.991329	+	0.131406i	$0.0419491\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.37367e35	−0.0906245
$385$	0	0
$386$	2.93264e36	1.79902
$387$	−2.20688e36	−1.30564
$388$	−2.44571e36	−1.39560
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	2.02304e36	1.00000
$393$	−3.72178e35	−0.177523
$394$	0	0
$395$	0	0
$396$	−4.50173e36	−1.93040
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	2.68435e36	1.00000
$401$	−2.67413e36	−0.961969	−0.480985	−	0.876729i	$-0.659721\pi$
$401$	−2.67413e36	−0.961969	−0.480985	+	0.876729i	$0.659721\pi$
$402$	4.69525e35	0.163115
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	4.36085e35	0.123120
$409$	−6.76719e36	−1.84621	−0.923107	−	0.384543i	$-0.874359\pi$
$409$	−6.76719e36	−1.84621	−0.923107	+	0.384543i	$0.874359\pi$
$410$	0	0
$411$	−3.99512e35	−0.101799
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.53550e35	0.115147
$418$	1.89732e37	3.81657
$419$	1.01140e37	1.96755	0.983776	−	0.179401i	$-0.0574158\pi$
$419$	1.01140e37	1.96755	0.983776	+	0.179401i	$0.0574158\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	6.60367e35	0.116255
$423$	0	0
$424$	0	0
$425$	−8.52175e36	−1.35857
$426$	0	0
$427$	0	0
$428$	−5.19879e36	−0.751086
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.42318e36	0.180505
$433$	1.12789e37	1.38496	0.692480	−	0.721437i	$-0.256518\pi$
$433$	1.12789e37	1.38496	0.692480	+	0.721437i	$0.256518\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.71574e36	0.179397
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−1.04366e37	−0.991787
$442$	0	0
$443$	−6.88239e36	−0.613885	−0.306943	−	0.951728i	$-0.599306\pi$
$443$	−6.88239e36	−0.613885	−0.306943	+	0.951728i	$0.599306\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.22325e37	−1.64263	−0.821315	−	0.570475i	$-0.806759\pi$
$449$	−2.22325e37	−1.64263	−0.821315	+	0.570475i	$0.806759\pi$
$450$	−1.38482e37	−0.991787
$451$	2.60863e37	1.81109
$452$	−2.97142e37	−1.99998
$453$	0	0
$454$	−2.75942e37	−1.74597
$455$	0	0
$456$	−2.98675e36	−0.177701
$457$	2.96149e37	1.70877	0.854386	−	0.519640i	$-0.173934\pi$
$457$	2.96149e37	1.70877	0.854386	+	0.519640i	$0.173934\pi$
$458$	0	0
$459$	−4.51803e36	−0.245229
$460$	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
$466$	2.20772e37	0.969485
$467$	−4.09651e37	−1.74573	−0.872867	−	0.487958i	$-0.837742\pi$
$467$	−4.09651e37	−1.74573	−0.872867	+	0.487958i	$0.837742\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	1.29148e37	0.474134
$473$	7.18936e37	2.56233
$474$	0	0
$475$	5.83654e37	1.96085
$476$	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
$482$	−6.46732e37	−1.77038
$483$	0	0
$484$	1.07942e38	2.78841
$485$	0	0
$486$	−1.10281e37	−0.268902
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	4.73779e36	0.105987
$490$	0	0
$491$	−8.95358e37	−1.89172	−0.945858	−	0.324580i	$-0.894777\pi$
$491$	−8.95358e37	−1.89172	−0.945858	+	0.324580i	$0.894777\pi$
$492$	−4.10648e36	−0.0843253
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	3.56961e36	0.0618601
$499$	−1.04150e38	−1.75490	−0.877452	−	0.479664i	$-0.840759\pi$
$499$	−1.04150e38	−1.75490	−0.877452	+	0.479664i	$0.840759\pi$
$500$	0	0
$501$	0	0
$502$	7.51984e37	1.16508
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.71981e36	−0.0906245
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	8.50706e37	1.00000
$513$	3.09440e37	0.353943
$514$	−1.40518e38	−1.56404
$515$	0	0
$516$	−1.13174e37	−0.119303
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.85217e38	−1.70589	−0.852946	−	0.521999i	$-0.825187\pi$
$521$	−1.85217e38	−1.70589	−0.852946	+	0.521999i	$0.825187\pi$
$522$	0	0
$523$	−1.56556e38	−1.36662	−0.683308	−	0.730130i	$-0.739460\pi$
$523$	−1.56556e38	−1.36662	−0.683308	+	0.730130i	$0.739460\pi$
$524$	2.30488e38	1.95889
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−2.30860e37	−0.176390
$529$	1.34394e38	1.00000
$530$	0	0
$531$	−6.66259e37	−0.470240
$532$	0	0
$533$	0	0
$534$	1.72975e37	0.112825
$535$	0	0
$536$	−2.90774e38	−1.79990
$537$	7.12020e36	0.0429390
$538$	0	0
$539$	3.39994e38	1.94638
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−2.70065e38	−1.35857
$545$	0	0
$546$	0	0
$547$	1.65293e38	0.769895	0.384947	−	0.922938i	$-0.374220\pi$
$547$	1.65293e38	0.769895	0.384947	+	0.922938i	$0.374220\pi$
$548$	2.47415e38	1.12331
$549$	0	0
$550$	4.51135e38	1.94638
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−3.42810e38	−1.27059
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	7.32888e37	0.239639
$562$	−6.18982e38	−1.97410
$563$	6.11783e38	1.90318	0.951590	−	0.307371i	$-0.0994493\pi$
$563$	6.11783e38	1.90318	0.951590	+	0.307371i	$0.0994493\pi$
$564$	0	0
$565$	0	0
$566$	3.10336e38	0.896194
$567$	0	0
$568$	0	0
$569$	2.20809e38	0.592169	0.296085	−	0.955162i	$-0.404319\pi$
$569$	2.20809e38	0.592169	0.296085	+	0.955162i	$0.404319\pi$
$570$	0	0
$571$	5.29747e38	1.35258	0.676290	−	0.736635i	$-0.263586\pi$
$571$	5.29747e38	1.35258	0.676290	+	0.736635i	$0.263586\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−4.38867e38	−0.991787
$577$	−9.06491e38	−1.99941	−0.999706	−	0.0242611i	$-0.992277\pi$
$577$	−9.06491e38	−1.99941	−0.999706	+	0.0242611i	$0.992277\pi$
$578$	3.92843e38	0.845726
$579$	−7.75857e37	−0.163035
$580$	0	0
$581$	0	0
$582$	6.47037e37	0.126476
$583$	0	0
$584$	−1.06255e39	−1.97956
$585$	0	0
$586$	0	0
$587$	−1.08948e39	−1.88923	−0.944614	−	0.328183i	$-0.893564\pi$
$587$	−1.08948e39	−1.88923	−0.944614	+	0.328183i	$0.893564\pi$
$588$	−5.35215e37	−0.0906245
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5.12719e38	−0.771106	−0.385553	−	0.922686i	$-0.625989\pi$
$593$	−5.12719e38	−0.771106	−0.385553	+	0.922686i	$0.625989\pi$
$594$	2.39181e38	0.351331
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−7.10171e37	−0.0906245
$601$	−1.42929e39	−1.78188	−0.890942	−	0.454118i	$-0.849954\pi$
$601$	−1.42929e39	−1.78188	−0.890942	+	0.454118i	$0.849954\pi$
$602$	0	0
$603$	1.50006e39	1.78512
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	1.84967e39	1.96085
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.39323e39	1.34742
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.82846e39	1.68938
$615$	0	0
$616$	0	0
$617$	1.77912e39	1.53537	0.767687	−	0.640825i	$-0.221408\pi$
$617$	1.77912e39	1.53537	0.767687	+	0.640825i	$0.221408\pi$
$618$	0	0
$619$	−2.30663e39	−1.90243	−0.951214	−	0.308531i	$-0.900163\pi$
$619$	−2.30663e39	−1.90243	−0.951214	+	0.308531i	$0.900163\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.38778e39	1.00000
$626$	−4.76300e38	−0.335614
$627$	−5.01955e38	−0.345875
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−1.74706e37	−0.0105355
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.49823e39	−1.26363	−0.631816	−	0.775118i	$-0.717690\pi$
$641$	−2.49823e39	−1.26363	−0.631816	+	0.775118i	$0.717690\pi$
$642$	1.37539e38	0.0680668
$643$	−2.26170e37	−0.0109517	−0.00547586	−	0.999985i	$-0.501743\pi$
$643$	−2.26170e37	−0.0109517	−0.00547586	+	0.999985i	$0.501743\pi$
$644$	0	0
$645$	0	0
$646$	−5.87197e39	−2.66397
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	2.24515e39	0.975429
$649$	2.17048e39	0.922847
$650$	0	0
$651$	0	0
$652$	−2.93408e39	−1.16952
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	2.54312e39	0.930491
$657$	5.48154e39	1.96330
$658$	0	0
$659$	−4.79298e39	−1.64517	−0.822583	−	0.568645i	$-0.807468\pi$
$659$	−4.79298e39	−1.64517	−0.822583	+	0.568645i	$0.807468\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−5.93908e39	−1.91297
$663$	0	0
$664$	−2.21064e39	−0.682598
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	5.14618e39	1.31606	0.658028	−	0.752993i	$-0.271391\pi$
$673$	5.14618e39	1.31606	0.658028	+	0.752993i	$0.271391\pi$
$674$	−3.31131e38	−0.0829395
$675$	7.35769e38	0.180505
$676$	4.16154e39	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	7.86118e38	0.181247
$679$	0	0
$680$	0	0
$681$	7.30032e38	0.158228
$682$	0	0
$683$	5.21481e39	1.08480	0.542399	−	0.840121i	$-0.317516\pi$
$683$	5.21481e39	1.08480	0.542399	+	0.840121i	$0.317516\pi$
$684$	−9.54221e39	−1.94475
$685$	0	0
$686$	0	0
$687$	0	0
$688$	7.00881e39	1.31645
$689$	0	0
$690$	0	0
$691$	−1.71660e38	−0.0303372	−0.0151686	−	0.999885i	$-0.504829\pi$
$691$	−1.71660e38	−0.0303372	−0.0151686	+	0.999885i	$0.504829\pi$
$692$	0	0
$693$	0	0
$694$	−2.19185e39	−0.364568
$695$	0	0
$696$	0	0
$697$	−8.07337e39	−1.26414
$698$	0	0
$699$	−5.84073e38	−0.0878591
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.42970e40	1.94638
$705$	0	0
$706$	1.11264e40	1.45576
$707$	0	0
$708$	−3.41674e38	−0.0429681
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	−1.07122e40	−1.24497
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.40950e39	−0.473813
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	2.97574e40	2.84494
$723$	1.71099e39	0.160439
$724$	0	0
$725$	0	0
$726$	−2.85571e39	−0.252698
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.13866e40	−0.951060
$730$	0	0
$731$	−2.22501e40	−1.78850
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.88677e40	−3.50330
$738$	−1.31196e40	−0.922849
$739$	2.89122e40	1.99553	0.997765	−	0.0668168i	$-0.0212843\pi$
$739$	2.89122e40	1.99553	0.997765	+	0.0668168i	$0.0212843\pi$
$740$	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
$747$	1.14044e40	0.676992
$748$	−4.53873e40	−2.64431
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−1.98944e39	−0.105585
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	4.09842e40	1.98266
$759$	0	0
$760$	0	0
$761$	3.83348e40	1.75472	0.877361	−	0.479831i	$-0.159302\pi$
$761$	3.83348e40	1.75472	0.877361	+	0.479831i	$0.159302\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−2.25062e39	−0.0906245
$769$	2.08676e40	0.825095	0.412548	−	0.910936i	$-0.364639\pi$
$769$	2.08676e40	0.825095	0.412548	+	0.910936i	$0.364639\pi$
$770$	0	0
$771$	3.71753e39	0.141740
$772$	4.80483e40	1.79902
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−3.61575e40	−1.30564
$775$	0	0
$776$	−4.00706e40	−1.39560
$777$	0	0
$778$	0	0
$779$	5.52945e40	1.82456
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.31455e40	1.00000
$785$	0	0
$786$	−6.09776e39	−0.177523
$787$	5.69473e40	1.62865	0.814323	−	0.580412i	$-0.197109\pi$
$787$	5.69473e40	1.62865	0.814323	+	0.580412i	$0.197109\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−7.37564e40	−1.93040
$793$	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
$800$	4.39805e40	1.00000
$801$	5.52628e40	1.23475
$802$	−4.38129e40	−0.961969
$803$	−1.78573e41	−3.85299
$804$	7.69270e39	0.163115
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.01161e41	−1.96667	−0.983337	−	0.181790i	$-0.941811\pi$
$809$	−1.01161e41	−1.96667	−0.983337	+	0.181790i	$0.941811\pi$
$810$	0	0
$811$	−3.33491e39	−0.0626315	−0.0313157	−	0.999510i	$-0.509970\pi$
$811$	−3.33491e39	−0.0626315	−0.0313157	+	0.999510i	$0.509970\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	7.14482e39	0.123120
$817$	1.52391e41	2.58137
$818$	−1.10874e41	−1.84621
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−6.54560e39	−0.101799
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	−1.19352e40	−0.176390
$826$	0	0
$827$	−1.28098e41	−1.83006	−0.915030	−	0.403386i	$-0.867833\pi$
$827$	−1.28098e41	−1.83006	−0.915030	+	0.403386i	$0.867833\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.05224e41	−1.35857
$834$	9.06936e39	0.115147
$835$	0	0
$836$	3.10858e41	3.81657
$837$	0	0
$838$	1.65708e41	1.96755
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	8.85409e40	1.00000
$842$	0	0
$843$	1.63757e40	0.178902
$844$	1.08195e40	0.116255
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.21022e39	−0.0812171
$850$	−1.39620e41	−1.35857
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−8.51770e40	−0.751086
$857$	2.64144e40	0.229145	0.114572	−	0.993415i	$-0.463450\pi$
$857$	2.64144e40	0.229145	0.114572	+	0.993415i	$0.463450\pi$
$858$	0	0
$859$	−1.43184e41	−1.20224	−0.601120	−	0.799159i	$-0.705278\pi$
$859$	−1.43184e41	−1.20224	−0.601120	+	0.799159i	$0.705278\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	2.33174e40	0.180505
$865$	0	0
$866$	1.84794e41	1.38496
$867$	−1.03930e40	−0.0766434
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.06719e41	1.38414
$874$	0	0
$875$	0	0
$876$	2.81107e40	0.179397
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.36723e41	−1.39501	−0.697504	−	0.716580i	$-0.745706\pi$
$881$	−2.36723e41	−1.39501	−0.697504	+	0.716580i	$0.745706\pi$
$882$	−1.70993e41	−0.991787
$883$	2.05927e41	1.17561	0.587805	−	0.809002i	$-0.299992\pi$
$883$	2.05927e41	1.17561	0.587805	+	0.809002i	$0.299992\pi$
$884$	0	0
$885$	0	0
$886$	−1.12761e41	−0.613885
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.77322e41	1.89856
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−3.64257e41	−1.64263
$899$	0	0
$900$	−2.26889e41	−0.991787
$901$	0	0
$902$	4.27398e41	1.81109
$903$	0	0
$904$	−4.86838e41	−1.99998
$905$	0	0
$906$	0	0
$907$	−2.14835e40	−0.0842564	−0.0421282	−	0.999112i	$-0.513414\pi$
$907$	−2.14835e40	−0.0842564	−0.0421282	+	0.999112i	$0.513414\pi$
$908$	−4.52104e41	−1.74597
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−4.89348e40	−0.177701
$913$	−3.71522e41	−1.32860
$914$	4.85211e41	1.70877
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−7.40235e40	−0.245229
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−4.83736e40	−0.153099
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.18705e41	1.17405	0.587027	−	0.809567i	$-0.300298\pi$
$929$	4.18705e41	1.17405	0.587027	+	0.809567i	$0.300298\pi$
$930$	0	0
$931$	7.20677e41	1.96085
$932$	3.61713e41	0.969485
$933$	0	0
$934$	−6.71173e41	−1.74573
$935$	0	0
$936$	0	0
$937$	−5.72583e41	−1.42392	−0.711958	−	0.702222i	$-0.752191\pi$
$937$	−5.72583e41	−1.42392	−0.711958	+	0.702222i	$0.752191\pi$
$938$	0	0
$939$	1.26010e40	0.0304148
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	2.11597e41	0.474134
$945$	0	0
$946$	1.17791e42	2.56233
$947$	−1.10786e41	−0.237458	−0.118729	−	0.992927i	$-0.537882\pi$
$947$	−1.10786e41	−0.237458	−0.118729	+	0.992927i	$0.537882\pi$
$948$	0	0
$949$	0	0
$950$	9.56259e41	1.96085
$951$	0	0
$952$	0	0
$953$	−5.12513e41	−1.00555	−0.502776	−	0.864417i	$-0.667688\pi$
$953$	−5.12513e41	−1.00555	−0.502776	+	0.864417i	$0.667688\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.72964e41	1.00000
$962$	0	0
$963$	4.39416e41	0.744918
$964$	−1.05961e42	−1.77038
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.76853e42	2.78841
$969$	1.55348e41	0.241420
$970$	0	0
$971$	−6.33455e41	−0.956416	−0.478208	−	0.878247i	$-0.658713\pi$
$971$	−6.33455e41	−0.956416	−0.478208	+	0.878247i	$0.658713\pi$
$972$	−1.80685e41	−0.268902
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.30017e42	−1.80085	−0.900425	−	0.435012i	$-0.856744\pi$
$977$	−1.30017e42	−1.80085	−0.900425	+	0.435012i	$0.856744\pi$
$978$	7.76239e40	0.105987
$979$	−1.80030e42	−2.42319
$980$	0	0
$981$	0	0
$982$	−1.46696e42	−1.89172
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−6.72805e40	−0.0843253
$985$	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
$993$	1.57124e41	0.173362
$994$	0	0
$995$	0	0
$996$	5.84845e40	0.0618601
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.70640e42	−1.75490
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.29.d.a.3.1	✓	1
4.3	odd	2		32.29.d.a.15.1		1
8.3	odd	2	CM	8.29.d.a.3.1	✓	1
8.5	even	2		32.29.d.a.15.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.29.d.a.3.1	✓	1	1.1	even	1	trivial
8.29.d.a.3.1	✓	1	8.3	odd	2	CM
32.29.d.a.15.1		1	4.3	odd	2
32.29.d.a.15.1		1	8.5	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:	\( k \)	\(=\)	\( 29 \)
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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