LMFDB - Embedded newform 54.21.b.c.53.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [54,21,Mod(53,54)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("54.53"); S:= CuspForms(chi, 21); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 21, names="a")

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 21 $
Character orbit:	$[\chi]$	$=$	54.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$136.897433155$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 64181120673558 x^{10} + \cdots + 59\!\cdots\!25 $ x^12 + 64181120673558x^10 + 1147499603855864639145548799x^8 + 4865528987041184601846573424159143560500x^6 + 7351376169346404374160247268269786722486327446859375x^4 + 3743587722766332549450519467149040800968992595997641252058593750*x^2 + 591904334199598449749405348007426564811862653246069965388281744211181640625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{63}\cdot 3^{90} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.9
Root			$-674291. i$ of defining polynomial
Character	$\chi$	$=$	54.53
Dual form			54.21.b.c.53.4

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+724.077i q^{2} -524288. q^{4} -2.02287e6i q^{5} +4.37005e8 q^{7} -3.79625e8i q^{8} +1.46472e9 q^{10} -3.44048e10i q^{11} -1.96478e11 q^{13} +3.16425e11i q^{14} +2.74878e11 q^{16} +2.59646e12i q^{17} +8.61761e12 q^{19} +1.06057e12i q^{20} +2.49117e13 q^{22} -6.26414e12i q^{23} +9.12754e13 q^{25} -1.42265e14i q^{26} -2.29116e14 q^{28} +4.94120e14i q^{29} +1.00983e15 q^{31} +1.99033e14i q^{32} -1.88004e15 q^{34} -8.84004e14i q^{35} -5.04313e15 q^{37} +6.23982e15i q^{38} -7.67933e14 q^{40} -2.43341e16i q^{41} -4.15395e16 q^{43} +1.80380e16i q^{44} +4.53572e15 q^{46} +2.35165e16i q^{47} +1.11181e17 q^{49} +6.60905e16i q^{50} +1.03011e17 q^{52} -1.50617e17i q^{53} -6.95965e16 q^{55} -1.65898e17i q^{56} -3.57781e17 q^{58} -7.23785e17i q^{59} -2.58232e16 q^{61} +7.31194e17i q^{62} -1.44115e17 q^{64} +3.97450e17i q^{65} +2.83393e17 q^{67} -1.36129e18i q^{68} +6.40088e17 q^{70} +2.76579e18i q^{71} +4.43282e18 q^{73} -3.65162e18i q^{74} -4.51811e18 q^{76} -1.50351e19i q^{77} -5.86582e17 q^{79} -5.56043e17i q^{80} +1.76198e19 q^{82} +1.82456e19i q^{83} +5.25230e18 q^{85} -3.00778e19i q^{86} -1.30609e19 q^{88} -4.04529e19i q^{89} -8.58619e19 q^{91} +3.28421e18i q^{92} -1.70278e19 q^{94} -1.74323e19i q^{95} +7.22477e19 q^{97} +8.05035e19i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6291456 q^{4} + 355856172 q^{7} - 5360984064 q^{10} + 374332013160 q^{13} + 3298534883328 q^{16} + 13268452496064 q^{19} + 64109345587200 q^{22} - 10850992436544 q^{25} - 186571120705536 q^{28} + 13\!\cdots\!76 q^{31}+ \cdots - 30\!\cdots\!60 q^{97}+O(q^{100}) $ 12 * q - 6291456 * q^4 + 355856172 * q^7 - 5360984064 * q^10 + 374332013160 * q^13 + 3298534883328 * q^16 + 13268452496064 * q^19 + 64109345587200 * q^22 - 10850992436544 * q^25 - 186571120705536 * q^28 + 1341631815613476 * q^31 - 1150892216451072 * q^34 - 1840381413631248 * q^37 + 2810699612946432 * q^40 - 58876514549555304 * q^43 - 52022884060053504 * q^46 + 301701288785631840 * q^49 - 196257782515630080 * q^52 + 1198488617517746484 * q^55 - 952998745417777152 * q^58 - 611525981019002448 * q^61 - 1729382256910270464 * q^64 - 3378608356010016504 * q^67 - 10123347229078487040 * q^70 + 35504687328043033068 * q^73 - 6956490422256402432 * q^76 + 15118277214382580040 * q^79 - 16372057824399605760 * q^82 - 2077217010606501720 * q^85 - 33611760579221913600 * q^88 - 154840755366303827352 * q^91 + 150175394882225012736 * q^94 - 3033444647600404860 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$.

$n$	$29$
$\chi(n)$	$-1$

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 $

$2$	724.077i	0.707107i
$2$	724.077i	0.707107i
$3$	0	0
$3$	0	0
$4$	−524288.	−0.500000
$5$	− 2.02287e6i	− 0.207142i	−0.994622	−	0.103571i	$-0.966973\pi$
$5$	− 2.02287e6i	− 0.207142i	0.994622	−	0.103571i	$-0.0330269\pi$
$6$	0	0
$7$	4.37005e8	1.54705	0.773527	−	0.633763i	$-0.218490\pi$
$7$	4.37005e8	1.54705	0.773527	+	0.633763i	$0.218490\pi$
$8$	− 3.79625e8i	− 0.353553i
$9$	0	0
$10$	1.46472e9	0.146472
$11$	− 3.44048e10i	− 1.32645i	−0.748419	−	0.663227i	$-0.769187\pi$
$11$	− 3.44048e10i	− 1.32645i	0.748419	−	0.663227i	$-0.230813\pi$
$12$	0	0
$13$	−1.96478e11	−1.42522	−0.712608	−	0.701562i	$-0.752486\pi$
$13$	−1.96478e11	−1.42522	−0.712608	+	0.701562i	$0.752486\pi$
$14$	3.16425e11i	1.09393i
$15$	0	0
$16$	2.74878e11	0.250000
$17$	2.59646e12i	1.28793i	0.765055	+	0.643965i	$0.222712\pi$
$17$	2.59646e12i	1.28793i	−0.765055	+	0.643965i	$0.777288\pi$
$18$	0	0
$19$	8.61761e12	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$19$	8.61761e12	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$20$	1.06057e12i	0.103571i
$21$	0	0
$22$	2.49117e13	0.937944
$23$	− 6.26414e12i	− 0.151211i	−0.997138	−	0.0756055i	$-0.975911\pi$
$23$	− 6.26414e12i	− 0.151211i	0.997138	−	0.0756055i	$-0.0240890\pi$
$24$	0	0
$25$	9.12754e13	0.957092
$26$	− 1.42265e14i	− 1.00778i
$27$	0	0
$28$	−2.29116e14	−0.773527
$29$	4.94120e14i	1.17450i	0.809406	+	0.587249i	$0.199789\pi$
$29$	4.94120e14i	1.17450i	−0.809406	+	0.587249i	$0.800211\pi$
$30$	0	0
$31$	1.00983e15	1.23206	0.616028	−	0.787724i	$-0.288741\pi$
$31$	1.00983e15	1.23206	0.616028	+	0.787724i	$0.288741\pi$
$32$	1.99033e14i	0.176777i
$33$	0	0
$34$	−1.88004e15	−0.910704
$35$	− 8.84004e14i	− 0.320460i
$36$	0	0
$37$	−5.04313e15	−1.04878	−0.524388	−	0.851479i	$-0.675706\pi$
$37$	−5.04313e15	−1.04878	−0.524388	+	0.851479i	$0.675706\pi$
$38$	6.23982e15i	0.993884i
$39$	0	0
$40$	−7.67933e14	−0.0732358
$41$	− 2.43341e16i	− 1.81291i	−0.422300	−	0.906456i	$-0.638777\pi$
$41$	− 2.43341e16i	− 1.81291i	0.422300	−	0.906456i	$-0.361223\pi$
$42$	0	0
$43$	−4.15395e16	−1.92211	−0.961053	−	0.276365i	$-0.910870\pi$
$43$	−4.15395e16	−1.92211	−0.961053	+	0.276365i	$0.910870\pi$
$44$	1.80380e16i	0.663227i
$45$	0	0
$46$	4.53572e15	0.106922
$47$	2.35165e16i	0.447089i	0.974694	+	0.223545i	$0.0717629\pi$
$47$	2.35165e16i	0.447089i	−0.974694	+	0.223545i	$0.928237\pi$
$48$	0	0
$49$	1.11181e17	1.39338
$50$	6.60905e16i	0.676766i
$51$	0	0
$52$	1.03011e17	0.712608
$53$	− 1.50617e17i	− 0.861224i	−0.902537	−	0.430612i	$-0.858298\pi$
$53$	− 1.50617e17i	− 0.861224i	0.902537	−	0.430612i	$-0.141702\pi$
$54$	0	0
$55$	−6.95965e16	−0.274764
$56$	− 1.65898e17i	− 0.546967i
$57$	0	0
$58$	−3.57781e17	−0.830496
$59$	− 7.23785e17i	− 1.41608i	−0.706170	−	0.708042i	$-0.749579\pi$
$59$	− 7.23785e17i	− 1.41608i	0.706170	−	0.708042i	$-0.250421\pi$
$60$	0	0
$61$	−2.58232e16	−0.0362003	−0.0181002	−	0.999836i	$-0.505762\pi$
$61$	−2.58232e16	−0.0362003	−0.0181002	+	0.999836i	$0.505762\pi$
$62$	7.31194e17i	0.871196i
$63$	0	0
$64$	−1.44115e17	−0.125000
$65$	3.97450e17i	0.295222i
$66$	0	0
$67$	2.83393e17	0.155468	0.0777339	−	0.996974i	$-0.475232\pi$
$67$	2.83393e17	0.155468	0.0777339	+	0.996974i	$0.475232\pi$
$68$	− 1.36129e18i	− 0.643965i
$69$	0	0
$70$	6.40088e17	0.226600
$71$	2.76579e18i	0.849643i	0.905277	+	0.424821i	$0.139663\pi$
$71$	2.76579e18i	0.849643i	−0.905277	+	0.424821i	$0.860337\pi$
$72$	0	0
$73$	4.43282e18	1.03146	0.515729	−	0.856752i	$-0.327521\pi$
$73$	4.43282e18	1.03146	0.515729	+	0.856752i	$0.327521\pi$
$74$	− 3.65162e18i	− 0.741597i
$75$	0	0
$76$	−4.51811e18	−0.702782
$77$	− 1.50351e19i	− 2.05210i
$78$	0	0
$79$	−5.86582e17	−0.0619524	−0.0309762	−	0.999520i	$-0.509862\pi$
$79$	−5.86582e17	−0.0619524	−0.0309762	+	0.999520i	$0.509862\pi$
$80$	− 5.56043e17i	− 0.0517855i
$81$	0	0
$82$	1.76198e19	1.28192
$83$	1.82456e19i	1.17592i	0.808891	+	0.587959i	$0.200068\pi$
$83$	1.82456e19i	1.17592i	−0.808891	+	0.587959i	$0.799932\pi$
$84$	0	0
$85$	5.25230e18	0.266785
$86$	− 3.00778e19i	− 1.35913i
$87$	0	0
$88$	−1.30609e19	−0.468972
$89$	− 4.04529e19i	− 1.29733i	−0.761075	−	0.648664i	$-0.775328\pi$
$89$	− 4.04529e19i	− 1.29733i	0.761075	−	0.648664i	$-0.224672\pi$
$90$	0	0
$91$	−8.58619e19	−2.20489
$92$	3.28421e18i	0.0756055i
$93$	0	0
$94$	−1.70278e19	−0.316140
$95$	− 1.74323e19i	− 0.291152i
$96$	0	0
$97$	7.22477e19	0.979730	0.489865	−	0.871798i	$-0.337046\pi$
$97$	7.22477e19	0.979730	0.489865	+	0.871798i	$0.337046\pi$
$98$	8.05035e19i	0.985268i
$99$	0	0
$100$	−4.78546e19	−0.478546
$101$	− 1.69955e20i	− 1.53858i	−0.638899	−	0.769291i	$-0.720610\pi$
$101$	− 1.69955e20i	− 1.53858i	0.638899	−	0.769291i	$-0.279390\pi$
$102$	0	0
$103$	1.19580e20	0.889788	0.444894	−	0.895583i	$-0.353241\pi$
$103$	1.19580e20	0.889788	0.444894	+	0.895583i	$0.353241\pi$
$104$	7.45881e19i	0.503890i
$105$	0	0
$106$	1.09059e20	0.608977
$107$	− 2.27746e20i	− 1.15774i	−0.815418	−	0.578872i	$-0.803493\pi$
$107$	− 2.27746e20i	− 1.15774i	0.815418	−	0.578872i	$-0.196507\pi$
$108$	0	0
$109$	2.62969e20	1.11081	0.555406	−	0.831579i	$-0.312563\pi$
$109$	2.62969e20	1.11081	0.555406	+	0.831579i	$0.312563\pi$
$110$	− 5.03932e19i	− 0.194288i
$111$	0	0
$112$	1.20123e20	0.386764
$113$	3.49988e20i	1.03102i	0.856882	+	0.515512i	$0.172398\pi$
$113$	3.49988e20i	1.03102i	−0.856882	+	0.515512i	$0.827602\pi$
$114$	0	0
$115$	−1.26716e19	−0.0313221
$116$	− 2.59061e20i	− 0.587249i
$117$	0	0
$118$	5.24076e20	1.00132
$119$	1.13467e21i	1.99250i
$120$	0	0
$121$	−5.10939e20	−0.759478
$122$	− 1.86980e19i	− 0.0255975i
$123$	0	0
$124$	−5.29441e20	−0.616028
$125$	− 3.77555e20i	− 0.405396i
$126$	0	0
$127$	−3.82486e20	−0.350411	−0.175206	−	0.984532i	$-0.556059\pi$
$127$	−3.82486e20	−0.350411	−0.175206	+	0.984532i	$0.556059\pi$
$128$	− 1.04351e20i	− 0.0883883i
$129$	0	0
$130$	−2.87785e20	−0.208754
$131$	− 1.36613e20i	− 0.0917864i	−0.998946	−	0.0458932i	$-0.985387\pi$
$131$	− 1.36613e20i	− 0.0917864i	0.998946	−	0.0458932i	$-0.0146134\pi$
$132$	0	0
$133$	3.76594e21	2.17449
$134$	2.05198e20i	0.109932i
$135$	0	0
$136$	9.85681e20	0.455352
$137$	1.08042e21i	0.463859i	0.972733	+	0.231929i	$0.0745038\pi$
$137$	1.08042e21i	0.463859i	−0.972733	+	0.231929i	$0.925496\pi$
$138$	0	0
$139$	2.69108e21	0.999491	0.499746	−	0.866172i	$-0.333427\pi$
$139$	2.69108e21	0.999491	0.499746	+	0.866172i	$0.333427\pi$
$140$	4.63473e20i	0.160230i
$141$	0	0
$142$	−2.00265e21	−0.600788
$143$	6.75979e21i	1.89048i
$144$	0	0
$145$	9.99541e20	0.243288
$146$	3.20971e21i	0.729351i
$147$	0	0
$148$	2.64405e21	0.524388
$149$	− 6.56676e21i	− 1.21755i	−0.793341	−	0.608777i	$-0.791660\pi$
$149$	− 6.56676e21i	− 1.21755i	0.793341	−	0.608777i	$-0.208340\pi$
$150$	0	0
$151$	3.32243e21	0.539121	0.269560	−	0.962983i	$-0.413122\pi$
$151$	3.32243e21	0.539121	0.269560	+	0.962983i	$0.413122\pi$
$152$	− 3.27146e21i	− 0.496942i
$153$	0	0
$154$	1.08865e22	1.45105
$155$	− 2.04275e21i	− 0.255211i
$156$	0	0
$157$	1.46150e22	1.60621	0.803104	−	0.595839i	$-0.203181\pi$
$157$	1.46150e22	1.60621	0.803104	+	0.595839i	$0.203181\pi$
$158$	− 4.24731e20i	− 0.0438070i
$159$	0	0
$160$	4.02618e20	0.0366179
$161$	− 2.73746e21i	− 0.233932i
$162$	0	0
$163$	−1.48548e21	−0.112200	−0.0560998	−	0.998425i	$-0.517867\pi$
$163$	−1.48548e21	−0.112200	−0.0560998	+	0.998425i	$0.517867\pi$
$164$	1.27581e22i	0.906456i
$165$	0	0
$166$	−1.32112e22	−0.831499
$167$	− 7.77950e21i	− 0.461091i	−0.973062	−	0.230546i	$-0.925949\pi$
$167$	− 7.77950e21i	− 0.461091i	0.973062	−	0.230546i	$-0.0740511\pi$
$168$	0	0
$169$	1.95987e22	1.03124
$170$	3.80307e21i	0.188645i
$171$	0	0
$172$	2.17787e22	0.961053
$173$	− 1.20656e22i	− 0.502444i	−0.967930	−	0.251222i	$-0.919168\pi$
$173$	− 1.20656e22i	− 0.502444i	0.967930	−	0.251222i	$-0.0808324\pi$
$174$	0	0
$175$	3.98878e22	1.48067
$176$	− 9.45711e21i	− 0.331613i
$177$	0	0
$178$	2.92911e22	0.917350
$179$	− 2.49361e22i	− 0.738410i	−0.929348	−	0.369205i	$-0.879630\pi$
$179$	− 2.49361e22i	− 0.738410i	0.929348	−	0.369205i	$-0.120370\pi$
$180$	0	0
$181$	4.79566e22	1.27076	0.635378	−	0.772201i	$-0.280844\pi$
$181$	4.79566e22	1.27076	0.635378	+	0.772201i	$0.280844\pi$
$182$	− 6.21707e22i	− 1.55909i
$183$	0	0
$184$	−2.37803e21	−0.0534611
$185$	1.02016e22i	0.217246i
$186$	0	0
$187$	8.93306e22	1.70838
$188$	− 1.23294e22i	− 0.223545i
$189$	0	0
$190$	1.26223e22	0.205875
$191$	4.12830e22i	0.638907i	0.947602	+	0.319454i	$0.103499\pi$
$191$	4.12830e22i	0.638907i	−0.947602	+	0.319454i	$0.896501\pi$
$192$	0	0
$193$	−9.88033e22	−1.37784	−0.688919	−	0.724838i	$-0.741915\pi$
$193$	−9.88033e22	−1.37784	−0.688919	+	0.724838i	$0.741915\pi$
$194$	5.23129e22i	0.692774i
$195$	0	0
$196$	−5.82908e22	−0.696689
$197$	− 6.33931e22i	− 0.720079i	−0.932937	−	0.360039i	$-0.882763\pi$
$197$	− 6.33931e22i	− 0.720079i	0.932937	−	0.360039i	$-0.117237\pi$
$198$	0	0
$199$	6.55536e22	0.673079	0.336540	−	0.941669i	$-0.390743\pi$
$199$	6.55536e22	0.673079	0.336540	+	0.941669i	$0.390743\pi$
$200$	− 3.46504e22i	− 0.338383i
$201$	0	0
$202$	1.23061e23	1.08794
$203$	2.15933e23i	1.81701i
$204$	0	0
$205$	−4.92248e22	−0.375530
$206$	8.65853e22i	0.629175i
$207$	0	0
$208$	−5.40075e22	−0.356304
$209$	− 2.96487e23i	− 1.86442i
$210$	0	0
$211$	9.18557e22	0.525148	0.262574	−	0.964912i	$-0.415429\pi$
$211$	9.18557e22	0.525148	0.262574	+	0.964912i	$0.415429\pi$
$212$	7.89668e22i	0.430612i
$213$	0	0
$214$	1.64906e23	0.818649
$215$	8.40292e22i	0.398149i
$216$	0	0
$217$	4.41300e23	1.90606
$218$	1.90410e23i	0.785462i
$219$	0	0
$220$	3.64886e22	0.137382
$221$	− 5.10148e23i	− 1.83558i
$222$	0	0
$223$	2.18805e23	0.719462	0.359731	−	0.933056i	$-0.382868\pi$
$223$	2.18805e23	0.719462	0.359731	+	0.933056i	$0.382868\pi$
$224$	8.69783e22i	0.273483i
$225$	0	0
$226$	−2.53419e23	−0.729045
$227$	5.79232e23i	1.59439i	0.603723	+	0.797194i	$0.293683\pi$
$227$	5.79232e23i	1.59439i	−0.603723	+	0.797194i	$0.706317\pi$
$228$	0	0
$229$	−5.45270e23	−1.37485	−0.687427	−	0.726253i	$-0.741260\pi$
$229$	−5.45270e23	−1.37485	−0.687427	+	0.726253i	$0.741260\pi$
$230$	− 9.17519e21i	− 0.0221481i
$231$	0	0
$232$	1.87580e23	0.415248
$233$	5.06658e23i	1.07437i	0.843463	+	0.537187i	$0.180513\pi$
$233$	5.06658e23i	1.07437i	−0.843463	+	0.537187i	$0.819487\pi$
$234$	0	0
$235$	4.75709e22	0.0926110
$236$	3.79472e23i	0.708042i
$237$	0	0
$238$	−8.21585e23	−1.40891
$239$	− 5.13059e22i	− 0.0843701i	−0.999110	−	0.0421850i	$-0.986568\pi$
$239$	− 5.13059e22i	− 0.0843701i	0.999110	−	0.0421850i	$-0.0134319\pi$
$240$	0	0
$241$	−4.12898e23	−0.624701	−0.312350	−	0.949967i	$-0.601116\pi$
$241$	−4.12898e23	−0.624701	−0.312350	+	0.949967i	$0.601116\pi$
$242$	− 3.69959e23i	− 0.537032i
$243$	0	0
$244$	1.35388e22	0.0181002
$245$	− 2.24905e23i	− 0.288627i
$246$	0	0
$247$	−1.69317e24	−2.00323
$248$	− 3.83356e23i	− 0.435598i
$249$	0	0
$250$	2.73379e23	0.286658
$251$	− 1.89093e23i	− 0.190519i	−0.995452	−	0.0952594i	$-0.969632\pi$
$251$	− 1.89093e23i	− 0.190519i	0.995452	−	0.0952594i	$-0.0303680\pi$
$252$	0	0
$253$	−2.15516e23	−0.200574
$254$	− 2.76949e23i	− 0.247778i
$255$	0	0
$256$	7.55579e22	0.0625000
$257$	− 1.34603e23i	− 0.107084i	−0.998566	−	0.0535420i	$-0.982949\pi$
$257$	− 1.34603e23i	− 0.107084i	0.998566	−	0.0535420i	$-0.0170511\pi$
$258$	0	0
$259$	−2.20387e24	−1.62251
$260$	− 2.08378e23i	− 0.147611i
$261$	0	0
$262$	9.89182e22	0.0649028
$263$	− 1.47156e24i	− 0.929437i	−0.885458	−	0.464718i	$-0.846156\pi$
$263$	− 1.47156e24i	− 0.929437i	0.885458	−	0.464718i	$-0.153844\pi$
$264$	0	0
$265$	−3.04679e23	−0.178396
$266$	2.72683e24i	1.53759i
$267$	0	0
$268$	−1.48579e23	−0.0777339
$269$	− 1.36350e24i	− 0.687279i	−0.939102	−	0.343640i	$-0.888340\pi$
$269$	− 1.36350e24i	− 0.687279i	0.939102	−	0.343640i	$-0.111660\pi$
$270$	0	0
$271$	−5.31859e23	−0.248945	−0.124472	−	0.992223i	$-0.539724\pi$
$271$	−5.31859e23	−0.248945	−0.124472	+	0.992223i	$0.539724\pi$
$272$	7.13709e23i	0.321983i
$273$	0	0
$274$	−7.82305e23	−0.327998
$275$	− 3.14031e24i	− 1.26954i
$276$	0	0
$277$	2.09322e24	0.787078	0.393539	−	0.919308i	$-0.371251\pi$
$277$	2.09322e24	0.787078	0.393539	+	0.919308i	$0.371251\pi$
$278$	1.94855e24i	0.706747i
$279$	0	0
$280$	−3.35590e23	−0.113300
$281$	6.39104e23i	0.208213i	0.994566	+	0.104107i	$0.0331983\pi$
$281$	6.39104e23i	0.208213i	−0.994566	+	0.104107i	$0.966802\pi$
$282$	0	0
$283$	−1.84721e24	−0.560597	−0.280299	−	0.959913i	$-0.590434\pi$
$283$	−1.84721e24	−0.560597	−0.280299	+	0.959913i	$0.590434\pi$
$284$	− 1.45007e24i	− 0.424821i
$285$	0	0
$286$	−4.89461e24	−1.33677
$287$	− 1.06341e25i	− 2.80467i
$288$	0	0
$289$	−2.67737e24	−0.658764
$290$	7.23745e23i	0.172031i
$291$	0	0
$292$	−2.32408e24	−0.515729
$293$	5.62853e24i	1.20703i	0.797351	+	0.603516i	$0.206234\pi$
$293$	5.62853e24i	1.20703i	−0.797351	+	0.603516i	$0.793766\pi$
$294$	0	0
$295$	−1.46412e24	−0.293331
$296$	1.91450e24i	0.370798i
$297$	0	0
$298$	4.75484e24	0.860941
$299$	1.23077e24i	0.215508i
$300$	0	0
$301$	−1.81530e25	−2.97360
$302$	2.40570e24i	0.381216i
$303$	0	0
$304$	2.36879e24	0.351391
$305$	5.22371e22i	0.00749861i
$306$	0	0
$307$	8.97493e24	1.20683	0.603417	−	0.797426i	$-0.293805\pi$
$307$	8.97493e24	1.20683	0.603417	+	0.797426i	$0.293805\pi$
$308$	7.88270e24i	1.02605i
$309$	0	0
$310$	1.47911e24	0.180461
$311$	− 1.00332e25i	− 1.18531i	−0.805455	−	0.592657i	$-0.798079\pi$
$311$	− 1.00332e25i	− 1.18531i	0.805455	−	0.592657i	$-0.201921\pi$
$312$	0	0
$313$	1.24338e25	1.37771	0.688857	−	0.724897i	$-0.258113\pi$
$313$	1.24338e25	1.37771	0.688857	+	0.724897i	$0.258113\pi$
$314$	1.05824e25i	1.13576i
$315$	0	0
$316$	3.07538e23	0.0309762
$317$	− 8.79433e24i	− 0.858243i	−0.903247	−	0.429122i	$-0.858823\pi$
$317$	− 8.79433e24i	− 0.858243i	0.903247	−	0.429122i	$-0.141177\pi$
$318$	0	0
$319$	1.70001e25	1.55792
$320$	2.91527e23i	0.0258928i
$321$	0	0
$322$	1.98213e24	0.165415
$323$	2.23753e25i	1.81027i
$324$	0	0
$325$	−1.79336e25	−1.36406
$326$	− 1.07560e24i	− 0.0793371i
$327$	0	0
$328$	−9.23783e24	−0.640961
$329$	1.02768e25i	0.691672i
$330$	0	0
$331$	−6.80745e24	−0.431225	−0.215613	−	0.976479i	$-0.569175\pi$
$331$	−6.80745e24	−0.431225	−0.215613	+	0.976479i	$0.569175\pi$
$332$	− 9.56594e24i	− 0.587959i
$333$	0	0
$334$	5.63296e24	0.326041
$335$	− 5.73267e23i	− 0.0322039i
$336$	0	0
$337$	2.86535e25	1.51663	0.758314	−	0.651889i	$-0.226023\pi$
$337$	2.86535e25	1.51663	0.758314	+	0.651889i	$0.226023\pi$
$338$	1.41910e25i	0.729199i
$339$	0	0
$340$	−2.75372e24	−0.133392
$341$	− 3.47429e25i	− 1.63427i
$342$	0	0
$343$	1.37170e25	0.608579
$344$	1.57695e25i	0.679567i
$345$	0	0
$346$	8.73642e24	0.355281
$347$	− 2.91446e25i	− 1.15150i	−0.817626	−	0.575749i	$-0.804710\pi$
$347$	− 2.91446e25i	− 1.15150i	0.817626	−	0.575749i	$-0.195290\pi$
$348$	0	0
$349$	2.34394e25	0.874362	0.437181	−	0.899373i	$-0.355977\pi$
$349$	2.34394e25	0.874362	0.437181	+	0.899373i	$0.355977\pi$
$350$	2.88818e25i	1.04699i
$351$	0	0
$352$	6.84768e24	0.234486
$353$	1.28443e24i	0.0427527i	0.999772	+	0.0213763i	$0.00680481\pi$
$353$	1.28443e24i	0.0427527i	−0.999772	+	0.0213763i	$0.993195\pi$
$354$	0	0
$355$	5.59485e24	0.175997
$356$	2.12090e25i	0.648664i
$357$	0	0
$358$	1.80556e25	0.522135
$359$	− 4.53333e25i	− 1.27489i	−0.770495	−	0.637446i	$-0.779991\pi$
$359$	− 4.53333e25i	− 1.27489i	0.770495	−	0.637446i	$-0.220009\pi$
$360$	0	0
$361$	3.66732e25	0.975613
$362$	3.47243e25i	0.898561i
$363$	0	0
$364$	4.50164e25	1.10244
$365$	− 8.96703e24i	− 0.213658i
$366$	0	0
$367$	1.72256e25	0.388610	0.194305	−	0.980941i	$-0.437755\pi$
$367$	1.72256e25	0.388610	0.194305	+	0.980941i	$0.437755\pi$
$368$	− 1.72187e24i	− 0.0378027i
$369$	0	0
$370$	−7.38675e24	−0.153616
$371$	− 6.58204e25i	− 1.33236i
$372$	0	0
$373$	3.32998e25	0.638783	0.319392	−	0.947623i	$-0.396521\pi$
$373$	3.32998e25	0.638783	0.319392	+	0.947623i	$0.396521\pi$
$374$	6.46823e25i	1.20801i
$375$	0	0
$376$	8.92746e24	0.158070
$377$	− 9.70838e25i	− 1.67391i
$378$	0	0
$379$	3.09816e25	0.506654	0.253327	−	0.967381i	$-0.418475\pi$
$379$	3.09816e25	0.506654	0.253327	+	0.967381i	$0.418475\pi$
$380$	9.13956e24i	0.145576i
$381$	0	0
$382$	−2.98921e25	−0.451775
$383$	3.01209e25i	0.443486i	0.975105	+	0.221743i	$0.0711746\pi$
$383$	3.01209e25i	0.443486i	−0.975105	+	0.221743i	$0.928825\pi$
$384$	0	0
$385$	−3.04140e25	−0.425075
$386$	− 7.15412e25i	− 0.974279i
$387$	0	0
$388$	−3.78786e25	−0.489865
$389$	− 1.97965e25i	− 0.249513i	−0.992187	−	0.124756i	$-0.960185\pi$
$389$	− 1.97965e25i	− 0.249513i	0.992187	−	0.124756i	$-0.0398149\pi$
$390$	0	0
$391$	1.62646e25	0.194749
$392$	− 4.22070e25i	− 0.492634i
$393$	0	0
$394$	4.59015e25	0.509173
$395$	1.18658e24i	0.0128329i
$396$	0	0
$397$	−1.41656e25	−0.145657	−0.0728284	−	0.997344i	$-0.523203\pi$
$397$	−1.41656e25	−0.145657	−0.0728284	+	0.997344i	$0.523203\pi$
$398$	4.74659e25i	0.475939i
$399$	0	0
$400$	2.50896e25	0.239273
$401$	5.88256e25i	0.547170i	0.961848	+	0.273585i	$0.0882095\pi$
$401$	5.88256e25i	0.547170i	−0.961848	+	0.273585i	$0.911791\pi$
$402$	0	0
$403$	−1.98409e26	−1.75595
$404$	8.91054e25i	0.769291i
$405$	0	0
$406$	−1.56352e26	−1.28482
$407$	1.73508e26i	1.39115i
$408$	0	0
$409$	9.40317e25	0.717863	0.358931	−	0.933364i	$-0.383141\pi$
$409$	9.40317e25	0.717863	0.358931	+	0.933364i	$0.383141\pi$
$410$	− 3.56425e25i	− 0.265540i
$411$	0	0
$412$	−6.26944e25	−0.444894
$413$	− 3.16297e26i	− 2.19076i
$414$	0	0
$415$	3.69085e25	0.243582
$416$	− 3.91056e25i	− 0.251945i
$417$	0	0
$418$	2.14680e26	1.31834
$419$	− 2.47253e26i	− 1.48252i	−0.671217	−	0.741261i	$-0.734228\pi$
$419$	− 2.47253e26i	− 1.48252i	0.671217	−	0.741261i	$-0.265772\pi$
$420$	0	0
$421$	7.68761e25	0.439511	0.219756	−	0.975555i	$-0.429474\pi$
$421$	7.68761e25	0.439511	0.219756	+	0.975555i	$0.429474\pi$
$422$	6.65106e25i	0.371335i
$423$	0	0
$424$	−5.71781e25	−0.304489
$425$	2.36993e26i	1.23267i
$426$	0	0
$427$	−1.12849e25	−0.0560039
$428$	1.19404e26i	0.578872i
$429$	0	0
$430$	−6.08436e25	−0.281534
$431$	1.10445e26i	0.499313i	0.968334	+	0.249657i	$0.0803178\pi$
$431$	1.10445e26i	0.499313i	−0.968334	+	0.249657i	$0.919682\pi$
$432$	0	0
$433$	−3.58985e26	−1.54952	−0.774760	−	0.632255i	$-0.782129\pi$
$433$	−3.58985e26	−1.54952	−0.774760	+	0.632255i	$0.782129\pi$
$434$	3.19535e26i	1.34779i
$435$	0	0
$436$	−1.37872e26	−0.555406
$437$	− 5.39819e25i	− 0.212537i
$438$	0	0
$439$	1.49169e26	0.561090	0.280545	−	0.959841i	$-0.409485\pi$
$439$	1.49169e26	0.561090	0.280545	+	0.959841i	$0.409485\pi$
$440$	2.64206e25i	0.0971438i
$441$	0	0
$442$	3.69386e26	1.29795
$443$	5.23216e26i	1.79740i	0.438569	+	0.898698i	$0.355486\pi$
$443$	5.23216e26i	1.79740i	−0.438569	+	0.898698i	$0.644514\pi$
$444$	0	0
$445$	−8.18311e25	−0.268731
$446$	1.58431e26i	0.508736i
$447$	0	0
$448$	−6.29790e25	−0.193382
$449$	− 2.97258e26i	− 0.892628i	−0.894876	−	0.446314i	$-0.852736\pi$
$449$	− 2.97258e26i	− 0.892628i	0.894876	−	0.446314i	$-0.147264\pi$
$450$	0	0
$451$	−8.37209e26	−2.40474
$452$	− 1.83495e26i	− 0.515512i
$453$	0	0
$454$	−4.19409e26	−1.12740
$455$	1.73688e26i	0.456725i
$456$	0	0
$457$	−7.27800e26	−1.83168	−0.915841	−	0.401541i	$-0.868475\pi$
$457$	−7.27800e26	−1.83168	−0.915841	+	0.401541i	$0.868475\pi$
$458$	− 3.94818e26i	− 0.972169i
$459$	0	0
$460$	6.64354e24	0.0156611
$461$	− 5.49413e26i	− 1.26733i	−0.773607	−	0.633665i	$-0.781550\pi$
$461$	− 5.49413e26i	− 1.26733i	0.773607	−	0.633665i	$-0.218450\pi$
$462$	0	0
$463$	−5.42201e26	−1.19771	−0.598853	−	0.800859i	$-0.704377\pi$
$463$	−5.42201e26	−1.19771	−0.598853	+	0.800859i	$0.704377\pi$
$464$	1.35823e26i	0.293625i
$465$	0	0
$466$	−3.66859e26	−0.759697
$467$	− 6.11109e26i	− 1.23865i	−0.785133	−	0.619327i	$-0.787405\pi$
$467$	− 6.11109e26i	− 1.23865i	0.785133	−	0.619327i	$-0.212595\pi$
$468$	0	0
$469$	1.23844e26	0.240517
$470$	3.44450e25i	0.0654859i
$471$	0	0
$472$	−2.74767e26	−0.500662
$473$	1.42916e27i	2.54958i
$474$	0	0
$475$	7.86576e26	1.34526
$476$	− 5.94891e26i	− 0.996249i
$477$	0	0
$478$	3.71494e25	0.0596586
$479$	1.41791e26i	0.222994i	0.993765	+	0.111497i	$0.0355645\pi$
$479$	1.41791e26i	0.222994i	−0.993765	+	0.111497i	$0.964436\pi$
$480$	0	0
$481$	9.90865e26	1.49473
$482$	− 2.98970e26i	− 0.441730i
$483$	0	0
$484$	2.67879e26	0.379739
$485$	− 1.46148e26i	− 0.202943i
$486$	0	0
$487$	9.47193e26	1.26226	0.631130	−	0.775677i	$-0.282591\pi$
$487$	9.47193e26	1.26226	0.631130	+	0.775677i	$0.282591\pi$
$488$	9.80315e24i	0.0127987i
$489$	0	0
$490$	1.62848e26	0.204090
$491$	4.90571e26i	0.602402i	0.953561	+	0.301201i	$0.0973876\pi$
$491$	4.90571e26i	0.602402i	−0.953561	+	0.301201i	$0.902612\pi$
$492$	0	0
$493$	−1.28296e27	−1.51267
$494$	− 1.22599e27i	− 1.41650i
$495$	0	0
$496$	2.77580e26	0.308014
$497$	1.20867e27i	1.31444i
$498$	0	0
$499$	−1.48885e27	−1.55542	−0.777708	−	0.628625i	$-0.783618\pi$
$499$	−1.48885e27	−1.55542	−0.777708	+	0.628625i	$0.783618\pi$
$500$	1.97947e26i	0.202698i
$501$	0	0
$502$	1.36918e26	0.134717
$503$	− 2.00385e27i	− 1.93279i	−0.257065	−	0.966394i	$-0.582755\pi$
$503$	− 2.00385e27i	− 1.93279i	0.257065	−	0.966394i	$-0.417245\pi$
$504$	0	0
$505$	−3.43797e26	−0.318705
$506$	− 1.56051e26i	− 0.141827i
$507$	0	0
$508$	2.00533e26	0.175206
$509$	− 8.96927e26i	− 0.768386i	−0.923253	−	0.384193i	$-0.874480\pi$
$509$	− 8.96927e26i	− 0.768386i	0.923253	−	0.384193i	$-0.125520\pi$
$510$	0	0
$511$	1.93716e27	1.59572
$512$	5.47097e25i	0.0441942i
$513$	0	0
$514$	9.74632e25	0.0757198
$515$	− 2.41895e26i	− 0.184313i
$516$	0	0
$517$	8.09080e26	0.593043
$518$	− 1.59577e27i	− 1.14729i
$519$	0	0
$520$	1.50882e26	0.104377
$521$	6.00174e26i	0.407286i	0.979045	+	0.203643i	$0.0652782\pi$
$521$	6.00174e26i	0.407286i	−0.979045	+	0.203643i	$0.934722\pi$
$522$	0	0
$523$	3.68249e26	0.240505	0.120253	−	0.992743i	$-0.461630\pi$
$523$	3.68249e26	0.240505	0.120253	+	0.992743i	$0.461630\pi$
$524$	7.16245e25i	0.0458932i
$525$	0	0
$526$	1.06552e27	0.657211
$527$	2.62198e27i	1.58680i
$528$	0	0
$529$	1.67692e27	0.977135
$530$	− 2.20611e26i	− 0.126145i
$531$	0	0
$532$	−1.97444e27	−1.08724
$533$	4.78112e27i	2.58379i
$534$	0	0
$535$	−4.60701e26	−0.239818
$536$	− 1.07583e26i	− 0.0549662i
$537$	0	0
$538$	9.87281e26	0.485980
$539$	− 3.82515e27i	− 1.84825i
$540$	0	0
$541$	3.59844e27	1.67549	0.837744	−	0.546063i	$-0.183874\pi$
$541$	3.59844e27	1.67549	0.837744	+	0.546063i	$0.183874\pi$
$542$	− 3.85107e26i	− 0.176031i
$543$	0	0
$544$	−5.16781e26	−0.227676
$545$	− 5.31954e26i	− 0.230096i
$546$	0	0
$547$	−5.33365e26	−0.222408	−0.111204	−	0.993798i	$-0.535471\pi$
$547$	−5.33365e26	−0.222408	−0.111204	+	0.993798i	$0.535471\pi$
$548$	− 5.66450e26i	− 0.231929i
$549$	0	0
$550$	2.27383e27	0.897699
$551$	4.25813e27i	1.65083i
$552$	0	0
$553$	−2.56339e26	−0.0958438
$554$	1.51566e27i	0.556548i
$555$	0	0
$556$	−1.41090e27	−0.499746
$557$	− 2.49124e26i	− 0.0866689i	−0.999061	−	0.0433345i	$-0.986202\pi$
$557$	− 2.49124e26i	− 0.0866689i	0.999061	−	0.0433345i	$-0.0137981\pi$
$558$	0	0
$559$	8.16162e27	2.73942
$560$	− 2.42993e26i	− 0.0801150i
$561$	0	0
$562$	−4.62761e26	−0.147229
$563$	7.35341e24i	0.00229829i	0.999999	+	0.00114915i	$0.000365785\pi$
$563$	7.35341e24i	0.00229829i	−0.999999	+	0.00114915i	$0.999634\pi$
$564$	0	0
$565$	7.07981e26	0.213569
$566$	− 1.33752e27i	− 0.396402i
$567$	0	0
$568$	1.04997e27	0.300394
$569$	8.95392e26i	0.251704i	0.992049	+	0.125852i	$0.0401665\pi$
$569$	8.95392e26i	0.251704i	−0.992049	+	0.125852i	$0.959834\pi$
$570$	0	0
$571$	−4.54342e27	−1.23317	−0.616583	−	0.787290i	$-0.711483\pi$
$571$	−4.54342e27	−1.23317	−0.616583	+	0.787290i	$0.711483\pi$
$572$	− 3.54408e27i	− 0.945242i
$573$	0	0
$574$	7.69992e27	1.98320
$575$	− 5.71762e26i	− 0.144723i
$576$	0	0
$577$	−6.50735e27	−1.59091	−0.795456	−	0.606012i	$-0.792768\pi$
$577$	−6.50735e27	−1.59091	−0.795456	+	0.606012i	$0.792768\pi$
$578$	− 1.93862e27i	− 0.465817i
$579$	0	0
$580$	−5.24048e26	−0.121644
$581$	7.97341e27i	1.81921i
$582$	0	0
$583$	−5.18195e27	−1.14237
$584$	− 1.68281e27i	− 0.364676i
$585$	0	0
$586$	−4.07549e27	−0.853500
$587$	8.38706e27i	1.72675i	0.504564	+	0.863374i	$0.331653\pi$
$587$	8.38706e27i	1.72675i	−0.504564	+	0.863374i	$0.668347\pi$
$588$	0	0
$589$	8.70231e27	1.73174
$590$	− 1.06014e27i	− 0.207416i
$591$	0	0
$592$	−1.38624e27	−0.262194
$593$	− 5.84892e27i	− 1.08775i	−0.839167	−	0.543875i	$-0.816957\pi$
$593$	− 5.84892e27i	− 1.08775i	0.839167	−	0.543875i	$-0.183043\pi$
$594$	0	0
$595$	2.29528e27	0.412730
$596$	3.44287e27i	0.608777i
$597$	0	0
$598$	−8.91171e26	−0.152387
$599$	9.02366e27i	1.51745i	0.651411	+	0.758725i	$0.274178\pi$
$599$	9.02366e27i	1.51745i	−0.651411	+	0.758725i	$0.725822\pi$
$600$	0	0
$601$	2.54045e27	0.413205	0.206602	−	0.978425i	$-0.433759\pi$
$601$	2.54045e27	0.413205	0.206602	+	0.978425i	$0.433759\pi$
$602$	− 1.31442e28i	− 2.10265i
$603$	0	0
$604$	−1.74191e27	−0.269560
$605$	1.03356e27i	0.157320i
$606$	0	0
$607$	−1.72349e27	−0.253818	−0.126909	−	0.991914i	$-0.540506\pi$
$607$	−1.72349e27	−0.253818	−0.126909	+	0.991914i	$0.540506\pi$
$608$	1.71519e27i	0.248471i
$609$	0	0
$610$	−3.78237e25	−0.00530232
$611$	− 4.62048e27i	− 0.637199i
$612$	0	0
$613$	−7.86108e27	−1.04925	−0.524623	−	0.851334i	$-0.675794\pi$
$613$	−7.86108e27	−1.04925	−0.524623	+	0.851334i	$0.675794\pi$
$614$	6.49855e27i	0.853360i
$615$	0	0
$616$	−5.70768e27	−0.725526
$617$	− 1.15044e28i	− 1.43884i	−0.694577	−	0.719418i	$-0.744408\pi$
$617$	− 1.15044e28i	− 1.43884i	0.694577	−	0.719418i	$-0.255592\pi$
$618$	0	0
$619$	7.80551e27	0.945138	0.472569	−	0.881294i	$-0.343327\pi$
$619$	7.80551e27	0.945138	0.472569	+	0.881294i	$0.343327\pi$
$620$	1.07099e27i	0.127605i
$621$	0	0
$622$	7.26478e27	0.838144
$623$	− 1.76781e28i	− 2.00704i
$624$	0	0
$625$	7.94096e27	0.873118
$626$	9.00301e27i	0.974191i
$627$	0	0
$628$	−7.66246e27	−0.803104
$629$	− 1.30943e28i	− 1.35075i
$630$	0	0
$631$	−3.60278e27	−0.360035	−0.180017	−	0.983663i	$-0.557615\pi$
$631$	−3.60278e27	−0.360035	−0.180017	+	0.983663i	$0.557615\pi$
$632$	2.22681e26i	0.0219035i
$633$	0	0
$634$	6.36778e27	0.606870
$635$	7.73720e26i	0.0725850i
$636$	0	0
$637$	−2.18446e28	−1.98587
$638$	1.23094e28i	1.10161i
$639$	0	0
$640$	−2.11088e26	−0.0183089
$641$	− 5.30941e27i	− 0.453384i	−0.973967	−	0.226692i	$-0.927209\pi$
$641$	− 5.30941e27i	− 0.453384i	0.973967	−	0.226692i	$-0.0727911\pi$
$642$	0	0
$643$	−1.69621e28	−1.40401	−0.702004	−	0.712173i	$-0.747711\pi$
$643$	−1.69621e28	−1.40401	−0.702004	+	0.712173i	$0.747711\pi$
$644$	1.43522e27i	0.116966i
$645$	0	0
$646$	−1.62014e28	−1.28005
$647$	2.03420e28i	1.58253i	0.611475	+	0.791264i	$0.290576\pi$
$647$	2.03420e28i	1.58253i	−0.611475	+	0.791264i	$0.709424\pi$
$648$	0	0
$649$	−2.49017e28	−1.87837
$650$	− 1.29853e28i	− 0.964539i
$651$	0	0
$652$	7.78820e26	0.0560998
$653$	− 1.76479e28i	− 1.25188i	−0.779873	−	0.625938i	$-0.784716\pi$
$653$	− 1.76479e28i	− 1.25188i	0.779873	−	0.625938i	$-0.215284\pi$
$654$	0	0
$655$	−2.76350e26	−0.0190128
$656$	− 6.68891e27i	− 0.453228i
$657$	0	0
$658$	−7.44122e27	−0.489086
$659$	1.78828e28i	1.15766i	0.815448	+	0.578830i	$0.196491\pi$
$659$	1.78828e28i	1.15766i	−0.815448	+	0.578830i	$0.803509\pi$
$660$	0	0
$661$	−1.98883e28	−1.24906	−0.624531	−	0.781000i	$-0.714710\pi$
$661$	−1.98883e28	−1.24906	−0.624531	+	0.781000i	$0.714710\pi$
$662$	− 4.92912e27i	− 0.304922i
$663$	0	0
$664$	6.92648e27	0.415750
$665$	− 7.61801e27i	− 0.450427i
$666$	0	0
$667$	3.09524e27	0.177597
$668$	4.07870e27i	0.230546i
$669$	0	0
$670$	4.15090e26	0.0227716
$671$	8.88443e26i	0.0480180i
$672$	0	0
$673$	−5.09355e26	−0.0267221	−0.0133610	−	0.999911i	$-0.504253\pi$
$673$	−5.09355e26	−0.0267221	−0.0133610	+	0.999911i	$0.504253\pi$
$674$	2.07474e28i	1.07242i
$675$	0	0
$676$	−1.02754e28	−0.515621
$677$	9.08552e27i	0.449224i	0.974448	+	0.224612i	$0.0721115\pi$
$677$	9.08552e27i	0.449224i	−0.974448	+	0.224612i	$0.927889\pi$
$678$	0	0
$679$	3.15726e28	1.51570
$680$	− 1.99391e27i	− 0.0943226i
$681$	0	0
$682$	2.51566e28	1.15560
$683$	− 2.99545e28i	− 1.35599i	−0.735069	−	0.677993i	$-0.762850\pi$
$683$	− 2.99545e28i	− 1.35599i	0.735069	−	0.677993i	$-0.237150\pi$
$684$	0	0
$685$	2.18554e27	0.0960846
$686$	9.93214e27i	0.430330i
$687$	0	0
$688$	−1.14183e28	−0.480526
$689$	2.95930e28i	1.22743i
$690$	0	0
$691$	−2.63516e28	−1.06176	−0.530879	−	0.847447i	$-0.678138\pi$
$691$	−2.63516e28	−1.06176	−0.530879	+	0.847447i	$0.678138\pi$
$692$	6.32584e27i	0.251222i
$693$	0	0
$694$	2.11030e28	0.814233
$695$	− 5.44372e27i	− 0.207037i
$696$	0	0
$697$	6.31825e28	2.33490
$698$	1.69719e28i	0.618268i
$699$	0	0
$700$	−2.09127e28	−0.740337
$701$	5.04547e28i	1.76085i	0.474189	+	0.880423i	$0.342741\pi$
$701$	5.04547e28i	1.76085i	−0.474189	+	0.880423i	$0.657259\pi$
$702$	0	0
$703$	−4.34597e28	−1.47412
$704$	4.95825e27i	0.165807i
$705$	0	0
$706$	−9.30026e26	−0.0302307
$707$	− 7.42712e28i	− 2.38027i
$708$	0	0
$709$	1.12224e28	0.349643	0.174821	−	0.984600i	$-0.444065\pi$
$709$	1.12224e28	0.349643	0.174821	+	0.984600i	$0.444065\pi$
$710$	4.05110e27i	0.124449i
$711$	0	0
$712$	−1.53570e28	−0.458675
$713$	− 6.32571e27i	− 0.186300i
$714$	0	0
$715$	1.36742e28	0.391599
$716$	1.30737e28i	0.369205i
$717$	0	0
$718$	3.28248e28	0.901484
$719$	− 9.66414e27i	− 0.261743i	−0.991399	−	0.130871i	$-0.958223\pi$
$719$	− 9.66414e27i	− 0.261743i	0.991399	−	0.130871i	$-0.0417775\pi$
$720$	0	0
$721$	5.22571e28	1.37655
$722$	2.65543e28i	0.689862i
$723$	0	0
$724$	−2.51431e28	−0.635378
$725$	4.51010e28i	1.12410i
$726$	0	0
$727$	−4.58365e28	−1.11139	−0.555696	−	0.831386i	$-0.687548\pi$
$727$	−4.58365e28	−1.11139	−0.555696	+	0.831386i	$0.687548\pi$
$728$	3.25953e28i	0.779546i
$729$	0	0
$730$	6.49282e27	0.151079
$731$	− 1.07856e29i	− 2.47554i
$732$	0	0
$733$	−6.80792e27	−0.152046	−0.0760230	−	0.997106i	$-0.524222\pi$
$733$	−6.80792e27	−0.152046	−0.0760230	+	0.997106i	$0.524222\pi$
$734$	1.24727e28i	0.274788i
$735$	0	0
$736$	1.24677e27	0.0267306
$737$	− 9.75006e27i	− 0.206221i
$738$	0	0
$739$	6.36178e28	1.30959	0.654793	−	0.755808i	$-0.272756\pi$
$739$	6.36178e28	1.30959	0.654793	+	0.755808i	$0.272756\pi$
$740$	− 5.34858e27i	− 0.108623i
$741$	0	0
$742$	4.76591e28	0.942121
$743$	5.30528e28i	1.03471i	0.855770	+	0.517356i	$0.173084\pi$
$743$	5.30528e28i	1.03471i	−0.855770	+	0.517356i	$0.826916\pi$
$744$	0	0
$745$	−1.32837e28	−0.252207
$746$	2.41116e28i	0.451688i
$747$	0	0
$748$	−4.68350e28	−0.854190
$749$	− 9.95260e28i	− 1.79109i
$750$	0	0
$751$	−4.38712e28	−0.768742	−0.384371	−	0.923179i	$-0.625582\pi$
$751$	−4.38712e28	−0.768742	−0.384371	+	0.923179i	$0.625582\pi$
$752$	6.46417e27i	0.111772i
$753$	0	0
$754$	7.02962e28	1.18364
$755$	− 6.72085e27i	− 0.111675i
$756$	0	0
$757$	3.02822e28	0.490037	0.245018	−	0.969518i	$-0.421206\pi$
$757$	3.02822e28	0.490037	0.245018	+	0.969518i	$0.421206\pi$
$758$	2.24331e28i	0.358259i
$759$	0	0
$760$	−6.61775e27	−0.102938
$761$	− 1.13048e29i	− 1.73547i	−0.497024	−	0.867737i	$-0.665574\pi$
$761$	− 1.13048e29i	− 1.73547i	0.497024	−	0.867737i	$-0.334426\pi$
$762$	0	0
$763$	1.14919e29	1.71849
$764$	− 2.16442e28i	− 0.319454i
$765$	0	0
$766$	−2.18098e28	−0.313592
$767$	1.42208e29i	2.01823i
$768$	0	0
$769$	2.18062e28	0.301521	0.150760	−	0.988570i	$-0.451828\pi$
$769$	2.18062e28	0.301521	0.150760	+	0.988570i	$0.451828\pi$
$770$	− 2.20221e28i	− 0.300574i
$771$	0	0
$772$	5.18014e28	0.688919
$773$	6.78621e28i	0.890907i	0.895305	+	0.445454i	$0.146958\pi$
$773$	6.78621e28i	0.890907i	−0.895305	+	0.445454i	$0.853042\pi$
$774$	0	0
$775$	9.21725e28	1.17919
$776$	− 2.74270e28i	− 0.346387i
$777$	0	0
$778$	1.43342e28	0.176432
$779$	− 2.09702e29i	− 2.54817i
$780$	0	0
$781$	9.51566e28	1.12701
$782$	1.17768e28i	0.137708i
$783$	0	0
$784$	3.05612e28	0.348345
$785$	− 2.95642e28i	− 0.332713i
$786$	0	0
$787$	9.08113e27	0.0996306	0.0498153	−	0.998758i	$-0.484137\pi$
$787$	9.08113e27	0.0996306	0.0498153	+	0.998758i	$0.484137\pi$
$788$	3.32363e28i	0.360039i
$789$	0	0
$790$	−8.59176e26	−0.00907426
$791$	1.52947e29i	1.59505i
$792$	0	0
$793$	5.07370e27	0.0515933
$794$	− 1.02570e28i	− 0.102995i
$795$	0	0
$796$	−3.43690e28	−0.336540
$797$	− 7.42199e27i	− 0.0717690i	−0.999356	−	0.0358845i	$-0.988575\pi$
$797$	− 7.42199e27i	− 0.0717690i	0.999356	−	0.0358845i	$-0.0114249\pi$
$798$	0	0
$799$	−6.10597e28	−0.575820
$800$	1.81668e28i	0.169192i
$801$	0	0
$802$	−4.25943e28	−0.386908
$803$	− 1.52510e29i	− 1.36818i
$804$	0	0
$805$	−5.53753e27	−0.0484571
$806$	− 1.43664e29i	− 1.24164i
$807$	0	0
$808$	−6.45192e28	−0.543971
$809$	1.40247e29i	1.16791i	0.811787	+	0.583954i	$0.198495\pi$
$809$	1.40247e29i	1.16791i	−0.811787	+	0.583954i	$0.801505\pi$
$810$	0	0
$811$	8.98295e28	0.729811	0.364905	−	0.931045i	$-0.381101\pi$
$811$	8.98295e28	0.729811	0.364905	+	0.931045i	$0.381101\pi$
$812$	− 1.13211e29i	− 0.908507i
$813$	0	0
$814$	−1.25633e29	−0.983693
$815$	3.00494e27i	0.0232413i
$816$	0	0
$817$	−3.57972e29	−2.70164
$818$	6.80863e28i	0.507606i
$819$	0	0
$820$	2.58080e28	0.187765
$821$	− 1.25263e29i	− 0.900307i	−0.892951	−	0.450154i	$-0.851369\pi$
$821$	− 1.25263e29i	− 0.900307i	0.892951	−	0.450154i	$-0.148631\pi$
$822$	0	0
$823$	4.35515e28	0.305496	0.152748	−	0.988265i	$-0.451188\pi$
$823$	4.35515e28	0.305496	0.152748	+	0.988265i	$0.451188\pi$
$824$	− 4.53956e28i	− 0.314588i
$825$	0	0
$826$	2.29024e29	1.54910
$827$	2.20901e29i	1.47619i	0.674697	+	0.738095i	$0.264274\pi$
$827$	2.20901e29i	1.47619i	−0.674697	+	0.738095i	$0.735726\pi$
$828$	0	0
$829$	−4.46162e28	−0.291036	−0.145518	−	0.989356i	$-0.546485\pi$
$829$	−4.46162e28	−0.291036	−0.145518	+	0.989356i	$0.546485\pi$
$830$	2.67246e28i	0.172238i
$831$	0	0
$832$	2.83155e28	0.178152
$833$	2.88677e29i	1.79457i
$834$	0	0
$835$	−1.57369e28	−0.0955114
$836$	1.55445e29i	0.932208i
$837$	0	0
$838$	1.79030e29	1.04830
$839$	2.92527e28i	0.169257i	0.996413	+	0.0846285i	$0.0269703\pi$
$839$	2.92527e28i	0.169257i	−0.996413	+	0.0846285i	$0.973030\pi$
$840$	0	0
$841$	−6.71600e28	−0.379447
$842$	5.56642e28i	0.310782i
$843$	0	0
$844$	−4.81588e28	−0.262574
$845$	− 3.96457e28i	− 0.213614i
$846$	0	0
$847$	−2.23283e29	−1.17495
$848$	− 4.14014e28i	− 0.215306i
$849$	0	0
$850$	−1.71601e29	−0.871628
$851$	3.15909e28i	0.158586i
$852$	0	0
$853$	1.18249e29	0.579839	0.289919	−	0.957051i	$-0.406371\pi$
$853$	1.18249e29	0.579839	0.289919	+	0.957051i	$0.406371\pi$
$854$	− 8.17112e27i	− 0.0396007i
$855$	0	0
$856$	−8.64581e28	−0.409325
$857$	− 3.33774e28i	− 0.156187i	−0.996946	−	0.0780934i	$-0.975117\pi$
$857$	− 3.33774e28i	− 0.156187i	0.996946	−	0.0780934i	$-0.0248832\pi$
$858$	0	0
$859$	−5.32644e27	−0.0243504	−0.0121752	−	0.999926i	$-0.503876\pi$
$859$	−5.32644e27	−0.0243504	−0.0121752	+	0.999926i	$0.503876\pi$
$860$	− 4.40555e28i	− 0.199074i
$861$	0	0
$862$	−7.99707e28	−0.353068
$863$	− 5.70074e28i	− 0.248785i	−0.992233	−	0.124392i	$-0.960302\pi$
$863$	− 5.70074e28i	− 0.248785i	0.992233	−	0.124392i	$-0.0396981\pi$
$864$	0	0
$865$	−2.44071e28	−0.104077
$866$	− 2.59933e29i	− 1.09568i
$867$	0	0
$868$	−2.31368e29	−0.953030
$869$	2.01812e28i	0.0821770i
$870$	0	0
$871$	−5.56805e28	−0.221575
$872$	− 9.98298e28i	− 0.392731i
$873$	0	0
$874$	3.90871e28	0.150286
$875$	− 1.64993e29i	− 0.627170i
$876$	0	0
$877$	−3.10649e29	−1.15418	−0.577090	−	0.816681i	$-0.695812\pi$
$877$	−3.10649e29	−1.15418	−0.577090	+	0.816681i	$0.695812\pi$
$878$	1.08010e29i	0.396750i
$879$	0	0
$880$	−1.91305e28	−0.0686911
$881$	3.03041e29i	1.07583i	0.843000	+	0.537914i	$0.180787\pi$
$881$	3.03041e29i	1.07583i	−0.843000	+	0.537914i	$0.819213\pi$
$882$	0	0
$883$	−3.20453e29	−1.11213	−0.556067	−	0.831138i	$-0.687690\pi$
$883$	−3.20453e29	−1.11213	−0.556067	+	0.831138i	$0.687690\pi$
$884$	2.67464e29i	0.917790i
$885$	0	0
$886$	−3.78849e29	−1.27095
$887$	2.87528e29i	0.953772i	0.878965	+	0.476886i	$0.158235\pi$
$887$	2.87528e29i	0.953772i	−0.878965	+	0.476886i	$0.841765\pi$
$888$	0	0
$889$	−1.67148e29	−0.542106
$890$	− 5.92521e28i	− 0.190022i
$891$	0	0
$892$	−1.14717e29	−0.359731
$893$	2.02656e29i	0.628413i
$894$	0	0
$895$	−5.04425e28	−0.152956
$896$	− 4.56017e28i	− 0.136742i
$897$	0	0
$898$	2.15238e29	0.631183
$899$	4.98976e29i	1.44705i
$900$	0	0
$901$	3.91072e29	1.10920
$902$	− 6.06204e29i	− 1.70041i
$903$	0	0
$904$	1.32864e29	0.364522
$905$	− 9.70100e28i	− 0.263227i
$906$	0	0
$907$	−5.42973e29	−1.44114	−0.720569	−	0.693383i	$-0.756119\pi$
$907$	−5.42973e29	−1.44114	−0.720569	+	0.693383i	$0.756119\pi$
$908$	− 3.03684e29i	− 0.797194i
$909$	0	0
$910$	−1.25763e29	−0.322953
$911$	3.84639e29i	0.976943i	0.872580	+	0.488472i	$0.162445\pi$
$911$	3.84639e29i	0.976943i	−0.872580	+	0.488472i	$0.837555\pi$
$912$	0	0
$913$	6.27735e29	1.55980
$914$	− 5.26983e29i	− 1.29519i
$915$	0	0
$916$	2.85879e29	0.687427
$917$	− 5.97004e28i	− 0.141999i
$918$	0	0
$919$	−1.81942e29	−0.423426	−0.211713	−	0.977332i	$-0.567904\pi$
$919$	−1.81942e29	−0.423426	−0.211713	+	0.977332i	$0.567904\pi$
$920$	4.81044e27i	0.0110741i
$921$	0	0
$922$	3.97818e29	0.896138
$923$	− 5.43418e29i	− 1.21093i
$924$	0	0
$925$	−4.60314e29	−1.00378
$926$	− 3.92595e29i	− 0.846906i
$927$	0	0
$928$	−9.83461e28	−0.207624
$929$	− 6.54217e28i	− 0.136636i	−0.997664	−	0.0683179i	$-0.978237\pi$
$929$	− 6.54217e28i	− 0.136636i	0.997664	−	0.0683179i	$-0.0217632\pi$
$930$	0	0
$931$	9.58113e29	1.95848
$932$	− 2.65635e29i	− 0.537187i
$933$	0	0
$934$	4.42490e29	0.875861
$935$	− 1.80704e29i	− 0.353877i
$936$	0	0
$937$	1.83998e29	0.352709	0.176355	−	0.984327i	$-0.443569\pi$
$937$	1.83998e29	0.352709	0.176355	+	0.984327i	$0.443569\pi$
$938$	8.96726e28i	0.170071i
$939$	0	0
$940$	−2.49408e28	−0.0463055
$941$	− 5.17567e29i	− 0.950758i	−0.879781	−	0.475379i	$-0.842311\pi$
$941$	− 5.17567e29i	− 0.950758i	0.879781	−	0.475379i	$-0.157689\pi$
$942$	0	0
$943$	−1.52432e29	−0.274132
$944$	− 1.98952e29i	− 0.354021i
$945$	0	0
$946$	−1.03482e30	−1.80283
$947$	5.11825e28i	0.0882311i	0.999026	+	0.0441156i	$0.0140470\pi$
$947$	5.11825e28i	0.0882311i	−0.999026	+	0.0441156i	$0.985953\pi$
$948$	0	0
$949$	−8.70953e29	−1.47005
$950$	5.69542e29i	0.951239i
$951$	0	0
$952$	4.30747e29	0.704455
$953$	5.65872e29i	0.915776i	0.889010	+	0.457888i	$0.151394\pi$
$953$	5.65872e29i	0.915776i	−0.889010	+	0.457888i	$0.848606\pi$
$954$	0	0
$955$	8.35102e28	0.132345
$956$	2.68991e28i	0.0421850i
$957$	0	0
$958$	−1.02667e29	−0.157680
$959$	4.72147e29i	0.717615i
$960$	0	0
$961$	3.47963e29	0.517963
$962$	7.17463e29i	1.05694i
$963$	0	0
$964$	2.16477e29	0.312350
$965$	1.99866e29i	0.285408i
$966$	0	0
$967$	1.18187e30	1.65312	0.826558	−	0.562851i	$-0.190296\pi$
$967$	1.18187e30	1.65312	0.826558	+	0.562851i	$0.190296\pi$
$968$	1.93965e29i	0.268516i
$969$	0	0
$970$	1.05822e29	0.143503
$971$	− 9.56785e29i	− 1.28417i	−0.766634	−	0.642084i	$-0.778070\pi$
$971$	− 9.56785e29i	− 1.28417i	0.766634	−	0.642084i	$-0.221930\pi$
$972$	0	0
$973$	1.17602e30	1.54627
$974$	6.85841e29i	0.892553i
$975$	0	0
$976$	−7.09824e27	−0.00905008
$977$	− 4.09410e28i	− 0.0516670i	−0.999666	−	0.0258335i	$-0.991776\pi$
$977$	− 4.09410e28i	− 0.0516670i	0.999666	−	0.0258335i	$-0.00822397\pi$
$978$	0	0
$979$	−1.39177e30	−1.72085
$980$	1.17915e29i	0.144314i
$981$	0	0
$982$	−3.55211e29	−0.425963
$983$	− 1.13576e30i	− 1.34819i	−0.738644	−	0.674095i	$-0.764534\pi$
$983$	− 1.13576e30i	− 1.34819i	0.738644	−	0.674095i	$-0.235466\pi$
$984$	0	0
$985$	−1.28236e29	−0.149159
$986$	− 9.28964e29i	− 1.06962i
$987$	0	0
$988$	8.87710e29	1.00162
$989$	2.60210e29i	0.290643i
$990$	0	0
$991$	7.51652e29	0.822774	0.411387	−	0.911461i	$-0.365045\pi$
$991$	7.51652e29	0.822774	0.411387	+	0.911461i	$0.365045\pi$
$992$	2.00989e29i	0.217799i
$993$	0	0
$994$	−8.75167e29	−0.929453
$995$	− 1.32607e29i	− 0.139423i
$996$	0	0
$997$	−2.03179e29	−0.209376	−0.104688	−	0.994505i	$-0.533384\pi$
$997$	−2.03179e29	−0.209376	−0.104688	+	0.994505i	$0.533384\pi$
$998$	− 1.07805e30i	− 1.09985i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.21.b.c.53.9	yes	12
3.2	odd	2	inner	54.21.b.c.53.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.21.b.c.53.4	✓	12	3.2	odd	2	inner
54.21.b.c.53.9	yes	12	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 21 \)
Character orbit:	\([\chi]\)	\(=\)	54.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+724.077i q^{2} -524288. q^{4} -2.02287e6i q^{5} +4.37005e8 q^{7} -3.79625e8i q^{8} +1.46472e9 q^{10} -3.44048e10i q^{11} -1.96478e11 q^{13} +3.16425e11i q^{14} +2.74878e11 q^{16} +2.59646e12i q^{17} +8.61761e12 q^{19} +1.06057e12i q^{20} +2.49117e13 q^{22} -6.26414e12i q^{23} +9.12754e13 q^{25} -1.42265e14i q^{26} -2.29116e14 q^{28} +4.94120e14i q^{29} +1.00983e15 q^{31} +1.99033e14i q^{32} -1.88004e15 q^{34} -8.84004e14i q^{35} -5.04313e15 q^{37} +6.23982e15i q^{38} -7.67933e14 q^{40} -2.43341e16i q^{41} -4.15395e16 q^{43} +1.80380e16i q^{44} +4.53572e15 q^{46} +2.35165e16i q^{47} +1.11181e17 q^{49} +6.60905e16i q^{50} +1.03011e17 q^{52} -1.50617e17i q^{53} -6.95965e16 q^{55} -1.65898e17i q^{56} -3.57781e17 q^{58} -7.23785e17i q^{59} -2.58232e16 q^{61} +7.31194e17i q^{62} -1.44115e17 q^{64} +3.97450e17i q^{65} +2.83393e17 q^{67} -1.36129e18i q^{68} +6.40088e17 q^{70} +2.76579e18i q^{71} +4.43282e18 q^{73} -3.65162e18i q^{74} -4.51811e18 q^{76} -1.50351e19i q^{77} -5.86582e17 q^{79} -5.56043e17i q^{80} +1.76198e19 q^{82} +1.82456e19i q^{83} +5.25230e18 q^{85} -3.00778e19i q^{86} -1.30609e19 q^{88} -4.04529e19i q^{89} -8.58619e19 q^{91} +3.28421e18i q^{92} -1.70278e19 q^{94} -1.74323e19i q^{95} +7.22477e19 q^{97} +8.05035e19i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6291456 q^{4} + 355856172 q^{7} - 5360984064 q^{10} + 374332013160 q^{13} + 3298534883328 q^{16} + 13268452496064 q^{19} + 64109345587200 q^{22} - 10850992436544 q^{25} - 186571120705536 q^{28} + 13\!\cdots\!76 q^{31}+ \cdots - 30\!\cdots\!60 q^{97}+O(q^{100}) \) 12 * q - 6291456 * q^4 + 355856172 * q^7 - 5360984064 * q^10 + 374332013160 * q^13 + 3298534883328 * q^16 + 13268452496064 * q^19 + 64109345587200 * q^22 - 10850992436544 * q^25 - 186571120705536 * q^28 + 1341631815613476 * q^31 - 1150892216451072 * q^34 - 1840381413631248 * q^37 + 2810699612946432 * q^40 - 58876514549555304 * q^43 - 52022884060053504 * q^46 + 301701288785631840 * q^49 - 196257782515630080 * q^52 + 1198488617517746484 * q^55 - 952998745417777152 * q^58 - 611525981019002448 * q^61 - 1729382256910270464 * q^64 - 3378608356010016504 * q^67 - 10123347229078487040 * q^70 + 35504687328043033068 * q^73 - 6956490422256402432 * q^76 + 15118277214382580040 * q^79 - 16372057824399605760 * q^82 - 2077217010606501720 * q^85 - 33611760579221913600 * q^88 - 154840755366303827352 * q^91 + 150175394882225012736 * q^94 - 3033444647600404860 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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