LMFDB - Embedded newform 252.9.bk.a.53.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(53,252)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bk (of order $6$, degree $2$, 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:	no
Analytic conductor:	$102.659409735$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.14
Character	$\chi$	$=$	252.53
Dual form			252.9.bk.a.233.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(173.159 - 99.9732i) q^{5} +(1613.03 - 1778.47i) q^{7} +O(q^{10})$	$q+(173.159 - 99.9732i) q^{5} +(1613.03 - 1778.47i) q^{7} +(22227.8 + 12833.2i) q^{11} +26477.6 q^{13} +(74222.8 + 42852.5i) q^{17} +(-6943.06 - 12025.7i) q^{19} +(-60908.1 + 35165.3i) q^{23} +(-175323. + 303669. i) q^{25} +405852. i q^{29} +(-130468. + 225977. i) q^{31} +(101510. - 469217. i) q^{35} +(209208. + 362359. i) q^{37} -927622. i q^{41} +2.87909e6 q^{43} +(-1.51450e6 + 874399. i) q^{47} +(-561100. - 5.73743e6i) q^{49} +(3.55025e6 + 2.04974e6i) q^{53} +5.13192e6 q^{55} +(-7.54515e6 - 4.35620e6i) q^{59} +(-4.68976e6 - 8.12291e6i) q^{61} +(4.58483e6 - 2.64705e6i) q^{65} +(-1.71554e7 + 2.97140e7i) q^{67} -9.42813e6i q^{71} +(-1.48278e7 + 2.56826e7i) q^{73} +(5.86775e7 - 1.88311e7i) q^{77} +(3.39845e6 + 5.88629e6i) q^{79} +6.35734e7i q^{83} +1.71364e7 q^{85} +(4.29135e7 - 2.47761e7i) q^{89} +(4.27091e7 - 4.70896e7i) q^{91} +(-2.40450e6 - 1.38824e6i) q^{95} +1.06436e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 1230 q^{7}+O(q^{10}) $ 44 * q + 1230 * q^7	$ 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) $ 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	173.159	−	99.9732i	0.277054	−	0.159957i	−0.355035	−	0.934853i	$-0.615531\pi$
$5$	173.159	−	99.9732i	0.277054	−	0.159957i	0.632089	+	0.774896i	$0.282198\pi$
$6$		0			0
$7$	1613.03	−	1778.47i	0.671814	−	0.740720i
$7$	1613.03	−	1778.47i	0.671814	−	0.740720i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	22227.8	+	12833.2i	1.51819	+	0.876527i	0.999771	+	0.0214000i	$0.00681235\pi$
$11$	22227.8	+	12833.2i	1.51819	+	0.876527i	0.518418	+	0.855127i	$0.326521\pi$
$12$		0			0
$13$	26477.6			0.927055			0.463527	−	0.886083i	$-0.346584\pi$
$13$	26477.6			0.927055			0.463527	+	0.886083i	$0.346584\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	74222.8	+	42852.5i	0.888672	+	0.513075i	0.873508	−	0.486810i	$-0.161840\pi$
$17$	74222.8	+	42852.5i	0.888672	+	0.513075i	0.0151639	+	0.999885i	$0.495173\pi$
$18$		0			0
$19$	−6943.06	−	12025.7i	−0.0532766	−	0.0922778i	0.838157	−	0.545429i	$-0.183633\pi$
$19$	−6943.06	−	12025.7i	−0.0532766	−	0.0922778i	−0.891434	+	0.453151i	$0.850300\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−60908.1	+	35165.3i	−0.217652	+	0.125662i	−0.604863	−	0.796330i	$-0.706772\pi$
$23$	−60908.1	+	35165.3i	−0.217652	+	0.125662i	0.387210	+	0.921991i	$0.373439\pi$
$24$		0			0
$25$	−175323.	+	303669.i	−0.448827	+	0.777392i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			405852.i			0.573820i	0.957958	+	0.286910i	$0.0926281\pi$
$29$			405852.i			0.573820i	−0.957958	+	0.286910i	$0.907372\pi$
$30$		0			0
$31$	−130468.	+	225977.i	−0.141272	+	0.244691i	−0.927976	−	0.372640i	$-0.878453\pi$
$31$	−130468.	+	225977.i	−0.141272	+	0.244691i	0.786704	+	0.617331i	$0.211786\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	101510.	−	469217.i	0.0676452	−	0.312681i
$36$		0			0
$37$	209208.	+	362359.i	0.111628	+	0.193345i	0.916427	−	0.400203i	$-0.131060\pi$
$37$	209208.	+	362359.i	0.111628	+	0.193345i	−0.804799	+	0.593547i	$0.797727\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	927622.i		−	0.328273i	−0.986438	−	0.164137i	$-0.947516\pi$
$41$		−	927622.i		−	0.328273i	0.986438	−	0.164137i	$-0.0524838\pi$
$42$		0			0
$43$	2.87909e6			0.842134			0.421067	−	0.907029i	$-0.361656\pi$
$43$	2.87909e6			0.842134			0.421067	+	0.907029i	$0.361656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.51450e6	+	874399.i	−0.310369	+	0.179192i	−0.647092	−	0.762412i	$-0.724015\pi$
$47$	−1.51450e6	+	874399.i	−0.310369	+	0.179192i	0.336722	+	0.941604i	$0.390682\pi$
$48$		0			0
$49$	−561100.	−	5.73743e6i	−0.0973321	−	0.995252i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.55025e6	+	2.04974e6i	0.449941	+	0.259773i	0.707805	−	0.706408i	$-0.249685\pi$
$53$	3.55025e6	+	2.04974e6i	0.449941	+	0.259773i	−0.257864	+	0.966181i	$0.583019\pi$
$54$		0			0
$55$	5.13192e6			0.560827
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.54515e6	−	4.35620e6i	−0.622673	−	0.359500i	0.155236	−	0.987877i	$-0.450386\pi$
$59$	−7.54515e6	−	4.35620e6i	−0.622673	−	0.359500i	−0.777909	+	0.628377i	$0.783720\pi$
$60$		0			0
$61$	−4.68976e6	−	8.12291e6i	−0.338713	−	0.586668i	0.645478	−	0.763779i	$-0.276658\pi$
$61$	−4.68976e6	−	8.12291e6i	−0.338713	−	0.586668i	−0.984191	+	0.177111i	$0.943325\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.58483e6	−	2.64705e6i	0.256844	−	0.148289i
$66$		0			0
$67$	−1.71554e7	+	2.97140e7i	−0.851338	+	1.47456i	0.0286634	+	0.999589i	$0.490875\pi$
$67$	−1.71554e7	+	2.97140e7i	−0.851338	+	1.47456i	−0.880001	+	0.474971i	$0.842458\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.42813e6i		−	0.371016i	−0.982643	−	0.185508i	$-0.940607\pi$
$71$		−	9.42813e6i		−	0.371016i	0.982643	−	0.185508i	$-0.0593930\pi$
$72$		0			0
$73$	−1.48278e7	+	2.56826e7i	−0.522139	+	0.904371i	0.477529	+	0.878616i	$0.341532\pi$
$73$	−1.48278e7	+	2.56826e7i	−0.522139	+	0.904371i	−0.999668	+	0.0257556i	$0.991801\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.86775e7	−	1.88311e7i	1.66920	−	0.535690i
$78$		0			0
$79$	3.39845e6	+	5.88629e6i	0.0872514	+	0.151124i	0.906348	−	0.422531i	$-0.138858\pi$
$79$	3.39845e6	+	5.88629e6i	0.0872514	+	0.151124i	−0.819097	+	0.573655i	$0.805525\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			6.35734e7i			1.33956i	0.742559	+	0.669781i	$0.233612\pi$
$83$			6.35734e7i			1.33956i	−0.742559	+	0.669781i	$0.766388\pi$
$84$		0			0
$85$	1.71364e7			0.328280
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.29135e7	−	2.47761e7i	0.683964	−	0.394887i	−0.117383	−	0.993087i	$-0.537450\pi$
$89$	4.29135e7	−	2.47761e7i	0.683964	−	0.394887i	0.801347	+	0.598200i	$0.204117\pi$
$90$		0			0
$91$	4.27091e7	−	4.70896e7i	0.622808	−	0.686688i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.40450e6	−	1.38824e6i	−0.0295210	−	0.0170440i
$96$		0			0
$97$	1.06436e8			1.20227			0.601133	−	0.799149i	$-0.294716\pi$
$97$	1.06436e8			1.20227			0.601133	+	0.799149i	$0.294716\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	881655.	+	509024.i	0.00847253	+	0.00489162i	0.504230	−	0.863569i	$-0.331776\pi$
$101$	881655.	+	509024.i	0.00847253	+	0.00489162i	−0.495758	+	0.868461i	$0.665110\pi$
$102$		0			0
$103$	−2.61597e7	−	4.53099e7i	−0.232425	−	0.402573i	0.726096	−	0.687593i	$-0.241333\pi$
$103$	−2.61597e7	−	4.53099e7i	−0.232425	−	0.402573i	−0.958521	+	0.285021i	$0.907999\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.11804e6	+	5.26430e6i	−0.0695611	+	0.0401611i	−0.534377	−	0.845246i	$-0.679454\pi$
$107$	−9.11804e6	+	5.26430e6i	−0.0695611	+	0.0401611i	0.464816	+	0.885407i	$0.346120\pi$
$108$		0			0
$109$	−1.21674e8	+	2.10745e8i	−0.861967	+	1.49297i	0.00806061	+	0.999968i	$0.497434\pi$
$109$	−1.21674e8	+	2.10745e8i	−0.861967	+	1.49297i	−0.870028	+	0.493003i	$0.835899\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2.24135e8i		−	1.37466i	−0.726343	−	0.687332i	$-0.758782\pi$
$113$		−	2.24135e8i		−	1.37466i	0.726343	−	0.687332i	$-0.241218\pi$
$114$		0			0
$115$	−7.03117e6	+	1.21783e7i	−0.0402010	+	0.0696301i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.95935e8	−	6.28806e7i	0.977067	−	0.313566i
$120$		0			0
$121$	2.22204e8	+	3.84869e8i	1.03660	+	1.79544i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.48215e8i			0.607087i
$126$		0			0
$127$	−1.73705e8			−0.667725			−0.333862	−	0.942622i	$-0.608352\pi$
$127$	−1.73705e8			−0.667725			−0.333862	+	0.942622i	$0.608352\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.26721e8	−	2.46367e8i	1.44897	−	0.836562i	0.450548	−	0.892752i	$-0.351229\pi$
$131$	4.26721e8	−	2.46367e8i	1.44897	−	0.836562i	0.998420	+	0.0561904i	$0.0178954\pi$
$132$		0			0
$133$	−3.25867e7	−	7.04980e6i	−0.104144	−	0.0225305i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.72570e8	+	1.57368e8i	0.773742	+	0.446720i	0.834208	−	0.551450i	$-0.185925\pi$
$137$	2.72570e8	+	1.57368e8i	0.773742	+	0.446720i	−0.0604659	+	0.998170i	$0.519259\pi$
$138$		0			0
$139$	3.67754e8			0.985141			0.492571	−	0.870272i	$-0.336057\pi$
$139$	3.67754e8			0.985141			0.492571	+	0.870272i	$0.336057\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.88539e8	+	3.39793e8i	1.40744	+	0.812589i
$144$		0			0
$145$	4.05743e7	+	7.02768e7i	0.0917866	+	0.158979i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.43555e8	−	3.13822e8i	1.10280	−	0.636705i	0.165848	−	0.986151i	$-0.446964\pi$
$149$	5.43555e8	−	3.13822e8i	1.10280	−	0.636705i	0.936956	+	0.349447i	$0.113630\pi$
$150$		0			0
$151$	−1.10865e8	+	1.92023e8i	−0.213248	+	0.369357i	−0.952729	−	0.303821i	$-0.901738\pi$
$151$	−1.10865e8	+	1.92023e8i	−0.213248	+	0.369357i	0.739481	+	0.673177i	$0.235071\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			5.21732e7i			0.0903900i
$156$		0			0
$157$	−2.98228e8	+	5.16546e8i	−0.490851	+	0.850179i	−0.999945	−	0.0105321i	$-0.996647\pi$
$157$	−2.98228e8	+	5.16546e8i	−0.490851	+	0.850179i	0.509093	+	0.860711i	$0.329981\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.57059e7	+	1.65046e8i	−0.0531418	+	0.245641i
$162$		0			0
$163$	3.17173e8	+	5.49359e8i	0.449309	+	0.778226i	0.998341	−	0.0575751i	$-0.0183369\pi$
$163$	3.17173e8	+	5.49359e8i	0.449309	+	0.778226i	−0.549032	+	0.835801i	$0.685004\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.34644e9i		−	1.73110i	−0.500825	−	0.865548i	$-0.666970\pi$
$167$		−	1.34644e9i		−	1.73110i	0.500825	−	0.865548i	$-0.333030\pi$
$168$		0			0
$169$	−1.14667e8			−0.140569
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.93330e8	−	5.73499e8i	1.10894	−	0.640248i	0.170388	−	0.985377i	$-0.445498\pi$
$173$	9.93330e8	−	5.73499e8i	1.10894	−	0.640248i	0.938555	+	0.345129i	$0.112165\pi$
$174$		0			0
$175$	2.57265e8	+	8.01632e8i	0.274301	+	0.854718i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.48754e8	+	2.59088e8i	0.437115	+	0.252369i	0.702373	−	0.711809i	$-0.252124\pi$
$179$	4.48754e8	+	2.59088e8i	0.437115	+	0.252369i	−0.265258	+	0.964177i	$0.585457\pi$
$180$		0			0
$181$	−6.69707e8			−0.623980			−0.311990	−	0.950085i	$-0.600996\pi$
$181$	−6.69707e8			−0.623980			−0.311990	+	0.950085i	$0.600996\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.24523e7	+	4.18304e7i	0.0618537	+	0.0357112i
$186$		0			0
$187$	1.09987e9	+	1.90504e9i	0.899448	+	1.55789i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.00025e9	−	5.77496e8i	0.751582	−	0.433926i	−0.0746833	−	0.997207i	$-0.523795\pi$
$191$	1.00025e9	−	5.77496e8i	0.751582	−	0.433926i	0.826265	+	0.563281i	$0.190461\pi$
$192$		0			0
$193$	9.70353e8	−	1.68070e9i	0.699359	−	1.21133i	−0.269330	−	0.963048i	$-0.586802\pi$
$193$	9.70353e8	−	1.68070e9i	0.699359	−	1.21133i	0.968689	−	0.248278i	$-0.0798646\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.42086e8i			0.293523i	0.989172	+	0.146761i	$0.0468849\pi$
$197$			4.42086e8i			0.293523i	−0.989172	+	0.146761i	$0.953115\pi$
$198$		0			0
$199$	−2.95221e8	+	5.11338e8i	−0.188250	+	0.326058i	−0.944667	−	0.328031i	$-0.893615\pi$
$199$	−2.95221e8	+	5.11338e8i	−0.188250	+	0.326058i	0.756417	+	0.654090i	$0.226948\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.21795e8	+	6.54650e8i	0.425040	+	0.385500i
$204$		0			0
$205$	−9.27374e7	−	1.60626e8i	−0.0525097	−	0.0909494i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.56408e8i		−	0.186794i
$210$		0			0
$211$	2.72443e9			1.37450			0.687252	−	0.726419i	$-0.258817\pi$
$211$	2.72443e9			1.37450			0.687252	+	0.726419i	$0.258817\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.98539e8	−	2.87832e8i	0.233317	−	0.134705i
$216$		0			0
$217$	1.91445e8	+	5.96540e8i	0.0863386	+	0.269030i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.96524e9	+	1.13463e9i	0.823847	+	0.475648i
$222$		0			0
$223$	−4.43205e9			−1.79220			−0.896098	−	0.443856i	$-0.853610\pi$
$223$	−4.43205e9			−1.79220			−0.896098	+	0.443856i	$0.853610\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.41259e9	+	8.15562e8i	0.532003	+	0.307152i	0.741832	−	0.670586i	$-0.233957\pi$
$227$	1.41259e9	+	8.15562e8i	0.532003	+	0.307152i	−0.209829	+	0.977738i	$0.567291\pi$
$228$		0			0
$229$	1.47388e9	+	2.55284e9i	0.535946	+	0.928285i	0.999117	+	0.0420162i	$0.0133781\pi$
$229$	1.47388e9	+	2.55284e9i	0.535946	+	0.928285i	−0.463171	+	0.886269i	$0.653289\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.51041e9	−	8.72038e8i	0.512474	−	0.295877i	−0.221376	−	0.975189i	$-0.571055\pi$
$233$	1.51041e9	−	8.72038e8i	0.512474	−	0.295877i	0.733850	+	0.679311i	$0.237721\pi$
$234$		0			0
$235$	−1.74833e8	+	3.02820e8i	−0.0573260	+	0.0992916i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			8.73772e8i			0.267798i	0.990995	+	0.133899i	$0.0427497\pi$
$239$			8.73772e8i			0.267798i	−0.990995	+	0.133899i	$0.957250\pi$
$240$		0			0
$241$	−8.94267e8	+	1.54892e9i	−0.265094	+	0.459155i	−0.967588	−	0.252534i	$-0.918736\pi$
$241$	−8.94267e8	+	1.54892e9i	−0.265094	+	0.459155i	0.702495	+	0.711689i	$0.252070\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.70749e8	−	9.37391e8i	−0.186164	−	0.260169i
$246$		0			0
$247$	−1.83836e8	−	3.18413e8i	−0.0493904	−	0.0855466i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	5.54699e9i		−	1.39753i	−0.715349	−	0.698767i	$-0.753732\pi$
$251$		−	5.54699e9i		−	1.39753i	0.715349	−	0.698767i	$-0.246268\pi$
$252$		0			0
$253$	−1.80514e9			−0.440583
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.40867e9	+	2.54535e9i	−1.01059	+	0.583465i	−0.911364	−	0.411600i	$-0.864970\pi$
$257$	−4.40867e9	+	2.54535e9i	−1.01059	+	0.583465i	−0.0992258	+	0.995065i	$0.531637\pi$
$258$		0			0
$259$	9.81901e8	+	2.12424e8i	0.218207	+	0.0472068i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.81391e8	+	1.04726e8i	0.0379135	+	0.0218894i	0.518837	−	0.854873i	$-0.326365\pi$
$263$	1.81391e8	+	1.04726e8i	0.0379135	+	0.0218894i	−0.480923	+	0.876763i	$0.659699\pi$
$264$		0			0
$265$	8.19675e8			0.166210
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.18658e9	−	1.83977e9i	−0.608577	−	0.351362i	0.163831	−	0.986488i	$-0.447615\pi$
$269$	−3.18658e9	−	1.83977e9i	−0.608577	−	0.351362i	−0.772408	+	0.635126i	$0.780948\pi$
$270$		0			0
$271$	−4.54663e9	−	7.87500e9i	−0.842971	−	1.46007i	−0.887372	−	0.461055i	$-0.847471\pi$
$271$	−4.54663e9	−	7.87500e9i	−0.842971	−	1.46007i	0.0444007	−	0.999014i	$-0.485862\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.79410e9	+	4.49993e9i	−1.36281	+	0.786819i
$276$		0			0
$277$	1.23434e9	−	2.13794e9i	0.209660	−	0.363142i	−0.741947	−	0.670458i	$-0.766098\pi$
$277$	1.23434e9	−	2.13794e9i	0.209660	−	0.363142i	0.951607	+	0.307316i	$0.0994309\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.51793e9i		−	0.564237i	−0.959380	−	0.282119i	$-0.908963\pi$
$281$		−	3.51793e9i		−	0.564237i	0.959380	−	0.282119i	$-0.0910371\pi$
$282$		0			0
$283$	4.27009e9	−	7.39601e9i	0.665719	−	1.15306i	−0.313371	−	0.949631i	$-0.601458\pi$
$283$	4.27009e9	−	7.39601e9i	0.665719	−	1.15306i	0.979090	−	0.203428i	$-0.0652084\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.64975e9	−	1.49628e9i	−0.243159	−	0.220539i
$288$		0			0
$289$	1.84799e8	+	3.20082e8i	0.0264916	+	0.0458849i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.26844e10i			1.72108i	0.509386	+	0.860538i	$0.329873\pi$
$293$			1.26844e10i			1.72108i	−0.509386	+	0.860538i	$0.670127\pi$
$294$		0			0
$295$	−1.74201e9			−0.230019
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.61270e9	+	9.31093e8i	−0.201776	+	0.116495i
$300$		0			0
$301$	4.64404e9	−	5.12037e9i	0.565757	−	0.623786i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.62415e9	−	9.37701e8i	−0.187683	−	0.108359i
$306$		0			0
$307$	7.64714e9			0.860886			0.430443	−	0.902618i	$-0.358357\pi$
$307$	7.64714e9			0.860886			0.430443	+	0.902618i	$0.358357\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.12282e9	−	3.53501e9i	−0.654500	−	0.377876i	0.135678	−	0.990753i	$-0.456679\pi$
$311$	−6.12282e9	−	3.53501e9i	−0.654500	−	0.377876i	−0.790178	+	0.612877i	$0.790012\pi$
$312$		0			0
$313$	−6.96692e9	−	1.20671e10i	−0.725877	−	1.25726i	−0.958612	−	0.284716i	$-0.908101\pi$
$313$	−6.96692e9	−	1.20671e10i	−0.725877	−	1.25726i	0.232735	−	0.972540i	$-0.425233\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.84976e9	+	3.37736e9i	−0.579296	+	0.334457i	−0.760854	−	0.648924i	$-0.775220\pi$
$317$	−5.84976e9	+	3.37736e9i	−0.579296	+	0.334457i	0.181557	+	0.983380i	$0.441886\pi$
$318$		0			0
$319$	−5.20839e9	+	9.02120e9i	−0.502969	+	0.871168i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1.19011e9i		−	0.109340i
$324$		0			0
$325$	−4.64214e9	+	8.04042e9i	−0.416088	+	0.720685i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.87841e8	+	4.10393e9i	−0.0757795	+	0.350280i
$330$		0			0
$331$	2.15768e9	+	3.73721e9i	0.179753	+	0.311340i	0.941796	−	0.336186i	$-0.109137\pi$
$331$	2.15768e9	+	3.73721e9i	0.179753	+	0.311340i	−0.762043	+	0.647526i	$0.775804\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			6.86033e9i			0.544710i
$336$		0			0
$337$	−1.74608e10			−1.35377			−0.676883	−	0.736091i	$-0.736670\pi$
$337$	−1.74608e10			−1.35377			−0.676883	+	0.736091i	$0.736670\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5.80003e9	+	3.34865e9i	−0.428956	+	0.247658i
$342$		0			0
$343$	−1.11089e10	−	8.25672e9i	−0.802592	−	0.596528i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.95296e10	−	1.12754e10i	−1.34702	−	0.777704i	−0.359196	−	0.933262i	$-0.616949\pi$
$347$	−1.95296e10	−	1.12754e10i	−1.34702	−	0.777704i	−0.987827	+	0.155558i	$0.950283\pi$
$348$		0			0
$349$	9.05754e9			0.610532			0.305266	−	0.952267i	$-0.401255\pi$
$349$	9.05754e9			0.610532			0.305266	+	0.952267i	$0.401255\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.82049e9	−	1.05106e9i	−0.117244	−	0.0676908i	0.440231	−	0.897884i	$-0.354896\pi$
$353$	−1.82049e9	−	1.05106e9i	−0.117244	−	0.0676908i	−0.557475	+	0.830194i	$0.688230\pi$
$354$		0			0
$355$	−9.42561e8	−	1.63256e9i	−0.0593466	−	0.102791i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.45590e10	+	1.41791e10i	−1.47854	+	0.853635i	−0.999705	−	0.0242715i	$-0.992273\pi$
$359$	−2.45590e10	+	1.41791e10i	−1.47854	+	0.853635i	−0.478833	+	0.877906i	$0.658940\pi$
$360$		0			0
$361$	8.39537e9	−	1.45412e10i	0.494323	−	0.856193i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			5.92954e9i			0.334079i
$366$		0			0
$367$	1.24614e10	−	2.15837e10i	0.686912	−	1.18977i	−0.285919	−	0.958254i	$-0.592299\pi$
$367$	1.24614e10	−	2.15837e10i	0.686912	−	1.18977i	0.972832	−	0.231513i	$-0.0743677\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.37204e9	−	3.00773e9i	0.494696	−	0.158761i
$372$		0			0
$373$	−1.51704e10	−	2.62758e10i	−0.783720	−	1.35744i	−0.929761	−	0.368164i	$-0.879987\pi$
$373$	−1.51704e10	−	2.62758e10i	−0.783720	−	1.35744i	0.146041	−	0.989279i	$-0.453347\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.07460e10i			0.531963i
$378$		0			0
$379$	2.00489e10			0.971704			0.485852	−	0.874041i	$-0.338509\pi$
$379$	2.00489e10			0.971704			0.485852	+	0.874041i	$0.338509\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.32712e9	−	4.23031e9i	0.340516	−	0.196597i	−0.319984	−	0.947423i	$-0.603678\pi$
$383$	7.32712e9	−	4.23031e9i	0.340516	−	0.196597i	0.660500	+	0.750826i	$0.270344\pi$
$384$		0			0
$385$	8.27791e9	−	9.12695e9i	0.376771	−	0.415416i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.74860e10	−	1.58691e10i	−1.20036	−	0.693031i	−0.239728	−	0.970840i	$-0.577058\pi$
$389$	−2.74860e10	−	1.58691e10i	−1.20036	−	0.693031i	−0.960636	+	0.277809i	$0.910392\pi$
$390$		0			0
$391$	−6.02769e9			−0.257895
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.17694e9	+	6.79508e8i	0.0483467	+	0.0279130i
$396$		0			0
$397$	−3.49364e9	−	6.05117e9i	−0.140642	−	0.243600i	0.787096	−	0.616830i	$-0.211583\pi$
$397$	−3.49364e9	−	6.05117e9i	−0.140642	−	0.243600i	−0.927739	+	0.373230i	$0.878250\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.01180e10	−	5.84161e9i	0.391306	−	0.225920i	−0.291420	−	0.956595i	$-0.594128\pi$
$401$	1.01180e10	−	5.84161e9i	0.391306	−	0.225920i	0.682726	+	0.730675i	$0.260794\pi$
$402$		0			0
$403$	−3.45448e9	+	5.98333e9i	−0.130967	+	0.226842i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.07393e10i			0.391378i
$408$		0			0
$409$	−2.19328e10	+	3.79886e10i	−0.783790	+	1.35756i	0.145929	+	0.989295i	$0.453383\pi$
$409$	−2.19328e10	+	3.79886e10i	−0.783790	+	1.35756i	−0.929719	+	0.368270i	$0.879950\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.99179e10	+	6.39216e9i	−0.684610	+	0.219709i
$414$		0			0
$415$	6.35563e9	+	1.10083e10i	0.214273	+	0.371131i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	5.63614e10i		−	1.82863i	−0.405003	−	0.914315i	$-0.632729\pi$
$419$		−	5.63614e10i		−	1.82863i	0.405003	−	0.914315i	$-0.367271\pi$
$420$		0			0
$421$	−1.90310e10			−0.605805			−0.302903	−	0.953022i	$-0.597956\pi$
$421$	−1.90310e10			−0.605805			−0.302903	+	0.953022i	$0.597956\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.60259e10	+	1.50261e10i	−0.797721	+	0.460564i
$426$		0			0
$427$	−2.20110e10	−	4.76186e9i	−0.662108	−	0.143240i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.54398e10	+	1.46877e10i	0.737233	+	0.425642i	0.821062	−	0.570838i	$-0.193382\pi$
$431$	2.54398e10	+	1.46877e10i	0.737233	+	0.425642i	−0.0838293	+	0.996480i	$0.526715\pi$
$432$		0			0
$433$	−5.04499e9			−0.143519			−0.0717594	−	0.997422i	$-0.522861\pi$
$433$	−5.04499e9			−0.143519			−0.0717594	+	0.997422i	$0.522861\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.45777e8	+	4.88310e8i	0.0231916	+	0.0133897i
$438$		0			0
$439$	3.05794e10	+	5.29651e10i	0.823325	+	1.42604i	0.903193	+	0.429236i	$0.141217\pi$
$439$	3.05794e10	+	5.29651e10i	0.823325	+	1.42604i	−0.0798674	+	0.996805i	$0.525450\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.57596e10	−	2.64193e10i	1.18814	−	0.685972i	0.230256	−	0.973130i	$-0.426044\pi$
$443$	4.57596e10	−	2.64193e10i	1.18814	−	0.685972i	0.957883	+	0.287158i	$0.0927104\pi$
$444$		0			0
$445$	4.95389e9	−	8.58039e9i	0.126330	−	0.218810i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			5.87526e10i			1.44558i	0.691068	+	0.722789i	$0.257140\pi$
$449$			5.87526e10i			1.44558i	−0.691068	+	0.722789i	$0.742860\pi$
$450$		0			0
$451$	1.19044e10	−	2.06190e10i	0.287741	−	0.498381i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.68774e9	−	1.24237e10i	0.0627108	−	0.289872i
$456$		0			0
$457$	−2.06415e10	−	3.57521e10i	−0.473234	−	0.819665i	0.526297	−	0.850301i	$-0.323580\pi$
$457$	−2.06415e10	−	3.57521e10i	−0.473234	−	0.819665i	−0.999531	+	0.0306360i	$0.990247\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			2.04131e10i			0.451965i	0.974131	+	0.225983i	$0.0725593\pi$
$461$			2.04131e10i			0.451965i	−0.974131	+	0.225983i	$0.927441\pi$
$462$		0			0
$463$	4.21217e10			0.916604			0.458302	−	0.888797i	$-0.348458\pi$
$463$	4.21217e10			0.916604			0.458302	+	0.888797i	$0.348458\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.39580e10	−	1.38322e10i	0.503713	−	0.290819i	−0.226532	−	0.974004i	$-0.572739\pi$
$467$	2.39580e10	−	1.38322e10i	0.503713	−	0.290819i	0.730246	+	0.683185i	$0.239406\pi$
$468$		0			0
$469$	2.51734e10	+	7.84399e10i	0.520296	+	1.62123i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.39959e10	+	3.69480e10i	1.27852	+	0.738153i
$474$		0			0
$475$	4.86912e9			0.0956481
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.75332e10	+	4.47638e10i	1.47281	+	0.850325i	0.999532	−	0.0305899i	$-0.00973860\pi$
$479$	7.75332e10	+	4.47638e10i	1.47281	+	0.850325i	0.473274	+	0.880915i	$0.343072\pi$
$480$		0			0
$481$	5.53933e9	+	9.59440e9i	0.103485	+	0.179241i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.84303e10	−	1.06407e10i	0.333092	−	0.192311i
$486$		0			0
$487$	2.07183e10	−	3.58852e10i	0.368332	−	0.637969i	−0.620973	−	0.783832i	$-0.713262\pi$
$487$	2.07183e10	−	3.58852e10i	0.368332	−	0.637969i	0.989305	+	0.145862i	$0.0465957\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	8.08752e10i		−	1.39152i	−0.718274	−	0.695760i	$-0.755068\pi$
$491$		−	8.08752e10i		−	1.39152i	0.718274	−	0.695760i	$-0.244932\pi$
$492$		0			0
$493$	−1.73918e10	+	3.01235e10i	−0.294413	+	0.509938i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.67676e10	−	1.52078e10i	−0.274819	−	0.249254i
$498$		0			0
$499$	1.50691e10	+	2.61004e10i	0.243044	+	0.420964i	0.961580	−	0.274525i	$-0.0885207\pi$
$499$	1.50691e10	+	2.61004e10i	0.243044	+	0.420964i	−0.718536	+	0.695490i	$0.755187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.05073e10i		−	1.25766i	−0.777543	−	0.628830i	$-0.783534\pi$
$503$		−	8.05073e10i		−	1.25766i	0.777543	−	0.628830i	$-0.216466\pi$
$504$		0			0
$505$	2.03555e8			0.00312980
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.56782e10	−	9.05181e9i	0.233574	−	0.134854i	−0.378646	−	0.925542i	$-0.623610\pi$
$509$	1.56782e10	−	9.05181e9i	0.233574	−	0.134854i	0.612220	+	0.790688i	$0.290277\pi$
$510$		0			0
$511$	2.17580e10	+	6.77975e10i	0.319106	+	0.994328i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.05955e9	−	5.23053e9i	−0.128789	−	0.0743562i
$516$		0			0
$517$	−4.48855e10			−0.628266
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.35278e10	+	1.35838e10i	0.319324	+	0.184362i	0.651091	−	0.759000i	$-0.274312\pi$
$521$	2.35278e10	+	1.35838e10i	0.319324	+	0.184362i	−0.331767	+	0.943361i	$0.607645\pi$
$522$		0			0
$523$	1.54832e10	+	2.68177e10i	0.206945	+	0.358439i	0.950751	−	0.309957i	$-0.100315\pi$
$523$	1.54832e10	+	2.68177e10i	0.206945	+	0.358439i	−0.743806	+	0.668396i	$0.766981\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.93674e10	+	1.11818e10i	−0.251089	+	0.144966i
$528$		0			0
$529$	−3.66823e10	+	6.35356e10i	−0.468418	+	0.811324i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2.45612e10i		−	0.304328i
$534$		0			0
$535$	−1.05258e9	+	1.82312e9i	−0.0128481	+	0.0222536i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.11577e10	−	1.34731e11i	0.724597	−	1.59630i
$540$		0			0
$541$	−6.25783e9	−	1.08389e10i	−0.0730524	−	0.126531i	0.827185	−	0.561929i	$-0.189941\pi$
$541$	−6.25783e9	−	1.08389e10i	−0.0730524	−	0.126531i	−0.900238	+	0.435399i	$0.856607\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			4.86564e10i			0.551511i
$546$		0			0
$547$	−2.71666e10			−0.303449			−0.151725	−	0.988423i	$-0.548483\pi$
$547$	−2.71666e10			−0.303449			−0.151725	+	0.988423i	$0.548483\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.88067e9	−	2.81786e9i	0.0529509	−	0.0305712i
$552$		0			0
$553$	1.59504e10	+	3.45069e9i	0.170557	+	0.0368983i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.79478e10	+	3.34562e10i	0.602027	+	0.347581i	0.769839	−	0.638238i	$-0.220337\pi$
$557$	5.79478e10	+	3.34562e10i	0.602027	+	0.347581i	−0.167811	+	0.985819i	$0.553670\pi$
$558$		0			0
$559$	7.62314e10			0.780705
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.82440e10	+	4.51742e10i	0.778784	+	0.449631i	0.835999	−	0.548730i	$-0.184889\pi$
$563$	7.82440e10	+	4.51742e10i	0.778784	+	0.449631i	−0.0572149	+	0.998362i	$0.518222\pi$
$564$		0			0
$565$	−2.24075e10	−	3.88110e10i	−0.219887	−	0.380856i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.44180e10	−	1.98712e10i	0.328349	−	0.189573i	−0.326759	−	0.945108i	$-0.605956\pi$
$569$	3.44180e10	−	1.98712e10i	0.328349	−	0.189573i	0.655108	+	0.755535i	$0.272623\pi$
$570$		0			0
$571$	−2.89573e10	+	5.01556e10i	−0.272405	+	0.471819i	−0.969477	−	0.245182i	$-0.921152\pi$
$571$	−2.89573e10	+	5.01556e10i	−0.272405	+	0.471819i	0.697072	+	0.717001i	$0.254486\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	2.46612e10i		−	0.225602i
$576$		0			0
$577$	−2.37708e9	+	4.11722e9i	−0.0214457	+	0.0371451i	−0.876549	−	0.481312i	$-0.840160\pi$
$577$	−2.37708e9	+	4.11722e9i	−0.0214457	+	0.0371451i	0.855103	+	0.518458i	$0.173494\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.13063e11	+	1.02545e11i	0.992241	+	0.899937i
$582$		0			0
$583$	5.26095e10	+	9.11224e10i	0.455397	+	0.788771i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.98005e9i		−	0.0166773i	−0.999965	−	0.00833863i	$-0.997346\pi$
$587$		−	1.98005e9i		−	0.0166773i	0.999965	−	0.00833863i	$-0.00265430\pi$
$588$		0			0
$589$	3.62339e9			0.0301060
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.24676e11	−	7.19820e10i	1.00824	−	0.582110i	0.0975670	−	0.995229i	$-0.468894\pi$
$593$	1.24676e11	−	7.19820e10i	1.00824	−	0.582110i	0.910677	+	0.413119i	$0.135561\pi$
$594$		0			0
$595$	2.76415e10	−	3.04766e10i	0.220543	−	0.243163i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.68023e10	−	9.70083e9i	−0.130515	−	0.0753532i	0.433321	−	0.901240i	$-0.357342\pi$
$599$	−1.68023e10	−	9.70083e9i	−0.130515	−	0.0753532i	−0.563836	+	0.825887i	$0.690675\pi$
$600$		0			0
$601$	7.28074e10			0.558056			0.279028	−	0.960283i	$-0.409988\pi$
$601$	7.28074e10			0.558056			0.279028	+	0.960283i	$0.409988\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.69532e10	+	4.44289e10i	0.574388	+	0.331623i
$606$		0			0
$607$	3.88928e10	+	6.73643e10i	0.286493	+	0.496221i	0.972970	−	0.230931i	$-0.0741771\pi$
$607$	3.88928e10	+	6.73643e10i	0.286493	+	0.496221i	−0.686477	+	0.727152i	$0.740844\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.01004e10	+	2.31520e10i	−0.287729	+	0.166121i
$612$		0			0
$613$	−4.62809e10	+	8.01609e10i	−0.327763	+	0.567702i	−0.982068	−	0.188529i	$-0.939628\pi$
$613$	−4.62809e10	+	8.01609e10i	−0.327763	+	0.567702i	0.654304	+	0.756231i	$0.272961\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.25873e11i			0.868546i	0.900781	+	0.434273i	$0.142995\pi$
$617$			1.25873e11i			0.868546i	−0.900781	+	0.434273i	$0.857005\pi$
$618$		0			0
$619$	−7.14248e10	+	1.23711e11i	−0.486504	+	0.842650i	−0.999880	−	0.0155142i	$-0.995061\pi$
$619$	−7.14248e10	+	1.23711e11i	−0.486504	+	0.842650i	0.513376	+	0.858164i	$0.328395\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.51570e10	−	1.16285e11i	0.166996	−	0.771917i
$624$		0			0
$625$	−5.36682e10	−	9.29560e10i	−0.351720	−	0.609196i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			3.58604e10i			0.229093i
$630$		0			0
$631$	−7.51592e10			−0.474094			−0.237047	−	0.971498i	$-0.576180\pi$
$631$	−7.51592e10			−0.474094			−0.237047	+	0.971498i	$0.576180\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.00785e10	+	1.73659e10i	−0.184996	+	0.106807i
$636$		0			0
$637$	−1.48566e10	−	1.51913e11i	−0.0902322	−	0.922653i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.99385e11	−	1.15115e11i	−1.18103	−	0.681866i	−0.224775	−	0.974411i	$-0.572165\pi$
$641$	−1.99385e11	−	1.15115e11i	−1.18103	−	0.681866i	−0.956252	+	0.292544i	$0.905498\pi$
$642$		0			0
$643$	2.71276e11			1.58696			0.793482	−	0.608594i	$-0.208266\pi$
$643$	2.71276e11			1.58696			0.793482	+	0.608594i	$0.208266\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.30537e11	−	7.53656e10i	−0.744932	−	0.430087i	0.0789278	−	0.996880i	$-0.474850\pi$
$647$	−1.30537e11	−	7.53656e10i	−0.744932	−	0.430087i	−0.823860	+	0.566794i	$0.808184\pi$
$648$		0			0
$649$	−1.11808e11	−	1.93657e11i	−0.630224	−	1.09158i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.55749e11	+	1.47657e11i	−1.40657	+	0.812084i	−0.995056	−	0.0993195i	$-0.968333\pi$
$653$	−2.55749e11	+	1.47657e11i	−1.40657	+	0.812084i	−0.411515	+	0.911403i	$0.635000\pi$
$654$		0			0
$655$	4.92603e10	−	8.53213e10i	0.267628	−	0.463545i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	2.93693e10i		−	0.155723i	−0.996964	−	0.0778615i	$-0.975191\pi$
$659$		−	2.93693e10i		−	0.155723i	0.996964	−	0.0778615i	$-0.0248092\pi$
$660$		0			0
$661$	−1.25565e11	+	2.17485e11i	−0.657754	+	1.13926i	0.323442	+	0.946248i	$0.395160\pi$
$661$	−1.25565e11	+	2.17485e11i	−0.657754	+	1.13926i	−0.981196	+	0.193015i	$0.938173\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.34747e9	+	2.03707e9i	−0.0324574	+	0.0104164i
$666$		0			0
$667$	−1.42719e10	−	2.47197e10i	−0.0721072	−	0.124893i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2.40739e11i		−	1.18756i
$672$		0			0
$673$	−1.36922e11			−0.667441			−0.333721	−	0.942672i	$-0.608304\pi$
$673$	−1.36922e11			−0.667441			−0.333721	+	0.942672i	$0.608304\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.32297e11	+	1.91852e11i	−1.58187	+	0.913294i	−0.587286	+	0.809380i	$0.699803\pi$
$677$	−3.32297e11	+	1.91852e11i	−1.58187	+	0.913294i	−0.994586	+	0.103914i	$0.966863\pi$
$678$		0			0
$679$	1.71683e11	−	1.89293e11i	0.807699	−	0.890542i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.91670e11	−	1.10660e11i	−0.880786	−	0.508522i	−0.00986828	−	0.999951i	$-0.503141\pi$
$683$	−1.91670e11	−	1.10660e11i	−0.880786	−	0.508522i	−0.870917	+	0.491429i	$0.836475\pi$
$684$		0			0
$685$	6.29305e10			0.285824
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.40021e10	+	5.42722e10i	0.417120	+	0.240824i
$690$		0			0
$691$	−9.83045e10	−	1.70268e11i	−0.431182	−	0.746830i	0.565793	−	0.824547i	$-0.308570\pi$
$691$	−9.83045e10	−	1.70268e11i	−0.431182	−	0.746830i	−0.996975	+	0.0777174i	$0.975237\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.36798e10	−	3.67656e10i	0.272937	−	0.157580i
$696$		0			0
$697$	3.97510e10	−	6.88507e10i	0.168429	−	0.291727i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			4.55475e11i			1.88622i	0.332482	+	0.943109i	$0.392114\pi$
$701$			4.55475e11i			1.88622i	−0.332482	+	0.943109i	$0.607886\pi$
$702$		0			0
$703$	2.90509e9	−	5.03176e9i	0.0118943	−	0.0206015i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.32742e9	−	7.46928e8i	0.00931529	−	0.00298952i
$708$		0			0
$709$	−1.02936e11	−	1.78291e11i	−0.407365	−	0.705576i	0.587229	−	0.809421i	$-0.300219\pi$
$709$	−1.02936e11	−	1.78291e11i	−0.407365	−	0.705576i	−0.994594	+	0.103845i	$0.966886\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.83518e10i		−	0.0710100i
$714$		0			0
$715$	1.35881e11			0.519917
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.11339e11	−	2.37487e11i	1.53916	−	0.888636i	0.540275	−	0.841488i	$-0.318320\pi$
$719$	4.11339e11	−	2.37487e11i	1.53916	−	0.888636i	0.998888	−	0.0471477i	$-0.0150132\pi$
$720$		0			0
$721$	−1.22778e11	−	2.65618e10i	−0.454340	−	0.0982917i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.23245e11	−	7.11553e10i	−0.446083	−	0.257546i
$726$		0			0
$727$	−1.69567e11			−0.607019			−0.303510	−	0.952828i	$-0.598158\pi$
$727$	−1.69567e11			−0.607019			−0.303510	+	0.952828i	$0.598158\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.13694e11	+	1.23376e11i	0.748381	+	0.432078i
$732$		0			0
$733$	5.53134e10	+	9.58056e10i	0.191608	+	0.331875i	0.945783	−	0.324798i	$-0.105296\pi$
$733$	5.53134e10	+	9.58056e10i	0.191608	+	0.331875i	−0.754175	+	0.656673i	$0.771963\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.62655e11	+	4.40319e11i	−2.58498	+	1.49244i
$738$		0			0
$739$	2.39650e11	−	4.15087e11i	0.803527	−	1.39175i	−0.113755	−	0.993509i	$-0.536288\pi$
$739$	2.39650e11	−	4.15087e11i	0.803527	−	1.39175i	0.917281	−	0.398240i	$-0.130379\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2.03809e11i		−	0.668755i	−0.942439	−	0.334378i	$-0.891474\pi$
$743$		−	2.03809e11i		−	0.668755i	0.942439	−	0.334378i	$-0.108526\pi$
$744$		0			0
$745$	6.27475e10	−	1.08682e11i	0.203691	−	0.352803i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.34523e9	+	2.47076e10i	−0.0169840	+	0.0785061i
$750$		0			0
$751$	−1.03368e11	−	1.79038e11i	−0.324956	−	0.562841i	0.656547	−	0.754285i	$-0.272016\pi$
$751$	−1.03368e11	−	1.79038e11i	−0.324956	−	0.562841i	−0.981503	+	0.191444i	$0.938683\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			4.43340e10i			0.136442i
$756$		0			0
$757$	−3.91878e11			−1.19335			−0.596674	−	0.802484i	$-0.703511\pi$
$757$	−3.91878e11			−1.19335			−0.596674	+	0.802484i	$0.703511\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.48747e11	+	2.59084e11i	−1.33802	+	0.772507i	−0.986514	−	0.163678i	$-0.947664\pi$
$761$	−4.48747e11	+	2.59084e11i	−1.33802	+	0.772507i	−0.351508	+	0.936185i	$0.614331\pi$
$762$		0			0
$763$	1.78541e11	+	5.56330e11i	0.526792	+	1.64147i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.99778e11	−	1.15342e11i	−0.577252	−	0.333277i
$768$		0			0
$769$	−5.97465e11			−1.70847			−0.854234	−	0.519888i	$-0.825974\pi$
$769$	−5.97465e11			−1.70847			−0.854234	+	0.519888i	$0.825974\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.43571e10	+	8.28910e9i	0.0402115	+	0.0232161i	0.519971	−	0.854184i	$-0.325943\pi$
$773$	1.43571e10	+	8.28910e9i	0.0402115	+	0.0232161i	−0.479760	+	0.877400i	$0.659276\pi$
$774$		0			0
$775$	−4.57481e10	−	7.92380e10i	−0.126814	−	0.219648i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.11553e10	+	6.44054e9i	−0.0302924	+	0.0174893i
$780$		0			0
$781$	1.20993e11	−	2.09567e11i	0.325205	−	0.563272i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1.19259e11i			0.314061i
$786$		0			0
$787$	−9.57485e10	+	1.65841e11i	−0.249593	+	0.432309i	−0.963413	−	0.268021i	$-0.913630\pi$
$787$	−9.57485e10	+	1.65841e11i	−0.249593	+	0.432309i	0.713820	+	0.700330i	$0.246964\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.98618e11	−	3.61536e11i	−1.01824	−	0.923519i
$792$		0			0
$793$	−1.24174e11	−	2.15075e11i	−0.314005	−	0.543873i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	5.08730e11i		−	1.26082i	−0.776261	−	0.630412i	$-0.782886\pi$
$797$		−	5.08730e11i		−	1.26082i	0.776261	−	0.630412i	$-0.217114\pi$
$798$		0			0
$799$	−1.49881e11			−0.367755
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.59181e11	+	3.80578e11i	−1.58541	+	0.915338i
$804$		0			0
$805$	1.03174e10	+	3.21487e10i	0.0245688	+	0.0765561i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.98510e11	+	2.30080e11i	0.930347	+	0.537136i	0.886921	−	0.461921i	$-0.152840\pi$
$809$	3.98510e11	+	2.30080e11i	0.930347	+	0.537136i	0.0434254	+	0.999057i	$0.486173\pi$
$810$		0			0
$811$	4.68837e11			1.08377			0.541886	−	0.840452i	$-0.317710\pi$
$811$	4.68837e11			1.08377			0.541886	+	0.840452i	$0.317710\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.09842e11	+	6.34175e10i	0.248966	+	0.143740i
$816$		0			0
$817$	−1.99897e10	−	3.46232e10i	−0.0448661	−	0.0777103i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.18419e11	+	2.41574e11i	−0.920955	+	0.531714i	−0.883940	−	0.467601i	$-0.845118\pi$
$821$	−4.18419e11	+	2.41574e11i	−0.920955	+	0.531714i	−0.0370153	+	0.999315i	$0.511785\pi$
$822$		0			0
$823$	1.67596e11	−	2.90285e11i	0.365313	−	0.632740i	−0.623514	−	0.781813i	$-0.714295\pi$
$823$	1.67596e11	−	2.90285e11i	0.365313	−	0.632740i	0.988826	+	0.149072i	$0.0476287\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			3.35019e11i			0.716221i	0.933679	+	0.358111i	$0.116579\pi$
$827$			3.35019e11i			0.716221i	−0.933679	+	0.358111i	$0.883421\pi$
$828$		0			0
$829$	2.89587e11	−	5.01580e11i	0.613143	−	1.06199i	−0.377564	−	0.925983i	$-0.623238\pi$
$829$	2.89587e11	−	5.01580e11i	0.613143	−	1.06199i	0.990707	−	0.136011i	$-0.0434283\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.04217e11	−	4.49892e11i	0.424142	−	0.934391i
$834$		0			0
$835$	−1.34608e11	−	2.33148e11i	−0.276901	−	0.479607i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	4.93362e11i		−	0.995676i	−0.867270	−	0.497838i	$-0.834127\pi$
$839$		−	4.93362e11i		−	0.995676i	0.867270	−	0.497838i	$-0.165873\pi$
$840$		0			0
$841$	3.35531e11			0.670731
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.98556e10	+	1.14636e10i	−0.0389453	+	0.0224851i
$846$		0			0
$847$	1.04290e12	+	2.25620e11i	2.02632	+	0.438374i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.54849e10	−	1.47137e10i	−0.0485920	−	0.0280546i
$852$		0			0
$853$	−9.37830e10			−0.177145			−0.0885723	−	0.996070i	$-0.528230\pi$
$853$	−9.37830e10			−0.177145			−0.0885723	+	0.996070i	$0.528230\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.00194e11	+	5.19727e11i	1.66883	+	0.963501i	0.968269	+	0.249912i	$0.0804016\pi$
$857$	9.00194e11	+	5.19727e11i	1.66883	+	0.963501i	0.700564	+	0.713589i	$0.252932\pi$
$858$		0			0
$859$	−1.47041e11	−	2.54683e11i	−0.270064	−	0.467764i	0.698814	−	0.715303i	$-0.253711\pi$
$859$	−1.47041e11	−	2.54683e11i	−0.270064	−	0.467764i	−0.968878	+	0.247539i	$0.920378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.68952e11	+	1.55280e11i	−0.484877	+	0.279944i	−0.722447	−	0.691427i	$-0.756983\pi$
$863$	−2.68952e11	+	1.55280e11i	−0.484877	+	0.279944i	0.237570	+	0.971371i	$0.423649\pi$
$864$		0			0
$865$	1.14669e11	−	1.98613e11i	0.204825	−	0.354767i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.74452e11i			0.305913i
$870$		0			0
$871$	−4.54234e11	+	7.86757e11i	−0.789237	+	1.36700i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.63595e11	+	2.39074e11i	0.449681	+	0.407849i
$876$		0			0
$877$	−2.71510e11	−	4.70270e11i	−0.458974	−	0.794966i	0.539933	−	0.841708i	$-0.318450\pi$
$877$	−2.71510e11	−	4.70270e11i	−0.458974	−	0.794966i	−0.998907	+	0.0467419i	$0.985116\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	3.65226e11i		−	0.606258i	−0.952949	−	0.303129i	$-0.901969\pi$
$881$		−	3.65226e11i		−	0.606258i	0.952949	−	0.303129i	$-0.0980314\pi$
$882$		0			0
$883$	−2.21358e11			−0.364127			−0.182063	−	0.983287i	$-0.558278\pi$
$883$	−2.21358e11			−0.364127			−0.182063	+	0.983287i	$0.558278\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.30382e10	+	3.06216e10i	−0.0856829	+	0.0494690i	−0.542229	−	0.840231i	$-0.682420\pi$
$887$	−5.30382e10	+	3.06216e10i	−0.0856829	+	0.0494690i	0.456546	+	0.889700i	$0.349086\pi$
$888$		0			0
$889$	−2.80191e11	+	3.08929e11i	−0.448587	+	0.494597i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.10306e10	+	1.21420e10i	0.0330709	+	0.0190935i
$894$		0			0
$895$	1.03607e11			0.161473
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.17132e10	−	5.29507e10i	−0.140408	−	0.0810648i
$900$		0			0
$901$	1.75673e11	+	3.04274e11i	0.266566	+	0.461707i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.15966e11	+	6.69528e10i	−0.172876	+	0.0998101i
$906$		0			0
$907$	−3.57947e11	+	6.19983e11i	−0.528920	+	0.916116i	0.470511	+	0.882394i	$0.344069\pi$
$907$	−3.57947e11	+	6.19983e11i	−0.528920	+	0.916116i	−0.999431	+	0.0337221i	$0.989264\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	5.81279e11i		−	0.843939i	−0.906610	−	0.421969i	$-0.861339\pi$
$911$		−	5.81279e11i		−	0.843939i	0.906610	−	0.421969i	$-0.138661\pi$
$912$		0			0
$913$	−8.15852e11	+	1.41310e12i	−1.17416	+	2.03371i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.50155e11	−	1.15631e12i	0.353779	−	1.63529i
$918$		0			0
$919$	−1.53168e11	−	2.65295e11i	−0.214736	−	0.371934i	0.738455	−	0.674303i	$-0.235556\pi$
$919$	−1.53168e11	−	2.65295e11i	−0.214736	−	0.371934i	−0.953191	+	0.302369i	$0.902223\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	2.49634e11i		−	0.343952i
$924$		0			0
$925$	−1.46716e11			−0.200406
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.51692e10	−	2.03049e10i	0.0472171	−	0.0272608i	−0.476206	−	0.879334i	$-0.657988\pi$
$929$	3.51692e10	−	2.03049e10i	0.0472171	−	0.0272608i	0.523423	+	0.852073i	$0.324655\pi$
$930$		0			0
$931$	−6.51011e10	+	4.65830e10i	−0.0866542	+	0.0620053i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.80905e11	+	2.19916e11i	0.498391	+	0.287746i
$936$		0			0
$937$	7.67613e11			0.995827			0.497914	−	0.867227i	$-0.334100\pi$
$937$	7.67613e11			0.995827			0.497914	+	0.867227i	$0.334100\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.18320e12	+	6.83120e11i	1.50903	+	0.871241i	0.999945	+	0.0105268i	$0.00335085\pi$
$941$	1.18320e12	+	6.83120e11i	1.50903	+	0.871241i	0.509089	+	0.860714i	$0.329982\pi$
$942$		0			0
$943$	3.26201e10	+	5.64997e10i	0.0412514	+	0.0714495i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.64775e11	−	3.26073e11i	0.702223	−	0.405429i	−0.105952	−	0.994371i	$-0.533789\pi$
$947$	5.64775e11	−	3.26073e11i	0.702223	−	0.405429i	0.808175	+	0.588942i	$0.200456\pi$
$948$		0			0
$949$	−3.92606e11	+	6.80013e11i	−0.484052	+	0.838402i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.50516e12i			1.82478i	0.409323	+	0.912390i	$0.365765\pi$
$953$			1.50516e12i			1.82478i	−0.409323	+	0.912390i	$0.634235\pi$
$954$		0			0
$955$	1.15468e11	−	1.99997e11i	0.138819	−	0.240442i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.19538e11	−	2.30918e11i	0.850705	−	0.273013i
$960$		0			0
$961$	3.92402e11	+	6.79660e11i	0.460084	+	0.796889i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	3.88037e11i		−	0.447470i
$966$		0			0
$967$	−1.17932e11			−0.134873			−0.0674366	−	0.997724i	$-0.521482\pi$
$967$	−1.17932e11			−0.134873			−0.0674366	+	0.997724i	$0.521482\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.49476e12	−	8.62999e11i	1.68149	−	0.970808i	0.720815	−	0.693128i	$-0.243768\pi$
$971$	1.49476e12	−	8.62999e11i	1.68149	−	0.970808i	0.960674	−	0.277680i	$-0.0895655\pi$
$972$		0			0
$973$	5.93197e11	−	6.54039e11i	0.661832	−	0.729714i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.44634e11	−	4.87650e11i	−0.927022	−	0.535217i	−0.0411538	−	0.999153i	$-0.513103\pi$
$977$	−8.44634e11	−	4.87650e11i	−0.927022	−	0.535217i	−0.885869	+	0.463936i	$0.846437\pi$
$978$		0			0
$979$	1.27183e12			1.38452
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.22955e12	+	7.09884e11i	1.31684	+	0.760279i	0.983219	−	0.182427i	$-0.0583954\pi$
$983$	1.22955e12	+	7.09884e11i	1.31684	+	0.760279i	0.333623	+	0.942707i	$0.391729\pi$
$984$		0			0
$985$	4.41967e10	+	7.65510e10i	0.0469510	+	0.0813216i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.75360e11	+	1.01244e11i	−0.183293	+	0.105824i
$990$		0			0
$991$	2.13877e11	−	3.70445e11i	0.221753	−	0.384087i	−0.733588	−	0.679595i	$-0.762156\pi$
$991$	2.13877e11	−	3.70445e11i	0.221753	−	0.384087i	0.955340	+	0.295508i	$0.0954889\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			1.18057e11i			0.120448i
$996$		0			0
$997$	−5.15688e11	+	8.93198e11i	−0.521923	+	0.903997i	0.477752	+	0.878495i	$0.341452\pi$
$997$	−5.15688e11	+	8.93198e11i	−0.521923	+	0.903997i	−0.999675	+	0.0255025i	$0.991881\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bk.a.53.14	yes	44
3.2	odd	2	inner	252.9.bk.a.53.9	✓	44
7.2	even	3	inner	252.9.bk.a.233.9	yes	44
21.2	odd	6	inner	252.9.bk.a.233.14	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bk.a.53.9	✓	44	3.2	odd	2	inner
252.9.bk.a.53.14	yes	44	1.1	even	1	trivial
252.9.bk.a.233.9	yes	44	7.2	even	3	inner
252.9.bk.a.233.14	yes	44	21.2	odd	6	inner

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(173.159 - 99.9732i) q^{5} +(1613.03 - 1778.47i) q^{7} +O(q^{10})\)	\(q+(173.159 - 99.9732i) q^{5} +(1613.03 - 1778.47i) q^{7} +(22227.8 + 12833.2i) q^{11} +26477.6 q^{13} +(74222.8 + 42852.5i) q^{17} +(-6943.06 - 12025.7i) q^{19} +(-60908.1 + 35165.3i) q^{23} +(-175323. + 303669. i) q^{25} +405852. i q^{29} +(-130468. + 225977. i) q^{31} +(101510. - 469217. i) q^{35} +(209208. + 362359. i) q^{37} -927622. i q^{41} +2.87909e6 q^{43} +(-1.51450e6 + 874399. i) q^{47} +(-561100. - 5.73743e6i) q^{49} +(3.55025e6 + 2.04974e6i) q^{53} +5.13192e6 q^{55} +(-7.54515e6 - 4.35620e6i) q^{59} +(-4.68976e6 - 8.12291e6i) q^{61} +(4.58483e6 - 2.64705e6i) q^{65} +(-1.71554e7 + 2.97140e7i) q^{67} -9.42813e6i q^{71} +(-1.48278e7 + 2.56826e7i) q^{73} +(5.86775e7 - 1.88311e7i) q^{77} +(3.39845e6 + 5.88629e6i) q^{79} +6.35734e7i q^{83} +1.71364e7 q^{85} +(4.29135e7 - 2.47761e7i) q^{89} +(4.27091e7 - 4.70896e7i) q^{91} +(-2.40450e6 - 1.38824e6i) q^{95} +1.06436e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 1230 q^{7}+O(q^{10}) \) 44 * q + 1230 * q^7	\( 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97}+O(q^{100}) \) 44 * q + 1230 * q^7 - 101380 * q^13 + 62770 * q^19 + 2247666 * q^25 + 1389254 * q^31 - 2136026 * q^37 + 11510140 * q^43 - 3824398 * q^49 - 42646528 * q^55 + 27346232 * q^61 + 14239194 * q^67 - 64344138 * q^73 + 7061786 * q^79 - 54198208 * q^85 - 45697066 * q^91 + 476543496 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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