LMFDB - Embedded newform 2352.2.bl.m.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(31,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2352.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.bl (of order $6$, degree $2$, not 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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2352.31
Dual form			2352.2.bl.m.607.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(2.12132 - 1.22474i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	$q+(-0.500000 + 0.866025i) q^{3} +(2.12132 - 1.22474i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(2.12132 + 1.22474i) q^{11} +4.89898i q^{13} +2.44949i q^{15} +(-2.12132 - 1.22474i) q^{17} +(1.00000 + 1.73205i) q^{19} +(-6.36396 + 3.67423i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +6.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(-2.12132 + 1.22474i) q^{33} +(-2.00000 - 3.46410i) q^{37} +(-4.24264 - 2.44949i) q^{39} -7.34847i q^{41} +4.89898i q^{43} +(-2.12132 - 1.22474i) q^{45} +(6.00000 + 10.3923i) q^{47} +(2.12132 - 1.22474i) q^{51} +(3.00000 - 5.19615i) q^{53} +6.00000 q^{55} -2.00000 q^{57} +(-6.00000 + 10.3923i) q^{59} +(6.00000 + 10.3923i) q^{65} -7.34847i q^{69} -12.2474i q^{71} +(12.7279 + 7.34847i) q^{73} +(0.500000 + 0.866025i) q^{75} +(8.48528 - 4.89898i) q^{79} +(-0.500000 + 0.866025i) q^{81} -6.00000 q^{85} +(-3.00000 + 5.19615i) q^{87} +(-10.6066 + 6.12372i) q^{89} +(-4.00000 - 6.92820i) q^{93} +(4.24264 + 2.44949i) q^{95} +4.89898i q^{97} -2.44949i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 2 * q^9	$ 4 q - 2 q^{3} - 2 q^{9} + 4 q^{19} + 2 q^{25} + 4 q^{27} + 24 q^{29} - 16 q^{31} - 8 q^{37} + 24 q^{47} + 12 q^{53} + 24 q^{55} - 8 q^{57} - 24 q^{59} + 24 q^{65} + 2 q^{75} - 2 q^{81} - 24 q^{85} - 12 q^{87} - 16 q^{93}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^9 + 4 * q^19 + 2 * q^25 + 4 * q^27 + 24 * q^29 - 16 * q^31 - 8 * q^37 + 24 * q^47 + 12 * q^53 + 24 * q^55 - 8 * q^57 - 24 * q^59 + 24 * q^65 + 2 * q^75 - 2 * q^81 - 24 * q^85 - 12 * q^87 - 16 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$	2.12132	−	1.22474i	0.948683	−	0.547723i	0.0560116	−	0.998430i	$-0.482162\pi$
$5$	2.12132	−	1.22474i	0.948683	−	0.547723i	0.892672	+	0.450708i	$0.148828\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	2.12132	+	1.22474i	0.639602	+	0.369274i	0.784461	−	0.620178i	$-0.212940\pi$
$11$	2.12132	+	1.22474i	0.639602	+	0.369274i	−0.144859	+	0.989452i	$0.546273\pi$
$12$		0			0
$13$			4.89898i			1.35873i	0.733799	+	0.679366i	$0.237745\pi$
$13$			4.89898i			1.35873i	−0.733799	+	0.679366i	$0.762255\pi$
$14$		0			0
$15$			2.44949i			0.632456i
$16$		0			0
$17$	−2.12132	−	1.22474i	−0.514496	−	0.297044i	0.220184	−	0.975458i	$-0.429334\pi$
$17$	−2.12132	−	1.22474i	−0.514496	−	0.297044i	−0.734680	+	0.678414i	$0.762668\pi$
$18$		0			0
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	$-0.0929851\pi$
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	−0.728219	+	0.685344i	$0.759652\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.36396	+	3.67423i	−1.32698	+	0.766131i	−0.984831	−	0.173516i	$-0.944487\pi$
$23$	−6.36396	+	3.67423i	−1.32698	+	0.766131i	−0.342147	+	0.939647i	$0.611154\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$		0			0
$33$	−2.12132	+	1.22474i	−0.369274	+	0.213201i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	0.653476	−	0.756948i	$-0.273310\pi$
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	−0.982274	+	0.187453i	$0.939977\pi$
$38$		0			0
$39$	−4.24264	−	2.44949i	−0.679366	−	0.392232i
$40$		0			0
$41$		−	7.34847i		−	1.14764i	−0.818982	−	0.573819i	$-0.805461\pi$
$41$		−	7.34847i		−	1.14764i	0.818982	−	0.573819i	$-0.194539\pi$
$42$		0			0
$43$			4.89898i			0.747087i	0.927613	+	0.373544i	$0.121857\pi$
$43$			4.89898i			0.747087i	−0.927613	+	0.373544i	$0.878143\pi$
$44$		0			0
$45$	−2.12132	−	1.22474i	−0.316228	−	0.182574i
$46$		0			0
$47$	6.00000	+	10.3923i	0.875190	+	1.51587i	0.856560	+	0.516047i	$0.172597\pi$
$47$	6.00000	+	10.3923i	0.875190	+	1.51587i	0.0186297	+	0.999826i	$0.494070\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	2.12132	−	1.22474i	0.297044	−	0.171499i
$52$		0			0
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$	−2.00000			−0.264906
$58$		0			0
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	+	10.3923i	0.744208	+	1.28901i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		−	7.34847i		−	0.884652i
$70$		0			0
$71$		−	12.2474i		−	1.45350i	−0.686900	−	0.726752i	$-0.741029\pi$
$71$		−	12.2474i		−	1.45350i	0.686900	−	0.726752i	$-0.258971\pi$
$72$		0			0
$73$	12.7279	+	7.34847i	1.48969	+	0.860073i	0.999931	−	0.0117840i	$-0.00375105\pi$
$73$	12.7279	+	7.34847i	1.48969	+	0.860073i	0.489760	+	0.871857i	$0.337084\pi$
$74$		0			0
$75$	0.500000	+	0.866025i	0.0577350	+	0.100000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.48528	−	4.89898i	0.954669	−	0.551178i	0.0601406	−	0.998190i	$-0.480845\pi$
$79$	8.48528	−	4.89898i	0.954669	−	0.551178i	0.894528	+	0.447012i	$0.147512\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$	−3.00000	+	5.19615i	−0.321634	+	0.557086i
$88$		0			0
$89$	−10.6066	+	6.12372i	−1.12430	+	0.649113i	−0.942495	−	0.334221i	$-0.891527\pi$
$89$	−10.6066	+	6.12372i	−1.12430	+	0.649113i	−0.181803	+	0.983335i	$0.558193\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−4.00000	−	6.92820i	−0.414781	−	0.718421i
$94$		0			0
$95$	4.24264	+	2.44949i	0.435286	+	0.251312i
$96$		0			0
$97$			4.89898i			0.497416i	0.968579	+	0.248708i	$0.0800060\pi$
$97$			4.89898i			0.497416i	−0.968579	+	0.248708i	$0.919994\pi$
$98$		0			0
$99$		−	2.44949i		−	0.246183i
$100$		0			0
$101$	2.12132	+	1.22474i	0.211079	+	0.121867i	0.601813	−	0.798637i	$-0.294445\pi$
$101$	2.12132	+	1.22474i	0.211079	+	0.121867i	−0.390734	+	0.920504i	$0.627779\pi$
$102$		0			0
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	0.927030	+	0.374987i	$0.122353\pi$
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	−0.138767	+	0.990325i	$0.544314\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.6066	−	6.12372i	1.02538	−	0.592003i	0.109721	−	0.993962i	$-0.465004\pi$
$107$	10.6066	−	6.12372i	1.02538	−	0.592003i	0.915657	+	0.401960i	$0.131671\pi$
$108$		0			0
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	0.814152	+	0.580651i	$0.197202\pi$
$110$		0			0
$111$	4.00000			0.379663
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	−9.00000	+	15.5885i	−0.839254	+	1.45363i
$116$		0			0
$117$	4.24264	−	2.44949i	0.392232	−	0.226455i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$		0			0
$123$	6.36396	+	3.67423i	0.573819	+	0.331295i
$124$		0			0
$125$			9.79796i			0.876356i
$126$		0			0
$127$		−	9.79796i		−	0.869428i	−0.900568	−	0.434714i	$-0.856849\pi$
$127$		−	9.79796i		−	0.869428i	0.900568	−	0.434714i	$-0.143151\pi$
$128$		0			0
$129$	−4.24264	−	2.44949i	−0.373544	−	0.215666i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.12132	−	1.22474i	0.182574	−	0.105409i
$136$		0			0
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	−0.965250	−	0.261329i	$-0.915839\pi$
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	0.708942	+	0.705266i	$0.249173\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	−12.0000			−1.01058
$142$		0			0
$143$	−6.00000	+	10.3923i	−0.501745	+	0.869048i
$144$		0			0
$145$	12.7279	−	7.34847i	1.05700	−	0.610257i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	0.953703	+	0.300750i	$0.0972370\pi$
$149$	9.00000	+	15.5885i	0.737309	+	1.27706i	−0.216394	+	0.976306i	$0.569430\pi$
$150$		0			0
$151$	−12.7279	−	7.34847i	−1.03578	−	0.598010i	−0.117147	−	0.993115i	$-0.537375\pi$
$151$	−12.7279	−	7.34847i	−1.03578	−	0.598010i	−0.918636	+	0.395105i	$0.870708\pi$
$152$		0			0
$153$			2.44949i			0.198030i
$154$		0			0
$155$			19.5959i			1.57398i
$156$		0			0
$157$	−16.9706	−	9.79796i	−1.35440	−	0.781962i	−0.365536	−	0.930797i	$-0.619114\pi$
$157$	−16.9706	−	9.79796i	−1.35440	−	0.781962i	−0.988862	+	0.148835i	$0.952448\pi$
$158$		0			0
$159$	3.00000	+	5.19615i	0.237915	+	0.412082i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−16.9706	+	9.79796i	−1.32924	+	0.767435i	−0.985182	−	0.171514i	$-0.945134\pi$
$163$	−16.9706	+	9.79796i	−1.32924	+	0.767435i	−0.344055	+	0.938949i	$0.611801\pi$
$164$		0			0
$165$	−3.00000	+	5.19615i	−0.233550	+	0.404520i
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$	1.00000	−	1.73205i	0.0764719	−	0.132453i
$172$		0			0
$173$	6.36396	−	3.67423i	0.483843	−	0.279347i	−0.238174	−	0.971223i	$-0.576549\pi$
$173$	6.36396	−	3.67423i	0.483843	−	0.279347i	0.722017	+	0.691876i	$0.243215\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−6.00000	−	10.3923i	−0.450988	−	0.781133i
$178$		0			0
$179$	−14.8492	−	8.57321i	−1.10988	−	0.640792i	−0.171085	−	0.985256i	$-0.554727\pi$
$179$	−14.8492	−	8.57321i	−1.10988	−	0.640792i	−0.938799	+	0.344464i	$0.888061\pi$
$180$		0			0
$181$		−	4.89898i		−	0.364138i	−0.983286	−	0.182069i	$-0.941721\pi$
$181$		−	4.89898i		−	0.364138i	0.983286	−	0.182069i	$-0.0582795\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.48528	−	4.89898i	−0.623850	−	0.360180i
$186$		0			0
$187$	−3.00000	−	5.19615i	−0.219382	−	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.12132	−	1.22474i	0.153493	−	0.0886194i	−0.421286	−	0.906928i	$-0.638421\pi$
$191$	2.12132	−	1.22474i	0.153493	−	0.0886194i	0.574779	+	0.818308i	$0.305088\pi$
$192$		0			0
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	0.420096	+	0.907480i	$0.361996\pi$
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	−0.995948	+	0.0899262i	$0.971337\pi$
$194$		0			0
$195$	−12.0000			−0.859338
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	1.00000	−	1.73205i	0.0708881	−	0.122782i	−0.828403	−	0.560133i	$-0.810750\pi$
$199$	1.00000	−	1.73205i	0.0708881	−	0.122782i	0.899291	+	0.437351i	$0.144083\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−9.00000	−	15.5885i	−0.628587	−	1.08875i
$206$		0			0
$207$	6.36396	+	3.67423i	0.442326	+	0.255377i
$208$		0			0
$209$			4.89898i			0.338869i
$210$		0			0
$211$		−	4.89898i		−	0.337260i	−0.985679	−	0.168630i	$-0.946066\pi$
$211$		−	4.89898i		−	0.337260i	0.985679	−	0.168630i	$-0.0539342\pi$
$212$		0			0
$213$	10.6066	+	6.12372i	0.726752	+	0.419591i
$214$		0			0
$215$	6.00000	+	10.3923i	0.409197	+	0.708749i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−12.7279	+	7.34847i	−0.860073	+	0.496564i
$220$		0			0
$221$	6.00000	−	10.3923i	0.403604	−	0.699062i
$222$		0			0
$223$	10.0000			0.669650			0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000			0.669650			0.334825	+	0.942280i	$0.391323\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	−0.993508	−	0.113761i	$-0.963710\pi$
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	0.595274	+	0.803523i	$0.297043\pi$
$228$		0			0
$229$	−21.2132	+	12.2474i	−1.40181	+	0.809334i	−0.994578	−	0.103992i	$-0.966839\pi$
$229$	−21.2132	+	12.2474i	−1.40181	+	0.809334i	−0.407230	+	0.913326i	$0.633505\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.0000	+	25.9808i	0.982683	+	1.70206i	0.651813	+	0.758380i	$0.274009\pi$
$233$	15.0000	+	25.9808i	0.982683	+	1.70206i	0.330870	+	0.943676i	$0.392658\pi$
$234$		0			0
$235$	25.4558	+	14.6969i	1.66056	+	0.958723i
$236$		0			0
$237$			9.79796i			0.636446i
$238$		0			0
$239$			12.2474i			0.792222i	0.918203	+	0.396111i	$0.129640\pi$
$239$			12.2474i			0.792222i	−0.918203	+	0.396111i	$0.870360\pi$
$240$		0			0
$241$	−4.24264	−	2.44949i	−0.273293	−	0.157786i	0.357090	−	0.934070i	$-0.383769\pi$
$241$	−4.24264	−	2.44949i	−0.273293	−	0.157786i	−0.630383	+	0.776284i	$0.717102\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−8.48528	+	4.89898i	−0.539906	+	0.311715i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−18.0000			−1.13165
$254$		0			0
$255$	3.00000	−	5.19615i	0.187867	−	0.325396i
$256$		0			0
$257$	19.0919	−	11.0227i	1.19092	−	0.687577i	0.232404	−	0.972619i	$-0.425341\pi$
$257$	19.0919	−	11.0227i	1.19092	−	0.687577i	0.958515	+	0.285042i	$0.0920075\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.00000	−	5.19615i	−0.185695	−	0.321634i
$262$		0			0
$263$	2.12132	+	1.22474i	0.130806	+	0.0755210i	0.563975	−	0.825792i	$-0.309271\pi$
$263$	2.12132	+	1.22474i	0.130806	+	0.0755210i	−0.433169	+	0.901313i	$0.642605\pi$
$264$		0			0
$265$		−	14.6969i		−	0.902826i
$266$		0			0
$267$		−	12.2474i		−	0.749532i
$268$		0			0
$269$	19.0919	+	11.0227i	1.16405	+	0.672066i	0.952272	−	0.305251i	$-0.0987404\pi$
$269$	19.0919	+	11.0227i	1.16405	+	0.672066i	0.211781	+	0.977317i	$0.432074\pi$
$270$		0			0
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	0.999870	−	0.0161307i	$-0.00513477\pi$
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	−0.513905	+	0.857847i	$0.671801\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.12132	−	1.22474i	0.127920	−	0.0738549i
$276$		0			0
$277$	4.00000	−	6.92820i	0.240337	−	0.416275i	−0.720473	−	0.693482i	$-0.756075\pi$
$277$	4.00000	−	6.92820i	0.240337	−	0.416275i	0.960810	+	0.277207i	$0.0894088\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	−0.579437	−	0.815017i	$-0.696728\pi$
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	0.995544	+	0.0942988i	$0.0300609\pi$
$284$		0			0
$285$	−4.24264	+	2.44949i	−0.251312	+	0.145095i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.50000	−	9.52628i	−0.323529	−	0.560369i
$290$		0			0
$291$	−4.24264	−	2.44949i	−0.248708	−	0.143592i
$292$		0			0
$293$		−	22.0454i		−	1.28791i	−0.765065	−	0.643953i	$-0.777293\pi$
$293$		−	22.0454i		−	1.28791i	0.765065	−	0.643953i	$-0.222707\pi$
$294$		0			0
$295$			29.3939i			1.71138i
$296$		0			0
$297$	2.12132	+	1.22474i	0.123091	+	0.0710669i
$298$		0			0
$299$	−18.0000	−	31.1769i	−1.04097	−	1.80301i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−2.12132	+	1.22474i	−0.121867	+	0.0703598i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−26.0000			−1.48390			−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000			−1.48390			−0.741949	+	0.670456i	$0.766098\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	−8.48528	+	4.89898i	−0.479616	+	0.276907i	−0.720257	−	0.693708i	$-0.755976\pi$
$313$	−8.48528	+	4.89898i	−0.479616	+	0.276907i	0.240640	+	0.970614i	$0.422643\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$		0			0
$319$	12.7279	+	7.34847i	0.712627	+	0.411435i
$320$		0			0
$321$			12.2474i			0.683586i
$322$		0			0
$323$		−	4.89898i		−	0.272587i
$324$		0			0
$325$	4.24264	+	2.44949i	0.235339	+	0.135873i
$326$		0			0
$327$	−1.00000	−	1.73205i	−0.0553001	−	0.0957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	21.2132	−	12.2474i	1.16598	−	0.673181i	0.213253	−	0.976997i	$-0.431594\pi$
$331$	21.2132	−	12.2474i	1.16598	−	0.673181i	0.952731	+	0.303816i	$0.0982610\pi$
$332$		0			0
$333$	−2.00000	+	3.46410i	−0.109599	+	0.189832i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	−3.00000	+	5.19615i	−0.162938	+	0.282216i
$340$		0			0
$341$	−16.9706	+	9.79796i	−0.919007	+	0.530589i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−9.00000	−	15.5885i	−0.484544	−	0.839254i
$346$		0			0
$347$	−14.8492	−	8.57321i	−0.797149	−	0.460234i	0.0453242	−	0.998972i	$-0.485568\pi$
$347$	−14.8492	−	8.57321i	−0.797149	−	0.460234i	−0.842473	+	0.538738i	$0.818901\pi$
$348$		0			0
$349$		−	19.5959i		−	1.04895i	−0.851427	−	0.524473i	$-0.824262\pi$
$349$		−	19.5959i		−	1.04895i	0.851427	−	0.524473i	$-0.175738\pi$
$350$		0			0
$351$			4.89898i			0.261488i
$352$		0			0
$353$	2.12132	+	1.22474i	0.112906	+	0.0651866i	0.555390	−	0.831590i	$-0.312569\pi$
$353$	2.12132	+	1.22474i	0.112906	+	0.0651866i	−0.442483	+	0.896777i	$0.645902\pi$
$354$		0			0
$355$	−15.0000	−	25.9808i	−0.796117	−	1.37892i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.6066	+	6.12372i	−0.559795	+	0.323198i	−0.753063	−	0.657948i	$-0.771425\pi$
$359$	−10.6066	+	6.12372i	−0.559795	+	0.323198i	0.193268	+	0.981146i	$0.438091\pi$
$360$		0			0
$361$	7.50000	−	12.9904i	0.394737	−	0.683704i
$362$		0			0
$363$	5.00000			0.262432
$364$		0			0
$365$	36.0000			1.88433
$366$		0			0
$367$	7.00000	−	12.1244i	0.365397	−	0.632886i	−0.623443	−	0.781869i	$-0.714267\pi$
$367$	7.00000	−	12.1244i	0.365397	−	0.632886i	0.988840	+	0.148983i	$0.0475999\pi$
$368$		0			0
$369$	−6.36396	+	3.67423i	−0.331295	+	0.191273i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$	−8.48528	−	4.89898i	−0.438178	−	0.252982i
$376$		0			0
$377$			29.3939i			1.51386i
$378$		0			0
$379$		−	34.2929i		−	1.76151i	−0.473576	−	0.880753i	$-0.657037\pi$
$379$		−	34.2929i		−	1.76151i	0.473576	−	0.880753i	$-0.342963\pi$
$380$		0			0
$381$	8.48528	+	4.89898i	0.434714	+	0.250982i
$382$		0			0
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	0.977613	−	0.210411i	$-0.0674801\pi$
$383$	6.00000	+	10.3923i	0.306586	+	0.531022i	−0.671027	+	0.741433i	$0.734147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.24264	−	2.44949i	0.215666	−	0.124515i
$388$		0			0
$389$	15.0000	−	25.9808i	0.760530	−	1.31728i	−0.182047	−	0.983290i	$-0.558272\pi$
$389$	15.0000	−	25.9808i	0.760530	−	1.31728i	0.942578	−	0.333987i	$-0.108394\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000	−	20.7846i	0.603786	−	1.04579i
$396$		0			0
$397$	8.48528	−	4.89898i	0.425864	−	0.245873i	−0.271719	−	0.962377i	$-0.587592\pi$
$397$	8.48528	−	4.89898i	0.425864	−	0.245873i	0.697583	+	0.716504i	$0.254259\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		0			0
$403$	−33.9411	−	19.5959i	−1.69073	−	0.976142i
$404$		0			0
$405$			2.44949i			0.121716i
$406$		0			0
$407$		−	9.79796i		−	0.485667i
$408$		0			0
$409$	−4.24264	−	2.44949i	−0.209785	−	0.121119i	0.391426	−	0.920209i	$-0.371982\pi$
$409$	−4.24264	−	2.44949i	−0.209785	−	0.121119i	−0.601212	+	0.799090i	$0.705315\pi$
$410$		0			0
$411$	−3.00000	−	5.19615i	−0.147979	−	0.256307i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.00000	−	3.46410i	0.0979404	−	0.169638i
$418$		0			0
$419$	−36.0000			−1.75872			−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000			−1.75872			−0.879358	+	0.476162i	$0.842028\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	6.00000	−	10.3923i	0.291730	−	0.505291i
$424$		0			0
$425$	−2.12132	+	1.22474i	−0.102899	+	0.0594089i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−6.00000	−	10.3923i	−0.289683	−	0.501745i
$430$		0			0
$431$	14.8492	+	8.57321i	0.715263	+	0.412957i	0.813007	−	0.582254i	$-0.197829\pi$
$431$	14.8492	+	8.57321i	0.715263	+	0.412957i	−0.0977438	+	0.995212i	$0.531163\pi$
$432$		0			0
$433$		−	9.79796i		−	0.470860i	−0.971891	−	0.235430i	$-0.924350\pi$
$433$		−	9.79796i		−	0.470860i	0.971891	−	0.235430i	$-0.0756498\pi$
$434$		0			0
$435$			14.6969i			0.704664i
$436$		0			0
$437$	−12.7279	−	7.34847i	−0.608859	−	0.351525i
$438$		0			0
$439$	−11.0000	−	19.0526i	−0.525001	−	0.909329i	−0.999576	−	0.0291138i	$-0.990731\pi$
$439$	−11.0000	−	19.0526i	−0.525001	−	0.909329i	0.474575	−	0.880215i	$-0.342602\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.12132	−	1.22474i	0.100787	−	0.0581894i	−0.448759	−	0.893653i	$-0.648134\pi$
$443$	2.12132	−	1.22474i	0.100787	−	0.0581894i	0.549546	+	0.835463i	$0.314801\pi$
$444$		0			0
$445$	−15.0000	+	25.9808i	−0.711068	+	1.23161i
$446$		0			0
$447$	−18.0000			−0.851371
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	9.00000	−	15.5885i	0.423793	−	0.734032i
$452$		0			0
$453$	12.7279	−	7.34847i	0.598010	−	0.345261i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−19.0000	−	32.9090i	−0.888783	−	1.53942i	−0.841316	−	0.540544i	$-0.818219\pi$
$457$	−19.0000	−	32.9090i	−0.888783	−	1.53942i	−0.0474665	−	0.998873i	$-0.515115\pi$
$458$		0			0
$459$	−2.12132	−	1.22474i	−0.0990148	−	0.0571662i
$460$		0			0
$461$			2.44949i			0.114084i	0.998372	+	0.0570421i	$0.0181669\pi$
$461$			2.44949i			0.114084i	−0.998372	+	0.0570421i	$0.981833\pi$
$462$		0			0
$463$		−	9.79796i		−	0.455350i	−0.973737	−	0.227675i	$-0.926888\pi$
$463$		−	9.79796i		−	0.455350i	0.973737	−	0.227675i	$-0.0731123\pi$
$464$		0			0
$465$	−16.9706	−	9.79796i	−0.786991	−	0.454369i
$466$		0			0
$467$	−12.0000	−	20.7846i	−0.555294	−	0.961797i	−0.997881	−	0.0650714i	$-0.979272\pi$
$467$	−12.0000	−	20.7846i	−0.555294	−	0.961797i	0.442587	−	0.896726i	$-0.354061\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	16.9706	−	9.79796i	0.781962	−	0.451466i
$472$		0			0
$473$	−6.00000	+	10.3923i	−0.275880	+	0.477839i
$474$		0			0
$475$	2.00000			0.0917663
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	12.0000	−	20.7846i	0.548294	−	0.949673i	−0.450098	−	0.892979i	$-0.648611\pi$
$479$	12.0000	−	20.7846i	0.548294	−	0.949673i	0.998392	−	0.0566937i	$-0.0180558\pi$
$480$		0			0
$481$	16.9706	−	9.79796i	0.773791	−	0.446748i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.00000	+	10.3923i	0.272446	+	0.471890i
$486$		0			0
$487$	4.24264	+	2.44949i	0.192252	+	0.110997i	0.593037	−	0.805176i	$-0.297929\pi$
$487$	4.24264	+	2.44949i	0.192252	+	0.110997i	−0.400784	+	0.916172i	$0.631262\pi$
$488$		0			0
$489$		−	19.5959i		−	0.886158i
$490$		0			0
$491$		−	36.7423i		−	1.65816i	−0.559131	−	0.829079i	$-0.688865\pi$
$491$		−	36.7423i		−	1.65816i	0.559131	−	0.829079i	$-0.311135\pi$
$492$		0			0
$493$	−12.7279	−	7.34847i	−0.573237	−	0.330958i
$494$		0			0
$495$	−3.00000	−	5.19615i	−0.134840	−	0.233550i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.7279	−	7.34847i	0.569780	−	0.328963i	−0.187281	−	0.982306i	$-0.559968\pi$
$499$	12.7279	−	7.34847i	0.569780	−	0.328963i	0.757061	+	0.653344i	$0.226634\pi$
$500$		0			0
$501$	−6.00000	+	10.3923i	−0.268060	+	0.464294i
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$	5.50000	−	9.52628i	0.244264	−	0.423077i
$508$		0			0
$509$	14.8492	−	8.57321i	0.658181	−	0.380001i	−0.133402	−	0.991062i	$-0.542590\pi$
$509$	14.8492	−	8.57321i	0.658181	−	0.380001i	0.791584	+	0.611061i	$0.209257\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	1.00000	+	1.73205i	0.0441511	+	0.0764719i
$514$		0			0
$515$	33.9411	+	19.5959i	1.49562	+	0.863499i
$516$		0			0
$517$			29.3939i			1.29274i
$518$		0			0
$519$			7.34847i			0.322562i
$520$		0			0
$521$	2.12132	+	1.22474i	0.0929367	+	0.0536570i	0.545748	−	0.837949i	$-0.316246\pi$
$521$	2.12132	+	1.22474i	0.0929367	+	0.0536570i	−0.452811	+	0.891606i	$0.649579\pi$
$522$		0			0
$523$	−10.0000	−	17.3205i	−0.437269	−	0.757373i	0.560208	−	0.828352i	$-0.310721\pi$
$523$	−10.0000	−	17.3205i	−0.437269	−	0.757373i	−0.997478	+	0.0709788i	$0.977388\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.9706	−	9.79796i	0.739249	−	0.426806i
$528$		0			0
$529$	15.5000	−	26.8468i	0.673913	−	1.16725i
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$	36.0000			1.55933
$534$		0			0
$535$	15.0000	−	25.9808i	0.648507	−	1.12325i
$536$		0			0
$537$	14.8492	−	8.57321i	0.640792	−	0.369961i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.00000	−	13.8564i	−0.343947	−	0.595733i	0.641215	−	0.767361i	$-0.278431\pi$
$541$	−8.00000	−	13.8564i	−0.343947	−	0.595733i	−0.985162	+	0.171628i	$0.945097\pi$
$542$		0			0
$543$	4.24264	+	2.44949i	0.182069	+	0.105118i
$544$		0			0
$545$			4.89898i			0.209849i
$546$		0			0
$547$			29.3939i			1.25679i	0.777894	+	0.628396i	$0.216288\pi$
$547$			29.3939i			1.25679i	−0.777894	+	0.628396i	$0.783712\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.00000	+	10.3923i	0.255609	+	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	8.48528	−	4.89898i	0.360180	−	0.207950i
$556$		0			0
$557$	−3.00000	+	5.19615i	−0.127114	+	0.220168i	−0.922557	−	0.385860i	$-0.873905\pi$
$557$	−3.00000	+	5.19615i	−0.127114	+	0.220168i	0.795443	+	0.606028i	$0.207238\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$	6.00000			0.253320
$562$		0			0
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	−0.494238	−	0.869326i	$-0.664553\pi$
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	0.999978	−	0.00664037i	$-0.00211371\pi$
$564$		0			0
$565$	12.7279	−	7.34847i	0.535468	−	0.309152i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	−16.9706	−	9.79796i	−0.710196	−	0.410032i	0.100938	−	0.994893i	$-0.467816\pi$
$571$	−16.9706	−	9.79796i	−0.710196	−	0.410032i	−0.811134	+	0.584861i	$0.801149\pi$
$572$		0			0
$573$			2.44949i			0.102329i
$574$		0			0
$575$			7.34847i			0.306452i
$576$		0			0
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$		0			0
$579$	−8.00000	−	13.8564i	−0.332469	−	0.575853i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.7279	−	7.34847i	0.527137	−	0.304342i
$584$		0			0
$585$	6.00000	−	10.3923i	0.248069	−	0.429669i
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	−9.00000	+	15.5885i	−0.370211	+	0.641223i
$592$		0			0
$593$	−40.3051	+	23.2702i	−1.65513	+	0.955591i	−0.680217	+	0.733011i	$0.738114\pi$
$593$	−40.3051	+	23.2702i	−1.65513	+	0.955591i	−0.974914	+	0.222580i	$0.928552\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.00000	+	1.73205i	0.0409273	+	0.0708881i
$598$		0			0
$599$	2.12132	+	1.22474i	0.0866748	+	0.0500417i	0.542711	−	0.839920i	$-0.317398\pi$
$599$	2.12132	+	1.22474i	0.0866748	+	0.0500417i	−0.456036	+	0.889961i	$0.650731\pi$
$600$		0			0
$601$			19.5959i			0.799334i	0.916660	+	0.399667i	$0.130874\pi$
$601$			19.5959i			0.799334i	−0.916660	+	0.399667i	$0.869126\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.6066	−	6.12372i	−0.431220	−	0.248965i
$606$		0			0
$607$	7.00000	+	12.1244i	0.284121	+	0.492112i	0.972396	−	0.233338i	$-0.0749648\pi$
$607$	7.00000	+	12.1244i	0.284121	+	0.492112i	−0.688274	+	0.725450i	$0.741632\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−50.9117	+	29.3939i	−2.05967	+	1.18915i
$612$		0			0
$613$	−13.0000	+	22.5167i	−0.525065	+	0.909439i	0.474509	+	0.880251i	$0.342626\pi$
$613$	−13.0000	+	22.5167i	−0.525065	+	0.909439i	−0.999574	+	0.0291886i	$0.990708\pi$
$614$		0			0
$615$	18.0000			0.725830
$616$		0			0
$617$	−42.0000			−1.69086			−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000			−1.69086			−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	−0.823029	−	0.567999i	$-0.807718\pi$
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	0.903416	+	0.428765i	$0.141051\pi$
$620$		0			0
$621$	−6.36396	+	3.67423i	−0.255377	+	0.147442i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$	−4.24264	−	2.44949i	−0.169435	−	0.0978232i
$628$		0			0
$629$			9.79796i			0.390670i
$630$		0			0
$631$			19.5959i			0.780101i	0.920793	+	0.390051i	$0.127542\pi$
$631$			19.5959i			0.780101i	−0.920793	+	0.390051i	$0.872458\pi$
$632$		0			0
$633$	4.24264	+	2.44949i	0.168630	+	0.0973585i
$634$		0			0
$635$	−12.0000	−	20.7846i	−0.476205	−	0.824812i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−10.6066	+	6.12372i	−0.419591	+	0.242251i
$640$		0			0
$641$	3.00000	−	5.19615i	0.118493	−	0.205236i	−0.800678	−	0.599095i	$-0.795527\pi$
$641$	3.00000	−	5.19615i	0.118493	−	0.205236i	0.919171	+	0.393860i	$0.128860\pi$
$642$		0			0
$643$	−14.0000			−0.552106			−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000			−0.552106			−0.276053	+	0.961142i	$0.589027\pi$
$644$		0			0
$645$	−12.0000			−0.472500
$646$		0			0
$647$	12.0000	−	20.7846i	0.471769	−	0.817127i	−0.527710	−	0.849425i	$-0.676949\pi$
$647$	12.0000	−	20.7846i	0.471769	−	0.817127i	0.999478	+	0.0322975i	$0.0102824\pi$
$648$		0			0
$649$	−25.4558	+	14.6969i	−0.999229	+	0.576905i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	0.918736	−	0.394872i	$-0.129211\pi$
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	−0.801337	+	0.598213i	$0.795878\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	14.6969i		−	0.573382i
$658$		0			0
$659$		−	7.34847i		−	0.286256i	−0.989704	−	0.143128i	$-0.954284\pi$
$659$		−	7.34847i		−	0.286256i	0.989704	−	0.143128i	$-0.0457160\pi$
$660$		0			0
$661$	25.4558	+	14.6969i	0.990118	+	0.571645i	0.905309	−	0.424753i	$-0.139639\pi$
$661$	25.4558	+	14.6969i	0.990118	+	0.571645i	0.0848081	+	0.996397i	$0.472972\pi$
$662$		0			0
$663$	6.00000	+	10.3923i	0.233021	+	0.403604i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−38.1838	+	22.0454i	−1.47848	+	0.853602i
$668$		0			0
$669$	−5.00000	+	8.66025i	−0.193311	+	0.334825i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−8.00000			−0.308377			−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000			−0.308377			−0.154189	+	0.988041i	$0.549276\pi$
$674$		0			0
$675$	0.500000	−	0.866025i	0.0192450	−	0.0333333i
$676$		0			0
$677$	10.6066	−	6.12372i	0.407645	−	0.235354i	−0.282132	−	0.959375i	$-0.591042\pi$
$677$	10.6066	−	6.12372i	0.407645	−	0.235354i	0.689777	+	0.724022i	$0.257708\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.00000	−	10.3923i	−0.229920	−	0.398234i
$682$		0			0
$683$	6.36396	+	3.67423i	0.243510	+	0.140591i	0.616789	−	0.787129i	$-0.288433\pi$
$683$	6.36396	+	3.67423i	0.243510	+	0.140591i	−0.373279	+	0.927719i	$0.621767\pi$
$684$		0			0
$685$			14.6969i			0.561541i
$686$		0			0
$687$		−	24.4949i		−	0.934539i
$688$		0			0
$689$	25.4558	+	14.6969i	0.969790	+	0.559909i
$690$		0			0
$691$	−10.0000	−	17.3205i	−0.380418	−	0.658903i	0.610704	−	0.791859i	$-0.290887\pi$
$691$	−10.0000	−	17.3205i	−0.380418	−	0.658903i	−0.991122	+	0.132956i	$0.957553\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.48528	+	4.89898i	−0.321865	+	0.185829i
$696$		0			0
$697$	−9.00000	+	15.5885i	−0.340899	+	0.590455i
$698$		0			0
$699$	−30.0000			−1.13470
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	4.00000	−	6.92820i	0.150863	−	0.261302i
$704$		0			0
$705$	−25.4558	+	14.6969i	−0.958723	+	0.553519i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	0.944509	−	0.328484i	$-0.106538\pi$
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	−0.756730	+	0.653727i	$0.773204\pi$
$710$		0			0
$711$	−8.48528	−	4.89898i	−0.318223	−	0.183726i
$712$		0			0
$713$		−	58.7878i		−	2.20162i
$714$		0			0
$715$			29.3939i			1.09927i
$716$		0			0
$717$	−10.6066	−	6.12372i	−0.396111	−	0.228695i
$718$		0			0
$719$	−12.0000	−	20.7846i	−0.447524	−	0.775135i	0.550700	−	0.834703i	$-0.314361\pi$
$719$	−12.0000	−	20.7846i	−0.447524	−	0.775135i	−0.998224	+	0.0595683i	$0.981028\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	4.24264	−	2.44949i	0.157786	−	0.0910975i
$724$		0			0
$725$	3.00000	−	5.19615i	0.111417	−	0.192980i
$726$		0			0
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	6.00000	−	10.3923i	0.221918	−	0.384373i
$732$		0			0
$733$	4.24264	−	2.44949i	0.156706	−	0.0904740i	−0.419597	−	0.907711i	$-0.637828\pi$
$733$	4.24264	−	2.44949i	0.156706	−	0.0904740i	0.576302	+	0.817237i	$0.304495\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	8.48528	+	4.89898i	0.312136	+	0.180212i	0.647882	−	0.761741i	$-0.275655\pi$
$739$	8.48528	+	4.89898i	0.312136	+	0.180212i	−0.335746	+	0.941953i	$0.608988\pi$
$740$		0			0
$741$		−	9.79796i		−	0.359937i
$742$		0			0
$743$			12.2474i			0.449315i	0.974438	+	0.224658i	$0.0721264\pi$
$743$			12.2474i			0.449315i	−0.974438	+	0.224658i	$0.927874\pi$
$744$		0			0
$745$	38.1838	+	22.0454i	1.39894	+	0.807681i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−29.6985	+	17.1464i	−1.08371	+	0.625682i	−0.931896	−	0.362726i	$-0.881846\pi$
$751$	−29.6985	+	17.1464i	−1.08371	+	0.625682i	−0.151817	+	0.988409i	$0.548513\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−36.0000			−1.31017
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$	9.00000	−	15.5885i	0.326679	−	0.565825i
$760$		0			0
$761$	36.0624	−	20.8207i	1.30726	−	0.754748i	0.325624	−	0.945499i	$-0.394426\pi$
$761$	36.0624	−	20.8207i	1.30726	−	0.754748i	0.981638	+	0.190751i	$0.0610923\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.00000	+	5.19615i	0.108465	+	0.187867i
$766$		0			0
$767$	−50.9117	−	29.3939i	−1.83831	−	1.06135i
$768$		0			0
$769$		−	9.79796i		−	0.353323i	−0.984272	−	0.176662i	$-0.943470\pi$
$769$		−	9.79796i		−	0.353323i	0.984272	−	0.176662i	$-0.0565299\pi$
$770$		0			0
$771$			22.0454i			0.793946i
$772$		0			0
$773$	−19.0919	−	11.0227i	−0.686687	−	0.396459i	0.115683	−	0.993286i	$-0.463094\pi$
$773$	−19.0919	−	11.0227i	−0.686687	−	0.396459i	−0.802370	+	0.596827i	$0.796428\pi$
$774$		0			0
$775$	4.00000	+	6.92820i	0.143684	+	0.248868i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.7279	−	7.34847i	0.456025	−	0.263286i
$780$		0			0
$781$	15.0000	−	25.9808i	0.536742	−	0.929665i
$782$		0			0
$783$	6.00000			0.214423
$784$		0			0
$785$	−48.0000			−1.71319
$786$		0			0
$787$	−10.0000	+	17.3205i	−0.356462	+	0.617409i	−0.987367	−	0.158450i	$-0.949350\pi$
$787$	−10.0000	+	17.3205i	−0.356462	+	0.617409i	0.630905	+	0.775860i	$0.282684\pi$
$788$		0			0
$789$	−2.12132	+	1.22474i	−0.0755210	+	0.0436021i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	12.7279	+	7.34847i	0.451413	+	0.260623i
$796$		0			0
$797$		−	36.7423i		−	1.30148i	−0.759300	−	0.650740i	$-0.774459\pi$
$797$		−	36.7423i		−	1.30148i	0.759300	−	0.650740i	$-0.225541\pi$
$798$		0			0
$799$		−	29.3939i		−	1.03988i
$800$		0			0
$801$	10.6066	+	6.12372i	0.374766	+	0.216371i
$802$		0			0
$803$	18.0000	+	31.1769i	0.635206	+	1.10021i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−19.0919	+	11.0227i	−0.672066	+	0.388018i
$808$		0			0
$809$	−9.00000	+	15.5885i	−0.316423	+	0.548061i	−0.979739	−	0.200279i	$-0.935815\pi$
$809$	−9.00000	+	15.5885i	−0.316423	+	0.548061i	0.663316	+	0.748340i	$0.269149\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		0			0
$815$	−24.0000	+	41.5692i	−0.840683	+	1.45611i
$816$		0			0
$817$	−8.48528	+	4.89898i	−0.296862	+	0.171394i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.00000	−	15.5885i	−0.314102	−	0.544041i	0.665144	−	0.746715i	$-0.268370\pi$
$821$	−9.00000	−	15.5885i	−0.314102	−	0.544041i	−0.979246	+	0.202674i	$0.935037\pi$
$822$		0			0
$823$	16.9706	+	9.79796i	0.591557	+	0.341535i	0.765713	−	0.643183i	$-0.222386\pi$
$823$	16.9706	+	9.79796i	0.591557	+	0.341535i	−0.174156	+	0.984718i	$0.555720\pi$
$824$		0			0
$825$			2.44949i			0.0852803i
$826$		0			0
$827$			46.5403i			1.61836i	0.587557	+	0.809182i	$0.300090\pi$
$827$			46.5403i			1.61836i	−0.587557	+	0.809182i	$0.699910\pi$
$828$		0			0
$829$	−4.24264	−	2.44949i	−0.147353	−	0.0850743i	0.424511	−	0.905423i	$-0.360446\pi$
$829$	−4.24264	−	2.44949i	−0.147353	−	0.0850743i	−0.571864	+	0.820349i	$0.693779\pi$
$830$		0			0
$831$	4.00000	+	6.92820i	0.138758	+	0.240337i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	25.4558	−	14.6969i	0.880936	−	0.508609i
$836$		0			0
$837$	−4.00000	+	6.92820i	−0.138260	+	0.239474i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	−3.00000	+	5.19615i	−0.103325	+	0.178965i
$844$		0			0
$845$	−23.3345	+	13.4722i	−0.802732	+	0.463458i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	7.00000	+	12.1244i	0.240239	+	0.416107i
$850$		0			0
$851$	25.4558	+	14.6969i	0.872615	+	0.503805i
$852$		0			0
$853$			29.3939i			1.00643i	0.864162	+	0.503214i	$0.167849\pi$
$853$			29.3939i			1.00643i	−0.864162	+	0.503214i	$0.832151\pi$
$854$		0			0
$855$		−	4.89898i		−	0.167542i
$856$		0			0
$857$	−19.0919	−	11.0227i	−0.652166	−	0.376528i	0.137119	−	0.990555i	$-0.456216\pi$
$857$	−19.0919	−	11.0227i	−0.652166	−	0.376528i	−0.789286	+	0.614026i	$0.789549\pi$
$858$		0			0
$859$	−11.0000	−	19.0526i	−0.375315	−	0.650065i	0.615059	−	0.788481i	$-0.289132\pi$
$859$	−11.0000	−	19.0526i	−0.375315	−	0.650065i	−0.990374	+	0.138416i	$0.955799\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	44.5477	−	25.7196i	1.51642	−	0.875507i	0.516608	−	0.856222i	$-0.327194\pi$
$863$	44.5477	−	25.7196i	1.51642	−	0.875507i	0.999814	−	0.0192849i	$-0.00613896\pi$
$864$		0			0
$865$	9.00000	−	15.5885i	0.306009	−	0.530023i
$866$		0			0
$867$	11.0000			0.373580
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$		0			0
$872$		0			0
$873$	4.24264	−	2.44949i	0.143592	−	0.0829027i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	0.989788	−	0.142548i	$-0.0455296\pi$
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	−0.618344	+	0.785907i	$0.712196\pi$
$878$		0			0
$879$	19.0919	+	11.0227i	0.643953	+	0.371787i
$880$		0			0
$881$		−	2.44949i		−	0.0825254i	−0.999148	−	0.0412627i	$-0.986862\pi$
$881$		−	2.44949i		−	0.0825254i	0.999148	−	0.0412627i	$-0.0131381\pi$
$882$		0			0
$883$		−	34.2929i		−	1.15405i	−0.816728	−	0.577023i	$-0.804214\pi$
$883$		−	34.2929i		−	1.15405i	0.816728	−	0.577023i	$-0.195786\pi$
$884$		0			0
$885$	−25.4558	−	14.6969i	−0.855689	−	0.494032i
$886$		0			0
$887$	−6.00000	−	10.3923i	−0.201460	−	0.348939i	0.747539	−	0.664218i	$-0.231235\pi$
$887$	−6.00000	−	10.3923i	−0.201460	−	0.348939i	−0.948999	+	0.315279i	$0.897902\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−2.12132	+	1.22474i	−0.0710669	+	0.0410305i
$892$		0			0
$893$	−12.0000	+	20.7846i	−0.401565	+	0.695530i
$894$		0			0
$895$	−42.0000			−1.40391
$896$		0			0
$897$	36.0000			1.20201
$898$		0			0
$899$	−24.0000	+	41.5692i	−0.800445	+	1.38641i
$900$		0			0
$901$	−12.7279	+	7.34847i	−0.424029	+	0.244813i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.00000	−	10.3923i	−0.199447	−	0.345452i
$906$		0			0
$907$	21.2132	+	12.2474i	0.704373	+	0.406670i	0.808974	−	0.587844i	$-0.200023\pi$
$907$	21.2132	+	12.2474i	0.704373	+	0.406670i	−0.104601	+	0.994514i	$0.533357\pi$
$908$		0			0
$909$		−	2.44949i		−	0.0812444i
$910$		0			0
$911$			7.34847i			0.243466i	0.992563	+	0.121733i	$0.0388451\pi$
$911$			7.34847i			0.243466i	−0.992563	+	0.121733i	$0.961155\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	12.7279	−	7.34847i	0.419855	−	0.242404i	−0.275160	−	0.961398i	$-0.588731\pi$
$919$	12.7279	−	7.34847i	0.419855	−	0.242404i	0.695015	+	0.718995i	$0.255398\pi$
$920$		0			0
$921$	13.0000	−	22.5167i	0.428365	−	0.741949i
$922$		0			0
$923$	60.0000			1.97492
$924$		0			0
$925$	−4.00000			−0.131519
$926$		0			0
$927$	8.00000	−	13.8564i	0.262754	−	0.455104i
$928$		0			0
$929$	10.6066	−	6.12372i	0.347991	−	0.200913i	−0.315809	−	0.948823i	$-0.602276\pi$
$929$	10.6066	−	6.12372i	0.347991	−	0.200913i	0.663800	+	0.747910i	$0.268943\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	12.0000	+	20.7846i	0.392862	+	0.680458i
$934$		0			0
$935$	−12.7279	−	7.34847i	−0.416248	−	0.240321i
$936$		0			0
$937$			39.1918i			1.28034i	0.768233	+	0.640171i	$0.221136\pi$
$937$			39.1918i			1.28034i	−0.768233	+	0.640171i	$0.778864\pi$
$938$		0			0
$939$		−	9.79796i		−	0.319744i
$940$		0			0
$941$	−10.6066	−	6.12372i	−0.345765	−	0.199628i	0.317053	−	0.948408i	$-0.397307\pi$
$941$	−10.6066	−	6.12372i	−0.345765	−	0.199628i	−0.662819	+	0.748780i	$0.730640\pi$
$942$		0			0
$943$	27.0000	+	46.7654i	0.879241	+	1.52289i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.3345	−	13.4722i	0.758270	−	0.437787i	−0.0704042	−	0.997519i	$-0.522429\pi$
$947$	23.3345	−	13.4722i	0.758270	−	0.437787i	0.828674	+	0.559731i	$0.189096\pi$
$948$		0			0
$949$	−36.0000	+	62.3538i	−1.16861	+	2.02409i
$950$		0			0
$951$	−18.0000			−0.583690
$952$		0			0
$953$	−18.0000			−0.583077			−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000			−0.583077			−0.291539	+	0.956559i	$0.594167\pi$
$954$		0			0
$955$	3.00000	−	5.19615i	0.0970777	−	0.168144i
$956$		0			0
$957$	−12.7279	+	7.34847i	−0.411435	+	0.237542i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	−	28.5788i	−0.532258	−	0.921898i
$962$		0			0
$963$	−10.6066	−	6.12372i	−0.341793	−	0.197334i
$964$		0			0
$965$			39.1918i			1.26163i
$966$		0			0
$967$		−	58.7878i		−	1.89049i	−0.326366	−	0.945243i	$-0.605824\pi$
$967$		−	58.7878i		−	1.89049i	0.326366	−	0.945243i	$-0.394176\pi$
$968$		0			0
$969$	4.24264	+	2.44949i	0.136293	+	0.0786889i
$970$		0			0
$971$	18.0000	+	31.1769i	0.577647	+	1.00051i	0.995748	+	0.0921142i	$0.0293625\pi$
$971$	18.0000	+	31.1769i	0.577647	+	1.00051i	−0.418101	+	0.908401i	$0.637304\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−4.24264	+	2.44949i	−0.135873	+	0.0784465i
$976$		0			0
$977$	−3.00000	+	5.19615i	−0.0959785	+	0.166240i	−0.910017	−	0.414572i	$-0.863931\pi$
$977$	−3.00000	+	5.19615i	−0.0959785	+	0.166240i	0.814038	+	0.580812i	$0.197265\pi$
$978$		0			0
$979$	−30.0000			−0.958804
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	−0.945705	−	0.325027i	$-0.894626\pi$
$983$	−6.00000	+	10.3923i	−0.191370	+	0.331463i	0.754334	+	0.656490i	$0.227960\pi$
$984$		0			0
$985$	38.1838	−	22.0454i	1.21664	−	0.702425i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000	−	31.1769i	−0.572367	−	0.991368i
$990$		0			0
$991$	−12.7279	−	7.34847i	−0.404316	−	0.233432i	0.284029	−	0.958816i	$-0.408329\pi$
$991$	−12.7279	−	7.34847i	−0.404316	−	0.233432i	−0.688344	+	0.725384i	$0.741662\pi$
$992$		0			0
$993$			24.4949i			0.777322i
$994$		0			0
$995$		−	4.89898i		−	0.155308i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$	−2.00000	−	3.46410i	−0.0632772	−	0.109599i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.bl.m.31.2		4
4.3	odd	2		2352.2.bl.n.31.2		4
7.2	even	3	inner	2352.2.bl.m.607.1		4
7.3	odd	6		336.2.b.b.223.1	✓	2
7.4	even	3		336.2.b.c.223.2	yes	2
7.5	odd	6		2352.2.bl.n.607.2		4
7.6	odd	2		2352.2.bl.n.31.1		4
21.11	odd	6		1008.2.b.d.559.1		2
21.17	even	6		1008.2.b.e.559.2		2
28.3	even	6		336.2.b.c.223.1	yes	2
28.11	odd	6		336.2.b.b.223.2	yes	2
28.19	even	6	inner	2352.2.bl.m.607.2		4
28.23	odd	6		2352.2.bl.n.607.1		4
28.27	even	2	inner	2352.2.bl.m.31.1		4
56.3	even	6		1344.2.b.a.895.2		2
56.11	odd	6		1344.2.b.d.895.1		2
56.45	odd	6		1344.2.b.d.895.2		2
56.53	even	6		1344.2.b.a.895.1		2
84.11	even	6		1008.2.b.e.559.1		2
84.59	odd	6		1008.2.b.d.559.2		2
168.11	even	6		4032.2.b.f.3583.2		2
168.53	odd	6		4032.2.b.d.3583.2		2
168.59	odd	6		4032.2.b.d.3583.1		2
168.101	even	6		4032.2.b.f.3583.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.b.b.223.1	✓	2	7.3	odd	6
336.2.b.b.223.2	yes	2	28.11	odd	6
336.2.b.c.223.1	yes	2	28.3	even	6
336.2.b.c.223.2	yes	2	7.4	even	3
1008.2.b.d.559.1		2	21.11	odd	6
1008.2.b.d.559.2		2	84.59	odd	6
1008.2.b.e.559.1		2	84.11	even	6
1008.2.b.e.559.2		2	21.17	even	6
1344.2.b.a.895.1		2	56.53	even	6
1344.2.b.a.895.2		2	56.3	even	6
1344.2.b.d.895.1		2	56.11	odd	6
1344.2.b.d.895.2		2	56.45	odd	6
2352.2.bl.m.31.1		4	28.27	even	2	inner
2352.2.bl.m.31.2		4	1.1	even	1	trivial
2352.2.bl.m.607.1		4	7.2	even	3	inner
2352.2.bl.m.607.2		4	28.19	even	6	inner
2352.2.bl.n.31.1		4	7.6	odd	2
2352.2.bl.n.31.2		4	4.3	odd	2
2352.2.bl.n.607.1		4	28.23	odd	6
2352.2.bl.n.607.2		4	7.5	odd	6
4032.2.b.d.3583.1		2	168.59	odd	6
4032.2.b.d.3583.2		2	168.53	odd	6
4032.2.b.f.3583.1		2	168.101	even	6
4032.2.b.f.3583.2		2	168.11	even	6

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(2.12132 - 1.22474i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(2.12132 - 1.22474i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(2.12132 + 1.22474i) q^{11} +4.89898i q^{13} +2.44949i q^{15} +(-2.12132 - 1.22474i) q^{17} +(1.00000 + 1.73205i) q^{19} +(-6.36396 + 3.67423i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +6.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(-2.12132 + 1.22474i) q^{33} +(-2.00000 - 3.46410i) q^{37} +(-4.24264 - 2.44949i) q^{39} -7.34847i q^{41} +4.89898i q^{43} +(-2.12132 - 1.22474i) q^{45} +(6.00000 + 10.3923i) q^{47} +(2.12132 - 1.22474i) q^{51} +(3.00000 - 5.19615i) q^{53} +6.00000 q^{55} -2.00000 q^{57} +(-6.00000 + 10.3923i) q^{59} +(6.00000 + 10.3923i) q^{65} -7.34847i q^{69} -12.2474i q^{71} +(12.7279 + 7.34847i) q^{73} +(0.500000 + 0.866025i) q^{75} +(8.48528 - 4.89898i) q^{79} +(-0.500000 + 0.866025i) q^{81} -6.00000 q^{85} +(-3.00000 + 5.19615i) q^{87} +(-10.6066 + 6.12372i) q^{89} +(-4.00000 - 6.92820i) q^{93} +(4.24264 + 2.44949i) q^{95} +4.89898i q^{97} -2.44949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 2 * q^9	\( 4 q - 2 q^{3} - 2 q^{9} + 4 q^{19} + 2 q^{25} + 4 q^{27} + 24 q^{29} - 16 q^{31} - 8 q^{37} + 24 q^{47} + 12 q^{53} + 24 q^{55} - 8 q^{57} - 24 q^{59} + 24 q^{65} + 2 q^{75} - 2 q^{81} - 24 q^{85} - 12 q^{87} - 16 q^{93}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^9 + 4 * q^19 + 2 * q^25 + 4 * q^27 + 24 * q^29 - 16 * q^31 - 8 * q^37 + 24 * q^47 + 12 * q^53 + 24 * q^55 - 8 * q^57 - 24 * q^59 + 24 * q^65 + 2 * q^75 - 2 * q^81 - 24 * q^85 - 12 * q^87 - 16 * q^93

Introduction

L-functions

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