LMFDB - Embedded newform 4536.2.k.a.3401.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4536,2,Mod(3401,4536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4536, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4536.3401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4536 = 2^{3} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4536.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$36.2201423569$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3401.2
Character	$\chi$	$=$	4536.3401
Dual form			4536.2.k.a.3401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.83667 q^{5} +(-2.13644 - 1.56064i) q^{7} +O(q^{10})$	$q-3.83667 q^{5} +(-2.13644 - 1.56064i) q^{7} -0.676306i q^{11} +4.87291i q^{13} -5.79523 q^{17} -4.22456i q^{19} +5.50300i q^{23} +9.72006 q^{25} +7.91246i q^{29} -2.05833i q^{31} +(8.19684 + 5.98768i) q^{35} -8.71513 q^{37} -9.68018 q^{41} -7.15091 q^{43} +1.33218 q^{47} +(2.12879 + 6.66845i) q^{49} +5.18304i q^{53} +2.59476i q^{55} -4.19238 q^{59} -2.75025i q^{61} -18.6957i q^{65} +6.55273 q^{67} -11.1515i q^{71} -3.65100i q^{73} +(-1.05547 + 1.44489i) q^{77} +11.2344 q^{79} -9.22558 q^{83} +22.2344 q^{85} +1.79702 q^{89} +(7.60487 - 10.4107i) q^{91} +16.2083i q^{95} +3.00814i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 48 q^{25} - 24 q^{43} - 12 q^{49} + 24 q^{79} - 12 q^{91}+O(q^{100}) $ 48 * q + 48 * q^25 - 24 * q^43 - 12 * q^49 + 24 * q^79 - 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4536\mathbb{Z}\right)^\times$.

$n$	$1135$	$2269$	$2593$	$3809$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.83667			−1.71581			−0.857906	−	0.513806i	$-0.828235\pi$
$5$	−3.83667			−1.71581			−0.857906	+	0.513806i	$0.828235\pi$
$6$		0			0
$7$	−2.13644	−	1.56064i	−0.807500	−	0.589868i
$7$	−2.13644	−	1.56064i	−0.807500	−	0.589868i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.676306i		−	0.203914i	−0.994789	−	0.101957i	$-0.967490\pi$
$11$		−	0.676306i		−	0.203914i	0.994789	−	0.101957i	$-0.0325104\pi$
$12$		0			0
$13$			4.87291i			1.35150i	0.737130	+	0.675751i	$0.236180\pi$
$13$			4.87291i			1.35150i	−0.737130	+	0.675751i	$0.763820\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.79523			−1.40555			−0.702775	−	0.711412i	$-0.748056\pi$
$17$	−5.79523			−1.40555			−0.702775	+	0.711412i	$0.748056\pi$
$18$		0			0
$19$		−	4.22456i		−	0.969181i	−0.874741	−	0.484590i	$-0.838969\pi$
$19$		−	4.22456i		−	0.969181i	0.874741	−	0.484590i	$-0.161031\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.50300i			1.14745i	0.819046	+	0.573727i	$0.194503\pi$
$23$			5.50300i			1.14745i	−0.819046	+	0.573727i	$0.805497\pi$
$24$		0			0
$25$	9.72006			1.94401
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.91246i			1.46931i	0.678443	+	0.734653i	$0.262655\pi$
$29$			7.91246i			1.46931i	−0.678443	+	0.734653i	$0.737345\pi$
$30$		0			0
$31$		−	2.05833i		−	0.369687i	−0.982768	−	0.184844i	$-0.940822\pi$
$31$		−	2.05833i		−	0.369687i	0.982768	−	0.184844i	$-0.0591779\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.19684	+	5.98768i	1.38552	+	1.01210i
$36$		0			0
$37$	−8.71513			−1.43276			−0.716380	−	0.697711i	$-0.754202\pi$
$37$	−8.71513			−1.43276			−0.716380	+	0.697711i	$0.754202\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.68018			−1.51179			−0.755895	−	0.654693i	$-0.772798\pi$
$41$	−9.68018			−1.51179			−0.755895	+	0.654693i	$0.772798\pi$
$42$		0			0
$43$	−7.15091			−1.09050			−0.545251	−	0.838273i	$-0.683566\pi$
$43$	−7.15091			−1.09050			−0.545251	+	0.838273i	$0.683566\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.33218			0.194319			0.0971594	−	0.995269i	$-0.469024\pi$
$47$	1.33218			0.194319			0.0971594	+	0.995269i	$0.469024\pi$
$48$		0			0
$49$	2.12879	+	6.66845i	0.304113	+	0.952636i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			5.18304i			0.711945i	0.934496	+	0.355972i	$0.115850\pi$
$53$			5.18304i			0.711945i	−0.934496	+	0.355972i	$0.884150\pi$
$54$		0			0
$55$			2.59476i			0.349878i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.19238			−0.545801			−0.272901	−	0.962042i	$-0.587983\pi$
$59$	−4.19238			−0.545801			−0.272901	+	0.962042i	$0.587983\pi$
$60$		0			0
$61$		−	2.75025i		−	0.352133i	−0.984378	−	0.176066i	$-0.943663\pi$
$61$		−	2.75025i		−	0.352133i	0.984378	−	0.176066i	$-0.0563374\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	18.6957i		−	2.31892i
$66$		0			0
$67$	6.55273			0.800543			0.400271	−	0.916397i	$-0.368916\pi$
$67$	6.55273			0.800543			0.400271	+	0.916397i	$0.368916\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.1515i		−	1.32344i	−0.749751	−	0.661721i	$-0.769827\pi$
$71$		−	11.1515i		−	1.32344i	0.749751	−	0.661721i	$-0.230173\pi$
$72$		0			0
$73$		−	3.65100i		−	0.427318i	−0.976908	−	0.213659i	$-0.931462\pi$
$73$		−	3.65100i		−	0.427318i	0.976908	−	0.213659i	$-0.0685381\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.05547	+	1.44489i	−0.120282	+	0.164660i
$78$		0			0
$79$	11.2344			1.26397			0.631985	−	0.774981i	$-0.282240\pi$
$79$	11.2344			1.26397			0.631985	+	0.774981i	$0.282240\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.22558			−1.01264			−0.506320	−	0.862346i	$-0.668994\pi$
$83$	−9.22558			−1.01264			−0.506320	+	0.862346i	$0.668994\pi$
$84$		0			0
$85$	22.2344			2.41166
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.79702			0.190484			0.0952419	−	0.995454i	$-0.469638\pi$
$89$	1.79702			0.190484			0.0952419	+	0.995454i	$0.469638\pi$
$90$		0			0
$91$	7.60487	−	10.4107i	0.797207	−	1.09134i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			16.2083i			1.66293i
$96$		0			0
$97$			3.00814i			0.305431i	0.988270	+	0.152715i	$0.0488018\pi$
$97$			3.00814i			0.305431i	−0.988270	+	0.152715i	$0.951198\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.30988			0.826864			0.413432	−	0.910535i	$-0.364330\pi$
$101$	8.30988			0.826864			0.413432	+	0.910535i	$0.364330\pi$
$102$		0			0
$103$		−	7.78968i		−	0.767540i	−0.923429	−	0.383770i	$-0.874625\pi$
$103$		−	7.78968i		−	0.767540i	0.923429	−	0.383770i	$-0.125375\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.50520i		−	0.532207i	−0.963944	−	0.266104i	$-0.914264\pi$
$107$		−	5.50520i		−	0.532207i	0.963944	−	0.266104i	$-0.0857364\pi$
$108$		0			0
$109$	14.2890			1.36864			0.684320	−	0.729182i	$-0.260099\pi$
$109$	14.2890			1.36864			0.684320	+	0.729182i	$0.260099\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	5.29141i		−	0.497774i	−0.968533	−	0.248887i	$-0.919935\pi$
$113$		−	5.29141i		−	0.497774i	0.968533	−	0.248887i	$-0.0800647\pi$
$114$		0			0
$115$		−	21.1132i		−	1.96882i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.3812	+	9.04429i	1.13498	+	0.829089i
$120$		0			0
$121$	10.5426			0.958419
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−18.1093			−1.61975
$126$		0			0
$127$	−2.23048			−0.197923			−0.0989615	−	0.995091i	$-0.531552\pi$
$127$	−2.23048			−0.197923			−0.0989615	+	0.995091i	$0.531552\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.949268			0.0829380			0.0414690	−	0.999140i	$-0.486796\pi$
$131$	0.949268			0.0829380			0.0414690	+	0.999140i	$0.486796\pi$
$132$		0			0
$133$	−6.59303	+	9.02554i	−0.571688	+	0.782613i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	12.0132i		−	1.02635i	−0.858283	−	0.513177i	$-0.828469\pi$
$137$		−	12.0132i		−	1.02635i	0.858283	−	0.513177i	$-0.171531\pi$
$138$		0			0
$139$			2.22784i			0.188963i	0.995527	+	0.0944816i	$0.0301194\pi$
$139$			2.22784i			0.188963i	−0.995527	+	0.0944816i	$0.969881\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.29557			0.275590
$144$		0			0
$145$		−	30.3575i		−	2.52105i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			6.51342i			0.533600i	0.963752	+	0.266800i	$0.0859664\pi$
$149$			6.51342i			0.533600i	−0.963752	+	0.266800i	$0.914034\pi$
$150$		0			0
$151$	2.34212			0.190599			0.0952997	−	0.995449i	$-0.469619\pi$
$151$	2.34212			0.190599			0.0952997	+	0.995449i	$0.469619\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			7.89714i			0.634314i
$156$		0			0
$157$		−	2.35248i		−	0.187748i	−0.995584	−	0.0938742i	$-0.970075\pi$
$157$		−	2.35248i		−	0.187748i	0.995584	−	0.0938742i	$-0.0299251\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.58822	−	11.7569i	0.676846	−	0.926570i
$162$		0			0
$163$	17.0419			1.33482			0.667412	−	0.744689i	$-0.267402\pi$
$163$	17.0419			1.33482			0.667412	+	0.744689i	$0.267402\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.79813			−0.448673			−0.224336	−	0.974512i	$-0.572021\pi$
$167$	−5.79813			−0.448673			−0.224336	+	0.974512i	$0.572021\pi$
$168$		0			0
$169$	−10.7452			−0.826555
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.94042			0.299585			0.149792	−	0.988717i	$-0.452139\pi$
$173$	3.94042			0.299585			0.149792	+	0.988717i	$0.452139\pi$
$174$		0			0
$175$	−20.7664	−	15.1695i	−1.56979	−	1.14671i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			6.52749i			0.487888i	0.969789	+	0.243944i	$0.0784412\pi$
$179$			6.52749i			0.487888i	−0.969789	+	0.243944i	$0.921559\pi$
$180$		0			0
$181$		−	6.39901i		−	0.475635i	−0.971310	−	0.237817i	$-0.923568\pi$
$181$		−	6.39901i		−	0.475635i	0.971310	−	0.237817i	$-0.0764320\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	33.4371			2.45835
$186$		0			0
$187$			3.91935i			0.286611i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			17.9851i			1.30135i	0.759355	+	0.650677i	$0.225515\pi$
$191$			17.9851i			1.30135i	−0.759355	+	0.650677i	$0.774485\pi$
$192$		0			0
$193$	−15.3454			−1.10459			−0.552293	−	0.833650i	$-0.686247\pi$
$193$	−15.3454			−1.10459			−0.552293	+	0.833650i	$0.686247\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	11.3420i		−	0.808086i	−0.914740	−	0.404043i	$-0.867605\pi$
$197$		−	11.3420i		−	0.808086i	0.914740	−	0.404043i	$-0.132395\pi$
$198$		0			0
$199$			11.2568i			0.797971i	0.916957	+	0.398986i	$0.130638\pi$
$199$			11.2568i			0.797971i	−0.916957	+	0.398986i	$0.869362\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.3485	−	16.9045i	0.866696	−	1.18646i
$204$		0			0
$205$	37.1397			2.59395
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.85709			−0.197629
$210$		0			0
$211$	7.59545			0.522892			0.261446	−	0.965218i	$-0.415801\pi$
$211$	7.59545			0.522892			0.261446	+	0.965218i	$0.415801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	27.4357			1.87110
$216$		0			0
$217$	−3.21232	+	4.39751i	−0.218067	+	0.298522i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	28.2396i		−	1.89960i
$222$		0			0
$223$		−	23.3497i		−	1.56361i	−0.623521	−	0.781807i	$-0.714298\pi$
$223$		−	23.3497i		−	1.56361i	0.623521	−	0.781807i	$-0.285702\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.6211			1.03681			0.518405	−	0.855135i	$-0.326526\pi$
$227$	15.6211			1.03681			0.518405	+	0.855135i	$0.326526\pi$
$228$		0			0
$229$		−	10.5730i		−	0.698684i	−0.936995	−	0.349342i	$-0.886405\pi$
$229$		−	10.5730i		−	0.698684i	0.936995	−	0.349342i	$-0.113595\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			5.08862i			0.333367i	0.986010	+	0.166683i	$0.0533057\pi$
$233$			5.08862i			0.333367i	−0.986010	+	0.166683i	$0.946694\pi$
$234$		0			0
$235$	−5.11115			−0.333415
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7.77477i		−	0.502908i	−0.967869	−	0.251454i	$-0.919091\pi$
$239$		−	7.77477i		−	0.502908i	0.967869	−	0.251454i	$-0.0809087\pi$
$240$		0			0
$241$		−	24.9058i		−	1.60432i	−0.597108	−	0.802161i	$-0.703684\pi$
$241$		−	24.9058i		−	1.60432i	0.597108	−	0.802161i	$-0.296316\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.16746	−	25.5847i	−0.521800	−	1.63454i
$246$		0			0
$247$	20.5859			1.30985
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.76999			−0.616676			−0.308338	−	0.951277i	$-0.599773\pi$
$251$	−9.76999			−0.616676			−0.308338	+	0.951277i	$0.599773\pi$
$252$		0			0
$253$	3.72171			0.233982
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.9817			−0.934533			−0.467267	−	0.884117i	$-0.654761\pi$
$257$	−14.9817			−0.934533			−0.467267	+	0.884117i	$0.654761\pi$
$258$		0			0
$259$	18.6194	+	13.6012i	1.15695	+	0.845138i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	16.5518i		−	1.02063i	−0.859988	−	0.510314i	$-0.829529\pi$
$263$		−	16.5518i		−	1.02063i	0.859988	−	0.510314i	$-0.170471\pi$
$264$		0			0
$265$		−	19.8856i		−	1.22156i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.269785			0.0164491			0.00822455	−	0.999966i	$-0.497382\pi$
$269$	0.269785			0.0164491			0.00822455	+	0.999966i	$0.497382\pi$
$270$		0			0
$271$			13.3371i			0.810172i	0.914279	+	0.405086i	$0.132758\pi$
$271$			13.3371i			0.810172i	−0.914279	+	0.405086i	$0.867242\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	6.57373i		−	0.396411i
$276$		0			0
$277$	2.06822			0.124267			0.0621336	−	0.998068i	$-0.480210\pi$
$277$	2.06822			0.124267			0.0621336	+	0.998068i	$0.480210\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5.84747i			0.348831i	0.984672	+	0.174415i	$0.0558036\pi$
$281$			5.84747i			0.348831i	−0.984672	+	0.174415i	$0.944196\pi$
$282$		0			0
$283$		−	23.7709i		−	1.41303i	−0.707698	−	0.706515i	$-0.750266\pi$
$283$		−	23.7709i		−	1.41303i	0.707698	−	0.706515i	$-0.249734\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	20.6812	+	15.1073i	1.22077	+	0.891756i
$288$		0			0
$289$	16.5847			0.975572
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.94224			−0.171887			−0.0859437	−	0.996300i	$-0.527391\pi$
$293$	−2.94224			−0.171887			−0.0859437	+	0.996300i	$0.527391\pi$
$294$		0			0
$295$	16.0848			0.936492
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−26.8156			−1.55079
$300$		0			0
$301$	15.2775	+	11.1600i	0.880581	+	0.643252i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			10.5518i			0.604194i
$306$		0			0
$307$			22.8040i			1.30149i	0.759296	+	0.650745i	$0.225543\pi$
$307$			22.8040i			1.30149i	−0.759296	+	0.650745i	$0.774457\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	30.5029			1.72966			0.864830	−	0.502065i	$-0.167426\pi$
$311$	30.5029			1.72966			0.864830	+	0.502065i	$0.167426\pi$
$312$		0			0
$313$		−	8.17785i		−	0.462239i	−0.972925	−	0.231120i	$-0.925761\pi$
$313$		−	8.17785i		−	0.462239i	0.972925	−	0.231120i	$-0.0742389\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			14.2541i			0.800593i	0.916386	+	0.400296i	$0.131093\pi$
$317$			14.2541i			0.800593i	−0.916386	+	0.400296i	$0.868907\pi$
$318$		0			0
$319$	5.35124			0.299612
$320$		0			0
$321$		0			0
$322$		0			0
$323$			24.4823i			1.36223i
$324$		0			0
$325$			47.3649i			2.62733i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.84613	−	2.07906i	−0.156912	−	0.114622i
$330$		0			0
$331$	−20.0509			−1.10210			−0.551048	−	0.834473i	$-0.685772\pi$
$331$	−20.0509			−1.10210			−0.551048	+	0.834473i	$0.685772\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−25.1407			−1.37358
$336$		0			0
$337$	−29.3860			−1.60076			−0.800378	−	0.599496i	$-0.795368\pi$
$337$	−29.3860			−1.60076			−0.800378	+	0.599496i	$0.795368\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.39206			−0.0753844
$342$		0			0
$343$	5.85904	−	17.5691i	0.316358	−	0.948640i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1.82412i			0.0979238i	0.998801	+	0.0489619i	$0.0155913\pi$
$347$			1.82412i			0.0979238i	−0.998801	+	0.0489619i	$0.984409\pi$
$348$		0			0
$349$			25.3405i			1.35645i	0.734855	+	0.678224i	$0.237250\pi$
$349$			25.3405i			1.35645i	−0.734855	+	0.678224i	$0.762750\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.7341			0.730992			0.365496	−	0.930813i	$-0.380899\pi$
$353$	13.7341			0.730992			0.365496	+	0.930813i	$0.380899\pi$
$354$		0			0
$355$			42.7847i			2.27078i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.77932i			0.305021i	0.988302	+	0.152510i	$0.0487357\pi$
$359$			5.77932i			0.305021i	−0.988302	+	0.152510i	$0.951264\pi$
$360$		0			0
$361$	1.15309			0.0606891
$362$		0			0
$363$		0			0
$364$		0			0
$365$			14.0077i			0.733197i
$366$		0			0
$367$			34.5495i			1.80347i	0.432291	+	0.901734i	$0.357706\pi$
$367$			34.5495i			1.80347i	−0.432291	+	0.901734i	$0.642294\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.08887	−	11.0733i	0.419953	−	0.574895i
$372$		0			0
$373$	−1.44074			−0.0745986			−0.0372993	−	0.999304i	$-0.511875\pi$
$373$	−1.44074			−0.0745986			−0.0372993	+	0.999304i	$0.511875\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−38.5567			−1.98577
$378$		0			0
$379$	−4.53029			−0.232705			−0.116353	−	0.993208i	$-0.537120\pi$
$379$	−4.53029			−0.232705			−0.116353	+	0.993208i	$0.537120\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.9831			−0.765600			−0.382800	−	0.923831i	$-0.625040\pi$
$383$	−14.9831			−0.765600			−0.382800	+	0.923831i	$0.625040\pi$
$384$		0			0
$385$	4.04950	−	5.54357i	0.206382	−	0.282526i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			37.3614i			1.89430i	0.320797	+	0.947148i	$0.396049\pi$
$389$			37.3614i			1.89430i	−0.320797	+	0.947148i	$0.603951\pi$
$390$		0			0
$391$		−	31.8912i		−	1.61281i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−43.1027			−2.16873
$396$		0			0
$397$		−	18.4269i		−	0.924820i	−0.886666	−	0.462410i	$-0.846985\pi$
$397$		−	18.4269i		−	0.924820i	0.886666	−	0.462410i	$-0.153015\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	31.9614i		−	1.59608i	−0.602607	−	0.798038i	$-0.705871\pi$
$401$		−	31.9614i		−	1.59608i	0.602607	−	0.798038i	$-0.294129\pi$
$402$		0			0
$403$	10.0301			0.499633
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5.89410i			0.292159i
$408$		0			0
$409$		−	1.93490i		−	0.0956745i	−0.998855	−	0.0478373i	$-0.984767\pi$
$409$		−	1.93490i		−	0.0956745i	0.998855	−	0.0478373i	$-0.0152329\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.95678	+	6.54281i	0.440734	+	0.321950i
$414$		0			0
$415$	35.3955			1.73750
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.0772			1.46937			0.734684	−	0.678410i	$-0.237330\pi$
$419$	30.0772			1.46937			0.734684	+	0.678410i	$0.237330\pi$
$420$		0			0
$421$	−2.35950			−0.114995			−0.0574974	−	0.998346i	$-0.518312\pi$
$421$	−2.35950			−0.114995			−0.0574974	+	0.998346i	$0.518312\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−56.3300			−2.73241
$426$		0			0
$427$	−4.29215	+	5.87575i	−0.207712	+	0.284347i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	33.3518i		−	1.60650i	−0.595643	−	0.803249i	$-0.703103\pi$
$431$		−	33.3518i		−	1.60650i	0.595643	−	0.803249i	$-0.296897\pi$
$432$		0			0
$433$		−	11.9121i		−	0.572459i	−0.958161	−	0.286229i	$-0.907598\pi$
$433$		−	11.9121i		−	0.572459i	0.958161	−	0.286229i	$-0.0924019\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	23.2478			1.11209
$438$		0			0
$439$			39.4859i			1.88456i	0.334830	+	0.942279i	$0.391321\pi$
$439$			39.4859i			1.88456i	−0.334830	+	0.942279i	$0.608679\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			29.6785i			1.41007i	0.709174	+	0.705034i	$0.249068\pi$
$443$			29.6785i			1.41007i	−0.709174	+	0.705034i	$0.750932\pi$
$444$		0			0
$445$	−6.89458			−0.326834
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	27.9245i		−	1.31784i	−0.752213	−	0.658920i	$-0.771014\pi$
$449$		−	27.9245i		−	1.31784i	0.752213	−	0.658920i	$-0.228986\pi$
$450$		0			0
$451$			6.54676i			0.308275i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−29.1774	+	39.9424i	−1.36786	+	1.87253i
$456$		0			0
$457$	−36.0779			−1.68766			−0.843828	−	0.536614i	$-0.819703\pi$
$457$	−36.0779			−1.68766			−0.843828	+	0.536614i	$0.819703\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.69884			0.125698			0.0628488	−	0.998023i	$-0.479981\pi$
$461$	2.69884			0.125698			0.0628488	+	0.998023i	$0.479981\pi$
$462$		0			0
$463$	−29.9986			−1.39415			−0.697077	−	0.716996i	$-0.745516\pi$
$463$	−29.9986			−1.39415			−0.697077	+	0.716996i	$0.745516\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.1160			−0.653211			−0.326605	−	0.945161i	$-0.605905\pi$
$467$	−14.1160			−0.653211			−0.326605	+	0.945161i	$0.605905\pi$
$468$		0			0
$469$	−13.9995	−	10.2265i	−0.646438	−	0.472214i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.83620i			0.222369i
$474$		0			0
$475$		−	41.0630i		−	1.88410i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	34.7032			1.58563			0.792815	−	0.609463i	$-0.208615\pi$
$479$	34.7032			1.58563			0.792815	+	0.609463i	$0.208615\pi$
$480$		0			0
$481$		−	42.4680i		−	1.93638i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	11.5413i		−	0.524062i
$486$		0			0
$487$	36.2125			1.64094			0.820472	−	0.571686i	$-0.193711\pi$
$487$	36.2125			1.64094			0.820472	+	0.571686i	$0.193711\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			33.8597i			1.52806i	0.645178	+	0.764032i	$0.276783\pi$
$491$			33.8597i			1.52806i	−0.645178	+	0.764032i	$0.723217\pi$
$492$		0			0
$493$		−	45.8545i		−	2.06518i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−17.4035	+	23.8246i	−0.780655	+	1.06868i
$498$		0			0
$499$	27.7172			1.24079			0.620395	−	0.784289i	$-0.286972\pi$
$499$	27.7172			1.24079			0.620395	+	0.784289i	$0.286972\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9.08643			−0.405144			−0.202572	−	0.979267i	$-0.564930\pi$
$503$	−9.08643			−0.405144			−0.202572	+	0.979267i	$0.564930\pi$
$504$		0			0
$505$	−31.8823			−1.41874
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−27.5351			−1.22047			−0.610236	−	0.792220i	$-0.708925\pi$
$509$	−27.5351			−1.22047			−0.610236	+	0.792220i	$0.708925\pi$
$510$		0			0
$511$	−5.69791	+	7.80017i	−0.252061	+	0.345059i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			29.8865i			1.31696i
$516$		0			0
$517$		−	0.900963i		−	0.0396243i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.47559			−0.152268			−0.0761341	−	0.997098i	$-0.524258\pi$
$521$	−3.47559			−0.152268			−0.0761341	+	0.997098i	$0.524258\pi$
$522$		0			0
$523$		−	0.315295i		−	0.0137869i	−0.999976	−	0.00689344i	$-0.997806\pi$
$523$		−	0.315295i		−	0.0137869i	0.999976	−	0.00689344i	$-0.00219427\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			11.9285i			0.519614i
$528$		0			0
$529$	−7.28301			−0.316653
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	47.1706i		−	2.04319i
$534$		0			0
$535$			21.1216i			0.913168i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.50991	−	1.43971i	0.194256	−	0.0620128i
$540$		0			0
$541$	39.0741			1.67993			0.839964	−	0.542643i	$-0.182576\pi$
$541$	39.0741			1.67993			0.839964	+	0.542643i	$0.182576\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−54.8223			−2.34833
$546$		0			0
$547$	1.21979			0.0521544			0.0260772	−	0.999660i	$-0.491698\pi$
$547$	1.21979			0.0521544			0.0260772	+	0.999660i	$0.491698\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	33.4266			1.42402
$552$		0			0
$553$	−24.0017	−	17.5329i	−1.02066	−	0.745575i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	0.456650i		−	0.0193489i	−0.999953	−	0.00967444i	$-0.996920\pi$
$557$		−	0.456650i		−	0.0193489i	0.999953	−	0.00967444i	$-0.00307952\pi$
$558$		0			0
$559$		−	34.8457i		−	1.47382i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	31.5360			1.32908			0.664541	−	0.747251i	$-0.268627\pi$
$563$	31.5360			1.32908			0.664541	+	0.747251i	$0.268627\pi$
$564$		0			0
$565$			20.3014i			0.854086i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	27.8717i		−	1.16844i	−0.811594	−	0.584221i	$-0.801400\pi$
$569$		−	27.8717i		−	1.16844i	0.811594	−	0.584221i	$-0.198600\pi$
$570$		0			0
$571$	26.5176			1.10973			0.554864	−	0.831941i	$-0.312770\pi$
$571$	26.5176			1.10973			0.554864	+	0.831941i	$0.312770\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			53.4895i			2.23067i
$576$		0			0
$577$			6.69146i			0.278569i	0.990252	+	0.139285i	$0.0444803\pi$
$577$			6.69146i			0.278569i	−0.990252	+	0.139285i	$0.955520\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	19.7099	+	14.3978i	0.817706	+	0.597323i
$582$		0			0
$583$	3.50532			0.145175
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.98775			−0.288415			−0.144208	−	0.989547i	$-0.546063\pi$
$587$	−6.98775			−0.288415			−0.144208	+	0.989547i	$0.546063\pi$
$588$		0			0
$589$	−8.69554			−0.358294
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.96766			−0.368258			−0.184129	−	0.982902i	$-0.558946\pi$
$593$	−8.96766			−0.368258			−0.184129	+	0.982902i	$0.558946\pi$
$594$		0			0
$595$	−47.5026	−	34.7000i	−1.94742	−	1.42256i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	28.3274i		−	1.15743i	−0.815531	−	0.578714i	$-0.803555\pi$
$599$		−	28.3274i		−	1.15743i	0.815531	−	0.578714i	$-0.196445\pi$
$600$		0			0
$601$		−	9.51462i		−	0.388110i	−0.980991	−	0.194055i	$-0.937836\pi$
$601$		−	9.51462i		−	0.388110i	0.980991	−	0.194055i	$-0.0621640\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−40.4485			−1.64447
$606$		0			0
$607$			14.7276i			0.597775i	0.954288	+	0.298888i	$0.0966156\pi$
$607$			14.7276i			0.597775i	−0.954288	+	0.298888i	$0.903384\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			6.49160i			0.262622i
$612$		0			0
$613$	−8.63782			−0.348878			−0.174439	−	0.984668i	$-0.555811\pi$
$613$	−8.63782			−0.348878			−0.174439	+	0.984668i	$0.555811\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	17.0907i		−	0.688044i	−0.938962	−	0.344022i	$-0.888210\pi$
$617$		−	17.0907i		−	0.688044i	0.938962	−	0.344022i	$-0.111790\pi$
$618$		0			0
$619$		−	11.1653i		−	0.448769i	−0.974501	−	0.224385i	$-0.927963\pi$
$619$		−	11.1653i		−	0.448769i	0.974501	−	0.224385i	$-0.0720372\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.83923	−	2.80451i	−0.153816	−	0.112360i
$624$		0			0
$625$	20.8792			0.835170
$626$		0			0
$627$		0			0
$628$		0			0
$629$	50.5062			2.01382
$630$		0			0
$631$	−37.1187			−1.47767			−0.738835	−	0.673886i	$-0.764624\pi$
$631$	−37.1187			−1.47767			−0.738835	+	0.673886i	$0.764624\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.55762			0.339599
$636$		0			0
$637$	−32.4947	+	10.3734i	−1.28749	+	0.411008i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18.6540i		−	0.736789i	−0.929670	−	0.368395i	$-0.879908\pi$
$641$		−	18.6540i		−	0.736789i	0.929670	−	0.368395i	$-0.120092\pi$
$642$		0			0
$643$			38.2617i			1.50890i	0.656360	+	0.754448i	$0.272095\pi$
$643$			38.2617i			1.50890i	−0.656360	+	0.754448i	$0.727905\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.28252			0.0897350			0.0448675	−	0.998993i	$-0.485713\pi$
$647$	2.28252			0.0897350			0.0448675	+	0.998993i	$0.485713\pi$
$648$		0			0
$649$			2.83533i			0.111296i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	40.1471i		−	1.57108i	−0.618813	−	0.785538i	$-0.712386\pi$
$653$		−	40.1471i		−	1.57108i	0.618813	−	0.785538i	$-0.287614\pi$
$654$		0			0
$655$	−3.64203			−0.142306
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	38.9552i		−	1.51748i	−0.651394	−	0.758740i	$-0.725815\pi$
$659$		−	38.9552i		−	1.51748i	0.651394	−	0.758740i	$-0.274185\pi$
$660$		0			0
$661$		−	8.46618i		−	0.329296i	−0.986352	−	0.164648i	$-0.947351\pi$
$661$		−	8.46618i		−	0.329296i	0.986352	−	0.164648i	$-0.0526488\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	25.2953	−	34.6280i	0.980910	−	1.34282i
$666$		0			0
$667$	−43.5423			−1.68596
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.86001			−0.0718048
$672$		0			0
$673$	17.4592			0.673003			0.336501	−	0.941683i	$-0.390756\pi$
$673$	17.4592			0.673003			0.336501	+	0.941683i	$0.390756\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	49.0022			1.88331			0.941654	−	0.336581i	$-0.109271\pi$
$677$	49.0022			1.88331			0.941654	+	0.336581i	$0.109271\pi$
$678$		0			0
$679$	4.69464	−	6.42673i	0.180164	−	0.246635i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			22.1852i			0.848893i	0.905453	+	0.424446i	$0.139531\pi$
$683$			22.1852i			0.848893i	−0.905453	+	0.424446i	$0.860469\pi$
$684$		0			0
$685$			46.0905i			1.76103i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−25.2565			−0.962194
$690$		0			0
$691$			30.9122i			1.17596i	0.808877	+	0.587978i	$0.200076\pi$
$691$			30.9122i			1.17596i	−0.808877	+	0.587978i	$0.799924\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	8.54751i		−	0.324226i
$696$		0			0
$697$	56.0989			2.12490
$698$		0			0
$699$		0			0
$700$		0			0
$701$			32.4719i			1.22645i	0.789910	+	0.613223i	$0.210127\pi$
$701$			32.4719i			1.22645i	−0.789910	+	0.613223i	$0.789873\pi$
$702$		0			0
$703$			36.8176i			1.38860i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−17.7536	−	12.9688i	−0.667693	−	0.487740i
$708$		0			0
$709$	51.3668			1.92912			0.964560	−	0.263863i	$-0.0849966\pi$
$709$	51.3668			1.92912			0.964560	+	0.263863i	$0.0849966\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.3270			0.424199
$714$		0			0
$715$	−12.6440			−0.472860
$716$		0			0
$717$		0			0
$718$		0			0
$719$	37.9805			1.41643			0.708216	−	0.705996i	$-0.249500\pi$
$719$	37.9805			1.41643			0.708216	+	0.705996i	$0.249500\pi$
$720$		0			0
$721$	−12.1569	+	16.6422i	−0.452747	+	0.619789i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			76.9095i			2.85635i
$726$		0			0
$727$		−	16.9162i		−	0.627385i	−0.949525	−	0.313693i	$-0.898434\pi$
$727$		−	16.9162i		−	0.627385i	0.949525	−	0.313693i	$-0.101566\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	41.4412			1.53276
$732$		0			0
$733$			30.5533i			1.12851i	0.825600	+	0.564256i	$0.190837\pi$
$733$			30.5533i			1.12851i	−0.825600	+	0.564256i	$0.809163\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	4.43165i		−	0.163242i
$738$		0			0
$739$	−23.0562			−0.848138			−0.424069	−	0.905630i	$-0.639399\pi$
$739$	−23.0562			−0.848138			−0.424069	+	0.905630i	$0.639399\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2.18162i		−	0.0800358i	−0.999199	−	0.0400179i	$-0.987259\pi$
$743$		−	2.18162i		−	0.0800358i	0.999199	−	0.0400179i	$-0.0127415\pi$
$744$		0			0
$745$		−	24.9899i		−	0.915558i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.59165	+	11.7615i	−0.313932	+	0.429758i
$750$		0			0
$751$	−18.9373			−0.691032			−0.345516	−	0.938413i	$-0.612296\pi$
$751$	−18.9373			−0.691032			−0.345516	+	0.938413i	$0.612296\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.98597			−0.327033
$756$		0			0
$757$	11.6383			0.423002			0.211501	−	0.977378i	$-0.432165\pi$
$757$	11.6383			0.423002			0.211501	+	0.977378i	$0.432165\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.27638			−0.300019			−0.150009	−	0.988685i	$-0.547930\pi$
$761$	−8.27638			−0.300019			−0.150009	+	0.988685i	$0.547930\pi$
$762$		0			0
$763$	−30.5277	−	22.3001i	−1.10518	−	0.807316i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	20.4291i		−	0.737651i
$768$		0			0
$769$			52.9232i			1.90846i	0.299074	+	0.954230i	$0.403322\pi$
$769$			52.9232i			1.90846i	−0.299074	+	0.954230i	$0.596678\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−11.1905			−0.402494			−0.201247	−	0.979540i	$-0.564499\pi$
$773$	−11.1905			−0.402494			−0.201247	+	0.979540i	$0.564499\pi$
$774$		0			0
$775$		−	20.0071i		−	0.718676i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			40.8945i			1.46520i
$780$		0			0
$781$	−7.54183			−0.269868
$782$		0			0
$783$		0			0
$784$		0			0
$785$			9.02570i			0.322141i
$786$		0			0
$787$			28.3849i			1.01181i	0.862589	+	0.505905i	$0.168841\pi$
$787$			28.3849i			1.01181i	−0.862589	+	0.505905i	$0.831159\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8.25800	+	11.3048i	−0.293621	+	0.401952i
$792$		0			0
$793$	13.4017			0.475908
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.702842			−0.0248960			−0.0124480	−	0.999923i	$-0.503962\pi$
$797$	−0.702842			−0.0248960			−0.0124480	+	0.999923i	$0.503962\pi$
$798$		0			0
$799$	−7.72031			−0.273125
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.46919			−0.0871360
$804$		0			0
$805$	−32.9502	+	45.1072i	−1.16134	+	1.58982i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	40.0190i		−	1.40699i	−0.710699	−	0.703496i	$-0.751621\pi$
$809$		−	40.0190i		−	1.40699i	0.710699	−	0.703496i	$-0.248379\pi$
$810$		0			0
$811$		−	16.5473i		−	0.581055i	−0.956867	−	0.290528i	$-0.906169\pi$
$811$		−	16.5473i		−	0.581055i	0.956867	−	0.290528i	$-0.0938309\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−65.3841			−2.29031
$816$		0			0
$817$			30.2094i			1.05689i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			20.4553i			0.713895i	0.934124	+	0.356948i	$0.116183\pi$
$821$			20.4553i			0.713895i	−0.934124	+	0.356948i	$0.883817\pi$
$822$		0			0
$823$	−2.99201			−0.104295			−0.0521474	−	0.998639i	$-0.516607\pi$
$823$	−2.99201			−0.104295			−0.0521474	+	0.998639i	$0.516607\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	37.3399i		−	1.29843i	−0.760603	−	0.649217i	$-0.775097\pi$
$827$		−	37.3399i		−	1.29843i	0.760603	−	0.649217i	$-0.224903\pi$
$828$		0			0
$829$			12.7334i			0.442250i	0.975246	+	0.221125i	$0.0709728\pi$
$829$			12.7334i			0.442250i	−0.975246	+	0.221125i	$0.929027\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.3368	−	38.6452i	−0.427446	−	1.33898i
$834$		0			0
$835$	22.2455			0.769838
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.71794			0.0938338			0.0469169	−	0.998899i	$-0.485060\pi$
$839$	2.71794			0.0938338			0.0469169	+	0.998899i	$0.485060\pi$
$840$		0			0
$841$	−33.6070			−1.15886
$842$		0			0
$843$		0			0
$844$		0			0
$845$	41.2259			1.41821
$846$		0			0
$847$	−22.5237	−	16.4532i	−0.773923	−	0.565340i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	47.9594i		−	1.64403i
$852$		0			0
$853$			50.3927i			1.72541i	0.505705	+	0.862707i	$0.331233\pi$
$853$			50.3927i			1.72541i	−0.505705	+	0.862707i	$0.668767\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.9527			−0.613251			−0.306626	−	0.951830i	$-0.599200\pi$
$857$	−17.9527			−0.613251			−0.306626	+	0.951830i	$0.599200\pi$
$858$		0			0
$859$			12.7080i			0.433591i	0.976217	+	0.216796i	$0.0695605\pi$
$859$			12.7080i			0.433591i	−0.976217	+	0.216796i	$0.930439\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.05813i		−	0.0700595i	−0.999386	−	0.0350298i	$-0.988847\pi$
$863$		−	2.05813i		−	0.0700595i	0.999386	−	0.0350298i	$-0.0111526\pi$
$864$		0			0
$865$	−15.1181			−0.514031
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	7.59789i		−	0.257741i
$870$		0			0
$871$			31.9308i			1.08193i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	38.6896	+	28.2622i	1.30795	+	0.955436i
$876$		0			0
$877$	−27.6154			−0.932506			−0.466253	−	0.884652i	$-0.654396\pi$
$877$	−27.6154			−0.932506			−0.466253	+	0.884652i	$0.654396\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.2629			−0.345765			−0.172882	−	0.984942i	$-0.555308\pi$
$881$	−10.2629			−0.345765			−0.172882	+	0.984942i	$0.555308\pi$
$882$		0			0
$883$	49.8978			1.67919			0.839597	−	0.543210i	$-0.182791\pi$
$883$	49.8978			1.67919			0.839597	+	0.543210i	$0.182791\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.0733			0.841880			0.420940	−	0.907089i	$-0.361700\pi$
$887$	25.0733			0.841880			0.420940	+	0.907089i	$0.361700\pi$
$888$		0			0
$889$	4.76529	+	3.48098i	0.159823	+	0.116748i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	5.62789i		−	0.188330i
$894$		0			0
$895$		−	25.0438i		−	0.837124i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	16.2865			0.543184
$900$		0			0
$901$		−	30.0369i		−	1.00067i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			24.5509i			0.816100i
$906$		0			0
$907$	−19.6948			−0.653954			−0.326977	−	0.945032i	$-0.606030\pi$
$907$	−19.6948			−0.653954			−0.326977	+	0.945032i	$0.606030\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	12.6631i		−	0.419548i	−0.977750	−	0.209774i	$-0.932727\pi$
$911$		−	12.6631i		−	0.419548i	0.977750	−	0.209774i	$-0.0672729\pi$
$912$		0			0
$913$			6.23931i			0.206491i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.02806	−	1.48147i	−0.0669724	−	0.0489224i
$918$		0			0
$919$	−34.9422			−1.15264			−0.576318	−	0.817226i	$-0.695511\pi$
$919$	−34.9422			−1.15264			−0.576318	+	0.817226i	$0.695511\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	54.3403			1.78863
$924$		0			0
$925$	−84.7116			−2.78530
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.8195			0.781492			0.390746	−	0.920499i	$-0.372217\pi$
$929$	23.8195			0.781492			0.390746	+	0.920499i	$0.372217\pi$
$930$		0			0
$931$	28.1713	−	8.99319i	0.923276	−	0.294740i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	15.0373i		−	0.491771i
$936$		0			0
$937$			30.0910i			0.983029i	0.870869	+	0.491514i	$0.163556\pi$
$937$			30.0910i			0.983029i	−0.870869	+	0.491514i	$0.836444\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.3345			0.728083			0.364042	−	0.931383i	$-0.381397\pi$
$941$	22.3345			0.728083			0.364042	+	0.931383i	$0.381397\pi$
$942$		0			0
$943$		−	53.2700i		−	1.73471i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	36.6828i		−	1.19203i	−0.802972	−	0.596016i	$-0.796749\pi$
$947$		−	36.6828i		−	1.19203i	0.802972	−	0.596016i	$-0.203251\pi$
$948$		0			0
$949$	17.7910			0.577520
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	23.4461i		−	0.759495i	−0.925090	−	0.379747i	$-0.876011\pi$
$953$		−	23.4461i		−	0.759495i	0.925090	−	0.379747i	$-0.123989\pi$
$954$		0			0
$955$		−	69.0028i		−	2.23288i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.7482	+	25.6654i	−0.605412	+	0.828780i
$960$		0			0
$961$	26.7633			0.863331
$962$		0			0
$963$		0			0
$964$		0			0
$965$	58.8753			1.89526
$966$		0			0
$967$	−42.4743			−1.36588			−0.682941	−	0.730473i	$-0.739300\pi$
$967$	−42.4743			−1.36588			−0.682941	+	0.730473i	$0.739300\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.9007			0.414002			0.207001	−	0.978341i	$-0.433630\pi$
$971$	12.9007			0.414002			0.207001	+	0.978341i	$0.433630\pi$
$972$		0			0
$973$	3.47687	−	4.75966i	0.111463	−	0.152588i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.88179i			0.156182i	0.996946	+	0.0780911i	$0.0248825\pi$
$977$			4.88179i			0.156182i	−0.996946	+	0.0780911i	$0.975118\pi$
$978$		0			0
$979$		−	1.21534i		−	0.0388423i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.9662			0.604929			0.302464	−	0.953161i	$-0.402191\pi$
$983$	18.9662			0.604929			0.302464	+	0.953161i	$0.402191\pi$
$984$		0			0
$985$			43.5157i			1.38652i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	39.3514i		−	1.25130i
$990$		0			0
$991$	−34.1369			−1.08439			−0.542197	−	0.840251i	$-0.682407\pi$
$991$	−34.1369			−1.08439			−0.542197	+	0.840251i	$0.682407\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	43.1885i		−	1.36917i
$996$		0			0
$997$		−	11.4608i		−	0.362967i	−0.983394	−	0.181484i	$-0.941910\pi$
$997$		−	11.4608i		−	0.362967i	0.983394	−	0.181484i	$-0.0580899\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4536.2.k.a.3401.2		48
3.2	odd	2	inner	4536.2.k.a.3401.47		48
7.6	odd	2	inner	4536.2.k.a.3401.48		48
9.2	odd	6		504.2.bu.a.41.10	✓	48
9.4	even	3		504.2.bu.a.209.15	yes	48
9.5	odd	6		1512.2.bu.a.1385.1		48
9.7	even	3		1512.2.bu.a.881.24		48
21.20	even	2	inner	4536.2.k.a.3401.1		48
36.7	odd	6		3024.2.cc.d.881.24		48
36.11	even	6		1008.2.cc.d.545.15		48
36.23	even	6		3024.2.cc.d.2897.1		48
36.31	odd	6		1008.2.cc.d.209.10		48
63.13	odd	6		504.2.bu.a.209.10	yes	48
63.20	even	6		504.2.bu.a.41.15	yes	48
63.34	odd	6		1512.2.bu.a.881.1		48
63.41	even	6		1512.2.bu.a.1385.24		48
252.83	odd	6		1008.2.cc.d.545.10		48
252.139	even	6		1008.2.cc.d.209.15		48
252.167	odd	6		3024.2.cc.d.2897.24		48
252.223	even	6		3024.2.cc.d.881.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bu.a.41.10	✓	48	9.2	odd	6
504.2.bu.a.41.15	yes	48	63.20	even	6
504.2.bu.a.209.10	yes	48	63.13	odd	6
504.2.bu.a.209.15	yes	48	9.4	even	3
1008.2.cc.d.209.10		48	36.31	odd	6
1008.2.cc.d.209.15		48	252.139	even	6
1008.2.cc.d.545.10		48	252.83	odd	6
1008.2.cc.d.545.15		48	36.11	even	6
1512.2.bu.a.881.1		48	63.34	odd	6
1512.2.bu.a.881.24		48	9.7	even	3
1512.2.bu.a.1385.1		48	9.5	odd	6
1512.2.bu.a.1385.24		48	63.41	even	6
3024.2.cc.d.881.1		48	252.223	even	6
3024.2.cc.d.881.24		48	36.7	odd	6
3024.2.cc.d.2897.1		48	36.23	even	6
3024.2.cc.d.2897.24		48	252.167	odd	6
4536.2.k.a.3401.1		48	21.20	even	2	inner
4536.2.k.a.3401.2		48	1.1	even	1	trivial
4536.2.k.a.3401.47		48	3.2	odd	2	inner
4536.2.k.a.3401.48		48	7.6	odd	2	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:	\( N \)	\(=\)	\( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4536.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.83667 q^{5} +(-2.13644 - 1.56064i) q^{7} +O(q^{10})\)	\(q-3.83667 q^{5} +(-2.13644 - 1.56064i) q^{7} -0.676306i q^{11} +4.87291i q^{13} -5.79523 q^{17} -4.22456i q^{19} +5.50300i q^{23} +9.72006 q^{25} +7.91246i q^{29} -2.05833i q^{31} +(8.19684 + 5.98768i) q^{35} -8.71513 q^{37} -9.68018 q^{41} -7.15091 q^{43} +1.33218 q^{47} +(2.12879 + 6.66845i) q^{49} +5.18304i q^{53} +2.59476i q^{55} -4.19238 q^{59} -2.75025i q^{61} -18.6957i q^{65} +6.55273 q^{67} -11.1515i q^{71} -3.65100i q^{73} +(-1.05547 + 1.44489i) q^{77} +11.2344 q^{79} -9.22558 q^{83} +22.2344 q^{85} +1.79702 q^{89} +(7.60487 - 10.4107i) q^{91} +16.2083i q^{95} +3.00814i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 48 q^{25} - 24 q^{43} - 12 q^{49} + 24 q^{79} - 12 q^{91}+O(q^{100}) \) 48 * q + 48 * q^25 - 24 * q^43 - 12 * q^49 + 24 * q^79 - 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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