LMFDB - Embedded newform 4536.2.k.a.3401.7

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			3401.7
Character	$\chi$	$=$	4536.3401
Dual form			4536.2.k.a.3401.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.20581 q^{5} +(2.51783 + 0.812729i) q^{7} +O(q^{10})$	$q-3.20581 q^{5} +(2.51783 + 0.812729i) q^{7} +4.77798i q^{11} +0.995090i q^{13} -6.51262 q^{17} -5.38881i q^{19} -7.58552i q^{23} +5.27720 q^{25} -3.39095i q^{29} -4.05595i q^{31} +(-8.07168 - 2.60545i) q^{35} +6.29763 q^{37} +6.97794 q^{41} -7.62728 q^{43} -7.56195 q^{47} +(5.67894 + 4.09263i) q^{49} +7.77397i q^{53} -15.3173i q^{55} +4.51260 q^{59} +6.07747i q^{61} -3.19007i q^{65} -0.986391 q^{67} +3.20090i q^{71} -2.63782i q^{73} +(-3.88321 + 12.0302i) q^{77} +15.9277 q^{79} +8.98141 q^{83} +20.8782 q^{85} -4.42081 q^{89} +(-0.808738 + 2.50547i) q^{91} +17.2755i q^{95} -3.62259i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.20581			−1.43368			−0.716840	−	0.697237i	$-0.754412\pi$
$5$	−3.20581			−1.43368			−0.716840	+	0.697237i	$0.754412\pi$
$6$		0			0
$7$	2.51783	+	0.812729i	0.951651	+	0.307183i
$7$	2.51783	+	0.812729i	0.951651	+	0.307183i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.77798i			1.44062i	0.693654	+	0.720308i	$0.256000\pi$
$11$			4.77798i			1.44062i	−0.693654	+	0.720308i	$0.744000\pi$
$12$		0			0
$13$			0.995090i			0.275988i	0.990433	+	0.137994i	$0.0440655\pi$
$13$			0.995090i			0.275988i	−0.990433	+	0.137994i	$0.955934\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.51262			−1.57954			−0.789771	−	0.613401i	$-0.789801\pi$
$17$	−6.51262			−1.57954			−0.789771	+	0.613401i	$0.789801\pi$
$18$		0			0
$19$		−	5.38881i		−	1.23628i	−0.786069	−	0.618138i	$-0.787887\pi$
$19$		−	5.38881i		−	1.23628i	0.786069	−	0.618138i	$-0.212113\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.58552i		−	1.58169i	−0.612016	−	0.790846i	$-0.709641\pi$
$23$		−	7.58552i		−	1.58169i	0.612016	−	0.790846i	$-0.290359\pi$
$24$		0			0
$25$	5.27720			1.05544
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.39095i		−	0.629684i	−0.949144	−	0.314842i	$-0.898048\pi$
$29$		−	3.39095i		−	0.629684i	0.949144	−	0.314842i	$-0.101952\pi$
$30$		0			0
$31$		−	4.05595i		−	0.728470i	−0.931307	−	0.364235i	$-0.881330\pi$
$31$		−	4.05595i		−	0.728470i	0.931307	−	0.364235i	$-0.118670\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.07168	−	2.60545i	−1.36436	−	0.440402i
$36$		0			0
$37$	6.29763			1.03532			0.517662	−	0.855585i	$-0.326802\pi$
$37$	6.29763			1.03532			0.517662	+	0.855585i	$0.326802\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.97794			1.08977			0.544885	−	0.838510i	$-0.316573\pi$
$41$	6.97794			1.08977			0.544885	+	0.838510i	$0.316573\pi$
$42$		0			0
$43$	−7.62728			−1.16315			−0.581575	−	0.813493i	$-0.697563\pi$
$43$	−7.62728			−1.16315			−0.581575	+	0.813493i	$0.697563\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.56195			−1.10302			−0.551512	−	0.834167i	$-0.685949\pi$
$47$	−7.56195			−1.10302			−0.551512	+	0.834167i	$0.685949\pi$
$48$		0			0
$49$	5.67894	+	4.09263i	0.811278	+	0.584661i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.77397i			1.06784i	0.845536	+	0.533919i	$0.179281\pi$
$53$			7.77397i			1.06784i	−0.845536	+	0.533919i	$0.820719\pi$
$54$		0			0
$55$		−	15.3173i		−	2.06538i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.51260			0.587490			0.293745	−	0.955884i	$-0.405098\pi$
$59$	4.51260			0.587490			0.293745	+	0.955884i	$0.405098\pi$
$60$		0			0
$61$			6.07747i			0.778140i	0.921208	+	0.389070i	$0.127204\pi$
$61$			6.07747i			0.778140i	−0.921208	+	0.389070i	$0.872796\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	3.19007i		−	0.395679i
$66$		0			0
$67$	−0.986391			−0.120507			−0.0602534	−	0.998183i	$-0.519191\pi$
$67$	−0.986391			−0.120507			−0.0602534	+	0.998183i	$0.519191\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.20090i			0.379877i	0.981796	+	0.189939i	$0.0608289\pi$
$71$			3.20090i			0.379877i	−0.981796	+	0.189939i	$0.939171\pi$
$72$		0			0
$73$		−	2.63782i		−	0.308733i	−0.988014	−	0.154367i	$-0.950666\pi$
$73$		−	2.63782i		−	0.308733i	0.988014	−	0.154367i	$-0.0493337\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.88321	+	12.0302i	−0.442532	+	1.37096i
$78$		0			0
$79$	15.9277			1.79201			0.896005	−	0.444045i	$-0.146457\pi$
$79$	15.9277			1.79201			0.896005	+	0.444045i	$0.146457\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.98141			0.985838			0.492919	−	0.870075i	$-0.335930\pi$
$83$	8.98141			0.985838			0.492919	+	0.870075i	$0.335930\pi$
$84$		0			0
$85$	20.8782			2.26456
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.42081			−0.468605			−0.234303	−	0.972164i	$-0.575281\pi$
$89$	−4.42081			−0.468605			−0.234303	+	0.972164i	$0.575281\pi$
$90$		0			0
$91$	−0.808738	+	2.50547i	−0.0847788	+	0.262644i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			17.2755i			1.77243i
$96$		0			0
$97$		−	3.62259i		−	0.367818i	−0.982943	−	0.183909i	$-0.941125\pi$
$97$		−	3.62259i		−	0.367818i	0.982943	−	0.183909i	$-0.0588753\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.6217			1.45491			0.727456	−	0.686155i	$-0.240703\pi$
$101$	14.6217			1.45491			0.727456	+	0.686155i	$0.240703\pi$
$102$		0			0
$103$			1.01982i			0.100485i	0.998737	+	0.0502427i	$0.0159995\pi$
$103$			1.01982i			0.100485i	−0.998737	+	0.0502427i	$0.984001\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			11.0089i			1.06428i	0.846658	+	0.532138i	$0.178611\pi$
$107$			11.0089i			1.06428i	−0.846658	+	0.532138i	$0.821389\pi$
$108$		0			0
$109$	−7.38735			−0.707579			−0.353790	−	0.935325i	$-0.615107\pi$
$109$	−7.38735			−0.707579			−0.353790	+	0.935325i	$0.615107\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.13218i		−	0.576867i	−0.957500	−	0.288433i	$-0.906866\pi$
$113$		−	6.13218i		−	0.576867i	0.957500	−	0.288433i	$-0.0931343\pi$
$114$		0			0
$115$			24.3177i			2.26764i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.3977	−	5.29300i	−1.50317	−	0.485208i
$120$		0			0
$121$	−11.8291			−1.07538
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.888649			−0.0794832
$126$		0			0
$127$	9.47289			0.840583			0.420291	−	0.907389i	$-0.361928\pi$
$127$	9.47289			0.840583			0.420291	+	0.907389i	$0.361928\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	15.1984			1.32789			0.663946	−	0.747781i	$-0.268880\pi$
$131$	15.1984			1.32789			0.663946	+	0.747781i	$0.268880\pi$
$132$		0			0
$133$	4.37964	−	13.5681i	0.379763	−	1.17650i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2.79570i		−	0.238853i	−0.992843	−	0.119427i	$-0.961894\pi$
$137$		−	2.79570i		−	0.238853i	0.992843	−	0.119427i	$-0.0381056\pi$
$138$		0			0
$139$		−	9.87941i		−	0.837961i	−0.907995	−	0.418980i	$-0.862388\pi$
$139$		−	9.87941i		−	0.837961i	0.907995	−	0.418980i	$-0.137612\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.75452			−0.397593
$144$		0			0
$145$			10.8707i			0.902766i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	23.7386i		−	1.94474i	−0.233448	−	0.972369i	$-0.575001\pi$
$149$		−	23.7386i		−	1.94474i	0.233448	−	0.972369i	$-0.424999\pi$
$150$		0			0
$151$	13.0575			1.06260			0.531302	−	0.847182i	$-0.321703\pi$
$151$	13.0575			1.06260			0.531302	+	0.847182i	$0.321703\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			13.0026i			1.04439i
$156$		0			0
$157$		−	0.309796i		−	0.0247244i	−0.999924	−	0.0123622i	$-0.996065\pi$
$157$		−	0.309796i		−	0.0247244i	0.999924	−	0.0123622i	$-0.00393511\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.16498	−	19.0991i	0.485868	−	1.50522i
$162$		0			0
$163$	−10.5524			−0.826525			−0.413262	−	0.910612i	$-0.635611\pi$
$163$	−10.5524			−0.826525			−0.413262	+	0.910612i	$0.635611\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.91309			0.612334			0.306167	−	0.951978i	$-0.400953\pi$
$167$	7.91309			0.612334			0.306167	+	0.951978i	$0.400953\pi$
$168$		0			0
$169$	12.0098			0.923830
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.86374			−0.141697			−0.0708487	−	0.997487i	$-0.522571\pi$
$173$	−1.86374			−0.141697			−0.0708487	+	0.997487i	$0.522571\pi$
$174$		0			0
$175$	13.2871	+	4.28893i	1.00441	+	0.324213i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	5.50222i		−	0.411255i	−0.978630	−	0.205628i	$-0.934076\pi$
$179$		−	5.50222i		−	0.411255i	0.978630	−	0.205628i	$-0.0659236\pi$
$180$		0			0
$181$			11.1382i			0.827900i	0.910300	+	0.413950i	$0.135851\pi$
$181$			11.1382i			0.827900i	−0.910300	+	0.413950i	$0.864149\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−20.1890			−1.48432
$186$		0			0
$187$		−	31.1172i		−	2.27552i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	4.20929i		−	0.304573i	−0.988336	−	0.152287i	$-0.951336\pi$
$191$		−	4.20929i		−	0.304573i	0.988336	−	0.152287i	$-0.0486637\pi$
$192$		0			0
$193$	−2.86218			−0.206024			−0.103012	−	0.994680i	$-0.532848\pi$
$193$	−2.86218			−0.206024			−0.103012	+	0.994680i	$0.532848\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.46947i		−	0.104695i	−0.998629	−	0.0523475i	$-0.983330\pi$
$197$		−	1.46947i		−	0.104695i	0.998629	−	0.0523475i	$-0.0166704\pi$
$198$		0			0
$199$		−	16.3790i		−	1.16108i	−0.814232	−	0.580539i	$-0.802842\pi$
$199$		−	16.3790i		−	1.16108i	0.814232	−	0.580539i	$-0.197158\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.75593	−	8.53784i	0.193428	−	0.599239i
$204$		0			0
$205$	−22.3699			−1.56238
$206$		0			0
$207$		0			0
$208$		0			0
$209$	25.7476			1.78100
$210$		0			0
$211$	10.3659			0.713620			0.356810	−	0.934177i	$-0.383864\pi$
$211$	10.3659			0.713620			0.356810	+	0.934177i	$0.383864\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	24.4516			1.66759
$216$		0			0
$217$	3.29639	−	10.2122i	0.223773	−	0.693249i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	6.48064i		−	0.435935i
$222$		0			0
$223$			5.63023i			0.377028i	0.982071	+	0.188514i	$0.0603671\pi$
$223$			5.63023i			0.377028i	−0.982071	+	0.188514i	$0.939633\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.05428			0.468209			0.234105	−	0.972211i	$-0.424784\pi$
$227$	7.05428			0.468209			0.234105	+	0.972211i	$0.424784\pi$
$228$		0			0
$229$			21.4479i			1.41731i	0.705553	+	0.708657i	$0.250699\pi$
$229$			21.4479i			1.41731i	−0.705553	+	0.708657i	$0.749301\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	11.6101i		−	0.760601i	−0.924863	−	0.380300i	$-0.875821\pi$
$233$		−	11.6101i		−	0.760601i	0.924863	−	0.380300i	$-0.124179\pi$
$234$		0			0
$235$	24.2422			1.58138
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.76161i		−	0.437372i	−0.975795	−	0.218686i	$-0.929823\pi$
$239$		−	6.76161i		−	0.437372i	0.975795	−	0.218686i	$-0.0701771\pi$
$240$		0			0
$241$		−	3.00503i		−	0.193571i	−0.995305	−	0.0967854i	$-0.969144\pi$
$241$		−	3.00503i		−	0.193571i	0.995305	−	0.0967854i	$-0.0308561\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−18.2056	−	13.1202i	−1.16311	−	0.838217i
$246$		0			0
$247$	5.36235			0.341198
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.9921			1.76685			0.883424	−	0.468575i	$-0.155232\pi$
$251$	27.9921			1.76685			0.883424	+	0.468575i	$0.155232\pi$
$252$		0			0
$253$	36.2435			2.27861
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.35385			0.146829			0.0734147	−	0.997302i	$-0.476610\pi$
$257$	2.35385			0.146829			0.0734147	+	0.997302i	$0.476610\pi$
$258$		0			0
$259$	15.8564	+	5.11827i	0.985267	+	0.318034i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			13.4832i			0.831409i	0.909500	+	0.415704i	$0.136465\pi$
$263$			13.4832i			0.831409i	−0.909500	+	0.415704i	$0.863535\pi$
$264$		0			0
$265$		−	24.9219i		−	1.53094i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.0797			−0.675543			−0.337772	−	0.941228i	$-0.609673\pi$
$269$	−11.0797			−0.675543			−0.337772	+	0.941228i	$0.609673\pi$
$270$		0			0
$271$		−	4.31052i		−	0.261845i	−0.991393	−	0.130923i	$-0.958206\pi$
$271$		−	4.31052i		−	0.261845i	0.991393	−	0.130923i	$-0.0417940\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			25.2144i			1.52048i
$276$		0			0
$277$	8.41822			0.505802			0.252901	−	0.967492i	$-0.418615\pi$
$277$	8.41822			0.505802			0.252901	+	0.967492i	$0.418615\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.6638i		−	1.11339i	−0.830718	−	0.556694i	$-0.812070\pi$
$281$		−	18.6638i		−	1.11339i	0.830718	−	0.556694i	$-0.187930\pi$
$282$		0			0
$283$		−	27.3708i		−	1.62702i	−0.581548	−	0.813512i	$-0.697553\pi$
$283$		−	27.3708i		−	1.62702i	0.581548	−	0.813512i	$-0.302447\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	17.5693	+	5.67117i	1.03708	+	0.334759i
$288$		0			0
$289$	25.4143			1.49496
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.32421			0.544726			0.272363	−	0.962195i	$-0.412195\pi$
$293$	9.32421			0.544726			0.272363	+	0.962195i	$0.412195\pi$
$294$		0			0
$295$	−14.4665			−0.842273
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.54828			0.436528
$300$		0			0
$301$	−19.2042	−	6.19891i	−1.10691	−	0.357299i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	19.4832i		−	1.11560i
$306$		0			0
$307$			33.6506i			1.92054i	0.279069	+	0.960271i	$0.409974\pi$
$307$			33.6506i			1.92054i	−0.279069	+	0.960271i	$0.590026\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.3903			−0.872703			−0.436351	−	0.899776i	$-0.643730\pi$
$311$	−15.3903			−0.872703			−0.436351	+	0.899776i	$0.643730\pi$
$312$		0			0
$313$		−	31.8730i		−	1.80157i	−0.434268	−	0.900784i	$-0.642993\pi$
$313$		−	31.8730i		−	1.80157i	0.434268	−	0.900784i	$-0.357007\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			19.9809i			1.12224i	0.827735	+	0.561119i	$0.189629\pi$
$317$			19.9809i			1.12224i	−0.827735	+	0.561119i	$0.810371\pi$
$318$		0			0
$319$	16.2019			0.907133
$320$		0			0
$321$		0			0
$322$		0			0
$323$			35.0953i			1.95275i
$324$		0			0
$325$			5.25129i			0.291289i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−19.0397	−	6.14582i	−1.04969	−	0.338830i
$330$		0			0
$331$	−8.96449			−0.492733			−0.246367	−	0.969177i	$-0.579237\pi$
$331$	−8.96449			−0.492733			−0.246367	+	0.969177i	$0.579237\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.16218			0.172768
$336$		0			0
$337$	5.44043			0.296359			0.148180	−	0.988960i	$-0.452659\pi$
$337$	5.44043			0.296359			0.148180	+	0.988960i	$0.452659\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	19.3793			1.04945
$342$		0			0
$343$	10.9724	+	14.9200i	0.592455	+	0.805604i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			16.1928i			0.869274i	0.900606	+	0.434637i	$0.143123\pi$
$347$			16.1928i			0.869274i	−0.900606	+	0.434637i	$0.856877\pi$
$348$		0			0
$349$			19.7536i			1.05739i	0.848813	+	0.528694i	$0.177318\pi$
$349$			19.7536i			1.05739i	−0.848813	+	0.528694i	$0.822682\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−20.6225			−1.09762			−0.548811	−	0.835946i	$-0.684919\pi$
$353$	−20.6225			−1.09762			−0.548811	+	0.835946i	$0.684919\pi$
$354$		0			0
$355$		−	10.2615i		−	0.544623i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	15.2442i		−	0.804557i	−0.915517	−	0.402278i	$-0.868218\pi$
$359$		−	15.2442i		−	0.804557i	0.915517	−	0.402278i	$-0.131782\pi$
$360$		0			0
$361$	−10.0392			−0.528380
$362$		0			0
$363$		0			0
$364$		0			0
$365$			8.45633i			0.442625i
$366$		0			0
$367$		−	3.32220i		−	0.173417i	−0.996234	−	0.0867087i	$-0.972365\pi$
$367$		−	3.32220i		−	0.173417i	0.996234	−	0.0867087i	$-0.0276350\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.31813	+	19.5735i	−0.328021	+	1.01621i
$372$		0			0
$373$	23.3770			1.21042			0.605208	−	0.796067i	$-0.293090\pi$
$373$	23.3770			1.21042			0.605208	+	0.796067i	$0.293090\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.37430			0.173785
$378$		0			0
$379$	13.9325			0.715663			0.357832	−	0.933786i	$-0.383516\pi$
$379$	13.9325			0.715663			0.357832	+	0.933786i	$0.383516\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.86264			−0.401762			−0.200881	−	0.979616i	$-0.564380\pi$
$383$	−7.86264			−0.401762			−0.200881	+	0.979616i	$0.564380\pi$
$384$		0			0
$385$	12.4488	−	38.5664i	0.634450	−	1.96552i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	19.0766i		−	0.967221i	−0.875283	−	0.483611i	$-0.839325\pi$
$389$		−	19.0766i		−	0.967221i	0.875283	−	0.483611i	$-0.160675\pi$
$390$		0			0
$391$			49.4017i			2.49835i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−51.0612			−2.56917
$396$		0			0
$397$			13.8919i			0.697214i	0.937269	+	0.348607i	$0.113345\pi$
$397$			13.8919i			0.697214i	−0.937269	+	0.348607i	$0.886655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			23.8647i			1.19174i	0.803079	+	0.595872i	$0.203193\pi$
$401$			23.8647i			1.19174i	−0.803079	+	0.595872i	$0.796807\pi$
$402$		0			0
$403$	4.03603			0.201049
$404$		0			0
$405$		0			0
$406$		0			0
$407$			30.0900i			1.49151i
$408$		0			0
$409$		−	24.1776i		−	1.19551i	−0.801680	−	0.597753i	$-0.796060\pi$
$409$		−	24.1776i		−	1.19551i	0.801680	−	0.597753i	$-0.203940\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.3620	+	3.66752i	0.559085	+	0.180467i
$414$		0			0
$415$	−28.7927			−1.41338
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.6877			−0.961806			−0.480903	−	0.876774i	$-0.659691\pi$
$419$	−19.6877			−0.961806			−0.480903	+	0.876774i	$0.659691\pi$
$420$		0			0
$421$	37.1768			1.81189			0.905944	−	0.423399i	$-0.139163\pi$
$421$	37.1768			1.81189			0.905944	+	0.423399i	$0.139163\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−34.3684			−1.66711
$426$		0			0
$427$	−4.93933	+	15.3020i	−0.239031	+	0.740517i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			8.86241i			0.426887i	0.976955	+	0.213444i	$0.0684680\pi$
$431$			8.86241i			0.426887i	−0.976955	+	0.213444i	$0.931532\pi$
$432$		0			0
$433$		−	35.2461i		−	1.69382i	−0.531737	−	0.846910i	$-0.678460\pi$
$433$		−	35.2461i		−	1.69382i	0.531737	−	0.846910i	$-0.321540\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−40.8769			−1.95541
$438$		0			0
$439$			8.67591i			0.414079i	0.978333	+	0.207039i	$0.0663828\pi$
$439$			8.67591i			0.414079i	−0.978333	+	0.207039i	$0.933617\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	15.8230i		−	0.751771i	−0.926666	−	0.375886i	$-0.877339\pi$
$443$		−	15.8230i		−	0.751771i	0.926666	−	0.375886i	$-0.122661\pi$
$444$		0			0
$445$	14.1723			0.671830
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	7.87563i		−	0.371674i	−0.982581	−	0.185837i	$-0.940500\pi$
$449$		−	7.87563i		−	0.371674i	0.982581	−	0.185837i	$-0.0594996\pi$
$450$		0			0
$451$			33.3405i			1.56994i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.59266	−	8.03205i	0.121546	−	0.376548i
$456$		0			0
$457$	−11.0626			−0.517489			−0.258744	−	0.965946i	$-0.583309\pi$
$457$	−11.0626			−0.517489			−0.258744	+	0.965946i	$0.583309\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−13.4601			−0.626899			−0.313450	−	0.949605i	$-0.601485\pi$
$461$	−13.4601			−0.626899			−0.313450	+	0.949605i	$0.601485\pi$
$462$		0			0
$463$	−17.1421			−0.796659			−0.398329	−	0.917242i	$-0.630410\pi$
$463$	−17.1421			−0.796659			−0.398329	+	0.917242i	$0.630410\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.67554			0.262633			0.131316	−	0.991341i	$-0.458080\pi$
$467$	5.67554			0.262633			0.131316	+	0.991341i	$0.458080\pi$
$468$		0			0
$469$	−2.48357	−	0.801669i	−0.114680	−	0.0370176i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	36.4430i		−	1.67565i
$474$		0			0
$475$		−	28.4378i		−	1.30482i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.4295			0.613610			0.306805	−	0.951772i	$-0.400740\pi$
$479$	13.4295			0.613610			0.306805	+	0.951772i	$0.400740\pi$
$480$		0			0
$481$			6.26671i			0.285737i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			11.6133i			0.527334i
$486$		0			0
$487$	−11.4841			−0.520394			−0.260197	−	0.965556i	$-0.583788\pi$
$487$	−11.4841			−0.520394			−0.260197	+	0.965556i	$0.583788\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			10.4461i			0.471424i	0.971823	+	0.235712i	$0.0757422\pi$
$491$			10.4461i			0.471424i	−0.971823	+	0.235712i	$0.924258\pi$
$492$		0			0
$493$			22.0840i			0.994613i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.60147	+	8.05933i	−0.116692	+	0.361510i
$498$		0			0
$499$	−16.8722			−0.755303			−0.377651	−	0.925948i	$-0.623268\pi$
$499$	−16.8722			−0.755303			−0.377651	+	0.925948i	$0.623268\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.1129			0.941379			0.470690	−	0.882299i	$-0.344005\pi$
$503$	21.1129			0.941379			0.470690	+	0.882299i	$0.344005\pi$
$504$		0			0
$505$	−46.8743			−2.08588
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22.4463			0.994916			0.497458	−	0.867488i	$-0.334267\pi$
$509$	22.4463			0.994916			0.497458	+	0.867488i	$0.334267\pi$
$510$		0			0
$511$	2.14383	−	6.64158i	0.0948375	−	0.293806i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	3.26933i		−	0.144064i
$516$		0			0
$517$		−	36.1309i		−	1.58903i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.8533			−1.22027			−0.610137	−	0.792296i	$-0.708886\pi$
$521$	−27.8533			−1.22027			−0.610137	+	0.792296i	$0.708886\pi$
$522$		0			0
$523$		−	3.02647i		−	0.132338i	−0.997808	−	0.0661691i	$-0.978922\pi$
$523$		−	3.02647i		−	0.132338i	0.997808	−	0.0661691i	$-0.0210777\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			26.4149i			1.15065i
$528$		0			0
$529$	−34.5402			−1.50175
$530$		0			0
$531$		0			0
$532$		0			0
$533$			6.94368i			0.300764i
$534$		0			0
$535$		−	35.2926i		−	1.52583i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−19.5545	+	27.1339i	−0.842273	+	1.16874i
$540$		0			0
$541$	−43.4800			−1.86935			−0.934677	−	0.355499i	$-0.884311\pi$
$541$	−43.4800			−1.86935			−0.934677	+	0.355499i	$0.884311\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	23.6824			1.01444
$546$		0			0
$547$	40.6797			1.73934			0.869670	−	0.493634i	$-0.164332\pi$
$547$	40.6797			1.73934			0.869670	+	0.493634i	$0.164332\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.2732			−0.778464
$552$		0			0
$553$	40.1033	+	12.9449i	1.70537	+	0.550474i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	13.6889i		−	0.580016i	−0.957024	−	0.290008i	$-0.906342\pi$
$557$		−	13.6889i		−	0.580016i	0.957024	−	0.290008i	$-0.0936580\pi$
$558$		0			0
$559$		−	7.58983i		−	0.321016i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.4141			0.565337			0.282668	−	0.959218i	$-0.408780\pi$
$563$	13.4141			0.565337			0.282668	+	0.959218i	$0.408780\pi$
$564$		0			0
$565$			19.6586i			0.827042i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	22.6749i		−	0.950581i	−0.879829	−	0.475291i	$-0.842343\pi$
$569$		−	22.6749i		−	0.950581i	0.879829	−	0.475291i	$-0.157657\pi$
$570$		0			0
$571$	−15.5476			−0.650645			−0.325323	−	0.945603i	$-0.605473\pi$
$571$	−15.5476			−0.650645			−0.325323	+	0.945603i	$0.605473\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	40.0303i		−	1.66938i
$576$		0			0
$577$		−	41.8101i		−	1.74058i	−0.492541	−	0.870289i	$-0.663932\pi$
$577$		−	41.8101i		−	1.74058i	0.492541	−	0.870289i	$-0.336068\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	22.6137	+	7.29945i	0.938173	+	0.302832i
$582$		0			0
$583$	−37.1439			−1.53834
$584$		0			0
$585$		0			0
$586$		0			0
$587$	26.7820			1.10541			0.552706	−	0.833376i	$-0.313595\pi$
$587$	26.7820			1.10541			0.552706	+	0.833376i	$0.313595\pi$
$588$		0			0
$589$	−21.8567			−0.900591
$590$		0			0
$591$		0			0
$592$		0			0
$593$	28.3434			1.16392			0.581962	−	0.813216i	$-0.302285\pi$
$593$	28.3434			1.16392			0.581962	+	0.813216i	$0.302285\pi$
$594$		0			0
$595$	52.5678	+	16.9683i	2.15507	+	0.695634i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8.93490i			0.365070i	0.983199	+	0.182535i	$0.0584303\pi$
$599$			8.93490i			0.365070i	−0.983199	+	0.182535i	$0.941570\pi$
$600$		0			0
$601$			22.1801i			0.904746i	0.891829	+	0.452373i	$0.149422\pi$
$601$			22.1801i			0.904746i	−0.891829	+	0.452373i	$0.850578\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	37.9219			1.54175
$606$		0			0
$607$			35.0899i			1.42425i	0.702050	+	0.712127i	$0.252268\pi$
$607$			35.0899i			1.42425i	−0.702050	+	0.712127i	$0.747732\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	7.52482i		−	0.304422i
$612$		0			0
$613$	15.4883			0.625566			0.312783	−	0.949825i	$-0.398739\pi$
$613$	15.4883			0.625566			0.312783	+	0.949825i	$0.398739\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	42.5180i		−	1.71171i	−0.517215	−	0.855856i	$-0.673031\pi$
$617$		−	42.5180i		−	1.71171i	0.517215	−	0.855856i	$-0.326969\pi$
$618$		0			0
$619$			30.0113i			1.20626i	0.797644	+	0.603128i	$0.206079\pi$
$619$			30.0113i			1.20626i	−0.797644	+	0.603128i	$0.793921\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.1309	−	3.59292i	−0.445948	−	0.143947i
$624$		0			0
$625$	−23.5372			−0.941486
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−41.0141			−1.63534
$630$		0			0
$631$	20.6833			0.823388			0.411694	−	0.911322i	$-0.364937\pi$
$631$	20.6833			0.823388			0.411694	+	0.911322i	$0.364937\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−30.3682			−1.20513
$636$		0			0
$637$	−4.07253	+	5.65106i	−0.161360	+	0.223903i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	29.9642i		−	1.18352i	−0.806116	−	0.591758i	$-0.798434\pi$
$641$		−	29.9642i		−	1.18352i	0.806116	−	0.591758i	$-0.201566\pi$
$642$		0			0
$643$			33.2508i			1.31129i	0.755071	+	0.655643i	$0.227602\pi$
$643$			33.2508i			1.31129i	−0.755071	+	0.655643i	$0.772398\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.06135			−0.356238			−0.178119	−	0.984009i	$-0.557001\pi$
$647$	−9.06135			−0.356238			−0.178119	+	0.984009i	$0.557001\pi$
$648$		0			0
$649$			21.5611i			0.846348i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			12.0228i			0.470490i	0.971936	+	0.235245i	$0.0755893\pi$
$653$			12.0228i			0.470490i	−0.971936	+	0.235245i	$0.924411\pi$
$654$		0			0
$655$	−48.7232			−1.90377
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	47.5347i		−	1.85169i	−0.377903	−	0.925845i	$-0.623355\pi$
$659$		−	47.5347i		−	1.85169i	0.377903	−	0.925845i	$-0.376645\pi$
$660$		0			0
$661$			21.3913i			0.832027i	0.909359	+	0.416013i	$0.136573\pi$
$661$			21.3913i			0.832027i	−0.909359	+	0.416013i	$0.863427\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−14.0403	+	43.4967i	−0.544459	+	1.68673i
$666$		0			0
$667$	−25.7221			−0.995966
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−29.0380			−1.12100
$672$		0			0
$673$	−17.5515			−0.676562			−0.338281	−	0.941045i	$-0.609845\pi$
$673$	−17.5515			−0.676562			−0.338281	+	0.941045i	$0.609845\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.7972			−0.645571			−0.322785	−	0.946472i	$-0.604619\pi$
$677$	−16.7972			−0.645571			−0.322785	+	0.946472i	$0.604619\pi$
$678$		0			0
$679$	2.94419	−	9.12107i	0.112987	−	0.350035i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			17.4518i			0.667773i	0.942613	+	0.333887i	$0.108360\pi$
$683$			17.4518i			0.667773i	−0.942613	+	0.333887i	$0.891640\pi$
$684$		0			0
$685$			8.96248i			0.342439i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.73580			−0.294710
$690$		0			0
$691$		−	43.7474i		−	1.66423i	−0.554603	−	0.832115i	$-0.687130\pi$
$691$		−	43.7474i		−	1.66423i	0.554603	−	0.832115i	$-0.312870\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			31.6715i			1.20137i
$696$		0			0
$697$	−45.4447			−1.72134
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	11.5880i		−	0.437671i	−0.975762	−	0.218836i	$-0.929774\pi$
$701$		−	11.5880i		−	0.437671i	0.975762	−	0.218836i	$-0.0702258\pi$
$702$		0			0
$703$		−	33.9367i		−	1.27995i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	36.8149	+	11.8835i	1.38457	+	0.446924i
$708$		0			0
$709$	−3.39627			−0.127549			−0.0637747	−	0.997964i	$-0.520314\pi$
$709$	−3.39627			−0.127549			−0.0637747	+	0.997964i	$0.520314\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−30.7665			−1.15221
$714$		0			0
$715$	15.2421			0.570022
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.13894			0.0797690			0.0398845	−	0.999204i	$-0.487301\pi$
$719$	2.13894			0.0797690			0.0398845	+	0.999204i	$0.487301\pi$
$720$		0			0
$721$	−0.828834	+	2.56772i	−0.0308674	+	0.0956270i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	17.8947i		−	0.664594i
$726$		0			0
$727$			35.2699i			1.30809i	0.756456	+	0.654045i	$0.226929\pi$
$727$			35.2699i			1.30809i	−0.756456	+	0.654045i	$0.773071\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	49.6736			1.83724
$732$		0			0
$733$			11.7330i			0.433370i	0.976242	+	0.216685i	$0.0695244\pi$
$733$			11.7330i			0.433370i	−0.976242	+	0.216685i	$0.930476\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	4.71296i		−	0.173604i
$738$		0			0
$739$	−37.8023			−1.39058			−0.695289	−	0.718730i	$-0.744724\pi$
$739$	−37.8023			−1.39058			−0.695289	+	0.718730i	$0.744724\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	11.2461i		−	0.412580i	−0.978491	−	0.206290i	$-0.933861\pi$
$743$		−	11.2461i		−	0.412580i	0.978491	−	0.206290i	$-0.0661391\pi$
$744$		0			0
$745$			76.1012i			2.78813i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.94729	+	27.7187i	−0.326927	+	1.01282i
$750$		0			0
$751$	−26.1970			−0.955943			−0.477972	−	0.878375i	$-0.658628\pi$
$751$	−26.1970			−0.955943			−0.477972	+	0.878375i	$0.658628\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−41.8598			−1.52344
$756$		0			0
$757$	−0.529802			−0.0192560			−0.00962799	−	0.999954i	$-0.503065\pi$
$757$	−0.529802			−0.0192560			−0.00962799	+	0.999954i	$0.503065\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	48.5364			1.75944			0.879721	−	0.475490i	$-0.157729\pi$
$761$	48.5364			1.75944			0.879721	+	0.475490i	$0.157729\pi$
$762$		0			0
$763$	−18.6001	−	6.00391i	−0.673368	−	0.217356i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			4.49044i			0.162140i
$768$		0			0
$769$		−	29.2547i		−	1.05495i	−0.849570	−	0.527476i	$-0.823139\pi$
$769$		−	29.2547i		−	1.05495i	0.849570	−	0.527476i	$-0.176861\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	43.5569			1.56663			0.783317	−	0.621623i	$-0.213526\pi$
$773$	43.5569			1.56663			0.783317	+	0.621623i	$0.213526\pi$
$774$		0			0
$775$		−	21.4041i		−	0.768857i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	37.6028i		−	1.34726i
$780$		0			0
$781$	−15.2939			−0.547257
$782$		0			0
$783$		0			0
$784$		0			0
$785$			0.993147i			0.0354469i
$786$		0			0
$787$			9.53736i			0.339970i	0.985447	+	0.169985i	$0.0543719\pi$
$787$			9.53736i			0.339970i	−0.985447	+	0.169985i	$0.945628\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.98380	−	15.4398i	0.177203	−	0.548975i
$792$		0			0
$793$	−6.04762			−0.214757
$794$		0			0
$795$		0			0
$796$		0			0
$797$	40.9195			1.44944			0.724722	−	0.689041i	$-0.241968\pi$
$797$	40.9195			1.44944			0.724722	+	0.689041i	$0.241968\pi$
$798$		0			0
$799$	49.2481			1.74227
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.6034			0.444766
$804$		0			0
$805$	−19.7637	+	61.2279i	−0.696580	+	2.15800i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	23.7588i		−	0.835314i	−0.908605	−	0.417657i	$-0.862851\pi$
$809$		−	23.7588i		−	0.835314i	0.908605	−	0.417657i	$-0.137149\pi$
$810$		0			0
$811$		−	26.9638i		−	0.946828i	−0.880840	−	0.473414i	$-0.843021\pi$
$811$		−	26.9638i		−	0.946828i	0.880840	−	0.473414i	$-0.156979\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	33.8288			1.18497
$816$		0			0
$817$			41.1019i			1.43797i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			36.8272i			1.28528i	0.766169	+	0.642639i	$0.222160\pi$
$821$			36.8272i			1.28528i	−0.766169	+	0.642639i	$0.777840\pi$
$822$		0			0
$823$	5.97774			0.208371			0.104185	−	0.994558i	$-0.466776\pi$
$823$	5.97774			0.208371			0.104185	+	0.994558i	$0.466776\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			5.21366i			0.181297i	0.995883	+	0.0906484i	$0.0288939\pi$
$827$			5.21366i			0.181297i	−0.995883	+	0.0906484i	$0.971106\pi$
$828$		0			0
$829$		−	0.453044i		−	0.0157349i	−0.999969	−	0.00786743i	$-0.997496\pi$
$829$		−	0.453044i		−	0.0157349i	0.999969	−	0.00786743i	$-0.00250431\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−36.9848	−	26.6537i	−1.28145	−	0.923497i
$834$		0			0
$835$	−25.3679			−0.877891
$836$		0			0
$837$		0			0
$838$		0			0
$839$	51.3338			1.77224			0.886119	−	0.463458i	$-0.153391\pi$
$839$	51.3338			1.77224			0.886119	+	0.463458i	$0.153391\pi$
$840$		0			0
$841$	17.5014			0.603498
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−38.5011			−1.32448
$846$		0			0
$847$	−29.7838	−	9.61388i	−1.02338	−	0.330337i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	47.7708i		−	1.63756i
$852$		0			0
$853$		−	15.9012i		−	0.544446i	−0.962234	−	0.272223i	$-0.912241\pi$
$853$		−	15.9012i		−	0.544446i	0.962234	−	0.272223i	$-0.0877588\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.6315			0.499802			0.249901	−	0.968271i	$-0.419602\pi$
$857$	14.6315			0.499802			0.249901	+	0.968271i	$0.419602\pi$
$858$		0			0
$859$		−	50.8206i		−	1.73398i	−0.498329	−	0.866988i	$-0.666053\pi$
$859$		−	50.8206i		−	1.73398i	0.498329	−	0.866988i	$-0.333947\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			3.00667i			0.102348i	0.998690	+	0.0511742i	$0.0162964\pi$
$863$			3.00667i			0.102348i	−0.998690	+	0.0511742i	$0.983704\pi$
$864$		0			0
$865$	5.97479			0.203149
$866$		0			0
$867$		0			0
$868$		0			0
$869$			76.1024i			2.58160i
$870$		0			0
$871$		−	0.981548i		−	0.0332585i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.23747	−	0.722231i	−0.0756402	−	0.0244159i
$876$		0			0
$877$	−42.2459			−1.42654			−0.713271	−	0.700889i	$-0.752787\pi$
$877$	−42.2459			−1.42654			−0.713271	+	0.700889i	$0.752787\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−29.4719			−0.992933			−0.496467	−	0.868056i	$-0.665370\pi$
$881$	−29.4719			−0.992933			−0.496467	+	0.868056i	$0.665370\pi$
$882$		0			0
$883$	−27.1571			−0.913910			−0.456955	−	0.889490i	$-0.651060\pi$
$883$	−27.1571			−0.913910			−0.456955	+	0.889490i	$0.651060\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.0378			0.370614			0.185307	−	0.982681i	$-0.440672\pi$
$887$	11.0378			0.370614			0.185307	+	0.982681i	$0.440672\pi$
$888$		0			0
$889$	23.8511	+	7.69889i	0.799941	+	0.258213i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			40.7499i			1.36364i
$894$		0			0
$895$			17.6391i			0.589609i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−13.7535			−0.458706
$900$		0			0
$901$		−	50.6289i		−	1.68669i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	35.7071i		−	1.18694i
$906$		0			0
$907$	2.33130			0.0774095			0.0387047	−	0.999251i	$-0.487677\pi$
$907$	2.33130			0.0774095			0.0387047	+	0.999251i	$0.487677\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	2.34036i		−	0.0775396i	−0.999248	−	0.0387698i	$-0.987656\pi$
$911$		−	2.34036i		−	0.0775396i	0.999248	−	0.0387698i	$-0.0123439\pi$
$912$		0			0
$913$			42.9130i			1.42021i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	38.2670	+	12.3522i	1.26369	+	0.407905i
$918$		0			0
$919$	3.54891			0.117068			0.0585339	−	0.998285i	$-0.481357\pi$
$919$	3.54891			0.117068			0.0585339	+	0.998285i	$0.481357\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.18519			−0.104842
$924$		0			0
$925$	33.2339			1.09272
$926$		0			0
$927$		0			0
$928$		0			0
$929$	41.3398			1.35631			0.678157	−	0.734917i	$-0.262779\pi$
$929$	41.3398			1.35631			0.678157	+	0.734917i	$0.262779\pi$
$930$		0			0
$931$	22.0544	−	30.6027i	0.722803	−	1.00296i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			99.7558i			3.26236i
$936$		0			0
$937$		−	1.12314i		−	0.0366913i	−0.999832	−	0.0183457i	$-0.994160\pi$
$937$		−	1.12314i		−	0.0366913i	0.999832	−	0.0183457i	$-0.00583993\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	40.7911			1.32975			0.664875	−	0.746954i	$-0.268485\pi$
$941$	40.7911			1.32975			0.664875	+	0.746954i	$0.268485\pi$
$942$		0			0
$943$		−	52.9313i		−	1.72368i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.7741i		−	0.675068i	−0.941313	−	0.337534i	$-0.890407\pi$
$947$		−	20.7741i		−	0.675068i	0.941313	−	0.337534i	$-0.109593\pi$
$948$		0			0
$949$	2.62486			0.0852067
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	50.6468i		−	1.64061i	−0.571926	−	0.820305i	$-0.693804\pi$
$953$		−	50.6468i		−	1.64061i	0.571926	−	0.820305i	$-0.306196\pi$
$954$		0			0
$955$			13.4942i			0.436661i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.27215	−	7.03911i	0.0733715	−	0.227305i
$960$		0			0
$961$	14.5493			0.469331
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.17559			0.295373
$966$		0			0
$967$	22.3111			0.717478			0.358739	−	0.933438i	$-0.383207\pi$
$967$	22.3111			0.717478			0.358739	+	0.933438i	$0.383207\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.74260			−0.248472			−0.124236	−	0.992253i	$-0.539648\pi$
$971$	−7.74260			−0.248472			−0.124236	+	0.992253i	$0.539648\pi$
$972$		0			0
$973$	8.02929	−	24.8747i	0.257407	−	0.797446i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.04243i			0.129329i	0.997907	+	0.0646644i	$0.0205977\pi$
$977$			4.04243i			0.129329i	−0.997907	+	0.0646644i	$0.979402\pi$
$978$		0			0
$979$		−	21.1226i		−	0.675080i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−33.0521			−1.05420			−0.527100	−	0.849804i	$-0.676720\pi$
$983$	−33.0521			−1.05420			−0.527100	+	0.849804i	$0.676720\pi$
$984$		0			0
$985$			4.71082i			0.150099i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			57.8569i			1.83974i
$990$		0			0
$991$	−45.7928			−1.45465			−0.727327	−	0.686291i	$-0.759238\pi$
$991$	−45.7928			−1.45465			−0.727327	+	0.686291i	$0.759238\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			52.5080i			1.66462i
$996$		0			0
$997$		−	49.8610i		−	1.57912i	−0.613676	−	0.789558i	$-0.710310\pi$
$997$		−	49.8610i		−	1.57912i	0.613676	−	0.789558i	$-0.289690\pi$
$998$		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.3401.7		48
3.2	odd	2	inner	4536.2.k.a.3401.42		48
7.6	odd	2	inner	4536.2.k.a.3401.41		48
9.2	odd	6		1512.2.bu.a.881.4		48
9.4	even	3		1512.2.bu.a.1385.21		48
9.5	odd	6		504.2.bu.a.209.8	yes	48
9.7	even	3		504.2.bu.a.41.17	yes	48
21.20	even	2	inner	4536.2.k.a.3401.8		48
36.7	odd	6		1008.2.cc.d.545.8		48
36.11	even	6		3024.2.cc.d.881.4		48
36.23	even	6		1008.2.cc.d.209.17		48
36.31	odd	6		3024.2.cc.d.2897.21		48
63.13	odd	6		1512.2.bu.a.1385.4		48
63.20	even	6		1512.2.bu.a.881.21		48
63.34	odd	6		504.2.bu.a.41.8	✓	48
63.41	even	6		504.2.bu.a.209.17	yes	48
252.83	odd	6		3024.2.cc.d.881.21		48
252.139	even	6		3024.2.cc.d.2897.4		48
252.167	odd	6		1008.2.cc.d.209.8		48
252.223	even	6		1008.2.cc.d.545.17		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bu.a.41.8	✓	48	63.34	odd	6
504.2.bu.a.41.17	yes	48	9.7	even	3
504.2.bu.a.209.8	yes	48	9.5	odd	6
504.2.bu.a.209.17	yes	48	63.41	even	6
1008.2.cc.d.209.8		48	252.167	odd	6
1008.2.cc.d.209.17		48	36.23	even	6
1008.2.cc.d.545.8		48	36.7	odd	6
1008.2.cc.d.545.17		48	252.223	even	6
1512.2.bu.a.881.4		48	9.2	odd	6
1512.2.bu.a.881.21		48	63.20	even	6
1512.2.bu.a.1385.4		48	63.13	odd	6
1512.2.bu.a.1385.21		48	9.4	even	3
3024.2.cc.d.881.4		48	36.11	even	6
3024.2.cc.d.881.21		48	252.83	odd	6
3024.2.cc.d.2897.4		48	252.139	even	6
3024.2.cc.d.2897.21		48	36.31	odd	6
4536.2.k.a.3401.7		48	1.1	even	1	trivial
4536.2.k.a.3401.8		48	21.20	even	2	inner
4536.2.k.a.3401.41		48	7.6	odd	2	inner
4536.2.k.a.3401.42		48	3.2	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)

\(f(q)\)	\(=\)	\(q-3.20581 q^{5} +(2.51783 + 0.812729i) q^{7} +O(q^{10})\)	\(q-3.20581 q^{5} +(2.51783 + 0.812729i) q^{7} +4.77798i q^{11} +0.995090i q^{13} -6.51262 q^{17} -5.38881i q^{19} -7.58552i q^{23} +5.27720 q^{25} -3.39095i q^{29} -4.05595i q^{31} +(-8.07168 - 2.60545i) q^{35} +6.29763 q^{37} +6.97794 q^{41} -7.62728 q^{43} -7.56195 q^{47} +(5.67894 + 4.09263i) q^{49} +7.77397i q^{53} -15.3173i q^{55} +4.51260 q^{59} +6.07747i q^{61} -3.19007i q^{65} -0.986391 q^{67} +3.20090i q^{71} -2.63782i q^{73} +(-3.88321 + 12.0302i) q^{77} +15.9277 q^{79} +8.98141 q^{83} +20.8782 q^{85} -4.42081 q^{89} +(-0.808738 + 2.50547i) q^{91} +17.2755i q^{95} -3.62259i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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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