LMFDB - Embedded newform 5292.2.i.j.2125.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1549,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 2]))

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

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

chi := DirichletCharacter("5292.1549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.i (of order $3$, 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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 1764)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2125.7
Character	$\chi$	$=$	5292.2125
Dual form			5292.2.i.j.1549.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0111913 - 0.0193839i) q^{5} +O(q^{10})$	$q+(0.0111913 - 0.0193839i) q^{5} +(-0.280281 - 0.485460i) q^{11} +(2.06442 + 3.57567i) q^{13} +(1.29152 - 2.23698i) q^{17} +(-1.69669 - 2.93876i) q^{19} +(2.24626 - 3.89064i) q^{23} +(2.49975 + 4.32969i) q^{25} +(-2.57275 + 4.45613i) q^{29} +0.815136 q^{31} +(-0.235546 - 0.407978i) q^{37} +(-3.02774 - 5.24420i) q^{41} +(0.811935 - 1.40631i) q^{43} +12.1108 q^{47} +(-1.45922 + 2.52745i) q^{53} -0.0125468 q^{55} -8.95882 q^{59} +2.11385 q^{61} +0.0924141 q^{65} +15.0909 q^{67} -2.16045 q^{71} +(-3.55205 + 6.15233i) q^{73} +2.31781 q^{79} +(3.83160 - 6.63653i) q^{83} +(-0.0289077 - 0.0500695i) q^{85} +(-0.502365 - 0.870122i) q^{89} -0.0759530 q^{95} +(-7.32986 + 12.6957i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} - 72 q^{65} - 24 q^{67} - 48 q^{71} - 24 q^{79} + 12 q^{85} + 64 q^{95}+O(q^{100}) $ 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 - 72 * q^65 - 24 * q^67 - 48 * q^71 - 24 * q^79 + 12 * q^85 + 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.0111913	−	0.0193839i	0.00500491	−	0.00866875i	−0.863512	−	0.504328i	$-0.831740\pi$
$5$	0.0111913	−	0.0193839i	0.00500491	−	0.00866875i	0.868517	+	0.495659i	$0.165074\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.280281	−	0.485460i	−0.0845078	−	0.146372i	0.820674	−	0.571397i	$-0.193598\pi$
$11$	−0.280281	−	0.485460i	−0.0845078	−	0.146372i	−0.905181	+	0.425025i	$0.860265\pi$
$12$		0			0
$13$	2.06442	+	3.57567i	0.572566	+	0.991713i	0.996301	+	0.0859272i	$0.0273852\pi$
$13$	2.06442	+	3.57567i	0.572566	+	0.991713i	−0.423736	+	0.905786i	$0.639281\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.29152	−	2.23698i	0.313240	−	0.542548i	−0.665822	−	0.746111i	$-0.731919\pi$
$17$	1.29152	−	2.23698i	0.313240	−	0.542548i	0.979062	+	0.203563i	$0.0652522\pi$
$18$		0			0
$19$	−1.69669	−	2.93876i	−0.389248	−	0.674198i	0.603100	−	0.797665i	$-0.293932\pi$
$19$	−1.69669	−	2.93876i	−0.389248	−	0.674198i	−0.992349	+	0.123467i	$0.960599\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.24626	−	3.89064i	0.468378	−	0.811255i	−0.530969	−	0.847391i	$-0.678172\pi$
$23$	2.24626	−	3.89064i	0.468378	−	0.811255i	0.999347	+	0.0361367i	$0.0115052\pi$
$24$		0			0
$25$	2.49975	+	4.32969i	0.499950	+	0.865939i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.57275	+	4.45613i	−0.477748	+	0.827483i	−0.999675	−	0.0255069i	$-0.991880\pi$
$29$	−2.57275	+	4.45613i	−0.477748	+	0.827483i	0.521927	+	0.852990i	$0.325213\pi$
$30$		0			0
$31$	0.815136			0.146403			0.0732014	−	0.997317i	$-0.476678\pi$
$31$	0.815136			0.146403			0.0732014	+	0.997317i	$0.476678\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.235546	−	0.407978i	−0.0387236	−	0.0670712i	0.846014	−	0.533161i	$-0.178996\pi$
$37$	−0.235546	−	0.407978i	−0.0387236	−	0.0670712i	−0.884738	+	0.466089i	$0.845663\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.02774	−	5.24420i	−0.472854	−	0.819007i	0.526663	−	0.850074i	$-0.323443\pi$
$41$	−3.02774	−	5.24420i	−0.472854	−	0.819007i	−0.999517	+	0.0310669i	$0.990110\pi$
$42$		0			0
$43$	0.811935	−	1.40631i	0.123819	−	0.214461i	−0.797452	−	0.603383i	$-0.793819\pi$
$43$	0.811935	−	1.40631i	0.123819	−	0.214461i	0.921271	+	0.388922i	$0.127152\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	12.1108			1.76654			0.883271	−	0.468863i	$-0.155336\pi$
$47$	12.1108			1.76654			0.883271	+	0.468863i	$0.155336\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.45922	+	2.52745i	−0.200440	+	0.347172i	−0.948670	−	0.316267i	$-0.897570\pi$
$53$	−1.45922	+	2.52745i	−0.200440	+	0.347172i	0.748230	+	0.663439i	$0.230904\pi$
$54$		0			0
$55$	−0.0125468			−0.00169182
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.95882			−1.16634			−0.583169	−	0.812351i	$-0.698188\pi$
$59$	−8.95882			−1.16634			−0.583169	+	0.812351i	$0.698188\pi$
$60$		0			0
$61$	2.11385			0.270650			0.135325	−	0.990801i	$-0.456792\pi$
$61$	2.11385			0.270650			0.135325	+	0.990801i	$0.456792\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.0924141			0.0114626
$66$		0			0
$67$	15.0909			1.84364			0.921822	−	0.387614i	$-0.126701\pi$
$67$	15.0909			1.84364			0.921822	+	0.387614i	$0.126701\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.16045			−0.256398			−0.128199	−	0.991748i	$-0.540920\pi$
$71$	−2.16045			−0.256398			−0.128199	+	0.991748i	$0.540920\pi$
$72$		0			0
$73$	−3.55205	+	6.15233i	−0.415736	+	0.720076i	−0.995505	−	0.0947047i	$-0.969809\pi$
$73$	−3.55205	+	6.15233i	−0.415736	+	0.720076i	0.579769	+	0.814781i	$0.303143\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.31781			0.260775			0.130387	−	0.991463i	$-0.458378\pi$
$79$	2.31781			0.260775			0.130387	+	0.991463i	$0.458378\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.83160	−	6.63653i	0.420573	−	0.728454i	−0.575423	−	0.817856i	$-0.695162\pi$
$83$	3.83160	−	6.63653i	0.420573	−	0.728454i	0.995996	+	0.0894025i	$0.0284958\pi$
$84$		0			0
$85$	−0.0289077	−	0.0500695i	−0.00313548	−	0.00543080i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.502365	−	0.870122i	−0.0532506	−	0.0922327i	0.838171	−	0.545407i	$-0.183625\pi$
$89$	−0.502365	−	0.870122i	−0.0532506	−	0.0922327i	−0.891422	+	0.453174i	$0.850292\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.0759530			−0.00779261
$96$		0			0
$97$	−7.32986	+	12.6957i	−0.744235	+	1.28905i	0.206316	+	0.978485i	$0.433852\pi$
$97$	−7.32986	+	12.6957i	−0.744235	+	1.28905i	−0.950551	+	0.310567i	$0.899481\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.57250	−	13.1160i	−0.753492	−	1.30509i	−0.946121	−	0.323814i	$-0.895035\pi$
$101$	−7.57250	−	13.1160i	−0.753492	−	1.30509i	0.192629	−	0.981272i	$-0.438299\pi$
$102$		0			0
$103$	4.45031	−	7.70817i	0.438502	−	0.759508i	−0.559072	−	0.829119i	$-0.688842\pi$
$103$	4.45031	−	7.70817i	0.438502	−	0.759508i	0.997574	+	0.0696109i	$0.0221758\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.85846	+	15.3433i	0.856380	+	1.48329i	0.875359	+	0.483473i	$0.160625\pi$
$107$	8.85846	+	15.3433i	0.856380	+	1.48329i	−0.0189792	+	0.999820i	$0.506042\pi$
$108$		0			0
$109$	5.97433	−	10.3478i	0.572237	−	0.991144i	−0.424098	−	0.905616i	$-0.639409\pi$
$109$	5.97433	−	10.3478i	0.572237	−	0.991144i	0.996336	−	0.0855281i	$-0.0272577\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.50835	+	13.0048i	0.706326	+	1.22339i	0.966211	+	0.257753i	$0.0829822\pi$
$113$	7.50835	+	13.0048i	0.706326	+	1.22339i	−0.259884	+	0.965640i	$0.583684\pi$
$114$		0			0
$115$	−0.0502773	−	0.0870828i	−0.00468838	−	0.00812051i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.34289	−	9.25415i	0.485717	−	0.841286i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.223815			0.0200186
$126$		0			0
$127$	4.17511			0.370481			0.185240	−	0.982693i	$-0.440694\pi$
$127$	4.17511			0.370481			0.185240	+	0.982693i	$0.440694\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.03947	−	5.26451i	0.265560	−	0.459963i	−0.702150	−	0.712029i	$-0.747777\pi$
$131$	3.03947	−	5.26451i	0.265560	−	0.459963i	0.967710	+	0.252066i	$0.0811100\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.04927	+	13.9418i	0.687696	+	1.19112i	0.972581	+	0.232563i	$0.0747111\pi$
$137$	8.04927	+	13.9418i	0.687696	+	1.19112i	−0.284886	+	0.958562i	$0.591956\pi$
$138$		0			0
$139$	1.83849	+	3.18435i	0.155938	+	0.270093i	0.933400	−	0.358837i	$-0.116827\pi$
$139$	1.83849	+	3.18435i	0.155938	+	0.270093i	−0.777462	+	0.628930i	$0.783493\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.15723	−	2.00438i	0.0967726	−	0.167615i
$144$		0			0
$145$	0.0575849	+	0.0997400i	0.00478217	+	0.00828296i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.57786	−	7.92909i	0.375033	−	0.649576i	−0.615299	−	0.788294i	$-0.710965\pi$
$149$	4.57786	−	7.92909i	0.375033	−	0.649576i	0.990332	+	0.138718i	$0.0442981\pi$
$150$		0			0
$151$	9.28603	+	16.0839i	0.755687	+	1.30889i	0.945032	+	0.326977i	$0.106030\pi$
$151$	9.28603	+	16.0839i	0.755687	+	1.30889i	−0.189346	+	0.981910i	$0.560637\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.00912245	−	0.0158005i	0.000732732	−	0.00126913i
$156$		0			0
$157$	−11.6619			−0.930721			−0.465361	−	0.885121i	$-0.654075\pi$
$157$	−11.6619			−0.930721			−0.465361	+	0.885121i	$0.654075\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.53069	−	11.3115i	−0.511523	−	0.885984i	−0.999911	−	0.0133569i	$-0.995748\pi$
$163$	−6.53069	−	11.3115i	−0.511523	−	0.885984i	0.488388	−	0.872627i	$-0.337585\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.86282	+	11.8867i	0.531061	+	0.919824i	0.999343	+	0.0362450i	$0.0115397\pi$
$167$	6.86282	+	11.8867i	0.531061	+	0.919824i	−0.468282	+	0.883579i	$0.655127\pi$
$168$		0			0
$169$	−2.02362	+	3.50502i	−0.155663	+	0.269617i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	24.9257			1.89507			0.947534	−	0.319656i	$-0.103567\pi$
$173$	24.9257			1.89507			0.947534	+	0.319656i	$0.103567\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.58202	+	7.93629i	−0.342476	+	0.593186i	−0.984892	−	0.173170i	$-0.944599\pi$
$179$	−4.58202	+	7.93629i	−0.342476	+	0.593186i	0.642416	+	0.766356i	$0.277932\pi$
$180$		0			0
$181$	−0.882658			−0.0656075			−0.0328037	−	0.999462i	$-0.510444\pi$
$181$	−0.882658			−0.0656075			−0.0328037	+	0.999462i	$0.510444\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.0105443			−0.000775231
$186$		0			0
$187$	−1.44795			−0.105885
$188$		0			0
$189$		0			0
$190$		0			0
$191$	16.6151			1.20222			0.601111	−	0.799165i	$-0.294725\pi$
$191$	16.6151			1.20222			0.601111	+	0.799165i	$0.294725\pi$
$192$		0			0
$193$	18.5869			1.33791			0.668957	−	0.743302i	$-0.266741\pi$
$193$	18.5869			1.33791			0.668957	+	0.743302i	$0.266741\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−23.4037			−1.66744			−0.833722	−	0.552184i	$-0.813795\pi$
$197$	−23.4037			−1.66744			−0.833722	+	0.552184i	$0.813795\pi$
$198$		0			0
$199$	4.03426	−	6.98754i	0.285981	−	0.495334i	−0.686865	−	0.726785i	$-0.741014\pi$
$199$	4.03426	−	6.98754i	0.285981	−	0.495334i	0.972847	+	0.231451i	$0.0743472\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.135538			−0.00946636
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.951102	+	1.64736i	−0.0657891	+	0.113950i
$210$		0			0
$211$	6.94647	+	12.0316i	0.478215	+	0.828292i	0.999688	−	0.0249755i	$-0.00795078\pi$
$211$	6.94647	+	12.0316i	0.478215	+	0.828292i	−0.521473	+	0.853268i	$0.674617\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.0181732	−	0.0314770i	−0.00123940	−	0.00214671i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.6650			0.717402
$222$		0			0
$223$	8.28588	−	14.3516i	0.554863	−	0.961052i	−0.443051	−	0.896497i	$-0.646104\pi$
$223$	8.28588	−	14.3516i	0.554863	−	0.961052i	0.997914	−	0.0645551i	$-0.0205628\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.06311	+	5.30547i	0.203306	+	0.352136i	0.949592	−	0.313490i	$-0.101498\pi$
$227$	3.06311	+	5.30547i	0.203306	+	0.352136i	−0.746286	+	0.665626i	$0.768165\pi$
$228$		0			0
$229$	14.4155	−	24.9683i	0.952600	−	1.64995i	0.212833	−	0.977089i	$-0.431731\pi$
$229$	14.4155	−	24.9683i	0.952600	−	1.64995i	0.739767	−	0.672863i	$-0.234936\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.44996	−	14.6358i	−0.553575	−	0.958821i	−0.998013	−	0.0630107i	$-0.979930\pi$
$233$	−8.44996	−	14.6358i	−0.553575	−	0.958821i	0.444438	−	0.895810i	$-0.353404\pi$
$234$		0			0
$235$	0.135536	−	0.234755i	0.00884138	−	0.0153137i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.63989	+	13.2327i	0.494184	+	0.855951i	0.999978	−	0.00670310i	$-0.00213368\pi$
$239$	7.63989	+	13.2327i	0.494184	+	0.855951i	−0.505794	+	0.862654i	$0.668800\pi$
$240$		0			0
$241$	−2.09714	−	3.63236i	−0.135089	−	0.233981i	0.790543	−	0.612407i	$-0.209799\pi$
$241$	−2.09714	−	3.63236i	−0.135089	−	0.233981i	−0.925631	+	0.378426i	$0.876465\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	7.00537	−	12.1336i	0.445741	−	0.772046i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−19.7104			−1.24411			−0.622056	−	0.782973i	$-0.713702\pi$
$251$	−19.7104			−1.24411			−0.622056	+	0.782973i	$0.713702\pi$
$252$		0			0
$253$	−2.51834			−0.158326
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.47060	+	4.27921i	−0.154112	+	0.266930i	−0.932735	−	0.360562i	$-0.882585\pi$
$257$	−2.47060	+	4.27921i	−0.154112	+	0.266930i	0.778623	+	0.627492i	$0.215918\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.5987	+	23.5537i	0.838533	+	1.45238i	0.891121	+	0.453766i	$0.149920\pi$
$263$	13.5987	+	23.5537i	0.838533	+	1.45238i	−0.0525879	+	0.998616i	$0.516747\pi$
$264$		0			0
$265$	0.0326613	+	0.0565710i	0.00200637	+	0.00347513i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.5884	−	18.3397i	0.645588	−	1.11819i	−0.338578	−	0.940938i	$-0.609946\pi$
$269$	10.5884	−	18.3397i	0.645588	−	1.11819i	0.984165	−	0.177253i	$-0.0567209\pi$
$270$		0			0
$271$	−11.7699	−	20.3860i	−0.714968	−	1.23836i	−0.962972	−	0.269602i	$-0.913108\pi$
$271$	−11.7699	−	20.3860i	−0.714968	−	1.23836i	0.248003	−	0.968759i	$-0.420226\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.40126	−	2.42706i	0.0844994	−	0.146357i
$276$		0			0
$277$	−10.1560	−	17.5907i	−0.610216	−	1.05693i	−0.991204	−	0.132345i	$-0.957749\pi$
$277$	−10.1560	−	17.5907i	−0.610216	−	1.05693i	0.380988	−	0.924580i	$-0.375584\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.63184	−	16.6828i	0.574587	−	0.995215i	−0.421499	−	0.906829i	$-0.638496\pi$
$281$	9.63184	−	16.6828i	0.574587	−	0.995215i	0.996086	−	0.0883856i	$-0.0281708\pi$
$282$		0			0
$283$	−17.9960			−1.06975			−0.534875	−	0.844931i	$-0.679641\pi$
$283$	−17.9960			−1.06975			−0.534875	+	0.844931i	$0.679641\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.16394	+	8.94421i	0.303761	+	0.526130i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.27396	+	9.13477i	0.308108	+	0.533659i	0.977949	−	0.208846i	$-0.0669708\pi$
$293$	5.27396	+	9.13477i	0.308108	+	0.533659i	−0.669840	+	0.742505i	$0.733637\pi$
$294$		0			0
$295$	−0.100261	+	0.173657i	−0.00583742	+	0.0101107i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	18.5489			1.07271
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.0236567	−	0.0409746i	0.00135458	−	0.00234620i
$306$		0			0
$307$	16.2337			0.926506			0.463253	−	0.886226i	$-0.346682\pi$
$307$	16.2337			0.926506			0.463253	+	0.886226i	$0.346682\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.7102			−0.890846			−0.445423	−	0.895320i	$-0.646947\pi$
$311$	−15.7102			−0.890846			−0.445423	+	0.895320i	$0.646947\pi$
$312$		0			0
$313$	33.0387			1.86746			0.933729	−	0.357979i	$-0.116534\pi$
$313$	33.0387			1.86746			0.933729	+	0.357979i	$0.116534\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.26125			−0.463998			−0.231999	−	0.972716i	$-0.574527\pi$
$317$	−8.26125			−0.463998			−0.231999	+	0.972716i	$0.574527\pi$
$318$		0			0
$319$	2.88437			0.161494
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.76527			−0.487713
$324$		0			0
$325$	−10.3210	+	17.8766i	−0.572508	+	0.991614i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.2767			1.00458			0.502290	−	0.864699i	$-0.332491\pi$
$331$	18.2767			1.00458			0.502290	+	0.864699i	$0.332491\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.168887	−	0.292520i	0.00922727	−	0.0159821i
$336$		0			0
$337$	10.6530	+	18.4516i	0.580307	+	1.00512i	0.995443	+	0.0953621i	$0.0304009\pi$
$337$	10.6530	+	18.4516i	0.580307	+	1.00512i	−0.415135	+	0.909760i	$0.636266\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.228467	−	0.395716i	−0.0123722	−	0.0214292i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.8755			1.17434			0.587170	−	0.809463i	$-0.300242\pi$
$347$	21.8755			1.17434			0.587170	+	0.809463i	$0.300242\pi$
$348$		0			0
$349$	2.46155	−	4.26354i	0.131764	−	0.228222i	−0.792593	−	0.609751i	$-0.791269\pi$
$349$	2.46155	−	4.26354i	0.131764	−	0.228222i	0.924357	+	0.381530i	$0.124603\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.30915	+	3.99956i	0.122903	+	0.212875i	0.920911	−	0.389772i	$-0.127446\pi$
$353$	2.30915	+	3.99956i	0.122903	+	0.212875i	−0.798008	+	0.602647i	$0.794113\pi$
$354$		0			0
$355$	−0.0241783	+	0.0418780i	−0.00128325	+	0.00222265i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	13.6695	+	23.6763i	0.721449	+	1.24959i	0.960419	+	0.278559i	$0.0898568\pi$
$359$	13.6695	+	23.6763i	0.721449	+	1.24959i	−0.238970	+	0.971027i	$0.576810\pi$
$360$		0			0
$361$	3.74245	−	6.48212i	0.196971	−	0.341164i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.0795042	+	0.137705i	0.00416144	+	0.00720783i
$366$		0			0
$367$	−12.0924	−	20.9446i	−0.631217	−	1.09330i	−0.987303	−	0.158847i	$-0.949222\pi$
$367$	−12.0924	−	20.9446i	−0.631217	−	1.09330i	0.356086	−	0.934453i	$-0.384111\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−7.11620	+	12.3256i	−0.368463	+	0.638196i	−0.989325	−	0.145723i	$-0.953449\pi$
$373$	−7.11620	+	12.3256i	−0.368463	+	0.638196i	0.620863	+	0.783919i	$0.286782\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−21.2449			−1.09417
$378$		0			0
$379$	14.5071			0.745178			0.372589	−	0.927996i	$-0.378470\pi$
$379$	14.5071			0.745178			0.372589	+	0.927996i	$0.378470\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.4659	+	21.5916i	−0.636979	+	1.10328i	0.349113	+	0.937080i	$0.386483\pi$
$383$	−12.4659	+	21.5916i	−0.636979	+	1.10328i	−0.986092	+	0.166199i	$0.946851\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	15.5536	+	26.9396i	0.788599	+	1.36589i	0.926825	+	0.375493i	$0.122527\pi$
$389$	15.5536	+	26.9396i	0.788599	+	1.36589i	−0.138226	+	0.990401i	$0.544140\pi$
$390$		0			0
$391$	−5.80219	−	10.0497i	−0.293430	−	0.508235i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.0259394	−	0.0449283i	0.00130515	−	0.00226059i
$396$		0			0
$397$	1.52243	+	2.63693i	0.0764086	+	0.132344i	0.901698	−	0.432367i	$-0.142321\pi$
$397$	1.52243	+	2.63693i	0.0764086	+	0.132344i	−0.825289	+	0.564710i	$0.808988\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.73300	−	9.92985i	0.286292	−	0.495873i	−0.686629	−	0.727008i	$-0.740910\pi$
$401$	5.73300	−	9.92985i	0.286292	−	0.495873i	0.972922	+	0.231135i	$0.0742438\pi$
$402$		0			0
$403$	1.68278	+	2.91466i	0.0838252	+	0.145190i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.132038	+	0.228697i	−0.00654489	+	0.0113361i
$408$		0			0
$409$	−5.12492			−0.253411			−0.126706	−	0.991940i	$-0.540440\pi$
$409$	−5.12492			−0.253411			−0.126706	+	0.991940i	$0.540440\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.0857613	−	0.148543i	−0.00420986	−	0.00729169i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.2951	+	29.9560i	0.844921	+	1.46345i	0.885690	+	0.464278i	$0.153686\pi$
$419$	17.2951	+	29.9560i	0.844921	+	1.46345i	−0.0407685	+	0.999169i	$0.512981\pi$
$420$		0			0
$421$	−16.3403	+	28.3023i	−0.796379	+	1.37937i	0.125581	+	0.992083i	$0.459920\pi$
$421$	−16.3403	+	28.3023i	−0.796379	+	1.37937i	−0.921960	+	0.387285i	$0.873413\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.9139			0.626417
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.0725515	−	0.125663i	0.00349468	−	0.00605297i	−0.864273	−	0.503023i	$-0.832221\pi$
$431$	0.0725515	−	0.125663i	0.00349468	−	0.00605297i	0.867767	+	0.496970i	$0.165554\pi$
$432$		0			0
$433$	−19.1583			−0.920690			−0.460345	−	0.887740i	$-0.652274\pi$
$433$	−19.1583			−0.920690			−0.460345	+	0.887740i	$0.652274\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.2449			−0.729262
$438$		0			0
$439$	−7.58500			−0.362012			−0.181006	−	0.983482i	$-0.557935\pi$
$439$	−7.58500			−0.362012			−0.181006	+	0.983482i	$0.557935\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.26484			0.155117			0.0775586	−	0.996988i	$-0.475288\pi$
$443$	3.26484			0.155117			0.0775586	+	0.996988i	$0.475288\pi$
$444$		0			0
$445$	−0.0224885			−0.00106606
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−19.8316			−0.935909			−0.467955	−	0.883752i	$-0.655009\pi$
$449$	−19.8316			−0.935909			−0.467955	+	0.883752i	$0.655009\pi$
$450$		0			0
$451$	−1.69724	+	2.93970i	−0.0799197	+	0.138425i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.46102			−0.208678			−0.104339	−	0.994542i	$-0.533273\pi$
$457$	−4.46102			−0.208678			−0.104339	+	0.994542i	$0.533273\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.88515	+	13.6575i	−0.367248	+	0.636092i	−0.989134	−	0.147015i	$-0.953033\pi$
$461$	−7.88515	+	13.6575i	−0.367248	+	0.636092i	0.621886	+	0.783108i	$0.286367\pi$
$462$		0			0
$463$	−8.34621	−	14.4561i	−0.387881	−	0.671830i	0.604283	−	0.796770i	$-0.293460\pi$
$463$	−8.34621	−	14.4561i	−0.387881	−	0.671830i	−0.992164	+	0.124940i	$0.960126\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.6974	−	25.4566i	−0.680112	−	1.17799i	−0.974946	−	0.222441i	$-0.928598\pi$
$467$	−14.6974	−	25.4566i	−0.680112	−	1.17799i	0.294834	−	0.955549i	$-0.404736\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.910278			−0.0418546
$474$		0			0
$475$	8.48262	−	14.6923i	0.389209	−	0.674131i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.16675	+	15.8773i	0.418839	+	0.725451i	0.995823	−	0.0913046i	$-0.0291037\pi$
$479$	9.16675	+	15.8773i	0.418839	+	0.725451i	−0.576984	+	0.816756i	$0.695770\pi$
$480$		0			0
$481$	0.972530	−	1.68447i	0.0443436	−	0.0768053i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.164062	+	0.284163i	0.00744965	+	0.0129032i
$486$		0			0
$487$	6.24766	−	10.8213i	0.283108	−	0.490358i	−0.689040	−	0.724723i	$-0.741968\pi$
$487$	6.24766	−	10.8213i	0.283108	−	0.490358i	0.972149	+	0.234365i	$0.0753010\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.75354	+	3.03721i	0.0791359	+	0.137067i	0.902877	−	0.429898i	$-0.141451\pi$
$491$	1.75354	+	3.03721i	0.0791359	+	0.137067i	−0.823741	+	0.566966i	$0.808117\pi$
$492$		0			0
$493$	6.64553	+	11.5104i	0.299299	+	0.518402i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	16.1819	−	28.0279i	0.724402	−	1.25470i	−0.234817	−	0.972040i	$-0.575449\pi$
$499$	16.1819	−	28.0279i	0.724402	−	1.25470i	0.959220	−	0.282662i	$-0.0912175\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.3277			0.683430			0.341715	−	0.939804i	$-0.388992\pi$
$503$	15.3277			0.683430			0.341715	+	0.939804i	$0.388992\pi$
$504$		0			0
$505$	−0.338985			−0.0150846
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8.47237	+	14.6746i	−0.375531	+	0.650439i	−0.990406	−	0.138185i	$-0.955873\pi$
$509$	−8.47237	+	14.6746i	−0.375531	+	0.650439i	0.614875	+	0.788625i	$0.289206\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.0996097	−	0.172529i	−0.00438933	−	0.00760254i
$516$		0			0
$517$	−3.39442	−	5.87931i	−0.149287	−	0.258572i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.36977	−	16.2289i	0.410497	−	0.711002i	−0.584447	−	0.811432i	$-0.698689\pi$
$521$	9.36977	−	16.2289i	0.410497	−	0.711002i	0.994944	+	0.100430i	$0.0320219\pi$
$522$		0			0
$523$	4.48382	+	7.76620i	0.196064	+	0.339592i	0.947249	−	0.320499i	$-0.103851\pi$
$523$	4.48382	+	7.76620i	0.196064	+	0.339592i	−0.751185	+	0.660092i	$0.770517\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.05277	−	1.82344i	0.0458592	−	0.0794305i
$528$		0			0
$529$	1.40861	+	2.43978i	0.0612439	+	0.106078i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.5010	−	21.6524i	0.541480	−	0.937871i
$534$		0			0
$535$	0.396551			0.0171444
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−17.5759	−	30.4423i	−0.755646	−	1.30882i	−0.945052	−	0.326919i	$-0.893990\pi$
$541$	−17.5759	−	30.4423i	−0.755646	−	1.30882i	0.189406	−	0.981899i	$-0.439344\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.133721	−	0.231612i	−0.00572799	−	0.00992117i
$546$		0			0
$547$	7.75354	−	13.4295i	0.331517	−	0.574205i	−0.651292	−	0.758827i	$-0.725773\pi$
$547$	7.75354	−	13.4295i	0.331517	−	0.574205i	0.982810	+	0.184622i	$0.0591061\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17.4607			0.743850
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.48253	−	2.56781i	0.0628166	−	0.108802i	−0.832907	−	0.553413i	$-0.813325\pi$
$557$	1.48253	−	2.56781i	0.0628166	−	0.108802i	0.895723	+	0.444612i	$0.146658\pi$
$558$		0			0
$559$	6.70468			0.283578
$560$		0			0
$561$		0			0
$562$		0			0
$563$	24.3713			1.02713			0.513564	−	0.858051i	$-0.328325\pi$
$563$	24.3713			1.02713			0.513564	+	0.858051i	$0.328325\pi$
$564$		0			0
$565$	0.336113			0.0141404
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−44.9602			−1.88483			−0.942414	−	0.334448i	$-0.891450\pi$
$569$	−44.9602			−1.88483			−0.942414	+	0.334448i	$0.891450\pi$
$570$		0			0
$571$	−15.6776			−0.656087			−0.328043	−	0.944663i	$-0.606389\pi$
$571$	−15.6776			−0.656087			−0.328043	+	0.944663i	$0.606389\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	22.4604			0.936662
$576$		0			0
$577$	13.7485	−	23.8132i	0.572360	−	0.991356i	−0.423963	−	0.905679i	$-0.639362\pi$
$577$	13.7485	−	23.8132i	0.572360	−	0.991356i	0.996323	−	0.0856765i	$-0.0273051\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.63597			0.0677549
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.3162	−	23.0644i	0.549619	−	0.951968i	−0.448681	−	0.893692i	$-0.648106\pi$
$587$	13.3162	−	23.0644i	0.549619	−	0.951968i	0.998300	−	0.0582763i	$-0.0185604\pi$
$588$		0			0
$589$	−1.38304	−	2.39549i	−0.0569871	−	0.0987045i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.87994	−	15.3805i	−0.364656	−	0.631602i	0.624065	−	0.781372i	$-0.285480\pi$
$593$	−8.87994	−	15.3805i	−0.364656	−	0.631602i	−0.988721	+	0.149770i	$0.952147\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−16.5809			−0.677475			−0.338738	−	0.940881i	$-0.610000\pi$
$599$	−16.5809			−0.677475			−0.338738	+	0.940881i	$0.610000\pi$
$600$		0			0
$601$	−12.3406	+	21.3746i	−0.503384	+	0.871886i	0.496608	+	0.867975i	$0.334578\pi$
$601$	−12.3406	+	21.3746i	−0.503384	+	0.871886i	−0.999992	+	0.00391177i	$0.998755\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.119588	−	0.207132i	−0.00486194	−	0.00842112i
$606$		0			0
$607$	6.19000	−	10.7214i	0.251244	−	0.435168i	−0.712624	−	0.701546i	$-0.752494\pi$
$607$	6.19000	−	10.7214i	0.251244	−	0.435168i	0.963869	+	0.266378i	$0.0858269\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25.0017	+	43.3042i	1.01146	+	1.75190i
$612$		0			0
$613$	2.74800	−	4.75968i	0.110991	−	0.192242i	−0.805179	−	0.593032i	$-0.797931\pi$
$613$	2.74800	−	4.75968i	0.110991	−	0.192242i	0.916170	+	0.400790i	$0.131264\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.4573	+	19.8446i	0.461254	+	0.798916i	0.999024	−	0.0441763i	$-0.0140663\pi$
$617$	11.4573	+	19.8446i	0.461254	+	0.798916i	−0.537770	+	0.843092i	$0.680733\pi$
$618$		0			0
$619$	−3.49256	−	6.04930i	−0.140378	−	0.243142i	0.787261	−	0.616620i	$-0.211498\pi$
$619$	−3.49256	−	6.04930i	−0.140378	−	0.243142i	−0.927639	+	0.373478i	$0.878165\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.4962	+	21.6441i	−0.499850	+	0.865765i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.21685			−0.0485191
$630$		0			0
$631$	−14.7028			−0.585310			−0.292655	−	0.956218i	$-0.594539\pi$
$631$	−14.7028			−0.585310			−0.292655	+	0.956218i	$0.594539\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.0467249	−	0.0809300i	0.00185422	−	0.00321161i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.8057	−	18.7161i	−0.426801	−	0.739241i	0.569786	−	0.821793i	$-0.307026\pi$
$641$	−10.8057	−	18.7161i	−0.426801	−	0.739241i	−0.996587	+	0.0825523i	$0.973693\pi$
$642$		0			0
$643$	−6.78105	−	11.7451i	−0.267418	−	0.463182i	0.700776	−	0.713381i	$-0.252837\pi$
$643$	−6.78105	−	11.7451i	−0.267418	−	0.463182i	−0.968194	+	0.250199i	$0.919504\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.2074	+	17.6798i	−0.401295	+	0.695063i	−0.993882	−	0.110443i	$-0.964773\pi$
$647$	−10.2074	+	17.6798i	−0.401295	+	0.695063i	0.592587	+	0.805506i	$0.298106\pi$
$648$		0			0
$649$	2.51098	+	4.34915i	0.0985647	+	0.170719i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24.3462	−	42.1689i	0.952741	−	1.65020i	0.213287	−	0.976990i	$-0.431583\pi$
$653$	24.3462	−	42.1689i	0.952741	−	1.65020i	0.739454	−	0.673207i	$-0.235084\pi$
$654$		0			0
$655$	−0.0680313	−	0.117834i	−0.00265820	−	0.00460414i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.65029	−	13.2507i	0.298013	−	0.516174i	−0.677668	−	0.735368i	$-0.737009\pi$
$659$	7.65029	−	13.2507i	0.298013	−	0.516174i	0.975681	+	0.219194i	$0.0703428\pi$
$660$		0			0
$661$	17.5499			0.682612			0.341306	−	0.939952i	$-0.389131\pi$
$661$	17.5499			0.682612			0.341306	+	0.939952i	$0.389131\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	11.5581	+	20.0193i	0.447533	+	0.775150i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.592470	−	1.02619i	−0.0228721	−	0.0396156i
$672$		0			0
$673$	3.79336	−	6.57029i	0.146223	−	0.253266i	−0.783605	−	0.621259i	$-0.786622\pi$
$673$	3.79336	−	6.57029i	0.146223	−	0.253266i	0.929829	+	0.367993i	$0.119955\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.5325			0.443232			0.221616	−	0.975134i	$-0.428867\pi$
$677$	11.5325			0.443232			0.221616	+	0.975134i	$0.428867\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.47937	+	12.9546i	−0.286190	+	0.495696i	−0.972897	−	0.231239i	$-0.925722\pi$
$683$	−7.47937	+	12.9546i	−0.286190	+	0.495696i	0.686707	+	0.726934i	$0.259056\pi$
$684$		0			0
$685$	0.360328			0.0137674
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.0498			−0.459060
$690$		0			0
$691$	−19.8961			−0.756882			−0.378441	−	0.925625i	$-0.623540\pi$
$691$	−19.8961			−0.756882			−0.378441	+	0.925625i	$0.623540\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.0823002			0.00312183
$696$		0			0
$697$	−15.6416			−0.592467
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−34.9655			−1.32063			−0.660315	−	0.750989i	$-0.729577\pi$
$701$	−34.9655			−1.32063			−0.660315	+	0.750989i	$0.729577\pi$
$702$		0			0
$703$	−0.799300	+	1.38443i	−0.0301462	+	0.0522147i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−19.3045			−0.724998			−0.362499	−	0.931984i	$-0.618076\pi$
$709$	−19.3045			−0.724998			−0.362499	+	0.931984i	$0.618076\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.83101	−	3.17140i	0.0685719	−	0.118770i
$714$		0			0
$715$	−0.0259019	−	0.0448634i	−0.000968676	−	0.00167780i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.7284	−	30.7066i	−0.661159	−	1.14516i	−0.980311	−	0.197458i	$-0.936731\pi$
$719$	−17.7284	−	30.7066i	−0.661159	−	1.14516i	0.319152	−	0.947704i	$-0.396602\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−25.7249			−0.955400
$726$		0			0
$727$	2.27159	−	3.93452i	0.0842487	−	0.145923i	−0.820822	−	0.571184i	$-0.806484\pi$
$727$	2.27159	−	3.93452i	0.0842487	−	0.145923i	0.905071	+	0.425261i	$0.139818\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.09726	−	3.63257i	−0.0775701	−	0.134355i
$732$		0			0
$733$	−16.6444	+	28.8290i	−0.614777	+	1.06482i	0.375647	+	0.926763i	$0.377421\pi$
$733$	−16.6444	+	28.8290i	−0.614777	+	1.06482i	−0.990424	+	0.138062i	$0.955913\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.22968	−	7.32602i	−0.155802	−	0.269858i
$738$		0			0
$739$	−20.6599	+	35.7841i	−0.759988	+	1.31634i	0.182868	+	0.983137i	$0.441462\pi$
$739$	−20.6599	+	35.7841i	−0.759988	+	1.31634i	−0.942856	+	0.333200i	$0.891871\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.7638	+	23.8396i	0.504946	+	0.874592i	0.999984	+	0.00572027i	$0.00182083\pi$
$743$	13.7638	+	23.8396i	0.504946	+	0.874592i	−0.495038	+	0.868871i	$0.664846\pi$
$744$		0			0
$745$	−0.102465	−	0.177474i	−0.00375401	−	0.00650214i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6.58398	+	11.4038i	−0.240253	+	0.416130i	−0.960786	−	0.277290i	$-0.910564\pi$
$751$	−6.58398	+	11.4038i	−0.240253	+	0.416130i	0.720533	+	0.693420i	$0.243897\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.415692			0.0151286
$756$		0			0
$757$	−13.7925			−0.501297			−0.250648	−	0.968078i	$-0.580644\pi$
$757$	−13.7925			−0.501297			−0.250648	+	0.968078i	$0.580644\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.6055	+	27.0294i	−0.565697	+	0.979816i	0.431287	+	0.902215i	$0.358060\pi$
$761$	−15.6055	+	27.0294i	−0.565697	+	0.979816i	−0.996984	+	0.0776016i	$0.975274\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.4947	−	32.0338i	−0.667806	−	1.15667i
$768$		0			0
$769$	19.1337	+	33.1405i	0.689977	+	1.19508i	0.971845	+	0.235622i	$0.0757128\pi$
$769$	19.1337	+	33.1405i	0.689977	+	1.19508i	−0.281867	+	0.959453i	$0.590954\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.73395	−	6.46739i	0.134301	−	0.232616i	−0.791029	−	0.611778i	$-0.790455\pi$
$773$	3.73395	−	6.46739i	0.134301	−	0.232616i	0.925330	+	0.379162i	$0.123788\pi$
$774$		0			0
$775$	2.03764	+	3.52929i	0.0731941	+	0.126776i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−10.2743	+	17.7956i	−0.368115	+	0.637594i
$780$		0			0
$781$	0.605533	+	1.04881i	0.0216677	+	0.0375295i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.130512	+	0.226053i	−0.00465817	+	0.00806819i
$786$		0			0
$787$	20.6901			0.737523			0.368761	−	0.929524i	$-0.379782\pi$
$787$	20.6901			0.737523			0.368761	+	0.929524i	$0.379782\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4.36386	+	7.55842i	0.154965	+	0.268407i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.8354	−	27.4276i	−0.560917	−	0.971537i	−0.997417	−	0.0718325i	$-0.977115\pi$
$797$	−15.8354	−	27.4276i	−0.560917	−	0.971537i	0.436500	−	0.899704i	$-0.356218\pi$
$798$		0			0
$799$	15.6414	−	27.0916i	0.553352	−	0.958433i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.98229			0.140532
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	24.0430	−	41.6438i	0.845308	−	1.46412i	−0.0400451	−	0.999198i	$-0.512750\pi$
$809$	24.0430	−	41.6438i	0.845308	−	1.46412i	0.885353	−	0.464919i	$-0.153917\pi$
$810$		0			0
$811$	17.6685			0.620427			0.310213	−	0.950667i	$-0.399600\pi$
$811$	17.6685			0.620427			0.310213	+	0.950667i	$0.399600\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.292348			−0.0102405
$816$		0			0
$817$	−5.51042			−0.192785
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.9979			−1.01203			−0.506017	−	0.862523i	$-0.668883\pi$
$821$	−28.9979			−1.01203			−0.506017	+	0.862523i	$0.668883\pi$
$822$		0			0
$823$	13.7745			0.480147			0.240074	−	0.970755i	$-0.422828\pi$
$823$	13.7745			0.480147			0.240074	+	0.970755i	$0.422828\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21.4150			−0.744673			−0.372337	−	0.928098i	$-0.621443\pi$
$827$	−21.4150			−0.744673			−0.372337	+	0.928098i	$0.621443\pi$
$828$		0			0
$829$	−9.40772	+	16.2947i	−0.326744	+	0.565937i	−0.981864	−	0.189588i	$-0.939285\pi$
$829$	−9.40772	+	16.2947i	−0.326744	+	0.565937i	0.655120	+	0.755525i	$0.272618\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	0.307216			0.0106316
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.4124	−	37.0874i	0.739239	−	1.28040i	−0.213600	−	0.976921i	$-0.568519\pi$
$839$	21.4124	−	37.0874i	0.739239	−	1.28040i	0.952839	−	0.303477i	$-0.0981477\pi$
$840$		0			0
$841$	1.26191	+	2.18569i	0.0435142	+	0.0753688i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.0452940	+	0.0784515i	0.00155816	+	0.00269881i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.11639			−0.0725491
$852$		0			0
$853$	−13.3281	+	23.0849i	−0.456345	+	0.790412i	−0.998764	−	0.0496954i	$-0.984175\pi$
$853$	−13.3281	+	23.0849i	−0.456345	+	0.790412i	0.542420	+	0.840108i	$0.317508\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.5907	+	27.0040i	0.532570	+	0.922438i	0.999277	+	0.0380257i	$0.0121069\pi$
$857$	15.5907	+	27.0040i	0.532570	+	0.922438i	−0.466707	+	0.884412i	$0.654560\pi$
$858$		0			0
$859$	−26.9668	+	46.7078i	−0.920094	+	1.59365i	−0.120827	+	0.992674i	$0.538555\pi$
$859$	−26.9668	+	46.7078i	−0.920094	+	1.59365i	−0.799267	+	0.600976i	$0.794779\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16.8755	−	29.2292i	−0.574448	−	0.994974i	−0.996101	−	0.0882164i	$-0.971883\pi$
$863$	−16.8755	−	29.2292i	−0.574448	−	0.994974i	0.421653	−	0.906757i	$-0.361450\pi$
$864$		0			0
$865$	0.278952	−	0.483158i	0.00948464	−	0.0164279i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.649639	−	1.12521i	−0.0220375	−	0.0381700i
$870$		0			0
$871$	31.1538	+	53.9600i	1.05561	+	1.82837i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.60808	+	2.78527i	−0.0543009	+	0.0940519i	−0.891898	−	0.452236i	$-0.850626\pi$
$877$	−1.60808	+	2.78527i	−0.0543009	+	0.0940519i	0.837597	+	0.546288i	$0.183960\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−35.9141			−1.20998			−0.604989	−	0.796234i	$-0.706822\pi$
$881$	−35.9141			−1.20998			−0.604989	+	0.796234i	$0.706822\pi$
$882$		0			0
$883$	4.58572			0.154322			0.0771609	−	0.997019i	$-0.475414\pi$
$883$	4.58572			0.154322			0.0771609	+	0.997019i	$0.475414\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−27.5154	+	47.6581i	−0.923877	+	1.60020i	−0.130521	+	0.991446i	$0.541665\pi$
$887$	−27.5154	+	47.6581i	−0.923877	+	1.60020i	−0.793357	+	0.608757i	$0.791668\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−20.5483	−	35.5907i	−0.687624	−	1.19100i
$894$		0			0
$895$	0.102558	+	0.177635i	0.00342812	+	0.00593768i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.09714	+	3.63236i	−0.0699436	+	0.121146i
$900$		0			0
$901$	3.76924	+	6.52851i	0.125572	+	0.217496i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.00987810	+	0.0171094i	−0.000328359	+	0.000568735i
$906$		0			0
$907$	6.38823	+	11.0647i	0.212118	+	0.367399i	0.952377	−	0.304923i	$-0.0986307\pi$
$907$	6.38823	+	11.0647i	0.212118	+	0.367399i	−0.740259	+	0.672321i	$0.765297\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.9053	−	27.5489i	0.526968	−	0.912735i	−0.472539	−	0.881310i	$-0.656662\pi$
$911$	15.9053	−	27.5489i	0.526968	−	0.912735i	0.999506	−	0.0314246i	$-0.0100044\pi$
$912$		0			0
$913$	−4.29570			−0.142167
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	29.2724	+	50.7013i	0.965607	+	1.67248i	0.707975	+	0.706237i	$0.249609\pi$
$919$	29.2724	+	50.7013i	0.965607	+	1.67248i	0.257631	+	0.966243i	$0.417058\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.46007	−	7.72506i	−0.146805	−	0.254274i
$924$		0			0
$925$	1.17761	−	2.03969i	0.0387197	−	0.0670644i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−12.7327			−0.417746			−0.208873	−	0.977943i	$-0.566980\pi$
$929$	−12.7327			−0.417746			−0.208873	+	0.977943i	$0.566980\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.0162045	+	0.0280670i	−0.000529944	+	0.000917891i
$936$		0			0
$937$	−28.2201			−0.921911			−0.460956	−	0.887423i	$-0.652493\pi$
$937$	−28.2201			−0.921911			−0.460956	+	0.887423i	$0.652493\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−48.6046			−1.58446			−0.792232	−	0.610220i	$-0.791081\pi$
$941$	−48.6046			−1.58446			−0.792232	+	0.610220i	$0.791081\pi$
$942$		0			0
$943$	−27.2044			−0.885898
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.2132			−0.396875			−0.198437	−	0.980114i	$-0.563587\pi$
$947$	−12.2132			−0.396875			−0.198437	+	0.980114i	$0.563587\pi$
$948$		0			0
$949$	−29.3316			−0.952145
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22.7206			−0.735993			−0.367996	−	0.929827i	$-0.619956\pi$
$953$	−22.7206			−0.735993			−0.367996	+	0.929827i	$0.619956\pi$
$954$		0			0
$955$	0.185944	−	0.322065i	0.00601701	−	0.0104218i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−30.3356			−0.978566
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.208012	−	0.360287i	0.00669613	−	0.0115980i
$966$		0			0
$967$	−26.9926	−	46.7526i	−0.868024	−	1.50346i	−0.864013	−	0.503469i	$-0.832057\pi$
$967$	−26.9926	−	46.7526i	−0.868024	−	1.50346i	−0.00401026	−	0.999992i	$-0.501277\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.3187	−	35.1930i	−0.652059	−	1.12940i	−0.982623	−	0.185615i	$-0.940572\pi$
$971$	−20.3187	−	35.1930i	−0.652059	−	1.12940i	0.330564	−	0.943784i	$-0.392761\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.0440			−0.641264			−0.320632	−	0.947204i	$-0.603895\pi$
$977$	−20.0440			−0.641264			−0.320632	+	0.947204i	$0.603895\pi$
$978$		0			0
$979$	−0.281607	+	0.487757i	−0.00900018	+	0.0155888i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.83700	−	8.37793i	−0.154276	−	0.267214i	0.778519	−	0.627621i	$-0.215971\pi$
$983$	−4.83700	−	8.37793i	−0.154276	−	0.267214i	−0.932795	+	0.360407i	$0.882638\pi$
$984$		0			0
$985$	−0.261918	+	0.453655i	−0.00834540	+	0.0144547i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.64764	−	6.31789i	−0.115988	−	0.200897i
$990$		0			0
$991$	−6.46878	+	11.2043i	−0.205488	+	0.355915i	−0.950288	−	0.311372i	$-0.899211\pi$
$991$	−6.46878	+	11.2043i	−0.205488	+	0.355915i	0.744800	+	0.667287i	$0.232545\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.0902974	−	0.156400i	−0.00286262	−	0.00495820i
$996$		0			0
$997$	−0.856372	−	1.48328i	−0.0271216	−	0.0469759i	0.852146	−	0.523304i	$-0.175301\pi$
$997$	−0.856372	−	1.48328i	−0.0271216	−	0.0469759i	−0.879268	+	0.476328i	$0.841967\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.i.j.2125.7		24
3.2	odd	2		1764.2.i.j.1537.11		24
7.2	even	3		5292.2.l.j.3313.6		24
7.3	odd	6		5292.2.j.i.3529.6		24
7.4	even	3		5292.2.j.i.3529.7		24
7.5	odd	6		5292.2.l.j.3313.7		24
7.6	odd	2	inner	5292.2.i.j.2125.6		24
9.4	even	3		5292.2.l.j.361.6		24
9.5	odd	6		1764.2.l.j.949.3		24
21.2	odd	6		1764.2.l.j.961.3		24
21.5	even	6		1764.2.l.j.961.10		24
21.11	odd	6		1764.2.j.i.1177.6	yes	24
21.17	even	6		1764.2.j.i.1177.7	yes	24
21.20	even	2		1764.2.i.j.1537.2		24
63.4	even	3		5292.2.j.i.1765.7		24
63.5	even	6		1764.2.i.j.373.2		24
63.13	odd	6		5292.2.l.j.361.7		24
63.23	odd	6		1764.2.i.j.373.11		24
63.31	odd	6		5292.2.j.i.1765.6		24
63.32	odd	6		1764.2.j.i.589.6	✓	24
63.40	odd	6	inner	5292.2.i.j.1549.6		24
63.41	even	6		1764.2.l.j.949.10		24
63.58	even	3	inner	5292.2.i.j.1549.7		24
63.59	even	6		1764.2.j.i.589.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.i.j.373.2		24	63.5	even	6
1764.2.i.j.373.11		24	63.23	odd	6
1764.2.i.j.1537.2		24	21.20	even	2
1764.2.i.j.1537.11		24	3.2	odd	2
1764.2.j.i.589.6	✓	24	63.32	odd	6
1764.2.j.i.589.7	yes	24	63.59	even	6
1764.2.j.i.1177.6	yes	24	21.11	odd	6
1764.2.j.i.1177.7	yes	24	21.17	even	6
1764.2.l.j.949.3		24	9.5	odd	6
1764.2.l.j.949.10		24	63.41	even	6
1764.2.l.j.961.3		24	21.2	odd	6
1764.2.l.j.961.10		24	21.5	even	6
5292.2.i.j.1549.6		24	63.40	odd	6	inner
5292.2.i.j.1549.7		24	63.58	even	3	inner
5292.2.i.j.2125.6		24	7.6	odd	2	inner
5292.2.i.j.2125.7		24	1.1	even	1	trivial
5292.2.j.i.1765.6		24	63.31	odd	6
5292.2.j.i.1765.7		24	63.4	even	3
5292.2.j.i.3529.6		24	7.3	odd	6
5292.2.j.i.3529.7		24	7.4	even	3
5292.2.l.j.361.6		24	9.4	even	3
5292.2.l.j.361.7		24	63.13	odd	6
5292.2.l.j.3313.6		24	7.2	even	3
5292.2.l.j.3313.7		24	7.5	odd	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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0111913 - 0.0193839i) q^{5} +O(q^{10})\)	\(q+(0.0111913 - 0.0193839i) q^{5} +(-0.280281 - 0.485460i) q^{11} +(2.06442 + 3.57567i) q^{13} +(1.29152 - 2.23698i) q^{17} +(-1.69669 - 2.93876i) q^{19} +(2.24626 - 3.89064i) q^{23} +(2.49975 + 4.32969i) q^{25} +(-2.57275 + 4.45613i) q^{29} +0.815136 q^{31} +(-0.235546 - 0.407978i) q^{37} +(-3.02774 - 5.24420i) q^{41} +(0.811935 - 1.40631i) q^{43} +12.1108 q^{47} +(-1.45922 + 2.52745i) q^{53} -0.0125468 q^{55} -8.95882 q^{59} +2.11385 q^{61} +0.0924141 q^{65} +15.0909 q^{67} -2.16045 q^{71} +(-3.55205 + 6.15233i) q^{73} +2.31781 q^{79} +(3.83160 - 6.63653i) q^{83} +(-0.0289077 - 0.0500695i) q^{85} +(-0.502365 - 0.870122i) q^{89} -0.0759530 q^{95} +(-7.32986 + 12.6957i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 4 q^{11} + 8 q^{23} - 12 q^{25} + 32 q^{29} - 12 q^{37} + 16 q^{53} - 72 q^{65} - 24 q^{67} - 48 q^{71} - 24 q^{79} + 12 q^{85} + 64 q^{95}+O(q^{100}) \) 24 * q + 4 * q^11 + 8 * q^23 - 12 * q^25 + 32 * q^29 - 12 * q^37 + 16 * q^53 - 72 * q^65 - 24 * q^67 - 48 * q^71 - 24 * q^79 + 12 * q^85 + 64 * q^95

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