LMFDB - Embedded newform 1216.2.m.a.303.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(303,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("1216.303");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.m (of order $4$, 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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$76$
Relative dimension:	$38$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 304)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			303.16
Character	$\chi$	$=$	1216.303
Dual form			1216.2.m.a.911.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.549728 - 0.549728i) q^{3} +(1.49999 - 1.49999i) q^{5} +3.48044 q^{7} -2.39560i q^{9} +(3.68459 + 3.68459i) q^{11} +(3.73303 - 3.73303i) q^{13} -1.64918 q^{15} +1.51477 q^{17} +(-4.16692 + 1.27937i) q^{19} +(-1.91329 - 1.91329i) q^{21} +0.482269 q^{23} +0.500032i q^{25} +(-2.96611 + 2.96611i) q^{27} +(-7.26730 + 7.26730i) q^{29} +2.39300 q^{31} -4.05104i q^{33} +(5.22064 - 5.22064i) q^{35} +(3.14537 + 3.14537i) q^{37} -4.10430 q^{39} +7.94060 q^{41} +(-3.02093 - 3.02093i) q^{43} +(-3.59339 - 3.59339i) q^{45} -3.58534i q^{47} +5.11344 q^{49} +(-0.832709 - 0.832709i) q^{51} +(-3.55950 - 3.55950i) q^{53} +11.0537 q^{55} +(2.99397 + 1.58737i) q^{57} +(-0.408827 + 0.408827i) q^{59} +(2.47804 + 2.47804i) q^{61} -8.33773i q^{63} -11.1991i q^{65} +(-10.6691 - 10.6691i) q^{67} +(-0.265116 - 0.265116i) q^{69} +5.59302i q^{71} -13.6019i q^{73} +(0.274881 - 0.274881i) q^{75} +(12.8240 + 12.8240i) q^{77} -5.02027 q^{79} -3.92569 q^{81} +(-10.3497 + 10.3497i) q^{83} +(2.27214 - 2.27214i) q^{85} +7.99007 q^{87} +7.42593 q^{89} +(12.9926 - 12.9926i) q^{91} +(-1.31550 - 1.31550i) q^{93} +(-4.33131 + 8.16940i) q^{95} +3.70124i q^{97} +(8.82679 - 8.82679i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q - 4 q^{5} + 8 q^{7} + 4 q^{11} - 8 q^{17} - 6 q^{19} + 8 q^{23} + 8 q^{39} + 4 q^{43} + 4 q^{45} + 44 q^{49} + 8 q^{55} + 28 q^{61} - 32 q^{77} - 52 q^{81} - 36 q^{83} - 56 q^{85} + 120 q^{87} - 16 q^{93}+ \cdots - 76 q^{99}+O(q^{100}) $ 76 * q - 4 * q^5 + 8 * q^7 + 4 * q^11 - 8 * q^17 - 6 * q^19 + 8 * q^23 + 8 * q^39 + 4 * q^43 + 4 * q^45 + 44 * q^49 + 8 * q^55 + 28 * q^61 - 32 * q^77 - 52 * q^81 - 36 * q^83 - 56 * q^85 + 120 * q^87 - 16 * q^93 - 76 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.549728	−	0.549728i	−0.317385	−	0.317385i	0.530377	−	0.847762i	$-0.322050\pi$
$3$	−0.549728	−	0.549728i	−0.317385	−	0.317385i	−0.847762	+	0.530377i	$0.822050\pi$
$4$		0			0
$5$	1.49999	−	1.49999i	0.670818	−	0.670818i	−0.287087	−	0.957905i	$-0.592687\pi$
$5$	1.49999	−	1.49999i	0.670818	−	0.670818i	0.957905	+	0.287087i	$0.0926868\pi$
$6$		0			0
$7$	3.48044			1.31548			0.657741	−	0.753244i	$-0.271512\pi$
$7$	3.48044			1.31548			0.657741	+	0.753244i	$0.271512\pi$
$8$		0			0
$9$		−	2.39560i		−	0.798533i
$10$		0			0
$11$	3.68459	+	3.68459i	1.11094	+	1.11094i	0.993023	+	0.117921i	$0.0376231\pi$
$11$	3.68459	+	3.68459i	1.11094	+	1.11094i	0.117921	+	0.993023i	$0.462377\pi$
$12$		0			0
$13$	3.73303	−	3.73303i	1.03536	−	1.03536i	0.0360050	−	0.999352i	$-0.488537\pi$
$13$	3.73303	−	3.73303i	1.03536	−	1.03536i	0.999352	−	0.0360050i	$-0.0114632\pi$
$14$		0			0
$15$	−1.64918			−0.425816
$16$		0			0
$17$	1.51477			0.367385			0.183693	−	0.982984i	$-0.441195\pi$
$17$	1.51477			0.367385			0.183693	+	0.982984i	$0.441195\pi$
$18$		0			0
$19$	−4.16692	+	1.27937i	−0.955957	+	0.293507i
$19$	−4.16692	+	1.27937i	−0.955957	+	0.293507i
$20$		0			0
$21$	−1.91329	−	1.91329i	−0.417515	−	0.417515i
$22$		0			0
$23$	0.482269			0.100560			0.0502800	−	0.998735i	$-0.483989\pi$
$23$	0.482269			0.100560			0.0502800	+	0.998735i	$0.483989\pi$
$24$		0			0
$25$			0.500032i			0.100006i
$26$		0			0
$27$	−2.96611	+	2.96611i	−0.570828	+	0.570828i
$28$		0			0
$29$	−7.26730	+	7.26730i	−1.34950	+	1.34950i	−0.463303	+	0.886200i	$0.653336\pi$
$29$	−7.26730	+	7.26730i	−1.34950	+	1.34950i	−0.886200	+	0.463303i	$0.846664\pi$
$30$		0			0
$31$	2.39300			0.429795			0.214897	−	0.976637i	$-0.431058\pi$
$31$	2.39300			0.429795			0.214897	+	0.976637i	$0.431058\pi$
$32$		0			0
$33$		−	4.05104i		−	0.705195i
$34$		0			0
$35$	5.22064	−	5.22064i	0.882449	−	0.882449i
$36$		0			0
$37$	3.14537	+	3.14537i	0.517096	+	0.517096i	0.916691	−	0.399596i	$-0.130850\pi$
$37$	3.14537	+	3.14537i	0.517096	+	0.517096i	−0.399596	+	0.916691i	$0.630850\pi$
$38$		0			0
$39$	−4.10430			−0.657214
$40$		0			0
$41$	7.94060			1.24011			0.620057	−	0.784557i	$-0.287110\pi$
$41$	7.94060			1.24011			0.620057	+	0.784557i	$0.287110\pi$
$42$		0			0
$43$	−3.02093	−	3.02093i	−0.460687	−	0.460687i	0.438193	−	0.898881i	$-0.355618\pi$
$43$	−3.02093	−	3.02093i	−0.460687	−	0.460687i	−0.898881	+	0.438193i	$0.855618\pi$
$44$		0			0
$45$	−3.59339	−	3.59339i	−0.535670	−	0.535670i
$46$		0			0
$47$		−	3.58534i		−	0.522975i	−0.965207	−	0.261488i	$-0.915787\pi$
$47$		−	3.58534i		−	0.522975i	0.965207	−	0.261488i	$-0.0842130\pi$
$48$		0			0
$49$	5.11344			0.730492
$50$		0			0
$51$	−0.832709	−	0.832709i	−0.116603	−	0.116603i
$52$		0			0
$53$	−3.55950	−	3.55950i	−0.488935	−	0.488935i	0.419035	−	0.907970i	$-0.362369\pi$
$53$	−3.55950	−	3.55950i	−0.488935	−	0.488935i	−0.907970	+	0.419035i	$0.862369\pi$
$54$		0			0
$55$	11.0537			1.49048
$56$		0			0
$57$	2.99397	+	1.58737i	0.396562	+	0.210252i
$58$		0			0
$59$	−0.408827	+	0.408827i	−0.0532248	+	0.0532248i	−0.733218	−	0.679993i	$-0.761983\pi$
$59$	−0.408827	+	0.408827i	−0.0532248	+	0.0532248i	0.679993	+	0.733218i	$0.261983\pi$
$60$		0			0
$61$	2.47804	+	2.47804i	0.317281	+	0.317281i	0.847722	−	0.530441i	$-0.177974\pi$
$61$	2.47804	+	2.47804i	0.317281	+	0.317281i	−0.530441	+	0.847722i	$0.677974\pi$
$62$		0			0
$63$		−	8.33773i		−	1.05046i
$64$		0			0
$65$		−	11.1991i		−	1.38907i
$66$		0			0
$67$	−10.6691	−	10.6691i	−1.30344	−	1.30344i	−0.926056	−	0.377387i	$-0.876823\pi$
$67$	−10.6691	−	10.6691i	−1.30344	−	1.30344i	−0.377387	−	0.926056i	$-0.623177\pi$
$68$		0			0
$69$	−0.265116	−	0.265116i	−0.0319163	−	0.0319163i
$70$		0			0
$71$			5.59302i			0.663769i	0.943320	+	0.331885i	$0.107684\pi$
$71$			5.59302i			0.663769i	−0.943320	+	0.331885i	$0.892316\pi$
$72$		0			0
$73$		−	13.6019i		−	1.59198i	−0.605311	−	0.795989i	$-0.706951\pi$
$73$		−	13.6019i		−	1.59198i	0.605311	−	0.795989i	$-0.293049\pi$
$74$		0			0
$75$	0.274881	−	0.274881i	0.0317406	−	0.0317406i
$76$		0			0
$77$	12.8240	+	12.8240i	1.46143	+	1.46143i
$78$		0			0
$79$	−5.02027			−0.564825			−0.282412	−	0.959293i	$-0.591135\pi$
$79$	−5.02027			−0.564825			−0.282412	+	0.959293i	$0.591135\pi$
$80$		0			0
$81$	−3.92569			−0.436188
$82$		0			0
$83$	−10.3497	+	10.3497i	−1.13603	+	1.13603i	−0.146870	+	0.989156i	$0.546920\pi$
$83$	−10.3497	+	10.3497i	−1.13603	+	1.13603i	−0.989156	+	0.146870i	$0.953080\pi$
$84$		0			0
$85$	2.27214	−	2.27214i	0.246448	−	0.246448i
$86$		0			0
$87$	7.99007			0.856625
$88$		0			0
$89$	7.42593			0.787147			0.393573	−	0.919293i	$-0.371239\pi$
$89$	7.42593			0.787147			0.393573	+	0.919293i	$0.371239\pi$
$90$		0			0
$91$	12.9926	−	12.9926i	1.36199	−	1.36199i
$92$		0			0
$93$	−1.31550	−	1.31550i	−0.136411	−	0.136411i
$94$		0			0
$95$	−4.33131	+	8.16940i	−0.444383	+	0.838163i
$96$		0			0
$97$			3.70124i			0.375804i	0.982188	+	0.187902i	$0.0601688\pi$
$97$			3.70124i			0.375804i	−0.982188	+	0.187902i	$0.939831\pi$
$98$		0			0
$99$	8.82679	−	8.82679i	0.887126	−	0.887126i
$100$		0			0
$101$	4.78718	−	4.78718i	0.476342	−	0.476342i	−0.427618	−	0.903960i	$-0.640647\pi$
$101$	4.78718	−	4.78718i	0.476342	−	0.476342i	0.903960	+	0.427618i	$0.140647\pi$
$102$		0			0
$103$			11.4253i			1.12577i	0.826536	+	0.562885i	$0.190308\pi$
$103$			11.4253i			1.12577i	−0.826536	+	0.562885i	$0.809692\pi$
$104$		0			0
$105$	−5.73986			−0.560153
$106$		0			0
$107$	11.7874	−	11.7874i	1.13953	−	1.13953i	0.150999	−	0.988534i	$-0.451751\pi$
$107$	11.7874	−	11.7874i	1.13953	−	1.13953i	0.988534	−	0.150999i	$-0.0482490\pi$
$108$		0			0
$109$	−10.3205	+	10.3205i	−0.988523	+	0.988523i	−0.999935	−	0.0114120i	$-0.996367\pi$
$109$	−10.3205	+	10.3205i	−0.988523	+	0.988523i	0.0114120	+	0.999935i	$0.496367\pi$
$110$		0			0
$111$		−	3.45819i		−	0.328237i
$112$		0			0
$113$		−	14.0443i		−	1.32118i	−0.750747	−	0.660590i	$-0.770306\pi$
$113$		−	14.0443i		−	1.32118i	0.750747	−	0.660590i	$-0.229694\pi$
$114$		0			0
$115$	0.723400	−	0.723400i	0.0674574	−	0.0674574i
$116$		0			0
$117$	−8.94285	−	8.94285i	−0.826766	−	0.826766i
$118$		0			0
$119$	5.27205			0.483288
$120$		0			0
$121$			16.1523i			1.46839i
$122$		0			0
$123$	−4.36517	−	4.36517i	−0.393594	−	0.393594i
$124$		0			0
$125$	8.25002	+	8.25002i	0.737904	+	0.737904i
$126$		0			0
$127$	5.68283			0.504269			0.252135	−	0.967692i	$-0.418867\pi$
$127$	5.68283			0.504269			0.252135	+	0.967692i	$0.418867\pi$
$128$		0			0
$129$			3.32138i			0.292431i
$130$		0			0
$131$	−0.994673	+	0.994673i	−0.0869050	+	0.0869050i	−0.749223	−	0.662318i	$-0.769573\pi$
$131$	−0.994673	+	0.994673i	−0.0869050	+	0.0869050i	0.662318	+	0.749223i	$0.269573\pi$
$132$		0			0
$133$	−14.5027	+	4.45276i	−1.25754	+	0.386104i
$134$		0			0
$135$			8.89830i			0.765844i
$136$		0			0
$137$		−	10.6023i		−	0.905817i	−0.891557	−	0.452909i	$-0.850386\pi$
$137$		−	10.6023i		−	0.905817i	0.891557	−	0.452909i	$-0.149614\pi$
$138$		0			0
$139$	−9.25182	−	9.25182i	−0.784729	−	0.784729i	0.195895	−	0.980625i	$-0.437239\pi$
$139$	−9.25182	−	9.25182i	−0.784729	−	0.784729i	−0.980625	+	0.195895i	$0.937239\pi$
$140$		0			0
$141$	−1.97096	+	1.97096i	−0.165985	+	0.165985i
$142$		0			0
$143$	27.5093			2.30045
$144$		0			0
$145$			21.8018i			1.81054i
$146$		0			0
$147$	−2.81100	−	2.81100i	−0.231847	−	0.231847i
$148$		0			0
$149$	0.265791	−	0.265791i	0.0217745	−	0.0217745i	−0.696136	−	0.717910i	$-0.745099\pi$
$149$	0.265791	−	0.265791i	0.0217745	−	0.0217745i	0.717910	+	0.696136i	$0.245099\pi$
$150$		0			0
$151$		−	8.26373i		−	0.672493i	−0.941774	−	0.336246i	$-0.890843\pi$
$151$		−	8.26373i		−	0.672493i	0.941774	−	0.336246i	$-0.109157\pi$
$152$		0			0
$153$		−	3.62878i		−	0.293369i
$154$		0			0
$155$	3.58948	−	3.58948i	0.288314	−	0.288314i
$156$		0			0
$157$	−11.1943	−	11.1943i	−0.893405	−	0.893405i	0.101437	−	0.994842i	$-0.467656\pi$
$157$	−11.1943	−	11.1943i	−0.893405	−	0.893405i	−0.994842	+	0.101437i	$0.967656\pi$
$158$		0			0
$159$			3.91351i			0.310362i
$160$		0			0
$161$	1.67851			0.132285
$162$		0			0
$163$	−5.31053	+	5.31053i	−0.415953	+	0.415953i	−0.883806	−	0.467853i	$-0.845028\pi$
$163$	−5.31053	+	5.31053i	−0.415953	+	0.415953i	0.467853	+	0.883806i	$0.345028\pi$
$164$		0			0
$165$	−6.07653	−	6.07653i	−0.473058	−	0.473058i
$166$		0			0
$167$			9.02701i			0.698531i	0.937024	+	0.349266i	$0.113569\pi$
$167$			9.02701i			0.698531i	−0.937024	+	0.349266i	$0.886431\pi$
$168$		0			0
$169$		−	14.8710i		−	1.14393i
$170$		0			0
$171$	3.06486	+	9.98227i	0.234375	+	0.763363i
$172$		0			0
$173$	−7.82463	+	7.82463i	−0.594895	+	0.594895i	−0.938950	−	0.344054i	$-0.888200\pi$
$173$	−7.82463	+	7.82463i	−0.594895	+	0.594895i	0.344054	+	0.938950i	$0.388200\pi$
$174$		0			0
$175$			1.74033i			0.131557i
$176$		0			0
$177$	0.449487			0.0337855
$178$		0			0
$179$	2.95338	+	2.95338i	0.220746	+	0.220746i	0.808813	−	0.588067i	$-0.200111\pi$
$179$	2.95338	+	2.95338i	0.220746	+	0.220746i	−0.588067	+	0.808813i	$0.700111\pi$
$180$		0			0
$181$	6.95777	+	6.95777i	0.517167	+	0.517167i	0.916713	−	0.399546i	$-0.130832\pi$
$181$	6.95777	+	6.95777i	0.517167	+	0.517167i	−0.399546	+	0.916713i	$0.630832\pi$
$182$		0			0
$183$		−	2.72450i		−	0.201401i
$184$		0			0
$185$	9.43607			0.693754
$186$		0			0
$187$	5.58129	+	5.58129i	0.408144	+	0.408144i
$188$		0			0
$189$	−10.3234	+	10.3234i	−0.750914	+	0.750914i
$190$		0			0
$191$			3.91455i			0.283247i	0.989921	+	0.141624i	$0.0452322\pi$
$191$			3.91455i			0.283247i	−0.989921	+	0.141624i	$0.954768\pi$
$192$		0			0
$193$			17.5680i			1.26457i	0.774736	+	0.632285i	$0.217883\pi$
$193$			17.5680i			1.26457i	−0.774736	+	0.632285i	$0.782117\pi$
$194$		0			0
$195$	−6.15643	+	6.15643i	−0.440871	+	0.440871i
$196$		0			0
$197$	−16.2889	+	16.2889i	−1.16053	+	1.16053i	−0.176176	+	0.984359i	$0.556373\pi$
$197$	−16.2889	+	16.2889i	−1.16053	+	1.16053i	−0.984359	+	0.176176i	$0.943627\pi$
$198$		0			0
$199$	−21.6100			−1.53189			−0.765945	−	0.642906i	$-0.777728\pi$
$199$	−21.6100			−1.53189			−0.765945	+	0.642906i	$0.777728\pi$
$200$		0			0
$201$			11.7302i			0.827387i
$202$		0			0
$203$	−25.2934	+	25.2934i	−1.77525	+	1.77525i
$204$		0			0
$205$	11.9109	−	11.9109i	0.831891	−	0.831891i
$206$		0			0
$207$		−	1.15532i		−	0.0803004i
$208$		0			0
$209$	−20.0673	−	10.6394i	−1.38809	−	0.735944i
$210$		0			0
$211$	−16.6981	−	16.6981i	−1.14954	−	1.14954i	−0.986642	−	0.162901i	$-0.947915\pi$
$211$	−16.6981	−	16.6981i	−1.14954	−	1.14954i	−0.162901	−	0.986642i	$-0.552085\pi$
$212$		0			0
$213$	3.07464	−	3.07464i	0.210671	−	0.210671i
$214$		0			0
$215$	−9.06275			−0.618075
$216$		0			0
$217$	8.32868			0.565387
$218$		0			0
$219$	−7.47732	+	7.47732i	−0.505271	+	0.505271i
$220$		0			0
$221$	5.65467	−	5.65467i	0.380374	−	0.380374i
$222$		0			0
$223$	−1.63480			−0.109474			−0.0547372	−	0.998501i	$-0.517432\pi$
$223$	−1.63480			−0.109474			−0.0547372	+	0.998501i	$0.517432\pi$
$224$		0			0
$225$	1.19788			0.0798584
$226$		0			0
$227$	0.463368	+	0.463368i	0.0307548	+	0.0307548i	0.722317	−	0.691562i	$-0.243077\pi$
$227$	0.463368	+	0.463368i	0.0307548	+	0.0307548i	−0.691562	+	0.722317i	$0.743077\pi$
$228$		0			0
$229$	−8.36381	+	8.36381i	−0.552696	+	0.552696i	−0.927218	−	0.374522i	$-0.877807\pi$
$229$	−8.36381	+	8.36381i	−0.552696	+	0.552696i	0.374522	+	0.927218i	$0.377807\pi$
$230$		0			0
$231$		−	14.0994i		−	0.927671i
$232$		0			0
$233$			9.02414i			0.591191i	0.955313	+	0.295596i	$0.0955181\pi$
$233$			9.02414i			0.591191i	−0.955313	+	0.295596i	$0.904482\pi$
$234$		0			0
$235$	−5.37799	−	5.37799i	−0.350821	−	0.350821i
$236$		0			0
$237$	2.75978	+	2.75978i	0.179267	+	0.179267i
$238$		0			0
$239$		−	0.575236i		−	0.0372089i	−0.999827	−	0.0186045i	$-0.994078\pi$
$239$		−	0.575236i		−	0.0372089i	0.999827	−	0.0186045i	$-0.00592233\pi$
$240$		0			0
$241$			10.2254i			0.658678i	0.944212	+	0.329339i	$0.106826\pi$
$241$			10.2254i			0.658678i	−0.944212	+	0.329339i	$0.893174\pi$
$242$		0			0
$243$	11.0564	+	11.0564i	0.709268	+	0.709268i
$244$		0			0
$245$	7.67014	−	7.67014i	0.490027	−	0.490027i
$246$		0			0
$247$	−10.7793	+	20.3312i	−0.685871	+	1.29364i
$248$		0			0
$249$	11.3790			0.721116
$250$		0			0
$251$	9.67088	+	9.67088i	0.610421	+	0.610421i	0.943056	−	0.332635i	$-0.107938\pi$
$251$	9.67088	+	9.67088i	0.610421	+	0.610421i	−0.332635	+	0.943056i	$0.607938\pi$
$252$		0			0
$253$	1.77696	+	1.77696i	0.111717	+	0.111717i
$254$		0			0
$255$	−2.49812			−0.156438
$256$		0			0
$257$		−	18.8850i		−	1.17802i	−0.808127	−	0.589008i	$-0.799519\pi$
$257$		−	18.8850i		−	1.17802i	0.808127	−	0.589008i	$-0.200481\pi$
$258$		0			0
$259$	10.9473	+	10.9473i	0.680230	+	0.680230i
$260$		0			0
$261$	17.4095	+	17.4095i	1.07762	+	1.07762i
$262$		0			0
$263$	12.0775			0.744732			0.372366	−	0.928086i	$-0.378547\pi$
$263$	12.0775			0.744732			0.372366	+	0.928086i	$0.378547\pi$
$264$		0			0
$265$	−10.6785			−0.655973
$266$		0			0
$267$	−4.08224	−	4.08224i	−0.249829	−	0.249829i
$268$		0			0
$269$	−8.54359	+	8.54359i	−0.520912	+	0.520912i	−0.917847	−	0.396935i	$-0.870074\pi$
$269$	−8.54359	+	8.54359i	−0.520912	+	0.520912i	0.396935	+	0.917847i	$0.370074\pi$
$270$		0			0
$271$		−	8.30882i		−	0.504725i	−0.967633	−	0.252362i	$-0.918793\pi$
$271$		−	8.30882i		−	0.504725i	0.967633	−	0.252362i	$-0.0812075\pi$
$272$		0			0
$273$	−14.2848			−0.864553
$274$		0			0
$275$	−1.84241	+	1.84241i	−0.111102	+	0.111102i
$276$		0			0
$277$	9.55608	−	9.55608i	0.574169	−	0.574169i	−0.359122	−	0.933291i	$-0.616924\pi$
$277$	9.55608	−	9.55608i	0.574169	−	0.574169i	0.933291	+	0.359122i	$0.116924\pi$
$278$		0			0
$279$		−	5.73266i		−	0.343205i
$280$		0			0
$281$	−9.32142			−0.556069			−0.278035	−	0.960571i	$-0.589683\pi$
$281$	−9.32142			−0.556069			−0.278035	+	0.960571i	$0.589683\pi$
$282$		0			0
$283$	−4.19436	−	4.19436i	−0.249329	−	0.249329i	0.571366	−	0.820695i	$-0.306414\pi$
$283$	−4.19436	−	4.19436i	−0.249329	−	0.249329i	−0.820695	+	0.571366i	$0.806414\pi$
$284$		0			0
$285$	6.87199	−	2.10991i	0.407061	−	0.124980i
$286$		0			0
$287$	27.6368			1.63135
$288$		0			0
$289$	−14.7055			−0.865028
$290$		0			0
$291$	2.03467	−	2.03467i	0.119275	−	0.119275i
$292$		0			0
$293$	13.7219	+	13.7219i	0.801640	+	0.801640i	0.983352	−	0.181712i	$-0.0581639\pi$
$293$	13.7219	+	13.7219i	0.801640	+	0.801640i	−0.181712	+	0.983352i	$0.558164\pi$
$294$		0			0
$295$			1.22648i			0.0714083i
$296$		0			0
$297$	−21.8578			−1.26832
$298$		0			0
$299$	1.80032	−	1.80032i	0.104115	−	0.104115i
$300$		0			0
$301$	−10.5142	−	10.5142i	−0.606026	−	0.606026i
$302$		0			0
$303$	−5.26329			−0.302368
$304$		0			0
$305$	7.43410			0.425675
$306$		0			0
$307$	9.90107	+	9.90107i	0.565084	+	0.565084i	0.930747	−	0.365663i	$-0.119158\pi$
$307$	9.90107	+	9.90107i	0.565084	+	0.565084i	−0.365663	+	0.930747i	$0.619158\pi$
$308$		0			0
$309$	6.28081	−	6.28081i	0.357303	−	0.357303i
$310$		0			0
$311$	−17.4462			−0.989283			−0.494642	−	0.869097i	$-0.664701\pi$
$311$	−17.4462			−0.989283			−0.494642	+	0.869097i	$0.664701\pi$
$312$		0			0
$313$			9.14849i			0.517103i	0.965997	+	0.258552i	$0.0832452\pi$
$313$			9.14849i			0.517103i	−0.965997	+	0.258552i	$0.916755\pi$
$314$		0			0
$315$	−12.5066	−	12.5066i	−0.704665	−	0.704665i
$316$		0			0
$317$	19.0979	−	19.0979i	1.07264	−	1.07264i	0.0754990	−	0.997146i	$-0.475945\pi$
$317$	19.0979	−	19.0979i	1.07264	−	1.07264i	0.997146	−	0.0754990i	$-0.0240550\pi$
$318$		0			0
$319$	−53.5539			−2.99845
$320$		0			0
$321$	−12.9597			−0.723342
$322$		0			0
$323$	−6.31191	+	1.93795i	−0.351204	+	0.107830i
$324$		0			0
$325$	1.86664	+	1.86664i	0.103542	+	0.103542i
$326$		0			0
$327$	11.3469			0.627485
$328$		0			0
$329$		−	12.4785i		−	0.687964i
$330$		0			0
$331$	−3.09582	+	3.09582i	−0.170162	+	0.170162i	−0.787050	−	0.616889i	$-0.788393\pi$
$331$	−3.09582	+	3.09582i	−0.170162	+	0.170162i	0.616889	+	0.787050i	$0.288393\pi$
$332$		0			0
$333$	7.53504	−	7.53504i	0.412918	−	0.412918i
$334$		0			0
$335$	−32.0073			−1.74875
$336$		0			0
$337$		−	12.1693i		−	0.662903i	−0.943472	−	0.331452i	$-0.892462\pi$
$337$		−	12.1693i		−	0.662903i	0.943472	−	0.331452i	$-0.107538\pi$
$338$		0			0
$339$	−7.72056	+	7.72056i	−0.419323	+	0.419323i
$340$		0			0
$341$	8.81720	+	8.81720i	0.477478	+	0.477478i
$342$		0			0
$343$	−6.56604			−0.354533
$344$		0			0
$345$	−0.795346			−0.0428200
$346$		0			0
$347$	8.14038	+	8.14038i	0.436999	+	0.436999i	0.891001	−	0.454002i	$-0.150004\pi$
$347$	8.14038	+	8.14038i	0.436999	+	0.436999i	−0.454002	+	0.891001i	$0.650004\pi$
$348$		0			0
$349$	11.7882	+	11.7882i	0.631010	+	0.631010i	0.948321	−	0.317311i	$-0.102780\pi$
$349$	11.7882	+	11.7882i	0.631010	+	0.631010i	−0.317311	+	0.948321i	$0.602780\pi$
$350$		0			0
$351$			22.1452i			1.18202i
$352$		0			0
$353$	4.06776			0.216505			0.108253	−	0.994123i	$-0.465474\pi$
$353$	4.06776			0.216505			0.108253	+	0.994123i	$0.465474\pi$
$354$		0			0
$355$	8.38950	+	8.38950i	0.445268	+	0.445268i
$356$		0			0
$357$	−2.89819	−	2.89819i	−0.153389	−	0.153389i
$358$		0			0
$359$	7.30143			0.385355			0.192677	−	0.981262i	$-0.438283\pi$
$359$	7.30143			0.385355			0.192677	+	0.981262i	$0.438283\pi$
$360$		0			0
$361$	15.7264	−	10.6621i	0.827707	−	0.561161i
$362$		0			0
$363$	8.87939	−	8.87939i	0.466047	−	0.466047i
$364$		0			0
$365$	−20.4027	−	20.4027i	−1.06793	−	1.06793i
$366$		0			0
$367$		−	8.15940i		−	0.425917i	−0.977061	−	0.212959i	$-0.931690\pi$
$367$		−	8.15940i		−	0.425917i	0.977061	−	0.212959i	$-0.0683100\pi$
$368$		0			0
$369$		−	19.0225i		−	0.990272i
$370$		0			0
$371$	−12.3886	−	12.3886i	−0.643185	−	0.643185i
$372$		0			0
$373$	7.35016	+	7.35016i	0.380577	+	0.380577i	0.871310	−	0.490733i	$-0.163271\pi$
$373$	7.35016	+	7.35016i	0.380577	+	0.380577i	−0.490733	+	0.871310i	$0.663271\pi$
$374$		0			0
$375$		−	9.07053i		−	0.468400i
$376$		0			0
$377$			54.2581i			2.79443i
$378$		0			0
$379$	20.2426	−	20.2426i	1.03979	−	1.03979i	0.0406169	−	0.999175i	$-0.487068\pi$
$379$	20.2426	−	20.2426i	1.03979	−	1.03979i	0.999175	−	0.0406169i	$-0.0129323\pi$
$380$		0			0
$381$	−3.12401	−	3.12401i	−0.160048	−	0.160048i
$382$		0			0
$383$	18.4669			0.943615			0.471807	−	0.881702i	$-0.343602\pi$
$383$	18.4669			0.943615			0.471807	+	0.881702i	$0.343602\pi$
$384$		0			0
$385$	38.4718			1.96070
$386$		0			0
$387$	−7.23693	+	7.23693i	−0.367874	+	0.367874i
$388$		0			0
$389$	−16.3368	+	16.3368i	−0.828309	+	0.828309i	−0.987283	−	0.158974i	$-0.949181\pi$
$389$	−16.3368	+	16.3368i	−0.828309	+	0.828309i	0.158974	+	0.987283i	$0.449181\pi$
$390$		0			0
$391$	0.730525			0.0369442
$392$		0			0
$393$	1.09360			0.0551648
$394$		0			0
$395$	−7.53038	+	7.53038i	−0.378895	+	0.378895i
$396$		0			0
$397$	−13.0533	−	13.0533i	−0.655126	−	0.655126i	0.299097	−	0.954223i	$-0.403315\pi$
$397$	−13.0533	−	13.0533i	−0.655126	−	0.655126i	−0.954223	+	0.299097i	$0.903315\pi$
$398$		0			0
$399$	10.4203	+	5.52473i	0.521670	+	0.276582i
$400$		0			0
$401$			26.3416i			1.31544i	0.753263	+	0.657719i	$0.228479\pi$
$401$			26.3416i			1.31544i	−0.753263	+	0.657719i	$0.771521\pi$
$402$		0			0
$403$	8.93313	−	8.93313i	0.444991	−	0.444991i
$404$		0			0
$405$	−5.88852	+	5.88852i	−0.292603	+	0.292603i
$406$		0			0
$407$			23.1788i			1.14893i
$408$		0			0
$409$	0.821310			0.0406112			0.0203056	−	0.999794i	$-0.493536\pi$
$409$	0.821310			0.0406112			0.0203056	+	0.999794i	$0.493536\pi$
$410$		0			0
$411$	−5.82839	+	5.82839i	−0.287493	+	0.287493i
$412$		0			0
$413$	−1.42290	+	1.42290i	−0.0700162	+	0.0700162i
$414$		0			0
$415$			31.0489i			1.52413i
$416$		0			0
$417$			10.1720i			0.498123i
$418$		0			0
$419$	−2.51426	+	2.51426i	−0.122830	+	0.122830i	−0.765849	−	0.643020i	$-0.777681\pi$
$419$	−2.51426	+	2.51426i	−0.122830	+	0.122830i	0.643020	+	0.765849i	$0.277681\pi$
$420$		0			0
$421$	−9.64923	−	9.64923i	−0.470275	−	0.470275i	0.431729	−	0.902003i	$-0.357904\pi$
$421$	−9.64923	−	9.64923i	−0.470275	−	0.470275i	−0.902003	+	0.431729i	$0.857904\pi$
$422$		0			0
$423$	−8.58903			−0.417613
$424$		0			0
$425$			0.757432i			0.0367409i
$426$		0			0
$427$	8.62467	+	8.62467i	0.417377	+	0.417377i
$428$		0			0
$429$	−15.1226	−	15.1226i	−0.730128	−	0.730128i
$430$		0			0
$431$	5.87725			0.283097			0.141549	−	0.989931i	$-0.454792\pi$
$431$	5.87725			0.283097			0.141549	+	0.989931i	$0.454792\pi$
$432$		0			0
$433$			2.62302i			0.126054i	0.998012	+	0.0630271i	$0.0200754\pi$
$433$			2.62302i			0.126054i	−0.998012	+	0.0630271i	$0.979925\pi$
$434$		0			0
$435$	11.9851	−	11.9851i	0.574639	−	0.574639i
$436$		0			0
$437$	−2.00957	+	0.617000i	−0.0961310	+	0.0295151i
$438$		0			0
$439$		−	34.6282i		−	1.65272i	−0.563146	−	0.826358i	$-0.690409\pi$
$439$		−	34.6282i		−	1.65272i	0.563146	−	0.826358i	$-0.309591\pi$
$440$		0			0
$441$		−	12.2498i		−	0.583322i
$442$		0			0
$443$	−3.58478	−	3.58478i	−0.170318	−	0.170318i	0.616801	−	0.787119i	$-0.288428\pi$
$443$	−3.58478	−	3.58478i	−0.170318	−	0.170318i	−0.787119	+	0.616801i	$0.788428\pi$
$444$		0			0
$445$	11.1389	−	11.1389i	0.528032	−	0.528032i
$446$		0			0
$447$	−0.292226			−0.0138218
$448$		0			0
$449$			6.98073i			0.329441i	0.986340	+	0.164721i	$0.0526722\pi$
$449$			6.98073i			0.329441i	−0.986340	+	0.164721i	$0.947328\pi$
$450$		0			0
$451$	29.2578	+	29.2578i	1.37770	+	1.37770i
$452$		0			0
$453$	−4.54280	+	4.54280i	−0.213439	+	0.213439i
$454$		0			0
$455$		−	38.9776i		−	1.82730i
$456$		0			0
$457$		−	1.61947i		−	0.0757556i	−0.999282	−	0.0378778i	$-0.987940\pi$
$457$		−	1.61947i		−	0.0757556i	0.999282	−	0.0378778i	$-0.0120598\pi$
$458$		0			0
$459$	−4.49297	+	4.49297i	−0.209714	+	0.209714i
$460$		0			0
$461$	11.2678	+	11.2678i	0.524792	+	0.524792i	0.919015	−	0.394223i	$-0.128986\pi$
$461$	11.2678	+	11.2678i	0.524792	+	0.524792i	−0.394223	+	0.919015i	$0.628986\pi$
$462$		0			0
$463$			26.2153i			1.21833i	0.793043	+	0.609165i	$0.208495\pi$
$463$			26.2153i			1.21833i	−0.793043	+	0.609165i	$0.791505\pi$
$464$		0			0
$465$	−3.94648			−0.183013
$466$		0			0
$467$	−7.45814	+	7.45814i	−0.345121	+	0.345121i	−0.858289	−	0.513167i	$-0.828472\pi$
$467$	−7.45814	+	7.45814i	−0.345121	+	0.345121i	0.513167	+	0.858289i	$0.328472\pi$
$468$		0			0
$469$	−37.1333	−	37.1333i	−1.71465	−	1.71465i
$470$		0			0
$471$			12.3077i			0.567107i
$472$		0			0
$473$		−	22.2617i		−	1.02360i
$474$		0			0
$475$	−0.639726	−	2.08359i	−0.0293526	−	0.0956018i
$476$		0			0
$477$	−8.52714	+	8.52714i	−0.390431	+	0.390431i
$478$		0			0
$479$		−	22.0408i		−	1.00707i	−0.863975	−	0.503534i	$-0.832033\pi$
$479$		−	22.0408i		−	1.00707i	0.863975	−	0.503534i	$-0.167967\pi$
$480$		0			0
$481$	23.4835			1.07076
$482$		0			0
$483$	−0.922721	−	0.922721i	−0.0419853	−	0.0419853i
$484$		0			0
$485$	5.55184	+	5.55184i	0.252096	+	0.252096i
$486$		0			0
$487$			21.9766i			0.995856i	0.867218	+	0.497928i	$0.165906\pi$
$487$			21.9766i			0.995856i	−0.867218	+	0.497928i	$0.834094\pi$
$488$		0			0
$489$	5.83869			0.264035
$490$		0			0
$491$	17.7224	+	17.7224i	0.799803	+	0.799803i	0.983064	−	0.183261i	$-0.0586655\pi$
$491$	17.7224	+	17.7224i	0.799803	+	0.799803i	−0.183261	+	0.983064i	$0.558665\pi$
$492$		0			0
$493$	−11.0083	+	11.0083i	−0.495787	+	0.495787i
$494$		0			0
$495$		−	26.4803i		−	1.19020i
$496$		0			0
$497$			19.4662i			0.873176i
$498$		0			0
$499$	30.3549	−	30.3549i	1.35887	−	1.35887i	0.483562	−	0.875310i	$-0.339343\pi$
$499$	30.3549	−	30.3549i	1.35887	−	1.35887i	0.875310	−	0.483562i	$-0.160657\pi$
$500$		0			0
$501$	4.96240	−	4.96240i	0.221704	−	0.221704i
$502$		0			0
$503$	−7.49320			−0.334105			−0.167053	−	0.985948i	$-0.553425\pi$
$503$	−7.49320			−0.334105			−0.167053	+	0.985948i	$0.553425\pi$
$504$		0			0
$505$		−	14.3615i		−	0.639077i
$506$		0			0
$507$	−8.17502	+	8.17502i	−0.363066	+	0.363066i
$508$		0			0
$509$	21.4192	−	21.4192i	0.949391	−	0.949391i	−0.0493888	−	0.998780i	$-0.515727\pi$
$509$	21.4192	−	21.4192i	0.949391	−	0.949391i	0.998780	+	0.0493888i	$0.0157273\pi$
$510$		0			0
$511$		−	47.3405i		−	2.09422i
$512$		0			0
$513$	8.56479	−	16.1543i	0.378145	−	0.713229i
$514$		0			0
$515$	17.1379	+	17.1379i	0.755186	+	0.755186i
$516$		0			0
$517$	13.2105	−	13.2105i	0.580996	−	0.580996i
$518$		0			0
$519$	8.60283			0.377622
$520$		0			0
$521$	−28.5238			−1.24965			−0.624824	−	0.780765i	$-0.714829\pi$
$521$	−28.5238			−1.24965			−0.624824	+	0.780765i	$0.714829\pi$
$522$		0			0
$523$	2.87573	−	2.87573i	0.125747	−	0.125747i	−0.641433	−	0.767179i	$-0.721660\pi$
$523$	2.87573	−	2.87573i	0.125747	−	0.125747i	0.767179	+	0.641433i	$0.221660\pi$
$524$		0			0
$525$	0.956708	−	0.956708i	0.0417541	−	0.0417541i
$526$		0			0
$527$	3.62483			0.157900
$528$		0			0
$529$	−22.7674			−0.989888
$530$		0			0
$531$	0.979386	+	0.979386i	0.0425017	+	0.0425017i
$532$		0			0
$533$	29.6425	−	29.6425i	1.28396	−	1.28396i
$534$		0			0
$535$		−	35.3621i		−	1.52884i
$536$		0			0
$537$		−	3.24711i		−	0.140123i
$538$		0			0
$539$	18.8409	+	18.8409i	0.811536	+	0.811536i
$540$		0			0
$541$	1.13344	+	1.13344i	0.0487304	+	0.0487304i	0.731052	−	0.682322i	$-0.239030\pi$
$541$	1.13344	+	1.13344i	0.0487304	+	0.0487304i	−0.682322	+	0.731052i	$0.739030\pi$
$542$		0			0
$543$		−	7.64975i		−	0.328282i
$544$		0			0
$545$			30.9613i			1.32624i
$546$		0			0
$547$	−13.4249	−	13.4249i	−0.574008	−	0.574008i	0.359238	−	0.933246i	$-0.383037\pi$
$547$	−13.4249	−	13.4249i	−0.574008	−	0.574008i	−0.933246	+	0.359238i	$0.883037\pi$
$548$		0			0
$549$	5.93640	−	5.93640i	0.253359	−	0.253359i
$550$		0			0
$551$	20.9847	−	39.5798i	0.893977	−	1.68616i
$552$		0			0
$553$	−17.4727			−0.743017
$554$		0			0
$555$	−5.18727	−	5.18727i	−0.220187	−	0.220187i
$556$		0			0
$557$	7.79578	+	7.79578i	0.330318	+	0.330318i	0.852707	−	0.522389i	$-0.174959\pi$
$557$	7.79578	+	7.79578i	0.330318	+	0.330318i	−0.522389	+	0.852707i	$0.674959\pi$
$558$		0			0
$559$	−22.5544			−0.953951
$560$		0			0
$561$		−	6.13638i		−	0.259078i
$562$		0			0
$563$	−4.84245	−	4.84245i	−0.204085	−	0.204085i	0.597663	−	0.801748i	$-0.296096\pi$
$563$	−4.84245	−	4.84245i	−0.204085	−	0.204085i	−0.801748	+	0.597663i	$0.796096\pi$
$564$		0			0
$565$	−21.0664	−	21.0664i	−0.886272	−	0.886272i
$566$		0			0
$567$	−13.6631			−0.573797
$568$		0			0
$569$	−20.2563			−0.849190			−0.424595	−	0.905383i	$-0.639583\pi$
$569$	−20.2563			−0.849190			−0.424595	+	0.905383i	$0.639583\pi$
$570$		0			0
$571$	−18.3379	−	18.3379i	−0.767419	−	0.767419i	0.210233	−	0.977651i	$-0.432578\pi$
$571$	−18.3379	−	18.3379i	−0.767419	−	0.767419i	−0.977651	+	0.210233i	$0.932578\pi$
$572$		0			0
$573$	2.15194	−	2.15194i	0.0898985	−	0.0898985i
$574$		0			0
$575$			0.241150i			0.0100566i
$576$		0			0
$577$	−5.18354			−0.215793			−0.107897	−	0.994162i	$-0.534412\pi$
$577$	−5.18354			−0.215793			−0.107897	+	0.994162i	$0.534412\pi$
$578$		0			0
$579$	9.65759	−	9.65759i	0.401356	−	0.401356i
$580$		0			0
$581$	−36.0214	+	36.0214i	−1.49442	+	1.49442i
$582$		0			0
$583$		−	26.2306i		−	1.08636i
$584$		0			0
$585$	−26.8284			−1.10922
$586$		0			0
$587$	1.88371	+	1.88371i	0.0777491	+	0.0777491i	0.744912	−	0.667163i	$-0.232491\pi$
$587$	1.88371	+	1.88371i	0.0777491	+	0.0777491i	−0.667163	+	0.744912i	$0.732491\pi$
$588$		0			0
$589$	−9.97142	+	3.06153i	−0.410865	+	0.126148i
$590$		0			0
$591$	17.9089			0.736673
$592$		0			0
$593$	44.2304			1.81632			0.908162	−	0.418619i	$-0.137486\pi$
$593$	44.2304			1.81632			0.908162	+	0.418619i	$0.137486\pi$
$594$		0			0
$595$	7.90805	−	7.90805i	0.324198	−	0.324198i
$596$		0			0
$597$	11.8796	+	11.8796i	0.486199	+	0.486199i
$598$		0			0
$599$			25.1207i			1.02640i	0.858268	+	0.513202i	$0.171541\pi$
$599$			25.1207i			1.02640i	−0.858268	+	0.513202i	$0.828459\pi$
$600$		0			0
$601$	−21.0826			−0.859976			−0.429988	−	0.902835i	$-0.641482\pi$
$601$	−21.0826			−0.859976			−0.429988	+	0.902835i	$0.641482\pi$
$602$		0			0
$603$	−25.5590	+	25.5590i	−1.04084	+	1.04084i
$604$		0			0
$605$	24.2284	+	24.2284i	0.985026	+	0.985026i
$606$		0			0
$607$	−15.2446			−0.618759			−0.309379	−	0.950939i	$-0.600121\pi$
$607$	−15.2446			−0.618759			−0.309379	+	0.950939i	$0.600121\pi$
$608$		0			0
$609$	27.8089			1.12687
$610$		0			0
$611$	−13.3842	−	13.3842i	−0.541466	−	0.541466i
$612$		0			0
$613$	21.4123	−	21.4123i	0.864833	−	0.864833i	−0.127062	−	0.991895i	$-0.540555\pi$
$613$	21.4123	−	21.4123i	0.864833	−	0.864833i	0.991895	+	0.127062i	$0.0405546\pi$
$614$		0			0
$615$	−13.0955			−0.528060
$616$		0			0
$617$			47.5231i			1.91321i	0.291390	+	0.956604i	$0.405882\pi$
$617$			47.5231i			1.91321i	−0.291390	+	0.956604i	$0.594118\pi$
$618$		0			0
$619$	1.43559	+	1.43559i	0.0577014	+	0.0577014i	0.735369	−	0.677667i	$-0.237009\pi$
$619$	1.43559	+	1.43559i	0.0577014	+	0.0577014i	−0.677667	+	0.735369i	$0.737009\pi$
$620$		0			0
$621$	−1.43046	+	1.43046i	−0.0574024	+	0.0574024i
$622$		0			0
$623$	25.8455			1.03548
$624$		0			0
$625$	22.2498			0.889992
$626$		0			0
$627$	5.18277	+	16.8803i	0.206980	+	0.674136i
$628$		0			0
$629$	4.76450	+	4.76450i	0.189973	+	0.189973i
$630$		0			0
$631$	0.490988			0.0195459			0.00977296	−	0.999952i	$-0.496889\pi$
$631$	0.490988			0.0195459			0.00977296	+	0.999952i	$0.496889\pi$
$632$		0			0
$633$			18.3588i			0.729696i
$634$		0			0
$635$	8.52421	−	8.52421i	0.338273	−	0.338273i
$636$		0			0
$637$	19.0886	−	19.0886i	0.756320	−	0.756320i
$638$		0			0
$639$	13.3986			0.530042
$640$		0			0
$641$		−	16.9893i		−	0.671039i	−0.942033	−	0.335519i	$-0.891088\pi$
$641$		−	16.9893i		−	0.671039i	0.942033	−	0.335519i	$-0.108912\pi$
$642$		0			0
$643$	−12.5699	+	12.5699i	−0.495709	+	0.495709i	−0.910099	−	0.414390i	$-0.863995\pi$
$643$	−12.5699	+	12.5699i	−0.495709	+	0.495709i	0.414390	+	0.910099i	$0.363995\pi$
$644$		0			0
$645$	4.98204	+	4.98204i	0.196168	+	0.196168i
$646$		0			0
$647$	37.3005			1.46643			0.733217	−	0.679995i	$-0.238018\pi$
$647$	37.3005			1.46643			0.733217	+	0.679995i	$0.238018\pi$
$648$		0			0
$649$	−3.01272			−0.118260
$650$		0			0
$651$	−4.57850	−	4.57850i	−0.179446	−	0.179446i
$652$		0			0
$653$	−19.3569	−	19.3569i	−0.757496	−	0.757496i	0.218370	−	0.975866i	$-0.429926\pi$
$653$	−19.3569	−	19.3569i	−0.757496	−	0.757496i	−0.975866	+	0.218370i	$0.929926\pi$
$654$		0			0
$655$			2.98401i			0.116595i
$656$		0			0
$657$	−32.5846			−1.27125
$658$		0			0
$659$	27.9807	+	27.9807i	1.08997	+	1.08997i	0.995530	+	0.0944429i	$0.0301070\pi$
$659$	27.9807	+	27.9807i	1.08997	+	1.08997i	0.0944429	+	0.995530i	$0.469893\pi$
$660$		0			0
$661$	27.6554	+	27.6554i	1.07567	+	1.07567i	0.996892	+	0.0787770i	$0.0251015\pi$
$661$	27.6554	+	27.6554i	1.07567	+	1.07567i	0.0787770	+	0.996892i	$0.474898\pi$
$662$		0			0
$663$	−6.21706			−0.241451
$664$		0			0
$665$	−15.0749	+	28.4331i	−0.584578	+	1.10259i
$666$		0			0
$667$	−3.50479	+	3.50479i	−0.135706	+	0.135706i
$668$		0			0
$669$	0.898696	+	0.898696i	0.0347456	+	0.0347456i
$670$		0			0
$671$			18.2611i			0.704963i
$672$		0			0
$673$		−	10.1211i		−	0.390141i	−0.980789	−	0.195070i	$-0.937506\pi$
$673$		−	10.1211i		−	0.390141i	0.980789	−	0.195070i	$-0.0624935\pi$
$674$		0			0
$675$	−1.48315	−	1.48315i	−0.0570865	−	0.0570865i
$676$		0			0
$677$	22.3139	+	22.3139i	0.857593	+	0.857593i	0.991054	−	0.133461i	$-0.0426090\pi$
$677$	22.3139	+	22.3139i	0.857593	+	0.857593i	−0.133461	+	0.991054i	$0.542609\pi$
$678$		0			0
$679$			12.8819i			0.494363i
$680$		0			0
$681$		−	0.509452i		−	0.0195223i
$682$		0			0
$683$	14.4876	−	14.4876i	0.554354	−	0.554354i	−0.373340	−	0.927695i	$-0.621788\pi$
$683$	14.4876	−	14.4876i	0.554354	−	0.554354i	0.927695	+	0.373340i	$0.121788\pi$
$684$		0			0
$685$	−15.9034	−	15.9034i	−0.607638	−	0.607638i
$686$		0			0
$687$	9.19563			0.350835
$688$		0			0
$689$	−26.5755			−1.01244
$690$		0			0
$691$	24.4373	−	24.4373i	0.929638	−	0.929638i	−0.0680444	−	0.997682i	$-0.521676\pi$
$691$	24.4373	−	24.4373i	0.929638	−	0.929638i	0.997682	+	0.0680444i	$0.0216759\pi$
$692$		0			0
$693$	30.7211	−	30.7211i	1.16700	−	1.16700i
$694$		0			0
$695$	−27.7554			−1.05282
$696$		0			0
$697$	12.0282			0.455599
$698$		0			0
$699$	4.96082	−	4.96082i	0.187635	−	0.187635i
$700$		0			0
$701$	16.1800	+	16.1800i	0.611110	+	0.611110i	0.943235	−	0.332125i	$-0.107766\pi$
$701$	16.1800	+	16.1800i	0.611110	+	0.611110i	−0.332125	+	0.943235i	$0.607766\pi$
$702$		0			0
$703$	−17.1306	−	9.08241i	−0.646092	−	0.342550i
$704$		0			0
$705$			5.91285i			0.222691i
$706$		0			0
$707$	16.6615	−	16.6615i	0.626619	−	0.626619i
$708$		0			0
$709$	11.6536	−	11.6536i	0.437660	−	0.437660i	−0.453563	−	0.891224i	$-0.649847\pi$
$709$	11.6536	−	11.6536i	0.437660	−	0.437660i	0.891224	+	0.453563i	$0.149847\pi$
$710$		0			0
$711$			12.0266i			0.451031i
$712$		0			0
$713$	1.15407			0.0432202
$714$		0			0
$715$	41.2639	−	41.2639i	1.54318	−	1.54318i
$716$		0			0
$717$	−0.316223	+	0.316223i	−0.0118096	+	0.0118096i
$718$		0			0
$719$			49.0662i			1.82986i	0.403611	+	0.914931i	$0.367755\pi$
$719$			49.0662i			1.82986i	−0.403611	+	0.914931i	$0.632245\pi$
$720$		0			0
$721$			39.7651i			1.48093i
$722$		0			0
$723$	5.62120	−	5.62120i	0.209055	−	0.209055i
$724$		0			0
$725$	−3.63388	−	3.63388i	−0.134959	−	0.134959i
$726$		0			0
$727$	10.6283			0.394182			0.197091	−	0.980385i	$-0.436851\pi$
$727$	10.6283			0.394182			0.197091	+	0.980385i	$0.436851\pi$
$728$		0			0
$729$		−	0.378928i		−	0.0140344i
$730$		0			0
$731$	−4.57600	−	4.57600i	−0.169250	−	0.169250i
$732$		0			0
$733$	36.3785	+	36.3785i	1.34367	+	1.34367i	0.892375	+	0.451295i	$0.149038\pi$
$733$	36.3785	+	36.3785i	1.34367	+	1.34367i	0.451295	+	0.892375i	$0.350962\pi$
$734$		0			0
$735$	−8.43297			−0.311055
$736$		0			0
$737$		−	78.6227i		−	2.89610i
$738$		0			0
$739$	9.52584	−	9.52584i	0.350414	−	0.350414i	−0.509850	−	0.860263i	$-0.670299\pi$
$739$	9.52584	−	9.52584i	0.350414	−	0.350414i	0.860263	+	0.509850i	$0.170299\pi$
$740$		0			0
$741$	17.1023	−	5.25092i	0.628268	−	0.192897i
$742$		0			0
$743$		−	15.1555i		−	0.556001i	−0.960581	−	0.278000i	$-0.910328\pi$
$743$		−	15.1555i		−	0.556001i	0.960581	−	0.278000i	$-0.0896716\pi$
$744$		0			0
$745$		−	0.797372i		−	0.0292134i
$746$		0			0
$747$	24.7937	+	24.7937i	0.907154	+	0.907154i
$748$		0			0
$749$	41.0254	−	41.0254i	1.49903	−	1.49903i
$750$		0			0
$751$	−40.2293			−1.46799			−0.733994	−	0.679156i	$-0.762346\pi$
$751$	−40.2293			−1.46799			−0.733994	+	0.679156i	$0.762346\pi$
$752$		0			0
$753$		−	10.6327i		−	0.387477i
$754$		0			0
$755$	−12.3955	−	12.3955i	−0.451120	−	0.451120i
$756$		0			0
$757$	−35.3310	+	35.3310i	−1.28413	+	1.28413i	−0.345827	+	0.938298i	$0.612402\pi$
$757$	−35.3310	+	35.3310i	−1.28413	+	1.28413i	−0.938298	+	0.345827i	$0.887598\pi$
$758$		0			0
$759$		−	1.95369i		−	0.0709144i
$760$		0			0
$761$			5.10254i			0.184967i	0.995714	+	0.0924834i	$0.0294805\pi$
$761$			5.10254i			0.184967i	−0.995714	+	0.0924834i	$0.970519\pi$
$762$		0			0
$763$	−35.9198	+	35.9198i	−1.30038	+	1.30038i
$764$		0			0
$765$	−5.44314	−	5.44314i	−0.196797	−	0.196797i
$766$		0			0
$767$			3.05233i			0.110213i
$768$		0			0
$769$	35.3943			1.27635			0.638175	−	0.769891i	$-0.279690\pi$
$769$	35.3943			1.27635			0.638175	+	0.769891i	$0.279690\pi$
$770$		0			0
$771$	−10.3816	+	10.3816i	−0.373885	+	0.373885i
$772$		0			0
$773$	−12.5110	−	12.5110i	−0.449991	−	0.449991i	0.445360	−	0.895351i	$-0.353076\pi$
$773$	−12.5110	−	12.5110i	−0.449991	−	0.449991i	−0.895351	+	0.445360i	$0.853076\pi$
$774$		0			0
$775$			1.19658i			0.0429823i
$776$		0			0
$777$		−	12.0360i		−	0.431790i
$778$		0			0
$779$	−33.0878	+	10.1590i	−1.18550	+	0.363983i
$780$		0			0
$781$	−20.6080	+	20.6080i	−0.737411	+	0.737411i
$782$		0			0
$783$		−	43.1112i		−	1.54067i
$784$		0			0
$785$	−33.5829			−1.19862
$786$		0			0
$787$	0.304154	+	0.304154i	0.0108419	+	0.0108419i	0.712507	−	0.701665i	$-0.247560\pi$
$787$	0.304154	+	0.304154i	0.0108419	+	0.0108419i	−0.701665	+	0.712507i	$0.747560\pi$
$788$		0			0
$789$	−6.63935	−	6.63935i	−0.236367	−	0.236367i
$790$		0			0
$791$		−	48.8805i		−	1.73799i
$792$		0			0
$793$	18.5012			0.656998
$794$		0			0
$795$	5.87025	+	5.87025i	0.208196	+	0.208196i
$796$		0			0
$797$	10.6768	−	10.6768i	0.378190	−	0.378190i	−0.492259	−	0.870449i	$-0.663829\pi$
$797$	10.6768	−	10.6768i	0.378190	−	0.378190i	0.870449	+	0.492259i	$0.163829\pi$
$798$		0			0
$799$		−	5.43095i		−	0.192133i
$800$		0			0
$801$		−	17.7895i		−	0.628563i
$802$		0			0
$803$	50.1173	−	50.1173i	1.76860	−	1.76860i
$804$		0			0
$805$	2.51775	−	2.51775i	0.0887390	−	0.0887390i
$806$		0			0
$807$	9.39329			0.330660
$808$		0			0
$809$		−	9.58027i		−	0.336824i	−0.985717	−	0.168412i	$-0.946136\pi$
$809$		−	9.58027i		−	0.336824i	0.985717	−	0.168412i	$-0.0538640\pi$
$810$		0			0
$811$	−22.4309	+	22.4309i	−0.787654	+	0.787654i	−0.981109	−	0.193455i	$-0.938031\pi$
$811$	−22.4309	+	22.4309i	−0.787654	+	0.787654i	0.193455	+	0.981109i	$0.438031\pi$
$812$		0			0
$813$	−4.56759	+	4.56759i	−0.160192	+	0.160192i
$814$		0			0
$815$			15.9315i			0.558058i
$816$		0			0
$817$	16.4528	+	8.72308i	0.575612	+	0.305182i
$818$		0			0
$819$	−31.1250	−	31.1250i	−1.08760	−	1.08760i
$820$		0			0
$821$	24.5153	−	24.5153i	0.855590	−	0.855590i	−0.135225	−	0.990815i	$-0.543176\pi$
$821$	24.5153	−	24.5153i	0.855590	−	0.855590i	0.990815	+	0.135225i	$0.0431756\pi$
$822$		0			0
$823$	23.3599			0.814274			0.407137	−	0.913367i	$-0.366527\pi$
$823$	23.3599			0.814274			0.407137	+	0.913367i	$0.366527\pi$
$824$		0			0
$825$	2.02565			0.0705240
$826$		0			0
$827$	24.9222	−	24.9222i	0.866631	−	0.866631i	−0.125467	−	0.992098i	$-0.540043\pi$
$827$	24.9222	−	24.9222i	0.866631	−	0.866631i	0.992098	+	0.125467i	$0.0400428\pi$
$828$		0			0
$829$	−8.85684	+	8.85684i	−0.307611	+	0.307611i	−0.843982	−	0.536371i	$-0.819795\pi$
$829$	−8.85684	+	8.85684i	−0.307611	+	0.307611i	0.536371	+	0.843982i	$0.319795\pi$
$830$		0			0
$831$	−10.5065			−0.364466
$832$		0			0
$833$	7.74568			0.268372
$834$		0			0
$835$	13.5405	+	13.5405i	0.468587	+	0.468587i
$836$		0			0
$837$	−7.09789	+	7.09789i	−0.245339	+	0.245339i
$838$		0			0
$839$			23.5109i			0.811686i	0.913943	+	0.405843i	$0.133022\pi$
$839$			23.5109i			0.811686i	−0.913943	+	0.405843i	$0.866978\pi$
$840$		0			0
$841$		−	76.6272i		−	2.64232i
$842$		0			0
$843$	5.12424	+	5.12424i	0.176488	+	0.176488i
$844$		0			0
$845$	−22.3065	−	22.3065i	−0.767366	−	0.767366i
$846$		0			0
$847$			56.2172i			1.93165i
$848$		0			0
$849$			4.61151i			0.158267i
$850$		0			0
$851$	1.51691	+	1.51691i	0.0519991	+	0.0519991i
$852$		0			0
$853$	−2.97132	+	2.97132i	−0.101736	+	0.101736i	−0.756143	−	0.654407i	$-0.772918\pi$
$853$	−2.97132	+	2.97132i	−0.101736	+	0.101736i	0.654407	+	0.756143i	$0.272918\pi$
$854$		0			0
$855$	19.5706	+	10.3761i	0.669301	+	0.354854i
$856$		0			0
$857$	−35.1401			−1.20036			−0.600181	−	0.799864i	$-0.704905\pi$
$857$	−35.1401			−1.20036			−0.600181	+	0.799864i	$0.704905\pi$
$858$		0			0
$859$	−34.6341	−	34.6341i	−1.18170	−	1.18170i	−0.979304	−	0.202395i	$-0.935128\pi$
$859$	−34.6341	−	34.6341i	−1.18170	−	1.18170i	−0.202395	−	0.979304i	$-0.564872\pi$
$860$		0			0
$861$	−15.1927	−	15.1927i	−0.517766	−	0.517766i
$862$		0			0
$863$	−22.4122			−0.762921			−0.381461	−	0.924385i	$-0.624579\pi$
$863$	−22.4122			−0.762921			−0.381461	+	0.924385i	$0.624579\pi$
$864$		0			0
$865$			23.4738i			0.798133i
$866$		0			0
$867$	8.08401	+	8.08401i	0.274547	+	0.274547i
$868$		0			0
$869$	−18.4976	−	18.4976i	−0.627489	−	0.627489i
$870$		0			0
$871$	−79.6564			−2.69906
$872$		0			0
$873$	8.86669			0.300092
$874$		0			0
$875$	28.7137	+	28.7137i	0.970699	+	0.970699i
$876$		0			0
$877$	19.6785	−	19.6785i	0.664496	−	0.664496i	−0.291940	−	0.956437i	$-0.594301\pi$
$877$	19.6785	−	19.6785i	0.664496	−	0.664496i	0.956437	+	0.291940i	$0.0943009\pi$
$878$		0			0
$879$		−	15.0866i		−	0.508857i
$880$		0			0
$881$	6.88025			0.231801			0.115901	−	0.993261i	$-0.463025\pi$
$881$	6.88025			0.231801			0.115901	+	0.993261i	$0.463025\pi$
$882$		0			0
$883$	−32.4715	+	32.4715i	−1.09275	+	1.09275i	−0.0975191	+	0.995234i	$0.531091\pi$
$883$	−32.4715	+	32.4715i	−1.09275	+	1.09275i	−0.995234	+	0.0975191i	$0.968909\pi$
$884$		0			0
$885$	0.674228	−	0.674228i	0.0226639	−	0.0226639i
$886$		0			0
$887$		−	11.4832i		−	0.385567i	−0.981241	−	0.192783i	$-0.938249\pi$
$887$		−	11.4832i		−	0.385567i	0.981241	−	0.192783i	$-0.0617515\pi$
$888$		0			0
$889$	19.7787			0.663357
$890$		0			0
$891$	−14.4646	−	14.4646i	−0.484581	−	0.484581i
$892$		0			0
$893$	4.58697	+	14.9398i	0.153497	+	0.499942i
$894$		0			0
$895$	8.86010			0.296161
$896$		0			0
$897$	−1.97938			−0.0660894
$898$		0			0
$899$	−17.3906	+	17.3906i	−0.580010	+	0.580010i
$900$		0			0
$901$	−5.39182	−	5.39182i	−0.179627	−	0.179627i
$902$		0			0
$903$			11.5598i			0.384687i
$904$		0			0
$905$	20.8732			0.693849
$906$		0			0
$907$	1.11054	−	1.11054i	0.0368750	−	0.0368750i	−0.688429	−	0.725304i	$-0.741699\pi$
$907$	1.11054	−	1.11054i	0.0368750	−	0.0368750i	0.725304	+	0.688429i	$0.241699\pi$
$908$		0			0
$909$	−11.4682	−	11.4682i	−0.380375	−	0.380375i
$910$		0			0
$911$	4.52560			0.149940			0.0749699	−	0.997186i	$-0.476114\pi$
$911$	4.52560			0.149940			0.0749699	+	0.997186i	$0.476114\pi$
$912$		0			0
$913$	−76.2686			−2.52412
$914$		0			0
$915$	−4.08673	−	4.08673i	−0.135103	−	0.135103i
$916$		0			0
$917$	−3.46190	+	3.46190i	−0.114322	+	0.114322i
$918$		0			0
$919$	34.2586			1.13009			0.565043	−	0.825062i	$-0.308860\pi$
$919$	34.2586			1.13009			0.565043	+	0.825062i	$0.308860\pi$
$920$		0			0
$921$		−	10.8858i		−	0.358699i
$922$		0			0
$923$	20.8789	+	20.8789i	0.687238	+	0.687238i
$924$		0			0
$925$	−1.57279	+	1.57279i	−0.0517129	+	0.0517129i
$926$		0			0
$927$	27.3705			0.898964
$928$		0			0
$929$	13.5486			0.444516			0.222258	−	0.974988i	$-0.428657\pi$
$929$	13.5486			0.444516			0.222258	+	0.974988i	$0.428657\pi$
$930$		0			0
$931$	−21.3073	+	6.54198i	−0.698319	+	0.214405i
$932$		0			0
$933$	9.59066	+	9.59066i	0.313984	+	0.313984i
$934$		0			0
$935$	16.7438			0.547581
$936$		0			0
$937$		−	32.7572i		−	1.07013i	−0.844811	−	0.535065i	$-0.820287\pi$
$937$		−	32.7572i		−	1.07013i	0.844811	−	0.535065i	$-0.179713\pi$
$938$		0			0
$939$	5.02918	−	5.02918i	0.164121	−	0.164121i
$940$		0			0
$941$	−24.7542	+	24.7542i	−0.806965	+	0.806965i	−0.984173	−	0.177208i	$-0.943293\pi$
$941$	−24.7542	+	24.7542i	−0.806965	+	0.806965i	0.177208	+	0.984173i	$0.443293\pi$
$942$		0			0
$943$	3.82950			0.124706
$944$		0			0
$945$			30.9700i			1.00745i
$946$		0			0
$947$	−27.4034	+	27.4034i	−0.890491	+	0.890491i	−0.994569	−	0.104078i	$-0.966811\pi$
$947$	−27.4034	+	27.4034i	−0.890491	+	0.890491i	0.104078	+	0.994569i	$0.466811\pi$
$948$		0			0
$949$	−50.7762	−	50.7762i	−1.64827	−	1.64827i
$950$		0			0
$951$	−20.9973			−0.680884
$952$		0			0
$953$	35.9912			1.16587			0.582935	−	0.812519i	$-0.301904\pi$
$953$	35.9912			1.16587			0.582935	+	0.812519i	$0.301904\pi$
$954$		0			0
$955$	5.87181	+	5.87181i	0.190007	+	0.190007i
$956$		0			0
$957$	29.4401	+	29.4401i	0.951663	+	0.951663i
$958$		0			0
$959$		−	36.9007i		−	1.19159i
$960$		0			0
$961$	−25.2736			−0.815276
$962$		0			0
$963$	−28.2379	−	28.2379i	−0.909955	−	0.909955i
$964$		0			0
$965$	26.3518	+	26.3518i	0.848296	+	0.848296i
$966$		0			0
$967$	20.6822			0.665093			0.332547	−	0.943087i	$-0.392092\pi$
$967$	20.6822			0.665093			0.332547	+	0.943087i	$0.392092\pi$
$968$		0			0
$969$	4.53518	+	2.40449i	0.145691	+	0.0772433i
$970$		0			0
$971$	5.17002	−	5.17002i	0.165914	−	0.165914i	−0.619267	−	0.785181i	$-0.712570\pi$
$971$	5.17002	−	5.17002i	0.165914	−	0.165914i	0.785181	+	0.619267i	$0.212570\pi$
$972$		0			0
$973$	−32.2004	−	32.2004i	−1.03230	−	1.03230i
$974$		0			0
$975$		−	2.05228i		−	0.0657256i
$976$		0			0
$977$			25.5684i			0.818006i	0.912533	+	0.409003i	$0.134123\pi$
$977$			25.5684i			0.818006i	−0.912533	+	0.409003i	$0.865877\pi$
$978$		0			0
$979$	27.3615	+	27.3615i	0.874476	+	0.874476i
$980$		0			0
$981$	24.7237	+	24.7237i	0.789368	+	0.789368i
$982$		0			0
$983$		−	20.9464i		−	0.668085i	−0.942558	−	0.334042i	$-0.891587\pi$
$983$		−	20.9464i		−	0.668085i	0.942558	−	0.334042i	$-0.108413\pi$
$984$		0			0
$985$			48.8665i			1.55702i
$986$		0			0
$987$	−6.85980	+	6.85980i	−0.218350	+	0.218350i
$988$		0			0
$989$	−1.45690	−	1.45690i	−0.0463267	−	0.0463267i
$990$		0			0
$991$	35.4955			1.12755			0.563776	−	0.825928i	$-0.309348\pi$
$991$	35.4955			1.12755			0.563776	+	0.825928i	$0.309348\pi$
$992$		0			0
$993$	3.40371			0.108014
$994$		0			0
$995$	−32.4148	+	32.4148i	−1.02762	+	1.02762i
$996$		0			0
$997$	−33.0690	+	33.0690i	−1.04731	+	1.04731i	−0.0484820	+	0.998824i	$0.515438\pi$
$997$	−33.0690	+	33.0690i	−1.04731	+	1.04731i	−0.998824	+	0.0484820i	$0.984562\pi$
$998$		0			0
$999$	−18.6590			−0.590345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.m.a.303.16		76
4.3	odd	2		304.2.m.a.227.19	yes	76
16.5	even	4		304.2.m.a.75.20	yes	76
16.11	odd	4	inner	1216.2.m.a.911.23		76
19.18	odd	2	inner	1216.2.m.a.303.23		76
76.75	even	2		304.2.m.a.227.20	yes	76
304.37	odd	4		304.2.m.a.75.19	✓	76
304.75	even	4	inner	1216.2.m.a.911.16		76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.m.a.75.19	✓	76	304.37	odd	4
304.2.m.a.75.20	yes	76	16.5	even	4
304.2.m.a.227.19	yes	76	4.3	odd	2
304.2.m.a.227.20	yes	76	76.75	even	2
1216.2.m.a.303.16		76	1.1	even	1	trivial
1216.2.m.a.303.23		76	19.18	odd	2	inner
1216.2.m.a.911.16		76	304.75	even	4	inner
1216.2.m.a.911.23		76	16.11	odd	4	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.549728 - 0.549728i) q^{3} +(1.49999 - 1.49999i) q^{5} +3.48044 q^{7} -2.39560i q^{9} +(3.68459 + 3.68459i) q^{11} +(3.73303 - 3.73303i) q^{13} -1.64918 q^{15} +1.51477 q^{17} +(-4.16692 + 1.27937i) q^{19} +(-1.91329 - 1.91329i) q^{21} +0.482269 q^{23} +0.500032i q^{25} +(-2.96611 + 2.96611i) q^{27} +(-7.26730 + 7.26730i) q^{29} +2.39300 q^{31} -4.05104i q^{33} +(5.22064 - 5.22064i) q^{35} +(3.14537 + 3.14537i) q^{37} -4.10430 q^{39} +7.94060 q^{41} +(-3.02093 - 3.02093i) q^{43} +(-3.59339 - 3.59339i) q^{45} -3.58534i q^{47} +5.11344 q^{49} +(-0.832709 - 0.832709i) q^{51} +(-3.55950 - 3.55950i) q^{53} +11.0537 q^{55} +(2.99397 + 1.58737i) q^{57} +(-0.408827 + 0.408827i) q^{59} +(2.47804 + 2.47804i) q^{61} -8.33773i q^{63} -11.1991i q^{65} +(-10.6691 - 10.6691i) q^{67} +(-0.265116 - 0.265116i) q^{69} +5.59302i q^{71} -13.6019i q^{73} +(0.274881 - 0.274881i) q^{75} +(12.8240 + 12.8240i) q^{77} -5.02027 q^{79} -3.92569 q^{81} +(-10.3497 + 10.3497i) q^{83} +(2.27214 - 2.27214i) q^{85} +7.99007 q^{87} +7.42593 q^{89} +(12.9926 - 12.9926i) q^{91} +(-1.31550 - 1.31550i) q^{93} +(-4.33131 + 8.16940i) q^{95} +3.70124i q^{97} +(8.82679 - 8.82679i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 4 q^{5} + 8 q^{7} + 4 q^{11} - 8 q^{17} - 6 q^{19} + 8 q^{23} + 8 q^{39} + 4 q^{43} + 4 q^{45} + 44 q^{49} + 8 q^{55} + 28 q^{61} - 32 q^{77} - 52 q^{81} - 36 q^{83} - 56 q^{85} + 120 q^{87} - 16 q^{93}+ \cdots - 76 q^{99}+O(q^{100}) \) 76 * q - 4 * q^5 + 8 * q^7 + 4 * q^11 - 8 * q^17 - 6 * q^19 + 8 * q^23 + 8 * q^39 + 4 * q^43 + 4 * q^45 + 44 * q^49 + 8 * q^55 + 28 * q^61 - 32 * q^77 - 52 * q^81 - 36 * q^83 - 56 * q^85 + 120 * q^87 - 16 * q^93 - 76 * q^99

Introduction

L-functions

Modular forms

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