LMFDB - Embedded newform 2340.2.bp.i.1513.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2340,2,Mod(1477,2340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2340, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2340.1477");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.bp (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 780)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1513.14
Character	$\chi$	$=$	2340.1513
Dual form			2340.2.bp.i.1477.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.23188 + 0.136841i) q^{5} +4.69969 q^{7} +O(q^{10})$	$q+(2.23188 + 0.136841i) q^{5} +4.69969 q^{7} +(-0.315352 - 0.315352i) q^{11} +(-2.17205 + 2.87788i) q^{13} +(1.07992 + 1.07992i) q^{17} +(3.49017 + 3.49017i) q^{19} +(-3.48094 + 3.48094i) q^{23} +(4.96255 + 0.610824i) q^{25} +2.87828i q^{29} +(-1.13365 + 1.13365i) q^{31} +(10.4891 + 0.643109i) q^{35} +8.92799 q^{37} +(-6.10267 + 6.10267i) q^{41} +(0.557675 - 0.557675i) q^{43} -6.16002 q^{47} +15.0871 q^{49} +(-8.42127 - 8.42127i) q^{53} +(-0.660673 - 0.746979i) q^{55} +(-1.44238 + 1.44238i) q^{59} -5.80499 q^{61} +(-5.24155 + 6.12586i) q^{65} -4.05741i q^{67} +(6.68803 - 6.68803i) q^{71} +0.734212i q^{73} +(-1.48205 - 1.48205i) q^{77} -12.9137i q^{79} +1.05600 q^{83} +(2.26246 + 2.55801i) q^{85} +(5.36534 - 5.36534i) q^{89} +(-10.2079 + 13.5251i) q^{91} +(7.31203 + 8.26723i) q^{95} +4.56277i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 4 q^{5}+O(q^{10}) $ 28 * q + 4 * q^5	$ 28 q + 4 q^{5} - 8 q^{11} - 4 q^{17} + 16 q^{19} - 8 q^{23} - 12 q^{25} + 24 q^{37} - 12 q^{41} - 16 q^{43} + 24 q^{47} + 36 q^{49} - 36 q^{53} + 40 q^{55} + 16 q^{59} + 8 q^{61} - 24 q^{65} - 8 q^{71} - 48 q^{77} + 24 q^{83} + 68 q^{85} - 36 q^{89} - 24 q^{91} + 72 q^{95}+O(q^{100}) $ 28 * q + 4 * q^5 - 8 * q^11 - 4 * q^17 + 16 * q^19 - 8 * q^23 - 12 * q^25 + 24 * q^37 - 12 * q^41 - 16 * q^43 + 24 * q^47 + 36 * q^49 - 36 * q^53 + 40 * q^55 + 16 * q^59 + 8 * q^61 - 24 * q^65 - 8 * q^71 - 48 * q^77 + 24 * q^83 + 68 * q^85 - 36 * q^89 - 24 * q^91 + 72 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.23188	+	0.136841i	0.998126	+	0.0611971i
$5$	2.23188	+	0.136841i	0.998126	+	0.0611971i
$6$		0			0
$7$	4.69969			1.77631			0.888157	−	0.459539i	$-0.151986\pi$
$7$	4.69969			1.77631			0.888157	+	0.459539i	$0.151986\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.315352	−	0.315352i	−0.0950822	−	0.0950822i	0.657966	−	0.753048i	$-0.271417\pi$
$11$	−0.315352	−	0.315352i	−0.0950822	−	0.0950822i	−0.753048	+	0.657966i	$0.771417\pi$
$12$		0			0
$13$	−2.17205	+	2.87788i	−0.602418	+	0.798181i
$13$	−2.17205	+	2.87788i	−0.602418	+	0.798181i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.07992	+	1.07992i	0.261918	+	0.261918i	0.825833	−	0.563915i	$-0.190705\pi$
$17$	1.07992	+	1.07992i	0.261918	+	0.261918i	−0.563915	+	0.825833i	$0.690705\pi$
$18$		0			0
$19$	3.49017	+	3.49017i	0.800700	+	0.800700i	0.983205	−	0.182505i	$-0.0584206\pi$
$19$	3.49017	+	3.49017i	0.800700	+	0.800700i	−0.182505	+	0.983205i	$0.558421\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.48094	+	3.48094i	−0.725827	+	0.725827i	−0.969786	−	0.243959i	$-0.921554\pi$
$23$	−3.48094	+	3.48094i	−0.725827	+	0.725827i	0.243959	+	0.969786i	$0.421554\pi$
$24$		0			0
$25$	4.96255	+	0.610824i	0.992510	+	0.122165i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.87828i			0.534483i	0.963630	+	0.267241i	$0.0861121\pi$
$29$			2.87828i			0.534483i	−0.963630	+	0.267241i	$0.913888\pi$
$30$		0			0
$31$	−1.13365	+	1.13365i	−0.203610	+	0.203610i	−0.801545	−	0.597935i	$-0.795988\pi$
$31$	−1.13365	+	1.13365i	−0.203610	+	0.203610i	0.597935	+	0.801545i	$0.295988\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.4891	+	0.643109i	1.77299	+	0.108705i
$36$		0			0
$37$	8.92799			1.46775			0.733876	−	0.679283i	$-0.237709\pi$
$37$	8.92799			1.46775			0.733876	+	0.679283i	$0.237709\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.10267	+	6.10267i	−0.953078	+	0.953078i	−0.998947	−	0.0458697i	$-0.985394\pi$
$41$	−6.10267	+	6.10267i	−0.953078	+	0.953078i	0.0458697	+	0.998947i	$0.485394\pi$
$42$		0			0
$43$	0.557675	−	0.557675i	0.0850447	−	0.0850447i	−0.663305	−	0.748349i	$-0.730847\pi$
$43$	0.557675	−	0.557675i	0.0850447	−	0.0850447i	0.748349	+	0.663305i	$0.230847\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.16002			−0.898531			−0.449265	−	0.893398i	$-0.648314\pi$
$47$	−6.16002			−0.898531			−0.449265	+	0.893398i	$0.648314\pi$
$48$		0			0
$49$	15.0871			2.15529
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8.42127	−	8.42127i	−1.15675	−	1.15675i	−0.985170	−	0.171581i	$-0.945113\pi$
$53$	−8.42127	−	8.42127i	−1.15675	−	1.15675i	−0.171581	−	0.985170i	$-0.554887\pi$
$54$		0			0
$55$	−0.660673	−	0.746979i	−0.0890852	−	0.100723i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.44238	+	1.44238i	−0.187782	+	0.187782i	−0.794736	−	0.606955i	$-0.792391\pi$
$59$	−1.44238	+	1.44238i	−0.187782	+	0.187782i	0.606955	+	0.794736i	$0.292391\pi$
$60$		0			0
$61$	−5.80499			−0.743253			−0.371626	−	0.928382i	$-0.621200\pi$
$61$	−5.80499			−0.743253			−0.371626	+	0.928382i	$0.621200\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.24155	+	6.12586i	−0.650135	+	0.759819i
$66$		0			0
$67$		−	4.05741i		−	0.495691i	−0.968800	−	0.247846i	$-0.920277\pi$
$67$		−	4.05741i		−	0.495691i	0.968800	−	0.247846i	$-0.0797226\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.68803	−	6.68803i	0.793723	−	0.793723i	−0.188374	−	0.982097i	$-0.560322\pi$
$71$	6.68803	−	6.68803i	0.793723	−	0.793723i	0.982097	+	0.188374i	$0.0603217\pi$
$72$		0			0
$73$			0.734212i			0.0859330i	0.999077	+	0.0429665i	$0.0136809\pi$
$73$			0.734212i			0.0859330i	−0.999077	+	0.0429665i	$0.986319\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.48205	−	1.48205i	−0.168896	−	0.168896i
$78$		0			0
$79$		−	12.9137i		−	1.45290i	−0.687218	−	0.726451i	$-0.741168\pi$
$79$		−	12.9137i		−	1.45290i	0.687218	−	0.726451i	$-0.258832\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.05600			0.115911			0.0579553	−	0.998319i	$-0.481542\pi$
$83$	1.05600			0.115911			0.0579553	+	0.998319i	$0.481542\pi$
$84$		0			0
$85$	2.26246	+	2.55801i	0.245398	+	0.277456i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.36534	−	5.36534i	0.568725	−	0.568725i	−0.363046	−	0.931771i	$-0.618263\pi$
$89$	5.36534	−	5.36534i	0.568725	−	0.568725i	0.931771	+	0.363046i	$0.118263\pi$
$90$		0			0
$91$	−10.2079	+	13.5251i	−1.07008	+	1.41782i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.31203	+	8.26723i	0.750199	+	0.848200i
$96$		0			0
$97$			4.56277i			0.463279i	0.972802	+	0.231640i	$0.0744090\pi$
$97$			4.56277i			0.463279i	−0.972802	+	0.231640i	$0.925591\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.68921i			0.864609i	0.901728	+	0.432304i	$0.142299\pi$
$101$			8.68921i			0.864609i	−0.901728	+	0.432304i	$0.857701\pi$
$102$		0			0
$103$	−3.72580	+	3.72580i	−0.367114	+	0.367114i	−0.866424	−	0.499310i	$-0.833587\pi$
$103$	−3.72580	+	3.72580i	−0.367114	+	0.367114i	0.499310	+	0.866424i	$0.333587\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.4478	−	12.4478i	1.20337	−	1.20337i	0.230240	−	0.973134i	$-0.426049\pi$
$107$	12.4478	−	12.4478i	1.20337	−	1.20337i	0.973134	−	0.230240i	$-0.0739512\pi$
$108$		0			0
$109$	−12.0020	−	12.0020i	−1.14958	−	1.14958i	−0.986635	−	0.162949i	$-0.947899\pi$
$109$	−12.0020	−	12.0020i	−1.14958	−	1.14958i	−0.162949	−	0.986635i	$-0.552101\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.3426	−	11.3426i	−1.06702	−	1.06702i	−0.997586	−	0.0694367i	$-0.977880\pi$
$113$	−11.3426	−	11.3426i	−1.06702	−	1.06702i	−0.0694367	−	0.997586i	$-0.522120\pi$
$114$		0			0
$115$	−8.24537	+	7.29270i	−0.768885	+	0.680048i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.07526	+	5.07526i	0.465249	+	0.465249i
$120$		0			0
$121$		−	10.8011i		−	0.981919i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.9922	+	2.04236i	0.983173	+	0.182674i
$126$		0			0
$127$	6.73937	+	6.73937i	0.598022	+	0.598022i	0.939786	−	0.341764i	$-0.111024\pi$
$127$	6.73937	+	6.73937i	0.598022	+	0.598022i	−0.341764	+	0.939786i	$0.611024\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.18022			0.452598			0.226299	−	0.974058i	$-0.427337\pi$
$131$	5.18022			0.452598			0.226299	+	0.974058i	$0.427337\pi$
$132$		0			0
$133$	16.4027	+	16.4027i	1.42229	+	1.42229i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16.8964			−1.44356			−0.721778	−	0.692124i	$-0.756675\pi$
$137$	−16.8964			−1.44356			−0.721778	+	0.692124i	$0.756675\pi$
$138$		0			0
$139$			3.05422i			0.259056i	0.991576	+	0.129528i	$0.0413462\pi$
$139$			3.05422i			0.259056i	−0.991576	+	0.129528i	$0.958654\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.59250	−	0.222587i	0.133172	−	0.0186136i
$144$		0			0
$145$	−0.393866	+	6.42396i	−0.0327088	+	0.533481i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.623594	−	0.623594i	−0.0510868	−	0.0510868i	0.681102	−	0.732189i	$-0.261501\pi$
$149$	−0.623594	−	0.623594i	−0.0510868	−	0.0510868i	−0.732189	+	0.681102i	$0.761501\pi$
$150$		0			0
$151$	5.84579	+	5.84579i	0.475724	+	0.475724i	0.903761	−	0.428037i	$-0.140795\pi$
$151$	5.84579	+	5.84579i	0.475724	+	0.475724i	−0.428037	+	0.903761i	$0.640795\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.68530	+	2.37504i	−0.215688	+	0.190768i
$156$		0			0
$157$	10.2125	−	10.2125i	0.815046	−	0.815046i	−0.170339	−	0.985385i	$-0.554486\pi$
$157$	10.2125	−	10.2125i	0.815046	−	0.815046i	0.985385	+	0.170339i	$0.0544864\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.3593	+	16.3593i	−1.28930	+	1.28930i
$162$		0			0
$163$			23.2373i			1.82009i	0.414511	+	0.910044i	$0.363953\pi$
$163$			23.2373i			1.82009i	−0.414511	+	0.910044i	$0.636047\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.5054			0.967697			0.483848	−	0.875152i	$-0.339239\pi$
$167$	12.5054			0.967697			0.483848	+	0.875152i	$0.339239\pi$
$168$		0			0
$169$	−3.56442	−	12.5018i	−0.274186	−	0.961677i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	18.3381	−	18.3381i	1.39422	−	1.39422i	0.578618	−	0.815599i	$-0.303592\pi$
$173$	18.3381	−	18.3381i	1.39422	−	1.39422i	0.815599	−	0.578618i	$-0.196408\pi$
$174$		0			0
$175$	23.3224	+	2.87068i	1.76301	+	0.217003i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.9880			1.04552			0.522758	−	0.852481i	$-0.324903\pi$
$179$	13.9880			1.04552			0.522758	+	0.852481i	$0.324903\pi$
$180$		0			0
$181$			14.6342i			1.08775i	0.839165	+	0.543877i	$0.183044\pi$
$181$			14.6342i			1.08775i	−0.839165	+	0.543877i	$0.816956\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	19.9262	+	1.22171i	1.46500	+	0.0898221i
$186$		0			0
$187$		−	0.681107i		−	0.0498074i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.6896			0.990548			0.495274	−	0.868737i	$-0.335068\pi$
$191$	13.6896			0.990548			0.495274	+	0.868737i	$0.335068\pi$
$192$		0			0
$193$		−	11.3452i		−	0.816647i	−0.912837	−	0.408323i	$-0.866114\pi$
$193$		−	11.3452i		−	0.816647i	0.912837	−	0.408323i	$-0.133886\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.7625i			0.909294i	0.890672	+	0.454647i	$0.150234\pi$
$197$			12.7625i			0.909294i	−0.890672	+	0.454647i	$0.849766\pi$
$198$		0			0
$199$	2.52762			0.179178			0.0895892	−	0.995979i	$-0.471445\pi$
$199$	2.52762			0.179178			0.0895892	+	0.995979i	$0.471445\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			13.5270i			0.949409i
$204$		0			0
$205$	−14.4555	+	12.7853i	−1.00962	+	0.892966i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2.20126i		−	0.152265i
$210$		0			0
$211$	10.4839			0.721738			0.360869	−	0.932616i	$-0.382480\pi$
$211$	10.4839			0.721738			0.360869	+	0.932616i	$0.382480\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.32098	−	1.16835i	0.0900898	−	0.0796808i
$216$		0			0
$217$	−5.32780	+	5.32780i	−0.361675	+	0.361675i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.45350	+	0.762243i	−0.366842	+	0.0512740i
$222$		0			0
$223$	1.90938			0.127861			0.0639306	−	0.997954i	$-0.479636\pi$
$223$	1.90938			0.127861			0.0639306	+	0.997954i	$0.479636\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			28.0353i			1.86077i	0.366584	+	0.930385i	$0.380527\pi$
$227$			28.0353i			1.86077i	−0.366584	+	0.930385i	$0.619473\pi$
$228$		0			0
$229$	−5.80903	+	5.80903i	−0.383871	+	0.383871i	−0.872495	−	0.488623i	$-0.837499\pi$
$229$	−5.80903	+	5.80903i	−0.383871	+	0.383871i	0.488623	+	0.872495i	$0.337499\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.4654	−	10.4654i	0.685613	−	0.685613i	−0.275646	−	0.961259i	$-0.588892\pi$
$233$	10.4654	−	10.4654i	0.685613	−	0.685613i	0.961259	+	0.275646i	$0.0888919\pi$
$234$		0			0
$235$	−13.7484	−	0.842942i	−0.896847	−	0.0549875i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.358095	+	0.358095i	0.0231633	+	0.0231633i	0.718594	−	0.695430i	$-0.244786\pi$
$239$	0.358095	+	0.358095i	0.0231633	+	0.0231633i	−0.695430	+	0.718594i	$0.744786\pi$
$240$		0			0
$241$	−8.96339	−	8.96339i	−0.577383	−	0.577383i	0.356799	−	0.934181i	$-0.383868\pi$
$241$	−8.96339	−	8.96339i	−0.577383	−	0.577383i	−0.934181	+	0.356799i	$0.883868\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	33.6725	+	2.06453i	2.15125	+	0.131898i
$246$		0			0
$247$	−17.6251	+	2.46349i	−1.12146	+	0.156748i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	10.3022i		−	0.650272i	−0.945667	−	0.325136i	$-0.894590\pi$
$251$		−	10.3022i		−	0.650272i	0.945667	−	0.325136i	$-0.105410\pi$
$252$		0			0
$253$	2.19544			0.138026
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.2281	−	20.2281i	−1.26179	−	1.26179i	−0.950222	−	0.311572i	$-0.899144\pi$
$257$	−20.2281	−	20.2281i	−1.26179	−	1.26179i	−0.311572	−	0.950222i	$-0.600856\pi$
$258$		0			0
$259$	41.9587			2.60719
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.83060	+	5.83060i	0.359530	+	0.359530i	0.863640	−	0.504110i	$-0.168179\pi$
$263$	5.83060	+	5.83060i	0.359530	+	0.359530i	−0.504110	+	0.863640i	$0.668179\pi$
$264$		0			0
$265$	−17.6429	−	19.9476i	−1.08379	−	1.22537i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	26.2933i		−	1.60313i	−0.597906	−	0.801566i	$-0.704001\pi$
$269$		−	26.2933i		−	1.60313i	0.597906	−	0.801566i	$-0.295999\pi$
$270$		0			0
$271$	−21.5497	−	21.5497i	−1.30905	−	1.30905i	−0.922100	−	0.386953i	$-0.873528\pi$
$271$	−21.5497	−	21.5497i	−1.30905	−	1.30905i	−0.386953	−	0.922100i	$-0.626472\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.37232	−	1.75757i	−0.0827543	−	0.105986i
$276$		0			0
$277$	−11.0701	−	11.0701i	−0.665137	−	0.665137i	0.291449	−	0.956586i	$-0.405863\pi$
$277$	−11.0701	−	11.0701i	−0.665137	−	0.665137i	−0.956586	+	0.291449i	$0.905863\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.6165	−	13.6165i	−0.812291	−	0.812291i	0.172686	−	0.984977i	$-0.444755\pi$
$281$	−13.6165	−	13.6165i	−0.812291	−	0.812291i	−0.984977	+	0.172686i	$0.944755\pi$
$282$		0			0
$283$	−12.6708	+	12.6708i	−0.753204	+	0.753204i	−0.975076	−	0.221872i	$-0.928783\pi$
$283$	−12.6708	+	12.6708i	−0.753204	+	0.753204i	0.221872	+	0.975076i	$0.428783\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−28.6807	+	28.6807i	−1.69297	+	1.69297i
$288$		0			0
$289$		−	14.6676i		−	0.862798i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			31.1560i			1.82015i	0.414439	+	0.910077i	$0.363978\pi$
$293$			31.1560i			1.82015i	−0.414439	+	0.910077i	$0.636022\pi$
$294$		0			0
$295$	−3.41659	+	3.02184i	−0.198921	+	0.175938i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.45697	−	17.5785i	−0.142090	−	1.01659i
$300$		0			0
$301$	2.62090	−	2.62090i	0.151066	−	0.151066i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.9560	−	0.794359i	−0.741860	−	0.0454849i
$306$		0			0
$307$	28.0732			1.60222			0.801110	−	0.598517i	$-0.204243\pi$
$307$	28.0732			1.60222			0.801110	+	0.598517i	$0.204243\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	12.7252i		−	0.721579i	−0.932647	−	0.360789i	$-0.882507\pi$
$311$		−	12.7252i		−	0.721579i	0.932647	−	0.360789i	$-0.117493\pi$
$312$		0			0
$313$	−14.1806	−	14.1806i	−0.801534	−	0.801534i	0.181801	−	0.983335i	$-0.441807\pi$
$313$	−14.1806	−	14.1806i	−0.801534	−	0.801534i	−0.983335	+	0.181801i	$0.941807\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			7.64077i			0.429148i	0.976708	+	0.214574i	$0.0688363\pi$
$317$			7.64077i			0.429148i	−0.976708	+	0.214574i	$0.931164\pi$
$318$		0			0
$319$	0.907670	−	0.907670i	0.0508198	−	0.0508198i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			7.53817i			0.419435i
$324$		0			0
$325$	−12.5368	+	12.9549i	−0.695415	+	0.718608i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−28.9502			−1.59607
$330$		0			0
$331$	15.0162	−	15.0162i	0.825365	−	0.825365i	−0.161507	−	0.986872i	$-0.551635\pi$
$331$	15.0162	−	15.0162i	0.825365	−	0.825365i	0.986872	+	0.161507i	$0.0516354\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.555219	−	9.05564i	0.0303349	−	0.494762i
$336$		0			0
$337$	9.44617	+	9.44617i	0.514566	+	0.514566i	0.915922	−	0.401356i	$-0.131461\pi$
$337$	9.44617	+	9.44617i	0.514566	+	0.514566i	−0.401356	+	0.915922i	$0.631461\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.714997			0.0387193
$342$		0			0
$343$	38.0066			2.05217
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.5249	+	20.5249i	−1.10183	+	1.10183i	−0.107642	+	0.994190i	$0.534330\pi$
$347$	−20.5249	+	20.5249i	−1.10183	+	1.10183i	−0.994190	+	0.107642i	$0.965670\pi$
$348$		0			0
$349$	1.54929	−	1.54929i	0.0829316	−	0.0829316i	−0.664424	−	0.747356i	$-0.731323\pi$
$349$	1.54929	−	1.54929i	0.0829316	−	0.0829316i	0.747356	+	0.664424i	$0.231323\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.75270			−0.519084			−0.259542	−	0.965732i	$-0.583572\pi$
$353$	−9.75270			−0.519084			−0.259542	+	0.965732i	$0.583572\pi$
$354$		0			0
$355$	15.8421	−	14.0117i	0.840809	−	0.743662i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.81152	−	2.81152i	0.148387	−	0.148387i	−0.629010	−	0.777397i	$-0.716540\pi$
$359$	2.81152	−	2.81152i	0.148387	−	0.148387i	0.777397	+	0.629010i	$0.216540\pi$
$360$		0			0
$361$			5.36257i			0.282240i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.100470	+	1.63867i	−0.00525885	+	0.0857719i
$366$		0			0
$367$	−23.2265	+	23.2265i	−1.21241	+	1.21241i	−0.242184	+	0.970230i	$0.577864\pi$
$367$	−23.2265	+	23.2265i	−1.21241	+	1.21241i	−0.970230	+	0.242184i	$0.922136\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−39.5773	−	39.5773i	−2.05475	−	2.05475i
$372$		0			0
$373$	−3.06758	−	3.06758i	−0.158833	−	0.158833i	0.623216	−	0.782050i	$-0.285826\pi$
$373$	−3.06758	−	3.06758i	−0.158833	−	0.158833i	−0.782050	+	0.623216i	$0.785826\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.28334	−	6.25175i	−0.426614	−	0.321982i
$378$		0			0
$379$	−16.7416	−	16.7416i	−0.859959	−	0.859959i	0.131374	−	0.991333i	$-0.458061\pi$
$379$	−16.7416	−	16.7416i	−0.859959	−	0.859959i	−0.991333	+	0.131374i	$0.958061\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.16648			0.263995			0.131997	−	0.991250i	$-0.457861\pi$
$383$	5.16648			0.263995			0.131997	+	0.991250i	$0.457861\pi$
$384$		0			0
$385$	−3.10496	−	3.51057i	−0.158243	−	0.178915i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.92391			−0.249652			−0.124826	−	0.992179i	$-0.539837\pi$
$389$	−4.92391			−0.249652			−0.124826	+	0.992179i	$0.539837\pi$
$390$		0			0
$391$	−7.51825			−0.380214
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.76712	−	28.8217i	0.0889134	−	1.45018i
$396$		0			0
$397$	−26.9445			−1.35231			−0.676154	−	0.736760i	$-0.736355\pi$
$397$	−26.9445			−1.35231			−0.676154	+	0.736760i	$0.736355\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	25.1188	+	25.1188i	1.25437	+	1.25437i	0.953739	+	0.300635i	$0.0971986\pi$
$401$	25.1188	+	25.1188i	1.25437	+	1.25437i	0.300635	+	0.953739i	$0.402801\pi$
$402$		0			0
$403$	−0.800171	−	5.72485i	−0.0398593	−	0.285175i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.81546	−	2.81546i	−0.139557	−	0.139557i
$408$		0			0
$409$	−0.765296	−	0.765296i	−0.0378414	−	0.0378414i	0.687933	−	0.725774i	$-0.258518\pi$
$409$	−0.765296	−	0.765296i	−0.0378414	−	0.0378414i	−0.725774	+	0.687933i	$0.758518\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.77873	+	6.77873i	−0.333559	+	0.333559i
$414$		0			0
$415$	2.35685	+	0.144503i	0.115693	+	0.00709340i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	6.89257i		−	0.336724i	−0.985725	−	0.168362i	$-0.946152\pi$
$419$		−	6.89257i		−	0.336724i	0.985725	−	0.168362i	$-0.0538477\pi$
$420$		0			0
$421$	19.8483	−	19.8483i	0.967349	−	0.967349i	−0.0321346	−	0.999484i	$-0.510231\pi$
$421$	19.8483	−	19.8483i	0.967349	−	0.967349i	0.999484	+	0.0321346i	$0.0102305\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.69949	+	6.01877i	0.227959	+	0.291953i
$426$		0			0
$427$	−27.2816			−1.32025
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.3880	−	18.3880i	0.885718	−	0.885718i	−0.108391	−	0.994108i	$-0.534570\pi$
$431$	18.3880	−	18.3880i	0.885718	−	0.885718i	0.994108	+	0.108391i	$0.0345697\pi$
$432$		0			0
$433$	−6.17848	+	6.17848i	−0.296919	+	0.296919i	−0.839806	−	0.542887i	$-0.817331\pi$
$433$	−6.17848	+	6.17848i	−0.296919	+	0.296919i	0.542887	+	0.839806i	$0.317331\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.2982			−1.16234
$438$		0			0
$439$	−15.7983			−0.754013			−0.377006	−	0.926211i	$-0.623046\pi$
$439$	−15.7983			−0.754013			−0.377006	+	0.926211i	$0.623046\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.28734	+	7.28734i	0.346232	+	0.346232i	0.858704	−	0.512472i	$-0.171270\pi$
$443$	7.28734	+	7.28734i	0.346232	+	0.346232i	−0.512472	+	0.858704i	$0.671270\pi$
$444$		0			0
$445$	12.7090	−	11.2406i	0.602463	−	0.532855i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.9047	+	11.9047i	−0.561819	+	0.561819i	−0.929824	−	0.368005i	$-0.880041\pi$
$449$	−11.9047	+	11.9047i	−0.561819	+	0.561819i	0.368005	+	0.929824i	$0.380041\pi$
$450$		0			0
$451$	3.84898			0.181241
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−24.6337	+	28.7896i	−1.15484	+	1.34968i
$456$		0			0
$457$			10.4884i			0.490628i	0.969444	+	0.245314i	$0.0788911\pi$
$457$			10.4884i			0.490628i	−0.969444	+	0.245314i	$0.921109\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.4899	−	17.4899i	0.814587	−	0.814587i	−0.170731	−	0.985318i	$-0.554613\pi$
$461$	17.4899	−	17.4899i	0.814587	−	0.814587i	0.985318	+	0.170731i	$0.0546128\pi$
$462$		0			0
$463$			16.3159i			0.758266i	0.925342	+	0.379133i	$0.123778\pi$
$463$			16.3159i			0.758266i	−0.925342	+	0.379133i	$0.876222\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.5684	−	13.5684i	−0.627870	−	0.627870i	0.319662	−	0.947532i	$-0.396431\pi$
$467$	−13.5684	−	13.5684i	−0.627870	−	0.627870i	−0.947532	+	0.319662i	$0.896431\pi$
$468$		0			0
$469$		−	19.0686i		−	0.880504i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.351728			−0.0161725
$474$		0			0
$475$	15.1883	+	19.4520i	0.696885	+	0.892520i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.7533	−	17.7533i	0.811171	−	0.811171i	−0.173639	−	0.984809i	$-0.555552\pi$
$479$	17.7533	−	17.7533i	0.811171	−	0.811171i	0.984809	+	0.173639i	$0.0555524\pi$
$480$		0			0
$481$	−19.3920	+	25.6937i	−0.884200	+	1.17153i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.624374	+	10.1835i	−0.0283513	+	0.462411i
$486$		0			0
$487$		−	33.7675i		−	1.53015i	−0.643941	−	0.765075i	$-0.722702\pi$
$487$		−	33.7675i		−	1.53015i	0.643941	−	0.765075i	$-0.277298\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			22.9938i			1.03770i	0.854866	+	0.518849i	$0.173639\pi$
$491$			22.9938i			1.03770i	−0.854866	+	0.518849i	$0.826361\pi$
$492$		0			0
$493$	−3.10830	+	3.10830i	−0.139991	+	0.139991i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	31.4317	−	31.4317i	1.40990	−	1.40990i
$498$		0			0
$499$	6.02065	+	6.02065i	0.269521	+	0.269521i	0.828907	−	0.559386i	$-0.188963\pi$
$499$	6.02065	+	6.02065i	0.269521	+	0.269521i	−0.559386	+	0.828907i	$0.688963\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.3721	+	10.3721i	0.462470	+	0.462470i	0.899464	−	0.436995i	$-0.143957\pi$
$503$	10.3721	+	10.3721i	0.462470	+	0.462470i	−0.436995	+	0.899464i	$0.643957\pi$
$504$		0			0
$505$	−1.18904	+	19.3932i	−0.0529115	+	0.862988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.39148	+	5.39148i	0.238973	+	0.238973i	0.816425	−	0.577452i	$-0.195953\pi$
$509$	5.39148	+	5.39148i	0.238973	+	0.238973i	−0.577452	+	0.816425i	$0.695953\pi$
$510$		0			0
$511$			3.45057i			0.152644i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.82536	+	7.80568i	−0.388892	+	0.343959i
$516$		0			0
$517$	1.94257	+	1.94257i	0.0854343	+	0.0854343i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−42.1874			−1.84826			−0.924131	−	0.382076i	$-0.875209\pi$
$521$	−42.1874			−1.84826			−0.924131	+	0.382076i	$0.875209\pi$
$522$		0			0
$523$	11.3298	+	11.3298i	0.495417	+	0.495417i	0.910008	−	0.414591i	$-0.136075\pi$
$523$	11.3298	+	11.3298i	0.495417	+	0.495417i	−0.414591	+	0.910008i	$0.636075\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.44849			−0.106658
$528$		0			0
$529$		−	1.23393i		−	0.0536492i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.30749	−	30.8181i	−0.186578	−	1.33488i
$534$		0			0
$535$	29.4853	−	26.0786i	1.27476	−	1.12748i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.75773	−	4.75773i	−0.204930	−	0.204930i
$540$		0			0
$541$	−11.3903	−	11.3903i	−0.489709	−	0.489709i	0.418506	−	0.908214i	$-0.362554\pi$
$541$	−11.3903	−	11.3903i	−0.489709	−	0.489709i	−0.908214	+	0.418506i	$0.862554\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−25.1446	−	28.4294i	−1.07708	−	1.21778i
$546$		0			0
$547$	11.3855	−	11.3855i	0.486809	−	0.486809i	−0.420489	−	0.907298i	$-0.638141\pi$
$547$	11.3855	−	11.3855i	0.486809	−	0.486809i	0.907298	+	0.420489i	$0.138141\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.0457	+	10.0457i	−0.427960	+	0.427960i
$552$		0			0
$553$		−	60.6902i		−	2.58081i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−34.1016			−1.44493			−0.722465	−	0.691408i	$-0.756991\pi$
$557$	−34.1016			−1.44493			−0.722465	+	0.691408i	$0.756991\pi$
$558$		0			0
$559$	0.393627	+	2.81622i	0.0166487	+	0.119113i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.7769	+	25.7769i	−1.08637	+	1.08637i	−0.0904666	+	0.995899i	$0.528836\pi$
$563$	−25.7769	+	25.7769i	−1.08637	+	1.08637i	−0.995899	+	0.0904666i	$0.971164\pi$
$564$		0			0
$565$	−23.7632	−	26.8674i	−0.999724	−	1.13032i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	26.2934			1.10228			0.551138	−	0.834414i	$-0.314194\pi$
$569$	26.2934			1.10228			0.551138	+	0.834414i	$0.314194\pi$
$570$		0			0
$571$		−	14.6838i		−	0.614496i	−0.951629	−	0.307248i	$-0.900592\pi$
$571$		−	14.6838i		−	0.614496i	0.951629	−	0.307248i	$-0.0994082\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.4006	+	15.1481i	−0.809061	+	0.631720i
$576$		0			0
$577$		−	23.1978i		−	0.965735i	−0.875693	−	0.482868i	$-0.839595\pi$
$577$		−	23.1978i		−	0.965735i	0.875693	−	0.482868i	$-0.160405\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.96285			0.205894
$582$		0			0
$583$			5.31133i			0.219973i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			7.91675i			0.326759i	0.986563	+	0.163380i	$0.0522395\pi$
$587$			7.91675i			0.326759i	−0.986563	+	0.163380i	$0.947760\pi$
$588$		0			0
$589$	−7.91326			−0.326060
$590$		0			0
$591$		0			0
$592$		0			0
$593$			43.6618i			1.79297i	0.443071	+	0.896487i	$0.353889\pi$
$593$			43.6618i			1.79297i	−0.443071	+	0.896487i	$0.646111\pi$
$594$		0			0
$595$	10.6329	+	12.0219i	0.435905	+	0.492849i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	15.5444i		−	0.635126i	−0.948237	−	0.317563i	$-0.897136\pi$
$599$		−	15.5444i		−	0.635126i	0.948237	−	0.317563i	$-0.102864\pi$
$600$		0			0
$601$	−7.39443			−0.301625			−0.150813	−	0.988562i	$-0.548189\pi$
$601$	−7.39443			−0.301625			−0.150813	+	0.988562i	$0.548189\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.47803	−	24.1067i	0.0600906	−	0.980078i
$606$		0			0
$607$	−29.2174	+	29.2174i	−1.18590	+	1.18590i	−0.207705	+	0.978192i	$0.566599\pi$
$607$	−29.2174	+	29.2174i	−1.18590	+	1.18590i	−0.978192	+	0.207705i	$0.933401\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.3798	−	17.7278i	0.541291	−	0.717190i
$612$		0			0
$613$	39.5039			1.59555			0.797774	−	0.602957i	$-0.206011\pi$
$613$	39.5039			1.59555			0.797774	+	0.602957i	$0.206011\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2.88852i			0.116288i	0.998308	+	0.0581438i	$0.0185182\pi$
$617$			2.88852i			0.116288i	−0.998308	+	0.0581438i	$0.981482\pi$
$618$		0			0
$619$	−13.1526	+	13.1526i	−0.528649	+	0.528649i	−0.920169	−	0.391520i	$-0.871949\pi$
$619$	−13.1526	+	13.1526i	−0.528649	+	0.528649i	0.391520	+	0.920169i	$0.371949\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	25.2154	−	25.2154i	1.01023	−	1.01023i
$624$		0			0
$625$	24.2538	+	6.06249i	0.970152	+	0.242499i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.64147	+	9.64147i	0.384431	+	0.384431i
$630$		0			0
$631$	−26.9044	−	26.9044i	−1.07105	−	1.07105i	−0.997275	−	0.0737713i	$-0.976497\pi$
$631$	−26.9044	−	26.9044i	−1.07105	−	1.07105i	−0.0737713	−	0.997275i	$-0.523503\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.1192	+	15.9637i	0.560304	+	0.633499i
$636$		0			0
$637$	−32.7698	+	43.4188i	−1.29839	+	1.72032i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	7.72592i		−	0.305156i	−0.988291	−	0.152578i	$-0.951243\pi$
$641$		−	7.72592i		−	0.305156i	0.988291	−	0.152578i	$-0.0487575\pi$
$642$		0			0
$643$	−17.3325			−0.683528			−0.341764	−	0.939786i	$-0.611024\pi$
$643$	−17.3325			−0.683528			−0.341764	+	0.939786i	$0.611024\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.2911	+	11.2911i	0.443898	+	0.443898i	0.893320	−	0.449421i	$-0.148370\pi$
$647$	11.2911	+	11.2911i	0.443898	+	0.443898i	−0.449421	+	0.893320i	$0.648370\pi$
$648$		0			0
$649$	0.909714			0.0357094
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.3452	+	19.3452i	0.757037	+	0.757037i	0.975782	−	0.218745i	$-0.0701963\pi$
$653$	19.3452	+	19.3452i	0.757037	+	0.757037i	−0.218745	+	0.975782i	$0.570196\pi$
$654$		0			0
$655$	11.5616	+	0.708866i	0.451750	+	0.0276977i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	41.8280i		−	1.62939i	−0.579890	−	0.814695i	$-0.696905\pi$
$659$		−	41.8280i		−	1.62939i	0.579890	−	0.814695i	$-0.303095\pi$
$660$		0			0
$661$	−14.9894	−	14.9894i	−0.583022	−	0.583022i	0.352711	−	0.935732i	$-0.385260\pi$
$661$	−14.9894	−	14.9894i	−0.583022	−	0.583022i	−0.935732	+	0.352711i	$0.885260\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	34.3643	+	38.8534i	1.33259	+	1.50667i
$666$		0			0
$667$	−10.0191	−	10.0191i	−0.387942	−	0.387942i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.83061	+	1.83061i	0.0706701	+	0.0706701i
$672$		0			0
$673$	−11.3096	+	11.3096i	−0.435952	+	0.435952i	−0.890647	−	0.454695i	$-0.849748\pi$
$673$	−11.3096	+	11.3096i	−0.435952	+	0.435952i	0.454695	+	0.890647i	$0.349748\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.2625	−	14.2625i	0.548153	−	0.548153i	−0.377753	−	0.925906i	$-0.623303\pi$
$677$	14.2625	−	14.2625i	0.548153	−	0.548153i	0.925906	+	0.377753i	$0.123303\pi$
$678$		0			0
$679$			21.4436i			0.822930i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	24.2502i		−	0.927910i	−0.885859	−	0.463955i	$-0.846430\pi$
$683$		−	24.2502i		−	0.927910i	0.885859	−	0.463955i	$-0.153570\pi$
$684$		0			0
$685$	−37.7107	−	2.31212i	−1.44085	−	0.0883415i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	42.5268	−	5.94404i	1.62014	−	0.226450i
$690$		0			0
$691$	18.8507	−	18.8507i	0.717116	−	0.717116i	−0.250898	−	0.968014i	$-0.580726\pi$
$691$	18.8507	−	18.8507i	0.717116	−	0.717116i	0.968014	+	0.250898i	$0.0807257\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.417942	+	6.81665i	−0.0158535	+	0.258570i
$696$		0			0
$697$	−13.1807			−0.499256
$698$		0			0
$699$		0			0
$700$		0			0
$701$			23.7436i			0.896783i	0.893837	+	0.448392i	$0.148003\pi$
$701$			23.7436i			0.896783i	−0.893837	+	0.448392i	$0.851997\pi$
$702$		0			0
$703$	31.1602	+	31.1602i	1.17523	+	1.17523i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			40.8366i			1.53582i
$708$		0			0
$709$	−6.02523	+	6.02523i	−0.226282	+	0.226282i	−0.811138	−	0.584855i	$-0.801151\pi$
$709$	−6.02523	+	6.02523i	−0.226282	+	0.226282i	0.584855	+	0.811138i	$0.301151\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.89234i		−	0.295571i
$714$		0			0
$715$	3.58473	−	0.278866i	0.134061	−	0.0104290i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.39301			0.201125			0.100563	−	0.994931i	$-0.467936\pi$
$719$	5.39301			0.201125			0.100563	+	0.994931i	$0.467936\pi$
$720$		0			0
$721$	−17.5101	+	17.5101i	−0.652109	+	0.652109i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.75812	+	14.2836i	−0.0652949	+	0.530479i
$726$		0			0
$727$	−27.4677	−	27.4677i	−1.01872	−	1.01872i	−0.999821	−	0.0188979i	$-0.993984\pi$
$727$	−27.4677	−	27.4677i	−1.01872	−	1.01872i	−0.0188979	−	0.999821i	$-0.506016\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.20448			0.0445495
$732$		0			0
$733$	−40.5951			−1.49941			−0.749707	−	0.661770i	$-0.769806\pi$
$733$	−40.5951			−1.49941			−0.749707	+	0.661770i	$0.769806\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.27951	+	1.27951i	−0.0471314	+	0.0471314i
$738$		0			0
$739$	−19.5815	+	19.5815i	−0.720315	+	0.720315i	−0.968669	−	0.248354i	$-0.920110\pi$
$739$	−19.5815	+	19.5815i	−0.720315	+	0.720315i	0.248354	+	0.968669i	$0.420110\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−33.1542			−1.21631			−0.608155	−	0.793818i	$-0.708090\pi$
$743$	−33.1542			−1.21631			−0.608155	+	0.793818i	$0.708090\pi$
$744$		0			0
$745$	−1.30645	−	1.47712i	−0.0478647	−	0.0541174i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	58.5008	−	58.5008i	2.13757	−	2.13757i
$750$		0			0
$751$			9.49285i			0.346399i	0.984887	+	0.173200i	$0.0554105\pi$
$751$			9.49285i			0.346399i	−0.984887	+	0.173200i	$0.944589\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.2471	+	13.8470i	0.445719	+	0.503945i
$756$		0			0
$757$	2.84463	−	2.84463i	0.103390	−	0.103390i	−0.653520	−	0.756909i	$-0.726708\pi$
$757$	2.84463	−	2.84463i	0.103390	−	0.103390i	0.756909	+	0.653520i	$0.226708\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−33.6477	−	33.6477i	−1.21973	−	1.21973i	−0.967727	−	0.252002i	$-0.918911\pi$
$761$	−33.6477	−	33.6477i	−1.21973	−	1.21973i	−0.252002	−	0.967727i	$-0.581089\pi$
$762$		0			0
$763$	−56.4057	−	56.4057i	−2.04202	−	2.04202i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.01808	−	7.28391i	−0.0367608	−	0.263007i
$768$		0			0
$769$	2.09238	+	2.09238i	0.0754532	+	0.0754532i	0.743826	−	0.668373i	$-0.233009\pi$
$769$	2.09238	+	2.09238i	0.0754532	+	0.0754532i	−0.668373	+	0.743826i	$0.733009\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	33.8949			1.21911			0.609557	−	0.792742i	$-0.291347\pi$
$773$	33.8949			1.21911			0.609557	+	0.792742i	$0.291347\pi$
$774$		0			0
$775$	−6.31825	+	4.93333i	−0.226958	+	0.177211i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−42.5987			−1.52626
$780$		0			0
$781$	−4.21817			−0.150938
$782$		0			0
$783$		0			0
$784$		0			0
$785$	24.1905	−	21.3956i	0.863397	−	0.763640i
$786$		0			0
$787$	13.3356			0.475363			0.237682	−	0.971343i	$-0.423612\pi$
$787$	13.3356			0.475363			0.237682	+	0.971343i	$0.423612\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−53.3067	−	53.3067i	−1.89537	−	1.89537i
$792$		0			0
$793$	12.6087	−	16.7061i	0.447748	−	0.593250i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.21618	−	2.21618i	−0.0785012	−	0.0785012i	0.666766	−	0.745267i	$-0.267678\pi$
$797$	−2.21618	−	2.21618i	−0.0785012	−	0.0785012i	−0.745267	+	0.666766i	$0.767678\pi$
$798$		0			0
$799$	−6.65230	−	6.65230i	−0.235341	−	0.235341i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.231535	−	0.231535i	0.00817069	−	0.00817069i
$804$		0			0
$805$	−38.7507	+	34.2734i	−1.36578	+	1.20798i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	10.3854i		−	0.365131i	−0.983194	−	0.182565i	$-0.941560\pi$
$809$		−	10.3854i		−	0.365131i	0.983194	−	0.182565i	$-0.0584401\pi$
$810$		0			0
$811$	−2.72201	+	2.72201i	−0.0955825	+	0.0955825i	−0.753281	−	0.657699i	$-0.771530\pi$
$811$	−2.72201	+	2.72201i	−0.0955825	+	0.0955825i	0.657699	+	0.753281i	$0.271530\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.17982	+	51.8629i	−0.111384	+	1.81668i
$816$		0			0
$817$	3.89276			0.136191
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.79821	+	3.79821i	−0.132558	+	0.132558i	−0.770273	−	0.637714i	$-0.779880\pi$
$821$	−3.79821	+	3.79821i	−0.132558	+	0.132558i	0.637714	+	0.770273i	$0.279880\pi$
$822$		0			0
$823$	−36.2441	+	36.2441i	−1.26339	+	1.26339i	−0.313951	+	0.949439i	$0.601653\pi$
$823$	−36.2441	+	36.2441i	−1.26339	+	1.26339i	−0.949439	+	0.313951i	$0.898347\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.2770			1.26147			0.630737	−	0.775996i	$-0.282753\pi$
$827$	36.2770			1.26147			0.630737	+	0.775996i	$0.282753\pi$
$828$		0			0
$829$	−11.4935			−0.399185			−0.199592	−	0.979879i	$-0.563962\pi$
$829$	−11.4935			−0.399185			−0.199592	+	0.979879i	$0.563962\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	16.2927	+	16.2927i	0.564510	+	0.564510i
$834$		0			0
$835$	27.9105	+	1.71125i	0.965883	+	0.0592202i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	20.5056	−	20.5056i	0.707931	−	0.707931i	−0.258169	−	0.966100i	$-0.583119\pi$
$839$	20.5056	−	20.5056i	0.707931	−	0.707931i	0.966100	+	0.258169i	$0.0831191\pi$
$840$		0			0
$841$	20.7155			0.714328
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.24459	−	28.3902i	−0.214821	−	0.976654i
$846$		0			0
$847$		−	50.7618i		−	1.74420i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−31.0778	+	31.0778i	−1.06533	+	1.06533i
$852$		0			0
$853$		−	10.6968i		−	0.366253i	−0.983089	−	0.183126i	$-0.941378\pi$
$853$		−	10.6968i		−	0.366253i	0.983089	−	0.183126i	$-0.0586217\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.4311	−	20.4311i	−0.697912	−	0.697912i	0.266048	−	0.963960i	$-0.414282\pi$
$857$	−20.4311	−	20.4311i	−0.697912	−	0.697912i	−0.963960	+	0.266048i	$0.914282\pi$
$858$		0			0
$859$			31.0854i			1.06062i	0.847803	+	0.530311i	$0.177925\pi$
$859$			31.0854i			1.06062i	−0.847803	+	0.530311i	$0.822075\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	32.6466			1.11130			0.555651	−	0.831415i	$-0.312469\pi$
$863$	32.6466			1.11130			0.555651	+	0.831415i	$0.312469\pi$
$864$		0			0
$865$	43.4377	−	38.4189i	1.47693	−	1.30628i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.07235	+	4.07235i	−0.138145	+	0.138145i
$870$		0			0
$871$	11.6768	+	8.81289i	0.395652	+	0.298613i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	51.6600	+	9.59847i	1.74643	+	0.324487i
$876$		0			0
$877$			24.8946i			0.840629i	0.907378	+	0.420315i	$0.138080\pi$
$877$			24.8946i			0.840629i	−0.907378	+	0.420315i	$0.861920\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			57.1910i			1.92681i	0.268046	+	0.963406i	$0.413622\pi$
$881$			57.1910i			1.92681i	−0.268046	+	0.963406i	$0.586378\pi$
$882$		0			0
$883$	18.8928	−	18.8928i	0.635793	−	0.635793i	−0.313722	−	0.949515i	$-0.601576\pi$
$883$	18.8928	−	18.8928i	0.635793	−	0.635793i	0.949515	+	0.313722i	$0.101576\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.5287	−	14.5287i	0.487826	−	0.487826i	−0.419794	−	0.907620i	$-0.637898\pi$
$887$	14.5287	−	14.5287i	0.487826	−	0.487826i	0.907620	+	0.419794i	$0.137898\pi$
$888$		0			0
$889$	31.6729	+	31.6729i	1.06228	+	1.06228i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21.4995	−	21.4995i	−0.719454	−	0.719454i
$894$		0			0
$895$	31.2196	+	1.91414i	1.04356	+	0.0639825i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.26296	−	3.26296i	−0.108826	−	0.108826i
$900$		0			0
$901$		−	18.1885i		−	0.605948i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.00256	+	32.6618i	−0.0665673	+	1.08572i
$906$		0			0
$907$	14.2325	+	14.2325i	0.472583	+	0.472583i	0.902750	−	0.430166i	$-0.141545\pi$
$907$	14.2325	+	14.2325i	0.472583	+	0.472583i	−0.430166	+	0.902750i	$0.641545\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−32.2436			−1.06828			−0.534139	−	0.845397i	$-0.679364\pi$
$911$	−32.2436			−1.06828			−0.534139	+	0.845397i	$0.679364\pi$
$912$		0			0
$913$	−0.333010	−	0.333010i	−0.0110210	−	0.0110210i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.3454			0.803956
$918$		0			0
$919$		−	9.73620i		−	0.321168i	−0.987022	−	0.160584i	$-0.948662\pi$
$919$		−	9.73620i		−	0.321168i	0.987022	−	0.160584i	$-0.0513377\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.72065	+	33.7741i	0.155382	+	1.11169i
$924$		0			0
$925$	44.3056	+	5.45343i	1.45676	+	0.179308i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.82032	+	9.82032i	0.322194	+	0.322194i	0.849608	−	0.527414i	$-0.176838\pi$
$929$	9.82032	+	9.82032i	0.322194	+	0.322194i	−0.527414	+	0.849608i	$0.676838\pi$
$930$		0			0
$931$	52.6564	+	52.6564i	1.72574	+	1.72574i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.0932032	−	1.52015i	0.00304807	−	0.0497141i
$936$		0			0
$937$	−6.29485	+	6.29485i	−0.205644	+	0.205644i	−0.802413	−	0.596769i	$-0.796451\pi$
$937$	−6.29485	+	6.29485i	−0.205644	+	0.205644i	0.596769	+	0.802413i	$0.296451\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−32.4276	+	32.4276i	−1.05711	+	1.05711i	−0.0588409	+	0.998267i	$0.518740\pi$
$941$	−32.4276	+	32.4276i	−1.05711	+	1.05711i	−0.998267	+	0.0588409i	$0.981260\pi$
$942$		0			0
$943$		−	42.4861i		−	1.38354i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.1923			1.33857			0.669285	−	0.743006i	$-0.266601\pi$
$947$	41.1923			1.33857			0.669285	+	0.743006i	$0.266601\pi$
$948$		0			0
$949$	−2.11298	−	1.59474i	−0.0685901	−	0.0517675i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−33.0574	+	33.0574i	−1.07083	+	1.07083i	−0.0735423	+	0.997292i	$0.523430\pi$
$953$	−33.0574	+	33.0574i	−1.07083	+	1.07083i	−0.997292	+	0.0735423i	$0.976570\pi$
$954$		0			0
$955$	30.5536	+	1.87330i	0.988691	+	0.0606186i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−79.4078			−2.56421
$960$		0			0
$961$			28.4297i			0.917086i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.55249	−	25.3211i	0.0499764	−	0.815116i
$966$		0			0
$967$		−	54.2975i		−	1.74609i	−0.487640	−	0.873045i	$-0.662142\pi$
$967$		−	54.2975i		−	1.74609i	0.487640	−	0.873045i	$-0.337858\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	48.8468			1.56757			0.783785	−	0.621033i	$-0.213287\pi$
$971$	48.8468			1.56757			0.783785	+	0.621033i	$0.213287\pi$
$972$		0			0
$973$			14.3539i			0.460165i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	56.8534i		−	1.81890i	−0.415811	−	0.909451i	$-0.636502\pi$
$977$		−	56.8534i		−	1.81890i	0.415811	−	0.909451i	$-0.363498\pi$
$978$		0			0
$979$	−3.38394			−0.108151
$980$		0			0
$981$		0			0
$982$		0			0
$983$			15.7995i			0.503924i	0.967737	+	0.251962i	$0.0810759\pi$
$983$			15.7995i			0.503924i	−0.967737	+	0.251962i	$0.918924\pi$
$984$		0			0
$985$	−1.74644	+	28.4844i	−0.0556461	+	0.907589i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			3.88247i			0.123455i
$990$		0			0
$991$	−17.8662			−0.567538			−0.283769	−	0.958893i	$-0.591585\pi$
$991$	−17.8662			−0.567538			−0.283769	+	0.958893i	$0.591585\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.64134	+	0.345882i	0.178842	+	0.0109652i
$996$		0			0
$997$	5.03661	−	5.03661i	0.159511	−	0.159511i	−0.622839	−	0.782350i	$-0.714021\pi$
$997$	5.03661	−	5.03661i	0.159511	−	0.159511i	0.782350	+	0.622839i	$0.214021\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.bp.i.1513.14		28
3.2	odd	2		780.2.bm.a.733.1	yes	28
5.2	odd	4		2340.2.u.i.577.7		28
13.8	odd	4		2340.2.u.i.73.7		28
15.2	even	4		780.2.r.a.577.4	yes	28
15.8	even	4		3900.2.r.b.1357.8		28
15.14	odd	2		3900.2.bm.b.2293.8		28
39.8	even	4		780.2.r.a.73.4	✓	28
65.47	even	4	inner	2340.2.bp.i.1477.14		28
195.8	odd	4		3900.2.bm.b.2257.8		28
195.47	odd	4		780.2.bm.a.697.1	yes	28
195.164	even	4		3900.2.r.b.3193.8		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
780.2.r.a.73.4	✓	28	39.8	even	4
780.2.r.a.577.4	yes	28	15.2	even	4
780.2.bm.a.697.1	yes	28	195.47	odd	4
780.2.bm.a.733.1	yes	28	3.2	odd	2
2340.2.u.i.73.7		28	13.8	odd	4
2340.2.u.i.577.7		28	5.2	odd	4
2340.2.bp.i.1477.14		28	65.47	even	4	inner
2340.2.bp.i.1513.14		28	1.1	even	1	trivial
3900.2.r.b.1357.8		28	15.8	even	4
3900.2.r.b.3193.8		28	195.164	even	4
3900.2.bm.b.2257.8		28	195.8	odd	4
3900.2.bm.b.2293.8		28	15.14	odd	2

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:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.bp (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.23188 + 0.136841i) q^{5} +4.69969 q^{7} +O(q^{10})\)	\(q+(2.23188 + 0.136841i) q^{5} +4.69969 q^{7} +(-0.315352 - 0.315352i) q^{11} +(-2.17205 + 2.87788i) q^{13} +(1.07992 + 1.07992i) q^{17} +(3.49017 + 3.49017i) q^{19} +(-3.48094 + 3.48094i) q^{23} +(4.96255 + 0.610824i) q^{25} +2.87828i q^{29} +(-1.13365 + 1.13365i) q^{31} +(10.4891 + 0.643109i) q^{35} +8.92799 q^{37} +(-6.10267 + 6.10267i) q^{41} +(0.557675 - 0.557675i) q^{43} -6.16002 q^{47} +15.0871 q^{49} +(-8.42127 - 8.42127i) q^{53} +(-0.660673 - 0.746979i) q^{55} +(-1.44238 + 1.44238i) q^{59} -5.80499 q^{61} +(-5.24155 + 6.12586i) q^{65} -4.05741i q^{67} +(6.68803 - 6.68803i) q^{71} +0.734212i q^{73} +(-1.48205 - 1.48205i) q^{77} -12.9137i q^{79} +1.05600 q^{83} +(2.26246 + 2.55801i) q^{85} +(5.36534 - 5.36534i) q^{89} +(-10.2079 + 13.5251i) q^{91} +(7.31203 + 8.26723i) q^{95} +4.56277i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 4 q^{5}+O(q^{10}) \) 28 * q + 4 * q^5	\( 28 q + 4 q^{5} - 8 q^{11} - 4 q^{17} + 16 q^{19} - 8 q^{23} - 12 q^{25} + 24 q^{37} - 12 q^{41} - 16 q^{43} + 24 q^{47} + 36 q^{49} - 36 q^{53} + 40 q^{55} + 16 q^{59} + 8 q^{61} - 24 q^{65} - 8 q^{71} - 48 q^{77} + 24 q^{83} + 68 q^{85} - 36 q^{89} - 24 q^{91} + 72 q^{95}+O(q^{100}) \) 28 * q + 4 * q^5 - 8 * q^11 - 4 * q^17 + 16 * q^19 - 8 * q^23 - 12 * q^25 + 24 * q^37 - 12 * q^41 - 16 * q^43 + 24 * q^47 + 36 * q^49 - 36 * q^53 + 40 * q^55 + 16 * q^59 + 8 * q^61 - 24 * q^65 - 8 * q^71 - 48 * q^77 + 24 * q^83 + 68 * q^85 - 36 * q^89 - 24 * q^91 + 72 * q^95

Introduction

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