LMFDB - Embedded newform 2340.2.y.a.53.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(53,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.53"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.1
Character	$\chi$	$=$	2340.53
Dual form			2340.2.y.a.1457.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.21583 + 0.300151i) q^{5} +(2.83763 - 2.83763i) q^{7} +0.275080i q^{11} +(0.707107 + 0.707107i) q^{13} +(-1.49149 - 1.49149i) q^{17} -0.767881i q^{19} +(2.48808 - 2.48808i) q^{23} +(4.81982 - 1.33017i) q^{25} -9.19836 q^{29} +0.0656918 q^{31} +(-5.43600 + 7.13943i) q^{35} +(2.85142 - 2.85142i) q^{37} -3.86419i q^{41} +(1.63941 + 1.63941i) q^{43} +(2.80313 + 2.80313i) q^{47} -9.10431i q^{49} +(-0.976883 + 0.976883i) q^{53} +(-0.0825655 - 0.609531i) q^{55} +7.13789 q^{59} -9.52683 q^{61} +(-1.77907 - 1.35459i) q^{65} +(0.223398 - 0.223398i) q^{67} -12.7488i q^{71} +(-9.60841 - 9.60841i) q^{73} +(0.780576 + 0.780576i) q^{77} -5.20411i q^{79} +(-5.65471 + 5.65471i) q^{83} +(3.75256 + 2.85722i) q^{85} -5.96742 q^{89} +4.01302 q^{91} +(0.230480 + 1.70149i) q^{95} +(0.0852795 - 0.0852795i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 8 q^{7} - 8 q^{17} - 8 q^{23} + 16 q^{25} + 32 q^{29} + 8 q^{35} + 16 q^{37} - 8 q^{43} - 40 q^{47} - 8 q^{53} + 8 q^{55} + 56 q^{59} + 8 q^{61} - 4 q^{65} - 16 q^{67} - 72 q^{77} - 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) $ 24 * q - 4 * q^5 - 8 * q^7 - 8 * q^17 - 8 * q^23 + 16 * q^25 + 32 * q^29 + 8 * q^35 + 16 * q^37 - 8 * q^43 - 40 * q^47 - 8 * q^53 + 8 * q^55 + 56 * q^59 + 8 * q^61 - 4 * q^65 - 16 * q^67 - 72 * q^77 - 32 * q^83 - 32 * q^85 + 128 * q^89 - 8 * q^91 + 16 * q^95 + 8 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.21583	+	0.300151i	−0.990950	+	0.134231i
$5$	−2.21583	+	0.300151i	−0.990950	+	0.134231i
$6$		0			0
$7$	2.83763	−	2.83763i	1.07252	−	1.07252i	0.0753685	−	0.997156i	$-0.475987\pi$
$7$	2.83763	−	2.83763i	1.07252	−	1.07252i	0.997156	−	0.0753685i	$-0.0240133\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.275080i			0.0829397i	0.999140	+	0.0414699i	$0.0132041\pi$
$11$			0.275080i			0.0829397i	−0.999140	+	0.0414699i	$0.986796\pi$
$12$		0			0
$13$	0.707107	+	0.707107i	0.196116	+	0.196116i
$13$	0.707107	+	0.707107i	0.196116	+	0.196116i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.49149	−	1.49149i	−0.361739	−	0.361739i	0.502714	−	0.864453i	$-0.332335\pi$
$17$	−1.49149	−	1.49149i	−0.361739	−	0.361739i	−0.864453	+	0.502714i	$0.832335\pi$
$18$		0			0
$19$		−	0.767881i		−	0.176164i	−0.996113	−	0.0880820i	$-0.971926\pi$
$19$		−	0.767881i		−	0.176164i	0.996113	−	0.0880820i	$-0.0280737\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.48808	−	2.48808i	0.518801	−	0.518801i	−0.398407	−	0.917209i	$-0.630437\pi$
$23$	2.48808	−	2.48808i	0.518801	−	0.518801i	0.917209	+	0.398407i	$0.130437\pi$
$24$		0			0
$25$	4.81982	−	1.33017i	0.963964	−	0.266033i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.19836			−1.70809			−0.854046	−	0.520197i	$-0.825858\pi$
$29$	−9.19836			−1.70809			−0.854046	+	0.520197i	$0.825858\pi$
$30$		0			0
$31$	0.0656918			0.0117986			0.00589930	−	0.999983i	$-0.498122\pi$
$31$	0.0656918			0.0117986			0.00589930	+	0.999983i	$0.498122\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.43600	+	7.13943i	−0.918851	+	1.20678i
$36$		0			0
$37$	2.85142	−	2.85142i	0.468770	−	0.468770i	−0.432746	−	0.901516i	$-0.642455\pi$
$37$	2.85142	−	2.85142i	0.468770	−	0.468770i	0.901516	+	0.432746i	$0.142455\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	3.86419i		−	0.603484i	−0.953390	−	0.301742i	$-0.902432\pi$
$41$		−	3.86419i		−	0.603484i	0.953390	−	0.301742i	$-0.0975682\pi$
$42$		0			0
$43$	1.63941	+	1.63941i	0.250008	+	0.250008i	0.820974	−	0.570966i	$-0.193431\pi$
$43$	1.63941	+	1.63941i	0.250008	+	0.250008i	−0.570966	+	0.820974i	$0.693431\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.80313	+	2.80313i	0.408878	+	0.408878i	0.881347	−	0.472469i	$-0.156637\pi$
$47$	2.80313	+	2.80313i	0.408878	+	0.408878i	−0.472469	+	0.881347i	$0.656637\pi$
$48$		0			0
$49$		−	9.10431i		−	1.30062i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.976883	+	0.976883i	−0.134185	+	0.134185i	−0.771009	−	0.636824i	$-0.780248\pi$
$53$	−0.976883	+	0.976883i	−0.134185	+	0.134185i	0.636824	+	0.771009i	$0.280248\pi$
$54$		0			0
$55$	−0.0825655	−	0.609531i	−0.0111331	−	0.0821891i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.13789			0.929274			0.464637	−	0.885501i	$-0.346185\pi$
$59$	7.13789			0.929274			0.464637	+	0.885501i	$0.346185\pi$
$60$		0			0
$61$	−9.52683			−1.21979			−0.609893	−	0.792484i	$-0.708788\pi$
$61$	−9.52683			−1.21979			−0.609893	+	0.792484i	$0.708788\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.77907	−	1.35459i	−0.220666	−	0.168016i
$66$		0			0
$67$	0.223398	−	0.223398i	0.0272924	−	0.0272924i	−0.693329	−	0.720621i	$-0.743857\pi$
$67$	0.223398	−	0.223398i	0.0272924	−	0.0272924i	0.720621	+	0.693329i	$0.243857\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.7488i		−	1.51301i	−0.653990	−	0.756503i	$-0.726906\pi$
$71$		−	12.7488i		−	1.51301i	0.653990	−	0.756503i	$-0.273094\pi$
$72$		0			0
$73$	−9.60841	−	9.60841i	−1.12458	−	1.12458i	−0.991044	−	0.133536i	$-0.957367\pi$
$73$	−9.60841	−	9.60841i	−1.12458	−	1.12458i	−0.133536	−	0.991044i	$-0.542633\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.780576	+	0.780576i	0.0889549	+	0.0889549i
$78$		0			0
$79$		−	5.20411i		−	0.585508i	−0.956188	−	0.292754i	$-0.905428\pi$
$79$		−	5.20411i		−	0.585508i	0.956188	−	0.292754i	$-0.0945718\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.65471	+	5.65471i	−0.620685	+	0.620685i	−0.945707	−	0.325021i	$-0.894628\pi$
$83$	−5.65471	+	5.65471i	−0.620685	+	0.620685i	0.325021	+	0.945707i	$0.394628\pi$
$84$		0			0
$85$	3.75256	+	2.85722i	0.407022	+	0.309909i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.96742			−0.632545			−0.316273	−	0.948668i	$-0.602431\pi$
$89$	−5.96742			−0.632545			−0.316273	+	0.948668i	$0.602431\pi$
$90$		0			0
$91$	4.01302			0.420679
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.230480	+	1.70149i	0.0236467	+	0.174570i
$96$		0			0
$97$	0.0852795	−	0.0852795i	0.00865882	−	0.00865882i	−0.702764	−	0.711423i	$-0.748051\pi$
$97$	0.0852795	−	0.0852795i	0.00865882	−	0.00865882i	0.711423	+	0.702764i	$0.248051\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	2.35876i		−	0.234706i	−0.993090	−	0.117353i	$-0.962559\pi$
$101$		−	2.35876i		−	0.234706i	0.993090	−	0.117353i	$-0.0374408\pi$
$102$		0			0
$103$	−11.3776	−	11.3776i	−1.12107	−	1.12107i	−0.991581	−	0.129485i	$-0.958668\pi$
$103$	−11.3776	−	11.3776i	−1.12107	−	1.12107i	−0.129485	−	0.991581i	$-0.541332\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.175898	−	0.175898i	−0.0170047	−	0.0170047i	0.698553	−	0.715558i	$-0.253827\pi$
$107$	−0.175898	−	0.175898i	−0.0170047	−	0.0170047i	−0.715558	+	0.698553i	$0.753827\pi$
$108$		0			0
$109$		−	0.611880i		−	0.0586075i	−0.999571	−	0.0293037i	$-0.990671\pi$
$109$		−	0.611880i		−	0.0586075i	0.999571	−	0.0293037i	$-0.00932900\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.02687	−	4.02687i	0.378816	−	0.378816i	−0.491859	−	0.870675i	$-0.663682\pi$
$113$	4.02687	−	4.02687i	0.378816	−	0.378816i	0.870675	+	0.491859i	$0.163682\pi$
$114$		0			0
$115$	−4.76637	+	6.25997i	−0.444467	+	0.583746i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.46460			−0.775948
$120$		0			0
$121$	10.9243			0.993121
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.2807	+	4.39410i	−0.919530	+	0.393020i
$126$		0			0
$127$	9.03483	−	9.03483i	0.801712	−	0.801712i	−0.181651	−	0.983363i	$-0.558144\pi$
$127$	9.03483	−	9.03483i	0.801712	−	0.801712i	0.983363	+	0.181651i	$0.0581442\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.2243i		−	1.06804i	−0.845472	−	0.534019i	$-0.820681\pi$
$131$		−	12.2243i		−	1.06804i	0.845472	−	0.534019i	$-0.179319\pi$
$132$		0			0
$133$	−2.17896	−	2.17896i	−0.188940	−	0.188940i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.7412	−	13.7412i	−1.17399	−	1.17399i	−0.981250	−	0.192740i	$-0.938263\pi$
$137$	−13.7412	−	13.7412i	−1.17399	−	1.17399i	−0.192740	−	0.981250i	$-0.561737\pi$
$138$		0			0
$139$		−	20.3062i		−	1.72235i	−0.508311	−	0.861173i	$-0.669730\pi$
$139$		−	20.3062i		−	1.72235i	0.508311	−	0.861173i	$-0.330270\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.194511	+	0.194511i	−0.0162658	+	0.0162658i
$144$		0			0
$145$	20.3820	−	2.76089i	1.69263	−	0.229280i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	16.9390			1.38770			0.693850	−	0.720120i	$-0.255913\pi$
$149$	16.9390			1.38770			0.693850	+	0.720120i	$0.255913\pi$
$150$		0			0
$151$	−17.2012			−1.39981			−0.699906	−	0.714235i	$-0.746775\pi$
$151$	−17.2012			−1.39981			−0.699906	+	0.714235i	$0.746775\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.145562	+	0.0197175i	−0.0116918	+	0.00158374i
$156$		0			0
$157$	8.63433	−	8.63433i	0.689095	−	0.689095i	−0.272937	−	0.962032i	$-0.587995\pi$
$157$	8.63433	−	8.63433i	0.689095	−	0.689095i	0.962032	+	0.272937i	$0.0879951\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	14.1205i		−	1.11285i
$162$		0			0
$163$	7.38433	+	7.38433i	0.578385	+	0.578385i	0.934458	−	0.356073i	$-0.115884\pi$
$163$	7.38433	+	7.38433i	0.578385	+	0.578385i	−0.356073	+	0.934458i	$0.615884\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.27677	−	3.27677i	−0.253564	−	0.253564i	0.568866	−	0.822430i	$-0.307382\pi$
$167$	−3.27677	−	3.27677i	−0.253564	−	0.253564i	−0.822430	+	0.568866i	$0.807382\pi$
$168$		0			0
$169$			1.00000i			0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.11049	−	7.11049i	0.540600	−	0.540600i	−0.383105	−	0.923705i	$-0.625145\pi$
$173$	7.11049	−	7.11049i	0.540600	−	0.540600i	0.923705	+	0.383105i	$0.125145\pi$
$174$		0			0
$175$	9.90235	−	17.4514i	0.748547	−	1.31920i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.99931			0.373666			0.186833	−	0.982392i	$-0.440178\pi$
$179$	4.99931			0.373666			0.186833	+	0.982392i	$0.440178\pi$
$180$		0			0
$181$	−2.77482			−0.206250			−0.103125	−	0.994668i	$-0.532884\pi$
$181$	−2.77482			−0.206250			−0.103125	+	0.994668i	$0.532884\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.46240	+	7.17411i	−0.401604	+	0.527451i
$186$		0			0
$187$	0.410279	−	0.410279i	0.0300026	−	0.0300026i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			13.5552i			0.980819i	0.871492	+	0.490409i	$0.163153\pi$
$191$			13.5552i			0.980819i	−0.871492	+	0.490409i	$0.836847\pi$
$192$		0			0
$193$	10.5223	+	10.5223i	0.757411	+	0.757411i	0.975850	−	0.218440i	$-0.0700967\pi$
$193$	10.5223	+	10.5223i	0.757411	+	0.757411i	−0.218440	+	0.975850i	$0.570097\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.4632	−	14.4632i	−1.03046	−	1.03046i	−0.999521	−	0.0309403i	$-0.990150\pi$
$197$	−14.4632	−	14.4632i	−1.03046	−	1.03046i	−0.0309403	−	0.999521i	$-0.509850\pi$
$198$		0			0
$199$		−	6.50993i		−	0.461477i	−0.973016	−	0.230738i	$-0.925886\pi$
$199$		−	6.50993i		−	0.461477i	0.973016	−	0.230738i	$-0.0741141\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−26.1016	+	26.1016i	−1.83197	+	1.83197i
$204$		0			0
$205$	1.15984	+	8.56238i	0.0810066	+	0.598023i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.211229			0.0146110
$210$		0			0
$211$	−2.11694			−0.145737			−0.0728683	−	0.997342i	$-0.523215\pi$
$211$	−2.11694			−0.145737			−0.0728683	+	0.997342i	$0.523215\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.12474	−	3.14060i	−0.281305	−	0.214187i
$216$		0			0
$217$	0.186409	−	0.186409i	0.0126543	−	0.0126543i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.10928i		−	0.141886i
$222$		0			0
$223$	17.0339	+	17.0339i	1.14067	+	1.14067i	0.988327	+	0.152345i	$0.0486824\pi$
$223$	17.0339	+	17.0339i	1.14067	+	1.14067i	0.152345	+	0.988327i	$0.451318\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.98651	+	7.98651i	0.530084	+	0.530084i	0.920597	−	0.390514i	$-0.127703\pi$
$227$	7.98651	+	7.98651i	0.530084	+	0.530084i	−0.390514	+	0.920597i	$0.627703\pi$
$228$		0			0
$229$			11.5307i			0.761970i	0.924581	+	0.380985i	$0.124415\pi$
$229$			11.5307i			0.761970i	−0.924581	+	0.380985i	$0.875585\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.45924	−	5.45924i	0.357647	−	0.357647i	−0.505298	−	0.862945i	$-0.668617\pi$
$233$	5.45924	−	5.45924i	0.357647	−	0.357647i	0.862945	+	0.505298i	$0.168617\pi$
$234$		0			0
$235$	−7.05262	−	5.36990i	−0.460062	−	0.350294i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.35874			0.605367			0.302683	−	0.953091i	$-0.402118\pi$
$239$	9.35874			0.605367			0.302683	+	0.953091i	$0.402118\pi$
$240$		0			0
$241$	21.6747			1.39619			0.698095	−	0.716005i	$-0.254031\pi$
$241$	21.6747			1.39619			0.698095	+	0.716005i	$0.254031\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.73267	+	20.1736i	0.174584	+	1.28885i
$246$		0			0
$247$	0.542974	−	0.542974i	0.0345486	−	0.0345486i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			9.32812i			0.588786i	0.955684	+	0.294393i	$0.0951174\pi$
$251$			9.32812i			0.588786i	−0.955684	+	0.294393i	$0.904883\pi$
$252$		0			0
$253$	0.684422	+	0.684422i	0.0430292	+	0.0430292i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.10658	−	5.10658i	−0.318539	−	0.318539i	0.529667	−	0.848206i	$-0.322317\pi$
$257$	−5.10658	−	5.10658i	−0.318539	−	0.318539i	−0.848206	+	0.529667i	$0.822317\pi$
$258$		0			0
$259$		−	16.1825i		−	1.00553i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.0644	+	18.0644i	−1.11390	+	1.11390i	−0.121282	+	0.992618i	$0.538701\pi$
$263$	−18.0644	+	18.0644i	−1.11390	+	1.11390i	−0.992618	+	0.121282i	$0.961299\pi$
$264$		0			0
$265$	1.87140	−	2.45782i	0.114959	−	0.150983i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.8098			0.963943			0.481972	−	0.876187i	$-0.339921\pi$
$269$	15.8098			0.963943			0.481972	+	0.876187i	$0.339921\pi$
$270$		0			0
$271$	−26.8065			−1.62838			−0.814189	−	0.580600i	$-0.802818\pi$
$271$	−26.8065			−1.62838			−0.814189	+	0.580600i	$0.802818\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.365902	+	1.32584i	0.0220647	+	0.0799509i
$276$		0			0
$277$	−20.1306	+	20.1306i	−1.20953	+	1.20953i	−0.238351	+	0.971179i	$0.576607\pi$
$277$	−20.1306	+	20.1306i	−1.20953	+	1.20953i	−0.971179	+	0.238351i	$0.923393\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			7.81305i			0.466087i	0.972466	+	0.233044i	$0.0748685\pi$
$281$			7.81305i			0.466087i	−0.972466	+	0.233044i	$0.925132\pi$
$282$		0			0
$283$	−9.73125	−	9.73125i	−0.578463	−	0.578463i	0.356017	−	0.934480i	$-0.384135\pi$
$283$	−9.73125	−	9.73125i	−0.578463	−	0.578463i	−0.934480	+	0.356017i	$0.884135\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.9651	−	10.9651i	−0.647252	−	0.647252i
$288$		0			0
$289$		−	12.5509i		−	0.738289i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.78127	−	4.78127i	0.279325	−	0.279325i	−0.553515	−	0.832839i	$-0.686714\pi$
$293$	4.78127	−	4.78127i	0.279325	−	0.279325i	0.832839	+	0.553515i	$0.186714\pi$
$294$		0			0
$295$	−15.8164	+	2.14244i	−0.920864	+	0.124738i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.51868			0.203491
$300$		0			0
$301$	9.30411			0.536280
$302$		0			0
$303$		0			0
$304$		0			0
$305$	21.1099	−	2.85949i	1.20875	−	0.163734i
$306$		0			0
$307$	−20.3856	+	20.3856i	−1.16347	+	1.16347i	−0.179754	+	0.983712i	$0.557530\pi$
$307$	−20.3856	+	20.3856i	−1.16347	+	1.16347i	−0.983712	+	0.179754i	$0.942470\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	0.328162i		−	0.0186084i	−0.999957	−	0.00930419i	$-0.997038\pi$
$311$		−	0.328162i		−	0.0186084i	0.999957	−	0.00930419i	$-0.00296166\pi$
$312$		0			0
$313$	24.3629	+	24.3629i	1.37707	+	1.37707i	0.849524	+	0.527551i	$0.176890\pi$
$313$	24.3629	+	24.3629i	1.37707	+	1.37707i	0.527551	+	0.849524i	$0.323110\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.45737	+	4.45737i	0.250351	+	0.250351i	0.821114	−	0.570764i	$-0.193353\pi$
$317$	4.45737	+	4.45737i	0.250351	+	0.250351i	−0.570764	+	0.821114i	$0.693353\pi$
$318$		0			0
$319$		−	2.53028i		−	0.141669i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.14529	+	1.14529i	−0.0637254	+	0.0637254i
$324$		0			0
$325$	4.34870	+	2.46756i	0.241222	+	0.136875i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	15.9085			0.877064
$330$		0			0
$331$	−20.4356			−1.12324			−0.561622	−	0.827394i	$-0.689822\pi$
$331$	−20.4356			−1.12324			−0.561622	+	0.827394i	$0.689822\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.427959	+	0.562066i	−0.0233819	+	0.0307089i
$336$		0			0
$337$	−18.3170	+	18.3170i	−0.997790	+	0.997790i	−0.999998	−	0.00220753i	$-0.999297\pi$
$337$	−18.3170	+	18.3170i	−0.997790	+	0.997790i	0.00220753	+	0.999998i	$0.499297\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			0.0180705i			0.000978573i
$342$		0			0
$343$	−5.97127	−	5.97127i	−0.322418	−	0.322418i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.6429	−	12.6429i	−0.678706	−	0.678706i	0.281001	−	0.959707i	$-0.409333\pi$
$347$	−12.6429	−	12.6429i	−0.678706	−	0.678706i	−0.959707	+	0.281001i	$0.909333\pi$
$348$		0			0
$349$		−	5.26764i		−	0.281971i	−0.990012	−	0.140985i	$-0.954973\pi$
$349$		−	5.26764i		−	0.281971i	0.990012	−	0.140985i	$-0.0450270\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.35043	+	6.35043i	−0.337999	+	0.337999i	−0.855614	−	0.517614i	$-0.826820\pi$
$353$	−6.35043	+	6.35043i	−0.337999	+	0.337999i	0.517614	+	0.855614i	$0.326820\pi$
$354$		0			0
$355$	3.82656	+	28.2492i	0.203093	+	1.49931i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.9800			0.579505			0.289752	−	0.957102i	$-0.406427\pi$
$359$	10.9800			0.579505			0.289752	+	0.957102i	$0.406427\pi$
$360$		0			0
$361$	18.4104			0.968966
$362$		0			0
$363$		0			0
$364$		0			0
$365$	24.1746	+	18.4067i	1.26536	+	0.963448i
$366$		0			0
$367$	−15.1472	+	15.1472i	−0.790680	+	0.790680i	−0.981605	−	0.190925i	$-0.938851\pi$
$367$	−15.1472	+	15.1472i	−0.790680	+	0.790680i	0.190925	+	0.981605i	$0.438851\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			5.54407i			0.287834i
$372$		0			0
$373$	−5.50028	−	5.50028i	−0.284794	−	0.284794i	0.550224	−	0.835017i	$-0.314542\pi$
$373$	−5.50028	−	5.50028i	−0.284794	−	0.284794i	−0.835017	+	0.550224i	$0.814542\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.50422	−	6.50422i	−0.334984	−	0.334984i
$378$		0			0
$379$			6.80175i			0.349382i	0.984623	+	0.174691i	$0.0558927\pi$
$379$			6.80175i			0.349382i	−0.984623	+	0.174691i	$0.944107\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.16450	+	7.16450i	−0.366089	+	0.366089i	−0.866049	−	0.499960i	$-0.833348\pi$
$383$	−7.16450	+	7.16450i	−0.366089	+	0.366089i	0.499960	+	0.866049i	$0.333348\pi$
$384$		0			0
$385$	−1.96392	−	1.49533i	−0.100090	−	0.0762093i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	25.1836			1.27686			0.638431	−	0.769679i	$-0.279584\pi$
$389$	25.1836			1.27686			0.638431	+	0.769679i	$0.279584\pi$
$390$		0			0
$391$	−7.42190			−0.375342
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.56202	+	11.5314i	0.0785937	+	0.580210i
$396$		0			0
$397$	−12.3821	+	12.3821i	−0.621439	+	0.621439i	−0.945899	−	0.324461i	$-0.894817\pi$
$397$	−12.3821	+	12.3821i	−0.621439	+	0.621439i	0.324461	+	0.945899i	$0.394817\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			13.7625i			0.687264i	0.939104	+	0.343632i	$0.111657\pi$
$401$			13.7625i			0.687264i	−0.939104	+	0.343632i	$0.888343\pi$
$402$		0			0
$403$	0.0464511	+	0.0464511i	0.00231390	+	0.00231390i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.784367	+	0.784367i	0.0388797	+	0.0388797i
$408$		0			0
$409$			13.6291i			0.673918i	0.941519	+	0.336959i	$0.109398\pi$
$409$			13.6291i			0.673918i	−0.941519	+	0.336959i	$0.890602\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	20.2547	−	20.2547i	0.996669	−	0.996669i
$414$		0			0
$415$	10.8326	−	14.2272i	0.531753	−	0.698384i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	20.2829			0.990886			0.495443	−	0.868641i	$-0.335006\pi$
$419$	20.2829			0.990886			0.495443	+	0.868641i	$0.335006\pi$
$420$		0			0
$421$	15.9040			0.775116			0.387558	−	0.921845i	$-0.373319\pi$
$421$	15.9040			0.775116			0.387558	+	0.921845i	$0.373319\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.17264	−	5.20478i	−0.444938	−	0.252469i
$426$		0			0
$427$	−27.0337	+	27.0337i	−1.30825	+	1.30825i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	33.2888i		−	1.60346i	−0.597684	−	0.801732i	$-0.703912\pi$
$431$		−	33.2888i		−	1.60346i	0.597684	−	0.801732i	$-0.296088\pi$
$432$		0			0
$433$	−1.75897	−	1.75897i	−0.0845309	−	0.0845309i	0.663577	−	0.748108i	$-0.269037\pi$
$433$	−1.75897	−	1.75897i	−0.0845309	−	0.0845309i	−0.748108	+	0.663577i	$0.769037\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.91055	−	1.91055i	−0.0913941	−	0.0913941i
$438$		0			0
$439$		−	27.3842i		−	1.30698i	−0.756936	−	0.653489i	$-0.773304\pi$
$439$		−	27.3842i		−	1.30698i	0.756936	−	0.653489i	$-0.226696\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.6664	+	14.6664i	−0.696822	+	0.696822i	−0.963724	−	0.266902i	$-0.914000\pi$
$443$	−14.6664	+	14.6664i	−0.696822	+	0.696822i	0.266902	+	0.963724i	$0.414000\pi$
$444$		0			0
$445$	13.2228	−	1.79113i	0.626821	−	0.0849075i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	39.6444			1.87094			0.935468	−	0.353411i	$-0.114978\pi$
$449$	39.6444			1.87094			0.935468	+	0.353411i	$0.114978\pi$
$450$		0			0
$451$	1.06296			0.0500528
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.89217	+	1.20451i	−0.416871	+	0.0564683i
$456$		0			0
$457$	13.1725	−	13.1725i	0.616186	−	0.616186i	−0.328365	−	0.944551i	$-0.606498\pi$
$457$	13.1725	−	13.1725i	0.616186	−	0.616186i	0.944551	+	0.328365i	$0.106498\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	14.9614i		−	0.696821i	−0.937342	−	0.348410i	$-0.886722\pi$
$461$		−	14.9614i		−	0.696821i	0.937342	−	0.348410i	$-0.113278\pi$
$462$		0			0
$463$	1.03042	+	1.03042i	0.0478877	+	0.0478877i	0.730645	−	0.682757i	$-0.239219\pi$
$463$	1.03042	+	1.03042i	0.0478877	+	0.0478877i	−0.682757	+	0.730645i	$0.739219\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.55703	+	2.55703i	0.118325	+	0.118325i	0.763790	−	0.645465i	$-0.223336\pi$
$467$	2.55703	+	2.55703i	0.118325	+	0.118325i	−0.645465	+	0.763790i	$0.723336\pi$
$468$		0			0
$469$		−	1.26784i		−	0.0585436i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.450970	+	0.450970i	−0.0207356	+	0.0207356i
$474$		0			0
$475$	−1.02141	−	3.70105i	−0.0468655	−	0.169816i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.9727			0.821192			0.410596	−	0.911817i	$-0.365321\pi$
$479$	17.9727			0.821192			0.410596	+	0.911817i	$0.365321\pi$
$480$		0			0
$481$	4.03251			0.183867
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.163368	+	0.214562i	−0.00741817	+	0.00974275i
$486$		0			0
$487$	3.48554	−	3.48554i	0.157945	−	0.157945i	−0.623711	−	0.781655i	$-0.714376\pi$
$487$	3.48554	−	3.48554i	0.157945	−	0.157945i	0.781655	+	0.623711i	$0.214376\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	19.5787i		−	0.883575i	−0.897120	−	0.441787i	$-0.854345\pi$
$491$		−	19.5787i		−	0.883575i	0.897120	−	0.441787i	$-0.145655\pi$
$492$		0			0
$493$	13.7193	+	13.7193i	0.617884	+	0.617884i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−36.1764	−	36.1764i	−1.62274	−	1.62274i
$498$		0			0
$499$			18.4910i			0.827770i	0.910329	+	0.413885i	$0.135828\pi$
$499$			18.4910i			0.827770i	−0.910329	+	0.413885i	$0.864172\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.4136	−	12.4136i	0.553494	−	0.553494i	−0.373954	−	0.927447i	$-0.621998\pi$
$503$	12.4136	−	12.4136i	0.553494	−	0.553494i	0.927447	+	0.373954i	$0.121998\pi$
$504$		0			0
$505$	0.707984	+	5.22662i	0.0315049	+	0.232581i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−23.4530			−1.03954			−0.519768	−	0.854307i	$-0.673982\pi$
$509$	−23.4530			−1.03954			−0.519768	+	0.854307i	$0.673982\pi$
$510$		0			0
$511$	−54.5303			−2.41228
$512$		0			0
$513$		0			0
$514$		0			0
$515$	28.6258	+	21.7958i	1.26140	+	0.960438i
$516$		0			0
$517$	−0.771085	+	0.771085i	−0.0339123	+	0.0339123i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	33.3172i		−	1.45965i	−0.683632	−	0.729827i	$-0.739600\pi$
$521$		−	33.3172i		−	1.45965i	0.683632	−	0.729827i	$-0.260400\pi$
$522$		0			0
$523$	26.4046	+	26.4046i	1.15459	+	1.15459i	0.985621	+	0.168972i	$0.0540449\pi$
$523$	26.4046	+	26.4046i	1.15459	+	1.15459i	0.168972	+	0.985621i	$0.445955\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.0979787	−	0.0979787i	−0.00426802	−	0.00426802i
$528$		0			0
$529$			10.6189i			0.461690i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.73239	−	2.73239i	0.118353	−	0.118353i
$534$		0			0
$535$	0.442556	+	0.336964i	0.0191334	+	0.0145682i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.50441			0.107873
$540$		0			0
$541$	−16.9850			−0.730241			−0.365121	−	0.930960i	$-0.618972\pi$
$541$	−16.9850			−0.730241			−0.365121	+	0.930960i	$0.618972\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.183656	+	1.35582i	0.00786697	+	0.0580771i
$546$		0			0
$547$	10.3513	−	10.3513i	0.442588	−	0.442588i	−0.450293	−	0.892881i	$-0.648680\pi$
$547$	10.3513	−	10.3513i	0.442588	−	0.442588i	0.892881	+	0.450293i	$0.148680\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			7.06324i			0.300904i
$552$		0			0
$553$	−14.7674	−	14.7674i	−0.627972	−	0.627972i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.5148	+	14.5148i	0.615011	+	0.615011i	0.944248	−	0.329236i	$-0.106791\pi$
$557$	14.5148	+	14.5148i	0.615011	+	0.615011i	−0.329236	+	0.944248i	$0.606791\pi$
$558$		0			0
$559$			2.31848i			0.0980614i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.2804	+	24.2804i	−1.02330	+	1.02330i	−0.0235762	+	0.999722i	$0.507505\pi$
$563$	−24.2804	+	24.2804i	−1.02330	+	1.02330i	−0.999722	+	0.0235762i	$0.992495\pi$
$564$		0			0
$565$	−7.71420	+	10.1315i	−0.324539	+	0.426237i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	45.5646			1.91017			0.955083	−	0.296338i	$-0.0957656\pi$
$569$	45.5646			1.91017			0.955083	+	0.296338i	$0.0957656\pi$
$570$		0			0
$571$	7.60606			0.318304			0.159152	−	0.987254i	$-0.449124\pi$
$571$	7.60606			0.318304			0.159152	+	0.987254i	$0.449124\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.68255	−	15.3017i	0.362087	−	0.638124i
$576$		0			0
$577$	−25.4575	+	25.4575i	−1.05981	+	1.05981i	−0.0617180	+	0.998094i	$0.519658\pi$
$577$	−25.4575	+	25.4575i	−1.05981	+	1.05981i	−0.998094	+	0.0617180i	$0.980342\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			32.0920i			1.33140i
$582$		0			0
$583$	−0.268721	−	0.268721i	−0.0111293	−	0.0111293i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	29.4487	+	29.4487i	1.21548	+	1.21548i	0.969198	+	0.246281i	$0.0792086\pi$
$587$	29.4487	+	29.4487i	1.21548	+	1.21548i	0.246281	+	0.969198i	$0.420791\pi$
$588$		0			0
$589$		−	0.0504435i		−	0.00207849i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	16.0624	−	16.0624i	0.659605	−	0.659605i	−0.295682	−	0.955287i	$-0.595547\pi$
$593$	16.0624	−	16.0624i	0.659605	−	0.659605i	0.955287	+	0.295682i	$0.0955468\pi$
$594$		0			0
$595$	18.7561	−	2.54065i	0.768926	−	0.104157i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−21.1660			−0.864818			−0.432409	−	0.901678i	$-0.642336\pi$
$599$	−21.1660			−0.864818			−0.432409	+	0.901678i	$0.642336\pi$
$600$		0			0
$601$	23.8702			0.973684			0.486842	−	0.873490i	$-0.338149\pi$
$601$	23.8702			0.973684			0.486842	+	0.873490i	$0.338149\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−24.2065	+	3.27895i	−0.984133	+	0.133308i
$606$		0			0
$607$	−14.8159	+	14.8159i	−0.601359	+	0.601359i	−0.940673	−	0.339314i	$-0.889805\pi$
$607$	−14.8159	+	14.8159i	−0.601359	+	0.601359i	0.339314	+	0.940673i	$0.389805\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			3.96422i			0.160375i
$612$		0			0
$613$	12.0901	+	12.0901i	0.488313	+	0.488313i	0.907774	−	0.419461i	$-0.137781\pi$
$613$	12.0901	+	12.0901i	0.488313	+	0.488313i	−0.419461	+	0.907774i	$0.637781\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.0166	+	11.0166i	0.443511	+	0.443511i	0.893190	−	0.449679i	$-0.148462\pi$
$617$	11.0166	+	11.0166i	0.443511	+	0.443511i	−0.449679	+	0.893190i	$0.648462\pi$
$618$		0			0
$619$			7.34736i			0.295315i	0.989039	+	0.147658i	$0.0471734\pi$
$619$			7.34736i			0.295315i	−0.989039	+	0.147658i	$0.952827\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−16.9333	+	16.9333i	−0.678420	+	0.678420i
$624$		0			0
$625$	21.4613	−	12.8223i	0.858453	−	0.512893i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.50571			−0.339145
$630$		0			0
$631$	−5.12594			−0.204061			−0.102030	−	0.994781i	$-0.532534\pi$
$631$	−5.12594			−0.204061			−0.102030	+	0.994781i	$0.532534\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17.3079	+	22.7315i	−0.686841	+	0.902071i
$636$		0			0
$637$	6.43772	−	6.43772i	0.255072	−	0.255072i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			38.2714i			1.51163i	0.654785	+	0.755815i	$0.272759\pi$
$641$			38.2714i			1.51163i	−0.654785	+	0.755815i	$0.727241\pi$
$642$		0			0
$643$	10.9026	+	10.9026i	0.429957	+	0.429957i	0.888614	−	0.458657i	$-0.151669\pi$
$643$	10.9026	+	10.9026i	0.429957	+	0.429957i	−0.458657	+	0.888614i	$0.651669\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−22.0086	−	22.0086i	−0.865246	−	0.865246i	0.126696	−	0.991942i	$-0.459563\pi$
$647$	−22.0086	−	22.0086i	−0.865246	−	0.865246i	−0.991942	+	0.126696i	$0.959563\pi$
$648$		0			0
$649$			1.96349i			0.0770737i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.42211	−	6.42211i	0.251317	−	0.251317i	−0.570194	−	0.821510i	$-0.693132\pi$
$653$	6.42211	−	6.42211i	0.251317	−	0.251317i	0.821510	+	0.570194i	$0.193132\pi$
$654$		0			0
$655$	3.66912	+	27.0869i	0.143364	+	1.05837i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	21.9219			0.853958			0.426979	−	0.904262i	$-0.359578\pi$
$659$	21.9219			0.853958			0.426979	+	0.904262i	$0.359578\pi$
$660$		0			0
$661$	−28.0841			−1.09235			−0.546173	−	0.837672i	$-0.683916\pi$
$661$	−28.0841			−1.09235			−0.546173	+	0.837672i	$0.683916\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.48223	+	4.17420i	0.212592	+	0.161868i
$666$		0			0
$667$	−22.8863	+	22.8863i	−0.886161	+	0.886161i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2.62064i		−	0.101169i
$672$		0			0
$673$	13.7754	+	13.7754i	0.531001	+	0.531001i	0.920870	−	0.389869i	$-0.127480\pi$
$673$	13.7754	+	13.7754i	0.531001	+	0.531001i	−0.389869	+	0.920870i	$0.627480\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.28103	−	8.28103i	−0.318266	−	0.318266i	0.529835	−	0.848101i	$-0.322254\pi$
$677$	−8.28103	−	8.28103i	−0.318266	−	0.318266i	−0.848101	+	0.529835i	$0.822254\pi$
$678$		0			0
$679$		−	0.483984i		−	0.0185736i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.3992	−	12.3992i	0.474442	−	0.474442i	−0.428907	−	0.903349i	$-0.641101\pi$
$683$	12.3992	−	12.3992i	0.474442	−	0.474442i	0.903349	+	0.428907i	$0.141101\pi$
$684$		0			0
$685$	34.5726	+	26.3238i	1.32095	+	1.00578i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.38152			−0.0526317
$690$		0			0
$691$	−28.6201			−1.08876			−0.544379	−	0.838839i	$-0.683235\pi$
$691$	−28.6201			−1.08876			−0.544379	+	0.838839i	$0.683235\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.09491	+	44.9951i	0.231193	+	1.70676i
$696$		0			0
$697$	−5.76339	+	5.76339i	−0.218304	+	0.218304i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	23.6612i		−	0.893671i	−0.894616	−	0.446835i	$-0.852551\pi$
$701$		−	23.6612i		−	0.893671i	0.894616	−	0.446835i	$-0.147449\pi$
$702$		0			0
$703$	−2.18955	−	2.18955i	−0.0825803	−	0.0825803i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.69330	−	6.69330i	−0.251727	−	0.251727i
$708$		0			0
$709$		−	17.4168i		−	0.654100i	−0.945007	−	0.327050i	$-0.893945\pi$
$709$		−	17.4168i		−	0.654100i	0.945007	−	0.327050i	$-0.106055\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.163447	−	0.163447i	0.00612113	−	0.00612113i
$714$		0			0
$715$	0.372621	−	0.489386i	0.0139352	−	0.0183020i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	50.0895			1.86802			0.934012	−	0.357240i	$-0.116282\pi$
$719$	50.0895			1.86802			0.934012	+	0.357240i	$0.116282\pi$
$720$		0			0
$721$	−64.5708			−2.40474
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−44.3344	+	12.2354i	−1.64654	+	0.454410i
$726$		0			0
$727$	21.3726	−	21.3726i	0.792665	−	0.792665i	−0.189262	−	0.981927i	$-0.560610\pi$
$727$	21.3726	−	21.3726i	0.792665	−	0.792665i	0.981927	+	0.189262i	$0.0606096\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	4.89034i		−	0.180876i
$732$		0			0
$733$	−12.6518	−	12.6518i	−0.467305	−	0.467305i	0.433735	−	0.901040i	$-0.357195\pi$
$733$	−12.6518	−	12.6518i	−0.467305	−	0.467305i	−0.901040	+	0.433735i	$0.857195\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.0614523	+	0.0614523i	0.00226363	+	0.00226363i
$738$		0			0
$739$			48.4462i			1.78212i	0.453882	+	0.891062i	$0.350039\pi$
$739$			48.4462i			1.78212i	−0.453882	+	0.891062i	$0.649961\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.9920	+	12.9920i	−0.476632	+	0.476632i	−0.904053	−	0.427421i	$-0.859422\pi$
$743$	−12.9920	+	12.9920i	−0.476632	+	0.476632i	0.427421	+	0.904053i	$0.359422\pi$
$744$		0			0
$745$	−37.5340	+	5.08426i	−1.37514	+	0.186273i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.998267			−0.0364759
$750$		0			0
$751$	31.5235			1.15031			0.575154	−	0.818045i	$-0.304942\pi$
$751$	31.5235			1.15031			0.575154	+	0.818045i	$0.304942\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	38.1149	−	5.16295i	1.38714	−	0.187899i
$756$		0			0
$757$	26.3914	−	26.3914i	0.959213	−	0.959213i	−0.0399876	−	0.999200i	$-0.512732\pi$
$757$	26.3914	−	26.3914i	0.959213	−	0.959213i	0.999200	+	0.0399876i	$0.0127319\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			50.8244i			1.84238i	0.389109	+	0.921192i	$0.372783\pi$
$761$			50.8244i			1.84238i	−0.389109	+	0.921192i	$0.627217\pi$
$762$		0			0
$763$	−1.73629	−	1.73629i	−0.0628579	−	0.0628579i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.04725	+	5.04725i	0.182246	+	0.182246i
$768$		0			0
$769$		−	35.6908i		−	1.28704i	−0.765427	−	0.643522i	$-0.777472\pi$
$769$		−	35.6908i		−	1.28704i	0.765427	−	0.643522i	$-0.222528\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.6436	+	12.6436i	−0.454758	+	0.454758i	−0.896930	−	0.442172i	$-0.854208\pi$
$773$	−12.6436	+	12.6436i	−0.454758	+	0.454758i	0.442172	+	0.896930i	$0.354208\pi$
$774$		0			0
$775$	0.316623	−	0.0873811i	0.0113734	−	0.00313882i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.96723			−0.106312
$780$		0			0
$781$	3.50694			0.125488
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.5406	+	21.7238i	−0.590360	+	0.775357i
$786$		0			0
$787$	−26.2854	+	26.2854i	−0.936974	+	0.936974i	−0.998128	−	0.0611546i	$-0.980522\pi$
$787$	−26.2854	+	26.2854i	−0.936974	+	0.936974i	0.0611546	+	0.998128i	$0.480522\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	22.8536i		−	0.812579i
$792$		0			0
$793$	−6.73649	−	6.73649i	−0.239220	−	0.239220i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	35.5031	+	35.5031i	1.25759	+	1.25759i	0.952243	+	0.305343i	$0.0987710\pi$
$797$	35.5031	+	35.5031i	1.25759	+	1.25759i	0.305343	+	0.952243i	$0.401229\pi$
$798$		0			0
$799$		−	8.36167i		−	0.295815i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.64308	−	2.64308i	0.0932723	−	0.0932723i
$804$		0			0
$805$	4.23829	+	31.2887i	0.149380	+	1.10278i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.9886			0.456654			0.228327	−	0.973585i	$-0.426675\pi$
$809$	12.9886			0.456654			0.228327	+	0.973585i	$0.426675\pi$
$810$		0			0
$811$	−23.8342			−0.836933			−0.418466	−	0.908232i	$-0.637432\pi$
$811$	−23.8342			−0.836933			−0.418466	+	0.908232i	$0.637432\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.5788	−	14.1460i	−0.650789	−	0.495513i
$816$		0			0
$817$	1.25888	−	1.25888i	0.0440425	−	0.0440425i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	52.9834i		−	1.84913i	−0.381019	−	0.924567i	$-0.624427\pi$
$821$		−	52.9834i		−	1.84913i	0.381019	−	0.924567i	$-0.375573\pi$
$822$		0			0
$823$	−39.2836	−	39.2836i	−1.36934	−	1.36934i	−0.861381	−	0.507959i	$-0.830400\pi$
$823$	−39.2836	−	39.2836i	−1.36934	−	1.36934i	−0.507959	−	0.861381i	$-0.669600\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.9550	+	20.9550i	0.728678	+	0.728678i	0.970356	−	0.241679i	$-0.0776980\pi$
$827$	20.9550	+	20.9550i	0.728678	+	0.728678i	−0.241679	+	0.970356i	$0.577698\pi$
$828$		0			0
$829$		−	2.38701i		−	0.0829044i	−0.999140	−	0.0414522i	$-0.986802\pi$
$829$		−	2.38701i		−	0.0829044i	0.999140	−	0.0414522i	$-0.0131984\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.5790	+	13.5790i	−0.470484	+	0.470484i
$834$		0			0
$835$	8.24430	+	6.27725i	0.285306	+	0.217233i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.3417			−0.909415			−0.454708	−	0.890641i	$-0.650256\pi$
$839$	−26.3417			−0.909415			−0.454708	+	0.890641i	$0.650256\pi$
$840$		0			0
$841$	55.6098			1.91758
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.300151	−	2.21583i	−0.0103255	−	0.0762269i
$846$		0			0
$847$	30.9992	−	30.9992i	1.06515	−	1.06515i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	14.1891i		−	0.486397i
$852$		0			0
$853$	33.9988	+	33.9988i	1.16410	+	1.16410i	0.983570	+	0.180526i	$0.0577800\pi$
$853$	33.9988	+	33.9988i	1.16410	+	1.16410i	0.180526	+	0.983570i	$0.442220\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23.9353	−	23.9353i	−0.817615	−	0.817615i	0.168147	−	0.985762i	$-0.446222\pi$
$857$	−23.9353	−	23.9353i	−0.817615	−	0.817615i	−0.985762	+	0.168147i	$0.946222\pi$
$858$		0			0
$859$			14.3395i			0.489258i	0.969617	+	0.244629i	$0.0786662\pi$
$859$			14.3395i			0.489258i	−0.969617	+	0.244629i	$0.921334\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.0874	−	14.0874i	0.479539	−	0.479539i	−0.425445	−	0.904984i	$-0.639882\pi$
$863$	14.0874	−	14.0874i	0.479539	−	0.479539i	0.904984	+	0.425445i	$0.139882\pi$
$864$		0			0
$865$	−13.6214	+	17.8899i	−0.463142	+	0.608273i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.43155			0.0485619
$870$		0			0
$871$	0.315933			0.0107050
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−16.7039	+	41.6416i	−0.564695	+	1.40774i
$876$		0			0
$877$	−17.9913	+	17.9913i	−0.607522	+	0.607522i	−0.942298	−	0.334776i	$-0.891339\pi$
$877$	−17.9913	+	17.9913i	−0.607522	+	0.607522i	0.334776	+	0.942298i	$0.391339\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	17.5229i		−	0.590361i	−0.955442	−	0.295180i	$-0.904620\pi$
$881$		−	17.5229i		−	0.590361i	0.955442	−	0.295180i	$-0.0953797\pi$
$882$		0			0
$883$	19.2582	+	19.2582i	0.648090	+	0.648090i	0.952531	−	0.304441i	$-0.0984697\pi$
$883$	19.2582	+	19.2582i	0.648090	+	0.648090i	−0.304441	+	0.952531i	$0.598470\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.29530	+	3.29530i	0.110645	+	0.110645i	0.760262	−	0.649617i	$-0.225071\pi$
$887$	3.29530	+	3.29530i	0.110645	+	0.110645i	−0.649617	+	0.760262i	$0.725071\pi$
$888$		0			0
$889$		−	51.2751i		−	1.71971i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.15247	−	2.15247i	0.0720296	−	0.0720296i
$894$		0			0
$895$	−11.0776	+	1.50055i	−0.370284	+	0.0501577i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.604257			−0.0201531
$900$		0			0
$901$	2.91402			0.0970801
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.14852	−	0.832863i	0.204384	−	0.0276853i
$906$		0			0
$907$	28.5872	−	28.5872i	0.949221	−	0.949221i	−0.0495507	−	0.998772i	$-0.515779\pi$
$907$	28.5872	−	28.5872i	0.949221	−	0.949221i	0.998772	+	0.0495507i	$0.0157789\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	55.0085i		−	1.82251i	−0.411838	−	0.911257i	$-0.635113\pi$
$911$		−	55.0085i		−	1.82251i	0.411838	−	0.911257i	$-0.364887\pi$
$912$		0			0
$913$	−1.55550	−	1.55550i	−0.0514795	−	0.0514795i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−34.6880	−	34.6880i	−1.14550	−	1.14550i
$918$		0			0
$919$		−	15.9879i		−	0.527394i	−0.964606	−	0.263697i	$-0.915058\pi$
$919$		−	15.9879i		−	0.527394i	0.964606	−	0.263697i	$-0.0849418\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.01477	−	9.01477i	0.296725	−	0.296725i
$924$		0			0
$925$	9.95045	−	17.5362i	0.327169	−	0.576586i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−38.3266			−1.25745			−0.628727	−	0.777626i	$-0.716424\pi$
$929$	−38.3266			−1.25745			−0.628727	+	0.777626i	$0.716424\pi$
$930$		0			0
$931$	−6.99103			−0.229122
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.785963	+	1.03225i	−0.0257038	+	0.0337583i
$936$		0			0
$937$	−0.329170	+	0.329170i	−0.0107535	+	0.0107535i	−0.712463	−	0.701710i	$-0.752420\pi$
$937$	−0.329170	+	0.329170i	−0.0107535	+	0.0107535i	0.701710	+	0.712463i	$0.252420\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	12.9998i		−	0.423782i	−0.977293	−	0.211891i	$-0.932038\pi$
$941$		−	12.9998i		−	0.423782i	0.977293	−	0.211891i	$-0.0679622\pi$
$942$		0			0
$943$	−9.61442	−	9.61442i	−0.313088	−	0.313088i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.3185	+	41.3185i	1.34267	+	1.34267i	0.893392	+	0.449277i	$0.148318\pi$
$947$	41.3185	+	41.3185i	1.34267	+	1.34267i	0.449277	+	0.893392i	$0.351682\pi$
$948$		0			0
$949$		−	13.5883i		−	0.441096i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.79980	+	5.79980i	−0.187874	+	0.187874i	−0.794776	−	0.606902i	$-0.792412\pi$
$953$	−5.79980	+	5.79980i	−0.187874	+	0.187874i	0.606902	+	0.794776i	$0.292412\pi$
$954$		0			0
$955$	−4.06860	−	30.0360i	−0.131657	−	0.971942i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−77.9849			−2.51826
$960$		0			0
$961$	−30.9957			−0.999861
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−26.4739	−	20.1573i	−0.852225	−	0.648888i
$966$		0			0
$967$	34.7490	−	34.7490i	1.11745	−	1.11745i	0.125337	−	0.992114i	$-0.459999\pi$
$967$	34.7490	−	34.7490i	1.11745	−	1.11745i	0.992114	−	0.125337i	$-0.0400013\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	59.5082i		−	1.90971i	−0.297076	−	0.954854i	$-0.596011\pi$
$971$		−	59.5082i		−	1.90971i	0.297076	−	0.954854i	$-0.403989\pi$
$972$		0			0
$973$	−57.6214	−	57.6214i	−1.84726	−	1.84726i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−29.6157	−	29.6157i	−0.947492	−	0.947492i	0.0511970	−	0.998689i	$-0.483696\pi$
$977$	−29.6157	−	29.6157i	−0.947492	−	0.947492i	−0.998689	+	0.0511970i	$0.983696\pi$
$978$		0			0
$979$		−	1.64152i		−	0.0524632i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.41764	−	2.41764i	0.0771106	−	0.0771106i	−0.667500	−	0.744610i	$-0.732635\pi$
$983$	2.41764	−	2.41764i	0.0771106	−	0.0771106i	0.744610	+	0.667500i	$0.232635\pi$
$984$		0			0
$985$	36.3892	+	27.7069i	1.15946	+	0.882816i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.15800			0.259409
$990$		0			0
$991$	57.0997			1.81383			0.906916	−	0.421311i	$-0.138430\pi$
$991$	57.0997			1.81383			0.906916	+	0.421311i	$0.138430\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.95396	+	14.4249i	0.0619447	+	0.457300i
$996$		0			0
$997$	13.2191	−	13.2191i	0.418652	−	0.418652i	−0.466087	−	0.884739i	$-0.654336\pi$
$997$	13.2191	−	13.2191i	0.418652	−	0.418652i	0.884739	+	0.466087i	$0.154336\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.y.a.53.1	✓	24
3.2	odd	2		2340.2.y.b.53.12	yes	24
5.2	odd	4		2340.2.y.b.1457.12	yes	24
15.2	even	4	inner	2340.2.y.a.1457.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.y.a.53.1	✓	24	1.1	even	1	trivial
2340.2.y.a.1457.1	yes	24	15.2	even	4	inner
2340.2.y.b.53.12	yes	24	3.2	odd	2
2340.2.y.b.1457.12	yes	24	5.2	odd	4

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-2.21583 + 0.300151i) q^{5} +(2.83763 - 2.83763i) q^{7} +0.275080i q^{11} +(0.707107 + 0.707107i) q^{13} +(-1.49149 - 1.49149i) q^{17} -0.767881i q^{19} +(2.48808 - 2.48808i) q^{23} +(4.81982 - 1.33017i) q^{25} -9.19836 q^{29} +0.0656918 q^{31} +(-5.43600 + 7.13943i) q^{35} +(2.85142 - 2.85142i) q^{37} -3.86419i q^{41} +(1.63941 + 1.63941i) q^{43} +(2.80313 + 2.80313i) q^{47} -9.10431i q^{49} +(-0.976883 + 0.976883i) q^{53} +(-0.0825655 - 0.609531i) q^{55} +7.13789 q^{59} -9.52683 q^{61} +(-1.77907 - 1.35459i) q^{65} +(0.223398 - 0.223398i) q^{67} -12.7488i q^{71} +(-9.60841 - 9.60841i) q^{73} +(0.780576 + 0.780576i) q^{77} -5.20411i q^{79} +(-5.65471 + 5.65471i) q^{83} +(3.75256 + 2.85722i) q^{85} -5.96742 q^{89} +4.01302 q^{91} +(0.230480 + 1.70149i) q^{95} +(0.0852795 - 0.0852795i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 8 q^{7} - 8 q^{17} - 8 q^{23} + 16 q^{25} + 32 q^{29} + 8 q^{35} + 16 q^{37} - 8 q^{43} - 40 q^{47} - 8 q^{53} + 8 q^{55} + 56 q^{59} + 8 q^{61} - 4 q^{65} - 16 q^{67} - 72 q^{77} - 32 q^{83}+ \cdots + 8 q^{97}+O(q^{100}) \) 24 * q - 4 * q^5 - 8 * q^7 - 8 * q^17 - 8 * q^23 + 16 * q^25 + 32 * q^29 + 8 * q^35 + 16 * q^37 - 8 * q^43 - 40 * q^47 - 8 * q^53 + 8 * q^55 + 56 * q^59 + 8 * q^61 - 4 * q^65 - 16 * q^67 - 72 * q^77 - 32 * q^83 - 32 * q^85 + 128 * q^89 - 8 * q^91 + 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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