LMFDB - Embedded newform 3087.2.c.c.3086.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3087,2,Mod(3086,3087)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3087, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3087.3086");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3087 = 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3087.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$24.6498191040$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3086.1
Character	$\chi$	$=$	3087.3086
Dual form			3087.2.c.c.3086.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.69345i q^{2} -0.867767 q^{4} -4.09038 q^{5} -1.91738i q^{8} +O(q^{10})$	$q-1.69345i q^{2} -0.867767 q^{4} -4.09038 q^{5} -1.91738i q^{8} +6.92685i q^{10} +3.45694i q^{11} -1.20247i q^{13} -4.98251 q^{16} -4.57558 q^{17} -6.51921i q^{19} +3.54950 q^{20} +5.85416 q^{22} +7.37649i q^{23} +11.7312 q^{25} -2.03633 q^{26} +3.34790i q^{29} +2.85297i q^{31} +4.60288i q^{32} +7.74852i q^{34} +0.551851 q^{37} -11.0399 q^{38} +7.84280i q^{40} -11.2559 q^{41} +2.39122 q^{43} -2.99982i q^{44} +12.4917 q^{46} +11.7968 q^{47} -19.8662i q^{50} +1.04347i q^{52} +2.27092i q^{53} -14.1402i q^{55} +5.66949 q^{58} +0.449599 q^{59} -3.26060i q^{61} +4.83135 q^{62} -2.17030 q^{64} +4.91857i q^{65} -3.57419 q^{67} +3.97054 q^{68} -11.6471i q^{71} +11.7283i q^{73} -0.934532i q^{74} +5.65716i q^{76} +9.80723 q^{79} +20.3804 q^{80} +19.0613i q^{82} +13.5773 q^{83} +18.7159 q^{85} -4.04940i q^{86} +6.62827 q^{88} +15.0461 q^{89} -6.40108i q^{92} -19.9772i q^{94} +26.6660i q^{95} +3.07919i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) $ 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3087\mathbb{Z}\right)^\times$.

$n$	$344$	$2404$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 1.69345i	− 1.19745i	−0.800955	−	0.598724i	$-0.795674\pi$
$2$	− 1.69345i	− 1.19745i	0.800955	−	0.598724i	$-0.204326\pi$
$3$	0	0
$3$	0	0
$4$	−0.867767	−0.433884
$5$	−4.09038	−1.82927	−0.914637	−	0.404277i	$-0.867523\pi$
$5$	−4.09038	−1.82927	−0.914637	+	0.404277i	$0.867523\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.91738i	− 0.677895i
$9$	0	0
$10$	6.92685i	2.19046i
$11$	3.45694i	1.04231i	0.853463	+	0.521154i	$0.174498\pi$
$11$	3.45694i	1.04231i	−0.853463	+	0.521154i	$0.825502\pi$
$12$	0	0
$13$	− 1.20247i	− 0.333506i	−0.985999	−	0.166753i	$-0.946672\pi$
$13$	− 1.20247i	− 0.333506i	0.985999	−	0.166753i	$-0.0533283\pi$
$14$	0	0
$15$	0	0
$16$	−4.98251	−1.24563
$17$	−4.57558	−1.10974	−0.554871	−	0.831936i	$-0.687232\pi$
$17$	−4.57558	−1.10974	−0.554871	+	0.831936i	$0.687232\pi$
$18$	0	0
$19$	− 6.51921i	− 1.49561i	−0.663919	−	0.747804i	$-0.731108\pi$
$19$	− 6.51921i	− 1.49561i	0.663919	−	0.747804i	$-0.268892\pi$
$20$	3.54950	0.793692
$21$	0	0
$22$	5.85416	1.24811
$23$	7.37649i	1.53810i	0.639186	+	0.769052i	$0.279271\pi$
$23$	7.37649i	1.53810i	−0.639186	+	0.769052i	$0.720729\pi$
$24$	0	0
$25$	11.7312	2.34624
$26$	−2.03633	−0.399357
$27$	0	0
$28$	0	0
$29$	3.34790i	0.621689i	0.950461	+	0.310844i	$0.100612\pi$
$29$	3.34790i	0.621689i	−0.950461	+	0.310844i	$0.899388\pi$
$30$	0	0
$31$	2.85297i	0.512408i	0.966623	+	0.256204i	$0.0824719\pi$
$31$	2.85297i	0.512408i	−0.966623	+	0.256204i	$0.917528\pi$
$32$	4.60288i	0.813681i
$33$	0	0
$34$	7.74852i	1.32886i
$35$	0	0
$36$	0	0
$37$	0.551851	0.0907238	0.0453619	−	0.998971i	$-0.485556\pi$
$37$	0.551851	0.0907238	0.0453619	+	0.998971i	$0.485556\pi$
$38$	−11.0399	−1.79092
$39$	0	0
$40$	7.84280i	1.24006i
$41$	−11.2559	−1.75787	−0.878936	−	0.476939i	$-0.841746\pi$
$41$	−11.2559	−1.75787	−0.878936	+	0.476939i	$0.841746\pi$
$42$	0	0
$43$	2.39122	0.364657	0.182329	−	0.983238i	$-0.441637\pi$
$43$	2.39122	0.364657	0.182329	+	0.983238i	$0.441637\pi$
$44$	− 2.99982i	− 0.452240i
$45$	0	0
$46$	12.4917	1.84180
$47$	11.7968	1.72073	0.860367	−	0.509675i	$-0.170234\pi$
$47$	11.7968	1.72073	0.860367	+	0.509675i	$0.170234\pi$
$48$	0	0
$49$	0	0
$50$	− 19.8662i	− 2.80950i
$51$	0	0
$52$	1.04347i	0.144703i
$53$	2.27092i	0.311935i	0.987762	+	0.155968i	$0.0498496\pi$
$53$	2.27092i	0.311935i	−0.987762	+	0.155968i	$0.950150\pi$
$54$	0	0
$55$	− 14.1402i	− 1.90667i
$56$	0	0
$57$	0	0
$58$	5.66949	0.744440
$59$	0.449599	0.0585328	0.0292664	−	0.999572i	$-0.490683\pi$
$59$	0.449599	0.0585328	0.0292664	+	0.999572i	$0.490683\pi$
$60$	0	0
$61$	− 3.26060i	− 0.417478i	−0.977971	−	0.208739i	$-0.933064\pi$
$61$	− 3.26060i	− 0.417478i	0.977971	−	0.208739i	$-0.0669358\pi$
$62$	4.83135	0.613582
$63$	0	0
$64$	−2.17030	−0.271287
$65$	4.91857i	0.610074i
$66$	0	0
$67$	−3.57419	−0.436656	−0.218328	−	0.975875i	$-0.570060\pi$
$67$	−3.57419	−0.436656	−0.218328	+	0.975875i	$0.570060\pi$
$68$	3.97054	0.481499
$69$	0	0
$70$	0	0
$71$	− 11.6471i	− 1.38225i	−0.722734	−	0.691126i	$-0.757115\pi$
$71$	− 11.6471i	− 1.38225i	0.722734	−	0.691126i	$-0.242885\pi$
$72$	0	0
$73$	11.7283i	1.37270i	0.727272	+	0.686349i	$0.240788\pi$
$73$	11.7283i	1.37270i	−0.727272	+	0.686349i	$0.759212\pi$
$74$	− 0.934532i	− 0.108637i
$75$	0	0
$76$	5.65716i	0.648920i
$77$	0	0
$78$	0	0
$79$	9.80723	1.10340	0.551700	−	0.834043i	$-0.313979\pi$
$79$	9.80723	1.10340	0.551700	+	0.834043i	$0.313979\pi$
$80$	20.3804	2.27860
$81$	0	0
$82$	19.0613i	2.10496i
$83$	13.5773	1.49030	0.745152	−	0.666894i	$-0.232377\pi$
$83$	13.5773	1.49030	0.745152	+	0.666894i	$0.232377\pi$
$84$	0	0
$85$	18.7159	2.03002
$86$	− 4.04940i	− 0.436658i
$87$	0	0
$88$	6.62827	0.706576
$89$	15.0461	1.59488	0.797441	−	0.603397i	$-0.206186\pi$
$89$	15.0461	1.59488	0.797441	+	0.603397i	$0.206186\pi$
$90$	0	0
$91$	0	0
$92$	− 6.40108i	− 0.667358i
$93$	0	0
$94$	− 19.9772i	− 2.06049i
$95$	26.6660i	2.73588i
$96$	0	0
$97$	3.07919i	0.312644i	0.987706	+	0.156322i	$0.0499638\pi$
$97$	3.07919i	0.312644i	−0.987706	+	0.156322i	$0.950036\pi$
$98$	0	0
$99$	0	0
$100$	−10.1800	−1.01800
$101$	8.82141	0.877763	0.438881	−	0.898545i	$-0.355375\pi$
$101$	8.82141	0.877763	0.438881	+	0.898545i	$0.355375\pi$
$102$	0	0
$103$	− 3.20386i	− 0.315685i	−0.987464	−	0.157843i	$-0.949546\pi$
$103$	− 3.20386i	− 0.315685i	0.987464	−	0.157843i	$-0.0504539\pi$
$104$	−2.30560	−0.226082
$105$	0	0
$106$	3.84569	0.373527
$107$	− 2.00944i	− 0.194260i	−0.995272	−	0.0971299i	$-0.969034\pi$
$107$	− 2.00944i	− 0.194260i	0.995272	−	0.0971299i	$-0.0309662\pi$
$108$	0	0
$109$	9.47942	0.907964	0.453982	−	0.891011i	$-0.350003\pi$
$109$	9.47942	0.907964	0.453982	+	0.891011i	$0.350003\pi$
$110$	−23.9457	−2.28313
$111$	0	0
$112$	0	0
$113$	− 18.9543i	− 1.78307i	−0.452955	−	0.891533i	$-0.649630\pi$
$113$	− 18.9543i	− 1.78307i	0.452955	−	0.891533i	$-0.350370\pi$
$114$	0	0
$115$	− 30.1726i	− 2.81361i
$116$	− 2.90519i	− 0.269741i
$117$	0	0
$118$	− 0.761373i	− 0.0700901i
$119$	0	0
$120$	0	0
$121$	−0.950461	−0.0864055
$122$	−5.52166	−0.499908
$123$	0	0
$124$	− 2.47571i	− 0.222325i
$125$	−27.5332	−2.46264
$126$	0	0
$127$	−3.24810	−0.288222	−0.144111	−	0.989562i	$-0.546032\pi$
$127$	−3.24810	−0.288222	−0.144111	+	0.989562i	$0.546032\pi$
$128$	12.8810i	1.13853i
$129$	0	0
$130$	8.32935	0.730532
$131$	−8.56118	−0.747994	−0.373997	−	0.927430i	$-0.622013\pi$
$131$	−8.56118	−0.747994	−0.373997	+	0.927430i	$0.622013\pi$
$132$	0	0
$133$	0	0
$134$	6.05270i	0.522874i
$135$	0	0
$136$	8.77312i	0.752289i
$137$	10.3494i	0.884205i	0.896965	+	0.442102i	$0.145767\pi$
$137$	10.3494i	0.884205i	−0.896965	+	0.442102i	$0.854233\pi$
$138$	0	0
$139$	8.29472i	0.703549i	0.936085	+	0.351774i	$0.114422\pi$
$139$	8.29472i	0.703549i	−0.936085	+	0.351774i	$0.885578\pi$
$140$	0	0
$141$	0	0
$142$	−19.7237	−1.65518
$143$	4.15688	0.347616
$144$	0	0
$145$	− 13.6942i	− 1.13724i
$146$	19.8613	1.64374
$147$	0	0
$148$	−0.478879	−0.0393636
$149$	− 13.6561i	− 1.11875i	−0.828913	−	0.559377i	$-0.811040\pi$
$149$	− 13.6561i	− 1.11875i	0.828913	−	0.559377i	$-0.188960\pi$
$150$	0	0
$151$	−2.66250	−0.216671	−0.108336	−	0.994114i	$-0.534552\pi$
$151$	−2.66250	−0.216671	−0.108336	+	0.994114i	$0.534552\pi$
$152$	−12.4998	−1.01387
$153$	0	0
$154$	0	0
$155$	− 11.6697i	− 0.937334i
$156$	0	0
$157$	− 10.5851i	− 0.844784i	−0.906413	−	0.422392i	$-0.861191\pi$
$157$	− 10.5851i	− 0.844784i	0.906413	−	0.422392i	$-0.138809\pi$
$158$	− 16.6080i	− 1.32126i
$159$	0	0
$160$	− 18.8275i	− 1.48845i
$161$	0	0
$162$	0	0
$163$	3.29793	0.258314	0.129157	−	0.991624i	$-0.458773\pi$
$163$	3.29793	0.258314	0.129157	+	0.991624i	$0.458773\pi$
$164$	9.76749	0.762712
$165$	0	0
$166$	− 22.9925i	− 1.78456i
$167$	8.11158	0.627693	0.313846	−	0.949474i	$-0.398382\pi$
$167$	8.11158	0.627693	0.313846	+	0.949474i	$0.398382\pi$
$168$	0	0
$169$	11.5541	0.888774
$170$	− 31.6944i	− 2.43085i
$171$	0	0
$172$	−2.07502	−0.158219
$173$	4.59143	0.349080	0.174540	−	0.984650i	$-0.444156\pi$
$173$	4.59143	0.349080	0.174540	+	0.984650i	$0.444156\pi$
$174$	0	0
$175$	0	0
$176$	− 17.2243i	− 1.29833i
$177$	0	0
$178$	− 25.4798i	− 1.90979i
$179$	− 11.1275i	− 0.831710i	−0.909431	−	0.415855i	$-0.863482\pi$
$179$	− 11.1275i	− 0.831710i	0.909431	−	0.415855i	$-0.136518\pi$
$180$	0	0
$181$	24.4095i	1.81434i	0.420762	+	0.907171i	$0.361763\pi$
$181$	24.4095i	1.81434i	−0.420762	+	0.907171i	$0.638237\pi$
$182$	0	0
$183$	0	0
$184$	14.1435	1.04267
$185$	−2.25728	−0.165959
$186$	0	0
$187$	− 15.8175i	− 1.15669i
$188$	−10.2368	−0.746599
$189$	0	0
$190$	45.1575	3.27607
$191$	− 1.76637i	− 0.127810i	−0.997956	−	0.0639051i	$-0.979645\pi$
$191$	− 1.76637i	− 0.127810i	0.997956	−	0.0639051i	$-0.0203555\pi$
$192$	0	0
$193$	−7.98123	−0.574502	−0.287251	−	0.957855i	$-0.592741\pi$
$193$	−7.98123	−0.574502	−0.287251	+	0.957855i	$0.592741\pi$
$194$	5.21445	0.374375
$195$	0	0
$196$	0	0
$197$	8.86958i	0.631932i	0.948771	+	0.315966i	$0.102329\pi$
$197$	8.86958i	0.631932i	−0.948771	+	0.315966i	$0.897671\pi$
$198$	0	0
$199$	14.2409i	1.00951i	0.863263	+	0.504755i	$0.168417\pi$
$199$	14.2409i	1.00951i	−0.863263	+	0.504755i	$0.831583\pi$
$200$	− 22.4931i	− 1.59051i
$201$	0	0
$202$	− 14.9386i	− 1.05108i
$203$	0	0
$204$	0	0
$205$	46.0408	3.21563
$206$	−5.42556	−0.378017
$207$	0	0
$208$	5.99134i	0.415425i
$209$	22.5365	1.55888
$210$	0	0
$211$	−2.28842	−0.157542	−0.0787708	−	0.996893i	$-0.525100\pi$
$211$	−2.28842	−0.157542	−0.0787708	+	0.996893i	$0.525100\pi$
$212$	− 1.97063i	− 0.135344i
$213$	0	0
$214$	−3.40288	−0.232616
$215$	−9.78099	−0.667058
$216$	0	0
$217$	0	0
$218$	− 16.0529i	− 1.08724i
$219$	0	0
$220$	12.2704i	0.827271i
$221$	5.50202i	0.370106i
$222$	0	0
$223$	− 1.47463i	− 0.0987482i	−0.998780	−	0.0493741i	$-0.984277\pi$
$223$	− 1.47463i	− 0.0987482i	0.998780	−	0.0493741i	$-0.0157227\pi$
$224$	0	0
$225$	0	0
$226$	−32.0981	−2.13513
$227$	8.38399	0.556465	0.278232	−	0.960514i	$-0.410251\pi$
$227$	8.38399	0.556465	0.278232	+	0.960514i	$0.410251\pi$
$228$	0	0
$229$	17.8353i	1.17859i	0.807919	+	0.589293i	$0.200594\pi$
$229$	17.8353i	1.17859i	−0.807919	+	0.589293i	$0.799406\pi$
$230$	−51.0958	−3.36916
$231$	0	0
$232$	6.41918	0.421440
$233$	23.3011i	1.52650i	0.646102	+	0.763251i	$0.276398\pi$
$233$	23.3011i	1.52650i	−0.646102	+	0.763251i	$0.723602\pi$
$234$	0	0
$235$	−48.2532	−3.14769
$236$	−0.390148	−0.0253964
$237$	0	0
$238$	0	0
$239$	− 9.54227i	− 0.617238i	−0.951186	−	0.308619i	$-0.900133\pi$
$239$	− 9.54227i	− 0.617238i	0.951186	−	0.308619i	$-0.0998668\pi$
$240$	0	0
$241$	15.7963i	1.01753i	0.860907	+	0.508763i	$0.169897\pi$
$241$	15.7963i	1.01753i	−0.860907	+	0.508763i	$0.830103\pi$
$242$	1.60956i	0.103466i
$243$	0	0
$244$	2.82945i	0.181137i
$245$	0	0
$246$	0	0
$247$	−7.83917	−0.498795
$248$	5.47021	0.347359
$249$	0	0
$250$	46.6260i	2.94889i
$251$	22.6289	1.42832	0.714161	−	0.699982i	$-0.246809\pi$
$251$	22.6289	1.42832	0.714161	+	0.699982i	$0.246809\pi$
$252$	0	0
$253$	−25.5001	−1.60318
$254$	5.50049i	0.345131i
$255$	0	0
$256$	17.4728	1.09205
$257$	20.1088	1.25435	0.627175	−	0.778878i	$-0.284211\pi$
$257$	20.1088	1.25435	0.627175	+	0.778878i	$0.284211\pi$
$258$	0	0
$259$	0	0
$260$	− 4.26818i	− 0.264701i
$261$	0	0
$262$	14.4979i	0.895684i
$263$	− 18.9465i	− 1.16829i	−0.811649	−	0.584146i	$-0.801430\pi$
$263$	− 18.9465i	− 1.16829i	0.811649	−	0.584146i	$-0.198570\pi$
$264$	0	0
$265$	− 9.28894i	− 0.570615i
$266$	0	0
$267$	0	0
$268$	3.10156	0.189458
$269$	−20.9273	−1.27596	−0.637979	−	0.770054i	$-0.720229\pi$
$269$	−20.9273	−1.27596	−0.637979	+	0.770054i	$0.720229\pi$
$270$	0	0
$271$	3.83719i	0.233093i	0.993185	+	0.116546i	$0.0371824\pi$
$271$	3.83719i	0.233093i	−0.993185	+	0.116546i	$0.962818\pi$
$272$	22.7979	1.38233
$273$	0	0
$274$	17.5261	1.05879
$275$	40.5541i	2.44550i
$276$	0	0
$277$	−5.61292	−0.337247	−0.168624	−	0.985680i	$-0.553932\pi$
$277$	−5.61292	−0.337247	−0.168624	+	0.985680i	$0.553932\pi$
$278$	14.0467	0.842464
$279$	0	0
$280$	0	0
$281$	10.1223i	0.603845i	0.953332	+	0.301923i	$0.0976284\pi$
$281$	10.1223i	0.603845i	−0.953332	+	0.301923i	$0.902372\pi$
$282$	0	0
$283$	19.2830i	1.14626i	0.819465	+	0.573129i	$0.194271\pi$
$283$	19.2830i	1.14626i	−0.819465	+	0.573129i	$0.805729\pi$
$284$	10.1069i	0.599737i
$285$	0	0
$286$	− 7.03947i	− 0.416252i
$287$	0	0
$288$	0	0
$289$	3.93597	0.231528
$290$	−23.1904	−1.36178
$291$	0	0
$292$	− 10.1775i	− 0.595592i
$293$	−9.43915	−0.551441	−0.275721	−	0.961238i	$-0.588916\pi$
$293$	−9.43915	−0.551441	−0.275721	+	0.961238i	$0.588916\pi$
$294$	0	0
$295$	−1.83903	−0.107073
$296$	− 1.05811i	− 0.0615012i
$297$	0	0
$298$	−23.1260	−1.33965
$299$	8.87003	0.512967
$300$	0	0
$301$	0	0
$302$	4.50881i	0.259453i
$303$	0	0
$304$	32.4820i	1.86297i
$305$	13.3371i	0.763681i
$306$	0	0
$307$	− 7.72168i	− 0.440700i	−0.975421	−	0.220350i	$-0.929280\pi$
$307$	− 7.72168i	− 0.440700i	0.975421	−	0.220350i	$-0.0707199\pi$
$308$	0	0
$309$	0	0
$310$	−19.7621	−1.12241
$311$	5.57500	0.316129	0.158065	−	0.987429i	$-0.449475\pi$
$311$	5.57500	0.316129	0.158065	+	0.987429i	$0.449475\pi$
$312$	0	0
$313$	− 3.82367i	− 0.216127i	−0.994144	−	0.108063i	$-0.965535\pi$
$313$	− 3.82367i	− 0.216127i	0.994144	−	0.108063i	$-0.0344649\pi$
$314$	−17.9253	−1.01159
$315$	0	0
$316$	−8.51040	−0.478747
$317$	− 25.9300i	− 1.45638i	−0.685377	−	0.728188i	$-0.740363\pi$
$317$	− 25.9300i	− 1.45638i	0.685377	−	0.728188i	$-0.259637\pi$
$318$	0	0
$319$	−11.5735	−0.647991
$320$	8.87733	0.496258
$321$	0	0
$322$	0	0
$323$	29.8292i	1.65974i
$324$	0	0
$325$	− 14.1065i	− 0.782485i
$326$	− 5.58487i	− 0.309317i
$327$	0	0
$328$	21.5818i	1.19165i
$329$	0	0
$330$	0	0
$331$	30.8806	1.69735	0.848676	−	0.528913i	$-0.177400\pi$
$331$	30.8806	1.69735	0.848676	+	0.528913i	$0.177400\pi$
$332$	−11.7820	−0.646619
$333$	0	0
$334$	− 13.7365i	− 0.751630i
$335$	14.6198	0.798764
$336$	0	0
$337$	−22.5445	−1.22808	−0.614038	−	0.789276i	$-0.710456\pi$
$337$	−22.5445	−1.22808	−0.614038	+	0.789276i	$0.710456\pi$
$338$	− 19.5662i	− 1.06426i
$339$	0	0
$340$	−16.2410	−0.880793
$341$	−9.86254	−0.534087
$342$	0	0
$343$	0	0
$344$	− 4.58487i	− 0.247199i
$345$	0	0
$346$	− 7.77535i	− 0.418005i
$347$	− 20.0475i	− 1.07621i	−0.842879	−	0.538103i	$-0.819141\pi$
$347$	− 20.0475i	− 1.07621i	0.842879	−	0.538103i	$-0.180859\pi$
$348$	0	0
$349$	31.8671i	1.70580i	0.522071	+	0.852902i	$0.325160\pi$
$349$	31.8671i	1.70580i	−0.522071	+	0.852902i	$0.674840\pi$
$350$	0	0
$351$	0	0
$352$	−15.9119	−0.848106
$353$	−13.8912	−0.739356	−0.369678	−	0.929160i	$-0.620532\pi$
$353$	−13.8912	−0.739356	−0.369678	+	0.929160i	$0.620532\pi$
$354$	0	0
$355$	47.6409i	2.52852i
$356$	−13.0565	−0.691994
$357$	0	0
$358$	−18.8439	−0.995930
$359$	14.8458i	0.783532i	0.920065	+	0.391766i	$0.128136\pi$
$359$	14.8458i	0.783532i	−0.920065	+	0.391766i	$0.871864\pi$
$360$	0	0
$361$	−23.5001	−1.23685
$362$	41.3362	2.17258
$363$	0	0
$364$	0	0
$365$	− 47.9734i	− 2.51104i
$366$	0	0
$367$	18.6516i	0.973608i	0.873511	+	0.486804i	$0.161837\pi$
$367$	18.6516i	0.973608i	−0.873511	+	0.486804i	$0.838163\pi$
$368$	− 36.7535i	− 1.91591i
$369$	0	0
$370$	3.82259i	0.198727i
$371$	0	0
$372$	0	0
$373$	15.0240	0.777911	0.388956	−	0.921256i	$-0.372836\pi$
$373$	15.0240	0.777911	0.388956	+	0.921256i	$0.372836\pi$
$374$	−26.7862	−1.38508
$375$	0	0
$376$	− 22.6188i	− 1.16648i
$377$	4.02575	0.207337
$378$	0	0
$379$	26.8252	1.37792	0.688959	−	0.724800i	$-0.258068\pi$
$379$	26.8252	1.37792	0.688959	+	0.724800i	$0.258068\pi$
$380$	− 23.1399i	− 1.18705i
$381$	0	0
$382$	−2.99126	−0.153046
$383$	−22.9415	−1.17226	−0.586128	−	0.810219i	$-0.699348\pi$
$383$	−22.9415	−1.17226	−0.586128	+	0.810219i	$0.699348\pi$
$384$	0	0
$385$	0	0
$386$	13.5158i	0.687936i
$387$	0	0
$388$	− 2.67202i	− 0.135651i
$389$	21.3909i	1.08456i	0.840198	+	0.542280i	$0.182439\pi$
$389$	21.3909i	1.08456i	−0.840198	+	0.542280i	$0.817561\pi$
$390$	0	0
$391$	− 33.7517i	− 1.70690i
$392$	0	0
$393$	0	0
$394$	15.0202	0.756706
$395$	−40.1153	−2.01842
$396$	0	0
$397$	− 19.2471i	− 0.965984i	−0.875625	−	0.482992i	$-0.839550\pi$
$397$	− 19.2471i	− 0.965984i	0.875625	−	0.482992i	$-0.160450\pi$
$398$	24.1162	1.20884
$399$	0	0
$400$	−58.4509	−2.92254
$401$	14.0401i	0.701130i	0.936538	+	0.350565i	$0.114010\pi$
$401$	14.0401i	0.701130i	−0.936538	+	0.350565i	$0.885990\pi$
$402$	0	0
$403$	3.43062	0.170891
$404$	−7.65493	−0.380847
$405$	0	0
$406$	0	0
$407$	1.90772i	0.0945621i
$408$	0	0
$409$	17.7053i	0.875472i	0.899104	+	0.437736i	$0.144219\pi$
$409$	17.7053i	0.875472i	−0.899104	+	0.437736i	$0.855781\pi$
$410$	− 77.9677i	− 3.85055i
$411$	0	0
$412$	2.78020i	0.136971i
$413$	0	0
$414$	0	0
$415$	−55.5364	−2.72617
$416$	5.53484	0.271368
$417$	0	0
$418$	− 38.1645i	− 1.86668i
$419$	4.11893	0.201223	0.100612	−	0.994926i	$-0.467920\pi$
$419$	4.11893	0.201223	0.100612	+	0.994926i	$0.467920\pi$
$420$	0	0
$421$	22.3244	1.08802	0.544011	−	0.839078i	$-0.316905\pi$
$421$	22.3244	1.08802	0.544011	+	0.839078i	$0.316905\pi$
$422$	3.87533i	0.188648i
$423$	0	0
$424$	4.35422	0.211460
$425$	−53.6771	−2.60372
$426$	0	0
$427$	0	0
$428$	1.74373i	0.0842862i
$429$	0	0
$430$	16.5636i	0.798768i
$431$	0.500332i	0.0241001i	0.999927	+	0.0120501i	$0.00383575\pi$
$431$	0.500332i	0.0241001i	−0.999927	+	0.0120501i	$0.996164\pi$
$432$	0	0
$433$	32.0367i	1.53959i	0.638294	+	0.769793i	$0.279641\pi$
$433$	32.0367i	1.53959i	−0.638294	+	0.769793i	$0.720359\pi$
$434$	0	0
$435$	0	0
$436$	−8.22594	−0.393951
$437$	48.0889	2.30040
$438$	0	0
$439$	− 13.1755i	− 0.628831i	−0.949285	−	0.314416i	$-0.898191\pi$
$439$	− 13.1755i	− 0.628831i	0.949285	−	0.314416i	$-0.101809\pi$
$440$	−27.1121	−1.29252
$441$	0	0
$442$	9.31738	0.443183
$443$	28.1080i	1.33545i	0.744408	+	0.667726i	$0.232732\pi$
$443$	28.1080i	1.33545i	−0.744408	+	0.667726i	$0.767268\pi$
$444$	0	0
$445$	−61.5442	−2.91748
$446$	−2.49720	−0.118246
$447$	0	0
$448$	0	0
$449$	30.5061i	1.43967i	0.694144	+	0.719836i	$0.255783\pi$
$449$	30.5061i	1.43967i	−0.694144	+	0.719836i	$0.744217\pi$
$450$	0	0
$451$	− 38.9109i	− 1.83224i
$452$	16.4479i	0.773644i
$453$	0	0
$454$	− 14.1978i	− 0.666338i
$455$	0	0
$456$	0	0
$457$	−34.2007	−1.59984	−0.799921	−	0.600106i	$-0.795125\pi$
$457$	−34.2007	−1.59984	−0.799921	+	0.600106i	$0.795125\pi$
$458$	30.2031	1.41130
$459$	0	0
$460$	26.1828i	1.22078i
$461$	−10.3998	−0.484366	−0.242183	−	0.970231i	$-0.577863\pi$
$461$	−10.3998	−0.484366	−0.242183	+	0.970231i	$0.577863\pi$
$462$	0	0
$463$	17.9414	0.833807	0.416904	−	0.908951i	$-0.363115\pi$
$463$	17.9414	0.833807	0.416904	+	0.908951i	$0.363115\pi$
$464$	− 16.6809i	− 0.774393i
$465$	0	0
$466$	39.4591	1.82791
$467$	−6.79035	−0.314220	−0.157110	−	0.987581i	$-0.550218\pi$
$467$	−6.79035	−0.314220	−0.157110	+	0.987581i	$0.550218\pi$
$468$	0	0
$469$	0	0
$470$	81.7143i	3.76920i
$471$	0	0
$472$	− 0.862051i	− 0.0396791i
$473$	8.26631i	0.380085i
$474$	0	0
$475$	− 76.4781i	− 3.50906i
$476$	0	0
$477$	0	0
$478$	−16.1593	−0.739111
$479$	41.8762	1.91337	0.956687	−	0.291118i	$-0.0940274\pi$
$479$	41.8762	1.91337	0.956687	+	0.291118i	$0.0940274\pi$
$480$	0	0
$481$	− 0.663587i	− 0.0302569i
$482$	26.7501	1.21844
$483$	0	0
$484$	0.824779	0.0374900
$485$	− 12.5950i	− 0.571912i
$486$	0	0
$487$	24.1029	1.09221	0.546104	−	0.837717i	$-0.316110\pi$
$487$	24.1029	1.09221	0.546104	+	0.837717i	$0.316110\pi$
$488$	−6.25181	−0.283006
$489$	0	0
$490$	0	0
$491$	− 33.2850i	− 1.50213i	−0.660227	−	0.751066i	$-0.729540\pi$
$491$	− 33.2850i	− 1.50213i	0.660227	−	0.751066i	$-0.270460\pi$
$492$	0	0
$493$	− 15.3186i	− 0.689914i
$494$	13.2752i	0.597281i
$495$	0	0
$496$	− 14.2149i	− 0.638270i
$497$	0	0
$498$	0	0
$499$	29.7177	1.33035	0.665174	−	0.746689i	$-0.268358\pi$
$499$	29.7177	1.33035	0.665174	+	0.746689i	$0.268358\pi$
$500$	23.8924	1.06850
$501$	0	0
$502$	− 38.3208i	− 1.71034i
$503$	−6.23775	−0.278127	−0.139064	−	0.990283i	$-0.544409\pi$
$503$	−6.23775	−0.278127	−0.139064	+	0.990283i	$0.544409\pi$
$504$	0	0
$505$	−36.0829	−1.60567
$506$	43.1831i	1.91972i
$507$	0	0
$508$	2.81859	0.125055
$509$	−18.3147	−0.811785	−0.405892	−	0.913921i	$-0.633039\pi$
$509$	−18.3147	−0.811785	−0.405892	+	0.913921i	$0.633039\pi$
$510$	0	0
$511$	0	0
$512$	− 3.82718i	− 0.169139i
$513$	0	0
$514$	− 34.0531i	− 1.50202i
$515$	13.1050i	0.577475i
$516$	0	0
$517$	40.7807i	1.79353i
$518$	0	0
$519$	0	0
$520$	9.43076	0.413566
$521$	−15.5542	−0.681442	−0.340721	−	0.940165i	$-0.610671\pi$
$521$	−15.5542	−0.681442	−0.340721	+	0.940165i	$0.610671\pi$
$522$	0	0
$523$	− 11.3888i	− 0.497998i	−0.968504	−	0.248999i	$-0.919898\pi$
$523$	− 11.3888i	− 0.497998i	0.968504	−	0.248999i	$-0.0801016\pi$
$524$	7.42911	0.324542
$525$	0	0
$526$	−32.0849	−1.39897
$527$	− 13.0540i	− 0.568641i
$528$	0	0
$529$	−31.4126	−1.36576
$530$	−15.7303	−0.683282
$531$	0	0
$532$	0	0
$533$	13.5349i	0.586261i
$534$	0	0
$535$	8.21937i	0.355354i
$536$	6.85307i	0.296007i
$537$	0	0
$538$	35.4392i	1.52789i
$539$	0	0
$540$	0	0
$541$	14.2426	0.612337	0.306168	−	0.951977i	$-0.400953\pi$
$541$	14.2426	0.612337	0.306168	+	0.951977i	$0.400953\pi$
$542$	6.49808	0.279117
$543$	0	0
$544$	− 21.0609i	− 0.902977i
$545$	−38.7744	−1.66091
$546$	0	0
$547$	−39.3128	−1.68090	−0.840448	−	0.541893i	$-0.817708\pi$
$547$	−39.3128	−1.68090	−0.840448	+	0.541893i	$0.817708\pi$
$548$	− 8.98083i	− 0.383642i
$549$	0	0
$550$	68.6763	2.92837
$551$	21.8256	0.929803
$552$	0	0
$553$	0	0
$554$	9.50518i	0.403837i
$555$	0	0
$556$	− 7.19789i	− 0.305258i
$557$	25.5093i	1.08086i	0.841388	+	0.540432i	$0.181739\pi$
$557$	25.5093i	1.08086i	−0.841388	+	0.540432i	$0.818261\pi$
$558$	0	0
$559$	− 2.87538i	− 0.121615i
$560$	0	0
$561$	0	0
$562$	17.1416	0.723074
$563$	−23.5938	−0.994359	−0.497179	−	0.867648i	$-0.665631\pi$
$563$	−23.5938	−0.994359	−0.497179	+	0.867648i	$0.665631\pi$
$564$	0	0
$565$	77.5301i	3.26172i
$566$	32.6548	1.37259
$567$	0	0
$568$	−22.3318	−0.937022
$569$	11.5562i	0.484460i	0.970219	+	0.242230i	$0.0778788\pi$
$569$	11.5562i	0.484460i	−0.970219	+	0.242230i	$0.922121\pi$
$570$	0	0
$571$	−21.6054	−0.904157	−0.452079	−	0.891978i	$-0.649317\pi$
$571$	−21.6054	−0.904157	−0.452079	+	0.891978i	$0.649317\pi$
$572$	−3.60721	−0.150825
$573$	0	0
$574$	0	0
$575$	86.5351i	3.60876i
$576$	0	0
$577$	− 45.4498i	− 1.89210i	−0.324019	−	0.946051i	$-0.605034\pi$
$577$	− 45.4498i	− 1.89210i	0.324019	−	0.946051i	$-0.394966\pi$
$578$	− 6.66537i	− 0.277243i
$579$	0	0
$580$	11.8833i	0.493429i
$581$	0	0
$582$	0	0
$583$	−7.85046	−0.325133
$584$	22.4877	0.930546
$585$	0	0
$586$	15.9847i	0.660323i
$587$	−26.9281	−1.11144	−0.555721	−	0.831369i	$-0.687558\pi$
$587$	−26.9281	−1.11144	−0.555721	+	0.831369i	$0.687558\pi$
$588$	0	0
$589$	18.5991	0.766362
$590$	3.11430i	0.128214i
$591$	0	0
$592$	−2.74961	−0.113008
$593$	26.6163	1.09300	0.546501	−	0.837458i	$-0.315959\pi$
$593$	26.6163	1.09300	0.546501	+	0.837458i	$0.315959\pi$
$594$	0	0
$595$	0	0
$596$	11.8503i	0.485409i
$597$	0	0
$598$	− 15.0209i	− 0.614252i
$599$	41.5907i	1.69935i	0.527308	+	0.849674i	$0.323201\pi$
$599$	41.5907i	1.69935i	−0.527308	+	0.849674i	$0.676799\pi$
$600$	0	0
$601$	41.5674i	1.69557i	0.530341	+	0.847784i	$0.322064\pi$
$601$	41.5674i	1.69557i	−0.530341	+	0.847784i	$0.677936\pi$
$602$	0	0
$603$	0	0
$604$	2.31043	0.0940102
$605$	3.88775	0.158059
$606$	0	0
$607$	39.4197i	1.60000i	0.600002	+	0.799998i	$0.295166\pi$
$607$	39.4197i	1.60000i	−0.600002	+	0.799998i	$0.704834\pi$
$608$	30.0071	1.21695
$609$	0	0
$610$	22.5857	0.914468
$611$	− 14.1853i	− 0.573875i
$612$	0	0
$613$	17.6403	0.712485	0.356242	−	0.934394i	$-0.384058\pi$
$613$	17.6403	0.712485	0.356242	+	0.934394i	$0.384058\pi$
$614$	−13.0763	−0.527715
$615$	0	0
$616$	0	0
$617$	7.78631i	0.313465i	0.987641	+	0.156733i	$0.0500961\pi$
$617$	7.78631i	0.313465i	−0.987641	+	0.156733i	$0.949904\pi$
$618$	0	0
$619$	11.7726i	0.473182i	0.971609	+	0.236591i	$0.0760301\pi$
$619$	11.7726i	0.473182i	−0.971609	+	0.236591i	$0.923970\pi$
$620$	10.1266i	0.406694i
$621$	0	0
$622$	− 9.44097i	− 0.378548i
$623$	0	0
$624$	0	0
$625$	53.9651	2.15860
$626$	−6.47519	−0.258800
$627$	0	0
$628$	9.18542i	0.366538i
$629$	−2.52504	−0.100680
$630$	0	0
$631$	32.3713	1.28868	0.644340	−	0.764739i	$-0.277132\pi$
$631$	32.3713	1.28868	0.644340	+	0.764739i	$0.277132\pi$
$632$	− 18.8042i	− 0.747990i
$633$	0	0
$634$	−43.9112	−1.74394
$635$	13.2860	0.527237
$636$	0	0
$637$	0	0
$638$	19.5991i	0.775936i
$639$	0	0
$640$	− 52.6883i	− 2.08269i
$641$	− 3.13659i	− 0.123888i	−0.998080	−	0.0619440i	$-0.980270\pi$
$641$	− 3.13659i	− 0.123888i	0.998080	−	0.0619440i	$-0.0197300\pi$
$642$	0	0
$643$	− 28.9234i	− 1.14063i	−0.821427	−	0.570314i	$-0.806822\pi$
$643$	− 28.9234i	− 1.14063i	0.821427	−	0.570314i	$-0.193178\pi$
$644$	0	0
$645$	0	0
$646$	50.5142	1.98745
$647$	−2.53033	−0.0994774	−0.0497387	−	0.998762i	$-0.515839\pi$
$647$	−2.53033	−0.0994774	−0.0497387	+	0.998762i	$0.515839\pi$
$648$	0	0
$649$	1.55424i	0.0610092i
$650$	−23.8886	−0.936986
$651$	0	0
$652$	−2.86183	−0.112078
$653$	− 22.0570i	− 0.863156i	−0.902076	−	0.431578i	$-0.857957\pi$
$653$	− 22.0570i	− 0.863156i	0.902076	−	0.431578i	$-0.142043\pi$
$654$	0	0
$655$	35.0185	1.36828
$656$	56.0826	2.18966
$657$	0	0
$658$	0	0
$659$	43.4940i	1.69429i	0.531364	+	0.847144i	$0.321680\pi$
$659$	43.4940i	1.69429i	−0.531364	+	0.847144i	$0.678320\pi$
$660$	0	0
$661$	− 28.7788i	− 1.11937i	−0.828707	−	0.559683i	$-0.810923\pi$
$661$	− 28.7788i	− 1.11937i	0.828707	−	0.559683i	$-0.189077\pi$
$662$	− 52.2947i	− 2.03249i
$663$	0	0
$664$	− 26.0328i	− 1.01027i
$665$	0	0
$666$	0	0
$667$	−24.6957	−0.956222
$668$	−7.03897	−0.272346
$669$	0	0
$670$	− 24.7578i	− 0.956479i
$671$	11.2717	0.435140
$672$	0	0
$673$	−16.3046	−0.628495	−0.314247	−	0.949341i	$-0.601752\pi$
$673$	−16.3046	−0.628495	−0.314247	+	0.949341i	$0.601752\pi$
$674$	38.1779i	1.47056i
$675$	0	0
$676$	−10.0262	−0.385624
$677$	6.51203	0.250278	0.125139	−	0.992139i	$-0.460062\pi$
$677$	6.51203	0.250278	0.125139	+	0.992139i	$0.460062\pi$
$678$	0	0
$679$	0	0
$680$	− 35.8854i	− 1.37614i
$681$	0	0
$682$	16.7017i	0.639541i
$683$	38.9298i	1.48961i	0.667284	+	0.744803i	$0.267457\pi$
$683$	38.9298i	1.48961i	−0.667284	+	0.744803i	$0.732543\pi$
$684$	0	0
$685$	− 42.3328i	− 1.61745i
$686$	0	0
$687$	0	0
$688$	−11.9143	−0.454228
$689$	2.73073	0.104032
$690$	0	0
$691$	− 10.1770i	− 0.387149i	−0.981086	−	0.193575i	$-0.937992\pi$
$691$	− 10.1770i	− 0.387149i	0.981086	−	0.193575i	$-0.0620082\pi$
$692$	−3.98429	−0.151460
$693$	0	0
$694$	−33.9494	−1.28870
$695$	− 33.9285i	− 1.28698i
$696$	0	0
$697$	51.5022	1.95079
$698$	53.9652	2.04261
$699$	0	0
$700$	0	0
$701$	− 8.75353i	− 0.330616i	−0.986242	−	0.165308i	$-0.947138\pi$
$701$	− 8.75353i	− 0.330616i	0.986242	−	0.165308i	$-0.0528618\pi$
$702$	0	0
$703$	− 3.59763i	− 0.135687i
$704$	− 7.50259i	− 0.282764i
$705$	0	0
$706$	23.5241i	0.885341i
$707$	0	0
$708$	0	0
$709$	19.4954	0.732166	0.366083	−	0.930582i	$-0.380699\pi$
$709$	19.4954	0.732166	0.366083	+	0.930582i	$0.380699\pi$
$710$	80.6774	3.02777
$711$	0	0
$712$	− 28.8490i	− 1.08116i
$713$	−21.0449	−0.788136
$714$	0	0
$715$	−17.0032	−0.635885
$716$	9.65610i	0.360865i
$717$	0	0
$718$	25.1406	0.938240
$719$	31.1638	1.16221	0.581106	−	0.813828i	$-0.302620\pi$
$719$	31.1638	1.16221	0.581106	+	0.813828i	$0.302620\pi$
$720$	0	0
$721$	0	0
$722$	39.7962i	1.48106i
$723$	0	0
$724$	− 21.1818i	− 0.787214i
$725$	39.2748i	1.45863i
$726$	0	0
$727$	− 10.0712i	− 0.373519i	−0.982406	−	0.186760i	$-0.940201\pi$
$727$	− 10.0712i	− 0.373519i	0.982406	−	0.186760i	$-0.0597986\pi$
$728$	0	0
$729$	0	0
$730$	−81.2404	−3.00684
$731$	−10.9412	−0.404676
$732$	0	0
$733$	13.4056i	0.495149i	0.968869	+	0.247574i	$0.0796334\pi$
$733$	13.4056i	0.495149i	−0.968869	+	0.247574i	$0.920367\pi$
$734$	31.5856	1.16585
$735$	0	0
$736$	−33.9531	−1.25153
$737$	− 12.3558i	− 0.455130i
$738$	0	0
$739$	10.6697	0.392490	0.196245	−	0.980555i	$-0.437125\pi$
$739$	10.6697	0.392490	0.196245	+	0.980555i	$0.437125\pi$
$740$	1.95880	0.0720068
$741$	0	0
$742$	0	0
$743$	− 33.4816i	− 1.22832i	−0.789181	−	0.614160i	$-0.789495\pi$
$743$	− 33.4816i	− 1.22832i	0.789181	−	0.614160i	$-0.210505\pi$
$744$	0	0
$745$	55.8588i	2.04651i
$746$	− 25.4423i	− 0.931509i
$747$	0	0
$748$	13.7259i	0.501870i
$749$	0	0
$750$	0	0
$751$	−1.65742	−0.0604801	−0.0302401	−	0.999543i	$-0.509627\pi$
$751$	−1.65742	−0.0604801	−0.0302401	+	0.999543i	$0.509627\pi$
$752$	−58.7775	−2.14340
$753$	0	0
$754$	− 6.81741i	− 0.248275i
$755$	10.8906	0.396351
$756$	0	0
$757$	−22.8336	−0.829902	−0.414951	−	0.909844i	$-0.636201\pi$
$757$	−22.8336	−0.829902	−0.414951	+	0.909844i	$0.636201\pi$
$758$	− 45.4271i	− 1.64999i
$759$	0	0
$760$	51.1288	1.85464
$761$	−1.41833	−0.0514144	−0.0257072	−	0.999670i	$-0.508184\pi$
$761$	−1.41833	−0.0514144	−0.0257072	+	0.999670i	$0.508184\pi$
$762$	0	0
$763$	0	0
$764$	1.53280i	0.0554548i
$765$	0	0
$766$	38.8502i	1.40372i
$767$	− 0.540631i	− 0.0195211i
$768$	0	0
$769$	15.9935i	0.576739i	0.957519	+	0.288370i	$0.0931132\pi$
$769$	15.9935i	0.576739i	−0.957519	+	0.288370i	$0.906887\pi$
$770$	0	0
$771$	0	0
$772$	6.92585	0.249267
$773$	31.6040	1.13672	0.568359	−	0.822781i	$-0.307578\pi$
$773$	31.6040	1.13672	0.568359	+	0.822781i	$0.307578\pi$
$774$	0	0
$775$	33.4687i	1.20223i
$776$	5.90397	0.211940
$777$	0	0
$778$	36.2244	1.29871
$779$	73.3794i	2.62909i
$780$	0	0
$781$	40.2632	1.44073
$782$	−57.1568	−2.04392
$783$	0	0
$784$	0	0
$785$	43.2971i	1.54534i
$786$	0	0
$787$	37.4919i	1.33644i	0.743964	+	0.668220i	$0.232944\pi$
$787$	37.4919i	1.33644i	−0.743964	+	0.668220i	$0.767056\pi$
$788$	− 7.69674i	− 0.274185i
$789$	0	0
$790$	67.9332i	2.41695i
$791$	0	0
$792$	0	0
$793$	−3.92079	−0.139231
$794$	−32.5940	−1.15672
$795$	0	0
$796$	− 12.3578i	− 0.438010i
$797$	−0.340686	−0.0120677	−0.00603385	−	0.999982i	$-0.501921\pi$
$797$	−0.340686	−0.0120677	−0.00603385	+	0.999982i	$0.501921\pi$
$798$	0	0
$799$	−53.9771	−1.90957
$800$	53.9973i	1.90909i
$801$	0	0
$802$	23.7762	0.839568
$803$	−40.5442	−1.43077
$804$	0	0
$805$	0	0
$806$	− 5.80957i	− 0.204633i
$807$	0	0
$808$	− 16.9140i	− 0.595031i
$809$	35.4866i	1.24764i	0.781567	+	0.623821i	$0.214421\pi$
$809$	35.4866i	1.24764i	−0.781567	+	0.623821i	$0.785579\pi$
$810$	0	0
$811$	− 9.34974i	− 0.328314i	−0.986434	−	0.164157i	$-0.947510\pi$
$811$	− 9.34974i	− 0.328314i	0.986434	−	0.164157i	$-0.0524903\pi$
$812$	0	0
$813$	0	0
$814$	3.23062	0.113233
$815$	−13.4898	−0.472526
$816$	0	0
$817$	− 15.5888i	− 0.545385i
$818$	29.9830	1.04833
$819$	0	0
$820$	−39.9527	−1.39521
$821$	− 51.5786i	− 1.80011i	−0.435780	−	0.900053i	$-0.643527\pi$
$821$	− 51.5786i	− 1.80011i	0.435780	−	0.900053i	$-0.356473\pi$
$822$	0	0
$823$	14.5487	0.507137	0.253568	−	0.967317i	$-0.418396\pi$
$823$	14.5487	0.507137	0.253568	+	0.967317i	$0.418396\pi$
$824$	−6.14300	−0.214002
$825$	0	0
$826$	0	0
$827$	3.52283i	0.122501i	0.998122	+	0.0612503i	$0.0195088\pi$
$827$	3.52283i	0.122501i	−0.998122	+	0.0612503i	$0.980491\pi$
$828$	0	0
$829$	− 15.7619i	− 0.547432i	−0.961811	−	0.273716i	$-0.911747\pi$
$829$	− 15.7619i	− 0.547432i	0.961811	−	0.273716i	$-0.0882529\pi$
$830$	94.0480i	3.26445i
$831$	0	0
$832$	2.60972i	0.0904758i
$833$	0	0
$834$	0	0
$835$	−33.1794	−1.14822
$836$	−19.5565	−0.676375
$837$	0	0
$838$	− 6.97520i	− 0.240954i
$839$	−15.1234	−0.522119	−0.261059	−	0.965323i	$-0.584072\pi$
$839$	−15.1234	−0.522119	−0.261059	+	0.965323i	$0.584072\pi$
$840$	0	0
$841$	17.7916	0.613503
$842$	− 37.8051i	− 1.30285i
$843$	0	0
$844$	1.98582	0.0683547
$845$	−47.2605	−1.62581
$846$	0	0
$847$	0	0
$848$	− 11.3149i	− 0.388556i
$849$	0	0
$850$	90.8994i	3.11782i
$851$	4.07072i	0.139543i
$852$	0	0
$853$	− 39.3743i	− 1.34815i	−0.738662	−	0.674076i	$-0.764542\pi$
$853$	− 39.3743i	− 1.34815i	0.738662	−	0.674076i	$-0.235458\pi$
$854$	0	0
$855$	0	0
$856$	−3.85285	−0.131688
$857$	12.0077	0.410174	0.205087	−	0.978744i	$-0.434252\pi$
$857$	12.0077	0.410174	0.205087	+	0.978744i	$0.434252\pi$
$858$	0	0
$859$	− 23.7120i	− 0.809042i	−0.914529	−	0.404521i	$-0.867438\pi$
$859$	− 23.7120i	− 0.809042i	0.914529	−	0.404521i	$-0.132562\pi$
$860$	8.48762	0.289426
$861$	0	0
$862$	0.847286	0.0288587
$863$	32.7887i	1.11614i	0.829794	+	0.558070i	$0.188458\pi$
$863$	32.7887i	1.11614i	−0.829794	+	0.558070i	$0.811542\pi$
$864$	0	0
$865$	−18.7807	−0.638563
$866$	54.2525	1.84358
$867$	0	0
$868$	0	0
$869$	33.9031i	1.15008i
$870$	0	0
$871$	4.29786i	0.145628i
$872$	− 18.1756i	− 0.615505i
$873$	0	0
$874$	− 81.4360i	− 2.75461i
$875$	0	0
$876$	0	0
$877$	−11.6410	−0.393088	−0.196544	−	0.980495i	$-0.562972\pi$
$877$	−11.6410	−0.393088	−0.196544	+	0.980495i	$0.562972\pi$
$878$	−22.3120	−0.752993
$879$	0	0
$880$	70.4538i	2.37500i
$881$	−3.83448	−0.129187	−0.0645935	−	0.997912i	$-0.520575\pi$
$881$	−3.83448	−0.129187	−0.0645935	+	0.997912i	$0.520575\pi$
$882$	0	0
$883$	−10.2343	−0.344410	−0.172205	−	0.985061i	$-0.555089\pi$
$883$	−10.2343	−0.344410	−0.172205	+	0.985061i	$0.555089\pi$
$884$	− 4.77447i	− 0.160583i
$885$	0	0
$886$	47.5994	1.59913
$887$	−46.5310	−1.56236	−0.781179	−	0.624307i	$-0.785382\pi$
$887$	−46.5310	−1.56236	−0.781179	+	0.624307i	$0.785382\pi$
$888$	0	0
$889$	0	0
$890$	104.222i	3.49353i
$891$	0	0
$892$	1.27963i	0.0428452i
$893$	− 76.9055i	− 2.57355i
$894$	0	0
$895$	45.5158i	1.52142i
$896$	0	0
$897$	0	0
$898$	51.6605	1.72393
$899$	−9.55143	−0.318558
$900$	0	0
$901$	− 10.3908i	− 0.346168i
$902$	−65.8937	−2.19402
$903$	0	0
$904$	−36.3425	−1.20873
$905$	− 99.8440i	− 3.31893i
$906$	0	0
$907$	−3.76516	−0.125020	−0.0625101	−	0.998044i	$-0.519911\pi$
$907$	−3.76516	−0.125020	−0.0625101	+	0.998044i	$0.519911\pi$
$908$	−7.27535	−0.241441
$909$	0	0
$910$	0	0
$911$	− 11.9172i	− 0.394833i	−0.980320	−	0.197417i	$-0.936745\pi$
$911$	− 11.9172i	− 0.394833i	0.980320	−	0.197417i	$-0.0632552\pi$
$912$	0	0
$913$	46.9360i	1.55336i
$914$	57.9171i	1.91573i
$915$	0	0
$916$	− 15.4769i	− 0.511370i
$917$	0	0
$918$	0	0
$919$	14.3084	0.471990	0.235995	−	0.971754i	$-0.424165\pi$
$919$	14.3084	0.471990	0.235995	+	0.971754i	$0.424165\pi$
$920$	−57.8523	−1.90733
$921$	0	0
$922$	17.6115i	0.580004i
$923$	−14.0053	−0.460989
$924$	0	0
$925$	6.47388	0.212860
$926$	− 30.3828i	− 0.998441i
$927$	0	0
$928$	−15.4099	−0.505856
$929$	4.43652	0.145557	0.0727787	−	0.997348i	$-0.476813\pi$
$929$	4.43652	0.145557	0.0727787	+	0.997348i	$0.476813\pi$
$930$	0	0
$931$	0	0
$932$	− 20.2199i	− 0.662325i
$933$	0	0
$934$	11.4991i	0.376262i
$935$	64.6997i	2.11591i
$936$	0	0
$937$	37.3421i	1.21991i	0.792434	+	0.609957i	$0.208813\pi$
$937$	37.3421i	1.21991i	−0.792434	+	0.609957i	$0.791187\pi$
$938$	0	0
$939$	0	0
$940$	41.8726	1.36573
$941$	44.1882	1.44049	0.720246	−	0.693718i	$-0.244029\pi$
$941$	44.1882	1.44049	0.720246	+	0.693718i	$0.244029\pi$
$942$	0	0
$943$	− 83.0288i	− 2.70379i
$944$	−2.24013	−0.0729102
$945$	0	0
$946$	13.9986	0.455133
$947$	5.13692i	0.166927i	0.996511	+	0.0834637i	$0.0265983\pi$
$947$	5.13692i	0.166927i	−0.996511	+	0.0834637i	$0.973402\pi$
$948$	0	0
$949$	14.1030	0.457804
$950$	−129.512	−4.20192
$951$	0	0
$952$	0	0
$953$	− 27.7105i	− 0.897632i	−0.893624	−	0.448816i	$-0.851846\pi$
$953$	− 27.7105i	− 0.897632i	0.893624	−	0.448816i	$-0.148154\pi$
$954$	0	0
$955$	7.22513i	0.233800i
$956$	8.28047i	0.267810i
$957$	0	0
$958$	− 70.9152i	− 2.29117i
$959$	0	0
$960$	0	0
$961$	22.8606	0.737438
$962$	−1.12375	−0.0362311
$963$	0	0
$964$	− 13.7075i	− 0.441488i
$965$	32.6463	1.05092
$966$	0	0
$967$	7.18813	0.231155	0.115577	−	0.993298i	$-0.463128\pi$
$967$	7.18813	0.231155	0.115577	+	0.993298i	$0.463128\pi$
$968$	1.82239i	0.0585739i
$969$	0	0
$970$	−21.3291	−0.684835
$971$	3.09861	0.0994390	0.0497195	−	0.998763i	$-0.484167\pi$
$971$	3.09861	0.0994390	0.0497195	+	0.998763i	$0.484167\pi$
$972$	0	0
$973$	0	0
$974$	− 40.8171i	− 1.30786i
$975$	0	0
$976$	16.2460i	0.520022i
$977$	3.83329i	0.122638i	0.998118	+	0.0613189i	$0.0195307\pi$
$977$	3.83329i	0.122638i	−0.998118	+	0.0613189i	$0.980469\pi$
$978$	0	0
$979$	52.0135i	1.66236i
$980$	0	0
$981$	0	0
$982$	−56.3665	−1.79873
$983$	3.71751	0.118570	0.0592851	−	0.998241i	$-0.481118\pi$
$983$	3.71751	0.118570	0.0592851	+	0.998241i	$0.481118\pi$
$984$	0	0
$985$	− 36.2800i	− 1.15598i
$986$	−25.9412	−0.826137
$987$	0	0
$988$	6.80258	0.216419
$989$	17.6388i	0.560881i
$990$	0	0
$991$	−35.0350	−1.11292	−0.556462	−	0.830873i	$-0.687841\pi$
$991$	−35.0350	−1.11292	−0.556462	+	0.830873i	$0.687841\pi$
$992$	−13.1319	−0.416937
$993$	0	0
$994$	0	0
$995$	− 58.2506i	− 1.84667i
$996$	0	0
$997$	− 42.7855i	− 1.35503i	−0.735509	−	0.677515i	$-0.763057\pi$
$997$	− 42.7855i	− 1.35503i	0.735509	−	0.677515i	$-0.236943\pi$
$998$	− 50.3254i	− 1.59302i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.c.c.3086.1	✓	24
3.2	odd	2	inner	3087.2.c.c.3086.24	yes	24
7.6	odd	2	inner	3087.2.c.c.3086.23	yes	24
21.20	even	2	inner	3087.2.c.c.3086.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3087.2.c.c.3086.1	✓	24	1.1	even	1	trivial
3087.2.c.c.3086.2	yes	24	21.20	even	2	inner
3087.2.c.c.3086.23	yes	24	7.6	odd	2	inner
3087.2.c.c.3086.24	yes	24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.69345i q^{2} -0.867767 q^{4} -4.09038 q^{5} -1.91738i q^{8} +O(q^{10})\)	\(q-1.69345i q^{2} -0.867767 q^{4} -4.09038 q^{5} -1.91738i q^{8} +6.92685i q^{10} +3.45694i q^{11} -1.20247i q^{13} -4.98251 q^{16} -4.57558 q^{17} -6.51921i q^{19} +3.54950 q^{20} +5.85416 q^{22} +7.37649i q^{23} +11.7312 q^{25} -2.03633 q^{26} +3.34790i q^{29} +2.85297i q^{31} +4.60288i q^{32} +7.74852i q^{34} +0.551851 q^{37} -11.0399 q^{38} +7.84280i q^{40} -11.2559 q^{41} +2.39122 q^{43} -2.99982i q^{44} +12.4917 q^{46} +11.7968 q^{47} -19.8662i q^{50} +1.04347i q^{52} +2.27092i q^{53} -14.1402i q^{55} +5.66949 q^{58} +0.449599 q^{59} -3.26060i q^{61} +4.83135 q^{62} -2.17030 q^{64} +4.91857i q^{65} -3.57419 q^{67} +3.97054 q^{68} -11.6471i q^{71} +11.7283i q^{73} -0.934532i q^{74} +5.65716i q^{76} +9.80723 q^{79} +20.3804 q^{80} +19.0613i q^{82} +13.5773 q^{83} +18.7159 q^{85} -4.04940i q^{86} +6.62827 q^{88} +15.0461 q^{89} -6.40108i q^{92} -19.9772i q^{94} +26.6660i q^{95} +3.07919i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) \) 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more