LMFDB - Embedded newform 6336.2.d.e.3455.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(3455,6336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6336.3455"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,8,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1768034304.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 9x^{6} - 2x^{5} + 34x^{4} - 18x^{3} + 51x^{2} + 18x + 9 $ x^8 - 2x^7 + 9x^6 - 2x^5 + 34x^4 - 18x^3 + 51x^2 + 18*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1584)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3455.7
Root			$1.44368 + 2.50053i$ of defining polynomial
Character	$\chi$	$=$	6336.3455
Dual form			6336.2.d.e.3455.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66913i q^{5} -2.66913i q^{7} +1.00000 q^{11} +3.60114 q^{13} -4.66385i q^{17} -3.04289i q^{19} -0.476882 q^{23} -2.12426 q^{25} +3.46410i q^{29} +1.19975i q^{31} +7.12426 q^{35} +8.37040 q^{37} +1.19975i q^{41} -10.7806i q^{43} -8.76926 q^{47} -0.124259 q^{49} +4.25907i q^{53} +2.66913i q^{55} -12.4484 q^{59} -0.398859 q^{61} +9.61191i q^{65} -11.9494i q^{67} -0.476882 q^{71} -14.4484 q^{73} -2.66913i q^{77} -11.9968i q^{79} +4.87574 q^{83} +12.4484 q^{85} -16.5091i q^{89} -9.61191i q^{91} +8.12188 q^{95} -3.12426 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 16 q^{25} + 56 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{59} - 24 q^{61} + 16 q^{71} + 40 q^{83} - 16 q^{85} - 8 q^{95} - 24 q^{97}+O(q^{100}) $ 8 * q + 8 * q^11 + 8 * q^13 + 16 * q^23 - 16 * q^25 + 56 * q^35 - 8 * q^37 - 16 * q^47 + 16 * q^59 - 24 * q^61 + 16 * q^71 + 40 * q^83 - 16 * q^85 - 8 * q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times$.

$n$	$1729$	$3521$	$4159$	$4357$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$5$	2.66913i	1.19367i	0.802363	+	0.596836i	$0.203576\pi$
$5$	2.66913i	1.19367i	−0.802363	+	0.596836i	$0.796424\pi$
$6$	0	0
$7$	− 2.66913i	− 1.00884i	−0.863459	−	0.504418i	$-0.831707\pi$
$7$	− 2.66913i	− 1.00884i	0.863459	−	0.504418i	$-0.168293\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.60114	0.998777	0.499388	−	0.866378i	$-0.333558\pi$
$13$	3.60114	0.998777	0.499388	+	0.866378i	$0.333558\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.66385i	− 1.13115i	−0.824697	−	0.565575i	$-0.808654\pi$
$17$	− 4.66385i	− 1.13115i	0.824697	−	0.565575i	$-0.191346\pi$
$18$	0	0
$19$	− 3.04289i	− 0.698088i	−0.937106	−	0.349044i	$-0.886506\pi$
$19$	− 3.04289i	− 0.698088i	0.937106	−	0.349044i	$-0.113494\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.476882	−0.0994368	−0.0497184	−	0.998763i	$-0.515832\pi$
$23$	−0.476882	−0.0994368	−0.0497184	+	0.998763i	$0.515832\pi$
$24$	0	0
$25$	−2.12426	−0.424852
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.46410i	0.643268i	0.946864	+	0.321634i	$0.104232\pi$
$29$	3.46410i	0.643268i	−0.946864	+	0.321634i	$0.895768\pi$
$30$	0	0
$31$	1.19975i	0.215481i	0.994179	+	0.107740i	$0.0343615\pi$
$31$	1.19975i	0.215481i	−0.994179	+	0.107740i	$0.965638\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.12426	1.20422
$36$	0	0
$37$	8.37040	1.37609	0.688043	−	0.725670i	$-0.258470\pi$
$37$	8.37040	1.37609	0.688043	+	0.725670i	$0.258470\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.19975i	0.187369i	0.995602	+	0.0936845i	$0.0298645\pi$
$41$	1.19975i	0.187369i	−0.995602	+	0.0936845i	$0.970136\pi$
$42$	0	0
$43$	− 10.7806i	− 1.64403i	−0.569463	−	0.822017i	$-0.692849\pi$
$43$	− 10.7806i	− 1.64403i	0.569463	−	0.822017i	$-0.307151\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.76926	−1.27913	−0.639564	−	0.768738i	$-0.720885\pi$
$47$	−8.76926	−1.27913	−0.639564	+	0.768738i	$0.720885\pi$
$48$	0	0
$49$	−0.124259	−0.0177512
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.25907i	0.585029i	0.956261	+	0.292514i	$0.0944919\pi$
$53$	4.25907i	0.585029i	−0.956261	+	0.292514i	$0.905508\pi$
$54$	0	0
$55$	2.66913i	0.359906i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.4484	−1.62065	−0.810323	−	0.585983i	$-0.800708\pi$
$59$	−12.4484	−1.62065	−0.810323	+	0.585983i	$0.800708\pi$
$60$	0	0
$61$	−0.398859	−0.0510687	−0.0255344	−	0.999674i	$-0.508129\pi$
$61$	−0.398859	−0.0510687	−0.0255344	+	0.999674i	$0.508129\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.61191i	1.19221i
$66$	0	0
$67$	− 11.9494i	− 1.45985i	−0.683528	−	0.729925i	$-0.739555\pi$
$67$	− 11.9494i	− 1.45985i	0.683528	−	0.729925i	$-0.260445\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.476882	−0.0565955	−0.0282977	−	0.999600i	$-0.509009\pi$
$71$	−0.476882	−0.0565955	−0.0282977	+	0.999600i	$0.509009\pi$
$72$	0	0
$73$	−14.4484	−1.69106	−0.845530	−	0.533928i	$-0.820715\pi$
$73$	−14.4484	−1.69106	−0.845530	+	0.533928i	$0.820715\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.66913i	− 0.304176i
$78$	0	0
$79$	− 11.9968i	− 1.34975i	−0.737933	−	0.674874i	$-0.764198\pi$
$79$	− 11.9968i	− 1.34975i	0.737933	−	0.674874i	$-0.235802\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.87574	0.535182	0.267591	−	0.963533i	$-0.413772\pi$
$83$	4.87574	0.535182	0.267591	+	0.963533i	$0.413772\pi$
$84$	0	0
$85$	12.4484	1.35022
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 16.5091i	− 1.74996i	−0.484158	−	0.874981i	$-0.660874\pi$
$89$	− 16.5091i	− 1.74996i	0.484158	−	0.874981i	$-0.339126\pi$
$90$	0	0
$91$	− 9.61191i	− 1.00760i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.12188	0.833287
$96$	0	0
$97$	−3.12426	−0.317220	−0.158610	−	0.987341i	$-0.550701\pi$
$97$	−3.12426	−0.317220	−0.158610	+	0.987341i	$0.550701\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.94727i	− 0.193761i	−0.995296	−	0.0968805i	$-0.969114\pi$
$101$	− 1.94727i	− 0.193761i	0.995296	−	0.0968805i	$-0.0308865\pi$
$102$	0	0
$103$	0.390195i	0.0384471i	0.999815	+	0.0192235i	$0.00611942\pi$
$103$	0.390195i	0.0384471i	−0.999815	+	0.0192235i	$0.993881\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.24614	−0.313816	−0.156908	−	0.987613i	$-0.550153\pi$
$107$	−3.24614	−0.313816	−0.156908	+	0.987613i	$0.550153\pi$
$108$	0	0
$109$	11.8935	1.13919	0.569596	−	0.821925i	$-0.307100\pi$
$109$	11.8935	1.13919	0.569596	+	0.821925i	$0.307100\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.84315i	0.173389i	0.996235	+	0.0866943i	$0.0276304\pi$
$113$	1.84315i	0.173389i	−0.996235	+	0.0866943i	$0.972370\pi$
$114$	0	0
$115$	− 1.27286i	− 0.118695i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.4484	−1.14114
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.67573i	0.686538i
$126$	0	0
$127$	4.25907i	0.377932i	0.981984	+	0.188966i	$0.0605135\pi$
$127$	4.25907i	0.377932i	−0.981984	+	0.188966i	$0.939486\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.3704	0.906066	0.453033	−	0.891494i	$-0.350342\pi$
$131$	10.3704	0.906066	0.453033	+	0.891494i	$0.350342\pi$
$132$	0	0
$133$	−8.12188	−0.704256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.9086i	− 1.61547i	−0.589545	−	0.807735i	$-0.700693\pi$
$137$	− 18.9086i	− 1.61547i	0.589545	−	0.807735i	$-0.299307\pi$
$138$	0	0
$139$	16.9613i	1.43864i	0.694678	+	0.719320i	$0.255547\pi$
$139$	16.9613i	1.43864i	−0.694678	+	0.719320i	$0.744453\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.60114	0.301143
$144$	0	0
$145$	−9.24614	−0.767850
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.0339i	0.903929i	0.892036	+	0.451964i	$0.149277\pi$
$149$	11.0339i	0.903929i	−0.892036	+	0.451964i	$0.850723\pi$
$150$	0	0
$151$	− 16.4927i	− 1.34215i	−0.741387	−	0.671077i	$-0.765832\pi$
$151$	− 16.4927i	− 1.34215i	0.741387	−	0.671077i	$-0.234168\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.20228	−0.257213
$156$	0	0
$157$	−23.6946	−1.89103	−0.945516	−	0.325576i	$-0.894442\pi$
$157$	−23.6946	−1.89103	−0.945516	+	0.325576i	$0.894442\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.27286i	0.100315i
$162$	0	0
$163$	− 12.2336i	− 0.958210i	−0.877758	−	0.479105i	$-0.840961\pi$
$163$	− 12.2336i	− 0.958210i	0.877758	−	0.479105i	$-0.159039\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.57268	−0.121698	−0.0608488	−	0.998147i	$-0.519381\pi$
$167$	−1.57268	−0.121698	−0.0608488	+	0.998147i	$0.519381\pi$
$168$	0	0
$169$	−0.0317864	−0.00244511
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.86360i	0.445801i	0.974841	+	0.222900i	$0.0715525\pi$
$173$	5.86360i	0.445801i	−0.974841	+	0.222900i	$0.928448\pi$
$174$	0	0
$175$	5.66992i	0.428606i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	25.4022	1.89865	0.949324	−	0.314298i	$-0.101769\pi$
$179$	25.4022	1.89865	0.949324	+	0.314298i	$0.101769\pi$
$180$	0	0
$181$	−7.41663	−0.551274	−0.275637	−	0.961262i	$-0.588889\pi$
$181$	−7.41663	−0.551274	−0.275637	+	0.961262i	$0.588889\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	22.3417i	1.64259i
$186$	0	0
$187$	− 4.66385i	− 0.341054i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.4662	1.69795	0.848977	−	0.528429i	$-0.177219\pi$
$191$	23.4662	1.69795	0.848977	+	0.528429i	$0.177219\pi$
$192$	0	0
$193$	12.1999	0.878168	0.439084	−	0.898446i	$-0.355303\pi$
$193$	12.1999	0.878168	0.439084	+	0.898446i	$0.355303\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.6238i	− 0.899408i	−0.893178	−	0.449704i	$-0.851529\pi$
$197$	− 12.6238i	− 0.899408i	0.893178	−	0.449704i	$-0.148471\pi$
$198$	0	0
$199$	− 16.4452i	− 1.16577i	−0.812554	−	0.582886i	$-0.801924\pi$
$199$	− 16.4452i	− 1.16577i	0.812554	−	0.582886i	$-0.198076\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.24614	0.648952
$204$	0	0
$205$	−3.20228	−0.223657
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.04289i	− 0.210481i
$210$	0	0
$211$	12.1623i	0.837290i	0.908150	+	0.418645i	$0.137495\pi$
$211$	12.1623i	0.837290i	−0.908150	+	0.418645i	$0.862505\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	28.7750	1.96244
$216$	0	0
$217$	3.20228	0.217385
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 16.7952i	− 1.12977i
$222$	0	0
$223$	16.5401i	1.10761i	0.832647	+	0.553804i	$0.186824\pi$
$223$	16.5401i	1.10761i	−0.832647	+	0.553804i	$0.813176\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.6507	1.43701	0.718504	−	0.695523i	$-0.244827\pi$
$227$	21.6507	1.43701	0.718504	+	0.695523i	$0.244827\pi$
$228$	0	0
$229$	−1.20228	−0.0794490	−0.0397245	−	0.999211i	$-0.512648\pi$
$229$	−1.20228	−0.0794490	−0.0397245	+	0.999211i	$0.512648\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 5.72846i	− 0.375284i	−0.982238	−	0.187642i	$-0.939916\pi$
$233$	− 5.72846i	− 0.375284i	0.982238	−	0.187642i	$-0.0600844\pi$
$234$	0	0
$235$	− 23.4063i	− 1.52686i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.3242	1.50871	0.754357	−	0.656464i	$-0.227949\pi$
$239$	23.3242	1.50871	0.754357	+	0.656464i	$0.227949\pi$
$240$	0	0
$241$	21.9431	1.41348	0.706739	−	0.707474i	$-0.250166\pi$
$241$	21.9431	1.41348	0.706739	+	0.707474i	$0.250166\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 0.331663i	− 0.0211891i
$246$	0	0
$247$	− 10.9579i	− 0.697234i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.5409	0.665335	0.332668	−	0.943044i	$-0.392051\pi$
$251$	10.5409	0.665335	0.332668	+	0.943044i	$0.392051\pi$
$252$	0	0
$253$	−0.476882	−0.0299813
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.12848i	0.0703929i	0.999380	+	0.0351965i	$0.0112057\pi$
$257$	1.12848i	0.0703929i	−0.999380	+	0.0351965i	$0.988794\pi$
$258$	0	0
$259$	− 22.3417i	− 1.38824i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.6189	−0.778116	−0.389058	−	0.921213i	$-0.627199\pi$
$263$	−12.6189	−0.778116	−0.389058	+	0.921213i	$0.627199\pi$
$264$	0	0
$265$	−11.3680	−0.698332
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 12.7444i	− 0.777037i	−0.921441	−	0.388518i	$-0.872987\pi$
$269$	− 12.7444i	− 0.777037i	0.921441	−	0.388518i	$-0.127013\pi$
$270$	0	0
$271$	− 15.1438i	− 0.919923i	−0.887939	−	0.459962i	$-0.847863\pi$
$271$	− 15.1438i	− 0.919923i	0.887939	−	0.459962i	$-0.152137\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.12426	−0.128098
$276$	0	0
$277$	12.3989	0.744975	0.372488	−	0.928037i	$-0.378505\pi$
$277$	12.3989	0.744975	0.372488	+	0.928037i	$0.378505\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.85509i	0.408940i	0.978873	+	0.204470i	$0.0655471\pi$
$281$	6.85509i	0.408940i	−0.978873	+	0.204470i	$0.934453\pi$
$282$	0	0
$283$	8.38116i	0.498208i	0.968477	+	0.249104i	$0.0801361\pi$
$283$	8.38116i	0.498208i	−0.968477	+	0.249104i	$0.919864\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.20228	0.189025
$288$	0	0
$289$	−4.75148	−0.279499
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 18.1301i	− 1.05917i	−0.848257	−	0.529585i	$-0.822348\pi$
$293$	− 18.1301i	− 1.05917i	0.848257	−	0.529585i	$-0.177652\pi$
$294$	0	0
$295$	− 33.2265i	− 1.93452i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.71732	−0.0993151
$300$	0	0
$301$	−28.7750	−1.65856
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.06461i	− 0.0609593i
$306$	0	0
$307$	16.1518i	0.921830i	0.887444	+	0.460915i	$0.152479\pi$
$307$	16.1518i	0.921830i	−0.887444	+	0.460915i	$0.847521\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.5231	1.33387	0.666937	−	0.745114i	$-0.267605\pi$
$311$	23.5231	1.33387	0.666937	+	0.745114i	$0.267605\pi$
$312$	0	0
$313$	12.1560	0.687100	0.343550	−	0.939134i	$-0.388370\pi$
$313$	12.1560	0.687100	0.343550	+	0.939134i	$0.388370\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.1438i	− 0.850563i	−0.905061	−	0.425282i	$-0.860175\pi$
$317$	− 15.1438i	− 0.850563i	0.905061	−	0.425282i	$-0.139825\pi$
$318$	0	0
$319$	3.46410i	0.193952i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14.1916	−0.789642
$324$	0	0
$325$	−7.64975	−0.424332
$326$	0	0
$327$	0	0
$328$	0	0
$329$	23.4063i	1.29043i
$330$	0	0
$331$	0.208252i	0.0114466i	0.999984	+	0.00572329i	$0.00182179\pi$
$331$	0.208252i	0.0114466i	−0.999984	+	0.00572329i	$0.998178\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	31.8945	1.74258
$336$	0	0
$337$	−15.9075	−0.866538	−0.433269	−	0.901265i	$-0.642640\pi$
$337$	−15.9075	−0.866538	−0.433269	+	0.901265i	$0.642640\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.19975i	0.0649699i
$342$	0	0
$343$	− 18.3523i	− 0.990928i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.1136	−1.61658	−0.808291	−	0.588784i	$-0.799607\pi$
$347$	−30.1136	−1.61658	−0.808291	+	0.588784i	$0.799607\pi$
$348$	0	0
$349$	−2.64738	−0.141711	−0.0708554	−	0.997487i	$-0.522573\pi$
$349$	−2.64738	−0.141711	−0.0708554	+	0.997487i	$0.522573\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.58090i	− 0.509940i	−0.966949	−	0.254970i	$-0.917934\pi$
$353$	− 9.58090i	− 0.509940i	0.966949	−	0.254970i	$-0.0820656\pi$
$354$	0	0
$355$	− 1.27286i	− 0.0675564i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.7052	0.565001	0.282501	−	0.959267i	$-0.408836\pi$
$359$	10.7052	0.565001	0.282501	+	0.959267i	$0.408836\pi$
$360$	0	0
$361$	9.74080	0.512674
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 38.5647i	− 2.01857i
$366$	0	0
$367$	− 6.57826i	− 0.343382i	−0.985151	−	0.171691i	$-0.945077\pi$
$367$	− 6.57826i	− 0.343382i	0.985151	−	0.171691i	$-0.0549231\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.3680	0.590198
$372$	0	0
$373$	12.5549	0.650068	0.325034	−	0.945702i	$-0.394624\pi$
$373$	12.5549	0.650068	0.325034	+	0.945702i	$0.394624\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.4747i	0.642481i
$378$	0	0
$379$	8.48528i	0.435860i	0.975964	+	0.217930i	$0.0699304\pi$
$379$	8.48528i	0.435860i	−0.975964	+	0.217930i	$0.930070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−27.6222	−1.41143	−0.705715	−	0.708496i	$-0.749374\pi$
$383$	−27.6222	−1.41143	−0.705715	+	0.708496i	$0.749374\pi$
$384$	0	0
$385$	7.12426	0.363086
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.9682i	1.62085i	0.585842	+	0.810426i	$0.300764\pi$
$389$	31.9682i	1.62085i	−0.585842	+	0.810426i	$0.699236\pi$
$390$	0	0
$391$	2.22411i	0.112478i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	32.0211	1.61116
$396$	0	0
$397$	−34.0650	−1.70967	−0.854836	−	0.518899i	$-0.826342\pi$
$397$	−34.0650	−1.70967	−0.854836	+	0.518899i	$0.826342\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.8092i	1.58848i	0.607606	+	0.794238i	$0.292130\pi$
$401$	31.8092i	1.58848i	−0.607606	+	0.794238i	$0.707870\pi$
$402$	0	0
$403$	4.32046i	0.215217i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.37040	0.414905
$408$	0	0
$409$	31.1892	1.54221	0.771104	−	0.636709i	$-0.219705\pi$
$409$	31.1892	1.54221	0.771104	+	0.636709i	$0.219705\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	33.2265i	1.63497i
$414$	0	0
$415$	13.0140i	0.638832i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.8968	−1.21629	−0.608145	−	0.793826i	$-0.708086\pi$
$419$	−24.8968	−1.21629	−0.608145	+	0.793826i	$0.708086\pi$
$420$	0	0
$421$	−6.03416	−0.294087	−0.147044	−	0.989130i	$-0.546976\pi$
$421$	−6.03416	−0.294087	−0.147044	+	0.989130i	$0.546976\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.90722i	0.480571i
$426$	0	0
$427$	1.06461i	0.0515200i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−37.5158	−1.80707	−0.903535	−	0.428514i	$-0.859037\pi$
$431$	−37.5158	−1.80707	−0.903535	+	0.428514i	$0.859037\pi$
$432$	0	0
$433$	21.9075	1.05281	0.526404	−	0.850234i	$-0.323540\pi$
$433$	21.9075	1.05281	0.526404	+	0.850234i	$0.323540\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.45110i	0.0694156i
$438$	0	0
$439$	− 27.1692i	− 1.29671i	−0.761336	−	0.648357i	$-0.775456\pi$
$439$	− 27.1692i	− 1.29671i	0.761336	−	0.648357i	$-0.224544\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.1098	−0.812912	−0.406456	−	0.913670i	$-0.633235\pi$
$443$	−17.1098	−0.812912	−0.406456	+	0.913670i	$0.633235\pi$
$444$	0	0
$445$	44.0650	2.08888
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 16.0027i	− 0.755214i	−0.925966	−	0.377607i	$-0.876747\pi$
$449$	− 16.0027i	− 0.755214i	0.925966	−	0.377607i	$-0.123253\pi$
$450$	0	0
$451$	1.19975i	0.0564939i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	25.6555	1.20275
$456$	0	0
$457$	16.4046	0.767373	0.383687	−	0.923463i	$-0.374654\pi$
$457$	16.4046	0.767373	0.383687	+	0.923463i	$0.374654\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.1554i	1.40448i	0.711942	+	0.702239i	$0.247816\pi$
$461$	30.1554i	1.40448i	−0.711942	+	0.702239i	$0.752184\pi$
$462$	0	0
$463$	32.5660i	1.51347i	0.653722	+	0.756735i	$0.273207\pi$
$463$	32.5660i	1.51347i	−0.653722	+	0.756735i	$0.726793\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.1999	1.02729	0.513645	−	0.858003i	$-0.328295\pi$
$467$	22.1999	1.02729	0.513645	+	0.858003i	$0.328295\pi$
$468$	0	0
$469$	−31.8945	−1.47275
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 10.7806i	− 0.495695i
$474$	0	0
$475$	6.46389i	0.296584i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.86743	0.131016	0.0655082	−	0.997852i	$-0.479133\pi$
$479$	2.86743	0.131016	0.0655082	+	0.997852i	$0.479133\pi$
$480$	0	0
$481$	30.1430	1.37440
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8.33905i	− 0.378657i
$486$	0	0
$487$	− 42.0288i	− 1.90451i	−0.305307	−	0.952254i	$-0.598759\pi$
$487$	− 42.0288i	− 1.90451i	0.305307	−	0.952254i	$-0.401241\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.3915	1.37155	0.685775	−	0.727814i	$-0.259464\pi$
$491$	30.3915	1.37155	0.685775	+	0.727814i	$0.259464\pi$
$492$	0	0
$493$	16.1560	0.727632
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.27286i	0.0570956i
$498$	0	0
$499$	− 25.5836i	− 1.14528i	−0.819807	−	0.572640i	$-0.805919\pi$
$499$	− 25.5836i	− 1.14528i	0.819807	−	0.572640i	$-0.194081\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−27.4802	−1.22528	−0.612641	−	0.790361i	$-0.709893\pi$
$503$	−27.4802	−1.22528	−0.612641	+	0.790361i	$0.709893\pi$
$504$	0	0
$505$	5.19753	0.231287
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 12.7772i	− 0.566340i	−0.959070	−	0.283170i	$-0.908614\pi$
$509$	− 12.7772i	− 0.566340i	0.959070	−	0.283170i	$-0.0913861\pi$
$510$	0	0
$511$	38.5647i	1.70600i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.04148	−0.0458932
$516$	0	0
$517$	−8.76926	−0.385672
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 34.2892i	− 1.50224i	−0.660167	−	0.751119i	$-0.729514\pi$
$521$	− 34.2892i	− 1.50224i	0.660167	−	0.751119i	$-0.270486\pi$
$522$	0	0
$523$	− 10.8135i	− 0.472842i	−0.971651	−	0.236421i	$-0.924026\pi$
$523$	− 10.8135i	− 0.472842i	0.971651	−	0.236421i	$-0.0759744\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.59544	0.243741
$528$	0	0
$529$	−22.7726	−0.990112
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.32046i	0.187140i
$534$	0	0
$535$	− 8.66437i	− 0.374593i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.124259	−0.00535220
$540$	0	0
$541$	22.5988	0.971597	0.485798	−	0.874071i	$-0.338529\pi$
$541$	22.5988	0.971597	0.485798	+	0.874071i	$0.338529\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	31.7454i	1.35982i
$546$	0	0
$547$	− 20.6148i	− 0.881423i	−0.897649	−	0.440712i	$-0.854726\pi$
$547$	− 20.6148i	− 0.881423i	0.897649	−	0.440712i	$-0.145274\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.5409	0.449057
$552$	0	0
$553$	−32.0211	−1.36168
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.6334i	0.450553i	0.974295	+	0.225276i	$0.0723285\pi$
$557$	10.6334i	0.450553i	−0.974295	+	0.225276i	$0.927672\pi$
$558$	0	0
$559$	− 38.8226i	− 1.64202i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.7301	1.84300	0.921502	−	0.388373i	$-0.126963\pi$
$563$	43.7301	1.84300	0.921502	+	0.388373i	$0.126963\pi$
$564$	0	0
$565$	−4.91960	−0.206969
$566$	0	0
$567$	0	0
$568$	0	0
$569$	33.6598i	1.41109i	0.708665	+	0.705545i	$0.249298\pi$
$569$	33.6598i	1.41109i	−0.708665	+	0.705545i	$0.750702\pi$
$570$	0	0
$571$	30.7849i	1.28831i	0.764896	+	0.644153i	$0.222790\pi$
$571$	30.7849i	1.28831i	−0.764896	+	0.644153i	$0.777210\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.01302	0.0422459
$576$	0	0
$577$	35.8651	1.49308	0.746541	−	0.665340i	$-0.231713\pi$
$577$	35.8651	1.49308	0.746541	+	0.665340i	$0.231713\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 13.0140i	− 0.539911i
$582$	0	0
$583$	4.25907i	0.176393i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.5499	−1.88005	−0.940023	−	0.341110i	$-0.889197\pi$
$587$	−45.5499	−1.88005	−0.940023	+	0.341110i	$0.889197\pi$
$588$	0	0
$589$	3.65070	0.150425
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 31.3880i	− 1.28895i	−0.764625	−	0.644476i	$-0.777076\pi$
$593$	− 31.3880i	− 1.28895i	0.764625	−	0.644476i	$-0.222924\pi$
$594$	0	0
$595$	− 33.2265i	− 1.36215i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.6353	0.883995	0.441997	−	0.897016i	$-0.354270\pi$
$599$	21.6353	0.883995	0.441997	+	0.897016i	$0.354270\pi$
$600$	0	0
$601$	−15.0901	−0.615538	−0.307769	−	0.951461i	$-0.599582\pi$
$601$	−15.0901	−0.615538	−0.307769	+	0.951461i	$0.599582\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.66913i	0.108516i
$606$	0	0
$607$	− 20.3067i	− 0.824224i	−0.911133	−	0.412112i	$-0.864791\pi$
$607$	− 20.3067i	− 0.824224i	0.911133	−	0.412112i	$-0.135209\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.5793	−1.27756
$612$	0	0
$613$	−6.94806	−0.280630	−0.140315	−	0.990107i	$-0.544811\pi$
$613$	−6.94806	−0.280630	−0.140315	+	0.990107i	$0.544811\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.68782i	− 0.309500i	−0.987954	−	0.154750i	$-0.950543\pi$
$617$	− 7.68782i	− 0.309500i	0.987954	−	0.154750i	$-0.0494572\pi$
$618$	0	0
$619$	14.4109i	0.579223i	0.957144	+	0.289612i	$0.0935261\pi$
$619$	14.4109i	0.579223i	−0.957144	+	0.289612i	$0.906474\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−44.0650	−1.76543
$624$	0	0
$625$	−31.1088	−1.24435
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 39.0383i	− 1.55656i
$630$	0	0
$631$	27.5637i	1.09729i	0.836054	+	0.548647i	$0.184857\pi$
$631$	27.5637i	1.09729i	−0.836054	+	0.548647i	$0.815143\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.3680	−0.451126
$636$	0	0
$637$	−0.447473	−0.0177295
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.79972i	0.229075i	0.993419	+	0.114538i	$0.0365386\pi$
$641$	5.79972i	0.229075i	−0.993419	+	0.114538i	$0.963461\pi$
$642$	0	0
$643$	− 21.7835i	− 0.859057i	−0.903053	−	0.429528i	$-0.858680\pi$
$643$	− 21.7835i	− 0.859057i	0.903053	−	0.429528i	$-0.141320\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0285	0.472888	0.236444	−	0.971645i	$-0.424018\pi$
$647$	12.0285	0.472888	0.236444	+	0.971645i	$0.424018\pi$
$648$	0	0
$649$	−12.4484	−0.488643
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 4.11285i	− 0.160948i	−0.996757	−	0.0804740i	$-0.974357\pi$
$653$	− 4.11285i	− 0.160948i	0.996757	−	0.0804740i	$-0.0256434\pi$
$654$	0	0
$655$	27.6799i	1.08155i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.40456	0.249486	0.124743	−	0.992189i	$-0.460189\pi$
$659$	6.40456	0.249486	0.124743	+	0.992189i	$0.460189\pi$
$660$	0	0
$661$	−2.86605	−0.111476	−0.0557381	−	0.998445i	$-0.517751\pi$
$661$	−2.86605	−0.111476	−0.0557381	+	0.998445i	$0.517751\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 21.6784i	− 0.840651i
$666$	0	0
$667$	− 1.65197i	− 0.0639644i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.398859	−0.0153978
$672$	0	0
$673$	−48.6400	−1.87493	−0.937467	−	0.348073i	$-0.886836\pi$
$673$	−48.6400	−1.87493	−0.937467	+	0.348073i	$0.886836\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.1231i	1.31146i	0.754997	+	0.655728i	$0.227638\pi$
$677$	34.1231i	1.31146i	−0.754997	+	0.655728i	$0.772362\pi$
$678$	0	0
$679$	8.33905i	0.320024i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.2817	−0.737793	−0.368897	−	0.929470i	$-0.620264\pi$
$683$	−19.2817	−0.737793	−0.368897	+	0.929470i	$0.620264\pi$
$684$	0	0
$685$	50.4695	1.92834
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.3375i	0.584313i
$690$	0	0
$691$	11.4430i	0.435311i	0.976026	+	0.217656i	$0.0698410\pi$
$691$	11.4430i	0.435311i	−0.976026	+	0.217656i	$0.930159\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−45.2720	−1.71726
$696$	0	0
$697$	5.59544	0.211942
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.9782i	1.54772i	0.633354	+	0.773862i	$0.281678\pi$
$701$	40.9782i	1.54772i	−0.633354	+	0.773862i	$0.718322\pi$
$702$	0	0
$703$	− 25.4702i	− 0.960628i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.19753	−0.195473
$708$	0	0
$709$	19.5385	0.733784	0.366892	−	0.930263i	$-0.380422\pi$
$709$	19.5385	0.733784	0.366892	+	0.930263i	$0.380422\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 0.572138i	− 0.0214267i
$714$	0	0
$715$	9.61191i	0.359465i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.7776	−0.625698	−0.312849	−	0.949803i	$-0.601283\pi$
$719$	−16.7776	−0.625698	−0.312849	+	0.949803i	$0.601283\pi$
$720$	0	0
$721$	1.04148	0.0387868
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.35865i	− 0.273293i
$726$	0	0
$727$	− 5.62248i	− 0.208526i	−0.994550	−	0.104263i	$-0.966752\pi$
$727$	− 5.62248i	− 0.208526i	0.994550	−	0.104263i	$-0.0332484\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−50.2793	−1.85965
$732$	0	0
$733$	16.3906	0.605399	0.302699	−	0.953086i	$-0.402112\pi$
$733$	16.3906	0.605399	0.302699	+	0.953086i	$0.402112\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11.9494i	− 0.440161i
$738$	0	0
$739$	− 35.7871i	− 1.31645i	−0.752821	−	0.658225i	$-0.771307\pi$
$739$	− 35.7871i	− 1.31645i	0.752821	−	0.658225i	$-0.228693\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	46.5914	1.70927	0.854636	−	0.519227i	$-0.173780\pi$
$743$	46.5914	1.70927	0.854636	+	0.519227i	$0.173780\pi$
$744$	0	0
$745$	−29.4508	−1.07899
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.66437i	0.316589i
$750$	0	0
$751$	− 24.9523i	− 0.910522i	−0.890358	−	0.455261i	$-0.849546\pi$
$751$	− 24.9523i	− 0.910522i	0.890358	−	0.455261i	$-0.150454\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	44.0211	1.60209
$756$	0	0
$757$	10.4629	0.380280	0.190140	−	0.981757i	$-0.439106\pi$
$757$	10.4629	0.380280	0.190140	+	0.981757i	$0.439106\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.1461i	1.02030i	0.860087	+	0.510148i	$0.170409\pi$
$761$	28.1461i	1.02030i	−0.860087	+	0.510148i	$0.829591\pi$
$762$	0	0
$763$	− 31.7454i	− 1.14926i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−44.8285	−1.61866
$768$	0	0
$769$	−37.2331	−1.34266	−0.671330	−	0.741159i	$-0.734276\pi$
$769$	−37.2331	−1.34266	−0.671330	+	0.741159i	$0.734276\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35.9285i	1.29226i	0.763229	+	0.646128i	$0.223613\pi$
$773$	35.9285i	1.29226i	−0.763229	+	0.646128i	$0.776387\pi$
$774$	0	0
$775$	− 2.54857i	− 0.0915474i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.65070	0.130800
$780$	0	0
$781$	−0.476882	−0.0170642
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 63.2439i	− 2.25727i
$786$	0	0
$787$	− 26.9745i	− 0.961538i	−0.876847	−	0.480769i	$-0.840358\pi$
$787$	− 26.9745i	− 0.961538i	0.876847	−	0.480769i	$-0.159642\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.91960	0.174921
$792$	0	0
$793$	−1.43635	−0.0510062
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.6748i	− 0.874027i	−0.899455	−	0.437013i	$-0.856036\pi$
$797$	− 24.6748i	− 0.874027i	0.899455	−	0.437013i	$-0.143964\pi$
$798$	0	0
$799$	40.8985i	1.44688i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−14.4484	−0.509874
$804$	0	0
$805$	−3.39743	−0.119744
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 22.4908i	− 0.790733i	−0.918523	−	0.395367i	$-0.870618\pi$
$809$	− 22.4908i	− 0.790733i	0.918523	−	0.395367i	$-0.129382\pi$
$810$	0	0
$811$	− 16.4549i	− 0.577810i	−0.957358	−	0.288905i	$-0.906709\pi$
$811$	− 16.4549i	− 0.577810i	0.957358	−	0.288905i	$-0.0932912\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	32.6531	1.14379
$816$	0	0
$817$	−32.8044	−1.14768
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 50.2327i	− 1.75313i	−0.481279	−	0.876567i	$-0.659828\pi$
$821$	− 50.2327i	− 1.75313i	0.481279	−	0.876567i	$-0.340172\pi$
$822$	0	0
$823$	36.3546i	1.26724i	0.773644	+	0.633620i	$0.218432\pi$
$823$	36.3546i	1.26724i	−0.773644	+	0.633620i	$0.781568\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.6059	1.27291	0.636455	−	0.771314i	$-0.280400\pi$
$827$	36.6059	1.27291	0.636455	+	0.771314i	$0.280400\pi$
$828$	0	0
$829$	−49.2331	−1.70994	−0.854968	−	0.518681i	$-0.826423\pi$
$829$	−49.2331	−1.70994	−0.854968	+	0.518681i	$0.826423\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.579524i	0.0200793i
$834$	0	0
$835$	− 4.19769i	− 0.145267i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−49.3737	−1.70457	−0.852285	−	0.523077i	$-0.824784\pi$
$839$	−49.3737	−1.70457	−0.852285	+	0.523077i	$0.824784\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 0.0848421i	− 0.00291866i
$846$	0	0
$847$	− 2.66913i	− 0.0917124i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.99169	−0.136833
$852$	0	0
$853$	44.7334	1.53164	0.765822	−	0.643053i	$-0.222332\pi$
$853$	44.7334	1.53164	0.765822	+	0.643053i	$0.222332\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.23487i	0.212979i	0.994314	+	0.106489i	$0.0339611\pi$
$857$	6.23487i	0.212979i	−0.994314	+	0.106489i	$0.966039\pi$
$858$	0	0
$859$	4.40096i	0.150159i	0.997178	+	0.0750793i	$0.0239210\pi$
$859$	4.40096i	0.150159i	−0.997178	+	0.0750793i	$0.976079\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.7693	−1.11548	−0.557739	−	0.830016i	$-0.688331\pi$
$863$	−32.7693	−1.11548	−0.557739	+	0.830016i	$0.688331\pi$
$864$	0	0
$865$	−15.6507	−0.532140
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 11.9968i	− 0.406965i
$870$	0	0
$871$	− 43.0314i	− 1.45806i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	20.4875	0.692605
$876$	0	0
$877$	−18.8034	−0.634946	−0.317473	−	0.948267i	$-0.602834\pi$
$877$	−18.8034	−0.634946	−0.317473	+	0.948267i	$0.602834\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 22.7518i	− 0.766528i	−0.923639	−	0.383264i	$-0.874800\pi$
$881$	− 22.7518i	− 0.766528i	0.923639	−	0.383264i	$-0.125200\pi$
$882$	0	0
$883$	28.8394i	0.970525i	0.874369	+	0.485262i	$0.161276\pi$
$883$	28.8394i	0.970525i	−0.874369	+	0.485262i	$0.838724\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.54920	0.219901	0.109950	−	0.993937i	$-0.464931\pi$
$887$	6.54920	0.219901	0.109950	+	0.993937i	$0.464931\pi$
$888$	0	0
$889$	11.3680	0.381271
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.6839i	0.892943i
$894$	0	0
$895$	67.8018i	2.26636i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.15605	−0.138612
$900$	0	0
$901$	19.8637	0.661755
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 19.7960i	− 0.658040i
$906$	0	0
$907$	8.70747i	0.289127i	0.989496	+	0.144563i	$0.0461778\pi$
$907$	8.70747i	0.289127i	−0.989496	+	0.144563i	$0.953822\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−50.6685	−1.67872	−0.839361	−	0.543575i	$-0.817070\pi$
$911$	−50.6685	−1.67872	−0.839361	+	0.543575i	$0.817070\pi$
$912$	0	0
$913$	4.87574	0.161363
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 27.6799i	− 0.914072i
$918$	0	0
$919$	33.2878i	1.09806i	0.835801	+	0.549032i	$0.185004\pi$
$919$	33.2878i	1.09806i	−0.835801	+	0.549032i	$0.814996\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.71732	−0.0565262
$924$	0	0
$925$	−17.7809	−0.584632
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 43.3758i	− 1.42311i	−0.702628	−	0.711557i	$-0.747990\pi$
$929$	− 43.3758i	− 1.42311i	0.702628	−	0.711557i	$-0.252010\pi$
$930$	0	0
$931$	0.378106i	0.0123919i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.4484	0.407107
$936$	0	0
$937$	5.70762	0.186460	0.0932300	−	0.995645i	$-0.470281\pi$
$937$	5.70762	0.186460	0.0932300	+	0.995645i	$0.470281\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	14.9576i	0.487603i	0.969825	+	0.243801i	$0.0783945\pi$
$941$	14.9576i	0.487603i	−0.969825	+	0.243801i	$0.921606\pi$
$942$	0	0
$943$	− 0.572138i	− 0.0186314i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.15605	−0.135053	−0.0675267	−	0.997717i	$-0.521511\pi$
$947$	−4.15605	−0.135053	−0.0675267	+	0.997717i	$0.521511\pi$
$948$	0	0
$949$	−52.0308	−1.68899
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 4.95177i	− 0.160404i	−0.996779	−	0.0802018i	$-0.974444\pi$
$953$	− 4.95177i	− 0.160404i	0.996779	−	0.0802018i	$-0.0255565\pi$
$954$	0	0
$955$	62.6343i	2.02680i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−50.4695	−1.62975
$960$	0	0
$961$	29.5606	0.953568
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.5631i	1.04824i
$966$	0	0
$967$	0.269637i	0.00867094i	0.999991	+	0.00433547i	$0.00138003\pi$
$967$	0.269637i	0.00867094i	−0.999991	+	0.00433547i	$0.998620\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.5385	−1.33303	−0.666517	−	0.745490i	$-0.732216\pi$
$971$	−41.5385	−1.33303	−0.666517	+	0.745490i	$0.732216\pi$
$972$	0	0
$973$	45.2720	1.45135
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.98870i	0.0956171i	0.998857	+	0.0478086i	$0.0152237\pi$
$977$	2.98870i	0.0956171i	−0.998857	+	0.0478086i	$0.984776\pi$
$978$	0	0
$979$	− 16.5091i	− 0.527633i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.1608	0.674923	0.337462	−	0.941339i	$-0.390432\pi$
$983$	21.1608	0.674923	0.337462	+	0.941339i	$0.390432\pi$
$984$	0	0
$985$	33.6946	1.07360
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.14110i	0.163477i
$990$	0	0
$991$	− 4.23340i	− 0.134479i	−0.997737	−	0.0672393i	$-0.978581\pi$
$991$	− 4.23340i	− 0.134479i	0.997737	−	0.0672393i	$-0.0214191\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	43.8945	1.39155
$996$	0	0
$997$	−42.3289	−1.34057	−0.670284	−	0.742104i	$-0.733828\pi$
$997$	−42.3289	−1.34057	−0.670284	+	0.742104i	$0.733828\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.d.e.3455.7		8
3.2	odd	2		6336.2.d.c.3455.2		8
4.3	odd	2		6336.2.d.c.3455.7		8
8.3	odd	2		1584.2.d.d.287.2	yes	8
8.5	even	2		1584.2.d.c.287.2	✓	8
12.11	even	2	inner	6336.2.d.e.3455.2		8
24.5	odd	2		1584.2.d.d.287.7	yes	8
24.11	even	2		1584.2.d.c.287.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1584.2.d.c.287.2	✓	8	8.5	even	2
1584.2.d.c.287.7	yes	8	24.11	even	2
1584.2.d.d.287.2	yes	8	8.3	odd	2
1584.2.d.d.287.7	yes	8	24.5	odd	2
6336.2.d.c.3455.2		8	3.2	odd	2
6336.2.d.c.3455.7		8	4.3	odd	2
6336.2.d.e.3455.2		8	12.11	even	2	inner
6336.2.d.e.3455.7		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.66913i q^{5} -2.66913i q^{7} +1.00000 q^{11} +3.60114 q^{13} -4.66385i q^{17} -3.04289i q^{19} -0.476882 q^{23} -2.12426 q^{25} +3.46410i q^{29} +1.19975i q^{31} +7.12426 q^{35} +8.37040 q^{37} +1.19975i q^{41} -10.7806i q^{43} -8.76926 q^{47} -0.124259 q^{49} +4.25907i q^{53} +2.66913i q^{55} -12.4484 q^{59} -0.398859 q^{61} +9.61191i q^{65} -11.9494i q^{67} -0.476882 q^{71} -14.4484 q^{73} -2.66913i q^{77} -11.9968i q^{79} +4.87574 q^{83} +12.4484 q^{85} -16.5091i q^{89} -9.61191i q^{91} +8.12188 q^{95} -3.12426 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 16 q^{25} + 56 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{59} - 24 q^{61} + 16 q^{71} + 40 q^{83} - 16 q^{85} - 8 q^{95} - 24 q^{97}+O(q^{100}) \) 8 * q + 8 * q^11 + 8 * q^13 + 16 * q^23 - 16 * q^25 + 56 * q^35 - 8 * q^37 - 16 * q^47 + 16 * q^59 - 24 * q^61 + 16 * q^71 + 40 * q^83 - 16 * q^85 - 8 * q^95 - 24 * q^97

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