LMFDB - Embedded newform 6336.2.d.e.3455.1

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			3455.1
Root			$-0.943680 - 1.63450i$ of defining polynomial
Character	$\chi$	$=$	6336.3455
Dual form			6336.2.d.e.3455.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.08334i q^{5} +4.08334i q^{7} +1.00000 q^{11} +5.74733 q^{13} -1.62868i q^{17} -6.07806i q^{19} +6.92637 q^{23} -11.6737 q^{25} +3.46410i q^{29} -1.83542i q^{31} +16.6737 q^{35} -5.47142 q^{37} -1.83542i q^{41} +5.75947i q^{43} +7.21875 q^{47} -9.67370 q^{49} +11.0115i q^{53} -4.08334i q^{55} +6.65046 q^{59} +1.74733 q^{61} -23.4683i q^{65} -11.9494i q^{67} +6.92637 q^{71} +4.65046 q^{73} +4.08334i q^{77} +0.825984i q^{79} -4.67370 q^{83} -6.65046 q^{85} -3.00416i q^{89} +23.4683i q^{91} -24.8188 q^{95} -12.6737 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 4.08334i	− 1.82613i	−0.407817	−	0.913064i	$-0.633710\pi$
$5$	− 4.08334i	− 1.82613i	0.407817	−	0.913064i	$-0.366290\pi$
$6$	0	0
$7$	4.08334i	1.54336i	0.636012	+	0.771680i	$0.280583\pi$
$7$	4.08334i	1.54336i	−0.636012	+	0.771680i	$0.719417\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.74733	1.59402	0.797011	−	0.603965i	$-0.206413\pi$
$13$	5.74733	1.59402	0.797011	+	0.603965i	$0.206413\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.62868i	− 0.395013i	−0.980302	−	0.197506i	$-0.936716\pi$
$17$	− 1.62868i	− 0.395013i	0.980302	−	0.197506i	$-0.0632844\pi$
$18$	0	0
$19$	− 6.07806i	− 1.39440i	−0.716875	−	0.697202i	$-0.754428\pi$
$19$	− 6.07806i	− 1.39440i	0.716875	−	0.697202i	$-0.245572\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.92637	1.44425	0.722124	−	0.691763i	$-0.243166\pi$
$23$	6.92637	1.44425	0.722124	+	0.691763i	$0.243166\pi$
$24$	0	0
$25$	−11.6737	−2.33474
$26$	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
$31$	− 1.83542i	− 0.329651i	−0.986323	−	0.164826i	$-0.947294\pi$
$31$	− 1.83542i	− 0.329651i	0.986323	−	0.164826i	$-0.0527062\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.6737	2.81837
$36$	0	0
$37$	−5.47142	−0.899496	−0.449748	−	0.893156i	$-0.648486\pi$
$37$	−5.47142	−0.899496	−0.449748	+	0.893156i	$0.648486\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.83542i	− 0.286645i	−0.989676	−	0.143322i	$-0.954221\pi$
$41$	− 1.83542i	− 0.286645i	0.989676	−	0.143322i	$-0.0457786\pi$
$42$	0	0
$43$	5.75947i	0.878311i	0.898411	+	0.439155i	$0.144722\pi$
$43$	5.75947i	0.878311i	−0.898411	+	0.439155i	$0.855278\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.21875	1.05296	0.526481	−	0.850187i	$-0.323511\pi$
$47$	7.21875	1.05296	0.526481	+	0.850187i	$0.323511\pi$
$48$	0	0
$49$	−9.67370	−1.38196
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.0115i	1.51255i	0.654252	+	0.756276i	$0.272983\pi$
$53$	11.0115i	1.51255i	−0.654252	+	0.756276i	$0.727017\pi$
$54$	0	0
$55$	− 4.08334i	− 0.550598i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.65046	0.865816	0.432908	−	0.901438i	$-0.357487\pi$
$59$	6.65046	0.865816	0.432908	+	0.901438i	$0.357487\pi$
$60$	0	0
$61$	1.74733	0.223723	0.111861	−	0.993724i	$-0.464319\pi$
$61$	1.74733	0.223723	0.111861	+	0.993724i	$0.464319\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 23.4683i	− 2.91089i
$66$	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
$71$	6.92637	0.822009	0.411005	−	0.911633i	$-0.365178\pi$
$71$	6.92637	0.822009	0.411005	+	0.911633i	$0.365178\pi$
$72$	0	0
$73$	4.65046	0.544295	0.272148	−	0.962255i	$-0.412266\pi$
$73$	4.65046	0.544295	0.272148	+	0.962255i	$0.412266\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.08334i	0.465340i
$78$	0	0
$79$	0.825984i	0.0929304i	0.998920	+	0.0464652i	$0.0147957\pi$
$79$	0.825984i	0.0929304i	−0.998920	+	0.0464652i	$0.985204\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.67370	−0.513005	−0.256503	−	0.966544i	$-0.582570\pi$
$83$	−4.67370	−0.513005	−0.256503	+	0.966544i	$0.582570\pi$
$84$	0	0
$85$	−6.65046	−0.721344
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.00416i	− 0.318440i	−0.987243	−	0.159220i	$-0.949102\pi$
$89$	− 3.00416i	− 0.318440i	0.987243	−	0.159220i	$-0.0508979\pi$
$90$	0	0
$91$	23.4683i	2.46015i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−24.8188	−2.54636
$96$	0	0
$97$	−12.6737	−1.28682	−0.643410	−	0.765522i	$-0.722481\pi$
$97$	−12.6737	−1.28682	−0.643410	+	0.765522i	$0.722481\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 18.4874i	− 1.83956i	−0.392429	−	0.919782i	$-0.628365\pi$
$101$	− 18.4874i	− 1.83956i	0.392429	−	0.919782i	$-0.371635\pi$
$102$	0	0
$103$	16.9303i	1.66819i	0.551618	+	0.834097i	$0.314011\pi$
$103$	16.9303i	1.66819i	−0.551618	+	0.834097i	$0.685989\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.1451	1.94750	0.973751	−	0.227615i	$-0.0730929\pi$
$107$	20.1451	1.94750	0.973751	+	0.227615i	$0.0730929\pi$
$108$	0	0
$109$	5.45495	0.522490	0.261245	−	0.965273i	$-0.415867\pi$
$109$	5.45495	0.522490	0.261245	+	0.965273i	$0.415867\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.91348i	0.744438i	0.928145	+	0.372219i	$0.121403\pi$
$113$	7.91348i	0.744438i	−0.928145	+	0.372219i	$0.878597\pi$
$114$	0	0
$115$	− 28.2828i	− 2.63738i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.65046	0.609647
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	27.2510i	2.43740i
$126$	0	0
$127$	11.0115i	0.977117i	0.872531	+	0.488558i	$0.162477\pi$
$127$	11.0115i	0.977117i	−0.872531	+	0.488558i	$0.837523\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.47142	−0.303299	−0.151650	−	0.988434i	$-0.548459\pi$
$131$	−3.47142	−0.303299	−0.151650	+	0.988434i	$0.548459\pi$
$132$	0	0
$133$	24.8188	2.15206
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.666687i	0.0569589i	0.999594	+	0.0284795i	$0.00906652\pi$
$137$	0.666687i	0.0569589i	−0.999594	+	0.0284795i	$0.990933\pi$
$138$	0	0
$139$	− 19.1541i	− 1.62463i	−0.583221	−	0.812314i	$-0.698208\pi$
$139$	− 19.1541i	− 1.62463i	0.583221	−	0.812314i	$-0.301792\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.74733	0.480616
$144$	0	0
$145$	14.1451	1.17469
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 12.9409i	− 1.06016i	−0.847948	−	0.530079i	$-0.822162\pi$
$149$	− 12.9409i	− 1.06016i	0.847948	−	0.530079i	$-0.177838\pi$
$150$	0	0
$151$	3.76475i	0.306371i	0.988197	+	0.153186i	$0.0489532\pi$
$151$	3.76475i	0.306371i	−0.988197	+	0.153186i	$0.951047\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.49466	−0.601985
$156$	0	0
$157$	18.7956	1.50005	0.750025	−	0.661409i	$-0.230041\pi$
$157$	18.7956	1.50005	0.750025	+	0.661409i	$0.230041\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	28.2828i	2.22899i
$162$	0	0
$163$	14.7763i	1.15737i	0.815551	+	0.578685i	$0.196434\pi$
$163$	14.7763i	1.15737i	−0.815551	+	0.578685i	$0.803566\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.97676	0.617260	0.308630	−	0.951182i	$-0.400129\pi$
$167$	7.97676	0.617260	0.308630	+	0.951182i	$0.400129\pi$
$168$	0	0
$169$	20.0318	1.54091
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 0.206741i	− 0.0157182i	−0.999969	−	0.00785912i	$-0.997498\pi$
$173$	− 0.206741i	− 0.0157182i	0.999969	−	0.00785912i	$-0.00250166\pi$
$174$	0	0
$175$	− 47.6677i	− 3.60334i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.50321	−0.635559	−0.317780	−	0.948165i	$-0.602937\pi$
$179$	−8.50321	−0.635559	−0.317780	+	0.948165i	$0.602937\pi$
$180$	0	0
$181$	−8.38132	−0.622979	−0.311489	−	0.950250i	$-0.600828\pi$
$181$	−8.38132	−0.622979	−0.311489	+	0.950250i	$0.600828\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	22.3417i	1.64259i
$186$	0	0
$187$	− 1.62868i	− 0.119101i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.47819	0.541103	0.270551	−	0.962706i	$-0.412794\pi$
$191$	7.47819	0.541103	0.270551	+	0.962706i	$0.412794\pi$
$192$	0	0
$193$	−25.9979	−1.87137	−0.935684	−	0.352840i	$-0.885216\pi$
$193$	−25.9979	−1.87137	−0.935684	+	0.352840i	$0.885216\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.15401i	− 0.153467i	−0.997052	−	0.0767336i	$-0.975551\pi$
$197$	− 2.15401i	− 0.153467i	0.997052	−	0.0767336i	$-0.0244491\pi$
$198$	0	0
$199$	− 9.01062i	− 0.638746i	−0.947629	−	0.319373i	$-0.896528\pi$
$199$	− 9.01062i	− 0.638746i	0.947629	−	0.319373i	$-0.103472\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−14.1451	−0.992793
$204$	0	0
$205$	−7.49466	−0.523450
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6.07806i	− 0.420428i
$210$	0	0
$211$	− 11.8124i	− 0.813199i	−0.913607	−	0.406599i	$-0.866715\pi$
$211$	− 11.8124i	− 0.813199i	0.913607	−	0.406599i	$-0.133285\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	23.5179	1.60391
$216$	0	0
$217$	7.49466	0.508770
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 9.36056i	− 0.629659i
$222$	0	0
$223$	− 16.5401i	− 1.10761i	−0.832647	−	0.553804i	$-0.813176\pi$
$223$	− 16.5401i	− 1.10761i	0.832647	−	0.553804i	$-0.186824\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.84420	0.454265	0.227133	−	0.973864i	$-0.427065\pi$
$227$	6.84420	0.454265	0.227133	+	0.973864i	$0.427065\pi$
$228$	0	0
$229$	−5.49466	−0.363097	−0.181549	−	0.983382i	$-0.558111\pi$
$229$	−5.49466	−0.363097	−0.181549	+	0.983382i	$0.558111\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 8.76362i	− 0.574124i	−0.957912	−	0.287062i	$-0.907321\pi$
$233$	− 8.76362i	− 0.574124i	0.957912	−	0.287062i	$-0.0926786\pi$
$234$	0	0
$235$	− 29.4766i	− 1.92284i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.32416	−0.344391	−0.172196	−	0.985063i	$-0.555086\pi$
$239$	−5.32416	−0.344391	−0.172196	+	0.985063i	$0.555086\pi$
$240$	0	0
$241$	−1.44818	−0.0932855	−0.0466428	−	0.998912i	$-0.514852\pi$
$241$	−1.44818	−0.0932855	−0.0466428	+	0.998912i	$0.514852\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	39.5010i	2.52363i
$246$	0	0
$247$	− 34.9326i	− 2.22271i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.0550	1.32898	0.664491	−	0.747297i	$-0.268649\pi$
$251$	21.0550	1.32898	0.664491	+	0.747297i	$0.268649\pi$
$252$	0	0
$253$	6.92637	0.435457
$254$	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
$263$	−17.8760	−1.10228	−0.551140	−	0.834413i	$-0.685807\pi$
$263$	−17.8760	−1.10228	−0.551140	+	0.834413i	$0.685807\pi$
$264$	0	0
$265$	44.9639	2.76211
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 19.4968i	− 1.18874i	−0.804191	−	0.594371i	$-0.797401\pi$
$269$	− 19.4968i	− 1.18874i	0.804191	−	0.594371i	$-0.202599\pi$
$270$	0	0
$271$	− 15.8260i	− 0.961360i	−0.876896	−	0.480680i	$-0.840390\pi$
$271$	− 15.8260i	− 0.961360i	0.876896	−	0.480680i	$-0.159610\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.6737	−0.703951
$276$	0	0
$277$	10.2527	0.616023	0.308012	−	0.951383i	$-0.400336\pi$
$277$	10.2527	0.616023	0.308012	+	0.951383i	$0.400336\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 23.1900i	− 1.38340i	−0.722186	−	0.691699i	$-0.756862\pi$
$281$	− 23.1900i	− 1.38340i	0.722186	−	0.691699i	$-0.243138\pi$
$282$	0	0
$283$	− 2.08863i	− 0.124156i	−0.998071	−	0.0620780i	$-0.980227\pi$
$283$	− 2.08863i	− 0.124156i	0.998071	−	0.0620780i	$-0.0197727\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.49466	0.442396
$288$	0	0
$289$	14.3474	0.843965
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.44523i	0.0844310i	0.999109	+	0.0422155i	$0.0134416\pi$
$293$	1.44523i	0.0844310i	−0.999109	+	0.0422155i	$0.986558\pi$
$294$	0	0
$295$	− 27.1561i	− 1.58109i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	39.8081	2.30216
$300$	0	0
$301$	−23.5179	−1.35555
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 7.13494i	− 0.408546i
$306$	0	0
$307$	− 0.388344i	− 0.0221640i	−0.999939	−	0.0110820i	$-0.996472\pi$
$307$	− 0.388344i	− 0.0221640i	0.999939	−	0.0110820i	$-0.00352758\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.9264	1.75367	0.876837	−	0.480788i	$-0.159649\pi$
$311$	30.9264	1.75367	0.876837	+	0.480788i	$0.159649\pi$
$312$	0	0
$313$	1.64191	0.0928065	0.0464032	−	0.998923i	$-0.485224\pi$
$313$	1.64191	0.0928065	0.0464032	+	0.998923i	$0.485224\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.8260i	− 0.888876i	−0.895810	−	0.444438i	$-0.853403\pi$
$317$	− 15.8260i	− 0.888876i	0.895810	−	0.444438i	$-0.146597\pi$
$318$	0	0
$319$	3.46410i	0.193952i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−9.89922	−0.550807
$324$	0	0
$325$	−67.0926	−3.72163
$326$	0	0
$327$	0	0
$328$	0	0
$329$	29.4766i	1.62510i
$330$	0	0
$331$	21.1478i	1.16239i	0.813765	+	0.581195i	$0.197414\pi$
$331$	21.1478i	1.16239i	−0.813765	+	0.581195i	$0.802586\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−48.7934	−2.66587
$336$	0	0
$337$	13.7055	0.746585	0.373293	−	0.927714i	$-0.378229\pi$
$337$	13.7055	0.746585	0.373293	+	0.927714i	$0.378229\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.83542i	− 0.0993936i
$342$	0	0
$343$	− 10.9176i	− 0.589497i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.0783	−1.66837	−0.834184	−	0.551486i	$-0.814061\pi$
$347$	−31.0783	−1.66837	−0.834184	+	0.551486i	$0.814061\pi$
$348$	0	0
$349$	−19.6001	−1.04917	−0.524584	−	0.851359i	$-0.675779\pi$
$349$	−19.6001	−1.04917	−0.524584	+	0.851359i	$0.675779\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.92405i	0.208856i	0.994532	+	0.104428i	$0.0333012\pi$
$353$	3.92405i	0.208856i	−0.994532	+	0.104428i	$0.966699\pi$
$354$	0	0
$355$	− 28.2828i	− 1.50109i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.2001	−1.22446	−0.612228	−	0.790681i	$-0.709727\pi$
$359$	−23.2001	−1.22446	−0.612228	+	0.790681i	$0.709727\pi$
$360$	0	0
$361$	−17.9428	−0.944360
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 18.9894i	− 0.993953i
$366$	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
$371$	−44.9639	−2.33441
$372$	0	0
$373$	−0.105415	−0.00545817	−0.00272908	−	0.999996i	$-0.500869\pi$
$373$	−0.105415	−0.00545817	−0.00272908	+	0.999996i	$0.500869\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.9093i	1.02538i
$378$	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
$383$	−1.12010	−0.0572347	−0.0286173	−	0.999590i	$-0.509110\pi$
$383$	−1.12010	−0.0572347	−0.0286173	+	0.999590i	$0.509110\pi$
$384$	0	0
$385$	16.6737	0.849770
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 27.4398i	− 1.39125i	−0.718403	−	0.695627i	$-0.755127\pi$
$389$	− 27.4398i	− 1.39125i	0.718403	−	0.695627i	$-0.244873\pi$
$390$	0	0
$391$	− 11.2808i	− 0.570497i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.37278	0.169703
$396$	0	0
$397$	22.2670	1.11755	0.558774	−	0.829320i	$-0.311272\pi$
$397$	22.2670	1.11755	0.558774	+	0.829320i	$0.311272\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 28.2809i	− 1.41228i	−0.708072	−	0.706141i	$-0.750435\pi$
$401$	− 28.2809i	− 1.41228i	0.708072	−	0.706141i	$-0.249565\pi$
$402$	0	0
$403$	− 10.5488i	− 0.525472i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.47142	−0.271208
$408$	0	0
$409$	−15.5933	−0.771039	−0.385520	−	0.922700i	$-0.625978\pi$
$409$	−15.5933	−0.771039	−0.385520	+	0.922700i	$0.625978\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	27.1561i	1.33627i
$414$	0	0
$415$	19.0843i	0.936813i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.3009	0.649793	0.324896	−	0.945750i	$-0.394671\pi$
$419$	13.3009	0.649793	0.324896	+	0.945750i	$0.394671\pi$
$420$	0	0
$421$	−28.4607	−1.38709	−0.693546	−	0.720413i	$-0.743952\pi$
$421$	−28.4607	−1.38709	−0.693546	+	0.720413i	$0.743952\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	19.0127i	0.922253i
$426$	0	0
$427$	7.13494i	0.345284i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.57506	−0.220373	−0.110186	−	0.993911i	$-0.535145\pi$
$431$	−4.57506	−0.220373	−0.110186	+	0.993911i	$0.535145\pi$
$432$	0	0
$433$	−7.70549	−0.370302	−0.185151	−	0.982710i	$-0.559277\pi$
$433$	−7.70549	−0.370302	−0.185151	+	0.982710i	$0.559277\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 42.0989i	− 2.01386i
$438$	0	0
$439$	20.0981i	0.959231i	0.877479	+	0.479616i	$0.159224\pi$
$439$	20.0981i	0.959231i	−0.877479	+	0.479616i	$0.840776\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.21083	0.390108	0.195054	−	0.980792i	$-0.437512\pi$
$443$	8.21083	0.390108	0.195054	+	0.980792i	$0.437512\pi$
$444$	0	0
$445$	−12.2670	−0.581512
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 17.3670i	− 0.819598i	−0.912176	−	0.409799i	$-0.865599\pi$
$449$	− 17.3670i	− 0.819598i	0.912176	−	0.409799i	$-0.134401\pi$
$450$	0	0
$451$	− 1.83542i	− 0.0864266i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	95.8292	4.49254
$456$	0	0
$457$	24.9893	1.16895	0.584475	−	0.811411i	$-0.301300\pi$
$457$	24.9893	1.16895	0.584475	+	0.811411i	$0.301300\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 37.3693i	− 1.74046i	−0.492642	−	0.870232i	$-0.663969\pi$
$461$	− 37.3693i	− 1.74046i	0.492642	−	0.870232i	$-0.336031\pi$
$462$	0	0
$463$	28.1665i	1.30901i	0.756058	+	0.654505i	$0.227123\pi$
$463$	28.1665i	1.30901i	−0.756058	+	0.654505i	$0.772877\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.9979	−0.740293	−0.370146	−	0.928973i	$-0.620692\pi$
$467$	−15.9979	−0.740293	−0.370146	+	0.928973i	$0.620692\pi$
$468$	0	0
$469$	48.7934	2.25307
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.75947i	0.264821i
$474$	0	0
$475$	70.9535i	3.25557i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.2234	1.24387	0.621934	−	0.783070i	$-0.286347\pi$
$479$	27.2234	1.24387	0.621934	+	0.783070i	$0.286347\pi$
$480$	0	0
$481$	−31.4460	−1.43382
$482$	0	0
$483$	0	0
$484$	0	0
$485$	51.7511i	2.34990i
$486$	0	0
$487$	− 22.4535i	− 1.01747i	−0.860924	−	0.508734i	$-0.830114\pi$
$487$	− 22.4535i	− 1.01747i	0.860924	−	0.508734i	$-0.169886\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.0986	−0.546004	−0.273002	−	0.962013i	$-0.588017\pi$
$491$	−12.0986	−0.546004	−0.273002	+	0.962013i	$0.588017\pi$
$492$	0	0
$493$	5.64191	0.254099
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.2828i	1.26866i
$498$	0	0
$499$	− 13.4429i	− 0.601788i	−0.953658	−	0.300894i	$-0.902715\pi$
$499$	− 13.4429i	− 0.601788i	0.953658	−	0.300894i	$-0.0972850\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.6822	0.520886	0.260443	−	0.965489i	$-0.416131\pi$
$503$	11.6822	0.520886	0.260443	+	0.965489i	$0.416131\pi$
$504$	0	0
$505$	−75.4904	−3.35928
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 33.0346i	− 1.46423i	−0.681178	−	0.732117i	$-0.738532\pi$
$509$	− 33.0346i	− 1.46423i	0.681178	−	0.732117i	$-0.261468\pi$
$510$	0	0
$511$	18.9894i	0.840043i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	69.1323	3.04633
$516$	0	0
$517$	7.21875	0.317480
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.20898i	− 0.0529665i	−0.999649	−	0.0264833i	$-0.991569\pi$
$521$	− 1.20898i	− 0.0529665i	0.999649	−	0.0264833i	$-0.00843087\pi$
$522$	0	0
$523$	− 7.77834i	− 0.340123i	−0.985433	−	0.170062i	$-0.945603\pi$
$523$	− 7.77834i	− 0.340123i	0.985433	−	0.170062i	$-0.0543967\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.98931	−0.130217
$528$	0	0
$529$	24.9746	1.08585
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 10.5488i	− 0.456918i
$534$	0	0
$535$	− 82.2595i	− 3.55639i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.67370	−0.416676
$540$	0	0
$541$	−17.7452	−0.762925	−0.381463	−	0.924384i	$-0.624580\pi$
$541$	−17.7452	−0.762925	−0.381463	+	0.924384i	$0.624580\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 22.2744i	− 0.954133i
$546$	0	0
$547$	16.8649i	0.721092i	0.932741	+	0.360546i	$0.117410\pi$
$547$	16.8649i	0.721092i	−0.932741	+	0.360546i	$0.882590\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21.0550	0.896974
$552$	0	0
$553$	−3.37278	−0.143425
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.0779i	1.91001i	0.296584	+	0.955007i	$0.404152\pi$
$557$	45.0779i	1.91001i	−0.296584	+	0.955007i	$0.595848\pi$
$558$	0	0
$559$	33.1016i	1.40005i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.46172	0.314474	0.157237	−	0.987561i	$-0.449741\pi$
$563$	7.46172	0.314474	0.157237	+	0.987561i	$0.449741\pi$
$564$	0	0
$565$	32.3135	1.35944
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 17.3249i	− 0.726296i	−0.931731	−	0.363148i	$-0.881702\pi$
$569$	− 17.3249i	− 0.726296i	0.931731	−	0.363148i	$-0.118298\pi$
$570$	0	0
$571$	− 18.8355i	− 0.788241i	−0.919059	−	0.394120i	$-0.871049\pi$
$571$	− 18.8355i	− 0.788241i	0.919059	−	0.394120i	$-0.128951\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−80.8564	−3.37194
$576$	0	0
$577$	17.7309	0.738145	0.369073	−	0.929401i	$-0.379675\pi$
$577$	17.7309	0.738145	0.369073	+	0.929401i	$0.379675\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 19.0843i	− 0.791751i
$582$	0	0
$583$	11.0115i	0.456052i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−35.0358	−1.44608	−0.723041	−	0.690805i	$-0.757256\pi$
$587$	−35.0358	−1.44608	−0.723041	+	0.690805i	$0.757256\pi$
$588$	0	0
$589$	−11.1558	−0.459667
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.6669i	1.05402i	0.849861	+	0.527008i	$0.176686\pi$
$593$	25.6669i	1.05402i	−0.849861	+	0.527008i	$0.823314\pi$
$594$	0	0
$595$	− 27.1561i	− 1.11329i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	46.2081	1.88801	0.944005	−	0.329931i	$-0.107025\pi$
$599$	46.2081	1.88801	0.944005	+	0.329931i	$0.107025\pi$
$600$	0	0
$601$	−2.21297	−0.0902688	−0.0451344	−	0.998981i	$-0.514372\pi$
$601$	−2.21297	−0.0902688	−0.0451344	+	0.998981i	$0.514372\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.08334i	− 0.166012i
$606$	0	0
$607$	− 0.0492955i	− 0.00200084i	−0.999999	−	0.00100042i	$-0.999682\pi$
$607$	− 0.0492955i	− 0.00200084i	0.999999	−	0.00100042i	$-0.000318444\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	41.4885	1.67845
$612$	0	0
$613$	18.5894	0.750818	0.375409	−	0.926859i	$-0.377502\pi$
$613$	18.5894	0.750818	0.375409	+	0.926859i	$0.377502\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.6160i	0.588419i	0.955741	+	0.294209i	$0.0950563\pi$
$617$	14.6160i	0.588419i	−0.955741	+	0.294209i	$0.904944\pi$
$618$	0	0
$619$	− 30.8100i	− 1.23836i	−0.785250	−	0.619179i	$-0.787465\pi$
$619$	− 30.8100i	− 1.23836i	0.785250	−	0.619179i	$-0.212535\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.2670	0.491467
$624$	0	0
$625$	52.9068	2.11627
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.91119i	0.355313i
$630$	0	0
$631$	45.4681i	1.81006i	0.425350	+	0.905029i	$0.360151\pi$
$631$	45.4681i	1.81006i	−0.425350	+	0.905029i	$0.639849\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	44.9639	1.78434
$636$	0	0
$637$	−55.5979	−2.20287
$638$	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
$643$	− 0.843927i	− 0.0332812i	−0.999862	−	0.0166406i	$-0.994703\pi$
$643$	− 0.843927i	− 0.0332812i	0.999862	−	0.0166406i	$-0.00529712\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.7241	0.932690	0.466345	−	0.884603i	$-0.345570\pi$
$647$	23.7241	0.932690	0.466345	+	0.884603i	$0.345570\pi$
$648$	0	0
$649$	6.65046	0.261053
$650$	0	0
$651$	0	0
$652$	0	0
$653$	49.2248i	1.92632i	0.268936	+	0.963158i	$0.413328\pi$
$653$	49.2248i	1.92632i	−0.268936	+	0.963158i	$0.586672\pi$
$654$	0	0
$655$	14.1750i	0.553863i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.9893	0.583901	0.291950	−	0.956433i	$-0.405696\pi$
$659$	14.9893	0.583901	0.291950	+	0.956433i	$0.405696\pi$
$660$	0	0
$661$	−43.4268	−1.68911	−0.844554	−	0.535471i	$-0.820134\pi$
$661$	−43.4268	−1.68911	−0.844554	+	0.535471i	$0.820134\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 101.344i	− 3.92994i
$666$	0	0
$667$	23.9937i	0.929038i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.74733	0.0674549
$672$	0	0
$673$	−25.2488	−0.973268	−0.486634	−	0.873606i	$-0.661775\pi$
$673$	−25.2488	−0.973268	−0.486634	+	0.873606i	$0.661775\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.7236i	1.14237i	0.820821	+	0.571186i	$0.193516\pi$
$677$	29.7236i	1.14237i	−0.820821	+	0.571186i	$0.806484\pi$
$678$	0	0
$679$	− 51.7511i	− 1.98602i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.11219	−0.0808206	−0.0404103	−	0.999183i	$-0.512866\pi$
$683$	−2.11219	−0.0808206	−0.0404103	+	0.999183i	$0.512866\pi$
$684$	0	0
$685$	2.72231	0.104014
$686$	0	0
$687$	0	0
$688$	0	0
$689$	63.2870i	2.41104i
$690$	0	0
$691$	26.3122i	1.00096i	0.865747	+	0.500482i	$0.166844\pi$
$691$	26.3122i	1.00096i	−0.865747	+	0.500482i	$0.833156\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−78.2127	−2.96678
$696$	0	0
$697$	−2.98931	−0.113228
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.53364i	0.246772i	0.992359	+	0.123386i	$0.0393754\pi$
$701$	6.53364i	0.246772i	−0.992359	+	0.123386i	$0.960625\pi$
$702$	0	0
$703$	33.2556i	1.25426i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	75.4904	2.83911
$708$	0	0
$709$	−12.4375	−0.467100	−0.233550	−	0.972345i	$-0.575034\pi$
$709$	−12.4375	−0.467100	−0.233550	+	0.972345i	$0.575034\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 12.7128i	− 0.476098i
$714$	0	0
$715$	− 23.4683i	− 0.877665i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.1158	1.23501	0.617506	−	0.786566i	$-0.288143\pi$
$719$	33.1158	1.23501	0.617506	+	0.786566i	$0.288143\pi$
$720$	0	0
$721$	−69.1323	−2.57462
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 40.4389i	− 1.50186i
$726$	0	0
$727$	34.8924i	1.29409i	0.762453	+	0.647043i	$0.223995\pi$
$727$	34.8924i	1.29409i	−0.762453	+	0.647043i	$0.776005\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.38033	0.346944
$732$	0	0
$733$	48.1498	1.77845	0.889226	−	0.457468i	$-0.151244\pi$
$733$	48.1498	1.77845	0.889226	+	0.457468i	$0.151244\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11.9494i	− 0.440161i
$738$	0	0
$739$	36.1371i	1.32932i	0.747144	+	0.664662i	$0.231424\pi$
$739$	36.1371i	1.32932i	−0.747144	+	0.664662i	$0.768576\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.0965	−1.25088	−0.625440	−	0.780272i	$-0.715080\pi$
$743$	−34.0965	−1.25088	−0.625440	+	0.780272i	$0.715080\pi$
$744$	0	0
$745$	−52.8421	−1.93598
$746$	0	0
$747$	0	0
$748$	0	0
$749$	82.2595i	3.00570i
$750$	0	0
$751$	38.1730i	1.39295i	0.717579	+	0.696477i	$0.245250\pi$
$751$	38.1730i	1.39295i	−0.717579	+	0.696477i	$0.754750\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.3728	0.559473
$756$	0	0
$757$	26.2341	0.953493	0.476747	−	0.879041i	$-0.341816\pi$
$757$	26.2341	0.953493	0.476747	+	0.879041i	$0.341816\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 16.7682i	− 0.607846i	−0.952697	−	0.303923i	$-0.901703\pi$
$761$	− 16.7682i	− 0.607846i	0.952697	−	0.303923i	$-0.0982966\pi$
$762$	0	0
$763$	22.2744i	0.806389i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	38.2224	1.38013
$768$	0	0
$769$	37.2331	1.34266	0.671330	−	0.741159i	$-0.265724\pi$
$769$	37.2331	1.34266	0.671330	+	0.741159i	$0.265724\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	36.6106i	1.31679i	0.752672	+	0.658396i	$0.228765\pi$
$773$	36.6106i	1.31679i	−0.752672	+	0.658396i	$0.771235\pi$
$774$	0	0
$775$	21.4262i	0.769650i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.1558	−0.399698
$780$	0	0
$781$	6.92637	0.247845
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 76.7488i	− 2.73928i
$786$	0	0
$787$	− 43.5146i	− 1.55113i	−0.631268	−	0.775565i	$-0.717465\pi$
$787$	− 43.5146i	− 1.55113i	0.631268	−	0.775565i	$-0.282535\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−32.3135	−1.14893
$792$	0	0
$793$	10.0425	0.356619
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 9.12345i	− 0.323169i	−0.986859	−	0.161585i	$-0.948340\pi$
$797$	− 9.12345i	− 0.323169i	0.986859	−	0.161585i	$-0.0516605\pi$
$798$	0	0
$799$	− 11.7570i	− 0.415934i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.65046	0.164111
$804$	0	0
$805$	115.488	4.07043
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.58637i	− 0.161248i	−0.996745	−	0.0806241i	$-0.974309\pi$
$809$	− 4.58637i	− 0.161248i	0.996745	−	0.0806241i	$-0.0256913\pi$
$810$	0	0
$811$	4.79126i	0.168244i	0.996455	+	0.0841220i	$0.0268085\pi$
$811$	4.79126i	0.168244i	−0.996455	+	0.0841220i	$0.973191\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	60.3367	2.11350
$816$	0	0
$817$	35.0064	1.22472
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.3272i	0.709423i	0.934976	+	0.354712i	$0.115421\pi$
$821$	20.3272i	0.709423i	−0.934976	+	0.354712i	$0.884579\pi$
$822$	0	0
$823$	21.4853i	0.748932i	0.927241	+	0.374466i	$0.122174\pi$
$823$	21.4853i	0.748932i	−0.927241	+	0.374466i	$0.877826\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.21198	−0.320332	−0.160166	−	0.987090i	$-0.551203\pi$
$827$	−9.21198	−0.320332	−0.160166	+	0.987090i	$0.551203\pi$
$828$	0	0
$829$	25.2331	0.876381	0.438190	−	0.898882i	$-0.355620\pi$
$829$	25.2331	0.876381	0.438190	+	0.898882i	$0.355620\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15.7554i	0.545891i
$834$	0	0
$835$	− 32.5719i	− 1.12720i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.77270	−0.130248	−0.0651241	−	0.997877i	$-0.520744\pi$
$839$	−3.77270	−0.130248	−0.0651241	+	0.997877i	$0.520744\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 81.7967i	− 2.81389i
$846$	0	0
$847$	4.08334i	0.140305i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−37.8971	−1.29910
$852$	0	0
$853$	−23.2942	−0.797577	−0.398788	−	0.917043i	$-0.630569\pi$
$853$	−23.2942	−0.797577	−0.398788	+	0.917043i	$0.630569\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 5.59919i	− 0.191265i	−0.995417	−	0.0956324i	$-0.969513\pi$
$857$	− 5.59919i	− 0.191265i	0.995417	−	0.0956324i	$-0.0304873\pi$
$858$	0	0
$859$	22.6120i	0.771510i	0.922601	+	0.385755i	$0.126059\pi$
$859$	22.6120i	0.771510i	−0.922601	+	0.385755i	$0.873941\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.7813	−0.571240	−0.285620	−	0.958343i	$-0.592200\pi$
$863$	−16.7813	−0.571240	−0.285620	+	0.958343i	$0.592200\pi$
$864$	0	0
$865$	−0.844195	−0.0287035
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.825984i	0.0280196i
$870$	0	0
$871$	− 68.6770i	− 2.32703i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−111.275	−3.76179
$876$	0	0
$877$	−25.2420	−0.852361	−0.426181	−	0.904638i	$-0.640141\pi$
$877$	−25.2420	−0.852361	−0.426181	+	0.904638i	$0.640141\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.4790i	1.66699i	0.552527	+	0.833495i	$0.313664\pi$
$881$	49.4790i	1.66699i	−0.552527	+	0.833495i	$0.686336\pi$
$882$	0	0
$883$	− 4.24079i	− 0.142714i	−0.997451	−	0.0713570i	$-0.977267\pi$
$883$	− 4.24079i	− 0.142714i	0.997451	−	0.0713570i	$-0.0227329\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.8421	−0.565501	−0.282750	−	0.959193i	$-0.591247\pi$
$887$	−16.8421	−0.565501	−0.282750	+	0.959193i	$0.591247\pi$
$888$	0	0
$889$	−44.9639	−1.50804
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 43.8760i	− 1.46825i
$894$	0	0
$895$	34.7215i	1.16061i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.35809	0.212054
$900$	0	0
$901$	17.9343	0.597478
$902$	0	0
$903$	0	0
$904$	0	0
$905$	34.2238i	1.13764i
$906$	0	0
$907$	20.8481i	0.692251i	0.938188	+	0.346126i	$0.112503\pi$
$907$	20.8481i	0.692251i	−0.938188	+	0.346126i	$0.887497\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38.9728	−1.29123	−0.645614	−	0.763664i	$-0.723398\pi$
$911$	−38.9728	−1.29123	−0.645614	+	0.763664i	$0.723398\pi$
$912$	0	0
$913$	−4.67370	−0.154677
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.1750i	− 0.468100i
$918$	0	0
$919$	5.59581i	0.184589i	0.995732	+	0.0922944i	$0.0294201\pi$
$919$	5.59581i	0.184589i	−0.995732	+	0.0922944i	$0.970580\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	39.8081	1.31030
$924$	0	0
$925$	63.8717	2.10009
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.2193i	0.991463i	0.868476	+	0.495731i	$0.165100\pi$
$929$	30.2193i	0.991463i	−0.868476	+	0.495731i	$0.834900\pi$
$930$	0	0
$931$	58.7974i	1.92701i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.65046	−0.217493
$936$	0	0
$937$	14.2924	0.466912	0.233456	−	0.972367i	$-0.424997\pi$
$937$	14.2924	0.466912	0.233456	+	0.972367i	$0.424997\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 28.5925i	− 0.932087i	−0.884762	−	0.466044i	$-0.845679\pi$
$941$	− 28.5925i	− 0.932087i	0.884762	−	0.466044i	$-0.154321\pi$
$942$	0	0
$943$	− 12.7128i	− 0.413986i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.35809	0.206610	0.103305	−	0.994650i	$-0.467058\pi$
$947$	6.35809	0.206610	0.103305	+	0.994650i	$0.467058\pi$
$948$	0	0
$949$	26.7277	0.867619
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 41.0672i	− 1.33030i	−0.746711	−	0.665148i	$-0.768368\pi$
$953$	− 41.0672i	− 1.33030i	0.746711	−	0.665148i	$-0.231632\pi$
$954$	0	0
$955$	− 30.5360i	− 0.988123i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.72231	−0.0879081
$960$	0	0
$961$	27.6312	0.891330
$962$	0	0
$963$	0	0
$964$	0	0
$965$	106.158i	3.41735i
$966$	0	0
$967$	− 0.412502i	− 0.0132652i	−0.999978	−	0.00663258i	$-0.997889\pi$
$967$	− 0.412502i	− 0.0132652i	0.999978	−	0.00663258i	$-0.00211123\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.56250	−0.306875	−0.153438	−	0.988158i	$-0.549034\pi$
$971$	−9.56250	−0.306875	−0.153438	+	0.988158i	$0.549034\pi$
$972$	0	0
$973$	78.2127	2.50738
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.71736i	− 0.0549431i	−0.999623	−	0.0274715i	$-0.991254\pi$
$977$	− 1.71736i	− 0.0549431i	0.999623	−	0.0274715i	$-0.00874556\pi$
$978$	0	0
$979$	− 3.00416i	− 0.0960132i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37.3174	−1.19024	−0.595120	−	0.803637i	$-0.702895\pi$
$983$	−37.3174	−1.19024	−0.595120	+	0.803637i	$0.702895\pi$
$984$	0	0
$985$	−8.79558	−0.280251
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39.8922i	1.26850i
$990$	0	0
$991$	31.8820i	1.01277i	0.862309	+	0.506383i	$0.169018\pi$
$991$	31.8820i	1.01277i	−0.862309	+	0.506383i	$0.830982\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−36.7934	−1.16643
$996$	0	0
$997$	34.2835	1.08577	0.542884	−	0.839808i	$-0.317332\pi$
$997$	34.2835	1.08577	0.542884	+	0.839808i	$0.317332\pi$
$998$	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.3455.1		8
3.2	odd	2		6336.2.d.c.3455.8		8
4.3	odd	2		6336.2.d.c.3455.1		8
8.3	odd	2		1584.2.d.d.287.8	yes	8
8.5	even	2		1584.2.d.c.287.8	yes	8
12.11	even	2	inner	6336.2.d.e.3455.8		8
24.5	odd	2		1584.2.d.d.287.1	yes	8
24.11	even	2		1584.2.d.c.287.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1584.2.d.c.287.1	✓	8	24.11	even	2
1584.2.d.c.287.8	yes	8	8.5	even	2
1584.2.d.d.287.1	yes	8	24.5	odd	2
1584.2.d.d.287.8	yes	8	8.3	odd	2
6336.2.d.c.3455.1		8	4.3	odd	2
6336.2.d.c.3455.8		8	3.2	odd	2
6336.2.d.e.3455.1		8	1.1	even	1	trivial
6336.2.d.e.3455.8		8	12.11	even	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-4.08334i q^{5} +4.08334i q^{7} +1.00000 q^{11} +5.74733 q^{13} -1.62868i q^{17} -6.07806i q^{19} +6.92637 q^{23} -11.6737 q^{25} +3.46410i q^{29} -1.83542i q^{31} +16.6737 q^{35} -5.47142 q^{37} -1.83542i q^{41} +5.75947i q^{43} +7.21875 q^{47} -9.67370 q^{49} +11.0115i q^{53} -4.08334i q^{55} +6.65046 q^{59} +1.74733 q^{61} -23.4683i q^{65} -11.9494i q^{67} +6.92637 q^{71} +4.65046 q^{73} +4.08334i q^{77} +0.825984i q^{79} -4.67370 q^{83} -6.65046 q^{85} -3.00416i q^{89} +23.4683i q^{91} -24.8188 q^{95} -12.6737 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 16 q^{25} + 56 q^{35} - 8 q^{37} - 16 q^{47} + 16 q^{59} - 24 q^{61} + 16 q^{71} + 40 q^{83} - 16 q^{85} - 8 q^{95} - 24 q^{97}+O(q^{100}) \) 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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