LMFDB - Embedded newform 576.4.k.c.145.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,4,Mod(145,576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("576.145"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.k (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			145.12
Character	$\chi$	$=$	576.145
Dual form			576.4.k.c.433.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(14.8202 - 14.8202i) q^{5} -10.4191i q^{7} +(23.1349 - 23.1349i) q^{11} +(7.26220 + 7.26220i) q^{13} +83.4233 q^{17} +(81.4449 + 81.4449i) q^{19} -86.6407i q^{23} -314.274i q^{25} +(37.1890 + 37.1890i) q^{29} -251.588 q^{31} +(-154.413 - 154.413i) q^{35} +(-102.360 + 102.360i) q^{37} +400.216i q^{41} +(332.983 - 332.983i) q^{43} -192.533 q^{47} +234.442 q^{49} +(-88.7392 + 88.7392i) q^{53} -685.726i q^{55} +(-528.529 + 528.529i) q^{59} +(-131.653 - 131.653i) q^{61} +215.254 q^{65} +(-480.467 - 480.467i) q^{67} -391.007i q^{71} -292.028i q^{73} +(-241.046 - 241.046i) q^{77} +805.236 q^{79} +(-232.313 - 232.313i) q^{83} +(1236.35 - 1236.35i) q^{85} +143.143i q^{89} +(75.6658 - 75.6658i) q^{91} +2414.05 q^{95} -733.540 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{19} - 744 q^{31} - 16 q^{37} + 376 q^{43} - 1176 q^{49} - 912 q^{61} - 1440 q^{67} + 328 q^{79} - 240 q^{85} + 104 q^{91}+O(q^{100}) $ 24 * q - 24 * q^19 - 744 * q^31 - 16 * q^37 + 376 * q^43 - 1176 * q^49 - 912 * q^61 - 1440 * q^67 + 328 * q^79 - 240 * q^85 + 104 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	14.8202	−	14.8202i	1.32555	−	1.32555i	0.416350	−	0.909204i	$-0.363309\pi$
$5$	14.8202	−	14.8202i	1.32555	−	1.32555i	0.909204	−	0.416350i	$-0.136691\pi$
$6$		0			0
$7$		−	10.4191i		−	0.562580i	−0.959623	−	0.281290i	$-0.909238\pi$
$7$		−	10.4191i		−	0.562580i	0.959623	−	0.281290i	$-0.0907623\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	23.1349	−	23.1349i	0.634131	−	0.634131i	−0.314970	−	0.949102i	$-0.601995\pi$
$11$	23.1349	−	23.1349i	0.634131	−	0.634131i	0.949102	+	0.314970i	$0.101995\pi$
$12$		0			0
$13$	7.26220	+	7.26220i	0.154936	+	0.154936i	0.780319	−	0.625382i	$-0.215057\pi$
$13$	7.26220	+	7.26220i	0.154936	+	0.154936i	−0.625382	+	0.780319i	$0.715057\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	83.4233			1.19018			0.595092	−	0.803657i	$-0.297116\pi$
$17$	83.4233			1.19018			0.595092	+	0.803657i	$0.297116\pi$
$18$		0			0
$19$	81.4449	+	81.4449i	0.983407	+	0.983407i	0.999865	−	0.0164576i	$-0.00523884\pi$
$19$	81.4449	+	81.4449i	0.983407	+	0.983407i	−0.0164576	+	0.999865i	$0.505239\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	86.6407i		−	0.785471i	−0.919651	−	0.392736i	$-0.871529\pi$
$23$		−	86.6407i		−	0.785471i	0.919651	−	0.392736i	$-0.128471\pi$
$24$		0			0
$25$		−	314.274i		−	2.51419i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	37.1890	+	37.1890i	0.238132	+	0.238132i	0.816076	−	0.577944i	$-0.196145\pi$
$29$	37.1890	+	37.1890i	0.238132	+	0.238132i	−0.577944	+	0.816076i	$0.696145\pi$
$30$		0			0
$31$	−251.588			−1.45763			−0.728814	−	0.684712i	$-0.759928\pi$
$31$	−251.588			−1.45763			−0.728814	+	0.684712i	$0.759928\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−154.413	−	154.413i	−0.745731	−	0.745731i
$36$		0			0
$37$	−102.360	+	102.360i	−0.454809	+	0.454809i	−0.896947	−	0.442138i	$-0.854220\pi$
$37$	−102.360	+	102.360i	−0.454809	+	0.454809i	0.442138	+	0.896947i	$0.354220\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			400.216i			1.52447i	0.647300	+	0.762235i	$0.275898\pi$
$41$			400.216i			1.52447i	−0.647300	+	0.762235i	$0.724102\pi$
$42$		0			0
$43$	332.983	−	332.983i	1.18092	−	1.18092i	0.201410	−	0.979507i	$-0.435448\pi$
$43$	332.983	−	332.983i	1.18092	−	1.18092i	0.979507	−	0.201410i	$-0.0645525\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−192.533			−0.597528			−0.298764	−	0.954327i	$-0.596574\pi$
$47$	−192.533			−0.597528			−0.298764	+	0.954327i	$0.596574\pi$
$48$		0			0
$49$	234.442			0.683503
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−88.7392	+	88.7392i	−0.229986	+	0.229986i	−0.812687	−	0.582701i	$-0.801996\pi$
$53$	−88.7392	+	88.7392i	−0.229986	+	0.229986i	0.582701	+	0.812687i	$0.301996\pi$
$54$		0			0
$55$		−	685.726i		−	1.68115i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−528.529	+	528.529i	−1.16625	+	1.16625i	−0.183166	+	0.983082i	$0.558635\pi$
$59$	−528.529	+	528.529i	−1.16625	+	1.16625i	−0.983082	+	0.183166i	$0.941365\pi$
$60$		0			0
$61$	−131.653	−	131.653i	−0.276336	−	0.276336i	0.555309	−	0.831644i	$-0.312600\pi$
$61$	−131.653	−	131.653i	−0.276336	−	0.276336i	−0.831644	+	0.555309i	$0.812600\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	215.254			0.410753
$66$		0			0
$67$	−480.467	−	480.467i	−0.876095	−	0.876095i	0.117033	−	0.993128i	$-0.462662\pi$
$67$	−480.467	−	480.467i	−0.876095	−	0.876095i	−0.993128	+	0.117033i	$0.962662\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	391.007i		−	0.653577i	−0.945097	−	0.326789i	$-0.894033\pi$
$71$		−	391.007i		−	0.653577i	0.945097	−	0.326789i	$-0.105967\pi$
$72$		0			0
$73$		−	292.028i		−	0.468209i	−0.972211	−	0.234104i	$-0.924784\pi$
$73$		−	292.028i		−	0.468209i	0.972211	−	0.234104i	$-0.0752158\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−241.046	−	241.046i	−0.356750	−	0.356750i
$78$		0			0
$79$	805.236			1.14679			0.573393	−	0.819280i	$-0.305627\pi$
$79$	805.236			1.14679			0.573393	+	0.819280i	$0.305627\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−232.313	−	232.313i	−0.307225	−	0.307225i	0.536607	−	0.843832i	$-0.319706\pi$
$83$	−232.313	−	232.313i	−0.307225	−	0.307225i	−0.843832	+	0.536607i	$0.819706\pi$
$84$		0			0
$85$	1236.35	−	1236.35i	1.57765	−	1.57765i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			143.143i			0.170484i	0.996360	+	0.0852420i	$0.0271664\pi$
$89$			143.143i			0.170484i	−0.996360	+	0.0852420i	$0.972834\pi$
$90$		0			0
$91$	75.6658	−	75.6658i	0.0871641	−	0.0871641i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2414.05			2.60712
$96$		0			0
$97$	−733.540			−0.767832			−0.383916	−	0.923368i	$-0.625425\pi$
$97$	−733.540			−0.767832			−0.383916	+	0.923368i	$0.625425\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	796.362	−	796.362i	0.784564	−	0.784564i	−0.196033	−	0.980597i	$-0.562806\pi$
$101$	796.362	−	796.362i	0.784564	−	0.784564i	0.980597	+	0.196033i	$0.0628061\pi$
$102$		0			0
$103$		−	583.858i		−	0.558536i	−0.960213	−	0.279268i	$-0.909908\pi$
$103$		−	583.858i		−	0.558536i	0.960213	−	0.279268i	$-0.0900919\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.5292	+	19.5292i	−0.0176445	+	0.0176445i	−0.715874	−	0.698230i	$-0.753972\pi$
$107$	−19.5292	+	19.5292i	−0.0176445	+	0.0176445i	0.698230	+	0.715874i	$0.253972\pi$
$108$		0			0
$109$	−1138.43	−	1138.43i	−1.00039	−	1.00039i	−1.00000		0.000387793i	$-0.999877\pi$
$109$	−1138.43	−	1138.43i	−1.00039	−	1.00039i	−0.000387793	−	1.00000i	$-0.500123\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	906.138			0.754356			0.377178	−	0.926141i	$-0.376894\pi$
$113$	906.138			0.754356			0.377178	+	0.926141i	$0.376894\pi$
$114$		0			0
$115$	−1284.03	−	1284.03i	−1.04118	−	1.04118i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	869.199i		−	0.669574i
$120$		0			0
$121$			260.550i			0.195755i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2805.07	−	2805.07i	−2.00714	−	2.00714i
$126$		0			0
$127$	1326.41			0.926772			0.463386	−	0.886157i	$-0.346634\pi$
$127$	1326.41			0.926772			0.463386	+	0.886157i	$0.346634\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1326.86	+	1326.86i	0.884950	+	0.884950i	0.994033	−	0.109082i	$-0.0347912\pi$
$131$	1326.86	+	1326.86i	0.884950	+	0.884950i	−0.109082	+	0.994033i	$0.534791\pi$
$132$		0			0
$133$	848.585	−	848.585i	0.553245	−	0.553245i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1978.79i			1.23401i	0.786959	+	0.617006i	$0.211654\pi$
$137$			1978.79i			1.23401i	−0.786959	+	0.617006i	$0.788346\pi$
$138$		0			0
$139$	−353.112	+	353.112i	−0.215472	+	0.215472i	−0.806587	−	0.591115i	$-0.798688\pi$
$139$	−353.112	+	353.112i	−0.215472	+	0.215472i	0.591115	+	0.806587i	$0.298688\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	336.021			0.196500
$144$		0			0
$145$	1102.29			0.631314
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−400.007	+	400.007i	−0.219932	+	0.219932i	−0.808470	−	0.588538i	$-0.799704\pi$
$149$	−400.007	+	400.007i	−0.219932	+	0.219932i	0.588538	+	0.808470i	$0.299704\pi$
$150$		0			0
$151$			1272.74i			0.685922i	0.939350	+	0.342961i	$0.111430\pi$
$151$			1272.74i			0.685922i	−0.939350	+	0.342961i	$0.888570\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3728.57	+	3728.57i	−1.93217	+	1.93217i
$156$		0			0
$157$	−380.500	−	380.500i	−0.193422	−	0.193422i	0.603751	−	0.797173i	$-0.293672\pi$
$157$	−380.500	−	380.500i	−0.193422	−	0.193422i	−0.797173	+	0.603751i	$0.793672\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−902.721			−0.441891
$162$		0			0
$163$	1904.08	+	1904.08i	0.914966	+	0.914966i	0.996658	−	0.0816918i	$-0.0260323\pi$
$163$	1904.08	+	1904.08i	0.914966	+	0.914966i	−0.0816918	+	0.996658i	$0.526032\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2648.33i			1.22715i	0.789636	+	0.613576i	$0.210269\pi$
$167$			2648.33i			1.22715i	−0.789636	+	0.613576i	$0.789731\pi$
$168$		0			0
$169$		−	2091.52i		−	0.951990i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1709.13	−	1709.13i	−0.751112	−	0.751112i	0.223574	−	0.974687i	$-0.428227\pi$
$173$	−1709.13	−	1709.13i	−0.751112	−	0.751112i	−0.974687	+	0.223574i	$0.928227\pi$
$174$		0			0
$175$	−3274.46			−1.41443
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2726.24	−	2726.24i	−1.13837	−	1.13837i	−0.988742	−	0.149629i	$-0.952192\pi$
$179$	−2726.24	−	2726.24i	−1.13837	−	1.13837i	−0.149629	−	0.988742i	$-0.547808\pi$
$180$		0			0
$181$	−3055.16	+	3055.16i	−1.25463	+	1.25463i	−0.301009	+	0.953621i	$0.597323\pi$
$181$	−3055.16	+	3055.16i	−1.25463	+	1.25463i	−0.953621	+	0.301009i	$0.902677\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			3033.99i			1.20575i
$186$		0			0
$187$	1929.99	−	1929.99i	0.754733	−	0.754733i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4033.32			1.52796			0.763981	−	0.645238i	$-0.223242\pi$
$191$	4033.32			1.52796			0.763981	+	0.645238i	$0.223242\pi$
$192$		0			0
$193$	159.728			0.0595724			0.0297862	−	0.999556i	$-0.490517\pi$
$193$	159.728			0.0595724			0.0297862	+	0.999556i	$0.490517\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	44.2361	−	44.2361i	0.0159984	−	0.0159984i	−0.699062	−	0.715061i	$-0.746399\pi$
$197$	44.2361	−	44.2361i	0.0159984	−	0.0159984i	0.715061	+	0.699062i	$0.246399\pi$
$198$		0			0
$199$		−	1589.83i		−	0.566333i	−0.959071	−	0.283167i	$-0.908615\pi$
$199$		−	1589.83i		−	0.566333i	0.959071	−	0.283167i	$-0.0913849\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	387.477	−	387.477i	0.133968	−	0.133968i
$204$		0			0
$205$	5931.27	+	5931.27i	2.02077	+	2.02077i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3768.44			1.24722
$210$		0			0
$211$	1952.43	+	1952.43i	0.637019	+	0.637019i	0.949819	−	0.312800i	$-0.101267\pi$
$211$	1952.43	+	1952.43i	0.637019	+	0.637019i	−0.312800	+	0.949819i	$0.601267\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	9869.72i		−	3.13074i
$216$		0			0
$217$			2621.32i			0.820033i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	605.837	+	605.837i	0.184403	+	0.184403i
$222$		0			0
$223$	−3886.74			−1.16715			−0.583577	−	0.812058i	$-0.698347\pi$
$223$	−3886.74			−1.16715			−0.583577	+	0.812058i	$0.698347\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1536.64	+	1536.64i	0.449296	+	0.449296i	0.895120	−	0.445824i	$-0.147089\pi$
$227$	1536.64	+	1536.64i	0.449296	+	0.449296i	−0.445824	+	0.895120i	$0.647089\pi$
$228$		0			0
$229$	−4221.34	+	4221.34i	−1.21814	+	1.21814i	−0.249857	+	0.968283i	$0.580384\pi$
$229$	−4221.34	+	4221.34i	−1.21814	+	1.21814i	−0.968283	+	0.249857i	$0.919616\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1991.46i			0.559935i	0.960009	+	0.279968i	$0.0903237\pi$
$233$			1991.46i			0.559935i	−0.960009	+	0.279968i	$0.909676\pi$
$234$		0			0
$235$	−2853.37	+	2853.37i	−0.792056	+	0.792056i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−225.264			−0.0609671			−0.0304836	−	0.999535i	$-0.509705\pi$
$239$	−225.264			−0.0609671			−0.0304836	+	0.999535i	$0.509705\pi$
$240$		0			0
$241$	−2055.57			−0.549422			−0.274711	−	0.961527i	$-0.588582\pi$
$241$	−2055.57			−0.549422			−0.274711	+	0.961527i	$0.588582\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3474.46	−	3474.46i	0.906021	−	0.906021i
$246$		0			0
$247$			1182.94i			0.304731i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2551.25	+	2551.25i	−0.641567	+	0.641567i	−0.950941	−	0.309373i	$-0.899881\pi$
$251$	−2551.25	+	2551.25i	−0.641567	+	0.641567i	0.309373	+	0.950941i	$0.399881\pi$
$252$		0			0
$253$	−2004.43	−	2004.43i	−0.498092	−	0.498092i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5083.08			1.23375			0.616875	−	0.787061i	$-0.288398\pi$
$257$	5083.08			1.23375			0.616875	+	0.787061i	$0.288398\pi$
$258$		0			0
$259$	1066.51	+	1066.51i	0.255867	+	0.255867i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			3233.91i			0.758219i	0.925352	+	0.379109i	$0.123770\pi$
$263$			3233.91i			0.758219i	−0.925352	+	0.379109i	$0.876230\pi$
$264$		0			0
$265$			2630.26i			0.609719i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	662.348	+	662.348i	0.150127	+	0.150127i	0.778175	−	0.628048i	$-0.216146\pi$
$269$	662.348	+	662.348i	0.150127	+	0.150127i	−0.628048	+	0.778175i	$0.716146\pi$
$270$		0			0
$271$	5617.02			1.25908			0.629539	−	0.776969i	$-0.283244\pi$
$271$	5617.02			1.25908			0.629539	+	0.776969i	$0.283244\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7270.70	−	7270.70i	−1.59433	−	1.59433i
$276$		0			0
$277$	3194.76	−	3194.76i	0.692978	−	0.692978i	−0.269908	−	0.962886i	$-0.586993\pi$
$277$	3194.76	−	3194.76i	0.692978	−	0.692978i	0.962886	+	0.269908i	$0.0869934\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			514.426i			0.109210i	0.998508	+	0.0546051i	$0.0173900\pi$
$281$			514.426i			0.109210i	−0.998508	+	0.0546051i	$0.982610\pi$
$282$		0			0
$283$	−4602.93	+	4602.93i	−0.966841	+	0.966841i	−0.999468	−	0.0326267i	$-0.989613\pi$
$283$	−4602.93	+	4602.93i	−0.966841	+	0.966841i	0.0326267	+	0.999468i	$0.489613\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4169.91			0.857637
$288$		0			0
$289$	2046.45			0.416539
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1506.83	+	1506.83i	−0.300444	+	0.300444i	−0.841187	−	0.540744i	$-0.818143\pi$
$293$	−1506.83	+	1506.83i	−0.300444	+	0.300444i	0.540744	+	0.841187i	$0.318143\pi$
$294$		0			0
$295$			15665.8i			3.09185i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	629.202	−	629.202i	0.121698	−	0.121698i
$300$		0			0
$301$	−3469.39	−	3469.39i	−0.664361	−	0.664361i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3902.25			−0.732597
$306$		0			0
$307$	−267.320	−	267.320i	−0.0496963	−	0.0496963i	0.681822	−	0.731518i	$-0.261188\pi$
$307$	−267.320	−	267.320i	−0.0496963	−	0.0496963i	−0.731518	+	0.681822i	$0.761188\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2639.31i		−	0.481228i	−0.970621	−	0.240614i	$-0.922651\pi$
$311$		−	2639.31i		−	0.481228i	0.970621	−	0.240614i	$-0.0773487\pi$
$312$		0			0
$313$			6802.92i			1.22851i	0.789107	+	0.614255i	$0.210543\pi$
$313$			6802.92i			1.22851i	−0.789107	+	0.614255i	$0.789457\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	448.870	+	448.870i	0.0795301	+	0.0795301i	0.745753	−	0.666223i	$-0.232090\pi$
$317$	448.870	+	448.870i	0.0795301	+	0.0795301i	−0.666223	+	0.745753i	$0.732090\pi$
$318$		0			0
$319$	1720.73			0.302014
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6794.40	+	6794.40i	1.17044	+	1.17044i
$324$		0			0
$325$	2282.32	−	2282.32i	0.389539	−	0.389539i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			2006.03i			0.336157i
$330$		0			0
$331$	1326.59	−	1326.59i	0.220290	−	0.220290i	−0.588331	−	0.808620i	$-0.700215\pi$
$331$	1326.59	−	1326.59i	0.220290	−	0.220290i	0.808620	+	0.588331i	$0.200215\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14241.2			−2.32262
$336$		0			0
$337$	−1888.30			−0.305230			−0.152615	−	0.988286i	$-0.548769\pi$
$337$	−1888.30			−0.305230			−0.152615	+	0.988286i	$0.548769\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5820.46	+	5820.46i	−0.924327	+	0.924327i
$342$		0			0
$343$		−	6016.44i		−	0.947106i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1714.34	+	1714.34i	−0.265218	+	0.265218i	−0.827170	−	0.561952i	$-0.810051\pi$
$347$	−1714.34	+	1714.34i	−0.265218	+	0.265218i	0.561952	+	0.827170i	$0.310051\pi$
$348$		0			0
$349$	3880.39	+	3880.39i	0.595165	+	0.595165i	0.939022	−	0.343857i	$-0.111734\pi$
$349$	3880.39	+	3880.39i	0.595165	+	0.595165i	−0.343857	+	0.939022i	$0.611734\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3413.42			−0.514669			−0.257334	−	0.966322i	$-0.582844\pi$
$353$	−3413.42			−0.514669			−0.257334	+	0.966322i	$0.582844\pi$
$354$		0			0
$355$	−5794.78	−	5794.78i	−0.866353	−	0.866353i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			9923.21i			1.45885i	0.684061	+	0.729425i	$0.260212\pi$
$359$			9923.21i			1.45885i	−0.684061	+	0.729425i	$0.739788\pi$
$360$		0			0
$361$			6407.53i			0.934179i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4327.89	−	4327.89i	−0.620637	−	0.620637i
$366$		0			0
$367$	7863.30			1.11842			0.559211	−	0.829025i	$-0.311104\pi$
$367$	7863.30			1.11842			0.559211	+	0.829025i	$0.311104\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	924.586	+	924.586i	0.129386	+	0.129386i
$372$		0			0
$373$	259.033	−	259.033i	0.0359576	−	0.0359576i	−0.688899	−	0.724857i	$-0.741906\pi$
$373$	259.033	−	259.033i	0.0359576	−	0.0359576i	0.724857	+	0.688899i	$0.241906\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			540.148i			0.0737905i
$378$		0			0
$379$	−1795.65	+	1795.65i	−0.243367	+	0.243367i	−0.818242	−	0.574874i	$-0.805051\pi$
$379$	−1795.65	+	1795.65i	−0.243367	+	0.243367i	0.574874	+	0.818242i	$0.305051\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5104.92			0.681069			0.340534	−	0.940232i	$-0.389392\pi$
$383$	5104.92			0.681069			0.340534	+	0.940232i	$0.389392\pi$
$384$		0			0
$385$	−7144.67			−0.945782
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3104.97	−	3104.97i	0.404700	−	0.404700i	−0.475186	−	0.879885i	$-0.657619\pi$
$389$	3104.97	−	3104.97i	0.404700	−	0.404700i	0.879885	+	0.475186i	$0.157619\pi$
$390$		0			0
$391$		−	7227.86i		−	0.934855i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11933.7	−	11933.7i	1.52013	−	1.52013i
$396$		0			0
$397$	−4196.92	−	4196.92i	−0.530573	−	0.530573i	0.390170	−	0.920743i	$-0.372416\pi$
$397$	−4196.92	−	4196.92i	−0.530573	−	0.530573i	−0.920743	+	0.390170i	$0.872416\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10440.1			−1.30013			−0.650066	−	0.759878i	$-0.725259\pi$
$401$	−10440.1			−1.30013			−0.650066	+	0.759878i	$0.725259\pi$
$402$		0			0
$403$	−1827.08	−	1827.08i	−0.225839	−	0.225839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			4736.20i			0.576818i
$408$		0			0
$409$		−	9673.99i		−	1.16955i	−0.811194	−	0.584777i	$-0.801182\pi$
$409$		−	9673.99i		−	1.16955i	0.811194	−	0.584777i	$-0.198818\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5506.82	+	5506.82i	0.656108	+	0.656108i
$414$		0			0
$415$	−6885.83			−0.814487
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2935.51	+	2935.51i	0.342265	+	0.342265i	0.857218	−	0.514953i	$-0.172191\pi$
$419$	2935.51	+	2935.51i	0.342265	+	0.342265i	−0.514953	+	0.857218i	$0.672191\pi$
$420$		0			0
$421$	2209.45	−	2209.45i	0.255776	−	0.255776i	−0.567557	−	0.823334i	$-0.692112\pi$
$421$	2209.45	−	2209.45i	0.255776	−	0.255776i	0.823334	+	0.567557i	$0.192112\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	26217.8i		−	2.99235i
$426$		0			0
$427$	−1371.71	+	1371.71i	−0.155461	+	0.155461i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4051.99			0.452848			0.226424	−	0.974029i	$-0.427297\pi$
$431$	4051.99			0.452848			0.226424	+	0.974029i	$0.427297\pi$
$432$		0			0
$433$	12206.3			1.35473			0.677365	−	0.735647i	$-0.263122\pi$
$433$	12206.3			1.35473			0.677365	+	0.735647i	$0.263122\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7056.44	−	7056.44i	0.772438	−	0.772438i
$438$		0			0
$439$		−	8630.90i		−	0.938339i	−0.883108	−	0.469169i	$-0.844553\pi$
$439$		−	8630.90i		−	0.938339i	0.883108	−	0.469169i	$-0.155447\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−342.023	+	342.023i	−0.0366817	+	0.0366817i	−0.725210	−	0.688528i	$-0.758257\pi$
$443$	−342.023	+	342.023i	−0.0366817	+	0.0366817i	0.688528	+	0.725210i	$0.258257\pi$
$444$		0			0
$445$	2121.39	+	2121.39i	0.225986	+	0.225986i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8495.76			0.892961			0.446481	−	0.894793i	$-0.352677\pi$
$449$	8495.76			0.892961			0.446481	+	0.894793i	$0.352677\pi$
$450$		0			0
$451$	9258.98	+	9258.98i	0.966715	+	0.966715i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	2242.76i		−	0.231082i
$456$		0			0
$457$		−	11832.0i		−	1.21111i	−0.795803	−	0.605556i	$-0.792951\pi$
$457$		−	11832.0i		−	1.21111i	0.795803	−	0.605556i	$-0.207049\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5997.14	+	5997.14i	0.605889	+	0.605889i	0.941869	−	0.335980i	$-0.109068\pi$
$461$	5997.14	+	5997.14i	0.605889	+	0.605889i	−0.335980	+	0.941869i	$0.609068\pi$
$462$		0			0
$463$	−12533.3			−1.25804			−0.629018	−	0.777391i	$-0.716543\pi$
$463$	−12533.3			−1.25804			−0.629018	+	0.777391i	$0.716543\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9707.05	+	9707.05i	0.961861	+	0.961861i	0.999299	−	0.0374384i	$-0.0119198\pi$
$467$	9707.05	+	9707.05i	0.961861	+	0.961861i	−0.0374384	+	0.999299i	$0.511920\pi$
$468$		0			0
$469$	−5006.05	+	5006.05i	−0.492874	+	0.492874i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	15407.1i		−	1.49771i
$474$		0			0
$475$	25596.0	−	25596.0i	2.47247	−	2.47247i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4387.69			−0.418536			−0.209268	−	0.977858i	$-0.567108\pi$
$479$	−4387.69			−0.418536			−0.209268	+	0.977858i	$0.567108\pi$
$480$		0			0
$481$	−1486.72			−0.140933
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10871.2	+	10871.2i	−1.01780	+	1.01780i
$486$		0			0
$487$		−	9320.66i		−	0.867268i	−0.901089	−	0.433634i	$-0.857231\pi$
$487$		−	9320.66i		−	0.867268i	0.901089	−	0.433634i	$-0.142769\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10257.4	−	10257.4i	0.942789	−	0.942789i	−0.0556603	−	0.998450i	$-0.517726\pi$
$491$	10257.4	−	10257.4i	0.942789	−	0.942789i	0.998450	+	0.0556603i	$0.0177264\pi$
$492$		0			0
$493$	3102.43	+	3102.43i	0.283421	+	0.283421i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4073.95			−0.367690
$498$		0			0
$499$	−9119.97	−	9119.97i	−0.818168	−	0.818168i	0.167674	−	0.985842i	$-0.446374\pi$
$499$	−9119.97	−	9119.97i	−0.818168	−	0.818168i	−0.985842	+	0.167674i	$0.946374\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			4904.40i			0.434744i	0.976089	+	0.217372i	$0.0697486\pi$
$503$			4904.40i			0.434744i	−0.976089	+	0.217372i	$0.930251\pi$
$504$		0			0
$505$		−	23604.4i		−	2.07996i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8343.01	+	8343.01i	0.726518	+	0.726518i	0.969924	−	0.243407i	$-0.0782650\pi$
$509$	8343.01	+	8343.01i	0.726518	+	0.726518i	−0.243407	+	0.969924i	$0.578265\pi$
$510$		0			0
$511$	−3042.67			−0.263405
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8652.87	−	8652.87i	−0.740371	−	0.740371i
$516$		0			0
$517$	−4454.23	+	4454.23i	−0.378911	+	0.378911i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			11593.6i			0.974905i	0.873150	+	0.487452i	$0.162074\pi$
$521$			11593.6i			0.974905i	−0.873150	+	0.487452i	$0.837926\pi$
$522$		0			0
$523$	−2107.92	+	2107.92i	−0.176239	+	0.176239i	−0.789714	−	0.613475i	$-0.789771\pi$
$523$	−2107.92	+	2107.92i	−0.176239	+	0.176239i	0.613475	+	0.789714i	$0.289771\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20988.3			−1.73485
$528$		0			0
$529$	4660.39			0.383035
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2906.45	+	2906.45i	−0.236196	+	0.236196i
$534$		0			0
$535$			578.851i			0.0467774i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5423.79	−	5423.79i	0.433431	−	0.433431i
$540$		0			0
$541$	−4161.70	−	4161.70i	−0.330731	−	0.330731i	0.522133	−	0.852864i	$-0.325137\pi$
$541$	−4161.70	−	4161.70i	−0.330731	−	0.330731i	−0.852864	+	0.522133i	$0.825137\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−33743.5			−2.65214
$546$		0			0
$547$	13475.0	+	13475.0i	1.05329	+	1.05329i	0.998498	+	0.0547947i	$0.0174504\pi$
$547$	13475.0	+	13475.0i	1.05329	+	1.05329i	0.0547947	+	0.998498i	$0.482550\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			6057.71i			0.468361i
$552$		0			0
$553$		−	8389.86i		−	0.645160i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13237.5	+	13237.5i	1.00698	+	1.00698i	0.999975	+	0.00700860i	$0.00223093\pi$
$557$	13237.5	+	13237.5i	1.00698	+	1.00698i	0.00700860	+	0.999975i	$0.497769\pi$
$558$		0			0
$559$	4836.38			0.365934
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3270.74	−	3270.74i	−0.244840	−	0.244840i	0.574009	−	0.818849i	$-0.305387\pi$
$563$	−3270.74	−	3270.74i	−0.244840	−	0.244840i	−0.818849	+	0.574009i	$0.805387\pi$
$564$		0			0
$565$	13429.1	−	13429.1i	0.999941	−	0.999941i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			7636.17i			0.562610i	0.959618	+	0.281305i	$0.0907672\pi$
$569$			7636.17i			0.562610i	−0.959618	+	0.281305i	$0.909233\pi$
$570$		0			0
$571$	−2614.60	+	2614.60i	−0.191624	+	0.191624i	−0.796398	−	0.604773i	$-0.793264\pi$
$571$	−2614.60	+	2614.60i	−0.191624	+	0.191624i	0.604773	+	0.796398i	$0.293264\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−27228.9			−1.97482
$576$		0			0
$577$	12989.0			0.937155			0.468578	−	0.883422i	$-0.344767\pi$
$577$	12989.0			0.937155			0.468578	+	0.883422i	$0.344767\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2420.50	+	2420.50i	−0.172839	+	0.172839i
$582$		0			0
$583$			4105.95i			0.291683i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7910.32	+	7910.32i	−0.556208	+	0.556208i	−0.928226	−	0.372018i	$-0.878666\pi$
$587$	−7910.32	+	7910.32i	−0.556208	+	0.556208i	0.372018	+	0.928226i	$0.378666\pi$
$588$		0			0
$589$	−20490.5	−	20490.5i	−1.43344	−	1.43344i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17584.0			1.21769			0.608844	−	0.793290i	$-0.291634\pi$
$593$	17584.0			1.21769			0.608844	+	0.793290i	$0.291634\pi$
$594$		0			0
$595$	−12881.7	−	12881.7i	−0.887557	−	0.887557i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			14241.2i			0.971417i	0.874121	+	0.485708i	$0.161438\pi$
$599$			14241.2i			0.971417i	−0.874121	+	0.485708i	$0.838562\pi$
$600$		0			0
$601$			6490.69i			0.440534i	0.975440	+	0.220267i	$0.0706928\pi$
$601$			6490.69i			0.440534i	−0.975440	+	0.220267i	$0.929307\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3861.39	+	3861.39i	0.259484	+	0.259484i
$606$		0			0
$607$	1302.36			0.0870856			0.0435428	−	0.999052i	$-0.486136\pi$
$607$	1302.36			0.0870856			0.0435428	+	0.999052i	$0.486136\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1398.21	−	1398.21i	−0.0925787	−	0.0925787i
$612$		0			0
$613$	17400.6	−	17400.6i	1.14650	−	1.14650i	0.159263	−	0.987236i	$-0.449088\pi$
$613$	17400.6	−	17400.6i	1.14650	−	1.14650i	0.987236	−	0.159263i	$-0.0509118\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	9223.56i		−	0.601826i	−0.953652	−	0.300913i	$-0.902709\pi$
$617$		−	9223.56i		−	0.601826i	0.953652	−	0.300913i	$-0.0972914\pi$
$618$		0			0
$619$	−16008.8	+	16008.8i	−1.03950	+	1.03950i	−0.0403107	+	0.999187i	$0.512835\pi$
$619$	−16008.8	+	16008.8i	−1.03950	+	1.03950i	−0.999187	+	0.0403107i	$0.987165\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1491.42			0.0959110
$624$		0			0
$625$	−43858.8			−2.80696
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8539.25	+	8539.25i	−0.541307	+	0.541307i
$630$		0			0
$631$		−	7184.18i		−	0.453245i	−0.973983	−	0.226622i	$-0.927232\pi$
$631$		−	7184.18i		−	0.453245i	0.973983	−	0.226622i	$-0.0727683\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	19657.6	−	19657.6i	1.22849	−	1.22849i
$636$		0			0
$637$	1702.56	+	1702.56i	0.105899	+	0.105899i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25937.6			1.59824			0.799120	−	0.601171i	$-0.205299\pi$
$641$	25937.6			1.59824			0.799120	+	0.601171i	$0.205299\pi$
$642$		0			0
$643$	−9092.95	−	9092.95i	−0.557684	−	0.557684i	0.370963	−	0.928647i	$-0.379028\pi$
$643$	−9092.95	−	9092.95i	−0.557684	−	0.557684i	−0.928647	+	0.370963i	$0.879028\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	28247.6i		−	1.71643i	−0.513291	−	0.858215i	$-0.671574\pi$
$647$		−	28247.6i		−	1.71643i	0.513291	−	0.858215i	$-0.328426\pi$
$648$		0			0
$649$			24455.0i			1.47911i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8837.90	+	8837.90i	0.529638	+	0.529638i	0.920465	−	0.390826i	$-0.127811\pi$
$653$	8837.90	+	8837.90i	0.529638	+	0.529638i	−0.390826	+	0.920465i	$0.627811\pi$
$654$		0			0
$655$	39328.6			2.34610
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2514.61	+	2514.61i	0.148642	+	0.148642i	0.777511	−	0.628869i	$-0.216482\pi$
$659$	2514.61	+	2514.61i	0.148642	+	0.148642i	−0.628869	+	0.777511i	$0.716482\pi$
$660$		0			0
$661$	−11065.3	+	11065.3i	−0.651120	+	0.651120i	−0.953263	−	0.302143i	$-0.902298\pi$
$661$	−11065.3	+	11065.3i	−0.651120	+	0.651120i	0.302143	+	0.953263i	$0.402298\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	25152.3i		−	1.46671i
$666$		0			0
$667$	3222.08	−	3222.08i	0.187046	−	0.187046i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6091.58			−0.350466
$672$		0			0
$673$	26512.1			1.51852			0.759262	−	0.650785i	$-0.225560\pi$
$673$	26512.1			1.51852			0.759262	+	0.650785i	$0.225560\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17699.6	−	17699.6i	1.00480	−	1.00480i	0.00481617	−	0.999988i	$-0.498467\pi$
$677$	17699.6	−	17699.6i	1.00480	−	1.00480i	0.999988	−	0.00481617i	$-0.00153304\pi$
$678$		0			0
$679$			7642.85i			0.431967i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4418.13	+	4418.13i	−0.247518	+	0.247518i	−0.819951	−	0.572433i	$-0.806000\pi$
$683$	−4418.13	+	4418.13i	−0.247518	+	0.247518i	0.572433	+	0.819951i	$0.306000\pi$
$684$		0			0
$685$	29326.0	+	29326.0i	1.63575	+	1.63575i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1288.88			−0.0712664
$690$		0			0
$691$	−9179.51	−	9179.51i	−0.505362	−	0.505362i	0.407737	−	0.913099i	$-0.366318\pi$
$691$	−9179.51	−	9179.51i	−0.505362	−	0.505362i	−0.913099	+	0.407737i	$0.866318\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			10466.4i			0.571239i
$696$		0			0
$697$			33387.4i			1.81440i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7741.65	−	7741.65i	−0.417116	−	0.417116i	0.467093	−	0.884208i	$-0.345301\pi$
$701$	−7741.65	−	7741.65i	−0.417116	−	0.417116i	−0.884208	+	0.467093i	$0.845301\pi$
$702$		0			0
$703$	−16673.5			−0.894526
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8297.40	−	8297.40i	−0.441380	−	0.441380i
$708$		0			0
$709$	1092.40	−	1092.40i	0.0578644	−	0.0578644i	−0.677582	−	0.735447i	$-0.736972\pi$
$709$	1092.40	−	1092.40i	0.0578644	−	0.0578644i	0.735447	+	0.677582i	$0.236972\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			21797.7i			1.14492i
$714$		0			0
$715$	4979.88	−	4979.88i	0.260471	−	0.260471i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24231.3			1.25685			0.628424	−	0.777871i	$-0.283700\pi$
$719$	24231.3			1.25685			0.628424	+	0.777871i	$0.283700\pi$
$720$		0			0
$721$	−6083.30			−0.314222
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11687.5	−	11687.5i	0.598709	−	0.598709i
$726$		0			0
$727$		−	20778.8i		−	1.06003i	−0.847989	−	0.530015i	$-0.822186\pi$
$727$		−	20778.8i		−	1.06003i	0.847989	−	0.530015i	$-0.177814\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	27778.6	−	27778.6i	1.40551	−	1.40551i
$732$		0			0
$733$	−9579.11	−	9579.11i	−0.482691	−	0.482691i	0.423299	−	0.905990i	$-0.360872\pi$
$733$	−9579.11	−	9579.11i	−0.482691	−	0.482691i	−0.905990	+	0.423299i	$0.860872\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22231.1			−1.11112
$738$		0			0
$739$	−1525.14	−	1525.14i	−0.0759178	−	0.0759178i	0.668128	−	0.744046i	$-0.267096\pi$
$739$	−1525.14	−	1525.14i	−0.0759178	−	0.0759178i	−0.744046	+	0.668128i	$0.767096\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.91705i		−	0.000390913i	−1.00000		0.000195457i	$-0.999938\pi$
$743$		−	7.91705i		−	0.000390913i	1.00000		0.000195457i	$-6.22158e-5\pi$
$744$		0			0
$745$			11856.3i			0.583064i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	203.477	+	203.477i	0.00992643	+	0.00992643i
$750$		0			0
$751$	3840.63			0.186613			0.0933067	−	0.995637i	$-0.470256\pi$
$751$	3840.63			0.186613			0.0933067	+	0.995637i	$0.470256\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18862.2	+	18862.2i	0.909227	+	0.909227i
$756$		0			0
$757$	−6448.92	+	6448.92i	−0.309630	+	0.309630i	−0.844766	−	0.535136i	$-0.820260\pi$
$757$	−6448.92	+	6448.92i	−0.309630	+	0.309630i	0.535136	+	0.844766i	$0.320260\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			22769.5i			1.08462i	0.840179	+	0.542309i	$0.182450\pi$
$761$			22769.5i			1.08462i	−0.840179	+	0.542309i	$0.817550\pi$
$762$		0			0
$763$	−11861.5	+	11861.5i	−0.562798	+	0.562798i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7676.57			−0.361388
$768$		0			0
$769$	−14324.3			−0.671714			−0.335857	−	0.941913i	$-0.609026\pi$
$769$	−14324.3			−0.671714			−0.335857	+	0.941913i	$0.609026\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8074.35	+	8074.35i	−0.375697	+	0.375697i	−0.869547	−	0.493850i	$-0.835589\pi$
$773$	−8074.35	+	8074.35i	−0.375697	+	0.375697i	0.493850	+	0.869547i	$0.335589\pi$
$774$		0			0
$775$			79067.4i			3.66475i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−32595.6	+	32595.6i	−1.49918	+	1.49918i
$780$		0			0
$781$	−9045.92	−	9045.92i	−0.414454	−	0.414454i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11278.1			−0.512782
$786$		0			0
$787$	16428.0	+	16428.0i	0.744084	+	0.744084i	0.973361	−	0.229277i	$-0.0736363\pi$
$787$	16428.0	+	16428.0i	0.744084	+	0.744084i	−0.229277	+	0.973361i	$0.573636\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	9441.17i		−	0.424386i
$792$		0			0
$793$		−	1912.19i		−	0.0856289i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8701.38	−	8701.38i	−0.386724	−	0.386724i	0.486794	−	0.873517i	$-0.338166\pi$
$797$	−8701.38	−	8701.38i	−0.386724	−	0.386724i	−0.873517	+	0.486794i	$0.838166\pi$
$798$		0			0
$799$	−16061.7			−0.711168
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6756.04	−	6756.04i	−0.296906	−	0.296906i
$804$		0			0
$805$	−13378.5	+	13378.5i	−0.585750	+	0.585750i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	43605.6i		−	1.89504i	−0.319691	−	0.947522i	$-0.603579\pi$
$809$		−	43605.6i		−	1.89504i	0.319691	−	0.947522i	$-0.396421\pi$
$810$		0			0
$811$	−8540.04	+	8540.04i	−0.369768	+	0.369768i	−0.867392	−	0.497625i	$-0.834206\pi$
$811$	−8540.04	+	8540.04i	−0.369768	+	0.369768i	0.497625	+	0.867392i	$0.334206\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	56437.7			2.42567
$816$		0			0
$817$	54239.5			2.32264
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−32464.0	+	32464.0i	−1.38003	+	1.38003i	−0.535477	+	0.844550i	$0.679868\pi$
$821$	−32464.0	+	32464.0i	−1.38003	+	1.38003i	−0.844550	+	0.535477i	$0.820132\pi$
$822$		0			0
$823$		−	9716.22i		−	0.411526i	−0.978602	−	0.205763i	$-0.934032\pi$
$823$		−	9716.22i		−	0.411526i	0.978602	−	0.205763i	$-0.0659676\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25049.7	+	25049.7i	−1.05328	+	1.05328i	−0.0547815	+	0.998498i	$0.517446\pi$
$827$	−25049.7	+	25049.7i	−1.05328	+	1.05328i	−0.998498	+	0.0547815i	$0.982554\pi$
$828$		0			0
$829$	11566.7	+	11566.7i	0.484594	+	0.484594i	0.906595	−	0.422001i	$-0.138672\pi$
$829$	11566.7	+	11566.7i	0.484594	+	0.484594i	−0.422001	+	0.906595i	$0.638672\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19557.9			0.813495
$834$		0			0
$835$	39248.7	+	39248.7i	1.62666	+	1.62666i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			7831.99i			0.322277i	0.986932	+	0.161138i	$0.0515166\pi$
$839$			7831.99i			0.322277i	−0.986932	+	0.161138i	$0.948483\pi$
$840$		0			0
$841$		−	21623.0i		−	0.886586i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−30996.7	−	30996.7i	−1.26191	−	1.26191i
$846$		0			0
$847$	2714.71			0.110128
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8868.58	+	8868.58i	0.357240	+	0.357240i
$852$		0			0
$853$	−28375.6	+	28375.6i	−1.13900	+	1.13900i	−0.150365	+	0.988631i	$0.548045\pi$
$853$	−28375.6	+	28375.6i	−1.13900	+	1.13900i	−0.988631	+	0.150365i	$0.951955\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	16182.0i		−	0.645000i	−0.946569	−	0.322500i	$-0.895477\pi$
$857$		−	16182.0i		−	0.645000i	0.946569	−	0.322500i	$-0.104523\pi$
$858$		0			0
$859$	−3869.82	+	3869.82i	−0.153710	+	0.153710i	−0.779773	−	0.626063i	$-0.784665\pi$
$859$	−3869.82	+	3869.82i	−0.153710	+	0.153710i	0.626063	+	0.779773i	$0.284665\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−40231.6			−1.58691			−0.793453	−	0.608632i	$-0.791719\pi$
$863$	−40231.6			−1.58691			−0.793453	+	0.608632i	$0.791719\pi$
$864$		0			0
$865$	−50659.0			−1.99128
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18629.1	−	18629.1i	0.727213	−	0.727213i
$870$		0			0
$871$		−	6978.49i		−	0.271478i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−29226.4	+	29226.4i	−1.12918	+	1.12918i
$876$		0			0
$877$	9968.05	+	9968.05i	0.383805	+	0.383805i	0.872471	−	0.488666i	$-0.162516\pi$
$877$	9968.05	+	9968.05i	0.383805	+	0.383805i	−0.488666	+	0.872471i	$0.662516\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2638.39			−0.100896			−0.0504481	−	0.998727i	$-0.516065\pi$
$881$	−2638.39			−0.100896			−0.0504481	+	0.998727i	$0.516065\pi$
$882$		0			0
$883$	−1931.38	−	1931.38i	−0.0736081	−	0.0736081i	0.669344	−	0.742952i	$-0.266575\pi$
$883$	−1931.38	−	1931.38i	−0.0736081	−	0.0736081i	−0.742952	+	0.669344i	$0.766575\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			49401.5i			1.87006i	0.354574	+	0.935028i	$0.384626\pi$
$887$			49401.5i			1.87006i	−0.354574	+	0.935028i	$0.615374\pi$
$888$		0			0
$889$		−	13820.1i		−	0.521383i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15680.8	−	15680.8i	−0.587613	−	0.587613i
$894$		0			0
$895$	−80806.5			−3.01795
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9356.29	−	9356.29i	−0.347108	−	0.347108i
$900$		0			0
$901$	−7402.92	+	7402.92i	−0.273726	+	0.273726i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			90555.8i			3.32616i
$906$		0			0
$907$	−22152.3	+	22152.3i	−0.810977	+	0.810977i	−0.984780	−	0.173803i	$-0.944394\pi$
$907$	−22152.3	+	22152.3i	−0.810977	+	0.810977i	0.173803	+	0.984780i	$0.444394\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	50952.6			1.85306			0.926529	−	0.376222i	$-0.122777\pi$
$911$	50952.6			1.85306			0.926529	+	0.376222i	$0.122777\pi$
$912$		0			0
$913$	−10749.1			−0.389642
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13824.8	−	13824.8i	0.497856	−	0.497856i
$918$		0			0
$919$		−	4966.56i		−	0.178272i	−0.996019	−	0.0891359i	$-0.971589\pi$
$919$		−	4966.56i		−	0.178272i	0.996019	−	0.0891359i	$-0.0284105\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2839.57	−	2839.57i	0.101263	−	0.101263i
$924$		0			0
$925$	32169.2	+	32169.2i	1.14348	+	1.14348i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−42097.4			−1.48673			−0.743364	−	0.668887i	$-0.766771\pi$
$929$	−42097.4			−1.48673			−0.743364	+	0.668887i	$0.766771\pi$
$930$		0			0
$931$	19094.1	+	19094.1i	0.672162	+	0.672162i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	57205.6i		−	2.00088i
$936$		0			0
$937$		−	14091.3i		−	0.491294i	−0.969359	−	0.245647i	$-0.921000\pi$
$937$		−	14091.3i		−	0.491294i	0.969359	−	0.245647i	$-0.0790004\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5592.37	−	5592.37i	−0.193737	−	0.193737i	0.603572	−	0.797309i	$-0.293744\pi$
$941$	−5592.37	−	5592.37i	−0.193737	−	0.193737i	−0.797309	+	0.603572i	$0.793744\pi$
$942$		0			0
$943$	34675.0			1.19743
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12885.8	+	12885.8i	0.442166	+	0.442166i	0.892739	−	0.450573i	$-0.148780\pi$
$947$	12885.8	+	12885.8i	0.442166	+	0.442166i	−0.450573	+	0.892739i	$0.648780\pi$
$948$		0			0
$949$	2120.76	−	2120.76i	0.0725425	−	0.0725425i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			18107.6i			0.615492i	0.951469	+	0.307746i	$0.0995746\pi$
$953$			18107.6i			0.615492i	−0.951469	+	0.307746i	$0.900425\pi$
$954$		0			0
$955$	59774.4	−	59774.4i	2.02540	−	2.02540i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	20617.3			0.694230
$960$		0			0
$961$	33505.3			1.12468
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2367.19	−	2367.19i	0.0789665	−	0.0789665i
$966$		0			0
$967$			16125.0i			0.536241i	0.963385	+	0.268120i	$0.0864025\pi$
$967$			16125.0i			0.536241i	−0.963385	+	0.268120i	$0.913597\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	893.116	−	893.116i	0.0295174	−	0.0295174i	−0.692194	−	0.721711i	$-0.743356\pi$
$971$	893.116	−	893.116i	0.0295174	−	0.0295174i	0.721711	+	0.692194i	$0.243356\pi$
$972$		0			0
$973$	3679.12	+	3679.12i	0.121220	+	0.121220i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	46541.5			1.52405			0.762024	−	0.647549i	$-0.224206\pi$
$977$	46541.5			1.52405			0.762024	+	0.647549i	$0.224206\pi$
$978$		0			0
$979$	3311.59	+	3311.59i	0.108109	+	0.108109i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	30204.7i		−	0.980041i	−0.871711	−	0.490020i	$-0.836989\pi$
$983$		−	30204.7i		−	0.980041i	0.871711	−	0.490020i	$-0.163011\pi$
$984$		0			0
$985$		−	1311.17i		−	0.0424136i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−28849.9	−	28849.9i	−0.927576	−	0.927576i
$990$		0			0
$991$	23193.1			0.743446			0.371723	−	0.928344i	$-0.378767\pi$
$991$	23193.1			0.743446			0.371723	+	0.928344i	$0.378767\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−23561.6	−	23561.6i	−0.750705	−	0.750705i
$996$		0			0
$997$	−17708.8	+	17708.8i	−0.562529	+	0.562529i	−0.930025	−	0.367496i	$-0.880215\pi$
$997$	−17708.8	+	17708.8i	−0.562529	+	0.562529i	0.367496	+	0.930025i	$0.380215\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.k.c.145.12		24
3.2	odd	2	inner	576.4.k.c.145.1		24
4.3	odd	2		144.4.k.c.109.9	yes	24
12.11	even	2		144.4.k.c.109.4	yes	24
16.5	even	4	inner	576.4.k.c.433.12		24
16.11	odd	4		144.4.k.c.37.9	yes	24
48.5	odd	4	inner	576.4.k.c.433.1		24
48.11	even	4		144.4.k.c.37.4	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.k.c.37.4	✓	24	48.11	even	4
144.4.k.c.37.9	yes	24	16.11	odd	4
144.4.k.c.109.4	yes	24	12.11	even	2
144.4.k.c.109.9	yes	24	4.3	odd	2
576.4.k.c.145.1		24	3.2	odd	2	inner
576.4.k.c.145.12		24	1.1	even	1	trivial
576.4.k.c.433.1		24	48.5	odd	4	inner
576.4.k.c.433.12		24	16.5	even	4	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(14.8202 - 14.8202i) q^{5} -10.4191i q^{7} +(23.1349 - 23.1349i) q^{11} +(7.26220 + 7.26220i) q^{13} +83.4233 q^{17} +(81.4449 + 81.4449i) q^{19} -86.6407i q^{23} -314.274i q^{25} +(37.1890 + 37.1890i) q^{29} -251.588 q^{31} +(-154.413 - 154.413i) q^{35} +(-102.360 + 102.360i) q^{37} +400.216i q^{41} +(332.983 - 332.983i) q^{43} -192.533 q^{47} +234.442 q^{49} +(-88.7392 + 88.7392i) q^{53} -685.726i q^{55} +(-528.529 + 528.529i) q^{59} +(-131.653 - 131.653i) q^{61} +215.254 q^{65} +(-480.467 - 480.467i) q^{67} -391.007i q^{71} -292.028i q^{73} +(-241.046 - 241.046i) q^{77} +805.236 q^{79} +(-232.313 - 232.313i) q^{83} +(1236.35 - 1236.35i) q^{85} +143.143i q^{89} +(75.6658 - 75.6658i) q^{91} +2414.05 q^{95} -733.540 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{19} - 744 q^{31} - 16 q^{37} + 376 q^{43} - 1176 q^{49} - 912 q^{61} - 1440 q^{67} + 328 q^{79} - 240 q^{85} + 104 q^{91}+O(q^{100}) \) 24 * q - 24 * q^19 - 744 * q^31 - 16 * q^37 + 376 * q^43 - 1176 * q^49 - 912 * q^61 - 1440 * q^67 + 328 * q^79 - 240 * q^85 + 104 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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