LMFDB - Embedded newform 576.4.k.c.145.8

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			145.8
Character	$\chi$	$=$	576.145
Dual form			576.4.k.c.433.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.16526 - 5.16526i) q^{5} -7.03833i q^{7} +(-22.2198 + 22.2198i) q^{11} +(37.2029 + 37.2029i) q^{13} +67.4608 q^{17} +(-37.2366 - 37.2366i) q^{19} +32.4348i q^{23} +71.6401i q^{25} +(94.7734 + 94.7734i) q^{29} +158.503 q^{31} +(-36.3548 - 36.3548i) q^{35} +(192.501 - 192.501i) q^{37} -283.277i q^{41} +(206.310 - 206.310i) q^{43} -584.883 q^{47} +293.462 q^{49} +(82.5251 - 82.5251i) q^{53} +229.542i q^{55} +(-79.7567 + 79.7567i) q^{59} +(132.394 + 132.394i) q^{61} +384.325 q^{65} +(596.157 + 596.157i) q^{67} -110.487i q^{71} -412.677i q^{73} +(156.390 + 156.390i) q^{77} +1142.41 q^{79} +(656.133 + 656.133i) q^{83} +(348.453 - 348.453i) q^{85} -474.926i q^{89} +(261.846 - 261.846i) q^{91} -384.673 q^{95} +1358.31 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	5.16526	−	5.16526i	0.461995	−	0.461995i	−0.437314	−	0.899309i	$-0.644070\pi$
$5$	5.16526	−	5.16526i	0.461995	−	0.461995i	0.899309	+	0.437314i	$0.144070\pi$
$6$		0			0
$7$		−	7.03833i		−	0.380034i	−0.981781	−	0.190017i	$-0.939146\pi$
$7$		−	7.03833i		−	0.380034i	0.981781	−	0.190017i	$-0.0608543\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−22.2198	+	22.2198i	−0.609047	+	0.609047i	−0.942697	−	0.333650i	$-0.891720\pi$
$11$	−22.2198	+	22.2198i	−0.609047	+	0.609047i	0.333650	+	0.942697i	$0.391720\pi$
$12$		0			0
$13$	37.2029	+	37.2029i	0.793710	+	0.793710i	0.982095	−	0.188386i	$-0.0603255\pi$
$13$	37.2029	+	37.2029i	0.793710	+	0.793710i	−0.188386	+	0.982095i	$0.560325\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	67.4608			0.962450			0.481225	−	0.876597i	$-0.340192\pi$
$17$	67.4608			0.962450			0.481225	+	0.876597i	$0.340192\pi$
$18$		0			0
$19$	−37.2366	−	37.2366i	−0.449613	−	0.449613i	0.445613	−	0.895226i	$-0.352986\pi$
$19$	−37.2366	−	37.2366i	−0.449613	−	0.449613i	−0.895226	+	0.445613i	$0.852986\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			32.4348i			0.294049i	0.989133	+	0.147024i	$0.0469696\pi$
$23$			32.4348i			0.294049i	−0.989133	+	0.147024i	$0.953030\pi$
$24$		0			0
$25$			71.6401i			0.573121i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	94.7734	+	94.7734i	0.606861	+	0.606861i	0.942125	−	0.335263i	$-0.108825\pi$
$29$	94.7734	+	94.7734i	0.606861	+	0.606861i	−0.335263	+	0.942125i	$0.608825\pi$
$30$		0			0
$31$	158.503			0.918324			0.459162	−	0.888353i	$-0.348150\pi$
$31$	158.503			0.918324			0.459162	+	0.888353i	$0.348150\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−36.3548	−	36.3548i	−0.175574	−	0.175574i
$36$		0			0
$37$	192.501	−	192.501i	0.855323	−	0.855323i	−0.135460	−	0.990783i	$-0.543251\pi$
$37$	192.501	−	192.501i	0.855323	−	0.855323i	0.990783	+	0.135460i	$0.0432511\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	283.277i		−	1.07903i	−0.841975	−	0.539517i	$-0.818607\pi$
$41$		−	283.277i		−	1.07903i	0.841975	−	0.539517i	$-0.181393\pi$
$42$		0			0
$43$	206.310	−	206.310i	0.731672	−	0.731672i	−0.239279	−	0.970951i	$-0.576911\pi$
$43$	206.310	−	206.310i	0.731672	−	0.731672i	0.970951	+	0.239279i	$0.0769109\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−584.883			−1.81519			−0.907596	−	0.419845i	$-0.862085\pi$
$47$	−584.883			−1.81519			−0.907596	+	0.419845i	$0.862085\pi$
$48$		0			0
$49$	293.462			0.855574
$50$		0			0
$51$		0			0
$52$		0			0
$53$	82.5251	−	82.5251i	0.213881	−	0.213881i	−0.592033	−	0.805914i	$-0.701674\pi$
$53$	82.5251	−	82.5251i	0.213881	−	0.213881i	0.805914	+	0.592033i	$0.201674\pi$
$54$		0			0
$55$			229.542i			0.562754i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−79.7567	+	79.7567i	−0.175990	+	0.175990i	−0.789605	−	0.613615i	$-0.789715\pi$
$59$	−79.7567	+	79.7567i	−0.175990	+	0.175990i	0.613615	+	0.789605i	$0.289715\pi$
$60$		0			0
$61$	132.394	+	132.394i	0.277889	+	0.277889i	0.832266	−	0.554377i	$-0.187043\pi$
$61$	132.394	+	132.394i	0.277889	+	0.277889i	−0.554377	+	0.832266i	$0.687043\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	384.325			0.733380
$66$		0			0
$67$	596.157	+	596.157i	1.08705	+	1.08705i	0.995831	+	0.0912161i	$0.0290754\pi$
$67$	596.157	+	596.157i	1.08705	+	1.08705i	0.0912161	+	0.995831i	$0.470925\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	110.487i		−	0.184682i	−0.995727	−	0.0923409i	$-0.970565\pi$
$71$		−	110.487i		−	0.184682i	0.995727	−	0.0923409i	$-0.0294350\pi$
$72$		0			0
$73$		−	412.677i		−	0.661646i	−0.943693	−	0.330823i	$-0.892674\pi$
$73$		−	412.677i		−	0.661646i	0.943693	−	0.330823i	$-0.107326\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	156.390	+	156.390i	0.231459	+	0.231459i
$78$		0			0
$79$	1142.41			1.62698			0.813491	−	0.581577i	$-0.197564\pi$
$79$	1142.41			1.62698			0.813491	+	0.581577i	$0.197564\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	656.133	+	656.133i	0.867711	+	0.867711i	0.992219	−	0.124508i	$-0.0397352\pi$
$83$	656.133	+	656.133i	0.867711	+	0.867711i	−0.124508	+	0.992219i	$0.539735\pi$
$84$		0			0
$85$	348.453	−	348.453i	0.444647	−	0.444647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	474.926i		−	0.565641i	−0.959173	−	0.282820i	$-0.908730\pi$
$89$		−	474.926i		−	0.565641i	0.959173	−	0.282820i	$-0.0912700\pi$
$90$		0			0
$91$	261.846	−	261.846i	0.301637	−	0.301637i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−384.673			−0.415438
$96$		0			0
$97$	1358.31			1.42181			0.710906	−	0.703287i	$-0.248285\pi$
$97$	1358.31			1.42181			0.710906	+	0.703287i	$0.248285\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1345.17	+	1345.17i	−1.32524	+	1.32524i	−0.415771	+	0.909470i	$0.636488\pi$
$101$	−1345.17	+	1345.17i	−1.32524	+	1.32524i	−0.909470	+	0.415771i	$0.863512\pi$
$102$		0			0
$103$		−	1077.73i		−	1.03099i	−0.856894	−	0.515493i	$-0.827609\pi$
$103$		−	1077.73i		−	1.03099i	0.856894	−	0.515493i	$-0.172391\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1505.13	−	1505.13i	1.35988	−	1.35988i	0.485816	−	0.874061i	$-0.338523\pi$
$107$	1505.13	−	1505.13i	1.35988	−	1.35988i	0.874061	−	0.485816i	$-0.161477\pi$
$108$		0			0
$109$	885.884	+	885.884i	0.778462	+	0.778462i	0.979569	−	0.201107i	$-0.0644540\pi$
$109$	885.884	+	885.884i	0.778462	+	0.778462i	−0.201107	+	0.979569i	$0.564454\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1002.89			−0.834902			−0.417451	−	0.908699i	$-0.637076\pi$
$113$	−1002.89			−0.834902			−0.417451	+	0.908699i	$0.637076\pi$
$114$		0			0
$115$	167.534	+	167.534i	0.135849	+	0.135849i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	474.812i		−	0.365764i
$120$		0			0
$121$			343.561i			0.258122i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1015.70	+	1015.70i	0.726774	+	0.726774i
$126$		0			0
$127$	−657.435			−0.459354			−0.229677	−	0.973267i	$-0.573767\pi$
$127$	−657.435			−0.459354			−0.229677	+	0.973267i	$0.573767\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1708.93	+	1708.93i	1.13977	+	1.13977i	0.988491	+	0.151279i	$0.0483392\pi$
$131$	1708.93	+	1708.93i	1.13977	+	1.13977i	0.151279	+	0.988491i	$0.451661\pi$
$132$		0			0
$133$	−262.083	+	262.083i	−0.170868	+	0.170868i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2990.96i		−	1.86522i	−0.360891	−	0.932608i	$-0.617527\pi$
$137$		−	2990.96i		−	1.86522i	0.360891	−	0.932608i	$-0.382473\pi$
$138$		0			0
$139$	898.789	−	898.789i	0.548448	−	0.548448i	−0.377543	−	0.925992i	$-0.623231\pi$
$139$	898.789	−	898.789i	0.548448	−	0.548448i	0.925992	+	0.377543i	$0.123231\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1653.28			−0.966814
$144$		0			0
$145$	979.059			0.560734
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1949.10	+	1949.10i	−1.07165	+	1.07165i	−0.0744264	+	0.997227i	$0.523713\pi$
$149$	−1949.10	+	1949.10i	−1.07165	+	1.07165i	−0.997227	+	0.0744264i	$0.976287\pi$
$150$		0			0
$151$			389.789i			0.210070i	0.994469	+	0.105035i	$0.0334955\pi$
$151$			389.789i			0.210070i	−0.994469	+	0.105035i	$0.966505\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	818.712	−	818.712i	0.424261	−	0.424261i
$156$		0			0
$157$	−573.333	−	573.333i	−0.291446	−	0.291446i	0.546206	−	0.837651i	$-0.316072\pi$
$157$	−573.333	−	573.333i	−0.291446	−	0.291446i	−0.837651	+	0.546206i	$0.816072\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	228.287			0.111749
$162$		0			0
$163$	−94.7841	−	94.7841i	−0.0455464	−	0.0455464i	0.683967	−	0.729513i	$-0.260253\pi$
$163$	−94.7841	−	94.7841i	−0.0455464	−	0.0455464i	−0.729513	+	0.683967i	$0.760253\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	396.734i		−	0.183834i	−0.995767	−	0.0919169i	$-0.970701\pi$
$167$		−	396.734i		−	0.183834i	0.995767	−	0.0919169i	$-0.0292994\pi$
$168$		0			0
$169$			571.109i			0.259950i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−193.034	−	193.034i	−0.0848329	−	0.0848329i	0.663417	−	0.748250i	$-0.269106\pi$
$173$	−193.034	−	193.034i	−0.0848329	−	0.0848329i	−0.748250	+	0.663417i	$0.769106\pi$
$174$		0			0
$175$	504.227			0.217805
$176$		0			0
$177$		0			0
$178$		0			0
$179$	114.931	+	114.931i	0.0479909	+	0.0479909i	0.730695	−	0.682704i	$-0.239196\pi$
$179$	114.931	+	114.931i	0.0479909	+	0.0479909i	−0.682704	+	0.730695i	$0.739196\pi$
$180$		0			0
$181$	764.431	−	764.431i	0.313921	−	0.313921i	−0.532505	−	0.846427i	$-0.678749\pi$
$181$	764.431	−	764.431i	0.313921	−	0.313921i	0.846427	+	0.532505i	$0.178749\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	1988.64i		−	0.790310i
$186$		0			0
$187$	−1498.97	+	1498.97i	−0.586178	+	0.586178i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2413.08			−0.914159			−0.457080	−	0.889426i	$-0.651105\pi$
$191$	−2413.08			−0.914159			−0.457080	+	0.889426i	$0.651105\pi$
$192$		0			0
$193$	−4996.88			−1.86364			−0.931822	−	0.362915i	$-0.881781\pi$
$193$	−4996.88			−1.86364			−0.931822	+	0.362915i	$0.881781\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1208.70	+	1208.70i	−0.437138	+	0.437138i	−0.891048	−	0.453910i	$-0.850029\pi$
$197$	−1208.70	+	1208.70i	−0.437138	+	0.437138i	0.453910	+	0.891048i	$0.350029\pi$
$198$		0			0
$199$			3020.23i			1.07587i	0.842986	+	0.537935i	$0.180795\pi$
$199$			3020.23i			1.07587i	−0.842986	+	0.537935i	$0.819205\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	667.047	−	667.047i	0.230628	−	0.230628i
$204$		0			0
$205$	−1463.20	−	1463.20i	−0.498508	−	0.498508i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1654.78			0.547672
$210$		0			0
$211$	−1874.16	−	1874.16i	−0.611482	−	0.611482i	0.331850	−	0.943332i	$-0.392327\pi$
$211$	−1874.16	−	1874.16i	−0.611482	−	0.611482i	−0.943332	+	0.331850i	$0.892327\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2131.29i		−	0.676058i
$216$		0			0
$217$		−	1115.60i		−	0.348994i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2509.74	+	2509.74i	0.763906	+	0.763906i
$222$		0			0
$223$	−974.233			−0.292554			−0.146277	−	0.989244i	$-0.546729\pi$
$223$	−974.233			−0.292554			−0.146277	+	0.989244i	$0.546729\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3640.99	+	3640.99i	1.06459	+	1.06459i	0.997765	+	0.0668201i	$0.0212854\pi$
$227$	3640.99	+	3640.99i	1.06459	+	1.06459i	0.0668201	+	0.997765i	$0.478715\pi$
$228$		0			0
$229$	−1480.42	+	1480.42i	−0.427202	+	0.427202i	−0.887674	−	0.460472i	$-0.847680\pi$
$229$	−1480.42	+	1480.42i	−0.427202	+	0.427202i	0.460472	+	0.887674i	$0.347680\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1623.75i		−	0.456545i	−0.973597	−	0.228273i	$-0.926692\pi$
$233$		−	1623.75i		−	0.456545i	0.973597	−	0.228273i	$-0.0733078\pi$
$234$		0			0
$235$	−3021.08	+	3021.08i	−0.838610	+	0.838610i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6868.93			−1.85905			−0.929527	−	0.368753i	$-0.879785\pi$
$239$	−6868.93			−1.85905			−0.929527	+	0.368753i	$0.879785\pi$
$240$		0			0
$241$	−2767.43			−0.739693			−0.369847	−	0.929093i	$-0.620590\pi$
$241$	−2767.43			−0.739693			−0.369847	+	0.929093i	$0.620590\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1515.81	−	1515.81i	0.395271	−	0.395271i
$246$		0			0
$247$		−	2770.61i		−	0.713725i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−35.4718	+	35.4718i	−0.00892016	+	0.00892016i	−0.711553	−	0.702633i	$-0.752008\pi$
$251$	−35.4718	+	35.4718i	−0.00892016	+	0.00892016i	0.702633	+	0.711553i	$0.252008\pi$
$252$		0			0
$253$	−720.694	−	720.694i	−0.179090	−	0.179090i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3415.73			0.829056			0.414528	−	0.910037i	$-0.363947\pi$
$257$	3415.73			0.829056			0.414528	+	0.910037i	$0.363947\pi$
$258$		0			0
$259$	−1354.89	−	1354.89i	−0.325052	−	0.325052i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			6780.71i			1.58980i	0.606742	+	0.794899i	$0.292476\pi$
$263$			6780.71i			1.58980i	−0.606742	+	0.794899i	$0.707524\pi$
$264$		0			0
$265$		−	852.528i		−	0.197624i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3834.98	+	3834.98i	0.869229	+	0.869229i	0.992387	−	0.123158i	$-0.0393022\pi$
$269$	3834.98	+	3834.98i	0.869229	+	0.869229i	−0.123158	+	0.992387i	$0.539302\pi$
$270$		0			0
$271$	−7031.32			−1.57610			−0.788048	−	0.615614i	$-0.788908\pi$
$271$	−7031.32			−1.57610			−0.788048	+	0.615614i	$0.788908\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1591.83	−	1591.83i	−0.349058	−	0.349058i
$276$		0			0
$277$	−3030.30	+	3030.30i	−0.657304	+	0.657304i	−0.954741	−	0.297437i	$-0.903868\pi$
$277$	−3030.30	+	3030.30i	−0.657304	+	0.657304i	0.297437	+	0.954741i	$0.403868\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	4719.06i		−	1.00183i	−0.865495	−	0.500917i	$-0.832996\pi$
$281$		−	4719.06i		−	1.00183i	0.865495	−	0.500917i	$-0.167004\pi$
$282$		0			0
$283$	2696.52	−	2696.52i	0.566400	−	0.566400i	−0.364718	−	0.931118i	$-0.618835\pi$
$283$	2696.52	−	2696.52i	0.566400	−	0.566400i	0.931118	+	0.364718i	$0.118835\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1993.79			−0.410070
$288$		0			0
$289$	−362.038			−0.0736898
$290$		0			0
$291$		0			0
$292$		0			0
$293$	865.453	−	865.453i	0.172561	−	0.172561i	−0.615543	−	0.788103i	$-0.711063\pi$
$293$	865.453	−	865.453i	0.172561	−	0.172561i	0.788103	+	0.615543i	$0.211063\pi$
$294$		0			0
$295$			823.929i			0.162614i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1206.67	+	1206.67i	−0.233389	+	0.233389i
$300$		0			0
$301$	−1452.08	−	1452.08i	−0.278061	−	0.278061i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1367.70			0.256767
$306$		0			0
$307$	−703.076	−	703.076i	−0.130706	−	0.130706i	0.638727	−	0.769433i	$-0.279461\pi$
$307$	−703.076	−	703.076i	−0.130706	−	0.130706i	−0.769433	+	0.638727i	$0.779461\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6484.26i		−	1.18228i	−0.806570	−	0.591139i	$-0.798678\pi$
$311$		−	6484.26i		−	1.18228i	0.806570	−	0.591139i	$-0.201322\pi$
$312$		0			0
$313$			7115.56i			1.28497i	0.766299	+	0.642484i	$0.222096\pi$
$313$			7115.56i			1.28497i	−0.766299	+	0.642484i	$0.777904\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5573.78	−	5573.78i	−0.987554	−	0.987554i	0.0123690	−	0.999924i	$-0.496063\pi$
$317$	−5573.78	−	5573.78i	−0.987554	−	0.987554i	−0.999924	+	0.0123690i	$0.996063\pi$
$318$		0			0
$319$	−4211.69			−0.739215
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2512.01	−	2512.01i	−0.432730	−	0.432730i
$324$		0			0
$325$	−2665.22	+	2665.22i	−0.454891	+	0.454891i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			4116.60i			0.689835i
$330$		0			0
$331$	−5774.02	+	5774.02i	−0.958819	+	0.958819i	−0.999185	−	0.0403663i	$-0.987148\pi$
$331$	−5774.02	+	5774.02i	−0.958819	+	0.958819i	0.0403663	+	0.999185i	$0.487148\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6158.62			1.00442
$336$		0			0
$337$	2111.30			0.341275			0.170638	−	0.985334i	$-0.445417\pi$
$337$	2111.30			0.341275			0.170638	+	0.985334i	$0.445417\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3521.91	+	3521.91i	−0.559303	+	0.559303i
$342$		0			0
$343$		−	4479.63i		−	0.705182i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2375.98	+	2375.98i	−0.367577	+	0.367577i	−0.866593	−	0.499016i	$-0.833695\pi$
$347$	−2375.98	+	2375.98i	−0.367577	+	0.367577i	0.499016	+	0.866593i	$0.333695\pi$
$348$		0			0
$349$	−212.228	−	212.228i	−0.0325511	−	0.0325511i	0.690644	−	0.723195i	$-0.257327\pi$
$349$	−212.228	−	212.228i	−0.0325511	−	0.0325511i	−0.723195	+	0.690644i	$0.757327\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8492.56			−1.28049			−0.640246	−	0.768170i	$-0.721167\pi$
$353$	−8492.56			−1.28049			−0.640246	+	0.768170i	$0.721167\pi$
$354$		0			0
$355$	−570.695	−	570.695i	−0.0853221	−	0.0853221i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	2300.75i		−	0.338242i	−0.985595	−	0.169121i	$-0.945907\pi$
$359$		−	2300.75i		−	0.338242i	0.985595	−	0.169121i	$-0.0540929\pi$
$360$		0			0
$361$		−	4085.88i		−	0.595696i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2131.58	−	2131.58i	−0.305677	−	0.305677i
$366$		0			0
$367$	4035.86			0.574033			0.287016	−	0.957926i	$-0.407337\pi$
$367$	4035.86			0.574033			0.287016	+	0.957926i	$0.407337\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−580.839	−	580.839i	−0.0812821	−	0.0812821i
$372$		0			0
$373$	1491.28	−	1491.28i	0.207013	−	0.207013i	−0.595984	−	0.802997i	$-0.703238\pi$
$373$	1491.28	−	1491.28i	0.207013	−	0.207013i	0.802997	+	0.595984i	$0.203238\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			7051.69i			0.963343i
$378$		0			0
$379$	4343.13	−	4343.13i	0.588632	−	0.588632i	−0.348629	−	0.937261i	$-0.613353\pi$
$379$	4343.13	−	4343.13i	0.588632	−	0.588632i	0.937261	+	0.348629i	$0.113353\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13342.2			1.78004			0.890022	−	0.455917i	$-0.150689\pi$
$383$	13342.2			1.78004			0.890022	+	0.455917i	$0.150689\pi$
$384$		0			0
$385$	1615.59			0.213866
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3336.92	−	3336.92i	0.434932	−	0.434932i	−0.455370	−	0.890302i	$-0.650493\pi$
$389$	3336.92	−	3336.92i	0.434932	−	0.434932i	0.890302	+	0.455370i	$0.150493\pi$
$390$		0			0
$391$			2188.08i			0.283007i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5900.87	−	5900.87i	0.751658	−	0.751658i
$396$		0			0
$397$	3228.02	+	3228.02i	0.408085	+	0.408085i	0.881070	−	0.472986i	$-0.156824\pi$
$397$	3228.02	+	3228.02i	0.408085	+	0.408085i	−0.472986	+	0.881070i	$0.656824\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	52.2560			0.00650759			0.00325379	−	0.999995i	$-0.498964\pi$
$401$	52.2560			0.00650759			0.00325379	+	0.999995i	$0.498964\pi$
$402$		0			0
$403$	5896.78	+	5896.78i	0.728882	+	0.728882i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8554.66i			1.04186i
$408$		0			0
$409$		−	14516.7i		−	1.75502i	−0.479558	−	0.877510i	$-0.659203\pi$
$409$		−	14516.7i		−	1.75502i	0.479558	−	0.877510i	$-0.340797\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	561.354	+	561.354i	0.0668824	+	0.0668824i
$414$		0			0
$415$	6778.20			0.801757
$416$		0			0
$417$		0			0
$418$		0			0
$419$	409.663	+	409.663i	0.0477646	+	0.0477646i	0.730586	−	0.682821i	$-0.239247\pi$
$419$	409.663	+	409.663i	0.0477646	+	0.0477646i	−0.682821	+	0.730586i	$0.739247\pi$
$420$		0			0
$421$	4465.76	−	4465.76i	0.516978	−	0.516978i	−0.399678	−	0.916656i	$-0.630878\pi$
$421$	4465.76	−	4465.76i	0.516978	−	0.516978i	0.916656	+	0.399678i	$0.130878\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			4832.90i			0.551600i
$426$		0			0
$427$	931.830	−	931.830i	0.105607	−	0.105607i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2298.21			0.256846			0.128423	−	0.991719i	$-0.459008\pi$
$431$	2298.21			0.256846			0.128423	+	0.991719i	$0.459008\pi$
$432$		0			0
$433$	−4951.41			−0.549538			−0.274769	−	0.961510i	$-0.588601\pi$
$433$	−4951.41			−0.549538			−0.274769	+	0.961510i	$0.588601\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1207.76	−	1207.76i	0.132208	−	0.132208i
$438$		0			0
$439$			2937.90i			0.319404i	0.987165	+	0.159702i	$0.0510533\pi$
$439$			2937.90i			0.319404i	−0.987165	+	0.159702i	$0.948947\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4175.48	+	4175.48i	−0.447818	+	0.447818i	−0.894628	−	0.446811i	$-0.852560\pi$
$443$	−4175.48	+	4175.48i	−0.447818	+	0.447818i	0.446811	+	0.894628i	$0.352560\pi$
$444$		0			0
$445$	−2453.12	−	2453.12i	−0.261323	−	0.261323i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6835.79			−0.718487			−0.359244	−	0.933244i	$-0.616965\pi$
$449$	−6835.79			−0.718487			−0.359244	+	0.933244i	$0.616965\pi$
$450$		0			0
$451$	6294.35	+	6294.35i	0.657183	+	0.657183i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	2705.01i		−	0.278709i
$456$		0			0
$457$			11521.0i			1.17928i	0.807668	+	0.589638i	$0.200729\pi$
$457$			11521.0i			1.17928i	−0.807668	+	0.589638i	$0.799271\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4875.43	−	4875.43i	−0.492563	−	0.492563i	0.416550	−	0.909113i	$-0.363239\pi$
$461$	−4875.43	−	4875.43i	−0.492563	−	0.492563i	−0.909113	+	0.416550i	$0.863239\pi$
$462$		0			0
$463$	13485.6			1.35362			0.676812	−	0.736156i	$-0.263361\pi$
$463$	13485.6			1.35362			0.676812	+	0.736156i	$0.263361\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6043.84	−	6043.84i	−0.598877	−	0.598877i	0.341137	−	0.940014i	$-0.389188\pi$
$467$	−6043.84	−	6043.84i	−0.598877	−	0.598877i	−0.940014	+	0.341137i	$0.889188\pi$
$468$		0			0
$469$	4195.95	−	4195.95i	0.413115	−	0.413115i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			9168.32i			0.891246i
$474$		0			0
$475$	2667.63	−	2667.63i	0.257683	−	0.257683i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12502.9			1.19263			0.596315	−	0.802750i	$-0.296631\pi$
$479$	12502.9			1.19263			0.596315	+	0.802750i	$0.296631\pi$
$480$		0			0
$481$	14323.2			1.35776
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7016.05	−	7016.05i	0.656871	−	0.656871i
$486$		0			0
$487$			10204.1i			0.949469i	0.880129	+	0.474734i	$0.157456\pi$
$487$			10204.1i			0.949469i	−0.880129	+	0.474734i	$0.842544\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3102.01	+	3102.01i	−0.285116	+	0.285116i	−0.835145	−	0.550030i	$-0.814616\pi$
$491$	−3102.01	+	3102.01i	−0.285116	+	0.285116i	0.550030	+	0.835145i	$0.314616\pi$
$492$		0			0
$493$	6393.49	+	6393.49i	0.584074	+	0.584074i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−777.645			−0.0701854
$498$		0			0
$499$	−13549.5	−	13549.5i	−1.21555	−	1.21555i	−0.969174	−	0.246379i	$-0.920759\pi$
$499$	−13549.5	−	13549.5i	−1.21555	−	1.21555i	−0.246379	−	0.969174i	$-0.579241\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			3300.94i			0.292607i	0.989240	+	0.146304i	$0.0467377\pi$
$503$			3300.94i			0.292607i	−0.989240	+	0.146304i	$0.953262\pi$
$504$		0			0
$505$			13896.3i			1.22451i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10081.2	−	10081.2i	−0.877880	−	0.877880i	0.115435	−	0.993315i	$-0.463174\pi$
$509$	−10081.2	−	10081.2i	−0.877880	−	0.877880i	−0.993315	+	0.115435i	$0.963174\pi$
$510$		0			0
$511$	−2904.55			−0.251448
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5566.74	−	5566.74i	−0.476311	−	0.476311i
$516$		0			0
$517$	12996.0	−	12996.0i	1.10554	−	1.10554i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			22396.4i			1.88331i	0.336579	+	0.941655i	$0.390730\pi$
$521$			22396.4i			1.88331i	−0.336579	+	0.941655i	$0.609270\pi$
$522$		0			0
$523$	10114.9	−	10114.9i	0.845688	−	0.845688i	−0.143904	−	0.989592i	$-0.545966\pi$
$523$	10114.9	−	10114.9i	0.845688	−	0.845688i	0.989592	+	0.143904i	$0.0459657\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10692.8			0.883841
$528$		0			0
$529$	11115.0			0.913535
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10538.7	−	10538.7i	0.856439	−	0.856439i
$534$		0			0
$535$		−	15548.8i		−	1.25651i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6520.66	+	6520.66i	−0.521085	+	0.521085i
$540$		0			0
$541$	10004.1	+	10004.1i	0.795024	+	0.795024i	0.982306	−	0.187282i	$-0.0599679\pi$
$541$	10004.1	+	10004.1i	0.795024	+	0.795024i	−0.187282	+	0.982306i	$0.559968\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9151.65			0.719291
$546$		0			0
$547$	11671.6	+	11671.6i	0.912325	+	0.912325i	0.996455	−	0.0841298i	$-0.0268110\pi$
$547$	11671.6	+	11671.6i	0.912325	+	0.912325i	−0.0841298	+	0.996455i	$0.526811\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	7058.07i		−	0.545706i
$552$		0			0
$553$		−	8040.69i		−	0.618309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	117.575	+	117.575i	0.00894401	+	0.00894401i	0.711565	−	0.702621i	$-0.247987\pi$
$557$	117.575	+	117.575i	0.00894401	+	0.00894401i	−0.702621	+	0.711565i	$0.747987\pi$
$558$		0			0
$559$	15350.6			1.16147
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6510.39	−	6510.39i	−0.487354	−	0.487354i	0.420116	−	0.907470i	$-0.361989\pi$
$563$	−6510.39	−	6510.39i	−0.487354	−	0.487354i	−0.907470	+	0.420116i	$0.861989\pi$
$564$		0			0
$565$	−5180.19	+	5180.19i	−0.385721	+	0.385721i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1080.96i			0.0796421i	0.999207	+	0.0398210i	$0.0126788\pi$
$569$			1080.96i			0.0796421i	−0.999207	+	0.0398210i	$0.987321\pi$
$570$		0			0
$571$	6064.11	−	6064.11i	0.444440	−	0.444440i	−0.449061	−	0.893501i	$-0.648242\pi$
$571$	6064.11	−	6064.11i	0.444440	−	0.444440i	0.893501	+	0.449061i	$0.148242\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2323.63			−0.168525
$576$		0			0
$577$	75.0142			0.00541227			0.00270614	−	0.999996i	$-0.499139\pi$
$577$	75.0142			0.00541227			0.00270614	+	0.999996i	$0.499139\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4618.08	−	4618.08i	0.329760	−	0.329760i
$582$		0			0
$583$			3667.38i			0.260527i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15487.5	+	15487.5i	−1.08899	+	1.08899i	−0.0933554	+	0.995633i	$0.529759\pi$
$587$	−15487.5	+	15487.5i	−1.08899	+	1.08899i	−0.995633	+	0.0933554i	$0.970241\pi$
$588$		0			0
$589$	−5902.12	−	5902.12i	−0.412891	−	0.412891i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6675.53			−0.462278			−0.231139	−	0.972921i	$-0.574245\pi$
$593$	−6675.53			−0.462278			−0.231139	+	0.972921i	$0.574245\pi$
$594$		0			0
$595$	−2452.53	−	2452.53i	−0.168981	−	0.168981i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			4338.77i			0.295955i	0.988991	+	0.147978i	$0.0472763\pi$
$599$			4338.77i			0.295955i	−0.988991	+	0.147978i	$0.952724\pi$
$600$		0			0
$601$		−	4339.37i		−	0.294520i	−0.989098	−	0.147260i	$-0.952955\pi$
$601$		−	4339.37i		−	0.294520i	0.989098	−	0.147260i	$-0.0470455\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1774.58	+	1774.58i	0.119251	+	0.119251i
$606$		0			0
$607$	−19248.1			−1.28708			−0.643539	−	0.765414i	$-0.722534\pi$
$607$	−19248.1			−1.28708			−0.643539	+	0.765414i	$0.722534\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21759.4	−	21759.4i	−1.44074	−	1.44074i
$612$		0			0
$613$	−9056.59	+	9056.59i	−0.596724	+	0.596724i	−0.939439	−	0.342715i	$-0.888653\pi$
$613$	−9056.59	+	9056.59i	−0.596724	+	0.596724i	0.342715	+	0.939439i	$0.388653\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12587.6i		−	0.821325i	−0.911787	−	0.410662i	$-0.865297\pi$
$617$		−	12587.6i		−	0.821325i	0.911787	−	0.410662i	$-0.134703\pi$
$618$		0			0
$619$	−11044.3	+	11044.3i	−0.717136	+	0.717136i	−0.968018	−	0.250881i	$-0.919280\pi$
$619$	−11044.3	+	11044.3i	−0.717136	+	0.717136i	0.250881	+	0.968018i	$0.419280\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3342.68			−0.214963
$624$		0			0
$625$	1537.68			0.0984117
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12986.3	−	12986.3i	0.823206	−	0.823206i
$630$		0			0
$631$		−	7554.44i		−	0.476605i	−0.971191	−	0.238302i	$-0.923409\pi$
$631$		−	7554.44i		−	0.476605i	0.971191	−	0.238302i	$-0.0765909\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3395.83	+	3395.83i	−0.212219	+	0.212219i
$636$		0			0
$637$	10917.6	+	10917.6i	0.679077	+	0.679077i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18511.6			1.14066			0.570330	−	0.821416i	$-0.306815\pi$
$641$	18511.6			1.14066			0.570330	+	0.821416i	$0.306815\pi$
$642$		0			0
$643$	7895.63	+	7895.63i	0.484251	+	0.484251i	0.906486	−	0.422236i	$-0.138755\pi$
$643$	7895.63	+	7895.63i	0.484251	+	0.484251i	−0.422236	+	0.906486i	$0.638755\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			25087.9i			1.52443i	0.647323	+	0.762216i	$0.275888\pi$
$647$			25087.9i			1.52443i	−0.647323	+	0.762216i	$0.724112\pi$
$648$		0			0
$649$		−	3544.36i		−	0.214373i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3966.20	−	3966.20i	−0.237687	−	0.237687i	0.578205	−	0.815892i	$-0.303753\pi$
$653$	−3966.20	−	3966.20i	−0.237687	−	0.237687i	−0.815892	+	0.578205i	$0.803753\pi$
$654$		0			0
$655$	17654.1			1.05314
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8121.04	+	8121.04i	0.480047	+	0.480047i	0.905147	−	0.425100i	$-0.139761\pi$
$659$	8121.04	+	8121.04i	0.480047	+	0.480047i	−0.425100	+	0.905147i	$0.639761\pi$
$660$		0			0
$661$	6417.92	−	6417.92i	0.377652	−	0.377652i	−0.492602	−	0.870255i	$-0.663954\pi$
$661$	6417.92	−	6417.92i	0.377652	−	0.377652i	0.870255	+	0.492602i	$0.163954\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			2707.46i			0.157881i
$666$		0			0
$667$	−3073.95	+	3073.95i	−0.178447	+	0.178447i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5883.52			−0.338496
$672$		0			0
$673$	−4207.12			−0.240969			−0.120485	−	0.992715i	$-0.538445\pi$
$673$	−4207.12			−0.240969			−0.120485	+	0.992715i	$0.538445\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2686.58	+	2686.58i	−0.152517	+	0.152517i	−0.779241	−	0.626724i	$-0.784395\pi$
$677$	−2686.58	+	2686.58i	−0.152517	+	0.152517i	0.626724	+	0.779241i	$0.284395\pi$
$678$		0			0
$679$		−	9560.26i		−	0.540338i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1806.99	−	1806.99i	0.101234	−	0.101234i	−0.654676	−	0.755910i	$-0.727195\pi$
$683$	1806.99	−	1806.99i	0.101234	−	0.101234i	0.755910	+	0.654676i	$0.227195\pi$
$684$		0			0
$685$	−15449.1	−	15449.1i	−0.861721	−	0.861721i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6140.34			0.339519
$690$		0			0
$691$	−3964.38	−	3964.38i	−0.218252	−	0.218252i	0.589509	−	0.807761i	$-0.299321\pi$
$691$	−3964.38	−	3964.38i	−0.218252	−	0.218252i	−0.807761	+	0.589509i	$0.799321\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	9284.97i		−	0.506761i
$696$		0			0
$697$		−	19110.1i		−	1.03852i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20882.0	+	20882.0i	1.12511	+	1.12511i	0.990961	+	0.134149i	$0.0428300\pi$
$701$	20882.0	+	20882.0i	1.12511	+	1.12511i	0.134149	+	0.990961i	$0.457170\pi$
$702$		0			0
$703$	−14336.1			−0.769129
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9467.74	+	9467.74i	0.503637	+	0.503637i
$708$		0			0
$709$	−8482.93	+	8482.93i	−0.449342	+	0.449342i	−0.895136	−	0.445794i	$-0.852921\pi$
$709$	−8482.93	+	8482.93i	−0.449342	+	0.449342i	0.445794	+	0.895136i	$0.352921\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			5141.02i			0.270032i
$714$		0			0
$715$	−8539.63	+	8539.63i	−0.446663	+	0.446663i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	23765.0			1.23266			0.616332	−	0.787486i	$-0.288618\pi$
$719$	23765.0			1.23266			0.616332	+	0.787486i	$0.288618\pi$
$720$		0			0
$721$	−7585.40			−0.391810
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6789.58	+	6789.58i	−0.347805	+	0.347805i
$726$		0			0
$727$			27979.3i			1.42737i	0.700468	+	0.713684i	$0.252975\pi$
$727$			27979.3i			1.42737i	−0.700468	+	0.713684i	$0.747025\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13917.8	−	13917.8i	0.704198	−	0.704198i
$732$		0			0
$733$	−17513.6	−	17513.6i	−0.882509	−	0.882509i	0.111280	−	0.993789i	$-0.464505\pi$
$733$	−17513.6	−	17513.6i	−0.882509	−	0.882509i	−0.993789	+	0.111280i	$0.964505\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−26493.0			−1.32413
$738$		0			0
$739$	9683.07	+	9683.07i	0.481999	+	0.481999i	0.905770	−	0.423770i	$-0.139294\pi$
$739$	9683.07	+	9683.07i	0.481999	+	0.481999i	−0.423770	+	0.905770i	$0.639294\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	19107.6i		−	0.943460i	−0.881743	−	0.471730i	$-0.843630\pi$
$743$		−	19107.6i		−	0.943460i	0.881743	−	0.471730i	$-0.156370\pi$
$744$		0			0
$745$			20135.2i			0.990197i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10593.6	−	10593.6i	−0.516800	−	0.516800i
$750$		0			0
$751$	−10823.0			−0.525883			−0.262942	−	0.964812i	$-0.584693\pi$
$751$	−10823.0			−0.525883			−0.262942	+	0.964812i	$0.584693\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2013.36	+	2013.36i	0.0970514	+	0.0970514i
$756$		0			0
$757$	−12578.0	+	12578.0i	−0.603904	+	0.603904i	−0.941346	−	0.337442i	$-0.890438\pi$
$757$	−12578.0	+	12578.0i	−0.603904	+	0.603904i	0.337442	+	0.941346i	$0.390438\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			30860.2i			1.47002i	0.678058	+	0.735009i	$0.262822\pi$
$761$			30860.2i			1.47002i	−0.678058	+	0.735009i	$0.737178\pi$
$762$		0			0
$763$	6235.15	−	6235.15i	0.295842	−	0.295842i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5934.36			−0.279371
$768$		0			0
$769$	−19357.6			−0.907741			−0.453870	−	0.891068i	$-0.649957\pi$
$769$	−19357.6			−0.907741			−0.453870	+	0.891068i	$0.649957\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2959.96	+	2959.96i	−0.137726	+	0.137726i	−0.772609	−	0.634882i	$-0.781049\pi$
$773$	−2959.96	+	2959.96i	−0.137726	+	0.137726i	0.634882	+	0.772609i	$0.281049\pi$
$774$		0			0
$775$			11355.2i			0.526311i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−10548.2	+	10548.2i	−0.485148	+	0.485148i
$780$		0			0
$781$	2455.00	+	2455.00i	0.112480	+	0.112480i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5922.83			−0.269293
$786$		0			0
$787$	−21346.3	−	21346.3i	−0.966853	−	0.966853i	0.0326147	−	0.999468i	$-0.489617\pi$
$787$	−21346.3	−	21346.3i	−0.966853	−	0.966853i	−0.999468	+	0.0326147i	$0.989617\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			7058.67i			0.317291i
$792$		0			0
$793$			9850.84i			0.441127i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22434.9	−	22434.9i	−0.997097	−	0.997097i	0.00289926	−	0.999996i	$-0.499077\pi$
$797$	−22434.9	−	22434.9i	−0.997097	−	0.997097i	−0.999996	+	0.00289926i	$0.999077\pi$
$798$		0			0
$799$	−39456.7			−1.74703
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9169.59	+	9169.59i	0.402974	+	0.402974i
$804$		0			0
$805$	1179.16	−	1179.16i	0.0516273	−	0.0516273i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	36085.5i		−	1.56823i	−0.620616	−	0.784115i	$-0.713117\pi$
$809$		−	36085.5i		−	1.56823i	0.620616	−	0.784115i	$-0.286883\pi$
$810$		0			0
$811$	−23874.2	+	23874.2i	−1.03371	+	1.03371i	−0.0342973	+	0.999412i	$0.510919\pi$
$811$	−23874.2	+	23874.2i	−1.03371	+	1.03371i	−0.999412	+	0.0342973i	$0.989081\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−979.170			−0.0420845
$816$		0			0
$817$	−15364.5			−0.657939
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−17774.8	+	17774.8i	−0.755596	+	0.755596i	−0.975518	−	0.219922i	$-0.929420\pi$
$821$	−17774.8	+	17774.8i	−0.755596	+	0.755596i	0.219922	+	0.975518i	$0.429420\pi$
$822$		0			0
$823$		−	44331.9i		−	1.87766i	−0.344384	−	0.938829i	$-0.611912\pi$
$823$		−	44331.9i		−	1.87766i	0.344384	−	0.938829i	$-0.388088\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29773.5	−	29773.5i	1.25191	−	1.25191i	0.297044	−	0.954864i	$-0.403999\pi$
$827$	29773.5	−	29773.5i	1.25191	−	1.25191i	0.954864	−	0.297044i	$-0.0960008\pi$
$828$		0			0
$829$	−9794.56	−	9794.56i	−0.410349	−	0.410349i	0.471511	−	0.881860i	$-0.343709\pi$
$829$	−9794.56	−	9794.56i	−0.410349	−	0.410349i	−0.881860	+	0.471511i	$0.843709\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19797.2			0.823447
$834$		0			0
$835$	−2049.24	−	2049.24i	−0.0849303	−	0.0849303i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	20248.1i		−	0.833183i	−0.909094	−	0.416591i	$-0.863225\pi$
$839$		−	20248.1i		−	0.833183i	0.909094	−	0.416591i	$-0.136775\pi$
$840$		0			0
$841$		−	6425.00i		−	0.263439i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2949.93	+	2949.93i	0.120095	+	0.120095i
$846$		0			0
$847$	2418.10			0.0980953
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6243.72	+	6243.72i	0.251507	+	0.251507i
$852$		0			0
$853$	−10983.4	+	10983.4i	−0.440873	+	0.440873i	−0.892305	−	0.451433i	$-0.850913\pi$
$853$	−10983.4	+	10983.4i	−0.440873	+	0.440873i	0.451433	+	0.892305i	$0.350913\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	26782.3i		−	1.06752i	−0.845635	−	0.533761i	$-0.820778\pi$
$857$		−	26782.3i		−	1.06752i	0.845635	−	0.533761i	$-0.179222\pi$
$858$		0			0
$859$	26073.4	−	26073.4i	1.03564	−	1.03564i	0.0362968	−	0.999341i	$-0.488444\pi$
$859$	26073.4	−	26073.4i	1.03564	−	1.03564i	0.999341	−	0.0362968i	$-0.0115562\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−30417.7			−1.19980			−0.599902	−	0.800073i	$-0.704794\pi$
$863$	−30417.7			−1.19980			−0.599902	+	0.800073i	$0.704794\pi$
$864$		0			0
$865$	−1994.14			−0.0783848
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−25384.2	+	25384.2i	−0.990910	+	0.990910i
$870$		0			0
$871$			44357.5i			1.72560i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7148.82	−	7148.82i	0.276199	−	0.276199i
$876$		0			0
$877$	11483.1	+	11483.1i	0.442142	+	0.442142i	0.892731	−	0.450590i	$-0.148786\pi$
$877$	11483.1	+	11483.1i	0.442142	+	0.442142i	−0.450590	+	0.892731i	$0.648786\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−34502.7			−1.31944			−0.659720	−	0.751511i	$-0.729325\pi$
$881$	−34502.7			−1.31944			−0.659720	+	0.751511i	$0.729325\pi$
$882$		0			0
$883$	−35667.9	−	35667.9i	−1.35937	−	1.35937i	−0.874699	−	0.484667i	$-0.838941\pi$
$883$	−35667.9	−	35667.9i	−1.35937	−	1.35937i	−0.484667	−	0.874699i	$-0.661059\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4004.04i			0.151570i	0.997124	+	0.0757850i	$0.0241463\pi$
$887$			4004.04i			0.151570i	−0.997124	+	0.0757850i	$0.975854\pi$
$888$		0			0
$889$			4627.25i			0.174570i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	21779.0	+	21779.0i	0.816134	+	0.816134i
$894$		0			0
$895$	1187.30			0.0443431
$896$		0			0
$897$		0			0
$898$		0			0
$899$	15021.9	+	15021.9i	0.557295	+	0.557295i
$900$		0			0
$901$	5567.21	−	5567.21i	0.205850	−	0.205850i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	7896.98i		−	0.290060i
$906$		0			0
$907$	−26847.6	+	26847.6i	−0.982865	+	0.982865i	−0.999856	−	0.0169907i	$-0.994591\pi$
$907$	−26847.6	+	26847.6i	−0.982865	+	0.982865i	0.0169907	+	0.999856i	$0.494591\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15303.2			−0.556552			−0.278276	−	0.960501i	$-0.589763\pi$
$911$	−15303.2			−0.556552			−0.278276	+	0.960501i	$0.589763\pi$
$912$		0			0
$913$	−29158.3			−1.05695
$914$		0			0
$915$		0			0
$916$		0			0
$917$	12028.0	−	12028.0i	0.433152	−	0.433152i
$918$		0			0
$919$		−	43252.1i		−	1.55251i	−0.630421	−	0.776254i	$-0.717118\pi$
$919$		−	43252.1i		−	1.55251i	0.630421	−	0.776254i	$-0.282882\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4110.44	−	4110.44i	0.146584	−	0.146584i
$924$		0			0
$925$	13790.8	+	13790.8i	0.490203	+	0.490203i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13328.9			−0.470729			−0.235365	−	0.971907i	$-0.575628\pi$
$929$	−13328.9			−0.470729			−0.235365	+	0.971907i	$0.575628\pi$
$930$		0			0
$931$	−10927.5	−	10927.5i	−0.384677	−	0.384677i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			15485.1i			0.541623i
$936$		0			0
$937$			15781.5i			0.550225i	0.961412	+	0.275112i	$0.0887151\pi$
$937$			15781.5i			0.550225i	−0.961412	+	0.275112i	$0.911285\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	24664.1	+	24664.1i	0.854440	+	0.854440i	0.990676	−	0.136236i	$-0.0435006\pi$
$941$	24664.1	+	24664.1i	0.854440	+	0.854440i	−0.136236	+	0.990676i	$0.543501\pi$
$942$		0			0
$943$	9188.01			0.317288
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16466.0	+	16466.0i	0.565017	+	0.565017i	0.930728	−	0.365711i	$-0.119174\pi$
$947$	16466.0	+	16466.0i	0.565017	+	0.565017i	−0.365711	+	0.930728i	$0.619174\pi$
$948$		0			0
$949$	15352.8	−	15352.8i	0.525155	−	0.525155i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	42180.3i		−	1.43374i	−0.697207	−	0.716870i	$-0.745574\pi$
$953$		−	42180.3i		−	1.43374i	0.697207	−	0.716870i	$-0.254426\pi$
$954$		0			0
$955$	−12464.2	+	12464.2i	−0.422337	+	0.422337i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−21051.3			−0.708846
$960$		0			0
$961$	−4667.69			−0.156681
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−25810.2	+	25810.2i	−0.860995	+	0.860995i
$966$		0			0
$967$			29742.4i			0.989091i	0.869152	+	0.494546i	$0.164666\pi$
$967$			29742.4i			0.989091i	−0.869152	+	0.494546i	$0.835334\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30803.6	−	30803.6i	1.01806	−	1.01806i	0.0182247	−	0.999834i	$-0.494199\pi$
$971$	30803.6	−	30803.6i	1.01806	−	1.01806i	0.999834	−	0.0182247i	$-0.00580144\pi$
$972$		0			0
$973$	−6325.98	−	6325.98i	−0.208429	−	0.208429i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	37576.0			1.23046			0.615232	−	0.788346i	$-0.289062\pi$
$977$	37576.0			1.23046			0.615232	+	0.788346i	$0.289062\pi$
$978$		0			0
$979$	10552.8	+	10552.8i	0.344502	+	0.344502i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	3002.36i		−	0.0974164i	−0.998813	−	0.0487082i	$-0.984490\pi$
$983$		−	3002.36i		−	0.0974164i	0.998813	−	0.0487082i	$-0.0155104\pi$
$984$		0			0
$985$			12486.5i			0.403911i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6691.61	+	6691.61i	0.215147	+	0.215147i
$990$		0			0
$991$	−4854.04			−0.155594			−0.0777971	−	0.996969i	$-0.524789\pi$
$991$	−4854.04			−0.155594			−0.0777971	+	0.996969i	$0.524789\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	15600.3	+	15600.3i	0.497047	+	0.497047i
$996$		0			0
$997$	−24273.6	+	24273.6i	−0.771067	+	0.771067i	−0.978293	−	0.207226i	$-0.933557\pi$
$997$	−24273.6	+	24273.6i	−0.771067	+	0.771067i	0.207226	+	0.978293i	$0.433557\pi$
$998$		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.145.8		24
3.2	odd	2	inner	576.4.k.c.145.5		24
4.3	odd	2		144.4.k.c.109.6	yes	24
12.11	even	2		144.4.k.c.109.7	yes	24
16.5	even	4	inner	576.4.k.c.433.8		24
16.11	odd	4		144.4.k.c.37.6	✓	24
48.5	odd	4	inner	576.4.k.c.433.5		24
48.11	even	4		144.4.k.c.37.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.k.c.37.6	✓	24	16.11	odd	4
144.4.k.c.37.7	yes	24	48.11	even	4
144.4.k.c.109.6	yes	24	4.3	odd	2
144.4.k.c.109.7	yes	24	12.11	even	2
576.4.k.c.145.5		24	3.2	odd	2	inner
576.4.k.c.145.8		24	1.1	even	1	trivial
576.4.k.c.433.5		24	48.5	odd	4	inner
576.4.k.c.433.8		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)

\(f(q)\)	\(=\)	\(q+(5.16526 - 5.16526i) q^{5} -7.03833i q^{7} +(-22.2198 + 22.2198i) q^{11} +(37.2029 + 37.2029i) q^{13} +67.4608 q^{17} +(-37.2366 - 37.2366i) q^{19} +32.4348i q^{23} +71.6401i q^{25} +(94.7734 + 94.7734i) q^{29} +158.503 q^{31} +(-36.3548 - 36.3548i) q^{35} +(192.501 - 192.501i) q^{37} -283.277i q^{41} +(206.310 - 206.310i) q^{43} -584.883 q^{47} +293.462 q^{49} +(82.5251 - 82.5251i) q^{53} +229.542i q^{55} +(-79.7567 + 79.7567i) q^{59} +(132.394 + 132.394i) q^{61} +384.325 q^{65} +(596.157 + 596.157i) q^{67} -110.487i q^{71} -412.677i q^{73} +(156.390 + 156.390i) q^{77} +1142.41 q^{79} +(656.133 + 656.133i) q^{83} +(348.453 - 348.453i) q^{85} -474.926i q^{89} +(261.846 - 261.846i) q^{91} -384.673 q^{95} +1358.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{19} - 744 q^{31} - 16 q^{37} + 376 q^{43} - 1176 q^{49} - 912 q^{61} - 1440 q^{67} + 328 q^{79} - 240 q^{85} + 104 q^{91}+O(q^{100}) \) 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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