LMFDB - Embedded newform 576.4.k.c.145.3

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			145.3
Character	$\chi$	$=$	576.145
Dual form			576.4.k.c.433.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.32567 + 8.32567i) q^{5} +22.1273i q^{7} +(-45.7813 + 45.7813i) q^{11} +(-55.7501 - 55.7501i) q^{13} +17.9113 q^{17} +(-44.2772 - 44.2772i) q^{19} -95.3934i q^{23} -13.6336i q^{25} +(137.497 + 137.497i) q^{29} +183.312 q^{31} +(-184.224 - 184.224i) q^{35} +(252.877 - 252.877i) q^{37} +158.116i q^{41} +(7.72872 - 7.72872i) q^{43} -32.9266 q^{47} -146.615 q^{49} +(212.424 - 212.424i) q^{53} -762.320i q^{55} +(-117.228 + 117.228i) q^{59} +(-172.590 - 172.590i) q^{61} +928.314 q^{65} +(-605.683 - 605.683i) q^{67} -443.271i q^{71} -703.319i q^{73} +(-1013.01 - 1013.01i) q^{77} -705.456 q^{79} +(170.914 + 170.914i) q^{83} +(-149.123 + 149.123i) q^{85} +1557.12i q^{89} +(1233.60 - 1233.60i) q^{91} +737.275 q^{95} +892.331 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−8.32567	+	8.32567i	−0.744671	+	0.744671i	−0.973473	−	0.228802i	$-0.926519\pi$
$5$	−8.32567	+	8.32567i	−0.744671	+	0.744671i	0.228802	+	0.973473i	$0.426519\pi$
$6$		0			0
$7$			22.1273i			1.19476i	0.801959	+	0.597380i	$0.203791\pi$
$7$			22.1273i			1.19476i	−0.801959	+	0.597380i	$0.796209\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−45.7813	+	45.7813i	−1.25487	+	1.25487i	−0.301360	+	0.953511i	$0.597440\pi$
$11$	−45.7813	+	45.7813i	−1.25487	+	1.25487i	−0.953511	+	0.301360i	$0.902560\pi$
$12$		0			0
$13$	−55.7501	−	55.7501i	−1.18941	−	1.18941i	−0.977232	−	0.212175i	$-0.931945\pi$
$13$	−55.7501	−	55.7501i	−1.18941	−	1.18941i	−0.212175	−	0.977232i	$-0.568055\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	17.9113			0.255537			0.127768	−	0.991804i	$-0.459219\pi$
$17$	17.9113			0.255537			0.127768	+	0.991804i	$0.459219\pi$
$18$		0			0
$19$	−44.2772	−	44.2772i	−0.534626	−	0.534626i	0.387320	−	0.921946i	$-0.373401\pi$
$19$	−44.2772	−	44.2772i	−0.534626	−	0.534626i	−0.921946	+	0.387320i	$0.873401\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	95.3934i		−	0.864822i	−0.901677	−	0.432411i	$-0.857663\pi$
$23$		−	95.3934i		−	0.864822i	0.901677	−	0.432411i	$-0.142337\pi$
$24$		0			0
$25$		−	13.6336i		−	0.109069i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	137.497	+	137.497i	0.880434	+	0.880434i	0.993579	−	0.113145i	$-0.0360924\pi$
$29$	137.497	+	137.497i	0.880434	+	0.880434i	−0.113145	+	0.993579i	$0.536092\pi$
$30$		0			0
$31$	183.312			1.06206			0.531029	−	0.847354i	$-0.321806\pi$
$31$	183.312			1.06206			0.531029	+	0.847354i	$0.321806\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−184.224	−	184.224i	−0.889702	−	0.889702i
$36$		0			0
$37$	252.877	−	252.877i	1.12359	−	1.12359i	0.132390	−	0.991198i	$-0.457735\pi$
$37$	252.877	−	252.877i	1.12359	−	1.12359i	0.991198	−	0.132390i	$-0.0422651\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			158.116i			0.602282i	0.953580	+	0.301141i	$0.0973674\pi$
$41$			158.116i			0.602282i	−0.953580	+	0.301141i	$0.902633\pi$
$42$		0			0
$43$	7.72872	−	7.72872i	0.0274097	−	0.0274097i	−0.693269	−	0.720679i	$-0.743830\pi$
$43$	7.72872	−	7.72872i	0.0274097	−	0.0274097i	0.720679	+	0.693269i	$0.243830\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−32.9266			−0.102188			−0.0510941	−	0.998694i	$-0.516271\pi$
$47$	−32.9266			−0.102188			−0.0510941	+	0.998694i	$0.516271\pi$
$48$		0			0
$49$	−146.615			−0.427450
$50$		0			0
$51$		0			0
$52$		0			0
$53$	212.424	−	212.424i	0.550542	−	0.550542i	−0.376056	−	0.926597i	$-0.622720\pi$
$53$	212.424	−	212.424i	0.550542	−	0.550542i	0.926597	+	0.376056i	$0.122720\pi$
$54$		0			0
$55$		−	762.320i		−	1.86893i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−117.228	+	117.228i	−0.258675	+	0.258675i	−0.824515	−	0.565840i	$-0.808552\pi$
$59$	−117.228	+	117.228i	−0.258675	+	0.258675i	0.565840	+	0.824515i	$0.308552\pi$
$60$		0			0
$61$	−172.590	−	172.590i	−0.362261	−	0.362261i	0.502384	−	0.864645i	$-0.332456\pi$
$61$	−172.590	−	172.590i	−0.362261	−	0.362261i	−0.864645	+	0.502384i	$0.832456\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	928.314			1.77143
$66$		0			0
$67$	−605.683	−	605.683i	−1.10442	−	1.10442i	−0.993871	−	0.110547i	$-0.964740\pi$
$67$	−605.683	−	605.683i	−1.10442	−	1.10442i	−0.110547	−	0.993871i	$-0.535260\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	443.271i		−	0.740937i	−0.928845	−	0.370469i	$-0.879197\pi$
$71$		−	443.271i		−	0.740937i	0.928845	−	0.370469i	$-0.120803\pi$
$72$		0			0
$73$		−	703.319i		−	1.12763i	−0.825900	−	0.563817i	$-0.809332\pi$
$73$		−	703.319i		−	1.12763i	0.825900	−	0.563817i	$-0.190668\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1013.01	−	1013.01i	−1.49927	−	1.49927i
$78$		0			0
$79$	−705.456			−1.00468			−0.502342	−	0.864669i	$-0.667528\pi$
$79$	−705.456			−1.00468			−0.502342	+	0.864669i	$0.667528\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	170.914	+	170.914i	0.226027	+	0.226027i	0.811031	−	0.585004i	$-0.198907\pi$
$83$	170.914	+	170.914i	0.226027	+	0.226027i	−0.585004	+	0.811031i	$0.698907\pi$
$84$		0			0
$85$	−149.123	+	149.123i	−0.190291	+	0.190291i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1557.12i			1.85455i	0.374382	+	0.927275i	$0.377855\pi$
$89$			1557.12i			1.85455i	−0.374382	+	0.927275i	$0.622145\pi$
$90$		0			0
$91$	1233.60	−	1233.60i	1.42106	−	1.42106i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	737.275			0.796241
$96$		0			0
$97$	892.331			0.934046			0.467023	−	0.884245i	$-0.345327\pi$
$97$	892.331			0.934046			0.467023	+	0.884245i	$0.345327\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−473.141	+	473.141i	−0.466132	+	0.466132i	−0.900659	−	0.434527i	$-0.856916\pi$
$101$	−473.141	+	473.141i	−0.466132	+	0.466132i	0.434527	+	0.900659i	$0.356916\pi$
$102$		0			0
$103$		−	226.421i		−	0.216601i	−0.994118	−	0.108300i	$-0.965459\pi$
$103$		−	226.421i		−	0.216601i	0.994118	−	0.108300i	$-0.0345409\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1226.93	+	1226.93i	−1.10852	+	1.10852i	−0.115173	+	0.993345i	$0.536742\pi$
$107$	−1226.93	+	1226.93i	−1.10852	+	1.10852i	−0.993345	+	0.115173i	$0.963258\pi$
$108$		0			0
$109$	−726.070	−	726.070i	−0.638026	−	0.638026i	0.312042	−	0.950068i	$-0.398987\pi$
$109$	−726.070	−	726.070i	−0.638026	−	0.638026i	−0.950068	+	0.312042i	$0.898987\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1239.16			−1.03160			−0.515798	−	0.856710i	$-0.672504\pi$
$113$	−1239.16			−1.03160			−0.515798	+	0.856710i	$0.672504\pi$
$114$		0			0
$115$	794.214	+	794.214i	0.644007	+	0.644007i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			396.328i			0.305305i
$120$		0			0
$121$		−	2860.85i		−	2.14940i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−927.200	−	927.200i	−0.663450	−	0.663450i
$126$		0			0
$127$	358.889			0.250758			0.125379	−	0.992109i	$-0.459985\pi$
$127$	358.889			0.250758			0.125379	+	0.992109i	$0.459985\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−33.2573	−	33.2573i	−0.0221810	−	0.0221810i	0.695929	−	0.718110i	$-0.254993\pi$
$131$	−33.2573	−	33.2573i	−0.0221810	−	0.0221810i	−0.718110	+	0.695929i	$0.754993\pi$
$132$		0			0
$133$	979.734	−	979.734i	0.638749	−	0.638749i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1266.67i		−	0.789917i	−0.918699	−	0.394959i	$-0.870759\pi$
$137$		−	1266.67i		−	0.789917i	0.918699	−	0.394959i	$-0.129241\pi$
$138$		0			0
$139$	−947.159	+	947.159i	−0.577964	+	0.577964i	−0.934342	−	0.356378i	$-0.884012\pi$
$139$	−947.159	+	947.159i	−0.577964	+	0.577964i	0.356378	+	0.934342i	$0.384012\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5104.62			2.98510
$144$		0			0
$145$	−2289.51			−1.31127
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1432.23	−	1432.23i	0.787467	−	0.787467i	−0.193611	−	0.981078i	$-0.562020\pi$
$149$	1432.23	−	1432.23i	0.787467	−	0.787467i	0.981078	+	0.193611i	$0.0620200\pi$
$150$		0			0
$151$		−	1445.01i		−	0.778763i	−0.921077	−	0.389382i	$-0.872689\pi$
$151$		−	1445.01i		−	0.778763i	0.921077	−	0.389382i	$-0.127311\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1526.19	+	1526.19i	−0.790883	+	0.790883i
$156$		0			0
$157$	1073.11	+	1073.11i	0.545498	+	0.545498i	0.925136	−	0.379637i	$-0.123951\pi$
$157$	1073.11	+	1073.11i	0.545498	+	0.545498i	−0.379637	+	0.925136i	$0.623951\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2110.79			1.03325
$162$		0			0
$163$	−481.558	−	481.558i	−0.231402	−	0.231402i	0.581876	−	0.813278i	$-0.302319\pi$
$163$	−481.558	−	481.558i	−0.231402	−	0.231402i	−0.813278	+	0.581876i	$0.802319\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4153.21i		−	1.92446i	−0.272236	−	0.962230i	$-0.587763\pi$
$167$		−	4153.21i		−	1.92446i	0.272236	−	0.962230i	$-0.412237\pi$
$168$		0			0
$169$			4019.14i			1.82938i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2892.19	−	2892.19i	−1.27103	−	1.27103i	−0.945546	−	0.325489i	$-0.894471\pi$
$173$	−2892.19	−	2892.19i	−1.27103	−	1.27103i	−0.325489	−	0.945546i	$-0.605529\pi$
$174$		0			0
$175$	301.674			0.130311
$176$		0			0
$177$		0			0
$178$		0			0
$179$	835.310	+	835.310i	0.348793	+	0.348793i	0.859660	−	0.510867i	$-0.170675\pi$
$179$	835.310	+	835.310i	0.348793	+	0.348793i	−0.510867	+	0.859660i	$0.670675\pi$
$180$		0			0
$181$	−1014.37	+	1014.37i	−0.416562	+	0.416562i	−0.884017	−	0.467455i	$-0.845171\pi$
$181$	−1014.37	+	1014.37i	−0.416562	+	0.416562i	0.467455	+	0.884017i	$0.345171\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			4210.74i			1.67341i
$186$		0			0
$187$	−820.002	+	820.002i	−0.320666	+	0.320666i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−254.976			−0.0965937			−0.0482968	−	0.998833i	$-0.515379\pi$
$191$	−254.976			−0.0965937			−0.0482968	+	0.998833i	$0.515379\pi$
$192$		0			0
$193$	1997.33			0.744927			0.372463	−	0.928047i	$-0.378513\pi$
$193$	1997.33			0.744927			0.372463	+	0.928047i	$0.378513\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1178.13	+	1178.13i	−0.426081	+	0.426081i	−0.887291	−	0.461210i	$-0.847415\pi$
$197$	−1178.13	+	1178.13i	−0.426081	+	0.426081i	0.461210	+	0.887291i	$0.347415\pi$
$198$		0			0
$199$			3489.67i			1.24310i	0.783375	+	0.621549i	$0.213496\pi$
$199$			3489.67i			1.24310i	−0.783375	+	0.621549i	$0.786504\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3042.43	+	3042.43i	−1.05191	+	1.05191i
$204$		0			0
$205$	−1316.42	−	1316.42i	−0.448501	−	0.448501i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4054.14			1.34177
$210$		0			0
$211$	−2270.28	−	2270.28i	−0.740723	−	0.740723i	0.231994	−	0.972717i	$-0.425475\pi$
$211$	−2270.28	−	2270.28i	−0.740723	−	0.740723i	−0.972717	+	0.231994i	$0.925475\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			128.694i			0.0408224i
$216$		0			0
$217$			4056.19i			1.26890i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−998.556	−	998.556i	−0.303937	−	0.303937i
$222$		0			0
$223$	1179.40			0.354164			0.177082	−	0.984196i	$-0.443334\pi$
$223$	1179.40			0.354164			0.177082	+	0.984196i	$0.443334\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4086.16	−	4086.16i	−1.19475	−	1.19475i	−0.975718	−	0.219031i	$-0.929710\pi$
$227$	−4086.16	−	4086.16i	−1.19475	−	1.19475i	−0.219031	−	0.975718i	$-0.570290\pi$
$228$		0			0
$229$	−119.993	+	119.993i	−0.0346262	+	0.0346262i	−0.724208	−	0.689582i	$-0.757794\pi$
$229$	−119.993	+	119.993i	−0.0346262	+	0.0346262i	0.689582	+	0.724208i	$0.257794\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6392.44i			1.79735i	0.438615	+	0.898675i	$0.355469\pi$
$233$			6392.44i			1.79735i	−0.438615	+	0.898675i	$0.644531\pi$
$234$		0			0
$235$	274.136	−	274.136i	0.0760965	−	0.0760965i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1600.52			−0.433176			−0.216588	−	0.976263i	$-0.569493\pi$
$239$	−1600.52			−0.433176			−0.216588	+	0.976263i	$0.569493\pi$
$240$		0			0
$241$	2393.18			0.639660			0.319830	−	0.947475i	$-0.396374\pi$
$241$	2393.18			0.639660			0.319830	+	0.947475i	$0.396374\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1220.67	−	1220.67i	0.318310	−	0.318310i
$246$		0			0
$247$			4936.92i			1.27178i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2510.04	−	2510.04i	0.631203	−	0.631203i	−0.317166	−	0.948370i	$-0.602731\pi$
$251$	2510.04	−	2510.04i	0.631203	−	0.631203i	0.948370	+	0.317166i	$0.102731\pi$
$252$		0			0
$253$	4367.23	+	4367.23i	1.08524	+	1.08524i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	73.9839			0.0179571			0.00897857	−	0.999960i	$-0.497142\pi$
$257$	73.9839			0.0179571			0.00897857	+	0.999960i	$0.497142\pi$
$258$		0			0
$259$	5595.48	+	5595.48i	1.34242	+	1.34242i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			1071.81i			0.251295i	0.992075	+	0.125648i	$0.0401008\pi$
$263$			1071.81i			0.251295i	−0.992075	+	0.125648i	$0.959899\pi$
$264$		0			0
$265$			3537.15i			0.819944i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1609.09	+	1609.09i	0.364715	+	0.364715i	0.865545	−	0.500831i	$-0.166972\pi$
$269$	1609.09	+	1609.09i	0.364715	+	0.364715i	−0.500831	+	0.865545i	$0.666972\pi$
$270$		0			0
$271$	2940.80			0.659192			0.329596	−	0.944122i	$-0.393087\pi$
$271$	2940.80			0.659192			0.329596	+	0.944122i	$0.393087\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	624.163	+	624.163i	0.136867	+	0.136867i
$276$		0			0
$277$	−3121.22	+	3121.22i	−0.677026	+	0.677026i	−0.959326	−	0.282300i	$-0.908903\pi$
$277$	−3121.22	+	3121.22i	−0.677026	+	0.677026i	0.282300	+	0.959326i	$0.408903\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2063.45i		−	0.438062i	−0.975718	−	0.219031i	$-0.929710\pi$
$281$		−	2063.45i		−	0.438062i	0.975718	−	0.219031i	$-0.0702895\pi$
$282$		0			0
$283$	−4484.94	+	4484.94i	−0.942056	+	0.942056i	−0.998411	−	0.0563547i	$-0.982052\pi$
$283$	−4484.94	+	4484.94i	−0.942056	+	0.942056i	0.0563547	+	0.998411i	$0.482052\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3498.67			−0.719582
$288$		0			0
$289$	−4592.19			−0.934701
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3574.93	−	3574.93i	0.712798	−	0.712798i	−0.254322	−	0.967120i	$-0.581852\pi$
$293$	3574.93	−	3574.93i	0.712798	−	0.712798i	0.967120	+	0.254322i	$0.0818523\pi$
$294$		0			0
$295$		−	1952.01i		−	0.385256i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5318.19	+	5318.19i	−1.02863	+	1.02863i
$300$		0			0
$301$	171.015	+	171.015i	0.0327480	+	0.0327480i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2873.86			0.539530
$306$		0			0
$307$	2829.72	+	2829.72i	0.526060	+	0.526060i	0.919395	−	0.393335i	$-0.128679\pi$
$307$	2829.72	+	2829.72i	0.526060	+	0.526060i	−0.393335	+	0.919395i	$0.628679\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2337.80i		−	0.426252i	−0.977025	−	0.213126i	$-0.931636\pi$
$311$		−	2337.80i		−	0.426252i	0.977025	−	0.213126i	$-0.0683644\pi$
$312$		0			0
$313$		−	7096.67i		−	1.28156i	−0.767726	−	0.640778i	$-0.778612\pi$
$313$		−	7096.67i		−	1.28156i	0.767726	−	0.640778i	$-0.221388\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1067.67	−	1067.67i	−0.189169	−	0.189169i	0.606168	−	0.795337i	$-0.292706\pi$
$317$	−1067.67	−	1067.67i	−0.189169	−	0.189169i	−0.795337	+	0.606168i	$0.792706\pi$
$318$		0			0
$319$	−12589.6			−2.20966
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−793.062	−	793.062i	−0.136617	−	0.136617i
$324$		0			0
$325$	−760.074	+	760.074i	−0.129727	+	0.129727i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	728.576i		−	0.122090i
$330$		0			0
$331$	−893.933	+	893.933i	−0.148444	+	0.148444i	−0.777423	−	0.628979i	$-0.783473\pi$
$331$	−893.933	+	893.933i	−0.148444	+	0.148444i	0.628979	+	0.777423i	$0.283473\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10085.4			1.64486
$336$		0			0
$337$	−6414.33			−1.03683			−0.518414	−	0.855130i	$-0.673477\pi$
$337$	−6414.33			−1.03683			−0.518414	+	0.855130i	$0.673477\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8392.25	+	8392.25i	−1.33274	+	1.33274i
$342$		0			0
$343$			4345.45i			0.684059i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3075.69	−	3075.69i	0.475826	−	0.475826i	−0.427968	−	0.903794i	$-0.640770\pi$
$347$	3075.69	−	3075.69i	0.475826	−	0.475826i	0.903794	+	0.427968i	$0.140770\pi$
$348$		0			0
$349$	318.660	+	318.660i	0.0488753	+	0.0488753i	0.731122	−	0.682247i	$-0.238997\pi$
$349$	318.660	+	318.660i	0.0488753	+	0.0488753i	−0.682247	+	0.731122i	$0.738997\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3794.30			−0.572097			−0.286049	−	0.958215i	$-0.592342\pi$
$353$	−3794.30			−0.572097			−0.286049	+	0.958215i	$0.592342\pi$
$354$		0			0
$355$	3690.53	+	3690.53i	0.551754	+	0.551754i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	10917.1i		−	1.60497i	−0.596673	−	0.802485i	$-0.703511\pi$
$359$		−	10917.1i		−	1.60497i	0.596673	−	0.802485i	$-0.296489\pi$
$360$		0			0
$361$		−	2938.05i		−	0.428350i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5855.60	+	5855.60i	0.839716	+	0.839716i
$366$		0			0
$367$	9008.05			1.28124			0.640622	−	0.767857i	$-0.278677\pi$
$367$	9008.05			1.28124			0.640622	+	0.767857i	$0.278677\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4700.36	+	4700.36i	0.657765	+	0.657765i
$372$		0			0
$373$	−2003.29	+	2003.29i	−0.278087	+	0.278087i	−0.832345	−	0.554258i	$-0.813002\pi$
$373$	−2003.29	+	2003.29i	−0.278087	+	0.278087i	0.554258	+	0.832345i	$0.313002\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	15331.0i		−	2.09439i
$378$		0			0
$379$	7231.62	−	7231.62i	0.980114	−	0.980114i	−0.0196920	−	0.999806i	$-0.506269\pi$
$379$	7231.62	−	7231.62i	0.980114	−	0.980114i	0.999806	+	0.0196920i	$0.00626857\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	328.058			0.0437676			0.0218838	−	0.999761i	$-0.493034\pi$
$383$	328.058			0.0437676			0.0218838	+	0.999761i	$0.493034\pi$
$384$		0			0
$385$	16868.0			2.23292
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6549.83	−	6549.83i	0.853701	−	0.853701i	−0.136886	−	0.990587i	$-0.543709\pi$
$389$	6549.83	−	6549.83i	0.853701	−	0.853701i	0.990587	+	0.136886i	$0.0437093\pi$
$390$		0			0
$391$		−	1708.62i		−	0.220994i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5873.40	−	5873.40i	0.748159	−	0.748159i
$396$		0			0
$397$	3107.99	+	3107.99i	0.392911	+	0.392911i	0.875724	−	0.482813i	$-0.160385\pi$
$397$	3107.99	+	3107.99i	0.392911	+	0.392911i	−0.482813	+	0.875724i	$0.660385\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9538.29			−1.18783			−0.593915	−	0.804528i	$-0.702418\pi$
$401$	−9538.29			−1.18783			−0.593915	+	0.804528i	$0.702418\pi$
$402$		0			0
$403$	−10219.7	−	10219.7i	−1.26322	−	1.26322i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			23154.1i			2.81991i
$408$		0			0
$409$			10394.6i			1.25668i	0.777939	+	0.628339i	$0.216265\pi$
$409$			10394.6i			1.25668i	−0.777939	+	0.628339i	$0.783735\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2593.94	−	2593.94i	−0.309055	−	0.309055i
$414$		0			0
$415$	−2845.94			−0.336631
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2148.02	+	2148.02i	0.250448	+	0.250448i	0.821154	−	0.570706i	$-0.193330\pi$
$419$	2148.02	+	2148.02i	0.250448	+	0.250448i	−0.570706	+	0.821154i	$0.693330\pi$
$420$		0			0
$421$	1345.70	−	1345.70i	0.155785	−	0.155785i	−0.624911	−	0.780696i	$-0.714865\pi$
$421$	1345.70	−	1345.70i	0.155785	−	0.155785i	0.780696	+	0.624911i	$0.214865\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	244.195i		−	0.0278711i
$426$		0			0
$427$	3818.95	−	3818.95i	0.432815	−	0.432815i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13119.7			−1.46625			−0.733123	−	0.680096i	$-0.761938\pi$
$431$	−13119.7			−1.46625			−0.733123	+	0.680096i	$0.761938\pi$
$432$		0			0
$433$	5191.00			0.576129			0.288064	−	0.957611i	$-0.406988\pi$
$433$	5191.00			0.576129			0.288064	+	0.957611i	$0.406988\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4223.76	+	4223.76i	−0.462356	+	0.462356i
$438$		0			0
$439$		−	14051.6i		−	1.52767i	−0.645412	−	0.763835i	$-0.723314\pi$
$439$		−	14051.6i		−	1.52767i	0.645412	−	0.763835i	$-0.276686\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4070.57	−	4070.57i	0.436566	−	0.436566i	−0.454289	−	0.890855i	$-0.650106\pi$
$443$	4070.57	−	4070.57i	0.436566	−	0.436566i	0.890855	+	0.454289i	$0.150106\pi$
$444$		0			0
$445$	−12964.1	−	12964.1i	−1.38103	−	1.38103i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18158.9			−1.90862			−0.954310	−	0.298817i	$-0.903408\pi$
$449$	−18158.9			−1.90862			−0.954310	+	0.298817i	$0.903408\pi$
$450$		0			0
$451$	−7238.74	−	7238.74i	−0.755785	−	0.755785i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			20541.0i			2.11644i
$456$		0			0
$457$		−	7489.80i		−	0.766648i	−0.923614	−	0.383324i	$-0.874779\pi$
$457$		−	7489.80i		−	0.766648i	0.923614	−	0.383324i	$-0.125221\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3672.25	+	3672.25i	0.371006	+	0.371006i	0.867844	−	0.496838i	$-0.165506\pi$
$461$	3672.25	+	3672.25i	0.371006	+	0.371006i	−0.496838	+	0.867844i	$0.665506\pi$
$462$		0			0
$463$	−1134.54			−0.113880			−0.0569401	−	0.998378i	$-0.518134\pi$
$463$	−1134.54			−0.113880			−0.0569401	+	0.998378i	$0.518134\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4263.91	+	4263.91i	0.422506	+	0.422506i	0.886066	−	0.463560i	$-0.153428\pi$
$467$	4263.91	+	4263.91i	0.422506	+	0.422506i	−0.463560	+	0.886066i	$0.653428\pi$
$468$		0			0
$469$	13402.1	−	13402.1i	1.31951	−	1.31951i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			707.661i			0.0687913i
$474$		0			0
$475$	−603.658	+	603.658i	−0.0583110	+	0.0583110i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−18470.1			−1.76184			−0.880920	−	0.473265i	$-0.843075\pi$
$479$	−18470.1			−1.76184			−0.880920	+	0.473265i	$0.843075\pi$
$480$		0			0
$481$	−28195.8			−2.67281
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7429.25	+	7429.25i	−0.695557	+	0.695557i
$486$		0			0
$487$			787.630i			0.0732873i	0.999328	+	0.0366437i	$0.0116667\pi$
$487$			787.630i			0.0732873i	−0.999328	+	0.0366437i	$0.988333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13874.6	−	13874.6i	1.27526	−	1.27526i	0.331967	−	0.943291i	$-0.392288\pi$
$491$	13874.6	−	13874.6i	1.27526	−	1.27526i	0.943291	−	0.331967i	$-0.107712\pi$
$492$		0			0
$493$	2462.75	+	2462.75i	0.224983	+	0.224983i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9808.36			0.885242
$498$		0			0
$499$	−4588.16	−	4588.16i	−0.411612	−	0.411612i	0.470688	−	0.882300i	$-0.344006\pi$
$499$	−4588.16	−	4588.16i	−0.411612	−	0.411612i	−0.882300	+	0.470688i	$0.844006\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	13498.3i		−	1.19654i	−0.801295	−	0.598270i	$-0.795855\pi$
$503$		−	13498.3i		−	1.19654i	0.801295	−	0.598270i	$-0.204145\pi$
$504$		0			0
$505$		−	7878.44i		−	0.694230i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4751.73	+	4751.73i	0.413785	+	0.413785i	0.883055	−	0.469270i	$-0.155483\pi$
$509$	4751.73	+	4751.73i	0.413785	+	0.413785i	−0.469270	+	0.883055i	$0.655483\pi$
$510$		0			0
$511$	15562.5			1.34725
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1885.10	+	1885.10i	0.161296	+	0.161296i
$516$		0			0
$517$	1507.42	−	1507.42i	0.128233	−	0.128233i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	8402.51i		−	0.706566i	−0.935517	−	0.353283i	$-0.885065\pi$
$521$		−	8402.51i		−	0.706566i	0.935517	−	0.353283i	$-0.114935\pi$
$522$		0			0
$523$	−14961.2	+	14961.2i	−1.25087	+	1.25087i	−0.295544	+	0.955329i	$0.595501\pi$
$523$	−14961.2	+	14961.2i	−1.25087	+	1.25087i	−0.955329	+	0.295544i	$0.904499\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3283.35			0.271395
$528$		0			0
$529$	3067.09			0.252083
$530$		0			0
$531$		0			0
$532$		0			0
$533$	8814.97	−	8814.97i	0.716358	−	0.716358i
$534$		0			0
$535$		−	20430.0i		−	1.65096i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6712.24	−	6712.24i	0.536394	−	0.536394i
$540$		0			0
$541$	−7348.43	−	7348.43i	−0.583981	−	0.583981i	0.352014	−	0.935995i	$-0.385497\pi$
$541$	−7348.43	−	7348.43i	−0.583981	−	0.583981i	−0.935995	+	0.352014i	$0.885497\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12090.0			0.950239
$546$		0			0
$547$	8665.46	+	8665.46i	0.677346	+	0.677346i	0.959399	−	0.282053i	$-0.0910154\pi$
$547$	8665.46	+	8665.46i	0.677346	+	0.677346i	−0.282053	+	0.959399i	$0.591015\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	12176.0i		−	0.941406i
$552$		0			0
$553$		−	15609.8i		−	1.20036i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9038.10	+	9038.10i	0.687534	+	0.687534i	0.961686	−	0.274152i	$-0.0883972\pi$
$557$	9038.10	+	9038.10i	0.687534	+	0.687534i	−0.274152	+	0.961686i	$0.588397\pi$
$558$		0			0
$559$	−861.753			−0.0652027
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11175.1	−	11175.1i	−0.836545	−	0.836545i	0.151857	−	0.988402i	$-0.451475\pi$
$563$	−11175.1	−	11175.1i	−0.836545	−	0.836545i	−0.988402	+	0.151857i	$0.951475\pi$
$564$		0			0
$565$	10316.8	−	10316.8i	0.768199	−	0.768199i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	12466.3i		−	0.918476i	−0.888313	−	0.459238i	$-0.848123\pi$
$569$		−	12466.3i		−	0.918476i	0.888313	−	0.459238i	$-0.151877\pi$
$570$		0			0
$571$	16591.8	−	16591.8i	1.21602	−	1.21602i	0.247000	−	0.969016i	$-0.420555\pi$
$571$	16591.8	−	16591.8i	1.21602	−	1.21602i	0.969016	−	0.247000i	$-0.0794446\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1300.55			−0.0943250
$576$		0			0
$577$	7047.00			0.508441			0.254221	−	0.967146i	$-0.418181\pi$
$577$	7047.00			0.508441			0.254221	+	0.967146i	$0.418181\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3781.85	+	3781.85i	−0.270048	+	0.270048i
$582$		0			0
$583$			19450.1i			1.38172i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−627.982	+	627.982i	−0.0441560	+	0.0441560i	−0.728840	−	0.684684i	$-0.759940\pi$
$587$	−627.982	+	627.982i	−0.0441560	+	0.0441560i	0.684684	+	0.728840i	$0.259940\pi$
$588$		0			0
$589$	−8116.54	−	8116.54i	−0.567804	−	0.567804i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3818.26			−0.264413			−0.132207	−	0.991222i	$-0.542206\pi$
$593$	−3818.26			−0.264413			−0.132207	+	0.991222i	$0.542206\pi$
$594$		0			0
$595$	−3299.69	−	3299.69i	−0.227352	−	0.227352i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8991.13i			0.613301i	0.951822	+	0.306651i	$0.0992083\pi$
$599$			8991.13i			0.613301i	−0.951822	+	0.306651i	$0.900792\pi$
$600$		0			0
$601$			7689.90i			0.521926i	0.965349	+	0.260963i	$0.0840400\pi$
$601$			7689.90i			0.521926i	−0.965349	+	0.260963i	$0.915960\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	23818.5	+	23818.5i	1.60059	+	1.60059i
$606$		0			0
$607$	−12759.4			−0.853190			−0.426595	−	0.904443i	$-0.640287\pi$
$607$	−12759.4			−0.853190			−0.426595	+	0.904443i	$0.640287\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1835.66	+	1835.66i	0.121543	+	0.121543i
$612$		0			0
$613$	−8199.74	+	8199.74i	−0.540268	+	0.540268i	−0.923608	−	0.383339i	$-0.874774\pi$
$613$	−8199.74	+	8199.74i	−0.540268	+	0.540268i	0.383339	+	0.923608i	$0.374774\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	13433.2i		−	0.876501i	−0.898853	−	0.438250i	$-0.855598\pi$
$617$		−	13433.2i		−	0.876501i	0.898853	−	0.438250i	$-0.144402\pi$
$618$		0			0
$619$	10658.0	−	10658.0i	0.692055	−	0.692055i	−0.270628	−	0.962684i	$-0.587231\pi$
$619$	10658.0	−	10658.0i	0.692055	−	0.692055i	0.962684	+	0.270628i	$0.0872315\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−34454.9			−2.21574
$624$		0			0
$625$	17143.3			1.09717
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4529.35	−	4529.35i	0.287118	−	0.287118i
$630$		0			0
$631$		−	11689.5i		−	0.737483i	−0.929532	−	0.368741i	$-0.879789\pi$
$631$		−	11689.5i		−	0.737483i	0.929532	−	0.368741i	$-0.120211\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2988.00	+	2988.00i	−0.186732	+	0.186732i
$636$		0			0
$637$	8173.82	+	8173.82i	0.508412	+	0.508412i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20613.2			−1.27016			−0.635080	−	0.772447i	$-0.719033\pi$
$641$	−20613.2			−1.27016			−0.635080	+	0.772447i	$0.719033\pi$
$642$		0			0
$643$	−8915.69	−	8915.69i	−0.546813	−	0.546813i	0.378705	−	0.925517i	$-0.376370\pi$
$643$	−8915.69	−	8915.69i	−0.546813	−	0.546813i	−0.925517	+	0.378705i	$0.876370\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			17814.8i			1.08249i	0.840864	+	0.541246i	$0.182047\pi$
$647$			17814.8i			1.08249i	−0.840864	+	0.541246i	$0.817953\pi$
$648$		0			0
$649$		−	10733.7i		−	0.649208i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8326.29	+	8326.29i	0.498978	+	0.498978i	0.911120	−	0.412142i	$-0.135219\pi$
$653$	8326.29	+	8326.29i	0.498978	+	0.498978i	−0.412142	+	0.911120i	$0.635219\pi$
$654$		0			0
$655$	553.779			0.0330350
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3581.86	−	3581.86i	−0.211729	−	0.211729i	0.593273	−	0.805002i	$-0.297836\pi$
$659$	−3581.86	−	3581.86i	−0.211729	−	0.211729i	−0.805002	+	0.593273i	$0.797836\pi$
$660$		0			0
$661$	−18328.8	+	18328.8i	−1.07853	+	1.07853i	−0.0818862	+	0.996642i	$0.526094\pi$
$661$	−18328.8	+	18328.8i	−1.07853	+	1.07853i	−0.996642	+	0.0818862i	$0.973906\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			16313.9i			0.951316i
$666$		0			0
$667$	13116.3	−	13116.3i	0.761418	−	0.761418i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15802.8			0.909181
$672$		0			0
$673$	−3149.76			−0.180408			−0.0902038	−	0.995923i	$-0.528752\pi$
$673$	−3149.76			−0.180408			−0.0902038	+	0.995923i	$0.528752\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5943.48	+	5943.48i	−0.337410	+	0.337410i	−0.855392	−	0.517982i	$-0.826684\pi$
$677$	−5943.48	+	5943.48i	−0.337410	+	0.337410i	0.517982	+	0.855392i	$0.326684\pi$
$678$		0			0
$679$			19744.8i			1.11596i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1617.17	+	1617.17i	−0.0905991	+	0.0905991i	−0.750954	−	0.660355i	$-0.770406\pi$
$683$	−1617.17	+	1617.17i	−0.0905991	+	0.0905991i	0.660355	+	0.750954i	$0.270406\pi$
$684$		0			0
$685$	10545.9	+	10545.9i	0.588228	+	0.588228i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23685.3			−1.30964
$690$		0			0
$691$	18016.0	+	18016.0i	0.991838	+	0.991838i	0.999967	−	0.00812867i	$-0.00258746\pi$
$691$	18016.0	+	18016.0i	0.991838	+	0.991838i	−0.00812867	+	0.999967i	$0.502587\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	15771.5i		−	0.860786i
$696$		0			0
$697$			2832.06i			0.153905i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	5447.99	+	5447.99i	0.293534	+	0.293534i	0.838475	−	0.544940i	$-0.183448\pi$
$701$	5447.99	+	5447.99i	0.293534	+	0.293534i	−0.544940	+	0.838475i	$0.683448\pi$
$702$		0			0
$703$	−22393.4			−1.20140
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10469.3	−	10469.3i	−0.556916	−	0.556916i
$708$		0			0
$709$	−6461.24	+	6461.24i	−0.342252	+	0.342252i	−0.857214	−	0.514961i	$-0.827806\pi$
$709$	−6461.24	+	6461.24i	−0.342252	+	0.342252i	0.514961	+	0.857214i	$0.327806\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	17486.8i		−	0.918491i
$714$		0			0
$715$	−42499.4	+	42499.4i	−2.22292	+	2.22292i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2418.70			−0.125455			−0.0627277	−	0.998031i	$-0.519980\pi$
$719$	−2418.70			−0.125455			−0.0627277	+	0.998031i	$0.519980\pi$
$720$		0			0
$721$	5010.07			0.258786
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1874.58	−	1874.58i	0.0960278	−	0.0960278i
$726$		0			0
$727$		−	11249.9i		−	0.573912i	−0.957944	−	0.286956i	$-0.907357\pi$
$727$		−	11249.9i		−	0.573912i	0.957944	−	0.286956i	$-0.0926433\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	138.431	−	138.431i	0.00700420	−	0.00700420i
$732$		0			0
$733$	8269.64	+	8269.64i	0.416707	+	0.416707i	0.884067	−	0.467360i	$-0.154795\pi$
$733$	8269.64	+	8269.64i	0.416707	+	0.416707i	−0.467360	+	0.884067i	$0.654795\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	55457.9			2.77180
$738$		0			0
$739$	12570.8	+	12570.8i	0.625745	+	0.625745i	0.946994	−	0.321250i	$-0.104103\pi$
$739$	12570.8	+	12570.8i	0.625745	+	0.625745i	−0.321250	+	0.946994i	$0.604103\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			14713.9i			0.726516i	0.931689	+	0.363258i	$0.118336\pi$
$743$			14713.9i			0.726516i	−0.931689	+	0.363258i	$0.881664\pi$
$744$		0			0
$745$			23848.5i			1.17281i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27148.5	−	27148.5i	−1.32441	−	1.32441i
$750$		0			0
$751$	1198.79			0.0582482			0.0291241	−	0.999576i	$-0.490728\pi$
$751$	1198.79			0.0582482			0.0291241	+	0.999576i	$0.490728\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12030.7	+	12030.7i	0.579922	+	0.579922i
$756$		0			0
$757$	7771.96	−	7771.96i	0.373153	−	0.373153i	−0.495471	−	0.868624i	$-0.665005\pi$
$757$	7771.96	−	7771.96i	0.373153	−	0.373153i	0.868624	+	0.495471i	$0.165005\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			23565.6i			1.12254i	0.827633	+	0.561270i	$0.189687\pi$
$761$			23565.6i			1.12254i	−0.827633	+	0.561270i	$0.810313\pi$
$762$		0			0
$763$	16065.9	−	16065.9i	0.762288	−	0.762288i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13071.0			0.615341
$768$		0			0
$769$	−38356.7			−1.79867			−0.899335	−	0.437260i	$-0.855949\pi$
$769$	−38356.7			−1.79867			−0.899335	+	0.437260i	$0.855949\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15175.4	+	15175.4i	−0.706110	+	0.706110i	−0.965715	−	0.259605i	$-0.916408\pi$
$773$	−15175.4	+	15175.4i	−0.706110	+	0.706110i	0.259605	+	0.965715i	$0.416408\pi$
$774$		0			0
$775$		−	2499.20i		−	0.115837i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7000.93	−	7000.93i	0.321995	−	0.321995i
$780$		0			0
$781$	20293.5	+	20293.5i	0.929780	+	0.929780i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17868.7			−0.812433
$786$		0			0
$787$	−23673.2	−	23673.2i	−1.07225	−	1.07225i	−0.997178	−	0.0750671i	$-0.976083\pi$
$787$	−23673.2	−	23673.2i	−1.07225	−	1.07225i	−0.0750671	−	0.997178i	$-0.523917\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	27419.2i		−	1.23251i
$792$		0			0
$793$			19243.8i			0.861751i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25228.1	−	25228.1i	−1.12124	−	1.12124i	−0.991556	−	0.129681i	$-0.958605\pi$
$797$	−25228.1	−	25228.1i	−1.12124	−	1.12124i	−0.129681	−	0.991556i	$-0.541395\pi$
$798$		0			0
$799$	−589.758			−0.0261128
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32198.8	+	32198.8i	1.41503	+	1.41503i
$804$		0			0
$805$	−17573.8	+	17573.8i	−0.769434	+	0.769434i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			12860.4i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$809$			12860.4i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$810$		0			0
$811$	−12943.3	+	12943.3i	−0.560421	+	0.560421i	−0.929427	−	0.369006i	$-0.879698\pi$
$811$	−12943.3	+	12943.3i	−0.560421	+	0.560421i	0.369006	+	0.929427i	$0.379698\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8018.59			0.344637
$816$		0			0
$817$	−684.413			−0.0293079
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5296.51	−	5296.51i	0.225152	−	0.225152i	−0.585512	−	0.810664i	$-0.699107\pi$
$821$	5296.51	−	5296.51i	0.225152	−	0.225152i	0.810664	+	0.585512i	$0.199107\pi$
$822$		0			0
$823$			11716.4i			0.496242i	0.968729	+	0.248121i	$0.0798130\pi$
$823$			11716.4i			0.496242i	−0.968729	+	0.248121i	$0.920187\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18597.5	+	18597.5i	−0.781980	+	0.781980i	−0.980165	−	0.198185i	$-0.936495\pi$
$827$	−18597.5	+	18597.5i	−0.781980	+	0.781980i	0.198185	+	0.980165i	$0.436495\pi$
$828$		0			0
$829$	−9219.95	−	9219.95i	−0.386275	−	0.386275i	0.487081	−	0.873357i	$-0.338061\pi$
$829$	−9219.95	−	9219.95i	−0.386275	−	0.386275i	−0.873357	+	0.487081i	$0.838061\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2626.07			−0.109229
$834$		0			0
$835$	34578.3	+	34578.3i	1.43309	+	1.43309i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			12228.0i			0.503166i	0.967836	+	0.251583i	$0.0809511\pi$
$839$			12228.0i			0.503166i	−0.967836	+	0.251583i	$0.919049\pi$
$840$		0			0
$841$			13421.9i			0.550327i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−33462.1	−	33462.1i	−1.36228	−	1.36228i
$846$		0			0
$847$	63302.7			2.56801
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24122.8	−	24122.8i	−0.971703	−	0.971703i
$852$		0			0
$853$	24926.0	−	24926.0i	1.00053	−	1.00053i	0.000527142	−	1.00000i	$-0.499832\pi$
$853$	24926.0	−	24926.0i	1.00053	−	1.00053i	1.00000		0.000527142i	$-0.000167794\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			15047.4i			0.599779i	0.953974	+	0.299889i	$0.0969498\pi$
$857$			15047.4i			0.599779i	−0.953974	+	0.299889i	$0.903050\pi$
$858$		0			0
$859$	−28014.3	+	28014.3i	−1.11273	+	1.11273i	−0.119951	+	0.992780i	$0.538274\pi$
$859$	−28014.3	+	28014.3i	−1.11273	+	1.11273i	−0.992780	+	0.119951i	$0.961726\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14011.6			0.552676			0.276338	−	0.961061i	$-0.410879\pi$
$863$	14011.6			0.552676			0.276338	+	0.961061i	$0.410879\pi$
$864$		0			0
$865$	48158.8			1.89300
$866$		0			0
$867$		0			0
$868$		0			0
$869$	32296.7	−	32296.7i	1.26075	−	1.26075i
$870$		0			0
$871$			67533.8i			2.62721i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	20516.4	−	20516.4i	0.792664	−	0.792664i
$876$		0			0
$877$	−11314.7	−	11314.7i	−0.435657	−	0.435657i	0.454890	−	0.890548i	$-0.349678\pi$
$877$	−11314.7	−	11314.7i	−0.435657	−	0.435657i	−0.890548	+	0.454890i	$0.849678\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16205.5			0.619723			0.309862	−	0.950782i	$-0.399717\pi$
$881$	16205.5			0.619723			0.309862	+	0.950782i	$0.399717\pi$
$882$		0			0
$883$	4148.30	+	4148.30i	0.158099	+	0.158099i	0.781724	−	0.623625i	$-0.214341\pi$
$883$	4148.30	+	4148.30i	0.158099	+	0.158099i	−0.623625	+	0.781724i	$0.714341\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			17188.7i			0.650665i	0.945600	+	0.325333i	$0.105476\pi$
$887$			17188.7i			0.650665i	−0.945600	+	0.325333i	$0.894524\pi$
$888$		0			0
$889$			7941.24i			0.299596i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1457.90	+	1457.90i	0.0546324	+	0.0546324i
$894$		0			0
$895$	−13909.0			−0.519472
$896$		0			0
$897$		0			0
$898$		0			0
$899$	25204.9	+	25204.9i	0.935072	+	0.935072i
$900$		0			0
$901$	3804.79	−	3804.79i	0.140684	−	0.140684i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	16890.7i		−	0.620403i
$906$		0			0
$907$	14533.5	−	14533.5i	0.532059	−	0.532059i	−0.389126	−	0.921185i	$-0.627223\pi$
$907$	14533.5	−	14533.5i	0.532059	−	0.532059i	0.921185	+	0.389126i	$0.127223\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	47429.5			1.72493			0.862463	−	0.506120i	$-0.168921\pi$
$911$	47429.5			1.72493			0.862463	+	0.506120i	$0.168921\pi$
$912$		0			0
$913$	−15649.3			−0.567268
$914$		0			0
$915$		0			0
$916$		0			0
$917$	735.893	−	735.893i	0.0265009	−	0.0265009i
$918$		0			0
$919$			17383.3i			0.623963i	0.950088	+	0.311982i	$0.100993\pi$
$919$			17383.3i			0.623963i	−0.950088	+	0.311982i	$0.899007\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24712.4	+	24712.4i	−0.881276	+	0.881276i
$924$		0			0
$925$	−3447.62	−	3447.62i	−0.122548	−	0.122548i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6381.95			−0.225387			−0.112694	−	0.993630i	$-0.535948\pi$
$929$	−6381.95			−0.225387			−0.112694	+	0.993630i	$0.535948\pi$
$930$		0			0
$931$	6491.72	+	6491.72i	0.228526	+	0.228526i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	13654.1i		−	0.477580i
$936$		0			0
$937$			25340.8i			0.883508i	0.897136	+	0.441754i	$0.145644\pi$
$937$			25340.8i			0.883508i	−0.897136	+	0.441754i	$0.854356\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6841.75	+	6841.75i	0.237019	+	0.237019i	0.815615	−	0.578596i	$-0.196399\pi$
$941$	6841.75	+	6841.75i	0.237019	+	0.237019i	−0.578596	+	0.815615i	$0.696399\pi$
$942$		0			0
$943$	15083.2			0.520866
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30955.9	−	30955.9i	−1.06223	−	1.06223i	−0.997931	−	0.0642999i	$-0.979519\pi$
$947$	−30955.9	−	30955.9i	−1.06223	−	1.06223i	−0.0642999	−	0.997931i	$-0.520481\pi$
$948$		0			0
$949$	−39210.1	+	39210.1i	−1.34122	+	1.34122i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	38750.3i		−	1.31715i	−0.752515	−	0.658576i	$-0.771159\pi$
$953$		−	38750.3i		−	1.31715i	0.752515	−	0.658576i	$-0.228841\pi$
$954$		0			0
$955$	2122.84	−	2122.84i	0.0719305	−	0.0719305i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	28027.9			0.943761
$960$		0			0
$961$	3812.25			0.127967
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16629.1	+	16629.1i	−0.554725	+	0.554725i
$966$		0			0
$967$			2130.28i			0.0708430i	0.999372	+	0.0354215i	$0.0112774\pi$
$967$			2130.28i			0.0708430i	−0.999372	+	0.0354215i	$0.988723\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−29449.7	+	29449.7i	−0.973313	+	0.973313i	−0.999653	−	0.0263404i	$-0.991615\pi$
$971$	−29449.7	+	29449.7i	−0.973313	+	0.973313i	0.0263404	+	0.999653i	$0.491615\pi$
$972$		0			0
$973$	−20958.0	−	20958.0i	−0.690528	−	0.690528i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1913.67			−0.0626650			−0.0313325	−	0.999509i	$-0.509975\pi$
$977$	−1913.67			−0.0626650			−0.0313325	+	0.999509i	$0.509975\pi$
$978$		0			0
$979$	−71287.2	−	71287.2i	−2.32722	−	2.32722i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	40958.4i		−	1.32896i	−0.747305	−	0.664481i	$-0.768653\pi$
$983$		−	40958.4i		−	1.32896i	0.747305	−	0.664481i	$-0.231347\pi$
$984$		0			0
$985$		−	19617.4i		−	0.634580i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−737.269	−	737.269i	−0.0237045	−	0.0237045i
$990$		0			0
$991$	−53772.7			−1.72366			−0.861830	−	0.507197i	$-0.830682\pi$
$991$	−53772.7			−1.72366			−0.861830	+	0.507197i	$0.830682\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29053.9	−	29053.9i	−0.925698	−	0.925698i
$996$		0			0
$997$	29569.4	−	29569.4i	0.939290	−	0.939290i	−0.0589701	−	0.998260i	$-0.518782\pi$
$997$	29569.4	−	29569.4i	0.939290	−	0.939290i	0.998260	+	0.0589701i	$0.0187817\pi$
$998$		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.145.3		24
3.2	odd	2	inner	576.4.k.c.145.10		24
4.3	odd	2		144.4.k.c.109.10	yes	24
12.11	even	2		144.4.k.c.109.3	yes	24
16.5	even	4	inner	576.4.k.c.433.3		24
16.11	odd	4		144.4.k.c.37.10	yes	24
48.5	odd	4	inner	576.4.k.c.433.10		24
48.11	even	4		144.4.k.c.37.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.k.c.37.3	✓	24	48.11	even	4
144.4.k.c.37.10	yes	24	16.11	odd	4
144.4.k.c.109.3	yes	24	12.11	even	2
144.4.k.c.109.10	yes	24	4.3	odd	2
576.4.k.c.145.3		24	1.1	even	1	trivial
576.4.k.c.145.10		24	3.2	odd	2	inner
576.4.k.c.433.3		24	16.5	even	4	inner
576.4.k.c.433.10		24	48.5	odd	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)

\(f(q)\)	\(=\)	\(q+(-8.32567 + 8.32567i) q^{5} +22.1273i q^{7} +(-45.7813 + 45.7813i) q^{11} +(-55.7501 - 55.7501i) q^{13} +17.9113 q^{17} +(-44.2772 - 44.2772i) q^{19} -95.3934i q^{23} -13.6336i q^{25} +(137.497 + 137.497i) q^{29} +183.312 q^{31} +(-184.224 - 184.224i) q^{35} +(252.877 - 252.877i) q^{37} +158.116i q^{41} +(7.72872 - 7.72872i) q^{43} -32.9266 q^{47} -146.615 q^{49} +(212.424 - 212.424i) q^{53} -762.320i q^{55} +(-117.228 + 117.228i) q^{59} +(-172.590 - 172.590i) q^{61} +928.314 q^{65} +(-605.683 - 605.683i) q^{67} -443.271i q^{71} -703.319i q^{73} +(-1013.01 - 1013.01i) q^{77} -705.456 q^{79} +(170.914 + 170.914i) q^{83} +(-149.123 + 149.123i) q^{85} +1557.12i q^{89} +(1233.60 - 1233.60i) q^{91} +737.275 q^{95} +892.331 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{19} - 744 q^{31} - 16 q^{37} + 376 q^{43} - 1176 q^{49} - 912 q^{61} - 1440 q^{67} + 328 q^{79} - 240 q^{85} + 104 q^{91}+O(q^{100}) \) 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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