LMFDB - Embedded newform 480.4.m.b.239.44

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,4,Mod(239,480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("480.239"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$64$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.44
Character	$\chi$	$=$	480.239
Dual form			480.4.m.b.239.42

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.64167 + 4.47455i) q^{3} +(10.4671 + 3.92924i) q^{5} -4.10158 q^{7} +(-13.0432 + 23.6406i) q^{9} +5.96978i q^{11} +33.0570 q^{13} +(10.0691 + 57.2155i) q^{15} +50.6136 q^{17} -74.1386 q^{19} +(-10.8350 - 18.3527i) q^{21} +184.927i q^{23} +(94.1221 + 82.2559i) q^{25} +(-140.237 + 4.08828i) q^{27} -98.3905 q^{29} +192.624i q^{31} +(-26.7121 + 15.7702i) q^{33} +(-42.9318 - 16.1161i) q^{35} +350.946 q^{37} +(87.3258 + 147.915i) q^{39} -292.535i q^{41} -80.8246i q^{43} +(-229.414 + 196.199i) q^{45} +63.1702i q^{47} -326.177 q^{49} +(133.705 + 226.473i) q^{51} -178.821i q^{53} +(-23.4567 + 62.4865i) q^{55} +(-195.850 - 331.737i) q^{57} -479.870i q^{59} +635.627i q^{61} +(53.4975 - 96.9635i) q^{63} +(346.013 + 129.889i) q^{65} -288.505i q^{67} +(-827.464 + 488.515i) q^{69} -1062.20 q^{71} +980.889i q^{73} +(-119.418 + 638.447i) q^{75} -24.4855i q^{77} +804.239i q^{79} +(-388.752 - 616.695i) q^{81} +300.045 q^{83} +(529.780 + 198.873i) q^{85} +(-259.915 - 440.253i) q^{87} +1112.49i q^{89} -135.586 q^{91} +(-861.905 + 508.849i) q^{93} +(-776.020 - 291.309i) q^{95} -1217.88i q^{97} +(-141.129 - 77.8648i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q - 112 q^{9} - 672 q^{19} + 496 q^{25} + 3520 q^{49} + 544 q^{51} + 1600 q^{75} - 2304 q^{81} + 2752 q^{91} - 256 q^{99}+O(q^{100}) $ 64 * q - 112 * q^9 - 672 * q^19 + 496 * q^25 + 3520 * q^49 + 544 * q^51 + 1600 * q^75 - 2304 * q^81 + 2752 * q^91 - 256 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	2.64167	+	4.47455i	0.508390	+	0.861127i
$3$	2.64167	+	4.47455i	0.508390	+	0.861127i
$4$		0			0
$5$	10.4671	+	3.92924i	0.936210	+	0.351442i
$5$	10.4671	+	3.92924i	0.936210	+	0.351442i
$6$		0			0
$7$	−4.10158			−0.221464			−0.110732	−	0.993850i	$-0.535320\pi$
$7$	−4.10158			−0.221464			−0.110732	+	0.993850i	$0.535320\pi$
$8$		0			0
$9$	−13.0432	+	23.6406i	−0.483080	+	0.875576i
$10$		0			0
$11$			5.96978i			0.163632i	0.996647	+	0.0818162i	$0.0260720\pi$
$11$			5.96978i			0.163632i	−0.996647	+	0.0818162i	$0.973928\pi$
$12$		0			0
$13$	33.0570			0.705259			0.352630	−	0.935763i	$-0.385288\pi$
$13$	33.0570			0.705259			0.352630	+	0.935763i	$0.385288\pi$
$14$		0			0
$15$	10.0691	+	57.2155i	0.173323	+	0.984865i
$16$		0			0
$17$	50.6136			0.722095			0.361047	−	0.932547i	$-0.382419\pi$
$17$	50.6136			0.722095			0.361047	+	0.932547i	$0.382419\pi$
$18$		0			0
$19$	−74.1386			−0.895188			−0.447594	−	0.894237i	$-0.647719\pi$
$19$	−74.1386			−0.895188			−0.447594	+	0.894237i	$0.647719\pi$
$20$		0			0
$21$	−10.8350	−	18.3527i	−0.112590	−	0.190709i
$22$		0			0
$23$			184.927i			1.67652i	0.545273	+	0.838259i	$0.316426\pi$
$23$			184.927i			1.67652i	−0.545273	+	0.838259i	$0.683574\pi$
$24$		0			0
$25$	94.1221	+	82.2559i	0.752977	+	0.658047i
$26$		0			0
$27$	−140.237	+	4.08828i	−0.999575	+	0.0291404i
$28$		0			0
$29$	−98.3905			−0.630023			−0.315011	−	0.949088i	$-0.602008\pi$
$29$	−98.3905			−0.630023			−0.315011	+	0.949088i	$0.602008\pi$
$30$		0			0
$31$			192.624i			1.11601i	0.829838	+	0.558004i	$0.188433\pi$
$31$			192.624i			1.11601i	−0.829838	+	0.558004i	$0.811567\pi$
$32$		0			0
$33$	−26.7121	+	15.7702i	−0.140908	+	0.0831890i
$34$		0			0
$35$	−42.9318	−	16.1161i	−0.207337	−	0.0778319i
$36$		0			0
$37$	350.946			1.55933			0.779665	−	0.626196i	$-0.215389\pi$
$37$	350.946			1.55933			0.779665	+	0.626196i	$0.215389\pi$
$38$		0			0
$39$	87.3258	+	147.915i	0.358547	+	0.607318i
$40$		0			0
$41$		−	292.535i		−	1.11430i	−0.830412	−	0.557150i	$-0.811895\pi$
$41$		−	292.535i		−	1.11430i	0.830412	−	0.557150i	$-0.188105\pi$
$42$		0			0
$43$		−	80.8246i		−	0.286643i	−0.989676	−	0.143321i	$-0.954222\pi$
$43$		−	80.8246i		−	0.286643i	0.989676	−	0.143321i	$-0.0457782\pi$
$44$		0			0
$45$	−229.414	+	196.199i	−0.759979	+	0.649948i
$46$		0			0
$47$			63.1702i			0.196049i	0.995184	+	0.0980246i	$0.0312524\pi$
$47$			63.1702i			0.196049i	−0.995184	+	0.0980246i	$0.968748\pi$
$48$		0			0
$49$	−326.177			−0.950954
$50$		0			0
$51$	133.705	+	226.473i	0.367105	+	0.621815i
$52$		0			0
$53$		−	178.821i		−	0.463453i	−0.972781	−	0.231726i	$-0.925563\pi$
$53$		−	178.821i		−	0.463453i	0.972781	−	0.231726i	$-0.0744374\pi$
$54$		0			0
$55$	−23.4567	+	62.4865i	−0.0575073	+	0.153194i
$56$		0			0
$57$	−195.850	−	331.737i	−0.455104	−	0.770871i
$58$		0			0
$59$		−	479.870i		−	1.05888i	−0.848348	−	0.529439i	$-0.822402\pi$
$59$		−	479.870i		−	1.05888i	0.848348	−	0.529439i	$-0.177598\pi$
$60$		0			0
$61$			635.627i			1.33416i	0.744986	+	0.667080i	$0.232456\pi$
$61$			635.627i			1.33416i	−0.744986	+	0.667080i	$0.767544\pi$
$62$		0			0
$63$	53.4975	−	96.9635i	0.106985	−	0.193909i
$64$		0			0
$65$	346.013	+	129.889i	0.660271	+	0.247858i
$66$		0			0
$67$		−	288.505i		−	0.526067i	−0.964787	−	0.263033i	$-0.915277\pi$
$67$		−	288.505i		−	0.526067i	0.964787	−	0.263033i	$-0.0847229\pi$
$68$		0			0
$69$	−827.464	+	488.515i	−1.44369	+	0.852324i
$70$		0			0
$71$	−1062.20			−1.77549			−0.887743	−	0.460340i	$-0.847727\pi$
$71$	−1062.20			−1.77549			−0.887743	+	0.460340i	$0.847727\pi$
$72$		0			0
$73$			980.889i			1.57266i	0.617805	+	0.786332i	$0.288022\pi$
$73$			980.889i			1.57266i	−0.617805	+	0.786332i	$0.711978\pi$
$74$		0			0
$75$	−119.418	+	638.447i	−0.183857	+	0.982953i
$76$		0			0
$77$		−	24.4855i		−	0.0362387i
$78$		0			0
$79$			804.239i			1.14537i	0.819777	+	0.572683i	$0.194097\pi$
$79$			804.239i			1.14537i	−0.819777	+	0.572683i	$0.805903\pi$
$80$		0			0
$81$	−388.752	−	616.695i	−0.533267	−	0.845947i
$82$		0			0
$83$	300.045			0.396798			0.198399	−	0.980121i	$-0.436426\pi$
$83$	300.045			0.396798			0.198399	+	0.980121i	$0.436426\pi$
$84$		0			0
$85$	529.780	+	198.873i	0.676032	+	0.253775i
$86$		0			0
$87$	−259.915	−	440.253i	−0.320297	−	0.542530i
$88$		0			0
$89$			1112.49i			1.32499i	0.749066	+	0.662495i	$0.230502\pi$
$89$			1112.49i			1.32499i	−0.749066	+	0.662495i	$0.769498\pi$
$90$		0			0
$91$	−135.586			−0.156190
$92$		0			0
$93$	−861.905	+	508.849i	−0.961026	+	0.567367i
$94$		0			0
$95$	−776.020	−	291.309i	−0.838083	−	0.314607i
$96$		0			0
$97$		−	1217.88i		−	1.27481i	−0.770529	−	0.637405i	$-0.780008\pi$
$97$		−	1217.88i		−	1.27481i	0.770529	−	0.637405i	$-0.219992\pi$
$98$		0			0
$99$	−141.129	−	77.8648i	−0.143273	−	0.0790475i
$100$		0			0
$101$	461.131			0.454300			0.227150	−	0.973860i	$-0.427059\pi$
$101$	461.131			0.454300			0.227150	+	0.973860i	$0.427059\pi$
$102$		0			0
$103$	1363.74			1.30460			0.652299	−	0.757962i	$-0.273805\pi$
$103$	1363.74			1.30460			0.652299	+	0.757962i	$0.273805\pi$
$104$		0			0
$105$	−41.2994	−	234.674i	−0.0383848	−	0.218112i
$106$		0			0
$107$	−626.784			−0.566295			−0.283147	−	0.959076i	$-0.591379\pi$
$107$	−626.784			−0.566295			−0.283147	+	0.959076i	$0.591379\pi$
$108$		0			0
$109$		−	472.473i		−	0.415181i	−0.978216	−	0.207590i	$-0.933438\pi$
$109$		−	472.473i		−	0.415181i	0.978216	−	0.207590i	$-0.0665621\pi$
$110$		0			0
$111$	927.084	+	1570.33i	0.792747	+	1.34278i
$112$		0			0
$113$	−507.424			−0.422429			−0.211214	−	0.977440i	$-0.567742\pi$
$113$	−507.424			−0.422429			−0.211214	+	0.977440i	$0.567742\pi$
$114$		0			0
$115$	−726.622	+	1935.65i	−0.589199	+	1.56957i
$116$		0			0
$117$	−431.168	+	781.487i	−0.340697	+	0.617508i
$118$		0			0
$119$	−207.596			−0.159918
$120$		0			0
$121$	1295.36			0.973224
$122$		0			0
$123$	1308.96	−	772.781i	0.959555	−	0.566499i
$124$		0			0
$125$	661.986	+	1230.81i	0.473679	+	0.880698i
$126$		0			0
$127$	2049.23			1.43181			0.715903	−	0.698199i	$-0.246015\pi$
$127$	2049.23			1.43181			0.715903	+	0.698199i	$0.246015\pi$
$128$		0			0
$129$	361.654	−	213.512i	0.246836	−	0.145726i
$130$		0			0
$131$		−	1677.53i		−	1.11883i	−0.828888	−	0.559415i	$-0.811026\pi$
$131$		−	1677.53i		−	1.11883i	0.828888	−	0.559415i	$-0.188974\pi$
$132$		0			0
$133$	304.085			0.198252
$134$		0			0
$135$	−1483.94	−	508.231i	−0.946053	−	0.324011i
$136$		0			0
$137$	−1839.77			−1.14731			−0.573657	−	0.819096i	$-0.694476\pi$
$137$	−1839.77			−1.14731			−0.573657	+	0.819096i	$0.694476\pi$
$138$		0			0
$139$	1598.25			0.975264			0.487632	−	0.873049i	$-0.337861\pi$
$139$	1598.25			0.975264			0.487632	+	0.873049i	$0.337861\pi$
$140$		0			0
$141$	−282.658	+	166.875i	−0.168823	+	0.0996694i
$142$		0			0
$143$			197.343i			0.115403i
$144$		0			0
$145$	−1029.87	−	386.600i	−0.589833	−	0.221417i
$146$		0			0
$147$	−861.652	−	1459.50i	−0.483455	−	0.818892i
$148$		0			0
$149$	533.530			0.293346			0.146673	−	0.989185i	$-0.453144\pi$
$149$	533.530			0.293346			0.146673	+	0.989185i	$0.453144\pi$
$150$		0			0
$151$		−	3056.41i		−	1.64720i	−0.567173	−	0.823599i	$-0.691963\pi$
$151$		−	3056.41i		−	1.64720i	0.567173	−	0.823599i	$-0.308037\pi$
$152$		0			0
$153$	−660.162	+	1196.53i	−0.348830	+	0.632249i
$154$		0			0
$155$	−756.866	+	2016.22i	−0.392213	+	1.04482i
$156$		0			0
$157$	1429.75			0.726794			0.363397	−	0.931634i	$-0.381617\pi$
$157$	1429.75			0.726794			0.363397	+	0.931634i	$0.381617\pi$
$158$		0			0
$159$	800.145	−	472.387i	0.399092	−	0.235615i
$160$		0			0
$161$		−	758.491i		−	0.371289i
$162$		0			0
$163$		−	600.115i		−	0.288372i	−0.989551	−	0.144186i	$-0.953944\pi$
$163$		−	600.115i		−	0.288372i	0.989551	−	0.144186i	$-0.0460564\pi$
$164$		0			0
$165$	−341.564	+	60.1105i	−0.161156	+	0.0283612i
$166$		0			0
$167$			1087.44i			0.503883i	0.967742	+	0.251942i	$0.0810691\pi$
$167$			1087.44i			0.503883i	−0.967742	+	0.251942i	$0.918931\pi$
$168$		0			0
$169$	−1104.23			−0.502609
$170$		0			0
$171$	967.002	−	1752.68i	0.432447	−	0.783805i
$172$		0			0
$173$		−	374.343i		−	0.164513i	−0.996611	−	0.0822567i	$-0.973787\pi$
$173$		−	374.343i		−	0.164513i	0.996611	−	0.0822567i	$-0.0262127\pi$
$174$		0			0
$175$	−386.049	−	337.379i	−0.166757	−	0.145734i
$176$		0			0
$177$	2147.20	−	1267.66i	0.911829	−	0.538322i
$178$		0			0
$179$			1886.45i			0.787708i	0.919173	+	0.393854i	$0.128858\pi$
$179$			1886.45i			0.787708i	−0.919173	+	0.393854i	$0.871142\pi$
$180$		0			0
$181$		−	986.271i		−	0.405022i	−0.979280	−	0.202511i	$-0.935090\pi$
$181$		−	986.271i		−	0.405022i	0.979280	−	0.202511i	$-0.0649101\pi$
$182$		0			0
$183$	−2844.15	+	1679.12i	−1.14888	+	0.678273i
$184$		0			0
$185$	3673.41	+	1378.95i	1.45986	+	0.548015i
$186$		0			0
$187$			302.152i			0.118158i
$188$		0			0
$189$	575.191	−	16.7684i	0.221370	−	0.00645355i
$190$		0			0
$191$	3513.46			1.33102			0.665510	−	0.746389i	$-0.268214\pi$
$191$	3513.46			1.33102			0.665510	+	0.746389i	$0.268214\pi$
$192$		0			0
$193$			3049.07i			1.13719i	0.822619	+	0.568593i	$0.192512\pi$
$193$			3049.07i			1.13719i	−0.822619	+	0.568593i	$0.807488\pi$
$194$		0			0
$195$	332.856	+	1891.37i	0.122238	+	0.694585i
$196$		0			0
$197$		−	3893.24i		−	1.40803i	−0.710186	−	0.704014i	$-0.751389\pi$
$197$		−	3893.24i		−	1.40803i	0.710186	−	0.704014i	$-0.248611\pi$
$198$		0			0
$199$		−	2888.21i		−	1.02884i	−0.857537	−	0.514422i	$-0.828006\pi$
$199$		−	2888.21i		−	1.02884i	0.857537	−	0.514422i	$-0.171994\pi$
$200$		0			0
$201$	1290.93	−	762.134i	0.453010	−	0.267447i
$202$		0			0
$203$	403.556			0.139528
$204$		0			0
$205$	1149.44	−	3062.01i	0.391612	−	1.04322i
$206$		0			0
$207$	−4371.77	−	2412.03i	−1.46792	−	0.809892i
$208$		0			0
$209$		−	442.591i		−	0.146482i
$210$		0			0
$211$	1800.81			0.587550			0.293775	−	0.955875i	$-0.405088\pi$
$211$	1800.81			0.587550			0.293775	+	0.955875i	$0.405088\pi$
$212$		0			0
$213$	−2805.97	−	4752.85i	−0.902638	−	1.52892i
$214$		0			0
$215$	317.579	−	846.003i	0.100738	−	0.268358i
$216$		0			0
$217$		−	790.062i		−	0.247156i
$218$		0			0
$219$	−4389.04	+	2591.19i	−1.35426	+	0.799526i
$220$		0			0
$221$	1673.14			0.509264
$222$		0			0
$223$	3394.93			1.01947			0.509734	−	0.860332i	$-0.329744\pi$
$223$	3394.93			1.01947			0.509734	+	0.860332i	$0.329744\pi$
$224$		0			0
$225$	−3172.22	+	1152.22i	−0.939918	+	0.341399i
$226$		0			0
$227$	1666.84			0.487366			0.243683	−	0.969855i	$-0.421644\pi$
$227$	1666.84			0.487366			0.243683	+	0.969855i	$0.421644\pi$
$228$		0			0
$229$		−	5278.56i		−	1.52322i	−0.648037	−	0.761609i	$-0.724410\pi$
$229$		−	5278.56i		−	1.52322i	0.648037	−	0.761609i	$-0.275590\pi$
$230$		0			0
$231$	109.562	−	64.6826i	0.0312061	−	0.0184234i
$232$		0			0
$233$	−469.587			−0.132033			−0.0660165	−	0.997819i	$-0.521029\pi$
$233$	−469.587			−0.132033			−0.0660165	+	0.997819i	$0.521029\pi$
$234$		0			0
$235$	−248.211	+	661.211i	−0.0689000	+	0.183543i
$236$		0			0
$237$	−3598.61	+	2124.53i	−0.986306	+	0.582292i
$238$		0			0
$239$	130.148			0.0352242			0.0176121	−	0.999845i	$-0.494394\pi$
$239$	130.148			0.0352242			0.0176121	+	0.999845i	$0.494394\pi$
$240$		0			0
$241$	5560.04			1.48611			0.743057	−	0.669228i	$-0.233375\pi$
$241$	5560.04			1.48611			0.743057	+	0.669228i	$0.233375\pi$
$242$		0			0
$243$	1732.48	−	3368.59i	0.457360	−	0.889281i
$244$		0			0
$245$	−3414.14	−	1281.63i	−0.890292	−	0.334205i
$246$		0			0
$247$	−2450.80			−0.631340
$248$		0			0
$249$	792.619	+	1342.57i	0.201728	+	0.341693i
$250$		0			0
$251$		−	6537.02i		−	1.64388i	−0.569576	−	0.821939i	$-0.692893\pi$
$251$		−	6537.02i		−	1.64388i	0.569576	−	0.821939i	$-0.307107\pi$
$252$		0			0
$253$	−1103.97			−0.274332
$254$		0			0
$255$	509.636	+	2895.88i	0.125155	+	0.711166i
$256$		0			0
$257$	2769.77			0.672270			0.336135	−	0.941814i	$-0.390880\pi$
$257$	2769.77			0.672270			0.336135	+	0.941814i	$0.390880\pi$
$258$		0			0
$259$	−1439.43			−0.345336
$260$		0			0
$261$	1283.32	−	2326.01i	0.304351	−	0.551633i
$262$		0			0
$263$		−	2001.24i		−	0.469207i	−0.972091	−	0.234604i	$-0.924621\pi$
$263$		−	2001.24i		−	0.469207i	0.972091	−	0.234604i	$-0.0753792\pi$
$264$		0			0
$265$	702.632	−	1871.75i	0.162877	−	0.433889i
$266$		0			0
$267$	−4977.91	+	2938.84i	−1.14098	+	0.673611i
$268$		0			0
$269$	2476.45			0.561307			0.280654	−	0.959809i	$-0.409449\pi$
$269$	2476.45			0.561307			0.280654	+	0.959809i	$0.409449\pi$
$270$		0			0
$271$			1616.36i			0.362313i	0.983454	+	0.181156i	$0.0579840\pi$
$271$			1616.36i			0.362313i	−0.983454	+	0.181156i	$0.942016\pi$
$272$		0			0
$273$	−358.173	−	606.686i	−0.0794052	−	0.134499i
$274$		0			0
$275$	−491.049	+	561.888i	−0.107678	+	0.123211i
$276$		0			0
$277$	5749.66			1.24716			0.623580	−	0.781759i	$-0.285677\pi$
$277$	5749.66			1.24716			0.623580	+	0.781759i	$0.285677\pi$
$278$		0			0
$279$	−4553.74	−	2512.43i	−0.977151	−	0.539122i
$280$		0			0
$281$		−	7754.89i		−	1.64633i	−0.567804	−	0.823164i	$-0.692207\pi$
$281$		−	7754.89i		−	1.64633i	0.567804	−	0.823164i	$-0.307793\pi$
$282$		0			0
$283$		−	4931.44i		−	1.03584i	−0.855428	−	0.517922i	$-0.826706\pi$
$283$		−	4931.44i		−	1.03584i	0.855428	−	0.517922i	$-0.173294\pi$
$284$		0			0
$285$	−746.512	−	4241.88i	−0.155156	−	0.881639i
$286$		0			0
$287$			1199.86i			0.246778i
$288$		0			0
$289$	−2351.26			−0.478579
$290$		0			0
$291$	5449.44	−	3217.22i	1.09777	−	0.648100i
$292$		0			0
$293$			688.926i			0.137363i	0.997639	+	0.0686817i	$0.0218793\pi$
$293$			688.926i			0.137363i	−0.997639	+	0.0686817i	$0.978121\pi$
$294$		0			0
$295$	1885.53	−	5022.87i	0.372134	−	0.991332i
$296$		0			0
$297$	−24.4061	−	837.181i	−0.00476831	−	0.163563i
$298$		0			0
$299$			6113.13i			1.18238i
$300$		0			0
$301$			331.508i			0.0634811i
$302$		0			0
$303$	1218.16	+	2063.35i	0.230961	+	0.391210i
$304$		0			0
$305$	−2497.53	+	6653.20i	−0.468880	+	1.24905i
$306$		0			0
$307$			6625.29i			1.23168i	0.787872	+	0.615839i	$0.211183\pi$
$307$			6625.29i			1.23168i	−0.787872	+	0.615839i	$0.788817\pi$
$308$		0			0
$309$	3602.56	+	6102.14i	0.663244	+	1.12343i
$310$		0			0
$311$	5355.88			0.976540			0.488270	−	0.872693i	$-0.337628\pi$
$311$	5355.88			0.976540			0.488270	+	0.872693i	$0.337628\pi$
$312$		0			0
$313$		−	57.6042i		−	0.0104025i	−0.999986	−	0.00520125i	$-0.998344\pi$
$313$		−	57.6042i		−	0.0104025i	0.999986	−	0.00520125i	$-0.00165562\pi$
$314$		0			0
$315$	940.959	−	804.726i	0.168308	−	0.143940i
$316$		0			0
$317$			7806.30i			1.38311i	0.722325	+	0.691554i	$0.243074\pi$
$317$			7806.30i			1.38311i	−0.722325	+	0.691554i	$0.756926\pi$
$318$		0			0
$319$		−	587.370i		−	0.103092i
$320$		0			0
$321$	−1655.76	−	2804.58i	−0.287898	−	0.487652i
$322$		0			0
$323$	−3752.43			−0.646410
$324$		0			0
$325$	3111.40	+	2719.14i	0.531044	+	0.464094i
$326$		0			0
$327$	2114.10	−	1248.12i	0.357523	−	0.211074i
$328$		0			0
$329$		−	259.097i		−	0.0434179i
$330$		0			0
$331$	6218.44			1.03262			0.516308	−	0.856403i	$-0.327306\pi$
$331$	6218.44			1.03262			0.516308	+	0.856403i	$0.327306\pi$
$332$		0			0
$333$	−4577.45	+	8296.57i	−0.753282	+	1.36531i
$334$		0			0
$335$	1133.61	−	3019.82i	0.184882	−	0.492509i
$336$		0			0
$337$			8547.36i			1.38162i	0.723038	+	0.690808i	$0.242745\pi$
$337$			8547.36i			1.38162i	−0.723038	+	0.690808i	$0.757255\pi$
$338$		0			0
$339$	−1340.45	−	2270.49i	−0.214758	−	0.363765i
$340$		0			0
$341$	−1149.92			−0.182615
$342$		0			0
$343$	2744.68			0.432067
$344$		0			0
$345$	−10580.7	+	1862.05i	−1.65114	+	0.290579i
$346$		0			0
$347$	11848.5			1.83303			0.916517	−	0.399996i	$-0.130988\pi$
$347$	11848.5			1.83303			0.916517	+	0.399996i	$0.130988\pi$
$348$		0			0
$349$			3765.09i			0.577480i	0.957408	+	0.288740i	$0.0932363\pi$
$349$			3765.09i			0.577480i	−0.957408	+	0.288740i	$0.906764\pi$
$350$		0			0
$351$	−4635.80	+	135.146i	−0.704960	+	0.0205515i
$352$		0			0
$353$	−7721.18			−1.16418			−0.582092	−	0.813123i	$-0.697766\pi$
$353$	−7721.18			−1.16418			−0.582092	+	0.813123i	$0.697766\pi$
$354$		0			0
$355$	−11118.2	−	4173.63i	−1.66223	−	0.623980i
$356$		0			0
$357$	−548.399	−	928.897i	−0.0813007	−	0.137710i
$358$		0			0
$359$	−1277.28			−0.187778			−0.0938892	−	0.995583i	$-0.529930\pi$
$359$	−1277.28			−0.187778			−0.0938892	+	0.995583i	$0.529930\pi$
$360$		0			0
$361$	−1362.46			−0.198639
$362$		0			0
$363$	3421.92	+	5796.16i	0.494777	+	0.838070i
$364$		0			0
$365$	−3854.15	+	10267.1i	−0.552700	+	1.47234i
$366$		0			0
$367$	4874.49			0.693315			0.346657	−	0.937992i	$-0.387317\pi$
$367$	4874.49			0.693315			0.346657	+	0.937992i	$0.387317\pi$
$368$		0			0
$369$	6915.70	+	3815.58i	0.975655	+	0.538296i
$370$		0			0
$371$			733.449i			0.102638i
$372$		0			0
$373$	9969.36			1.38390			0.691949	−	0.721946i	$-0.256752\pi$
$373$	9969.36			1.38390			0.691949	+	0.721946i	$0.256752\pi$
$374$		0			0
$375$	−3758.58	+	6213.49i	−0.517580	+	0.855635i
$376$		0			0
$377$	−3252.50			−0.444330
$378$		0			0
$379$	3910.73			0.530028			0.265014	−	0.964245i	$-0.414623\pi$
$379$	3910.73			0.530028			0.265014	+	0.964245i	$0.414623\pi$
$380$		0			0
$381$	5413.38	+	9169.36i	0.727916	+	1.23297i
$382$		0			0
$383$		−	2301.53i		−	0.307056i	−0.988144	−	0.153528i	$-0.950936\pi$
$383$		−	2301.53i		−	0.307056i	0.988144	−	0.153528i	$-0.0490636\pi$
$384$		0			0
$385$	96.2095	−	256.293i	0.0127358	−	0.0339270i
$386$		0			0
$387$	1910.74	+	1054.21i	0.250978	+	0.138471i
$388$		0			0
$389$	−13438.1			−1.75152			−0.875758	−	0.482751i	$-0.839638\pi$
$389$	−13438.1			−1.75152			−0.875758	+	0.482751i	$0.839638\pi$
$390$		0			0
$391$			9359.82i			1.21060i
$392$		0			0
$393$	7506.20	−	4431.49i	0.963455	−	0.568801i
$394$		0			0
$395$	−3160.05	+	8418.08i	−0.402530	+	1.07230i
$396$		0			0
$397$	−10501.3			−1.32757			−0.663786	−	0.747922i	$-0.731051\pi$
$397$	−10501.3			−1.32757			−0.663786	+	0.747922i	$0.731051\pi$
$398$		0			0
$399$	803.293	+	1360.64i	0.100789	+	0.170720i
$400$		0			0
$401$		−	13566.8i		−	1.68951i	−0.535155	−	0.844754i	$-0.679747\pi$
$401$		−	13566.8i		−	1.68951i	0.535155	−	0.844754i	$-0.320253\pi$
$402$		0			0
$403$			6367.58i			0.787076i
$404$		0			0
$405$	−1645.97	−	7982.54i	−0.201948	−	0.979396i
$406$		0			0
$407$			2095.07i			0.255157i
$408$		0			0
$409$	−7673.26			−0.927674			−0.463837	−	0.885921i	$-0.653528\pi$
$409$	−7673.26			−0.927674			−0.463837	+	0.885921i	$0.653528\pi$
$410$		0			0
$411$	−4860.06	−	8232.13i	−0.583282	−	0.987983i
$412$		0			0
$413$			1968.22i			0.234504i
$414$		0			0
$415$	3140.61	+	1178.95i	0.371486	+	0.139451i
$416$		0			0
$417$	4222.05	+	7151.44i	0.495814	+	0.839827i
$418$		0			0
$419$		−	1239.80i		−	0.144554i	−0.997385	−	0.0722768i	$-0.976973\pi$
$419$		−	1239.80i		−	0.144554i	0.997385	−	0.0722768i	$-0.0230265\pi$
$420$		0			0
$421$			1701.96i			0.197027i	0.995136	+	0.0985134i	$0.0314087\pi$
$421$			1701.96i			0.197027i	−0.995136	+	0.0985134i	$0.968591\pi$
$422$		0			0
$423$	−1493.38	−	823.939i	−0.171656	−	0.0947075i
$424$		0			0
$425$	4763.86	+	4163.27i	0.543721	+	0.475172i
$426$		0			0
$427$		−	2607.07i		−	0.295469i
$428$		0			0
$429$	−883.021	+	521.315i	−0.0993769	+	0.0586698i
$430$		0			0
$431$	−9669.03			−1.08061			−0.540303	−	0.841471i	$-0.681690\pi$
$431$	−9669.03			−1.08061			−0.540303	+	0.841471i	$0.681690\pi$
$432$		0			0
$433$		−	11871.2i		−	1.31753i	−0.752348	−	0.658766i	$-0.771079\pi$
$433$		−	11871.2i		−	1.31753i	0.752348	−	0.658766i	$-0.228921\pi$
$434$		0			0
$435$	−990.708	−	5629.46i	−0.109197	−	0.620487i
$436$		0			0
$437$		−	13710.2i		−	1.50080i
$438$		0			0
$439$			16373.4i			1.78009i	0.455868	+	0.890047i	$0.349329\pi$
$439$			16373.4i			1.78009i	−0.455868	+	0.890047i	$0.650671\pi$
$440$		0			0
$441$	4254.38	−	7711.01i	0.459387	−	0.832632i
$442$		0			0
$443$	−13008.8			−1.39519			−0.697594	−	0.716493i	$-0.745746\pi$
$443$	−13008.8			−1.39519			−0.697594	+	0.716493i	$0.745746\pi$
$444$		0			0
$445$	−4371.26	+	11644.6i	−0.465657	+	1.24047i
$446$		0			0
$447$	1409.41	+	2387.31i	0.149134	+	0.252608i
$448$		0			0
$449$			9615.94i			1.01070i	0.862914	+	0.505350i	$0.168637\pi$
$449$			9615.94i			1.01070i	−0.862914	+	0.505350i	$0.831363\pi$
$450$		0			0
$451$	1746.37			0.182336
$452$		0			0
$453$	13676.0	−	8074.02i	1.41845	−	0.837418i
$454$		0			0
$455$	−1419.20	−	532.750i	−0.146226	−	0.0548917i
$456$		0			0
$457$			3926.38i			0.401900i	0.979601	+	0.200950i	$0.0644030\pi$
$457$			3926.38i			0.401900i	−0.979601	+	0.200950i	$0.935597\pi$
$458$		0			0
$459$	−7097.88	+	206.923i	−0.721788	+	0.0210421i
$460$		0			0
$461$	360.528			0.0364240			0.0182120	−	0.999834i	$-0.494203\pi$
$461$	360.528			0.0364240			0.0182120	+	0.999834i	$0.494203\pi$
$462$		0			0
$463$	−14036.6			−1.40894			−0.704468	−	0.709735i	$-0.748815\pi$
$463$	−14036.6			−1.40894			−0.704468	+	0.709735i	$0.748815\pi$
$464$		0			0
$465$	−11021.1	+	1939.56i	−1.09912	+	0.193430i
$466$		0			0
$467$	−4186.44			−0.414829			−0.207415	−	0.978253i	$-0.566505\pi$
$467$	−4186.44			−0.414829			−0.207415	+	0.978253i	$0.566505\pi$
$468$		0			0
$469$			1183.32i			0.116505i
$470$		0			0
$471$	3776.93	+	6397.49i	0.369494	+	0.625862i
$472$		0			0
$473$	482.505			0.0469040
$474$		0			0
$475$	−6978.08	−	6098.34i	−0.674056	−	0.589076i
$476$		0			0
$477$	4227.44	+	2332.40i	0.405788	+	0.223885i
$478$		0			0
$479$	−3466.07			−0.330624			−0.165312	−	0.986241i	$-0.552863\pi$
$479$	−3466.07			−0.330624			−0.165312	+	0.986241i	$0.552863\pi$
$480$		0			0
$481$	11601.2			1.09973
$482$		0			0
$483$	3393.91	−	2003.68i	0.319727	−	0.188759i
$484$		0			0
$485$	4785.33	−	12747.7i	0.448022	−	1.19349i
$486$		0			0
$487$	2368.19			0.220355			0.110178	−	0.993912i	$-0.464858\pi$
$487$	2368.19			0.220355			0.110178	+	0.993912i	$0.464858\pi$
$488$		0			0
$489$	2685.25	−	1585.31i	0.248325	−	0.146605i
$490$		0			0
$491$		−	19310.1i		−	1.77485i	−0.460953	−	0.887424i	$-0.652492\pi$
$491$		−	19310.1i		−	1.77485i	0.460953	−	0.887424i	$-0.347508\pi$
$492$		0			0
$493$	−4979.90			−0.454936
$494$		0			0
$495$	−1171.27	−	1369.55i	−0.106353	−	0.124357i
$496$		0			0
$497$	4356.68			0.393207
$498$		0			0
$499$	2257.79			0.202551			0.101275	−	0.994858i	$-0.467708\pi$
$499$	2257.79			0.202551			0.101275	+	0.994858i	$0.467708\pi$
$500$		0			0
$501$	−4865.79	+	2872.65i	−0.433907	+	0.256169i
$502$		0			0
$503$		−	171.197i		−	0.0151755i	−0.999971	−	0.00758776i	$-0.997585\pi$
$503$		−	171.197i		−	0.0151755i	0.999971	−	0.00758776i	$-0.00241528\pi$
$504$		0			0
$505$	4826.73	+	1811.90i	0.425320	+	0.159660i
$506$		0			0
$507$	−2917.02	−	4940.94i	−0.255521	−	0.432810i
$508$		0			0
$509$	872.727			0.0759979			0.0379989	−	0.999278i	$-0.487902\pi$
$509$	872.727			0.0759979			0.0379989	+	0.999278i	$0.487902\pi$
$510$		0			0
$511$		−	4023.19i		−	0.348289i
$512$		0			0
$513$	10396.9	−	303.099i	0.894808	−	0.0260861i
$514$		0			0
$515$	14274.5	+	5358.48i	1.22138	+	0.458491i
$516$		0			0
$517$	−377.112			−0.0320800
$518$		0			0
$519$	1675.02	−	988.892i	0.141667	−	0.0836368i
$520$		0			0
$521$			783.069i			0.0658481i	0.999458	+	0.0329240i	$0.0104819\pi$
$521$			783.069i			0.0658481i	−0.999458	+	0.0329240i	$0.989518\pi$
$522$		0			0
$523$			10952.6i			0.915727i	0.889023	+	0.457863i	$0.151385\pi$
$523$			10952.6i			0.915727i	−0.889023	+	0.457863i	$0.848615\pi$
$524$		0			0
$525$	489.804	−	2618.64i	0.0407177	−	0.217689i
$526$		0			0
$527$			9749.40i			0.805864i
$528$		0			0
$529$	−22030.9			−1.81071
$530$		0			0
$531$	11344.4	+	6259.03i	0.927128	+	0.511523i
$532$		0			0
$533$		−	9670.35i		−	0.785871i
$534$		0			0
$535$	−6560.64	−	2462.79i	−0.530170	−	0.199020i
$536$		0			0
$537$	−8441.00	+	4983.37i	−0.678317	+	0.400463i
$538$		0			0
$539$		−	1947.20i		−	0.155607i
$540$		0			0
$541$			8779.37i			0.697698i	0.937179	+	0.348849i	$0.113427\pi$
$541$			8779.37i			0.697698i	−0.937179	+	0.348849i	$0.886573\pi$
$542$		0			0
$543$	4413.12	−	2605.40i	0.348775	−	0.205909i
$544$		0			0
$545$	1856.46	−	4945.44i	0.145912	−	0.388696i
$546$		0			0
$547$		−	4135.81i		−	0.323281i	−0.986850	−	0.161640i	$-0.948322\pi$
$547$		−	4135.81i		−	0.323281i	0.986850	−	0.161640i	$-0.0516784\pi$
$548$		0			0
$549$	−15026.6	−	8290.59i	−1.16816	−	0.644506i
$550$		0			0
$551$	7294.54			0.563989
$552$		0			0
$553$		−	3298.65i		−	0.253658i
$554$		0			0
$555$	3533.73	+	20079.6i	0.270267	+	1.53573i
$556$		0			0
$557$			13645.6i			1.03803i	0.854765	+	0.519015i	$0.173701\pi$
$557$			13645.6i			1.03803i	−0.854765	+	0.519015i	$0.826299\pi$
$558$		0			0
$559$		−	2671.82i		−	0.202157i
$560$		0			0
$561$	−1351.99	+	798.186i	−0.101749	+	0.0600703i
$562$		0			0
$563$	−10861.6			−0.813076			−0.406538	−	0.913634i	$-0.633264\pi$
$563$	−10861.6			−0.813076			−0.406538	+	0.913634i	$0.633264\pi$
$564$		0			0
$565$	−5311.28	−	1993.79i	−0.395482	−	0.148459i
$566$		0			0
$567$	1594.50	+	2529.42i	0.118100	+	0.187347i
$568$		0			0
$569$			4479.34i			0.330024i	0.986292	+	0.165012i	$0.0527663\pi$
$569$			4479.34i			0.330024i	−0.986292	+	0.165012i	$0.947234\pi$
$570$		0			0
$571$	−9122.59			−0.668596			−0.334298	−	0.942467i	$-0.608499\pi$
$571$	−9122.59			−0.668596			−0.334298	+	0.942467i	$0.608499\pi$
$572$		0			0
$573$	9281.40	+	15721.1i	0.676677	+	1.14618i
$574$		0			0
$575$	−15211.3	+	17405.7i	−1.10323	+	1.26238i
$576$		0			0
$577$		−	12703.2i		−	0.916533i	−0.888815	−	0.458267i	$-0.848471\pi$
$577$		−	12703.2i		−	0.916533i	0.888815	−	0.458267i	$-0.151529\pi$
$578$		0			0
$579$	−13643.2	+	8054.64i	−0.979262	+	0.578134i
$580$		0			0
$581$	−1230.66			−0.0878765
$582$		0			0
$583$	1067.52			0.0758359
$584$		0			0
$585$	−7583.75	+	6485.77i	−0.535982	+	0.458382i
$586$		0			0
$587$	−584.550			−0.0411021			−0.0205511	−	0.999789i	$-0.506542\pi$
$587$	−584.550			−0.0411021			−0.0205511	+	0.999789i	$0.506542\pi$
$588$		0			0
$589$		−	14280.9i		−	0.999038i
$590$		0			0
$591$	17420.5	−	10284.6i	1.21249	−	0.715827i
$592$		0			0
$593$	9207.43			0.637612			0.318806	−	0.947820i	$-0.396718\pi$
$593$	9207.43			0.637612			0.318806	+	0.947820i	$0.396718\pi$
$594$		0			0
$595$	−2172.93	−	815.694i	−0.149717	−	0.0562020i
$596$		0			0
$597$	12923.5	−	7629.71i	0.885966	−	0.523054i
$598$		0			0
$599$	14077.4			0.960246			0.480123	−	0.877201i	$-0.340592\pi$
$599$	14077.4			0.960246			0.480123	+	0.877201i	$0.340592\pi$
$600$		0			0
$601$	9056.37			0.614670			0.307335	−	0.951601i	$-0.400563\pi$
$601$	9056.37			0.614670			0.307335	+	0.951601i	$0.400563\pi$
$602$		0			0
$603$	6820.41	+	3763.01i	0.460611	+	0.254132i
$604$		0			0
$605$	13558.7	+	5089.79i	0.911142	+	0.342032i
$606$		0			0
$607$	−21455.8			−1.43470			−0.717350	−	0.696713i	$-0.754645\pi$
$607$	−21455.8			−1.43470			−0.717350	+	0.696713i	$0.754645\pi$
$608$		0			0
$609$	1066.06	+	1805.73i	0.0709344	+	0.120151i
$610$		0			0
$611$			2088.22i			0.138266i
$612$		0			0
$613$	−9740.71			−0.641800			−0.320900	−	0.947113i	$-0.603985\pi$
$613$	−9740.71			−0.641800			−0.320900	+	0.947113i	$0.603985\pi$
$614$		0			0
$615$	16737.5	−	2945.58i	1.09744	−	0.193134i
$616$		0			0
$617$	3739.84			0.244020			0.122010	−	0.992529i	$-0.461066\pi$
$617$	3739.84			0.244020			0.122010	+	0.992529i	$0.461066\pi$
$618$		0			0
$619$	21751.2			1.41237			0.706183	−	0.708029i	$-0.250415\pi$
$619$	21751.2			1.41237			0.706183	+	0.708029i	$0.250415\pi$
$620$		0			0
$621$	−756.032	−	25933.5i	−0.0488543	−	1.67581i
$622$		0			0
$623$		−	4562.98i		−	0.293438i
$624$		0			0
$625$	2092.94	+	15484.2i	0.133948	+	0.990988i
$626$		0			0
$627$	1980.40	−	1169.18i	0.126139	−	0.0744698i
$628$		0			0
$629$	17762.7			1.12598
$630$		0			0
$631$		−	5974.16i		−	0.376906i	−0.982082	−	0.188453i	$-0.939653\pi$
$631$		−	5974.16i		−	0.376906i	0.982082	−	0.188453i	$-0.0603472\pi$
$632$		0			0
$633$	4757.15	+	8057.82i	0.298704	+	0.505955i
$634$		0			0
$635$	21449.5	+	8051.91i	1.34047	+	0.503197i
$636$		0			0
$637$	−10782.4			−0.670669
$638$		0			0
$639$	13854.4	−	25110.9i	0.857702	−	1.55457i
$640$		0			0
$641$			7044.53i			0.434075i	0.976163	+	0.217038i	$0.0696394\pi$
$641$			7044.53i			0.434075i	−0.976163	+	0.217038i	$0.930361\pi$
$642$		0			0
$643$			2177.00i			0.133519i	0.997769	+	0.0667595i	$0.0212660\pi$
$643$			2177.00i			0.133519i	−0.997769	+	0.0667595i	$0.978734\pi$
$644$		0			0
$645$	4624.42	−	813.834i	0.282304	−	0.0496817i
$646$		0			0
$647$		−	15827.7i		−	0.961747i	−0.876790	−	0.480873i	$-0.840320\pi$
$647$		−	15827.7i		−	0.961747i	0.876790	−	0.480873i	$-0.159680\pi$
$648$		0			0
$649$	2864.72			0.173267
$650$		0			0
$651$	3535.17	−	2087.08i	0.212833	−	0.125652i
$652$		0			0
$653$			18484.0i			1.10771i	0.832613	+	0.553856i	$0.186844\pi$
$653$			18484.0i			1.10771i	−0.832613	+	0.553856i	$0.813156\pi$
$654$		0			0
$655$	6591.43	−	17559.0i	0.393204	−	1.04746i
$656$		0			0
$657$	−23188.8	−	12793.9i	−1.37699	−	0.759722i
$658$		0			0
$659$		−	11168.2i		−	0.660169i	−0.943951	−	0.330084i	$-0.892923\pi$
$659$		−	11168.2i		−	0.660169i	0.943951	−	0.330084i	$-0.107077\pi$
$660$		0			0
$661$			6525.85i			0.384003i	0.981395	+	0.192002i	$0.0614979\pi$
$661$			6525.85i			0.384003i	−0.981395	+	0.192002i	$0.938502\pi$
$662$		0			0
$663$	4419.87	+	7486.53i	0.258905	+	0.438541i
$664$		0			0
$665$	3182.90	+	1194.82i	0.185606	+	0.0696742i
$666$		0			0
$667$		−	18195.0i		−	1.05624i
$668$		0			0
$669$	8968.28	+	15190.8i	0.518287	+	0.877892i
$670$		0			0
$671$	−3794.55			−0.218312
$672$		0			0
$673$		−	10707.9i		−	0.613313i	−0.951820	−	0.306656i	$-0.900790\pi$
$673$		−	10707.9i		−	0.613313i	0.951820	−	0.306656i	$-0.0992103\pi$
$674$		0			0
$675$	−13535.6	−	11150.5i	−0.771833	−	0.635826i
$676$		0			0
$677$		−	5626.13i		−	0.319394i	−0.987166	−	0.159697i	$-0.948948\pi$
$677$		−	5626.13i		−	0.319394i	0.987166	−	0.159697i	$-0.0510517\pi$
$678$		0			0
$679$			4995.21i			0.282325i
$680$		0			0
$681$	4403.24	+	7458.35i	0.247772	+	0.419684i
$682$		0			0
$683$	−8947.12			−0.501247			−0.250624	−	0.968085i	$-0.580636\pi$
$683$	−8947.12			−0.501247			−0.250624	+	0.968085i	$0.580636\pi$
$684$		0			0
$685$	−19257.1	−	7228.90i	−1.07413	−	0.403215i
$686$		0			0
$687$	23619.2	−	13944.2i	1.31168	−	0.774388i
$688$		0			0
$689$		−	5911.30i		−	0.326854i
$690$		0			0
$691$	2216.99			0.122053			0.0610263	−	0.998136i	$-0.480563\pi$
$691$	2216.99			0.122053			0.0610263	+	0.998136i	$0.480563\pi$
$692$		0			0
$693$	578.851	+	319.368i	0.0317298	+	0.0175062i
$694$		0			0
$695$	16729.1	+	6279.91i	0.913052	+	0.342749i
$696$		0			0
$697$		−	14806.3i		−	0.804631i
$698$		0			0
$699$	−1240.49	−	2101.19i	−0.0671242	−	0.113697i
$700$		0			0
$701$	34614.1			1.86499			0.932493	−	0.361188i	$-0.117629\pi$
$701$	34614.1			1.86499			0.932493	+	0.361188i	$0.117629\pi$
$702$		0			0
$703$	−26018.7			−1.39589
$704$		0			0
$705$	−3614.31	+	636.069i	−0.193082	+	0.0339798i
$706$		0			0
$707$	−1891.37			−0.100611
$708$		0			0
$709$		−	21114.8i		−	1.11845i	−0.829015	−	0.559226i	$-0.811098\pi$
$709$		−	21114.8i		−	1.11845i	0.829015	−	0.559226i	$-0.188902\pi$
$710$		0			0
$711$	−19012.7	−	10489.8i	−1.00286	−	0.553304i
$712$		0			0
$713$	−35621.3			−1.87101
$714$		0			0
$715$	−775.409	+	2065.62i	−0.0405576	+	0.108042i
$716$		0			0
$717$	343.809	+	582.355i	0.0179076	+	0.0303325i
$718$		0			0
$719$	−26234.0			−1.36073			−0.680363	−	0.732875i	$-0.738178\pi$
$719$	−26234.0			−1.36073			−0.680363	+	0.732875i	$0.738178\pi$
$720$		0			0
$721$	−5593.50			−0.288922
$722$		0			0
$723$	14687.8	+	24878.7i	0.755525	+	1.27973i
$724$		0			0
$725$	−9260.72	−	8093.20i	−0.474393	−	0.414585i
$726$		0			0
$727$	1445.23			0.0737287			0.0368643	−	0.999320i	$-0.488263\pi$
$727$	1445.23			0.0737287			0.0368643	+	0.999320i	$0.488263\pi$
$728$		0			0
$729$	19649.6	−	1146.65i	0.998302	−	0.0582560i
$730$		0			0
$731$		−	4090.83i		−	0.206983i
$732$		0			0
$733$	−19855.7			−1.00053			−0.500265	−	0.865872i	$-0.666764\pi$
$733$	−19855.7			−1.00053			−0.500265	+	0.865872i	$0.666764\pi$
$734$		0			0
$735$	−3284.32	−	18662.4i	−0.164822	−	0.936561i
$736$		0			0
$737$	1722.31			0.0860815
$738$		0			0
$739$	6549.70			0.326028			0.163014	−	0.986624i	$-0.447878\pi$
$739$	6549.70			0.326028			0.163014	+	0.986624i	$0.447878\pi$
$740$		0			0
$741$	−6474.21	−	10966.2i	−0.320966	−	0.543664i
$742$		0			0
$743$		−	22234.5i		−	1.09785i	−0.835870	−	0.548927i	$-0.815037\pi$
$743$		−	22234.5i		−	1.09785i	0.835870	−	0.548927i	$-0.184963\pi$
$744$		0			0
$745$	5584.54	+	2096.37i	0.274633	+	0.103094i
$746$		0			0
$747$	−3913.53	+	7093.23i	−0.191685	+	0.347427i
$748$		0			0
$749$	2570.80			0.125414
$750$		0			0
$751$		−	9278.78i		−	0.450849i	−0.974261	−	0.225424i	$-0.927623\pi$
$751$		−	9278.78i		−	0.450849i	0.974261	−	0.225424i	$-0.0723769\pi$
$752$		0			0
$753$	29250.2	−	17268.7i	1.41559	−	0.835730i
$754$		0			0
$755$	12009.4	−	31991.8i	0.578895	−	1.54212i
$756$		0			0
$757$	−990.407			−0.0475521			−0.0237761	−	0.999717i	$-0.507569\pi$
$757$	−990.407			−0.0475521			−0.0237761	+	0.999717i	$0.507569\pi$
$758$		0			0
$759$	−2916.33	−	4939.77i	−0.139468	−	0.236235i
$760$		0			0
$761$			10150.9i			0.483533i	0.970334	+	0.241766i	$0.0777268\pi$
$761$			10150.9i			0.483533i	−0.970334	+	0.241766i	$0.922273\pi$
$762$		0			0
$763$			1937.88i			0.0919477i
$764$		0			0
$765$	−11611.5	+	9930.36i	−0.548777	+	0.469324i
$766$		0			0
$767$		−	15863.1i		−	0.746784i
$768$		0			0
$769$	10878.0			0.510103			0.255051	−	0.966927i	$-0.417908\pi$
$769$	10878.0			0.510103			0.255051	+	0.966927i	$0.417908\pi$
$770$		0			0
$771$	7316.81	+	12393.5i	0.341775	+	0.578910i
$772$		0			0
$773$		−	32400.4i		−	1.50758i	−0.657113	−	0.753792i	$-0.728222\pi$
$773$		−	32400.4i		−	1.50758i	0.657113	−	0.753792i	$-0.271778\pi$
$774$		0			0
$775$	−15844.5	+	18130.2i	−0.734386	+	0.840329i
$776$		0			0
$777$	−3802.51	−	6440.81i	−0.175565	−	0.297378i
$778$		0			0
$779$			21688.2i			0.997508i
$780$		0			0
$781$		−	6341.07i		−	0.290527i
$782$		0			0
$783$	13797.9	−	402.248i	0.629755	−	0.0183591i
$784$		0			0
$785$	14965.4	+	5617.84i	0.680431	+	0.255426i
$786$		0			0
$787$		−	25346.2i		−	1.14802i	−0.818848	−	0.574011i	$-0.805387\pi$
$787$		−	25346.2i		−	1.14802i	0.818848	−	0.574011i	$-0.194613\pi$
$788$		0			0
$789$	8954.63	−	5286.60i	0.404047	−	0.238540i
$790$		0			0
$791$	2081.24			0.0935529
$792$		0			0
$793$			21012.0i			0.940929i
$794$		0			0
$795$	10231.3	−	1800.58i	0.456438	−	0.0803269i
$796$		0			0
$797$			11138.7i			0.495047i	0.968882	+	0.247523i	$0.0796167\pi$
$797$			11138.7i			0.495047i	−0.968882	+	0.247523i	$0.920383\pi$
$798$		0			0
$799$			3197.27i			0.141566i
$800$		0			0
$801$	−26300.0	−	14510.4i	−1.16013	−	0.640076i
$802$		0			0
$803$	−5855.69			−0.257339
$804$		0			0
$805$	2980.30	−	7939.23i	0.130487	−	0.347604i
$806$		0			0
$807$	6541.96	+	11081.0i	0.285363	+	0.483357i
$808$		0			0
$809$			2494.78i			0.108420i	0.998530	+	0.0542100i	$0.0172641\pi$
$809$			2494.78i			0.108420i	−0.998530	+	0.0542100i	$0.982736\pi$
$810$		0			0
$811$	−33345.1			−1.44378			−0.721891	−	0.692007i	$-0.756727\pi$
$811$	−33345.1			−1.44378			−0.721891	+	0.692007i	$0.756727\pi$
$812$		0			0
$813$	−7232.47	+	4269.88i	−0.311997	+	0.184196i
$814$		0			0
$815$	2358.00	−	6281.49i	0.101346	−	0.269977i
$816$		0			0
$817$			5992.22i			0.256599i
$818$		0			0
$819$	1768.47	−	3205.33i	0.0754522	−	0.136756i
$820$		0			0
$821$	−20815.1			−0.884838			−0.442419	−	0.896809i	$-0.645880\pi$
$821$	−20815.1			−0.884838			−0.442419	+	0.896809i	$0.645880\pi$
$822$		0			0
$823$	−20846.1			−0.882927			−0.441463	−	0.897279i	$-0.645541\pi$
$823$	−20846.1			−0.882927			−0.441463	+	0.897279i	$0.645541\pi$
$824$		0			0
$825$	−3811.39	−	712.901i	−0.160843	−	0.0300849i
$826$		0			0
$827$	−8542.64			−0.359198			−0.179599	−	0.983740i	$-0.557480\pi$
$827$	−8542.64			−0.359198			−0.179599	+	0.983740i	$0.557480\pi$
$828$		0			0
$829$			24981.9i			1.04663i	0.852139	+	0.523315i	$0.175305\pi$
$829$			24981.9i			1.04663i	−0.852139	+	0.523315i	$0.824695\pi$
$830$		0			0
$831$	15188.7	+	25727.1i	0.634043	+	1.07396i
$832$		0			0
$833$	−16509.0			−0.686679
$834$		0			0
$835$	−4272.81	+	11382.4i	−0.177086	+	0.471740i
$836$		0			0
$837$	−787.501	−	27012.9i	−0.0325209	−	1.11554i
$838$		0			0
$839$	11325.0			0.466010			0.233005	−	0.972476i	$-0.425144\pi$
$839$	11325.0			0.466010			0.233005	+	0.972476i	$0.425144\pi$
$840$		0			0
$841$	−14708.3			−0.603071
$842$		0			0
$843$	34699.6	−	20485.9i	1.41770	−	0.836976i
$844$		0			0
$845$	−11558.2	−	4338.80i	−0.470548	−	0.176638i
$846$		0			0
$847$	−5313.02			−0.215534
$848$		0			0
$849$	22066.0	−	13027.2i	0.891993	−	0.526612i
$850$		0			0
$851$			64899.4i			2.61424i
$852$		0			0
$853$	25405.7			1.01978			0.509892	−	0.860238i	$-0.329685\pi$
$853$	25405.7			1.01978			0.509892	+	0.860238i	$0.329685\pi$
$854$		0			0
$855$	17008.4	−	14545.9i	0.680324	−	0.581826i
$856$		0			0
$857$	37636.5			1.50016			0.750080	−	0.661347i	$-0.230015\pi$
$857$	37636.5			1.50016			0.750080	+	0.661347i	$0.230015\pi$
$858$		0			0
$859$	−9419.30			−0.374135			−0.187068	−	0.982347i	$-0.559898\pi$
$859$	−9419.30			−0.374135			−0.187068	+	0.982347i	$0.559898\pi$
$860$		0			0
$861$	−5368.81	+	3169.62i	−0.212507	+	0.125459i
$862$		0			0
$863$			30381.0i			1.19836i	0.800616	+	0.599178i	$0.204506\pi$
$863$			30381.0i			1.19836i	−0.800616	+	0.599178i	$0.795494\pi$
$864$		0			0
$865$	1470.89	−	3918.31i	0.0578169	−	0.154019i
$866$		0			0
$867$	−6211.25	−	10520.8i	−0.243305	−	0.412118i
$868$		0			0
$869$	−4801.13			−0.187419
$870$		0			0
$871$		−	9537.11i		−	0.371013i
$872$		0			0
$873$	28791.2	+	15884.9i	1.11619	+	0.615835i
$874$		0			0
$875$	−2715.19	−	5048.27i	−0.104903	−	0.195043i
$876$		0			0
$877$	−16307.0			−0.627876			−0.313938	−	0.949443i	$-0.601648\pi$
$877$	−16307.0			−0.627876			−0.313938	+	0.949443i	$0.601648\pi$
$878$		0			0
$879$	−3082.63	+	1819.91i	−0.118287	+	0.0698341i
$880$		0			0
$881$			24793.0i			0.948125i	0.880491	+	0.474063i	$0.157213\pi$
$881$			24793.0i			0.948125i	−0.880491	+	0.474063i	$0.842787\pi$
$882$		0			0
$883$		−	20857.8i		−	0.794927i	−0.917618	−	0.397463i	$-0.869891\pi$
$883$		−	20857.8i		−	0.794927i	0.917618	−	0.397463i	$-0.130109\pi$
$884$		0			0
$885$	27456.0	−	4831.88i	1.04285	−	0.183528i
$886$		0			0
$887$			15190.5i			0.575025i	0.957777	+	0.287512i	$0.0928282\pi$
$887$			15190.5i			0.575025i	−0.957777	+	0.287512i	$0.907172\pi$
$888$		0			0
$889$	−8405.06			−0.317094
$890$		0			0
$891$	3681.53	−	2320.76i	0.138424	−	0.0872598i
$892$		0			0
$893$		−	4683.35i		−	0.175501i
$894$		0			0
$895$	−7412.31	+	19745.7i	−0.276834	+	0.737460i
$896$		0			0
$897$	−27353.5	+	16148.9i	−1.01818	+	0.601109i
$898$		0			0
$899$		−	18952.4i		−	0.703111i
$900$		0			0
$901$		−	9050.80i		−	0.334657i
$902$		0			0
$903$	−1483.35	+	875.735i	−0.0546653	+	0.0322731i
$904$		0			0
$905$	3875.30	−	10323.4i	0.142342	−	0.379185i
$906$		0			0
$907$			30391.4i			1.11260i	0.830981	+	0.556301i	$0.187780\pi$
$907$			30391.4i			1.11260i	−0.830981	+	0.556301i	$0.812220\pi$
$908$		0			0
$909$	−6014.61	+	10901.4i	−0.219463	+	0.397774i
$910$		0			0
$911$	−9799.08			−0.356375			−0.178188	−	0.983997i	$-0.557023\pi$
$911$	−9799.08			−0.356375			−0.178188	+	0.983997i	$0.557023\pi$
$912$		0			0
$913$			1791.20i			0.0649289i
$914$		0			0
$915$	−36367.7	+	6400.22i	−1.31397	+	0.231240i
$916$		0			0
$917$			6880.53i			0.247781i
$918$		0			0
$919$		−	7062.68i		−	0.253511i	−0.991934	−	0.126755i	$-0.959544\pi$
$919$		−	7062.68i		−	0.253511i	0.991934	−	0.126755i	$-0.0404563\pi$
$920$		0			0
$921$	−29645.2	+	17501.8i	−1.06063	+	0.626172i
$922$		0			0
$923$	−35113.0			−1.25218
$924$		0			0
$925$	33031.8	+	28867.4i	1.17414	+	1.02611i
$926$		0			0
$927$	−17787.5	+	32239.7i	−0.630225	+	1.14228i
$928$		0			0
$929$		−	36154.0i		−	1.27683i	−0.769693	−	0.638414i	$-0.779591\pi$
$929$		−	36154.0i		−	1.27683i	0.769693	−	0.638414i	$-0.220409\pi$
$930$		0			0
$931$	24182.3			0.851282
$932$		0			0
$933$	14148.5	+	23965.1i	0.496463	+	0.840925i
$934$		0			0
$935$	−1187.23	+	3162.67i	−0.0415257	+	0.110621i
$936$		0			0
$937$		−	702.135i		−	0.0244800i	−0.999925	−	0.0122400i	$-0.996104\pi$
$937$		−	702.135i		−	0.0244800i	0.999925	−	0.0122400i	$-0.00389621\pi$
$938$		0			0
$939$	257.753	−	152.171i	0.00895787	−	0.00528852i
$940$		0			0
$941$	−41463.7			−1.43643			−0.718213	−	0.695823i	$-0.755040\pi$
$941$	−41463.7			−1.43643			−0.718213	+	0.695823i	$0.755040\pi$
$942$		0			0
$943$	54097.6			1.86814
$944$		0			0
$945$	6086.49	+	2084.55i	0.209517	+	0.0717570i
$946$		0			0
$947$	5311.23			0.182251			0.0911255	−	0.995839i	$-0.470954\pi$
$947$	5311.23			0.182251			0.0911255	+	0.995839i	$0.470954\pi$
$948$		0			0
$949$			32425.3i			1.10914i
$950$		0			0
$951$	−34929.7	+	20621.7i	−1.19103	+	0.703158i
$952$		0			0
$953$	−42641.9			−1.44943			−0.724715	−	0.689048i	$-0.758029\pi$
$953$	−42641.9			−1.44943			−0.724715	+	0.689048i	$0.758029\pi$
$954$		0			0
$955$	36775.9	+	13805.2i	1.24611	+	0.467777i
$956$		0			0
$957$	2628.21	−	1551.64i	0.0887754	−	0.0524110i
$958$		0			0
$959$	7545.95			0.254089
$960$		0			0
$961$	−7312.98			−0.245476
$962$		0			0
$963$	8175.25	−	14817.5i	0.273566	−	0.495834i
$964$		0			0
$965$	−11980.5	+	31915.1i	−0.399655	+	1.06464i
$966$		0			0
$967$	3381.87			0.112465			0.0562324	−	0.998418i	$-0.482091\pi$
$967$	3381.87			0.112465			0.0562324	+	0.998418i	$0.482091\pi$
$968$		0			0
$969$	−9912.67	−	16790.4i	−0.328628	−	0.556642i
$970$		0			0
$971$		−	12946.7i		−	0.427888i	−0.976846	−	0.213944i	$-0.931369\pi$
$971$		−	12946.7i		−	0.427888i	0.976846	−	0.213944i	$-0.0686309\pi$
$972$		0			0
$973$	−6555.34			−0.215986
$974$		0			0
$975$	−3947.62	+	21105.2i	−0.129667	+	0.693237i
$976$		0			0
$977$	−13219.6			−0.432888			−0.216444	−	0.976295i	$-0.569446\pi$
$977$	−13219.6			−0.432888			−0.216444	+	0.976295i	$0.569446\pi$
$978$		0			0
$979$	−6641.34			−0.216811
$980$		0			0
$981$	11169.5	+	6162.54i	0.363522	+	0.200566i
$982$		0			0
$983$			23218.2i			0.753354i	0.926345	+	0.376677i	$0.122933\pi$
$983$			23218.2i			0.753354i	−0.926345	+	0.376677i	$0.877067\pi$
$984$		0			0
$985$	15297.5	−	40751.0i	0.494841	−	1.31821i
$986$		0			0
$987$	1159.34	−	684.449i	0.0373883	−	0.0220732i
$988$		0			0
$989$	14946.6			0.480561
$990$		0			0
$991$			19934.1i			0.638977i	0.947590	+	0.319489i	$0.103511\pi$
$991$			19934.1i			0.638977i	−0.947590	+	0.319489i	$0.896489\pi$
$992$		0			0
$993$	16427.1	+	27824.7i	0.524972	+	0.889214i
$994$		0			0
$995$	11348.5	−	30231.3i	0.361579	−	0.963214i
$996$		0			0
$997$	48021.5			1.52543			0.762716	−	0.646734i	$-0.223866\pi$
$997$	48021.5			1.52543			0.762716	+	0.646734i	$0.223866\pi$
$998$		0			0
$999$	−49215.5	+	1434.77i	−1.55867	+	0.0454395i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.m.b.239.44		64
3.2	odd	2	inner	480.4.m.b.239.23		64
4.3	odd	2		120.4.m.b.59.7	yes	64
5.4	even	2	inner	480.4.m.b.239.22		64
8.3	odd	2	inner	480.4.m.b.239.43		64
8.5	even	2		120.4.m.b.59.5	✓	64
12.11	even	2		120.4.m.b.59.57	yes	64
15.14	odd	2	inner	480.4.m.b.239.41		64
20.19	odd	2		120.4.m.b.59.58	yes	64
24.5	odd	2		120.4.m.b.59.59	yes	64
24.11	even	2	inner	480.4.m.b.239.24		64
40.19	odd	2	inner	480.4.m.b.239.21		64
40.29	even	2		120.4.m.b.59.60	yes	64
60.59	even	2		120.4.m.b.59.8	yes	64
120.29	odd	2		120.4.m.b.59.6	yes	64
120.59	even	2	inner	480.4.m.b.239.42		64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.m.b.59.5	✓	64	8.5	even	2
120.4.m.b.59.6	yes	64	120.29	odd	2
120.4.m.b.59.7	yes	64	4.3	odd	2
120.4.m.b.59.8	yes	64	60.59	even	2
120.4.m.b.59.57	yes	64	12.11	even	2
120.4.m.b.59.58	yes	64	20.19	odd	2
120.4.m.b.59.59	yes	64	24.5	odd	2
120.4.m.b.59.60	yes	64	40.29	even	2
480.4.m.b.239.21		64	40.19	odd	2	inner
480.4.m.b.239.22		64	5.4	even	2	inner
480.4.m.b.239.23		64	3.2	odd	2	inner
480.4.m.b.239.24		64	24.11	even	2	inner
480.4.m.b.239.41		64	15.14	odd	2	inner
480.4.m.b.239.42		64	120.59	even	2	inner
480.4.m.b.239.43		64	8.3	odd	2	inner
480.4.m.b.239.44		64	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.64167 + 4.47455i) q^{3} +(10.4671 + 3.92924i) q^{5} -4.10158 q^{7} +(-13.0432 + 23.6406i) q^{9} +5.96978i q^{11} +33.0570 q^{13} +(10.0691 + 57.2155i) q^{15} +50.6136 q^{17} -74.1386 q^{19} +(-10.8350 - 18.3527i) q^{21} +184.927i q^{23} +(94.1221 + 82.2559i) q^{25} +(-140.237 + 4.08828i) q^{27} -98.3905 q^{29} +192.624i q^{31} +(-26.7121 + 15.7702i) q^{33} +(-42.9318 - 16.1161i) q^{35} +350.946 q^{37} +(87.3258 + 147.915i) q^{39} -292.535i q^{41} -80.8246i q^{43} +(-229.414 + 196.199i) q^{45} +63.1702i q^{47} -326.177 q^{49} +(133.705 + 226.473i) q^{51} -178.821i q^{53} +(-23.4567 + 62.4865i) q^{55} +(-195.850 - 331.737i) q^{57} -479.870i q^{59} +635.627i q^{61} +(53.4975 - 96.9635i) q^{63} +(346.013 + 129.889i) q^{65} -288.505i q^{67} +(-827.464 + 488.515i) q^{69} -1062.20 q^{71} +980.889i q^{73} +(-119.418 + 638.447i) q^{75} -24.4855i q^{77} +804.239i q^{79} +(-388.752 - 616.695i) q^{81} +300.045 q^{83} +(529.780 + 198.873i) q^{85} +(-259.915 - 440.253i) q^{87} +1112.49i q^{89} -135.586 q^{91} +(-861.905 + 508.849i) q^{93} +(-776.020 - 291.309i) q^{95} -1217.88i q^{97} +(-141.129 - 77.8648i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 112 q^{9} - 672 q^{19} + 496 q^{25} + 3520 q^{49} + 544 q^{51} + 1600 q^{75} - 2304 q^{81} + 2752 q^{91} - 256 q^{99}+O(q^{100}) \) 64 * q - 112 * q^9 - 672 * q^19 + 496 * q^25 + 3520 * q^49 + 544 * q^51 + 1600 * q^75 - 2304 * q^81 + 2752 * q^91 - 256 * q^99

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