LMFDB - Embedded newform 480.3.l.a.161.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,3,Mod(161,480)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("480.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	480.l (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$13.0790526893$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2434 x^{12} - 8780 x^{11} + 25532 x^{10} - 57568 x^{9} + \cdots + 486 $ x^16 - 8x^15 + 88x^14 - 476x^13 + 2434x^12 - 8780x^11 + 25532x^10 - 57568x^9 + 104312x^8 - 150688x^7 + 173858x^6 - 157972x^5 + 110493x^4 - 57264x^3 + 20682x^2 - 4644*x + 486
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.8
Root			$0.500000 + 0.0650247i$ of defining polynomial
Character	$\chi$	$=$	480.161
Dual form			480.3.l.a.161.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.460323 + 2.96447i) q^{3} +2.23607i q^{5} +10.9698 q^{7} +(-8.57621 - 2.72923i) q^{9} +O(q^{10})$	\(q+(-0.460323 + 2.96447i) q^{3} +2.23607i q^{5} +10.9698 q^{7} +(-8.57621 - 2.72923i) q^{9} -1.78654i q^{11} +22.4485 q^{13} +(-6.62876 - 1.02931i) q^{15} +15.5788i q^{17} +26.3809 q^{19} +(-5.04966 + 32.5197i) q^{21} -30.1861i q^{23} -5.00000 q^{25} +(12.0386 - 24.1676i) q^{27} +44.8246i q^{29} -51.3786 q^{31} +(5.29614 + 0.822384i) q^{33} +24.5293i q^{35} -36.3424 q^{37} +(-10.3336 + 66.5481i) q^{39} -7.93212i q^{41} +26.4872 q^{43} +(6.10275 - 19.1770i) q^{45} +31.2242i q^{47} +71.3370 q^{49} +(-46.1830 - 7.17130i) q^{51} +90.5039i q^{53} +3.99481 q^{55} +(-12.1437 + 78.2054i) q^{57} +17.8064i q^{59} -11.6132 q^{61} +(-94.0794 - 29.9392i) q^{63} +50.1965i q^{65} +32.3116 q^{67} +(89.4859 + 13.8954i) q^{69} -34.1717i q^{71} -68.9145 q^{73} +(2.30162 - 14.8224i) q^{75} -19.5980i q^{77} +29.5267 q^{79} +(66.1026 + 46.8129i) q^{81} -76.1626i q^{83} -34.8353 q^{85} +(-132.881 - 20.6338i) q^{87} -112.955i q^{89} +246.257 q^{91} +(23.6508 - 152.311i) q^{93} +58.9894i q^{95} -79.5471 q^{97} +(-4.87587 + 15.3217i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{9}+O(q^{10}) $ 16 * q - 24 * q^9	$ 16 q - 24 q^{9} - 16 q^{13} + 88 q^{21} - 80 q^{25} - 64 q^{33} + 16 q^{37} + 40 q^{45} + 320 q^{49} - 272 q^{57} - 368 q^{61} + 104 q^{69} + 256 q^{73} + 192 q^{81} + 416 q^{93} - 544 q^{97}+O(q^{100}) $ 16 * q - 24 * q^9 - 16 * q^13 + 88 * q^21 - 80 * q^25 - 64 * q^33 + 16 * q^37 + 40 * q^45 + 320 * q^49 - 272 * q^57 - 368 * q^61 + 104 * q^69 + 256 * q^73 + 192 * q^81 + 416 * q^93 - 544 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−0.460323	+	2.96447i	−0.153441	+	0.988158i
$3$	−0.460323	+	2.96447i	−0.153441	+	0.988158i
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$	10.9698			1.56712			0.783559	−	0.621318i	$-0.213402\pi$
$7$	10.9698			1.56712			0.783559	+	0.621318i	$0.213402\pi$
$8$		0			0
$9$	−8.57621	−	2.72923i	−0.952912	−	0.303248i
$10$		0			0
$11$		−	1.78654i		−	0.162412i	−0.996697	−	0.0812062i	$-0.974123\pi$
$11$		−	1.78654i		−	0.162412i	0.996697	−	0.0812062i	$-0.0258772\pi$
$12$		0			0
$13$	22.4485			1.72681			0.863406	−	0.504510i	$-0.168327\pi$
$13$	22.4485			1.72681			0.863406	+	0.504510i	$0.168327\pi$
$14$		0			0
$15$	−6.62876	−	1.02931i	−0.441918	−	0.0686209i
$16$		0			0
$17$			15.5788i			0.916402i	0.888849	+	0.458201i	$0.151506\pi$
$17$			15.5788i			0.916402i	−0.888849	+	0.458201i	$0.848494\pi$
$18$		0			0
$19$	26.3809			1.38847			0.694233	−	0.719750i	$-0.255744\pi$
$19$	26.3809			1.38847			0.694233	+	0.719750i	$0.255744\pi$
$20$		0			0
$21$	−5.04966	+	32.5197i	−0.240460	+	1.54856i
$22$		0			0
$23$		−	30.1861i		−	1.31244i	−0.754570	−	0.656220i	$-0.772154\pi$
$23$		−	30.1861i		−	1.31244i	0.754570	−	0.656220i	$-0.227846\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	12.0386	−	24.1676i	0.445873	−	0.895096i
$28$		0			0
$29$			44.8246i			1.54567i	0.634604	+	0.772837i	$0.281163\pi$
$29$			44.8246i			1.54567i	−0.634604	+	0.772837i	$0.718837\pi$
$30$		0			0
$31$	−51.3786			−1.65738			−0.828688	−	0.559711i	$-0.810912\pi$
$31$	−51.3786			−1.65738			−0.828688	+	0.559711i	$0.810912\pi$
$32$		0			0
$33$	5.29614	+	0.822384i	0.160489	+	0.0249207i
$34$		0			0
$35$			24.5293i			0.700836i
$36$		0			0
$37$	−36.3424			−0.982226			−0.491113	−	0.871096i	$-0.663410\pi$
$37$	−36.3424			−0.982226			−0.491113	+	0.871096i	$0.663410\pi$
$38$		0			0
$39$	−10.3336	+	66.5481i	−0.264964	+	1.70636i
$40$		0			0
$41$		−	7.93212i		−	0.193466i	−0.995310	−	0.0967332i	$-0.969161\pi$
$41$		−	7.93212i		−	0.193466i	0.995310	−	0.0967332i	$-0.0308393\pi$
$42$		0			0
$43$	26.4872			0.615982			0.307991	−	0.951389i	$-0.400343\pi$
$43$	26.4872			0.615982			0.307991	+	0.951389i	$0.400343\pi$
$44$		0			0
$45$	6.10275	−	19.1770i	0.135617	−	0.426155i
$46$		0			0
$47$			31.2242i			0.664344i	0.943219	+	0.332172i	$0.107781\pi$
$47$			31.2242i			0.664344i	−0.943219	+	0.332172i	$0.892219\pi$
$48$		0			0
$49$	71.3370			1.45586
$50$		0			0
$51$	−46.1830	−	7.17130i	−0.905550	−	0.140614i
$52$		0			0
$53$			90.5039i			1.70762i	0.520585	+	0.853810i	$0.325714\pi$
$53$			90.5039i			1.70762i	−0.520585	+	0.853810i	$0.674286\pi$
$54$		0			0
$55$	3.99481			0.0726330
$56$		0			0
$57$	−12.1437	+	78.2054i	−0.213048	+	1.37202i
$58$		0			0
$59$			17.8064i			0.301803i	0.988549	+	0.150902i	$0.0482177\pi$
$59$			17.8064i			0.301803i	−0.988549	+	0.150902i	$0.951782\pi$
$60$		0			0
$61$	−11.6132			−0.190381			−0.0951903	−	0.995459i	$-0.530346\pi$
$61$	−11.6132			−0.190381			−0.0951903	+	0.995459i	$0.530346\pi$
$62$		0			0
$63$	−94.0794	−	29.9392i	−1.49332	−	0.475225i
$64$		0			0
$65$			50.1965i			0.772253i
$66$		0			0
$67$	32.3116			0.482262			0.241131	−	0.970493i	$-0.422482\pi$
$67$	32.3116			0.482262			0.241131	+	0.970493i	$0.422482\pi$
$68$		0			0
$69$	89.4859	+	13.8954i	1.29690	+	0.201382i
$70$		0			0
$71$		−	34.1717i		−	0.481292i	−0.970613	−	0.240646i	$-0.922641\pi$
$71$		−	34.1717i		−	0.481292i	0.970613	−	0.240646i	$-0.0773593\pi$
$72$		0			0
$73$	−68.9145			−0.944034			−0.472017	−	0.881590i	$-0.656474\pi$
$73$	−68.9145			−0.944034			−0.472017	+	0.881590i	$0.656474\pi$
$74$		0			0
$75$	2.30162	−	14.8224i	0.0306882	−	0.197632i
$76$		0			0
$77$		−	19.5980i		−	0.254519i
$78$		0			0
$79$	29.5267			0.373756			0.186878	−	0.982383i	$-0.440163\pi$
$79$	29.5267			0.373756			0.186878	+	0.982383i	$0.440163\pi$
$80$		0			0
$81$	66.1026	+	46.8129i	0.816081	+	0.577937i
$82$		0			0
$83$		−	76.1626i		−	0.917622i	−0.888534	−	0.458811i	$-0.848275\pi$
$83$		−	76.1626i		−	0.917622i	0.888534	−	0.458811i	$-0.151725\pi$
$84$		0			0
$85$	−34.8353			−0.409827
$86$		0			0
$87$	−132.881	−	20.6338i	−1.52737	−	0.237170i
$88$		0			0
$89$		−	112.955i		−	1.26915i	−0.772860	−	0.634577i	$-0.781174\pi$
$89$		−	112.955i		−	1.26915i	0.772860	−	0.634577i	$-0.218826\pi$
$90$		0			0
$91$	246.257			2.70612
$92$		0			0
$93$	23.6508	−	152.311i	0.254309	−	1.63775i
$94$		0			0
$95$			58.9894i			0.620941i
$96$		0			0
$97$	−79.5471			−0.820073			−0.410037	−	0.912069i	$-0.634484\pi$
$97$	−79.5471			−0.820073			−0.410037	+	0.912069i	$0.634484\pi$
$98$		0			0
$99$	−4.87587	+	15.3217i	−0.0492512	+	0.154765i
$100$		0			0
$101$		−	59.3021i		−	0.587150i	−0.955936	−	0.293575i	$-0.905155\pi$
$101$		−	59.3021i		−	0.587150i	0.955936	−	0.293575i	$-0.0948450\pi$
$102$		0			0
$103$	19.9969			0.194144			0.0970721	−	0.995277i	$-0.469052\pi$
$103$	19.9969			0.194144			0.0970721	+	0.995277i	$0.469052\pi$
$104$		0			0
$105$	−72.7164	−	11.2914i	−0.692537	−	0.107537i
$106$		0			0
$107$		−	39.2253i		−	0.366591i	−0.983058	−	0.183296i	$-0.941323\pi$
$107$		−	39.2253i		−	0.366591i	0.983058	−	0.183296i	$-0.0586766\pi$
$108$		0			0
$109$	6.15707			0.0564869			0.0282434	−	0.999601i	$-0.491009\pi$
$109$	6.15707			0.0564869			0.0282434	+	0.999601i	$0.491009\pi$
$110$		0			0
$111$	16.7292	−	107.736i	0.150714	−	0.970595i
$112$		0			0
$113$			88.1167i			0.779794i	0.920859	+	0.389897i	$0.127489\pi$
$113$			88.1167i			0.779794i	−0.920859	+	0.389897i	$0.872511\pi$
$114$		0			0
$115$	67.4982			0.586941
$116$		0			0
$117$	−192.523	−	61.2673i	−1.64550	−	0.523652i
$118$		0			0
$119$			170.897i			1.43611i
$120$		0			0
$121$	117.808			0.973622
$122$		0			0
$123$	23.5146	+	3.65134i	0.191175	+	0.0296857i
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−4.62127			−0.0363880			−0.0181940	−	0.999834i	$-0.505792\pi$
$127$	−4.62127			−0.0363880			−0.0181940	+	0.999834i	$0.505792\pi$
$128$		0			0
$129$	−12.1927	+	78.5207i	−0.0945170	+	0.608688i
$130$		0			0
$131$			78.8114i			0.601613i	0.953685	+	0.300807i	$0.0972559\pi$
$131$			78.8114i			0.601613i	−0.953685	+	0.300807i	$0.902744\pi$
$132$		0			0
$133$	289.393			2.17589
$134$		0			0
$135$	54.0404	+	26.9190i	0.400299	+	0.199400i
$136$		0			0
$137$		−	127.646i		−	0.931721i	−0.884858	−	0.465861i	$-0.845745\pi$
$137$		−	127.646i		−	0.931721i	0.884858	−	0.465861i	$-0.154255\pi$
$138$		0			0
$139$	74.4324			0.535485			0.267742	−	0.963491i	$-0.413722\pi$
$139$	74.4324			0.535485			0.267742	+	0.963491i	$0.413722\pi$
$140$		0			0
$141$	−92.5632	−	14.3732i	−0.656477	−	0.101938i
$142$		0			0
$143$		−	40.1051i		−	0.280455i
$144$		0			0
$145$	−100.231			−0.691247
$146$		0			0
$147$	−32.8381	+	211.477i	−0.223388	+	1.43862i
$148$		0			0
$149$		−	120.249i		−	0.807038i	−0.914971	−	0.403519i	$-0.867787\pi$
$149$		−	120.249i		−	0.807038i	0.914971	−	0.403519i	$-0.132213\pi$
$150$		0			0
$151$	15.1093			0.100062			0.0500310	−	0.998748i	$-0.484068\pi$
$151$	15.1093			0.100062			0.0500310	+	0.998748i	$0.484068\pi$
$152$		0			0
$153$	42.5182	−	133.607i	0.277897	−	0.873250i
$154$		0			0
$155$		−	114.886i		−	0.741201i
$156$		0			0
$157$	−265.567			−1.69151			−0.845754	−	0.533573i	$-0.820849\pi$
$157$	−265.567			−1.69151			−0.845754	+	0.533573i	$0.820849\pi$
$158$		0			0
$159$	−268.296	−	41.6610i	−1.68740	−	0.262019i
$160$		0			0
$161$		−	331.136i		−	2.05675i
$162$		0			0
$163$	24.5655			0.150709			0.0753544	−	0.997157i	$-0.475991\pi$
$163$	24.5655			0.150709			0.0753544	+	0.997157i	$0.475991\pi$
$164$		0			0
$165$	−1.83891	+	11.8425i	−0.0111449	+	0.0717729i
$166$		0			0
$167$		−	258.690i		−	1.54904i	−0.632549	−	0.774520i	$-0.717991\pi$
$167$		−	258.690i		−	1.54904i	0.632549	−	0.774520i	$-0.282009\pi$
$168$		0			0
$169$	334.937			1.98188
$170$		0			0
$171$	−226.248	−	71.9995i	−1.32309	−	0.421050i
$172$		0			0
$173$		−	172.804i		−	0.998866i	−0.866353	−	0.499433i	$-0.833542\pi$
$173$		−	172.804i		−	0.998866i	0.866353	−	0.499433i	$-0.166458\pi$
$174$		0			0
$175$	−54.8491			−0.313423
$176$		0			0
$177$	−52.7866	−	8.19670i	−0.298229	−	0.0463090i
$178$		0			0
$179$		−	274.427i		−	1.53311i	−0.642177	−	0.766556i	$-0.721969\pi$
$179$		−	274.427i		−	1.53311i	0.642177	−	0.766556i	$-0.278031\pi$
$180$		0			0
$181$	−136.602			−0.754710			−0.377355	−	0.926069i	$-0.623166\pi$
$181$	−136.602			−0.754710			−0.377355	+	0.926069i	$0.623166\pi$
$182$		0			0
$183$	5.34584	−	34.4271i	0.0292122	−	0.188126i
$184$		0			0
$185$		−	81.2640i		−	0.439265i
$186$		0			0
$187$	27.8321			0.148835
$188$		0			0
$189$	132.061	−	265.114i	0.698735	−	1.40272i
$190$		0			0
$191$			201.748i			1.05627i	0.849160	+	0.528135i	$0.177109\pi$
$191$			201.748i			1.05627i	−0.849160	+	0.528135i	$0.822891\pi$
$192$		0			0
$193$	−29.5623			−0.153173			−0.0765863	−	0.997063i	$-0.524402\pi$
$193$	−29.5623			−0.153173			−0.0765863	+	0.997063i	$0.524402\pi$
$194$		0			0
$195$	−148.806	−	23.1066i	−0.763108	−	0.118495i
$196$		0			0
$197$			287.440i			1.45908i	0.683936	+	0.729542i	$0.260267\pi$
$197$			287.440i			1.45908i	−0.683936	+	0.729542i	$0.739733\pi$
$198$		0			0
$199$	201.363			1.01187			0.505937	−	0.862571i	$-0.331147\pi$
$199$	201.363			1.01187			0.505937	+	0.862571i	$0.331147\pi$
$200$		0			0
$201$	−14.8738	+	95.7868i	−0.0739988	+	0.476551i
$202$		0			0
$203$			491.717i			2.42225i
$204$		0			0
$205$	17.7368			0.0865208
$206$		0			0
$207$	−82.3849	+	258.882i	−0.397995	+	1.25064i
$208$		0			0
$209$		−	47.1304i		−	0.225504i
$210$		0			0
$211$	−126.205			−0.598128			−0.299064	−	0.954233i	$-0.596674\pi$
$211$	−126.205			−0.598128			−0.299064	+	0.954233i	$0.596674\pi$
$212$		0			0
$213$	101.301	+	15.7300i	0.475593	+	0.0738500i
$214$		0			0
$215$			59.2273i			0.275476i
$216$		0			0
$217$	−563.614			−2.59730
$218$		0			0
$219$	31.7229	−	204.295i	0.144854	−	0.932854i
$220$		0			0
$221$			349.722i			1.58245i
$222$		0			0
$223$	−5.22218			−0.0234178			−0.0117089	−	0.999931i	$-0.503727\pi$
$223$	−5.22218			−0.0234178			−0.0117089	+	0.999931i	$0.503727\pi$
$224$		0			0
$225$	42.8810	+	13.6462i	0.190582	+	0.0606496i
$226$		0			0
$227$		−	170.760i		−	0.752248i	−0.926569	−	0.376124i	$-0.877257\pi$
$227$		−	170.760i		−	0.752248i	0.926569	−	0.376124i	$-0.122743\pi$
$228$		0			0
$229$	352.033			1.53726			0.768630	−	0.639694i	$-0.220939\pi$
$229$	352.033			1.53726			0.768630	+	0.639694i	$0.220939\pi$
$230$		0			0
$231$	58.0977	+	9.02140i	0.251505	+	0.0390537i
$232$		0			0
$233$		−	266.708i		−	1.14467i	−0.820021	−	0.572334i	$-0.806038\pi$
$233$		−	266.708i		−	1.14467i	0.820021	−	0.572334i	$-0.193962\pi$
$234$		0			0
$235$	−69.8194			−0.297104
$236$		0			0
$237$	−13.5918	+	87.5311i	−0.0573495	+	0.369330i
$238$		0			0
$239$			6.98824i			0.0292395i	0.999893	+	0.0146197i	$0.00465378\pi$
$239$			6.98824i			0.0292395i	−0.999893	+	0.0146197i	$0.995346\pi$
$240$		0			0
$241$	−11.2449			−0.0466593			−0.0233296	−	0.999728i	$-0.507427\pi$
$241$	−11.2449			−0.0466593			−0.0233296	+	0.999728i	$0.507427\pi$
$242$		0			0
$243$	−169.204	+	174.410i	−0.696313	+	0.717738i
$244$		0			0
$245$			159.514i			0.651079i
$246$		0			0
$247$	592.212			2.39762
$248$		0			0
$249$	225.782	+	35.0594i	0.906756	+	0.140801i
$250$		0			0
$251$			278.539i			1.10972i	0.831945	+	0.554858i	$0.187228\pi$
$251$			278.539i			1.10972i	−0.831945	+	0.554858i	$0.812772\pi$
$252$		0			0
$253$	−53.9286			−0.213156
$254$		0			0
$255$	16.0355	−	103.268i	0.0628843	−	0.404974i
$256$		0			0
$257$		−	296.546i		−	1.15388i	−0.816788	−	0.576938i	$-0.804247\pi$
$257$		−	296.546i		−	1.15388i	0.816788	−	0.576938i	$-0.195753\pi$
$258$		0			0
$259$	−398.669			−1.53926
$260$		0			0
$261$	122.337	−	384.425i	0.468723	−	1.47289i
$262$		0			0
$263$		−	393.054i		−	1.49450i	−0.664543	−	0.747250i	$-0.731374\pi$
$263$		−	393.054i		−	1.49450i	0.664543	−	0.747250i	$-0.268626\pi$
$264$		0			0
$265$	−202.373			−0.763671
$266$		0			0
$267$	334.851	+	51.9956i	1.25412	+	0.194740i
$268$		0			0
$269$			235.224i			0.874437i	0.899355	+	0.437219i	$0.144036\pi$
$269$			235.224i			0.874437i	−0.899355	+	0.437219i	$0.855964\pi$
$270$		0			0
$271$	−203.239			−0.749958			−0.374979	−	0.927033i	$-0.622350\pi$
$271$	−203.239			−0.749958			−0.374979	+	0.927033i	$0.622350\pi$
$272$		0			0
$273$	−113.358	+	730.021i	−0.415229	+	2.67407i
$274$		0			0
$275$			8.93268i			0.0324825i
$276$		0			0
$277$	−214.698			−0.775082			−0.387541	−	0.921852i	$-0.626675\pi$
$277$	−214.698			−0.775082			−0.387541	+	0.921852i	$0.626675\pi$
$278$		0			0
$279$	440.634	+	140.224i	1.57933	+	0.502596i
$280$		0			0
$281$			343.205i			1.22137i	0.791874	+	0.610685i	$0.209106\pi$
$281$			343.205i			1.22137i	−0.791874	+	0.610685i	$0.790894\pi$
$282$		0			0
$283$	402.851			1.42350			0.711751	−	0.702432i	$-0.247903\pi$
$283$	402.851			1.42350			0.711751	+	0.702432i	$0.247903\pi$
$284$		0			0
$285$	−174.873	−	27.1542i	−0.613588	−	0.0952779i
$286$		0			0
$287$		−	87.0139i		−	0.303184i
$288$		0			0
$289$	46.3000			0.160208
$290$		0			0
$291$	36.6174	−	235.815i	0.125833	−	0.810362i
$292$		0			0
$293$			121.592i			0.414989i	0.978236	+	0.207494i	$0.0665309\pi$
$293$			121.592i			0.414989i	−0.978236	+	0.207494i	$0.933469\pi$
$294$		0			0
$295$	−39.8163			−0.134971
$296$		0			0
$297$	−43.1763	−	21.5073i	−0.145375	−	0.0724152i
$298$		0			0
$299$		−	677.634i		−	2.26634i
$300$		0			0
$301$	290.560			0.965316
$302$		0			0
$303$	175.800	+	27.2981i	0.580196	+	0.0900929i
$304$		0			0
$305$		−	25.9680i		−	0.0851408i
$306$		0			0
$307$	−551.075			−1.79503			−0.897517	−	0.440980i	$-0.854631\pi$
$307$	−551.075			−1.79503			−0.897517	+	0.440980i	$0.854631\pi$
$308$		0			0
$309$	−9.20502	+	59.2802i	−0.0297897	+	0.191845i
$310$		0			0
$311$			30.9019i			0.0993631i	0.998765	+	0.0496816i	$0.0158206\pi$
$311$			30.9019i			0.0993631i	−0.998765	+	0.0496816i	$0.984179\pi$
$312$		0			0
$313$	471.595			1.50669			0.753346	−	0.657624i	$-0.228438\pi$
$313$	471.595			1.50669			0.753346	+	0.657624i	$0.228438\pi$
$314$		0			0
$315$	66.9460	−	210.368i	0.212527	−	0.667835i
$316$		0			0
$317$			399.347i			1.25977i	0.776689	+	0.629885i	$0.216898\pi$
$317$			399.347i			1.25977i	−0.776689	+	0.629885i	$0.783102\pi$
$318$		0			0
$319$	80.0807			0.251037
$320$		0			0
$321$	116.282	+	18.0563i	0.362250	+	0.0562502i
$322$		0			0
$323$			410.983i			1.27239i
$324$		0			0
$325$	−112.243			−0.345362
$326$		0			0
$327$	−2.83424	+	18.2525i	−0.00866741	+	0.0558179i
$328$		0			0
$329$			342.524i			1.04111i
$330$		0			0
$331$	−43.3950			−0.131103			−0.0655514	−	0.997849i	$-0.520881\pi$
$331$	−43.3950			−0.131103			−0.0655514	+	0.997849i	$0.520881\pi$
$332$		0			0
$333$	311.680	+	99.1868i	0.935975	+	0.297858i
$334$		0			0
$335$			72.2509i			0.215674i
$336$		0			0
$337$	233.859			0.693945			0.346972	−	0.937875i	$-0.387210\pi$
$337$	233.859			0.693945			0.346972	+	0.937875i	$0.387210\pi$
$338$		0			0
$339$	−261.220	−	40.5621i	−0.770559	−	0.119652i
$340$		0			0
$341$			91.7898i			0.269178i
$342$		0			0
$343$	245.032			0.714380
$344$		0			0
$345$	−31.0710	+	200.097i	−0.0900608	+	0.579990i
$346$		0			0
$347$			200.902i			0.578969i	0.957183	+	0.289485i	$0.0934839\pi$
$347$			200.902i			0.578969i	−0.957183	+	0.289485i	$0.906516\pi$
$348$		0			0
$349$	−106.570			−0.305358			−0.152679	−	0.988276i	$-0.548790\pi$
$349$	−106.570			−0.305358			−0.152679	+	0.988276i	$0.548790\pi$
$350$		0			0
$351$	270.248	−	542.528i	0.769938	−	1.54566i
$352$		0			0
$353$		−	269.597i		−	0.763732i	−0.924218	−	0.381866i	$-0.875282\pi$
$353$		−	269.597i		−	0.763732i	0.924218	−	0.381866i	$-0.124718\pi$
$354$		0			0
$355$	76.4104			0.215240
$356$		0			0
$357$	−506.620	−	78.6678i	−1.41910	−	0.220358i
$358$		0			0
$359$		−	31.2384i		−	0.0870150i	−0.999053	−	0.0435075i	$-0.986147\pi$
$359$		−	31.2384i		−	0.0870150i	0.999053	−	0.0435075i	$-0.0138532\pi$
$360$		0			0
$361$	334.950			0.927840
$362$		0			0
$363$	−54.2299	+	349.240i	−0.149394	+	0.962092i
$364$		0			0
$365$		−	154.097i		−	0.422185i
$366$		0			0
$367$	−502.908			−1.37032			−0.685160	−	0.728392i	$-0.740268\pi$
$367$	−502.908			−1.37032			−0.685160	+	0.728392i	$0.740268\pi$
$368$		0			0
$369$	−21.6486	+	68.0275i	−0.0586683	+	0.184356i
$370$		0			0
$371$			992.811i			2.67604i
$372$		0			0
$373$	−347.493			−0.931617			−0.465808	−	0.884886i	$-0.654236\pi$
$373$	−347.493			−0.931617			−0.465808	+	0.884886i	$0.654236\pi$
$374$		0			0
$375$	33.1438	+	5.14657i	0.0883835	+	0.0137242i
$376$		0			0
$377$			1006.25i			2.66909i
$378$		0			0
$379$	−262.136			−0.691652			−0.345826	−	0.938299i	$-0.612401\pi$
$379$	−262.136			−0.691652			−0.345826	+	0.938299i	$0.612401\pi$
$380$		0			0
$381$	2.12728	−	13.6996i	0.00558341	−	0.0359570i
$382$		0			0
$383$		−	75.5290i		−	0.197204i	−0.995127	−	0.0986018i	$-0.968563\pi$
$383$		−	75.5290i		−	0.197204i	0.995127	−	0.0986018i	$-0.0314370\pi$
$384$		0			0
$385$	43.8224			0.113824
$386$		0			0
$387$	−227.160	−	72.2898i	−0.586977	−	0.186795i
$388$		0			0
$389$		−	76.1862i		−	0.195852i	−0.995194	−	0.0979258i	$-0.968779\pi$
$389$		−	76.1862i		−	0.195852i	0.995194	−	0.0979258i	$-0.0312208\pi$
$390$		0			0
$391$	470.264			1.20272
$392$		0			0
$393$	−233.634	−	36.2787i	−0.594489	−	0.0923122i
$394$		0			0
$395$			66.0237i			0.167149i
$396$		0			0
$397$	516.613			1.30129			0.650647	−	0.759381i	$-0.274498\pi$
$397$	516.613			1.30129			0.650647	+	0.759381i	$0.274498\pi$
$398$		0			0
$399$	−133.214	+	857.899i	−0.333871	+	2.15012i
$400$		0			0
$401$			113.732i			0.283621i	0.989894	+	0.141810i	$0.0452923\pi$
$401$			113.732i			0.283621i	−0.989894	+	0.141810i	$0.954708\pi$
$402$		0			0
$403$	−1153.38			−2.86197
$404$		0			0
$405$	−104.677	+	147.810i	−0.258461	+	0.364963i
$406$		0			0
$407$			64.9269i			0.159526i
$408$		0			0
$409$	101.838			0.248994			0.124497	−	0.992220i	$-0.460268\pi$
$409$	101.838			0.248994			0.124497	+	0.992220i	$0.460268\pi$
$410$		0			0
$411$	378.403	+	58.7583i	0.920688	+	0.142964i
$412$		0			0
$413$			195.333i			0.472961i
$414$		0			0
$415$	170.305			0.410373
$416$		0			0
$417$	−34.2629	+	220.653i	−0.0821653	+	0.529143i
$418$		0			0
$419$		−	391.559i		−	0.934507i	−0.884123	−	0.467254i	$-0.845243\pi$
$419$		−	391.559i		−	0.934507i	0.884123	−	0.467254i	$-0.154757\pi$
$420$		0			0
$421$	−342.755			−0.814145			−0.407073	−	0.913396i	$-0.633450\pi$
$421$	−342.755			−0.814145			−0.407073	+	0.913396i	$0.633450\pi$
$422$		0			0
$423$	85.2180	−	267.785i	0.201461	−	0.633061i
$424$		0			0
$425$		−	77.8942i		−	0.183280i
$426$		0			0
$427$	−127.395			−0.298349
$428$		0			0
$429$	118.891	+	18.4613i	0.277134	+	0.0430334i
$430$		0			0
$431$		−	367.168i		−	0.851898i	−0.904747	−	0.425949i	$-0.859940\pi$
$431$		−	367.168i		−	0.851898i	0.904747	−	0.425949i	$-0.140060\pi$
$432$		0			0
$433$	367.630			0.849029			0.424515	−	0.905421i	$-0.360445\pi$
$433$	367.630			0.849029			0.424515	+	0.905421i	$0.360445\pi$
$434$		0			0
$435$	46.1386	−	297.131i	0.106066	−	0.683061i
$436$		0			0
$437$		−	796.336i		−	1.82228i
$438$		0			0
$439$	−492.913			−1.12281			−0.561404	−	0.827542i	$-0.689739\pi$
$439$	−492.913			−1.12281			−0.561404	+	0.827542i	$0.689739\pi$
$440$		0			0
$441$	−611.800	−	194.695i	−1.38730	−	0.441485i
$442$		0			0
$443$		−	733.159i		−	1.65499i	−0.561476	−	0.827493i	$-0.689766\pi$
$443$		−	733.159i		−	1.65499i	0.561476	−	0.827493i	$-0.310234\pi$
$444$		0			0
$445$	252.574			0.567583
$446$		0			0
$447$	356.474	+	55.3532i	0.797481	+	0.123833i
$448$		0			0
$449$		−	704.511i		−	1.56907i	−0.620087	−	0.784533i	$-0.712903\pi$
$449$		−	704.511i		−	1.56907i	0.620087	−	0.784533i	$-0.287097\pi$
$450$		0			0
$451$	−14.1710			−0.0314213
$452$		0			0
$453$	−6.95518	+	44.7913i	−0.0153536	+	0.0988770i
$454$		0			0
$455$			550.646i			1.21021i
$456$		0			0
$457$	−755.476			−1.65312			−0.826560	−	0.562848i	$-0.809705\pi$
$457$	−755.476			−1.65312			−0.826560	+	0.562848i	$0.809705\pi$
$458$		0			0
$459$	376.503	+	187.547i	0.820268	+	0.408598i
$460$		0			0
$461$			243.102i			0.527335i	0.964614	+	0.263668i	$0.0849322\pi$
$461$			243.102i			0.527335i	−0.964614	+	0.263668i	$0.915068\pi$
$462$		0			0
$463$	−733.057			−1.58328			−0.791638	−	0.610990i	$-0.790771\pi$
$463$	−733.057			−1.58328			−0.791638	+	0.610990i	$0.790771\pi$
$464$		0			0
$465$	340.577	+	52.8847i	0.732423	+	0.113731i
$466$		0			0
$467$		−	580.434i		−	1.24290i	−0.783454	−	0.621450i	$-0.786544\pi$
$467$		−	580.434i		−	1.24290i	0.783454	−	0.621450i	$-0.213456\pi$
$468$		0			0
$469$	354.452			0.755761
$470$		0			0
$471$	122.246	−	787.265i	0.259547	−	1.67148i
$472$		0			0
$473$		−	47.3204i		−	0.100043i
$474$		0			0
$475$	−131.904			−0.277693
$476$		0			0
$477$	247.006	−	776.180i	0.517832	−	1.62721i
$478$		0			0
$479$			866.763i			1.80953i	0.425915	+	0.904763i	$0.359952\pi$
$479$			866.763i			1.80953i	−0.425915	+	0.904763i	$0.640048\pi$
$480$		0			0
$481$	−815.834			−1.69612
$482$		0			0
$483$	981.645	+	152.430i	2.03239	+	0.315589i
$484$		0			0
$485$		−	177.873i		−	0.366748i
$486$		0			0
$487$	−388.362			−0.797458			−0.398729	−	0.917069i	$-0.630549\pi$
$487$	−388.362			−0.797458			−0.398729	+	0.917069i	$0.630549\pi$
$488$		0			0
$489$	−11.3081	+	72.8239i	−0.0231249	+	0.148924i
$490$		0			0
$491$		−	412.624i		−	0.840375i	−0.907437	−	0.420188i	$-0.861964\pi$
$491$		−	412.624i		−	0.840375i	0.907437	−	0.420188i	$-0.138036\pi$
$492$		0			0
$493$	−698.314			−1.41646
$494$		0			0
$495$	−34.2603	−	10.9028i	−0.0692128	−	0.0220258i
$496$		0			0
$497$		−	374.858i		−	0.754241i
$498$		0			0
$499$	382.658			0.766849			0.383425	−	0.923572i	$-0.374745\pi$
$499$	382.658			0.766849			0.383425	+	0.923572i	$0.374745\pi$
$500$		0			0
$501$	766.879	+	119.081i	1.53070	+	0.237686i
$502$		0			0
$503$		−	313.069i		−	0.622404i	−0.950344	−	0.311202i	$-0.899268\pi$
$503$		−	313.069i		−	0.622404i	0.950344	−	0.311202i	$-0.100732\pi$
$504$		0			0
$505$	132.604			0.262581
$506$		0			0
$507$	−154.179	+	992.913i	−0.304101	+	1.95841i
$508$		0			0
$509$			591.748i			1.16257i	0.813700	+	0.581285i	$0.197450\pi$
$509$			591.748i			1.16257i	−0.813700	+	0.581285i	$0.802550\pi$
$510$		0			0
$511$	−755.979			−1.47941
$512$		0			0
$513$	317.588	−	637.562i	0.619079	−	1.24281i
$514$		0			0
$515$			44.7143i			0.0868240i
$516$		0			0
$517$	55.7831			0.107898
$518$		0			0
$519$	512.272	+	79.5456i	0.987037	+	0.153267i
$520$		0			0
$521$		−	204.448i		−	0.392414i	−0.980562	−	0.196207i	$-0.937137\pi$
$521$		−	204.448i		−	0.392414i	0.980562	−	0.196207i	$-0.0628625\pi$
$522$		0			0
$523$	464.580			0.888299			0.444150	−	0.895953i	$-0.353506\pi$
$523$	464.580			0.888299			0.444150	+	0.895953i	$0.353506\pi$
$524$		0			0
$525$	25.2483	−	162.599i	0.0480920	−	0.309712i
$526$		0			0
$527$		−	800.419i		−	1.51882i
$528$		0			0
$529$	−382.202			−0.722499
$530$		0			0
$531$	48.5978	−	152.711i	0.0915212	−	0.287592i
$532$		0			0
$533$		−	178.065i		−	0.334080i
$534$		0			0
$535$	87.7104			0.163945
$536$		0			0
$537$	813.532	+	126.325i	1.51496	+	0.235242i
$538$		0			0
$539$		−	127.446i		−	0.236449i
$540$		0			0
$541$	260.310			0.481165			0.240582	−	0.970629i	$-0.422662\pi$
$541$	260.310			0.481165			0.240582	+	0.970629i	$0.422662\pi$
$542$		0			0
$543$	62.8813	−	404.954i	0.115803	−	0.745772i
$544$		0			0
$545$			13.7676i			0.0252617i
$546$		0			0
$547$	736.820			1.34702			0.673511	−	0.739178i	$-0.264786\pi$
$547$	736.820			1.34702			0.673511	+	0.739178i	$0.264786\pi$
$548$		0			0
$549$	99.5974	+	31.6952i	0.181416	+	0.0577326i
$550$		0			0
$551$			1182.51i			2.14612i
$552$		0			0
$553$	323.903			0.585719
$554$		0			0
$555$	240.905	+	37.4077i	0.434063	+	0.0674013i
$556$		0			0
$557$		−	33.1907i		−	0.0595883i	−0.999556	−	0.0297941i	$-0.990515\pi$
$557$		−	33.1907i		−	0.0595883i	0.999556	−	0.0297941i	$-0.00948517\pi$
$558$		0			0
$559$	594.600			1.06369
$560$		0			0
$561$	−12.8118	+	82.5076i	−0.0228374	+	0.147072i
$562$		0			0
$563$			373.141i			0.662772i	0.943495	+	0.331386i	$0.107516\pi$
$563$			373.141i			0.662772i	−0.943495	+	0.331386i	$0.892484\pi$
$564$		0			0
$565$	−197.035			−0.348734
$566$		0			0
$567$	725.134	+	513.529i	1.27890	+	0.905695i
$568$		0			0
$569$		−	894.360i		−	1.57181i	−0.618347	−	0.785905i	$-0.712197\pi$
$569$		−	894.360i		−	1.57181i	0.618347	−	0.785905i	$-0.287803\pi$
$570$		0			0
$571$	−180.286			−0.315738			−0.157869	−	0.987460i	$-0.550462\pi$
$571$	−180.286			−0.315738			−0.157869	+	0.987460i	$0.550462\pi$
$572$		0			0
$573$	−598.076	−	92.8691i	−1.04376	−	0.162075i
$574$		0			0
$575$			150.931i			0.262488i
$576$		0			0
$577$	386.559			0.669947			0.334973	−	0.942228i	$-0.391273\pi$
$577$	386.559			0.669947			0.334973	+	0.942228i	$0.391273\pi$
$578$		0			0
$579$	13.6082	−	87.6367i	0.0235030	−	0.151359i
$580$		0			0
$581$		−	835.491i		−	1.43802i
$582$		0			0
$583$	161.688			0.277338
$584$		0			0
$585$	136.998	−	430.495i	0.234184	−	0.735889i
$586$		0			0
$587$		−	316.118i		−	0.538531i	−0.963066	−	0.269266i	$-0.913219\pi$
$587$		−	316.118i		−	0.538531i	0.963066	−	0.269266i	$-0.0867809\pi$
$588$		0			0
$589$	−1355.41			−2.30121
$590$		0			0
$591$	−852.107	−	132.315i	−1.44181	−	0.223884i
$592$		0			0
$593$		−	482.116i		−	0.813012i	−0.913648	−	0.406506i	$-0.866747\pi$
$593$		−	482.116i		−	0.813012i	0.913648	−	0.406506i	$-0.133253\pi$
$594$		0			0
$595$	−382.137			−0.642247
$596$		0			0
$597$	−92.6919	+	596.935i	−0.155263	+	0.999890i
$598$		0			0
$599$			323.435i			0.539959i	0.962866	+	0.269979i	$0.0870169\pi$
$599$			323.435i			0.539959i	−0.962866	+	0.269979i	$0.912983\pi$
$600$		0			0
$601$	1.11704			0.00185863			0.000929316	−	1.00000i	$-0.499704\pi$
$601$	1.11704			0.00185863			0.000929316		1.00000i	$0.499704\pi$
$602$		0			0
$603$	−277.111	−	88.1858i	−0.459553	−	0.146245i
$604$		0			0
$605$			263.427i			0.435417i
$606$		0			0
$607$	−325.831			−0.536790			−0.268395	−	0.963309i	$-0.586493\pi$
$607$	−325.831			−0.536790			−0.268395	+	0.963309i	$0.586493\pi$
$608$		0			0
$609$	−1457.68	−	226.349i	−2.39357	−	0.371673i
$610$		0			0
$611$			700.937i			1.14720i
$612$		0			0
$613$	84.6761			0.138134			0.0690670	−	0.997612i	$-0.477998\pi$
$613$	84.6761			0.138134			0.0690670	+	0.997612i	$0.477998\pi$
$614$		0			0
$615$	−8.16464	+	52.5802i	−0.0132758	+	0.0854962i
$616$		0			0
$617$		−	1059.47i		−	1.71713i	−0.512706	−	0.858564i	$-0.671357\pi$
$617$		−	1059.47i		−	1.71713i	0.512706	−	0.858564i	$-0.328643\pi$
$618$		0			0
$619$	−233.336			−0.376956			−0.188478	−	0.982077i	$-0.560355\pi$
$619$	−233.336			−0.376956			−0.188478	+	0.982077i	$0.560355\pi$
$620$		0			0
$621$	−729.526	−	363.397i	−1.17476	−	0.585181i
$622$		0			0
$623$		−	1239.09i		−	1.98891i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	139.717	+	21.6952i	0.222834	+	0.0346016i
$628$		0			0
$629$		−	566.172i		−	0.900114i
$630$		0			0
$631$	−179.423			−0.284347			−0.142174	−	0.989842i	$-0.545409\pi$
$631$	−179.423			−0.284347			−0.142174	+	0.989842i	$0.545409\pi$
$632$		0			0
$633$	58.0951	−	374.132i	0.0917774	−	0.591045i
$634$		0			0
$635$		−	10.3335i		−	0.0162732i
$636$		0			0
$637$	1601.41			2.51399
$638$		0			0
$639$	−93.2626	+	293.064i	−0.145951	+	0.458629i
$640$		0			0
$641$		−	469.934i		−	0.733126i	−0.930393	−	0.366563i	$-0.880534\pi$
$641$		−	469.934i		−	0.733126i	0.930393	−	0.366563i	$-0.119466\pi$
$642$		0			0
$643$	408.297			0.634988			0.317494	−	0.948260i	$-0.397159\pi$
$643$	408.297			0.634988			0.317494	+	0.948260i	$0.397159\pi$
$644$		0			0
$645$	−175.578	−	27.2637i	−0.272213	−	0.0422693i
$646$		0			0
$647$		−	1100.12i		−	1.70035i	−0.526502	−	0.850174i	$-0.676497\pi$
$647$		−	1100.12i		−	1.70035i	0.526502	−	0.850174i	$-0.323503\pi$
$648$		0			0
$649$	31.8118			0.0490166
$650$		0			0
$651$	259.445	−	1670.82i	0.398533	−	2.56654i
$652$		0			0
$653$			580.056i			0.888295i	0.895954	+	0.444147i	$0.146493\pi$
$653$			580.056i			0.888295i	−0.895954	+	0.444147i	$0.853507\pi$
$654$		0			0
$655$	−176.228			−0.269050
$656$		0			0
$657$	591.025	+	188.084i	0.899581	+	0.286276i
$658$		0			0
$659$		−	24.8268i		−	0.0376734i	−0.999823	−	0.0188367i	$-0.994004\pi$
$659$		−	24.8268i		−	0.0376734i	0.999823	−	0.0188367i	$-0.00599626\pi$
$660$		0			0
$661$	680.704			1.02981			0.514904	−	0.857248i	$-0.327827\pi$
$661$	680.704			1.02981			0.514904	+	0.857248i	$0.327827\pi$
$662$		0			0
$663$	−1036.74	−	160.985i	−1.56371	−	0.242813i
$664$		0			0
$665$			647.103i			0.973088i
$666$		0			0
$667$	1353.08			2.02861
$668$		0			0
$669$	2.40389	−	15.4810i	0.00359326	−	0.0231405i
$670$		0			0
$671$			20.7474i			0.0309202i
$672$		0			0
$673$	−803.058			−1.19325			−0.596625	−	0.802520i	$-0.703492\pi$
$673$	−803.058			−1.19325			−0.596625	+	0.802520i	$0.703492\pi$
$674$		0			0
$675$	−60.1928	+	120.838i	−0.0891745	+	0.179019i
$676$		0			0
$677$			229.259i			0.338639i	0.985561	+	0.169319i	$0.0541570\pi$
$677$			229.259i			0.338639i	−0.985561	+	0.169319i	$0.945843\pi$
$678$		0			0
$679$	−872.618			−1.28515
$680$		0			0
$681$	506.214	+	78.6049i	0.743339	+	0.115426i
$682$		0			0
$683$			1098.59i			1.60848i	0.594302	+	0.804242i	$0.297429\pi$
$683$			1098.59i			1.60848i	−0.594302	+	0.804242i	$0.702571\pi$
$684$		0			0
$685$	285.425			0.416678
$686$		0			0
$687$	−162.049	+	1043.59i	−0.235879	+	1.51906i
$688$		0			0
$689$			2031.68i			2.94874i
$690$		0			0
$691$	−1139.91			−1.64965			−0.824824	−	0.565389i	$-0.808726\pi$
$691$	−1139.91			−1.64965			−0.824824	+	0.565389i	$0.808726\pi$
$692$		0			0
$693$	−53.4874	+	168.076i	−0.0771824	+	0.242534i
$694$		0			0
$695$			166.436i			0.239476i
$696$		0			0
$697$	123.573			0.177293
$698$		0			0
$699$	790.647	+	122.772i	1.13111	+	0.175639i
$700$		0			0
$701$		−	850.938i		−	1.21389i	−0.794743	−	0.606946i	$-0.792395\pi$
$701$		−	850.938i		−	1.21389i	0.794743	−	0.606946i	$-0.207605\pi$
$702$		0			0
$703$	−958.744			−1.36379
$704$		0			0
$705$	32.1395	−	206.978i	0.0455879	−	0.293585i
$706$		0			0
$707$		−	650.534i		−	0.920132i
$708$		0			0
$709$	−1057.04			−1.49088			−0.745441	−	0.666571i	$-0.767761\pi$
$709$	−1057.04			−1.49088			−0.745441	+	0.666571i	$0.767761\pi$
$710$		0			0
$711$	−253.227	−	80.5852i	−0.356156	−	0.113341i
$712$		0			0
$713$			1550.92i			2.17521i
$714$		0			0
$715$	89.6778			0.125423
$716$		0			0
$717$	−20.7164	−	3.21685i	−0.0288932	−	0.00448654i
$718$		0			0
$719$			671.914i			0.934512i	0.884122	+	0.467256i	$0.154757\pi$
$719$			671.914i			0.934512i	−0.884122	+	0.467256i	$0.845243\pi$
$720$		0			0
$721$	219.362			0.304247
$722$		0			0
$723$	5.17628	−	33.3352i	0.00715945	−	0.0461067i
$724$		0			0
$725$		−	224.123i		−	0.309135i
$726$		0			0
$727$	−1318.52			−1.81364			−0.906821	−	0.421515i	$-0.861498\pi$
$727$	−1318.52			−1.81364			−0.906821	+	0.421515i	$0.861498\pi$
$728$		0			0
$729$	−439.146	−	581.886i	−0.602395	−	0.798198i
$730$		0			0
$731$			412.640i			0.564487i
$732$		0			0
$733$	−550.561			−0.751107			−0.375553	−	0.926801i	$-0.622547\pi$
$733$	−550.561			−0.751107			−0.375553	+	0.926801i	$0.622547\pi$
$734$		0			0
$735$	−472.876	−	73.4281i	−0.643369	−	0.0999022i
$736$		0			0
$737$		−	57.7258i		−	0.0783253i
$738$		0			0
$739$	887.354			1.20075			0.600375	−	0.799719i	$-0.295018\pi$
$739$	887.354			1.20075			0.600375	+	0.799719i	$0.295018\pi$
$740$		0			0
$741$	−272.609	+	1755.60i	−0.367893	+	2.36923i
$742$		0			0
$743$		−	174.512i		−	0.234874i	−0.993080	−	0.117437i	$-0.962532\pi$
$743$		−	174.512i		−	0.234874i	0.993080	−	0.117437i	$-0.0374679\pi$
$744$		0			0
$745$	268.884			0.360918
$746$		0			0
$747$	−207.866	+	653.186i	−0.278267	+	0.874413i
$748$		0			0
$749$		−	430.294i		−	0.574491i
$750$		0			0
$751$	974.125			1.29710			0.648552	−	0.761170i	$-0.275375\pi$
$751$	974.125			1.29710			0.648552	+	0.761170i	$0.275375\pi$
$752$		0			0
$753$	−825.721	−	128.218i	−1.09658	−	0.170276i
$754$		0			0
$755$			33.7855i			0.0447490i
$756$		0			0
$757$	1449.71			1.91507			0.957533	−	0.288322i	$-0.0930974\pi$
$757$	1449.71			1.91507			0.957533	+	0.288322i	$0.0930974\pi$
$758$		0			0
$759$	24.8246	−	159.870i	0.0327069	−	0.210632i
$760$		0			0
$761$			468.047i			0.615042i	0.951541	+	0.307521i	$0.0994994\pi$
$761$			468.047i			0.615042i	−0.951541	+	0.307521i	$0.900501\pi$
$762$		0			0
$763$	67.5419			0.0885216
$764$		0			0
$765$	298.755	+	95.0737i	0.390529	+	0.124279i
$766$		0			0
$767$			399.728i			0.521157i
$768$		0			0
$769$	707.833			0.920459			0.460230	−	0.887800i	$-0.347767\pi$
$769$	707.833			0.920459			0.460230	+	0.887800i	$0.347767\pi$
$770$		0			0
$771$	879.104	+	136.507i	1.14021	+	0.177052i
$772$		0			0
$773$		−	610.281i		−	0.789496i	−0.918789	−	0.394748i	$-0.870832\pi$
$773$		−	610.281i		−	0.789496i	0.918789	−	0.394748i	$-0.129168\pi$
$774$		0			0
$775$	256.893			0.331475
$776$		0			0
$777$	183.517	−	1181.84i	0.236186	−	1.52104i
$778$		0			0
$779$		−	209.256i		−	0.268622i
$780$		0			0
$781$	−61.0490			−0.0781678
$782$		0			0
$783$	1083.30	+	539.623i	1.38353	+	0.689174i
$784$		0			0
$785$		−	593.825i		−	0.756465i
$786$		0			0
$787$	−561.570			−0.713557			−0.356779	−	0.934189i	$-0.616125\pi$
$787$	−561.570			−0.713557			−0.356779	+	0.934189i	$0.616125\pi$
$788$		0			0
$789$	1165.20	+	180.932i	1.47680	+	0.229318i
$790$		0			0
$791$			966.624i			1.22203i
$792$		0			0
$793$	−260.700			−0.328752
$794$		0			0
$795$	93.1569	−	599.929i	0.117178	−	0.754627i
$796$		0			0
$797$		−	789.976i		−	0.991187i	−0.868554	−	0.495594i	$-0.834951\pi$
$797$		−	789.976i		−	0.991187i	0.868554	−	0.495594i	$-0.165049\pi$
$798$		0			0
$799$	−486.436			−0.608806
$800$		0			0
$801$	−308.279	+	968.722i	−0.384868	+	1.20939i
$802$		0			0
$803$			123.118i			0.153323i
$804$		0			0
$805$	740.443			0.919805
$806$		0			0
$807$	−697.314	−	108.279i	−0.864082	−	0.134175i
$808$		0			0
$809$			892.712i			1.10348i	0.834018	+	0.551738i	$0.186035\pi$
$809$			892.712i			1.10348i	−0.834018	+	0.551738i	$0.813965\pi$
$810$		0			0
$811$	622.708			0.767827			0.383913	−	0.923369i	$-0.374576\pi$
$811$	622.708			0.767827			0.383913	+	0.923369i	$0.374576\pi$
$812$		0			0
$813$	93.5554	−	602.495i	0.115074	−	0.741077i
$814$		0			0
$815$			54.9302i			0.0673990i
$816$		0			0
$817$	698.756			0.855271
$818$		0			0
$819$	−2111.95	−	672.091i	−2.57869	−	0.820624i
$820$		0			0
$821$			111.933i			0.136338i	0.997674	+	0.0681690i	$0.0217157\pi$
$821$			111.933i			0.136338i	−0.997674	+	0.0681690i	$0.978284\pi$
$822$		0			0
$823$	1171.63			1.42361			0.711804	−	0.702378i	$-0.247879\pi$
$823$	1171.63			1.42361			0.711804	+	0.702378i	$0.247879\pi$
$824$		0			0
$825$	−26.4807	−	4.11192i	−0.0320978	−	0.00498414i
$826$		0			0
$827$		−	108.790i		−	0.131548i	−0.997835	−	0.0657738i	$-0.979048\pi$
$827$		−	108.790i		−	0.131548i	0.997835	−	0.0657738i	$-0.0209516\pi$
$828$		0			0
$829$	−1303.48			−1.57235			−0.786177	−	0.618001i	$-0.787943\pi$
$829$	−1303.48			−1.57235			−0.786177	+	0.618001i	$0.787943\pi$
$830$		0			0
$831$	98.8304	−	636.466i	0.118929	−	0.765904i
$832$		0			0
$833$			1111.35i			1.33415i
$834$		0			0
$835$	578.448			0.692752
$836$		0			0
$837$	−618.525	+	1241.70i	−0.738978	+	1.48351i
$838$		0			0
$839$		−	1190.36i		−	1.41878i	−0.704816	−	0.709390i	$-0.748971\pi$
$839$		−	1190.36i		−	1.41878i	0.704816	−	0.709390i	$-0.251029\pi$
$840$		0			0
$841$	−1168.24			−1.38911
$842$		0			0
$843$	−1017.42	−	157.985i	−1.20691	−	0.187408i
$844$		0			0
$845$			748.942i			0.886322i
$846$		0			0
$847$	1292.34			1.52578
$848$		0			0
$849$	−185.442	+	1194.24i	−0.218424	+	1.40664i
$850$		0			0
$851$			1097.04i			1.28911i
$852$		0			0
$853$	−291.614			−0.341869			−0.170935	−	0.985282i	$-0.554679\pi$
$853$	−291.614			−0.341869			−0.170935	+	0.985282i	$0.554679\pi$
$854$		0			0
$855$	160.996	−	505.905i	0.188299	−	0.591702i
$856$		0			0
$857$			1537.66i			1.79423i	0.441796	+	0.897115i	$0.354341\pi$
$857$			1537.66i			1.79423i	−0.441796	+	0.897115i	$0.645659\pi$
$858$		0			0
$859$	488.132			0.568256			0.284128	−	0.958786i	$-0.408296\pi$
$859$	488.132			0.568256			0.284128	+	0.958786i	$0.408296\pi$
$860$		0			0
$861$	257.950	+	40.0545i	0.299594	+	0.0465209i
$862$		0			0
$863$			1070.75i			1.24073i	0.784313	+	0.620365i	$0.213016\pi$
$863$			1070.75i			1.24073i	−0.784313	+	0.620365i	$0.786984\pi$
$864$		0			0
$865$	386.401			0.446706
$866$		0			0
$867$	−21.3130	+	137.255i	−0.0245824	+	0.158311i
$868$		0			0
$869$		−	52.7505i		−	0.0607025i
$870$		0			0
$871$	725.348			0.832776
$872$		0			0
$873$	682.212	+	217.102i	0.781457	+	0.248686i
$874$		0			0
$875$		−	122.646i		−	0.140167i
$876$		0			0
$877$	655.738			0.747705			0.373853	−	0.927488i	$-0.378037\pi$
$877$	655.738			0.747705			0.373853	+	0.927488i	$0.378037\pi$
$878$		0			0
$879$	−360.455	−	55.9715i	−0.410074	−	0.0636763i
$880$		0			0
$881$			1051.89i			1.19398i	0.802249	+	0.596989i	$0.203636\pi$
$881$			1051.89i			1.19398i	−0.802249	+	0.596989i	$0.796364\pi$
$882$		0			0
$883$	1342.09			1.51992			0.759960	−	0.649970i	$-0.225219\pi$
$883$	1342.09			1.51992			0.759960	+	0.649970i	$0.225219\pi$
$884$		0			0
$885$	18.3284	−	118.034i	0.0207100	−	0.133372i
$886$		0			0
$887$		−	773.466i		−	0.872002i	−0.899946	−	0.436001i	$-0.856394\pi$
$887$		−	773.466i		−	0.872002i	0.899946	−	0.436001i	$-0.143606\pi$
$888$		0			0
$889$	−50.6945			−0.0570242
$890$		0			0
$891$	83.6329	−	118.095i	0.0938641	−	0.132542i
$892$		0			0
$893$			823.721i			0.922420i
$894$		0			0
$895$	613.638			0.685629
$896$		0			0
$897$	2008.83	+	311.931i	2.23950	+	0.347749i
$898$		0			0
$899$		−	2303.03i		−	2.56176i
$900$		0			0
$901$	−1409.94			−1.56487
$902$		0			0
$903$	−133.752	+	861.358i	−0.148119	+	0.953885i
$904$		0			0
$905$		−	305.452i		−	0.337516i
$906$		0			0
$907$	−1397.06			−1.54031			−0.770156	−	0.637855i	$-0.779822\pi$
$907$	−1397.06			−1.54031			−0.770156	+	0.637855i	$0.779822\pi$
$908$		0			0
$909$	−161.849	+	508.587i	−0.178052	+	0.559502i
$910$		0			0
$911$		−	1548.07i		−	1.69931i	−0.527339	−	0.849655i	$-0.676810\pi$
$911$		−	1548.07i		−	1.69931i	0.527339	−	0.849655i	$-0.323190\pi$
$912$		0			0
$913$	−136.067			−0.149033
$914$		0			0
$915$	76.9813	+	11.9537i	0.0841326	+	0.0130641i
$916$		0			0
$917$			864.546i			0.942799i
$918$		0			0
$919$	469.880			0.511294			0.255647	−	0.966770i	$-0.417711\pi$
$919$	469.880			0.511294			0.255647	+	0.966770i	$0.417711\pi$
$920$		0			0
$921$	253.673	−	1633.65i	0.275432	−	1.77378i
$922$		0			0
$923$		−	767.106i		−	0.831101i
$924$		0			0
$925$	181.712			0.196445
$926$		0			0
$927$	−171.497	−	54.5761i	−0.185002	−	0.0588739i
$928$		0			0
$929$		−	656.649i		−	0.706834i	−0.935466	−	0.353417i	$-0.885020\pi$
$929$		−	656.649i		−	0.706834i	0.935466	−	0.353417i	$-0.114980\pi$
$930$		0			0
$931$	1881.93			2.02141
$932$		0			0
$933$	−91.6080	−	14.2249i	−0.0981865	−	0.0152464i
$934$		0			0
$935$			62.2345i			0.0665610i
$936$		0			0
$937$	−759.554			−0.810623			−0.405312	−	0.914179i	$-0.632837\pi$
$937$	−759.554			−0.810623			−0.405312	+	0.914179i	$0.632837\pi$
$938$		0			0
$939$	−217.086	+	1398.03i	−0.231189	+	1.48885i
$940$		0			0
$941$			227.955i			0.242247i	0.992637	+	0.121124i	$0.0386497\pi$
$941$			227.955i			0.242247i	−0.992637	+	0.121124i	$0.961350\pi$
$942$		0			0
$943$	−239.440			−0.253913
$944$		0			0
$945$	592.814	+	295.297i	0.627316	+	0.312484i
$946$		0			0
$947$			150.055i			0.158453i	0.996857	+	0.0792266i	$0.0252451\pi$
$947$			150.055i			0.158453i	−0.996857	+	0.0792266i	$0.974755\pi$
$948$		0			0
$949$	−1547.03			−1.63017
$950$		0			0
$951$	−1183.85	−	183.829i	−1.24485	−	0.193300i
$952$		0			0
$953$			216.231i			0.226895i	0.993544	+	0.113447i	$0.0361893\pi$
$953$			216.231i			0.226895i	−0.993544	+	0.113447i	$0.963811\pi$
$954$		0			0
$955$	−451.122			−0.472379
$956$		0			0
$957$	−36.8630	+	237.397i	−0.0385193	+	0.248064i
$958$		0			0
$959$		−	1400.25i		−	1.46012i
$960$		0			0
$961$	1678.76			1.74689
$962$		0			0
$963$	−107.055	+	336.404i	−0.111168	+	0.349329i
$964$		0			0
$965$		−	66.1034i		−	0.0685009i
$966$		0			0
$967$	976.244			1.00956			0.504780	−	0.863248i	$-0.331574\pi$
$967$	976.244			1.00956			0.504780	+	0.863248i	$0.331574\pi$
$968$		0			0
$969$	−1218.35	−	189.185i	−1.25733	−	0.195237i
$970$		0			0
$971$			602.949i			0.620956i	0.950580	+	0.310478i	$0.100489\pi$
$971$			602.949i			0.620956i	−0.950580	+	0.310478i	$0.899511\pi$
$972$		0			0
$973$	816.510			0.839167
$974$		0			0
$975$	51.6679	−	332.741i	0.0529928	−	0.341272i
$976$		0			0
$977$		−	325.588i		−	0.333253i	−0.986020	−	0.166626i	$-0.946713\pi$
$977$		−	325.588i		−	0.333253i	0.986020	−	0.166626i	$-0.0532874\pi$
$978$		0			0
$979$	−201.797			−0.206126
$980$		0			0
$981$	−52.8043	−	16.8041i	−0.0538270	−	0.0171295i
$982$		0			0
$983$			1185.68i			1.20619i	0.797670	+	0.603094i	$0.206065\pi$
$983$			1185.68i			1.20619i	−0.797670	+	0.603094i	$0.793935\pi$
$984$		0			0
$985$	−642.735			−0.652523
$986$		0			0
$987$	−1015.40	−	157.672i	−1.02878	−	0.159748i
$988$		0			0
$989$		−	799.547i		−	0.808440i
$990$		0			0
$991$	710.407			0.716858			0.358429	−	0.933557i	$-0.383312\pi$
$991$	710.407			0.716858			0.358429	+	0.933557i	$0.383312\pi$
$992$		0			0
$993$	19.9757	−	128.643i	0.0201166	−	0.129550i
$994$		0			0
$995$			450.261i			0.452523i
$996$		0			0
$997$	381.686			0.382834			0.191417	−	0.981509i	$-0.438692\pi$
$997$	381.686			0.382834			0.191417	+	0.981509i	$0.438692\pi$
$998$		0			0
$999$	−437.510	+	878.308i	−0.437948	+	0.879187i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.3.l.a.161.8	yes	16
3.2	odd	2	inner	480.3.l.a.161.7	✓	16
4.3	odd	2	inner	480.3.l.a.161.9	yes	16
8.3	odd	2		960.3.l.i.641.8		16
8.5	even	2		960.3.l.i.641.9		16
12.11	even	2	inner	480.3.l.a.161.10	yes	16
24.5	odd	2		960.3.l.i.641.10		16
24.11	even	2		960.3.l.i.641.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.l.a.161.7	✓	16	3.2	odd	2	inner
480.3.l.a.161.8	yes	16	1.1	even	1	trivial
480.3.l.a.161.9	yes	16	4.3	odd	2	inner
480.3.l.a.161.10	yes	16	12.11	even	2	inner
960.3.l.i.641.7		16	24.11	even	2
960.3.l.i.641.8		16	8.3	odd	2
960.3.l.i.641.9		16	8.5	even	2
960.3.l.i.641.10		16	24.5	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	480.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.460323 + 2.96447i) q^{3} +2.23607i q^{5} +10.9698 q^{7} +(-8.57621 - 2.72923i) q^{9} +O(q^{10})\)	\(q+(-0.460323 + 2.96447i) q^{3} +2.23607i q^{5} +10.9698 q^{7} +(-8.57621 - 2.72923i) q^{9} -1.78654i q^{11} +22.4485 q^{13} +(-6.62876 - 1.02931i) q^{15} +15.5788i q^{17} +26.3809 q^{19} +(-5.04966 + 32.5197i) q^{21} -30.1861i q^{23} -5.00000 q^{25} +(12.0386 - 24.1676i) q^{27} +44.8246i q^{29} -51.3786 q^{31} +(5.29614 + 0.822384i) q^{33} +24.5293i q^{35} -36.3424 q^{37} +(-10.3336 + 66.5481i) q^{39} -7.93212i q^{41} +26.4872 q^{43} +(6.10275 - 19.1770i) q^{45} +31.2242i q^{47} +71.3370 q^{49} +(-46.1830 - 7.17130i) q^{51} +90.5039i q^{53} +3.99481 q^{55} +(-12.1437 + 78.2054i) q^{57} +17.8064i q^{59} -11.6132 q^{61} +(-94.0794 - 29.9392i) q^{63} +50.1965i q^{65} +32.3116 q^{67} +(89.4859 + 13.8954i) q^{69} -34.1717i q^{71} -68.9145 q^{73} +(2.30162 - 14.8224i) q^{75} -19.5980i q^{77} +29.5267 q^{79} +(66.1026 + 46.8129i) q^{81} -76.1626i q^{83} -34.8353 q^{85} +(-132.881 - 20.6338i) q^{87} -112.955i q^{89} +246.257 q^{91} +(23.6508 - 152.311i) q^{93} +58.9894i q^{95} -79.5471 q^{97} +(-4.87587 + 15.3217i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{9}+O(q^{10}) \) 16 * q - 24 * q^9	\( 16 q - 24 q^{9} - 16 q^{13} + 88 q^{21} - 80 q^{25} - 64 q^{33} + 16 q^{37} + 40 q^{45} + 320 q^{49} - 272 q^{57} - 368 q^{61} + 104 q^{69} + 256 q^{73} + 192 q^{81} + 416 q^{93} - 544 q^{97}+O(q^{100}) \) 16 * q - 24 * q^9 - 16 * q^13 + 88 * q^21 - 80 * q^25 - 64 * q^33 + 16 * q^37 + 40 * q^45 + 320 * q^49 - 272 * q^57 - 368 * q^61 + 104 * q^69 + 256 * q^73 + 192 * q^81 + 416 * q^93 - 544 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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