LMFDB - Embedded newform 578.4.b.f.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [578,4,Mod(577,578)] 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("578.577"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 578 = 2 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	578.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,8,0,16,0,0,0,32,-58,0,0,0,90] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$34.1031039833$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$2.30278i$ of defining polynomial
Character	$\chi$	$=$	578.577
Dual form			578.4.b.f.577.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -8.90833i q^{3} +4.00000 q^{4} -1.30278i q^{5} -17.8167i q^{6} +32.9361i q^{7} +8.00000 q^{8} -52.3583 q^{9} -2.60555i q^{10} +39.2389i q^{11} -35.6333i q^{12} +67.5694 q^{13} +65.8722i q^{14} -11.6056 q^{15} +16.0000 q^{16} -104.717 q^{18} +26.0639 q^{19} -5.21110i q^{20} +293.405 q^{21} +78.4777i q^{22} -50.0917i q^{23} -71.2666i q^{24} +123.303 q^{25} +135.139 q^{26} +225.900i q^{27} +131.744i q^{28} +215.902i q^{29} -23.2111 q^{30} +93.8999i q^{31} +32.0000 q^{32} +349.553 q^{33} +42.9083 q^{35} -209.433 q^{36} +193.761i q^{37} +52.1278 q^{38} -601.930i q^{39} -10.4222i q^{40} -282.955i q^{41} +586.811 q^{42} -124.522 q^{43} +156.955i q^{44} +68.2111i q^{45} -100.183i q^{46} +202.817 q^{47} -142.533i q^{48} -741.786 q^{49} +246.606 q^{50} +270.278 q^{52} +569.258 q^{53} +451.800i q^{54} +51.1194 q^{55} +263.489i q^{56} -232.186i q^{57} +431.805i q^{58} -137.566 q^{59} -46.4222 q^{60} -3.34988i q^{61} +187.800i q^{62} -1724.48i q^{63} +64.0000 q^{64} -88.0278i q^{65} +699.105 q^{66} -764.560 q^{67} -446.233 q^{69} +85.8167 q^{70} +492.934i q^{71} -418.866 q^{72} -342.739i q^{73} +387.522i q^{74} -1098.42i q^{75} +104.256 q^{76} -1292.37 q^{77} -1203.86i q^{78} -645.643i q^{79} -20.8444i q^{80} +598.717 q^{81} -565.911i q^{82} +1274.67 q^{83} +1173.62 q^{84} -249.045 q^{86} +1923.33 q^{87} +313.911i q^{88} -830.700 q^{89} +136.422i q^{90} +2225.47i q^{91} -200.367i q^{92} +836.491 q^{93} +405.633 q^{94} -33.9554i q^{95} -285.066i q^{96} -956.748i q^{97} -1483.57 q^{98} -2054.48i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 58 q^{9} + 90 q^{13} - 32 q^{15} + 64 q^{16} - 116 q^{18} + 198 q^{19} + 640 q^{21} + 486 q^{25} + 180 q^{26} - 64 q^{30} + 128 q^{32} + 742 q^{33} + 150 q^{35} - 232 q^{36}+ \cdots - 2372 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 - 58 * q^9 + 90 * q^13 - 32 * q^15 + 64 * q^16 - 116 * q^18 + 198 * q^19 + 640 * q^21 + 486 * q^25 + 180 * q^26 - 64 * q^30 + 128 * q^32 + 742 * q^33 + 150 * q^35 - 232 * q^36 + 396 * q^38 + 1280 * q^42 - 700 * q^43 + 768 * q^47 - 1186 * q^49 + 972 * q^50 + 360 * q^52 + 1866 * q^53 + 154 * q^55 + 1440 * q^59 - 128 * q^60 + 256 * q^64 + 1484 * q^66 - 44 * q^67 - 660 * q^69 + 300 * q^70 - 464 * q^72 + 792 * q^76 - 2898 * q^77 + 2092 * q^81 - 14 * q^83 + 2560 * q^84 - 1400 * q^86 + 5018 * q^87 - 2544 * q^89 + 1810 * q^93 + 1536 * q^94 - 2372 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/578\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	− 8.90833i	− 1.71441i	−0.514977	−	0.857204i	$-0.672199\pi$
$3$	− 8.90833i	− 1.71441i	0.514977	−	0.857204i	$-0.327801\pi$
$4$	4.00000	0.500000
$5$	− 1.30278i	− 0.116524i	−0.998301	−	0.0582619i	$-0.981444\pi$
$5$	− 1.30278i	− 0.116524i	0.998301	−	0.0582619i	$-0.0185558\pi$
$6$	− 17.8167i	− 1.21227i
$7$	32.9361i	1.77838i	0.457537	+	0.889191i	$0.348732\pi$
$7$	32.9361i	1.77838i	−0.457537	+	0.889191i	$0.651268\pi$
$8$	8.00000	0.353553
$9$	−52.3583	−1.93920
$10$	− 2.60555i	− 0.0823948i
$11$	39.2389i	1.07554i	0.843091	+	0.537771i	$0.180733\pi$
$11$	39.2389i	1.07554i	−0.843091	+	0.537771i	$0.819267\pi$
$12$	− 35.6333i	− 0.857204i
$13$	67.5694	1.44157	0.720784	−	0.693160i	$-0.243782\pi$
$13$	67.5694	1.44157	0.720784	+	0.693160i	$0.243782\pi$
$14$	65.8722i	1.25751i
$15$	−11.6056	−0.199769
$16$	16.0000	0.250000
$17$	0	0
$17$	0	0
$18$	−104.717	−1.37122
$19$	26.0639	0.314709	0.157355	−	0.987542i	$-0.449703\pi$
$19$	26.0639	0.314709	0.157355	+	0.987542i	$0.449703\pi$
$20$	− 5.21110i	− 0.0582619i
$21$	293.405	3.04887
$22$	78.4777i	0.760523i
$23$	− 50.0917i	− 0.454123i	−0.973880	−	0.227062i	$-0.927088\pi$
$23$	− 50.0917i	− 0.454123i	0.973880	−	0.227062i	$-0.0729119\pi$
$24$	− 71.2666i	− 0.606135i
$25$	123.303	0.986422
$26$	135.139	1.01934
$27$	225.900i	1.61017i
$28$	131.744i	0.889191i
$29$	215.902i	1.38249i	0.722623	+	0.691243i	$0.242936\pi$
$29$	215.902i	1.38249i	−0.722623	+	0.691243i	$0.757064\pi$
$30$	−23.2111	−0.141258
$31$	93.8999i	0.544030i	0.962293	+	0.272015i	$0.0876900\pi$
$31$	93.8999i	0.544030i	−0.962293	+	0.272015i	$0.912310\pi$
$32$	32.0000	0.176777
$33$	349.553	1.84392
$34$	0	0
$35$	42.9083	0.207224
$36$	−209.433	−0.969598
$37$	193.761i	0.860923i	0.902609	+	0.430461i	$0.141649\pi$
$37$	193.761i	0.860923i	−0.902609	+	0.430461i	$0.858351\pi$
$38$	52.1278	0.222533
$39$	− 601.930i	− 2.47144i
$40$	− 10.4222i	− 0.0411974i
$41$	− 282.955i	− 1.07781i	−0.842367	−	0.538905i	$-0.818838\pi$
$41$	− 282.955i	− 1.07781i	0.842367	−	0.538905i	$-0.181162\pi$
$42$	586.811	2.15588
$43$	−124.522	−0.441616	−0.220808	−	0.975317i	$-0.570869\pi$
$43$	−124.522	−0.441616	−0.220808	+	0.975317i	$0.570869\pi$
$44$	156.955i	0.537771i
$45$	68.2111i	0.225962i
$46$	− 100.183i	− 0.321114i
$47$	202.817	0.629444	0.314722	−	0.949184i	$-0.398089\pi$
$47$	202.817	0.629444	0.314722	+	0.949184i	$0.398089\pi$
$48$	− 142.533i	− 0.428602i
$49$	−741.786	−2.16264
$50$	246.606	0.697506
$51$	0	0
$52$	270.278	0.720784
$53$	569.258	1.47535	0.737676	−	0.675155i	$-0.235923\pi$
$53$	569.258	1.47535	0.737676	+	0.675155i	$0.235923\pi$
$54$	451.800i	1.13856i
$55$	51.1194	0.125326
$56$	263.489i	0.628753i
$57$	− 232.186i	− 0.539540i
$58$	431.805i	0.977565i
$59$	−137.566	−0.303552	−0.151776	−	0.988415i	$-0.548499\pi$
$59$	−137.566	−0.303552	−0.151776	+	0.988415i	$0.548499\pi$
$60$	−46.4222	−0.0998847
$61$	− 3.34988i	− 0.00703129i	−0.999994	−	0.00351565i	$-0.998881\pi$
$61$	− 3.34988i	− 0.00703129i	0.999994	−	0.00351565i	$-0.00111907\pi$
$62$	187.800i	0.384687i
$63$	− 1724.48i	− 3.44863i
$64$	64.0000	0.125000
$65$	− 88.0278i	− 0.167977i
$66$	699.105	1.30385
$67$	−764.560	−1.39412	−0.697059	−	0.717014i	$-0.745508\pi$
$67$	−764.560	−1.39412	−0.697059	+	0.717014i	$0.745508\pi$
$68$	0	0
$69$	−446.233	−0.778553
$70$	85.8167	0.146529
$71$	492.934i	0.823950i	0.911195	+	0.411975i	$0.135161\pi$
$71$	492.934i	0.823950i	−0.911195	+	0.411975i	$0.864839\pi$
$72$	−418.866	−0.685609
$73$	− 342.739i	− 0.549515i	−0.961514	−	0.274757i	$-0.911402\pi$
$73$	− 342.739i	− 0.549515i	0.961514	−	0.274757i	$-0.0885976\pi$
$74$	387.522i	0.608764i
$75$	− 1098.42i	− 1.69113i
$76$	104.256	0.157355
$77$	−1292.37	−1.91272
$78$	− 1203.86i	− 1.74757i
$79$	− 645.643i	− 0.919501i	−0.888048	−	0.459750i	$-0.847939\pi$
$79$	− 645.643i	− 0.919501i	0.888048	−	0.459750i	$-0.152061\pi$
$80$	− 20.8444i	− 0.0291309i
$81$	598.717	0.821285
$82$	− 565.911i	− 0.762127i
$83$	1274.67	1.68570	0.842849	−	0.538150i	$-0.180876\pi$
$83$	1274.67	1.68570	0.842849	+	0.538150i	$0.180876\pi$
$84$	1173.62	1.52444
$85$	0	0
$86$	−249.045	−0.312269
$87$	1923.33	2.37014
$88$	313.911i	0.380261i
$89$	−830.700	−0.989371	−0.494685	−	0.869072i	$-0.664717\pi$
$89$	−830.700	−0.989371	−0.494685	+	0.869072i	$0.664717\pi$
$90$	136.422i	0.159780i
$91$	2225.47i	2.56366i
$92$	− 200.367i	− 0.227062i
$93$	836.491	0.932689
$94$	405.633	0.445084
$95$	− 33.9554i	− 0.0366711i
$96$	− 285.066i	− 0.303067i
$97$	− 956.748i	− 1.00147i	−0.865599	−	0.500737i	$-0.833062\pi$
$97$	− 956.748i	− 1.00147i	0.865599	−	0.500737i	$-0.166938\pi$
$98$	−1483.57	−1.52922
$99$	− 2054.48i	− 2.08569i
$100$	493.211	0.493211
$101$	1229.79	1.21158	0.605788	−	0.795626i	$-0.292858\pi$
$101$	1229.79	1.21158	0.605788	+	0.795626i	$0.292858\pi$
$102$	0	0
$103$	483.810	0.462827	0.231414	−	0.972855i	$-0.425665\pi$
$103$	483.810	0.462827	0.231414	+	0.972855i	$0.425665\pi$
$104$	540.555	0.509671
$105$	− 382.241i	− 0.355266i
$106$	1138.52	1.04323
$107$	− 729.239i	− 0.658862i	−0.944180	−	0.329431i	$-0.893143\pi$
$107$	− 729.239i	− 0.658862i	0.944180	−	0.329431i	$-0.106857\pi$
$108$	903.600i	0.805083i
$109$	2235.27i	1.96422i	0.188316	+	0.982108i	$0.439697\pi$
$109$	2235.27i	1.96422i	−0.188316	+	0.982108i	$0.560303\pi$
$110$	102.239	0.0886190
$111$	1726.09	1.47597
$112$	526.977i	0.444595i
$113$	241.664i	0.201185i	0.994928	+	0.100592i	$0.0320738\pi$
$113$	241.664i	0.201185i	−0.994928	+	0.100592i	$0.967926\pi$
$114$	− 464.372i	− 0.381512i
$115$	−65.2582	−0.0529162
$116$	863.610i	0.691243i
$117$	−3537.82	−2.79548
$118$	−275.132	−0.214644
$119$	0	0
$120$	−92.8444	−0.0706291
$121$	−208.688	−0.156790
$122$	− 6.69977i	− 0.00497187i
$123$	−2520.66	−1.84781
$124$	375.600i	0.272015i
$125$	− 323.483i	− 0.231465i
$126$	− 3448.95i	− 2.43855i
$127$	−951.589	−0.664881	−0.332440	−	0.943124i	$-0.607872\pi$
$127$	−951.589	−0.664881	−0.332440	+	0.943124i	$0.607872\pi$
$128$	128.000	0.0883883
$129$	1109.29i	0.757109i
$130$	− 176.056i	− 0.118778i
$131$	74.6493i	0.0497874i	0.999690	+	0.0248937i	$0.00792472\pi$
$131$	74.6493i	0.0497874i	−0.999690	+	0.0248937i	$0.992075\pi$
$132$	1398.21	0.921959
$133$	858.443i	0.559673i
$134$	−1529.12	−0.985790
$135$	294.297	0.187623
$136$	0	0
$137$	−544.661	−0.339661	−0.169830	−	0.985473i	$-0.554322\pi$
$137$	−544.661	−0.339661	−0.169830	+	0.985473i	$0.554322\pi$
$138$	−892.466	−0.550520
$139$	1385.49i	0.845437i	0.906261	+	0.422718i	$0.138924\pi$
$139$	1385.49i	0.845437i	−0.906261	+	0.422718i	$0.861076\pi$
$140$	171.633	0.103612
$141$	− 1806.76i	− 1.07912i
$142$	985.867i	0.582621i
$143$	2651.35i	1.55047i
$144$	−837.733	−0.484799
$145$	281.272	0.161092
$146$	− 685.478i	− 0.388566i
$147$	6608.07i	3.70765i
$148$	775.045i	0.430461i
$149$	481.628	0.264809	0.132404	−	0.991196i	$-0.457730\pi$
$149$	481.628	0.264809	0.132404	+	0.991196i	$0.457730\pi$
$150$	− 2196.84i	− 1.19581i
$151$	202.793	0.109292	0.0546460	−	0.998506i	$-0.482597\pi$
$151$	202.793	0.109292	0.0546460	+	0.998506i	$0.482597\pi$
$152$	208.511	0.111266
$153$	0	0
$154$	−2584.75	−1.35250
$155$	122.331	0.0633924
$156$	− 2407.72i	− 1.23572i
$157$	2186.07	1.11126	0.555628	−	0.831431i	$-0.312478\pi$
$157$	2186.07	1.11126	0.555628	+	0.831431i	$0.312478\pi$
$158$	− 1291.29i	− 0.650185i
$159$	− 5071.14i	− 2.52935i
$160$	− 41.6888i	− 0.0205987i
$161$	1649.82	0.807604
$162$	1197.43	0.580736
$163$	354.790i	0.170486i	0.996360	+	0.0852432i	$0.0271667\pi$
$163$	354.790i	0.170486i	−0.996360	+	0.0852432i	$0.972833\pi$
$164$	− 1131.82i	− 0.538905i
$165$	− 455.389i	− 0.214860i
$166$	2549.34	1.19197
$167$	1120.78i	0.519333i	0.965698	+	0.259666i	$0.0836126\pi$
$167$	1120.78i	0.519333i	−0.965698	+	0.259666i	$0.916387\pi$
$168$	2347.24	1.07794
$169$	2368.62	1.07812
$170$	0	0
$171$	−1364.66	−0.610283
$172$	−498.089	−0.220808
$173$	− 1731.68i	− 0.761025i	−0.924776	−	0.380512i	$-0.875748\pi$
$173$	− 1731.68i	− 0.761025i	0.924776	−	0.380512i	$-0.124252\pi$
$174$	3846.66	1.67595
$175$	4061.11i	1.75423i
$176$	627.822i	0.268885i
$177$	1225.48i	0.520412i
$178$	−1661.40	−0.699591
$179$	−4198.04	−1.75294	−0.876470	−	0.481457i	$-0.840108\pi$
$179$	−4198.04	−1.75294	−0.876470	+	0.481457i	$0.840108\pi$
$180$	272.844i	0.112981i
$181$	47.4660i	0.0194924i	0.999953	+	0.00974619i	$0.00310236\pi$
$181$	47.4660i	0.0194924i	−0.999953	+	0.00974619i	$0.996898\pi$
$182$	4450.94i	1.81278i
$183$	−29.8419	−0.0120545
$184$	− 400.733i	− 0.160557i
$185$	252.427	0.100318
$186$	1672.98	0.659511
$187$	0	0
$188$	811.267	0.314722
$189$	−7440.26	−2.86349
$190$	− 67.9109i	− 0.0259304i
$191$	1349.60	0.511276	0.255638	−	0.966773i	$-0.417715\pi$
$191$	1349.60	0.511276	0.255638	+	0.966773i	$0.417715\pi$
$192$	− 570.133i	− 0.214301i
$193$	2853.94i	1.06441i	0.846615	+	0.532205i	$0.178636\pi$
$193$	2853.94i	1.06441i	−0.846615	+	0.532205i	$0.821364\pi$
$194$	− 1913.50i	− 0.708149i
$195$	−784.180	−0.287981
$196$	−2967.14	−1.08132
$197$	− 4980.68i	− 1.80131i	−0.434530	−	0.900657i	$-0.643086\pi$
$197$	− 4980.68i	− 1.80131i	0.434530	−	0.900657i	$-0.356914\pi$
$198$	− 4108.96i	− 1.47480i
$199$	2954.72i	1.05254i	0.850319	+	0.526268i	$0.176409\pi$
$199$	2954.72i	1.05254i	−0.850319	+	0.526268i	$0.823591\pi$
$200$	986.422	0.348753
$201$	6810.95i	2.39009i
$202$	2459.59	0.856713
$203$	−7110.98	−2.45859
$204$	0	0
$205$	−368.627	−0.125591
$206$	967.620	0.327268
$207$	2622.71i	0.880634i
$208$	1081.11	0.360392
$209$	1022.72i	0.338483i
$210$	− 764.483i	− 0.251211i
$211$	2142.09i	0.698899i	0.936955	+	0.349449i	$0.113631\pi$
$211$	2142.09i	0.698899i	−0.936955	+	0.349449i	$0.886369\pi$
$212$	2277.03	0.737676
$213$	4391.21	1.41259
$214$	− 1458.48i	− 0.465885i
$215$	162.225i	0.0514587i
$216$	1807.20i	0.569279i
$217$	−3092.70	−0.967492
$218$	4470.53i	1.38891i
$219$	−3053.23	−0.942093
$220$	204.478	0.0626631
$221$	0	0
$222$	3452.18	1.04367
$223$	3170.14	0.951966	0.475983	−	0.879454i	$-0.342092\pi$
$223$	3170.14	0.951966	0.475983	+	0.879454i	$0.342092\pi$
$224$	1053.95i	0.314376i
$225$	−6455.92	−1.91287
$226$	483.329i	0.142259i
$227$	− 2228.79i	− 0.651674i	−0.945426	−	0.325837i	$-0.894354\pi$
$227$	− 2228.79i	− 0.651674i	0.945426	−	0.325837i	$-0.105646\pi$
$228$	− 928.744i	− 0.269770i
$229$	1977.95	0.570772	0.285386	−	0.958413i	$-0.407878\pi$
$229$	1977.95	0.570772	0.285386	+	0.958413i	$0.407878\pi$
$230$	−130.516	−0.0374174
$231$	11512.9i	3.27919i
$232$	1727.22i	0.488782i
$233$	− 4826.83i	− 1.35715i	−0.734530	−	0.678576i	$-0.762597\pi$
$233$	− 4826.83i	− 1.35715i	0.734530	−	0.678576i	$-0.237403\pi$
$234$	−7075.64	−1.97670
$235$	− 264.225i	− 0.0733452i
$236$	−550.264	−0.151776
$237$	−5751.60	−1.57640
$238$	0	0
$239$	149.720	0.0405212	0.0202606	−	0.999795i	$-0.493550\pi$
$239$	149.720	0.0405212	0.0202606	+	0.999795i	$0.493550\pi$
$240$	−185.689	−0.0499423
$241$	− 7345.74i	− 1.96340i	−0.190422	−	0.981702i	$-0.560986\pi$
$241$	− 7345.74i	− 1.96340i	0.190422	−	0.981702i	$-0.439014\pi$
$242$	−417.376	−0.110868
$243$	765.735i	0.202148i
$244$	− 13.3995i	− 0.00351565i
$245$	966.380i	0.251999i
$246$	−5041.32	−1.30660
$247$	1761.12	0.453674
$248$	751.199i	0.192344i
$249$	− 11355.2i	− 2.88998i
$250$	− 646.966i	− 0.163671i
$251$	1960.43	0.492992	0.246496	−	0.969144i	$-0.420721\pi$
$251$	1960.43	0.492992	0.246496	+	0.969144i	$0.420721\pi$
$252$	− 6897.91i	− 1.72431i
$253$	1965.54	0.488429
$254$	−1903.18	−0.470142
$255$	0	0
$256$	256.000	0.0625000
$257$	4022.21	0.976259	0.488129	−	0.872771i	$-0.337679\pi$
$257$	4022.21	0.976259	0.488129	+	0.872771i	$0.337679\pi$
$258$	2218.57i	0.535357i
$259$	−6381.73	−1.53105
$260$	− 352.111i	− 0.0839885i
$261$	− 11304.3i	− 2.68091i
$262$	149.299i	0.0352050i
$263$	−970.195	−0.227471	−0.113735	−	0.993511i	$-0.536282\pi$
$263$	−970.195	−0.227471	−0.113735	+	0.993511i	$0.536282\pi$
$264$	2796.42	0.651923
$265$	− 741.616i	− 0.171914i
$266$	1716.89i	0.395748i
$267$	7400.15i	1.69619i
$268$	−3058.24	−0.697059
$269$	− 1617.17i	− 0.366545i	−0.983062	−	0.183272i	$-0.941331\pi$
$269$	− 1617.17i	− 0.366545i	0.983062	−	0.183272i	$-0.0586690\pi$
$270$	588.594	0.132669
$271$	−4493.83	−1.00731	−0.503655	−	0.863905i	$-0.668012\pi$
$271$	−4493.83	−1.00731	−0.503655	+	0.863905i	$0.668012\pi$
$272$	0	0
$273$	19825.2	4.39515
$274$	−1089.32	−0.240176
$275$	4838.26i	1.06094i
$276$	−1784.93	−0.389276
$277$	− 2694.17i	− 0.584394i	−0.956358	−	0.292197i	$-0.905614\pi$
$277$	− 2694.17i	− 0.584394i	0.956358	−	0.292197i	$-0.0943863\pi$
$278$	2770.98i	0.597814i
$279$	− 4916.44i	− 1.05498i
$280$	343.267	0.0732647
$281$	−2980.66	−0.632779	−0.316390	−	0.948629i	$-0.602471\pi$
$281$	−2980.66	−0.632779	−0.316390	+	0.948629i	$0.602471\pi$
$282$	− 3613.51i	− 0.763055i
$283$	− 2863.55i	− 0.601486i	−0.953705	−	0.300743i	$-0.902765\pi$
$283$	− 2863.55i	− 0.601486i	0.953705	−	0.300743i	$-0.0972346\pi$
$284$	1971.73i	0.411975i
$285$	−302.486	−0.0628692
$286$	5302.69i	1.09635i
$287$	9319.44	1.91676
$288$	−1675.47	−0.342805
$289$	0	0
$290$	562.545	0.113910
$291$	−8523.02	−1.71694
$292$	− 1370.96i	− 0.274757i
$293$	−754.928	−0.150524	−0.0752618	−	0.997164i	$-0.523979\pi$
$293$	−754.928	−0.150524	−0.0752618	+	0.997164i	$0.523979\pi$
$294$	13216.1i	2.62170i
$295$	179.218i	0.0353711i
$296$	1550.09i	0.304382i
$297$	−8864.06	−1.73180
$298$	963.256	0.187248
$299$	− 3384.66i	− 0.654649i
$300$	− 4393.69i	− 0.845565i
$301$	− 4101.28i	− 0.785361i
$302$	405.586	0.0772810
$303$	− 10955.4i	− 2.07713i
$304$	417.023	0.0786773
$305$	−4.36415	−0.000819313 0
$306$	0	0
$307$	−7983.00	−1.48408	−0.742042	−	0.670354i	$-0.766142\pi$
$307$	−7983.00	−1.48408	−0.742042	+	0.670354i	$0.766142\pi$
$308$	−5169.50	−0.956362
$309$	− 4309.94i	− 0.793475i
$310$	244.661	0.0448252
$311$	3528.06i	0.643272i	0.946863	+	0.321636i	$0.104233\pi$
$311$	3528.06i	0.643272i	−0.946863	+	0.321636i	$0.895767\pi$
$312$	− 4815.44i	− 0.873784i
$313$	2801.14i	0.505846i	0.967486	+	0.252923i	$0.0813920\pi$
$313$	2801.14i	0.505846i	−0.967486	+	0.252923i	$0.918608\pi$
$314$	4372.13	0.785776
$315$	−2246.61	−0.401847
$316$	− 2582.57i	− 0.459750i
$317$	− 9021.98i	− 1.59850i	−0.600998	−	0.799251i	$-0.705230\pi$
$317$	− 9021.98i	− 1.59850i	0.600998	−	0.799251i	$-0.294770\pi$
$318$	− 10142.3i	− 1.78852i
$319$	−8471.77	−1.48692
$320$	− 83.3776i	− 0.0145655i
$321$	−6496.30	−1.12956
$322$	3299.65	0.571063
$323$	0	0
$324$	2394.87	0.410642
$325$	8331.49	1.42199
$326$	709.579i	0.120552i
$327$	19912.5	3.36747
$328$	− 2263.64i	− 0.381063i
$329$	6679.99i	1.11939i
$330$	− 910.777i	− 0.151929i
$331$	−10122.6	−1.68093	−0.840464	−	0.541867i	$-0.817718\pi$
$331$	−10122.6	−1.68093	−0.840464	+	0.541867i	$0.817718\pi$
$332$	5098.67	0.842849
$333$	− 10145.0i	− 1.66950i
$334$	2241.56i	0.367224i
$335$	996.050i	0.162448i
$336$	4694.49	0.762218
$337$	1939.05i	0.313433i	0.987644	+	0.156716i	$0.0500908\pi$
$337$	1939.05i	0.313433i	−0.987644	+	0.156716i	$0.949909\pi$
$338$	4737.25	0.762344
$339$	2152.83	0.344913
$340$	0	0
$341$	−3684.53	−0.585127
$342$	−2729.32	−0.431535
$343$	− 13134.4i	− 2.06762i
$344$	−996.178	−0.156135
$345$	581.341i	0.0907199i
$346$	− 3463.36i	− 0.538126i
$347$	− 6610.95i	− 1.02275i	−0.859358	−	0.511375i	$-0.829136\pi$
$347$	− 6610.95i	− 1.02275i	0.859358	−	0.511375i	$-0.170864\pi$
$348$	7693.32	1.18507
$349$	−4923.51	−0.755156	−0.377578	−	0.925978i	$-0.623243\pi$
$349$	−4923.51	−0.755156	−0.377578	+	0.925978i	$0.623243\pi$
$350$	8122.22i	1.24043i
$351$	15263.9i	2.32116i
$352$	1255.64i	0.190131i
$353$	2041.87	0.307869	0.153934	−	0.988081i	$-0.450806\pi$
$353$	2041.87	0.307869	0.153934	+	0.988081i	$0.450806\pi$
$354$	2450.97i	0.367987i
$355$	642.182	0.0960098
$356$	−3322.80	−0.494685
$357$	0	0
$358$	−8396.08	−1.23952
$359$	−505.112	−0.0742585	−0.0371293	−	0.999310i	$-0.511821\pi$
$359$	−505.112	−0.0742585	−0.0371293	+	0.999310i	$0.511821\pi$
$360$	545.689i	0.0798898i
$361$	−6179.67	−0.900958
$362$	94.9320i	0.0137832i
$363$	1859.06i	0.268803i
$364$	8901.88i	1.28183i
$365$	−446.512	−0.0640316
$366$	−59.6837	−0.00852382
$367$	9550.08i	1.35834i	0.733982	+	0.679169i	$0.237660\pi$
$367$	9550.08i	1.35834i	−0.733982	+	0.679169i	$0.762340\pi$
$368$	− 801.467i	− 0.113531i
$369$	14815.1i	2.09008i
$370$	504.855	0.0709355
$371$	18749.1i	2.62374i
$372$	3345.96	0.466345
$373$	1068.56	0.148332	0.0741662	−	0.997246i	$-0.476370\pi$
$373$	1068.56	0.148332	0.0741662	+	0.997246i	$0.476370\pi$
$374$	0	0
$375$	−2881.69	−0.396826
$376$	1622.53	0.222542
$377$	14588.4i	1.99295i
$378$	−14880.5	−2.02479
$379$	− 8247.52i	− 1.11780i	−0.829234	−	0.558901i	$-0.811223\pi$
$379$	− 8247.52i	− 1.11780i	0.829234	−	0.558901i	$-0.188777\pi$
$380$	− 135.822i	− 0.0183355i
$381$	8477.06i	1.13988i
$382$	2699.20	0.361526
$383$	−4557.54	−0.608040	−0.304020	−	0.952666i	$-0.598329\pi$
$383$	−4557.54	−0.608040	−0.304020	+	0.952666i	$0.598329\pi$
$384$	− 1140.27i	− 0.151534i
$385$	1683.67i	0.222878i
$386$	5707.88i	0.752652i
$387$	6519.77	0.856379
$388$	− 3826.99i	− 0.500737i
$389$	−8456.75	−1.10225	−0.551123	−	0.834424i	$-0.685801\pi$
$389$	−8456.75	−1.10225	−0.551123	+	0.834424i	$0.685801\pi$
$390$	−1568.36	−0.203633
$391$	0	0
$392$	−5934.28	−0.764609
$393$	665.001	0.0853559
$394$	− 9961.36i	− 1.27372i
$395$	−841.129	−0.107144
$396$	− 8217.92i	− 1.04284i
$397$	3414.24i	0.431627i	0.976435	+	0.215814i	$0.0692404\pi$
$397$	3414.24i	0.431627i	−0.976435	+	0.215814i	$0.930760\pi$
$398$	5909.44i	0.744255i
$399$	7647.29	0.959508
$400$	1972.84	0.246606
$401$	5788.46i	0.720852i	0.932788	+	0.360426i	$0.117369\pi$
$401$	5788.46i	0.720852i	−0.932788	+	0.360426i	$0.882631\pi$
$402$	13621.9i	1.69005i
$403$	6344.76i	0.784256i
$404$	4919.18	0.605788
$405$	− 779.993i	− 0.0956992i
$406$	−14222.0	−1.73848
$407$	−7602.97	−0.925958
$408$	0	0
$409$	−12554.4	−1.51779	−0.758897	−	0.651211i	$-0.774261\pi$
$409$	−12554.4	−1.51779	−0.758897	+	0.651211i	$0.774261\pi$
$410$	−737.255	−0.0888059
$411$	4852.02i	0.582317i
$412$	1935.24	0.231414
$413$	− 4530.89i	− 0.539832i
$414$	5245.43i	0.622702i
$415$	− 1660.61i	− 0.196424i
$416$	2162.22	0.254836
$417$	12342.4	1.44942
$418$	2045.44i	0.239343i
$419$	5105.66i	0.595293i	0.954676	+	0.297647i	$0.0962017\pi$
$419$	5105.66i	0.595293i	−0.954676	+	0.297647i	$0.903798\pi$
$420$	− 1528.97i	− 0.177633i
$421$	10075.1	1.16634	0.583170	−	0.812350i	$-0.301812\pi$
$421$	10075.1	1.16634	0.583170	+	0.812350i	$0.301812\pi$
$422$	4284.18i	0.494196i
$423$	−10619.1	−1.22061
$424$	4554.07	0.521616
$425$	0	0
$426$	8782.43	0.998850
$427$	110.332	0.0125043
$428$	− 2916.96i	− 0.329431i
$429$	23619.1	2.65813
$430$	324.449i	0.0363868i
$431$	13974.2i	1.56175i	0.624690	+	0.780873i	$0.285226\pi$
$431$	13974.2i	1.56175i	−0.624690	+	0.780873i	$0.714774\pi$
$432$	3614.40i	0.402541i
$433$	−1828.71	−0.202962	−0.101481	−	0.994838i	$-0.532358\pi$
$433$	−1828.71	−0.202962	−0.101481	+	0.994838i	$0.532358\pi$
$434$	−6185.39	−0.684120
$435$	− 2505.67i	− 0.276178i
$436$	8941.06i	0.982108i
$437$	− 1305.59i	− 0.142917i
$438$	−6106.47	−0.666160
$439$	− 9759.54i	− 1.06104i	−0.847672	−	0.530521i	$-0.821996\pi$
$439$	− 9759.54i	− 1.06104i	0.847672	−	0.530521i	$-0.178004\pi$
$440$	408.955	0.0443095
$441$	38838.6	4.19378
$442$	0	0
$443$	8806.00	0.944438	0.472219	−	0.881481i	$-0.343453\pi$
$443$	8806.00	0.944438	0.472219	+	0.881481i	$0.343453\pi$
$444$	6904.35	0.737986
$445$	1082.22i	0.115285i
$446$	6340.29	0.673142
$447$	− 4290.50i	− 0.453991i
$448$	2107.91i	0.222298i
$449$	− 17257.7i	− 1.81390i	−0.421235	−	0.906952i	$-0.638403\pi$
$449$	− 17257.7i	− 1.81390i	0.421235	−	0.906952i	$-0.361597\pi$
$450$	−12911.8	−1.35260
$451$	11102.8	1.15923
$452$	966.658i	0.100592i
$453$	− 1806.55i	− 0.187371i
$454$	− 4457.58i	− 0.460803i
$455$	2899.29	0.298727
$456$	− 1857.49i	− 0.190756i
$457$	8456.02	0.865549	0.432775	−	0.901502i	$-0.357535\pi$
$457$	8456.02	0.865549	0.432775	+	0.901502i	$0.357535\pi$
$458$	3955.90	0.403597
$459$	0	0
$460$	−261.033	−0.0264581
$461$	−13792.9	−1.39349	−0.696743	−	0.717321i	$-0.745368\pi$
$461$	−13792.9	−1.39349	−0.696743	+	0.717321i	$0.745368\pi$
$462$	23025.8i	2.31874i
$463$	−12478.0	−1.25249	−0.626243	−	0.779628i	$-0.715408\pi$
$463$	−12478.0	−1.25249	−0.626243	+	0.779628i	$0.715408\pi$
$464$	3454.44i	0.345621i
$465$	− 1089.76i	− 0.108680i
$466$	− 9653.67i	− 0.959651i
$467$	16066.1	1.59197	0.795985	−	0.605316i	$-0.206953\pi$
$467$	16066.1	1.59197	0.795985	+	0.605316i	$0.206953\pi$
$468$	−14151.3	−1.39774
$469$	− 25181.6i	− 2.47927i
$470$	− 528.449i	− 0.0518629i
$471$	− 19474.2i	− 1.90515i
$472$	−1100.53	−0.107322
$473$	− 4886.11i	− 0.474976i
$474$	−11503.2	−1.11468
$475$	3213.75	0.310436
$476$	0	0
$477$	−29805.4	−2.86100
$478$	299.439	0.0286528
$479$	− 13352.5i	− 1.27368i	−0.770997	−	0.636838i	$-0.780242\pi$
$479$	− 13352.5i	− 1.27368i	0.770997	−	0.636838i	$-0.219758\pi$
$480$	−371.378	−0.0353146
$481$	13092.3i	1.24108i
$482$	− 14691.5i	− 1.38834i
$483$	− 14697.2i	− 1.38456i
$484$	−834.752	−0.0783952
$485$	−1246.43	−0.116696
$486$	1531.47i	0.142940i
$487$	11085.4i	1.03147i	0.856748	+	0.515736i	$0.172481\pi$
$487$	11085.4i	1.03147i	−0.856748	+	0.515736i	$0.827519\pi$
$488$	− 26.7991i	− 0.00248594i
$489$	3160.58	0.292283
$490$	1932.76i	0.178190i
$491$	13437.6	1.23510	0.617548	−	0.786533i	$-0.288126\pi$
$491$	13437.6	1.23510	0.617548	+	0.786533i	$0.288126\pi$
$492$	−10082.6	−0.923903
$493$	0	0
$494$	3522.25	0.320796
$495$	−2676.53	−0.243032
$496$	1502.40i	0.136007i
$497$	−16235.3	−1.46530
$498$	− 22710.3i	− 2.04352i
$499$	13165.2i	1.18107i	0.807012	+	0.590535i	$0.201083\pi$
$499$	13165.2i	1.18107i	−0.807012	+	0.590535i	$0.798917\pi$
$500$	− 1293.93i	− 0.115733i
$501$	9984.28	0.890349
$502$	3920.86	0.348598
$503$	− 7873.67i	− 0.697951i	−0.937132	−	0.348976i	$-0.886530\pi$
$503$	− 7873.67i	− 0.697951i	0.937132	−	0.348976i	$-0.113470\pi$
$504$	− 13795.8i	− 1.21927i
$505$	− 1602.15i	− 0.141177i
$506$	3931.08	0.345371
$507$	− 21100.5i	− 1.84833i
$508$	−3806.35	−0.332440
$509$	8039.79	0.700113	0.350056	−	0.936729i	$-0.386162\pi$
$509$	8039.79	0.700113	0.350056	+	0.936729i	$0.386162\pi$
$510$	0	0
$511$	11288.5	0.977247
$512$	512.000	0.0441942
$513$	5887.84i	0.506734i
$514$	8044.42	0.690319
$515$	− 630.296i	− 0.0539304i
$516$	4437.14i	0.378555i
$517$	7958.29i	0.676993i
$518$	−12763.5	−1.08261
$519$	−15426.4	−1.30471
$520$	− 704.222i	− 0.0593888i
$521$	− 11806.6i	− 0.992813i	−0.868090	−	0.496407i	$-0.834653\pi$
$521$	− 11806.6i	− 0.992813i	0.868090	−	0.496407i	$-0.165347\pi$
$522$	− 22608.6i	− 1.89569i
$523$	−2991.18	−0.250086	−0.125043	−	0.992151i	$-0.539907\pi$
$523$	−2991.18	−0.250086	−0.125043	+	0.992151i	$0.539907\pi$
$524$	298.597i	0.0248937i
$525$	36177.7	3.00747
$526$	−1940.39	−0.160846
$527$	0	0
$528$	5592.84	0.460980
$529$	9657.82	0.793772
$530$	− 1483.23i	− 0.121561i
$531$	7202.72	0.588647
$532$	3433.77i	0.279836i
$533$	− 19119.1i	− 1.55374i
$534$	14800.3i	1.19938i
$535$	−950.035	−0.0767730
$536$	−6116.48	−0.492895
$537$	37397.5i	3.00525i
$538$	− 3234.34i	− 0.259186i
$539$	− 29106.8i	− 2.32601i
$540$	1177.19	0.0938113
$541$	− 17360.1i	− 1.37961i	−0.723995	−	0.689805i	$-0.757696\pi$
$541$	− 17360.1i	− 1.37961i	0.723995	−	0.689805i	$-0.242304\pi$
$542$	−8987.66	−0.712275
$543$	422.843	0.0334179
$544$	0	0
$545$	2912.05	0.228878
$546$	39650.4	3.10784
$547$	3124.54i	0.244234i	0.992516	+	0.122117i	$0.0389683\pi$
$547$	3124.54i	0.244234i	−0.992516	+	0.122117i	$0.961032\pi$
$548$	−2178.64	−0.169830
$549$	175.394i	0.0136351i
$550$	9676.52i	0.750197i
$551$	5627.26i	0.435081i
$552$	−3569.86	−0.275260
$553$	21265.0	1.63522
$554$	− 5388.35i	− 0.413229i
$555$	− 2248.70i	− 0.171986i
$556$	5541.96i	0.422718i
$557$	−7434.65	−0.565559	−0.282779	−	0.959185i	$-0.591256\pi$
$557$	−7434.65	−0.565559	−0.282779	+	0.959185i	$0.591256\pi$
$558$	− 9832.88i	− 0.745984i
$559$	−8413.89	−0.636619
$560$	686.533	0.0518059
$561$	0	0
$562$	−5961.31	−0.447443
$563$	22605.6	1.69221	0.846105	−	0.533017i	$-0.178942\pi$
$563$	22605.6	1.69221	0.846105	+	0.533017i	$0.178942\pi$
$564$	− 7227.03i	− 0.539562i
$565$	314.834	0.0234428
$566$	− 5727.11i	− 0.425315i
$567$	19719.4i	1.46056i
$568$	3943.47i	0.291310i
$569$	22968.0	1.69222	0.846108	−	0.533012i	$-0.178940\pi$
$569$	22968.0	1.69222	0.846108	+	0.533012i	$0.178940\pi$
$570$	−604.972	−0.0444553
$571$	− 20889.0i	− 1.53096i	−0.643461	−	0.765479i	$-0.722502\pi$
$571$	− 20889.0i	− 1.53096i	0.643461	−	0.765479i	$-0.277498\pi$
$572$	10605.4i	0.775233i
$573$	− 12022.7i	− 0.876535i
$574$	18638.9	1.35535
$575$	− 6176.44i	− 0.447957i
$576$	−3350.93	−0.242399
$577$	−14649.6	−1.05697	−0.528485	−	0.848942i	$-0.677240\pi$
$577$	−14649.6	−1.05697	−0.528485	+	0.848942i	$0.677240\pi$
$578$	0	0
$579$	25423.8	1.82483
$580$	1125.09	0.0805462
$581$	41982.6i	2.99781i
$582$	−17046.0	−1.21406
$583$	22337.0i	1.58680i
$584$	− 2741.91i	− 0.194283i
$585$	4608.98i	0.325740i
$586$	−1509.86	−0.106436
$587$	−13888.4	−0.976548	−0.488274	−	0.872690i	$-0.662373\pi$
$587$	−13888.4	−0.976548	−0.488274	+	0.872690i	$0.662373\pi$
$588$	26432.3i	1.85382i
$589$	2447.40i	0.171211i
$590$	358.435i	0.0250111i
$591$	−44369.5	−3.08819
$592$	3100.18i	0.215231i
$593$	−14653.3	−1.01474	−0.507368	−	0.861729i	$-0.669382\pi$
$593$	−14653.3	−1.01474	−0.507368	+	0.861729i	$0.669382\pi$
$594$	−17728.1	−1.22457
$595$	0	0
$596$	1926.51	0.132404
$597$	26321.6	1.80448
$598$	− 6769.33i	− 0.462907i
$599$	−7698.71	−0.525143	−0.262572	−	0.964913i	$-0.584571\pi$
$599$	−7698.71	−0.525143	−0.262572	+	0.964913i	$0.584571\pi$
$600$	− 8787.37i	− 0.597905i
$601$	19478.7i	1.32205i	0.750363	+	0.661026i	$0.229879\pi$
$601$	19478.7i	1.32205i	−0.750363	+	0.661026i	$0.770121\pi$
$602$	− 8202.55i	− 0.555334i
$603$	40031.1	2.70347
$604$	811.173	0.0546460
$605$	271.874i	0.0182698i
$606$	− 21910.8i	− 1.46876i
$607$	3136.14i	0.209707i	0.994488	+	0.104853i	$0.0334373\pi$
$607$	3136.14i	0.209707i	−0.994488	+	0.104853i	$0.966563\pi$
$608$	834.045	0.0556332
$609$	63346.9i	4.21502i
$610$	−8.72830	−0.000579342 0
$611$	13704.2	0.907385
$612$	0	0
$613$	2865.35	0.188793	0.0943967	−	0.995535i	$-0.469908\pi$
$613$	2865.35	0.188793	0.0943967	+	0.995535i	$0.469908\pi$
$614$	−15966.0	−1.04941
$615$	3283.85i	0.215313i
$616$	−10339.0	−0.676250
$617$	− 15952.4i	− 1.04087i	−0.853900	−	0.520437i	$-0.825769\pi$
$617$	− 15952.4i	− 1.04087i	0.853900	−	0.520437i	$-0.174231\pi$
$618$	− 8619.88i	− 0.561072i
$619$	− 14278.9i	− 0.927171i	−0.886052	−	0.463585i	$-0.846563\pi$
$619$	− 14278.9i	− 0.927171i	0.886052	−	0.463585i	$-0.153437\pi$
$620$	489.322	0.0316962
$621$	11315.7	0.731214
$622$	7056.11i	0.454862i
$623$	− 27360.0i	− 1.75948i
$624$	− 9630.88i	− 0.617859i
$625$	14991.4	0.959451
$626$	5602.28i	0.357687i
$627$	9110.71	0.580298
$628$	8744.27	0.555628
$629$	0	0
$630$	−4493.21	−0.284149
$631$	−22216.4	−1.40162	−0.700808	−	0.713349i	$-0.747177\pi$
$631$	−22216.4	−1.40162	−0.700808	+	0.713349i	$0.747177\pi$
$632$	− 5165.15i	− 0.325093i
$633$	19082.5	1.19820
$634$	− 18044.0i	− 1.13031i
$635$	1239.71i	0.0774744i
$636$	− 20284.6i	− 1.26468i
$637$	−50122.0	−3.11759
$638$	−16943.5	−1.05141
$639$	− 25809.2i	− 1.59780i
$640$	− 166.755i	− 0.0102993i
$641$	22674.5i	1.39718i	0.715523	+	0.698589i	$0.246188\pi$
$641$	22674.5i	1.39718i	−0.715523	+	0.698589i	$0.753812\pi$
$642$	−12992.6	−0.798718
$643$	− 22509.4i	− 1.38053i	−0.723555	−	0.690267i	$-0.757493\pi$
$643$	− 22509.4i	− 1.38053i	0.723555	−	0.690267i	$-0.242507\pi$
$644$	6599.29	0.403802
$645$	1445.15	0.0882213
$646$	0	0
$647$	−75.5368	−0.00458989	−0.00229494	−	0.999997i	$-0.500731\pi$
$647$	−75.5368	−0.00458989	−0.00229494	+	0.999997i	$0.500731\pi$
$648$	4789.73	0.290368
$649$	− 5397.94i	− 0.326483i
$650$	16663.0	1.00550
$651$	27550.7i	1.65868i
$652$	1419.16i	0.0852432i
$653$	− 1134.04i	− 0.0679610i	−0.999422	−	0.0339805i	$-0.989182\pi$
$653$	− 1134.04i	− 0.0679610i	0.999422	−	0.0339805i	$-0.0108184\pi$
$654$	39825.0	2.38116
$655$	97.2513	0.00580141
$656$	− 4527.29i	− 0.269453i
$657$	17945.2i	1.06562i
$658$	13360.0i	0.791529i
$659$	−33374.8	−1.97283	−0.986416	−	0.164264i	$-0.947475\pi$
$659$	−33374.8	−1.97283	−0.986416	+	0.164264i	$0.947475\pi$
$660$	− 1821.55i	− 0.107430i
$661$	4598.54	0.270594	0.135297	−	0.990805i	$-0.456801\pi$
$661$	4598.54	0.270594	0.135297	+	0.990805i	$0.456801\pi$
$662$	−20245.2	−1.18860
$663$	0	0
$664$	10197.3	0.595984
$665$	1118.36	0.0652152
$666$	− 20290.0i	− 1.18051i
$667$	10814.9	0.627819
$668$	4483.12i	0.259666i
$669$	− 28240.7i	− 1.63206i
$670$	1992.10i	0.114868i
$671$	131.446	0.00756245
$672$	9388.97	0.538969
$673$	− 2426.84i	− 0.139001i	−0.997582	−	0.0695007i	$-0.977859\pi$
$673$	− 2426.84i	− 0.139001i	0.997582	−	0.0695007i	$-0.0221406\pi$
$674$	3878.10i	0.221630i
$675$	27854.1i	1.58830i
$676$	9474.49	0.539058
$677$	− 24830.9i	− 1.40964i	−0.709385	−	0.704821i	$-0.751027\pi$
$677$	− 24830.9i	− 1.40964i	0.709385	−	0.704821i	$-0.248973\pi$
$678$	4305.65	0.243890
$679$	31511.5	1.78100
$680$	0	0
$681$	−19854.8	−1.11724
$682$	−7369.05	−0.413747
$683$	− 6611.10i	− 0.370376i	−0.982703	−	0.185188i	$-0.940711\pi$
$683$	− 6611.10i	− 0.370376i	0.982703	−	0.185188i	$-0.0592894\pi$
$684$	−5458.65	−0.305141
$685$	709.571i	0.0395786i
$686$	− 26268.9i	− 1.46203i
$687$	− 17620.2i	− 0.978536i
$688$	−1992.36	−0.110404
$689$	38464.4	2.12682
$690$	1162.68i	0.0641487i
$691$	1055.68i	0.0581186i	0.999578	+	0.0290593i	$0.00925116\pi$
$691$	1055.68i	0.0581186i	−0.999578	+	0.0290593i	$0.990749\pi$
$692$	− 6926.73i	− 0.380512i
$693$	67666.5	3.70915
$694$	− 13221.9i	− 0.723193i
$695$	1804.98	0.0985135
$696$	15386.6	0.837973
$697$	0	0
$698$	−9847.01	−0.533976
$699$	−42999.0	−2.32671
$700$	16244.4i	0.877117i
$701$	13811.1	0.744137	0.372068	−	0.928205i	$-0.378649\pi$
$701$	13811.1	0.744137	0.372068	+	0.928205i	$0.378649\pi$
$702$	30527.8i	1.64131i
$703$	5050.17i	0.270940i
$704$	2511.29i	0.134443i
$705$	−2353.80	−0.125744
$706$	4083.74	0.217696
$707$	40504.6i	2.15464i
$708$	4901.93i	0.260206i
$709$	− 15046.7i	− 0.797025i	−0.917163	−	0.398512i	$-0.869527\pi$
$709$	− 15046.7i	− 0.797025i	0.917163	−	0.398512i	$-0.130473\pi$
$710$	1284.36	0.0678892
$711$	33804.8i	1.78309i
$712$	−6645.60	−0.349795
$713$	4703.60	0.247057
$714$	0	0
$715$	3454.11	0.180666
$716$	−16792.2	−0.876470
$717$	− 1333.75i	− 0.0694698i
$718$	−1010.22	−0.0525087
$719$	− 36534.1i	− 1.89498i	−0.319779	−	0.947492i	$-0.603609\pi$
$719$	− 36534.1i	− 1.89498i	0.319779	−	0.947492i	$-0.396391\pi$
$720$	1091.38i	0.0564906i
$721$	15934.8i	0.823084i
$722$	−12359.3	−0.637074
$723$	−65438.2	−3.36608
$724$	189.864i	0.00974619i
$725$	26621.4i	1.36371i
$726$	3718.12i	0.190072i
$727$	18950.4	0.966756	0.483378	−	0.875412i	$-0.339410\pi$
$727$	18950.4	0.966756	0.483378	+	0.875412i	$0.339410\pi$
$728$	17803.8i	0.906390i
$729$	22986.8	1.16785
$730$	−893.025	−0.0452772
$731$	0	0
$732$	−119.367	−0.00602725
$733$	−35398.9	−1.78375	−0.891874	−	0.452283i	$-0.850610\pi$
$733$	−35398.9	−1.78375	−0.891874	+	0.452283i	$0.850610\pi$
$734$	19100.2i	0.960491i
$735$	8608.83	0.432029
$736$	− 1602.93i	− 0.0802784i
$737$	− 30000.5i	− 1.49943i
$738$	29630.1i	1.47791i
$739$	−19350.5	−0.963220	−0.481610	−	0.876386i	$-0.659948\pi$
$739$	−19350.5	−0.963220	−0.481610	+	0.876386i	$0.659948\pi$
$740$	1009.71	0.0501590
$741$	− 15688.7i	− 0.777783i
$742$	37498.3i	1.85526i
$743$	17426.3i	0.860443i	0.902723	+	0.430221i	$0.141564\pi$
$743$	17426.3i	0.860443i	−0.902723	+	0.430221i	$0.858436\pi$
$744$	6691.93	0.329755
$745$	− 627.454i	− 0.0308565i
$746$	2137.12	0.104887
$747$	−66739.4	−3.26890
$748$	0	0
$749$	24018.3	1.17171
$750$	−5763.38	−0.280599
$751$	− 11096.1i	− 0.539152i	−0.962979	−	0.269576i	$-0.913116\pi$
$751$	− 11096.1i	− 0.539152i	0.962979	−	0.269576i	$-0.0868835\pi$
$752$	3245.07	0.157361
$753$	− 17464.1i	− 0.845190i
$754$	29176.8i	1.40923i
$755$	− 264.194i	− 0.0127351i
$756$	−29761.0	−1.43174
$757$	−13378.2	−0.642326	−0.321163	−	0.947024i	$-0.604074\pi$
$757$	−13378.2	−0.642326	−0.321163	+	0.947024i	$0.604074\pi$
$758$	− 16495.0i	− 0.790405i
$759$	− 17509.7i	− 0.837366i
$760$	− 271.643i	− 0.0129652i
$761$	−6331.90	−0.301618	−0.150809	−	0.988563i	$-0.548188\pi$
$761$	−6331.90	−0.301618	−0.150809	+	0.988563i	$0.548188\pi$
$762$	16954.1i	0.806015i
$763$	−73620.9	−3.49313
$764$	5398.40	0.255638
$765$	0	0
$766$	−9115.08	−0.429949
$767$	−9295.26	−0.437591
$768$	− 2280.53i	− 0.107151i
$769$	−14799.5	−0.693996	−0.346998	−	0.937866i	$-0.612799\pi$
$769$	−14799.5	−0.693996	−0.346998	+	0.937866i	$0.612799\pi$
$770$	3367.35i	0.157598i
$771$	− 35831.2i	− 1.67371i
$772$	11415.8i	0.532205i
$773$	−9139.21	−0.425245	−0.212623	−	0.977134i	$-0.568200\pi$
$773$	−9139.21	−0.425245	−0.212623	+	0.977134i	$0.568200\pi$
$774$	13039.5	0.605551
$775$	11578.1i	0.536643i
$776$	− 7653.98i	− 0.354075i
$777$	56850.6i	2.62484i
$778$	−16913.5	−0.779406
$779$	− 7374.93i	− 0.339197i
$780$	−3136.72	−0.143991
$781$	−19342.1	−0.886193
$782$	0	0
$783$	−48772.4	−2.22603
$784$	−11868.6	−0.540660
$785$	− 2847.95i	− 0.129488i
$786$	1330.00	0.0603557
$787$	14159.4i	0.641332i	0.947192	+	0.320666i	$0.103907\pi$
$787$	14159.4i	0.641332i	−0.947192	+	0.320666i	$0.896093\pi$
$788$	− 19922.7i	− 0.900657i
$789$	8642.82i	0.389978i
$790$	−1682.26	−0.0757621
$791$	−7959.48	−0.357783
$792$	− 16435.8i	− 0.737402i
$793$	− 226.350i	− 0.0101361i
$794$	6828.49i	0.305207i
$795$	−6606.56	−0.294730
$796$	11818.9i	0.526268i
$797$	35152.0	1.56229	0.781146	−	0.624349i	$-0.214636\pi$
$797$	35152.0	1.56229	0.781146	+	0.624349i	$0.214636\pi$
$798$	15294.6	0.678474
$799$	0	0
$800$	3945.69	0.174376
$801$	43494.0	1.91858
$802$	11576.9i	0.509720i
$803$	13448.7	0.591026
$804$	27243.8i	1.19504i
$805$	− 2149.35i	− 0.0941051i
$806$	12689.5i	0.554553i
$807$	−14406.3	−0.628408
$808$	9838.35	0.428356
$809$	16641.9i	0.723234i	0.932327	+	0.361617i	$0.117775\pi$
$809$	16641.9i	0.723234i	−0.932327	+	0.361617i	$0.882225\pi$
$810$	− 1559.99i	− 0.0676696i
$811$	− 5699.29i	− 0.246768i	−0.992359	−	0.123384i	$-0.960625\pi$
$811$	− 5699.29i	− 0.246768i	0.992359	−	0.123384i	$-0.0393747\pi$
$812$	−28443.9	−1.22929
$813$	40032.5i	1.72694i
$814$	−15205.9	−0.654751
$815$	462.211	0.0198657
$816$	0	0
$817$	−3245.54	−0.138980
$818$	−25108.9	−1.07324
$819$	− 116522.i	− 4.97143i
$820$	−1474.51	−0.0627953
$821$	9843.36i	0.418436i	0.977869	+	0.209218i	$0.0670918\pi$
$821$	9843.36i	0.418436i	−0.977869	+	0.209218i	$0.932908\pi$
$822$	9704.04i	0.411761i
$823$	25466.1i	1.07860i	0.842112	+	0.539302i	$0.181312\pi$
$823$	25466.1i	1.07860i	−0.842112	+	0.539302i	$0.818688\pi$
$824$	3870.48	0.163634
$825$	43100.8	1.81888
$826$	− 9061.78i	− 0.381719i
$827$	20037.3i	0.842521i	0.906940	+	0.421260i	$0.138412\pi$
$827$	20037.3i	0.842521i	−0.906940	+	0.421260i	$0.861588\pi$
$828$	10490.9i	0.440317i
$829$	−21911.2	−0.917982	−0.458991	−	0.888441i	$-0.651789\pi$
$829$	−21911.2	−0.917982	−0.458991	+	0.888441i	$0.651789\pi$
$830$	− 3321.21i	− 0.138893i
$831$	−24000.6	−1.00189
$832$	4324.44	0.180196
$833$	0	0
$834$	24684.8	1.02490
$835$	1460.13	0.0605146
$836$	4090.87i	0.169241i
$837$	−21212.0	−0.875978
$838$	10211.3i	0.420936i
$839$	39532.6i	1.62672i	0.581761	+	0.813360i	$0.302364\pi$
$839$	39532.6i	1.62672i	−0.581761	+	0.813360i	$0.697636\pi$
$840$	− 3057.93i	− 0.125606i
$841$	−22224.9	−0.911266
$842$	20150.1	0.824726
$843$	26552.7i	1.08484i
$844$	8568.37i	0.349449i
$845$	− 3085.78i	− 0.125626i
$846$	−21238.3	−0.863105
$847$	− 6873.37i	− 0.278833i
$848$	9108.13	0.368838
$849$	−25509.5	−1.03119
$850$	0	0
$851$	9705.82	0.390965
$852$	17564.9	0.706293
$853$	− 32412.3i	− 1.30103i	−0.759495	−	0.650513i	$-0.774554\pi$
$853$	− 32412.3i	− 1.30103i	0.759495	−	0.650513i	$-0.225446\pi$
$854$	220.664	0.00884189
$855$	1777.85i	0.0711124i
$856$	− 5833.91i	− 0.232943i
$857$	− 41373.5i	− 1.64912i	−0.565777	−	0.824559i	$-0.691424\pi$
$857$	− 41373.5i	− 1.64912i	0.565777	−	0.824559i	$-0.308576\pi$
$858$	47238.1	1.87958
$859$	19281.0	0.765843	0.382922	−	0.923781i	$-0.374918\pi$
$859$	19281.0	0.765843	0.382922	+	0.923781i	$0.374918\pi$
$860$	648.898i	0.0257294i
$861$	− 83020.7i	− 3.28610i
$862$	27948.4i	1.10432i
$863$	36539.7	1.44128	0.720640	−	0.693309i	$-0.243848\pi$
$863$	36539.7	1.44128	0.720640	+	0.693309i	$0.243848\pi$
$864$	7228.80i	0.284640i
$865$	−2255.99	−0.0886775
$866$	−3657.43	−0.143515
$867$	0	0
$868$	−12370.8	−0.483746
$869$	25334.3	0.988962
$870$	− 5011.33i	− 0.195288i
$871$	−51660.9	−2.00971
$872$	17882.1i	0.694456i
$873$	50093.7i	1.94205i
$874$	− 2611.17i	− 0.101057i
$875$	10654.3	0.411634
$876$	−12212.9	−0.471047
$877$	− 42217.1i	− 1.62551i	−0.582608	−	0.812753i	$-0.697968\pi$
$877$	− 42217.1i	− 1.62551i	0.582608	−	0.812753i	$-0.302032\pi$
$878$	− 19519.1i	− 0.750270i
$879$	6725.15i	0.258059i
$880$	817.911	0.0313316
$881$	− 18566.8i	− 0.710023i	−0.934862	−	0.355012i	$-0.884477\pi$
$881$	− 18566.8i	− 0.710023i	0.934862	−	0.355012i	$-0.115523\pi$
$882$	77677.2	2.96545
$883$	1627.71	0.0620347	0.0310174	−	0.999519i	$-0.490125\pi$
$883$	1627.71	0.0620347	0.0310174	+	0.999519i	$0.490125\pi$
$884$	0	0
$885$	1596.53	0.0606404
$886$	17612.0	0.667818
$887$	32502.7i	1.23037i	0.788384	+	0.615183i	$0.210918\pi$
$887$	32502.7i	1.23037i	−0.788384	+	0.615183i	$0.789082\pi$
$888$	13808.7	0.521835
$889$	− 31341.6i	− 1.18241i
$890$	2164.43i	0.0815190i
$891$	23493.0i	0.883326i
$892$	12680.6	0.475983
$893$	5286.20	0.198092
$894$	− 8581.00i	− 0.321020i
$895$	5469.10i	0.204259i
$896$	4215.82i	0.157188i
$897$	−30151.7	−1.12234
$898$	− 34515.5i	− 1.28262i
$899$	−20273.2	−0.752113
$900$	−25823.7	−0.956433
$901$	0	0
$902$	22205.7	0.819699
$903$	−36535.5	−1.34643
$904$	1933.32i	0.0711295i
$905$	61.8375	0.00227133
$906$	− 3613.10i	− 0.132491i
$907$	9352.99i	0.342404i	0.985236	+	0.171202i	$0.0547651\pi$
$907$	9352.99i	0.342404i	−0.985236	+	0.171202i	$0.945235\pi$
$908$	− 8915.16i	− 0.325837i
$909$	−64389.9	−2.34948
$910$	5798.58	0.211232
$911$	− 20763.9i	− 0.755148i	−0.925979	−	0.377574i	$-0.876758\pi$
$911$	− 20763.9i	− 0.755148i	0.925979	−	0.377574i	$-0.123242\pi$
$912$	− 3714.97i	− 0.134885i
$913$	50016.5i	1.81304i
$914$	16912.0	0.612036
$915$	38.8773i	0.00140464i
$916$	7911.81	0.285386
$917$	−2458.66	−0.0885409
$918$	0	0
$919$	7827.21	0.280953	0.140477	−	0.990084i	$-0.455137\pi$
$919$	7827.21	0.280953	0.140477	+	0.990084i	$0.455137\pi$
$920$	−522.066	−0.0187087
$921$	71115.1i	2.54433i
$922$	−27585.7	−0.985343
$923$	33307.2i	1.18778i
$924$	46051.6i	1.63959i
$925$	23891.3i	0.849233i
$926$	−24956.0	−0.885641
$927$	−25331.5	−0.897513
$928$	6908.88i	0.244391i
$929$	− 36622.2i	− 1.29336i	−0.762759	−	0.646682i	$-0.776156\pi$
$929$	− 36622.2i	− 1.29336i	0.762759	−	0.646682i	$-0.223844\pi$
$930$	− 2179.52i	− 0.0768487i
$931$	−19333.8	−0.680603
$932$	− 19307.3i	− 0.678576i
$933$	31429.1	1.10283
$934$	32132.2	1.12569
$935$	0	0
$936$	−28302.5	−0.988352
$937$	17041.0	0.594136	0.297068	−	0.954856i	$-0.403991\pi$
$937$	17041.0	0.594136	0.297068	+	0.954856i	$0.403991\pi$
$938$	− 50363.2i	− 1.75311i
$939$	24953.5	0.867227
$940$	− 1056.90i	− 0.0366726i
$941$	− 41668.5i	− 1.44352i	−0.692143	−	0.721760i	$-0.743333\pi$
$941$	− 41668.5i	− 1.44352i	0.692143	−	0.721760i	$-0.256667\pi$
$942$	− 38948.4i	− 1.34714i
$943$	−14173.7	−0.489459
$944$	−2201.06	−0.0758880
$945$	9692.99i	0.333664i
$946$	− 9772.22i	− 0.335859i
$947$	− 52253.7i	− 1.79305i	−0.442995	−	0.896524i	$-0.646084\pi$
$947$	− 52253.7i	− 1.79305i	0.442995	−	0.896524i	$-0.353916\pi$
$948$	−23006.4	−0.788200
$949$	− 23158.7i	− 0.792163i
$950$	6427.51	0.219511
$951$	−80370.7	−2.74048
$952$	0	0
$953$	55894.2	1.89988	0.949942	−	0.312425i	$-0.101141\pi$
$953$	55894.2	1.89988	0.949942	+	0.312425i	$0.101141\pi$
$954$	−59610.8	−2.02303
$955$	− 1758.23i	− 0.0595758i
$956$	598.879	0.0202606
$957$	75469.3i	2.54919i
$958$	− 26705.0i	− 0.900625i
$959$	− 17939.0i	− 0.604046i
$960$	−742.755	−0.0249712
$961$	20973.8	0.704032
$962$	26184.6i	0.877575i
$963$	38181.7i	1.27766i
$964$	− 29382.9i	− 0.981702i
$965$	3718.04	0.124029
$966$	− 29394.3i	− 0.979034i
$967$	27667.3	0.920082	0.460041	−	0.887898i	$-0.347835\pi$
$967$	27667.3	0.920082	0.460041	+	0.887898i	$0.347835\pi$
$968$	−1669.50	−0.0554338
$969$	0	0
$970$	−2492.86	−0.0825162
$971$	42803.0	1.41464	0.707319	−	0.706894i	$-0.249904\pi$
$971$	42803.0	1.41464	0.707319	+	0.706894i	$0.249904\pi$
$972$	3062.94i	0.101074i
$973$	−45632.6	−1.50351
$974$	22170.8i	0.729360i
$975$	− 74219.7i	− 2.43788i
$976$	− 53.5982i	− 0.00175782i
$977$	−30623.7	−1.00280	−0.501401	−	0.865215i	$-0.667182\pi$
$977$	−30623.7	−1.00280	−0.501401	+	0.865215i	$0.667182\pi$
$978$	6321.16	0.206675
$979$	− 32595.7i	− 1.06411i
$980$	3865.52i	0.126000i
$981$	− 117035.i	− 3.80900i
$982$	26875.3	0.873344
$983$	17370.2i	0.563603i	0.959473	+	0.281802i	$0.0909320\pi$
$983$	17370.2i	0.563603i	−0.959473	+	0.281802i	$0.909068\pi$
$984$	−20165.3	−0.653298
$985$	−6488.71	−0.209896
$986$	0	0
$987$	59507.5	1.91909
$988$	7044.49	0.226837
$989$	6237.53i	0.200548i
$990$	−5353.05	−0.171850
$991$	− 56024.6i	− 1.79584i	−0.440155	−	0.897922i	$-0.645077\pi$
$991$	− 56024.6i	− 1.79584i	0.440155	−	0.897922i	$-0.354923\pi$
$992$	3004.80i	0.0961718i
$993$	90175.3i	2.88180i
$994$	−32470.6	−1.03612
$995$	3849.34	0.122645
$996$	− 45420.6i	− 1.44499i
$997$	11709.6i	0.371962i	0.982553	+	0.185981i	$0.0595463\pi$
$997$	11709.6i	0.371962i	−0.982553	+	0.185981i	$0.940454\pi$
$998$	26330.3i	0.835142i
$999$	−43770.6	−1.38623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	578.4.b.f.577.1		4
17.4	even	4		578.4.a.e.1.1	✓	2
17.13	even	4		578.4.a.g.1.2	yes	2
17.16	even	2	inner	578.4.b.f.577.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
578.4.a.e.1.1	✓	2	17.4	even	4
578.4.a.g.1.2	yes	2	17.13	even	4
578.4.b.f.577.1		4	1.1	even	1	trivial
578.4.b.f.577.4		4	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 578 = 2 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	578.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -8.90833i q^{3} +4.00000 q^{4} -1.30278i q^{5} -17.8167i q^{6} +32.9361i q^{7} +8.00000 q^{8} -52.3583 q^{9} -2.60555i q^{10} +39.2389i q^{11} -35.6333i q^{12} +67.5694 q^{13} +65.8722i q^{14} -11.6056 q^{15} +16.0000 q^{16} -104.717 q^{18} +26.0639 q^{19} -5.21110i q^{20} +293.405 q^{21} +78.4777i q^{22} -50.0917i q^{23} -71.2666i q^{24} +123.303 q^{25} +135.139 q^{26} +225.900i q^{27} +131.744i q^{28} +215.902i q^{29} -23.2111 q^{30} +93.8999i q^{31} +32.0000 q^{32} +349.553 q^{33} +42.9083 q^{35} -209.433 q^{36} +193.761i q^{37} +52.1278 q^{38} -601.930i q^{39} -10.4222i q^{40} -282.955i q^{41} +586.811 q^{42} -124.522 q^{43} +156.955i q^{44} +68.2111i q^{45} -100.183i q^{46} +202.817 q^{47} -142.533i q^{48} -741.786 q^{49} +246.606 q^{50} +270.278 q^{52} +569.258 q^{53} +451.800i q^{54} +51.1194 q^{55} +263.489i q^{56} -232.186i q^{57} +431.805i q^{58} -137.566 q^{59} -46.4222 q^{60} -3.34988i q^{61} +187.800i q^{62} -1724.48i q^{63} +64.0000 q^{64} -88.0278i q^{65} +699.105 q^{66} -764.560 q^{67} -446.233 q^{69} +85.8167 q^{70} +492.934i q^{71} -418.866 q^{72} -342.739i q^{73} +387.522i q^{74} -1098.42i q^{75} +104.256 q^{76} -1292.37 q^{77} -1203.86i q^{78} -645.643i q^{79} -20.8444i q^{80} +598.717 q^{81} -565.911i q^{82} +1274.67 q^{83} +1173.62 q^{84} -249.045 q^{86} +1923.33 q^{87} +313.911i q^{88} -830.700 q^{89} +136.422i q^{90} +2225.47i q^{91} -200.367i q^{92} +836.491 q^{93} +405.633 q^{94} -33.9554i q^{95} -285.066i q^{96} -956.748i q^{97} -1483.57 q^{98} -2054.48i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 58 q^{9} + 90 q^{13} - 32 q^{15} + 64 q^{16} - 116 q^{18} + 198 q^{19} + 640 q^{21} + 486 q^{25} + 180 q^{26} - 64 q^{30} + 128 q^{32} + 742 q^{33} + 150 q^{35} - 232 q^{36}+ \cdots - 2372 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 - 58 * q^9 + 90 * q^13 - 32 * q^15 + 64 * q^16 - 116 * q^18 + 198 * q^19 + 640 * q^21 + 486 * q^25 + 180 * q^26 - 64 * q^30 + 128 * q^32 + 742 * q^33 + 150 * q^35 - 232 * q^36 + 396 * q^38 + 1280 * q^42 - 700 * q^43 + 768 * q^47 - 1186 * q^49 + 972 * q^50 + 360 * q^52 + 1866 * q^53 + 154 * q^55 + 1440 * q^59 - 128 * q^60 + 256 * q^64 + 1484 * q^66 - 44 * q^67 - 660 * q^69 + 300 * q^70 - 464 * q^72 + 792 * q^76 - 2898 * q^77 + 2092 * q^81 - 14 * q^83 + 2560 * q^84 - 1400 * q^86 + 5018 * q^87 - 2544 * q^89 + 1810 * q^93 + 1536 * q^94 - 2372 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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