LMFDB - Embedded newform 289.4.b.d.288.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [289,4,Mod(288,289)] 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("289.288"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(289, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	289.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$17.0515519917$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 43x^{6} + 505x^{4} + 1528x^{2} + 1296 $ x^8 + 43x^6 + 505x^4 + 1528*x^2 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.6
Root			$-1.58184i$ of defining polynomial
Character	$\chi$	$=$	289.288
Dual form			289.4.b.d.288.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.58184 q^{2} +4.98387i q^{3} -5.49778 q^{4} +14.4471i q^{5} +7.88368i q^{6} +29.4104i q^{7} -21.3513 q^{8} +2.16104 q^{9} +22.8530i q^{10} -13.5324i q^{11} -27.4002i q^{12} +45.7656 q^{13} +46.5225i q^{14} -72.0024 q^{15} +10.2079 q^{16} +3.41842 q^{18} -113.386 q^{19} -79.4269i q^{20} -146.578 q^{21} -21.4060i q^{22} -144.513i q^{23} -106.412i q^{24} -83.7182 q^{25} +72.3938 q^{26} +145.335i q^{27} -161.692i q^{28} -1.26688i q^{29} -113.896 q^{30} -30.0858i q^{31} +186.958 q^{32} +67.4435 q^{33} -424.894 q^{35} -11.8809 q^{36} +398.512i q^{37} -179.358 q^{38} +228.090i q^{39} -308.464i q^{40} -184.595i q^{41} -231.862 q^{42} -135.605 q^{43} +74.3979i q^{44} +31.2207i q^{45} -228.596i q^{46} +247.542 q^{47} +50.8748i q^{48} -521.971 q^{49} -132.429 q^{50} -251.609 q^{52} +635.182 q^{53} +229.896i q^{54} +195.503 q^{55} -627.951i q^{56} -565.100i q^{57} -2.00399i q^{58} -625.420 q^{59} +395.853 q^{60} +166.873i q^{61} -47.5909i q^{62} +63.5570i q^{63} +214.074 q^{64} +661.179i q^{65} +106.685 q^{66} +159.984 q^{67} +720.232 q^{69} -672.115 q^{70} +19.4559i q^{71} -46.1411 q^{72} +336.039i q^{73} +630.382i q^{74} -417.241i q^{75} +623.370 q^{76} +397.992 q^{77} +360.801i q^{78} -1072.15i q^{79} +147.474i q^{80} -665.982 q^{81} -291.999i q^{82} -47.1944 q^{83} +805.852 q^{84} -214.506 q^{86} +6.31394 q^{87} +288.934i q^{88} -626.312 q^{89} +49.3862i q^{90} +1345.98i q^{91} +794.499i q^{92} +149.944 q^{93} +391.572 q^{94} -1638.09i q^{95} +931.774i q^{96} +692.045i q^{97} -825.674 q^{98} -29.2439i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9} - 44 q^{13} - 108 q^{15} + 126 q^{16} - 668 q^{18} - 44 q^{19} - 704 q^{21} - 756 q^{25} + 896 q^{26} + 626 q^{30} - 662 q^{32} + 188 q^{33} - 484 q^{35}+ \cdots - 3754 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9 - 44 * q^13 - 108 * q^15 + 126 * q^16 - 668 * q^18 - 44 * q^19 - 704 * q^21 - 756 * q^25 + 896 * q^26 + 626 * q^30 - 662 * q^32 + 188 * q^33 - 484 * q^35 - 282 * q^36 - 1048 * q^38 - 2910 * q^42 + 228 * q^43 + 20 * q^47 - 2012 * q^49 + 1610 * q^50 - 3074 * q^52 - 100 * q^53 - 2632 * q^55 - 1992 * q^59 + 434 * q^60 - 300 * q^64 + 2180 * q^66 - 1736 * q^67 - 2256 * q^69 - 2104 * q^70 - 78 * q^72 + 1746 * q^76 + 1788 * q^77 + 2160 * q^81 - 1700 * q^83 + 886 * q^84 + 4822 * q^86 + 768 * q^87 + 1568 * q^89 + 3100 * q^93 - 2238 * q^94 - 3754 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.58184	0.559265	0.279632	−	0.960107i	$-0.409787\pi$
$2$	1.58184	0.559265	0.279632	+	0.960107i	$0.409787\pi$
$3$	4.98387i	0.959146i	0.877502	+	0.479573i	$0.159208\pi$
$3$	4.98387i	0.959146i	−0.877502	+	0.479573i	$0.840792\pi$
$4$	−5.49778	−0.687223
$5$	14.4471i	1.29219i	0.763259	+	0.646093i	$0.223598\pi$
$5$	14.4471i	1.29219i	−0.763259	+	0.646093i	$0.776402\pi$
$6$	7.88368i	0.536417i
$7$	29.4104i	1.58801i	0.607910	+	0.794006i	$0.292008\pi$
$7$	29.4104i	1.58801i	−0.607910	+	0.794006i	$0.707992\pi$
$8$	−21.3513	−0.943604
$9$	2.16104	0.0800385
$10$	22.8530i	0.722674i
$11$	− 13.5324i	− 0.370923i	−0.982651	−	0.185462i	$-0.940622\pi$
$11$	− 13.5324i	− 0.370923i	0.982651	−	0.185462i	$-0.0593781\pi$
$12$	− 27.4002i	− 0.659147i
$13$	45.7656	0.976391	0.488196	−	0.872734i	$-0.337655\pi$
$13$	45.7656	0.976391	0.488196	+	0.872734i	$0.337655\pi$
$14$	46.5225i	0.888119i
$15$	−72.0024	−1.23940
$16$	10.2079	0.159498
$17$	0	0
$17$	0	0
$18$	3.41842	0.0447627
$19$	−113.386	−1.36908	−0.684539	−	0.728976i	$-0.739996\pi$
$19$	−113.386	−1.36908	−0.684539	+	0.728976i	$0.739996\pi$
$20$	− 79.4269i	− 0.888020i
$21$	−146.578	−1.52314
$22$	− 21.4060i	− 0.207444i
$23$	− 144.513i	− 1.31013i	−0.755573	−	0.655065i	$-0.772641\pi$
$23$	− 144.513i	− 1.31013i	0.755573	−	0.655065i	$-0.227359\pi$
$24$	− 106.412i	− 0.905055i
$25$	−83.7182	−0.669745
$26$	72.3938	0.546061
$27$	145.335i	1.03591i
$28$	− 161.692i	− 1.09132i
$29$	− 1.26688i	− 0.00811217i	−0.999992	−	0.00405608i	$-0.998709\pi$
$29$	− 1.26688i	− 0.00811217i	0.999992	−	0.00405608i	$-0.00129109\pi$
$30$	−113.896	−0.693150
$31$	− 30.0858i	− 0.174309i	−0.996195	−	0.0871544i	$-0.972223\pi$
$31$	− 30.0858i	− 0.174309i	0.996195	−	0.0871544i	$-0.0277773\pi$
$32$	186.958	1.03281
$33$	67.4435	0.355770
$34$	0	0
$35$	−424.894	−2.05201
$36$	−11.8809	−0.0550043
$37$	398.512i	1.77067i	0.464950	+	0.885337i	$0.346072\pi$
$37$	398.512i	1.77067i	−0.464950	+	0.885337i	$0.653928\pi$
$38$	−179.358	−0.765677
$39$	228.090i	0.936502i
$40$	− 308.464i	− 1.21931i
$41$	− 184.595i	− 0.703143i	−0.936161	−	0.351571i	$-0.885647\pi$
$41$	− 184.595i	− 0.703143i	0.936161	−	0.351571i	$-0.114353\pi$
$42$	−231.862	−0.851836
$43$	−135.605	−0.480921	−0.240460	−	0.970659i	$-0.577298\pi$
$43$	−135.605	−0.480921	−0.240460	+	0.970659i	$0.577298\pi$
$44$	74.3979i	0.254907i
$45$	31.2207i	0.103425i
$46$	− 228.596i	− 0.732709i
$47$	247.542	0.768249	0.384124	−	0.923281i	$-0.374503\pi$
$47$	247.542	0.768249	0.384124	+	0.923281i	$0.374503\pi$
$48$	50.8748i	0.152982i
$49$	−521.971	−1.52178
$50$	−132.429	−0.374565
$51$	0	0
$52$	−251.609	−0.670998
$53$	635.182	1.64621	0.823103	−	0.567892i	$-0.192241\pi$
$53$	635.182	1.64621	0.823103	+	0.567892i	$0.192241\pi$
$54$	229.896i	0.579351i
$55$	195.503	0.479302
$56$	− 627.951i	− 1.49845i
$57$	− 565.100i	− 1.31315i
$58$	− 2.00399i	− 0.00453685i
$59$	−625.420	−1.38005	−0.690023	−	0.723788i	$-0.742400\pi$
$59$	−625.420	−1.38005	−0.690023	+	0.723788i	$0.742400\pi$
$60$	395.853	0.851741
$61$	166.873i	0.350261i	0.984545	+	0.175130i	$0.0560347\pi$
$61$	166.873i	0.350261i	−0.984545	+	0.175130i	$0.943965\pi$
$62$	− 47.5909i	− 0.0974847i
$63$	63.5570i	0.127102i
$64$	214.074	0.418114
$65$	661.179i	1.26168i
$66$	106.685	0.198970
$67$	159.984	0.291719	0.145859	−	0.989305i	$-0.453405\pi$
$67$	159.984	0.291719	0.145859	+	0.989305i	$0.453405\pi$
$68$	0	0
$69$	720.232	1.25661
$70$	−672.115	−1.14762
$71$	19.4559i	0.0325211i	0.999868	+	0.0162605i	$0.00517611\pi$
$71$	19.4559i	0.0325211i	−0.999868	+	0.0162605i	$0.994824\pi$
$72$	−46.1411	−0.0755247
$73$	336.039i	0.538772i	0.963032	+	0.269386i	$0.0868207\pi$
$73$	336.039i	0.538772i	−0.963032	+	0.269386i	$0.913179\pi$
$74$	630.382i	0.990276i
$75$	− 417.241i	− 0.642384i
$76$	623.370	0.940862
$77$	397.992	0.589031
$78$	360.801i	0.523753i
$79$	− 1072.15i	− 1.52691i	−0.645859	−	0.763457i	$-0.723500\pi$
$79$	− 1072.15i	− 1.52691i	0.645859	−	0.763457i	$-0.276500\pi$
$80$	147.474i	0.206101i
$81$	−665.982	−0.913555
$82$	− 291.999i	− 0.393243i
$83$	−47.1944	−0.0624128	−0.0312064	−	0.999513i	$-0.509935\pi$
$83$	−47.1944	−0.0624128	−0.0312064	+	0.999513i	$0.509935\pi$
$84$	805.852	1.04673
$85$	0	0
$86$	−214.506	−0.268962
$87$	6.31394	0.00778075
$88$	288.934i	0.350005i
$89$	−626.312	−0.745944	−0.372972	−	0.927843i	$-0.621661\pi$
$89$	−626.312	−0.745944	−0.372972	+	0.927843i	$0.621661\pi$
$90$	49.3862i	0.0578418i
$91$	1345.98i	1.55052i
$92$	794.499i	0.900351i
$93$	149.944	0.167188
$94$	391.572	0.429655
$95$	− 1638.09i	− 1.76910i
$96$	931.774i	0.990612i
$97$	692.045i	0.724398i	0.932101	+	0.362199i	$0.117974\pi$
$97$	692.045i	0.724398i	−0.932101	+	0.362199i	$0.882026\pi$
$98$	−825.674	−0.851079
$99$	− 29.2439i	− 0.0296882i
$100$	460.264	0.460264
$101$	−551.668	−0.543496	−0.271748	−	0.962369i	$-0.587602\pi$
$101$	−551.668	−0.543496	−0.271748	+	0.962369i	$0.587602\pi$
$102$	0	0
$103$	−175.990	−0.168358	−0.0841789	−	0.996451i	$-0.526827\pi$
$103$	−175.990	−0.168358	−0.0841789	+	0.996451i	$0.526827\pi$
$104$	−977.156	−0.921327
$105$	− 2117.62i	− 1.96817i
$106$	1004.76	0.920665
$107$	148.521i	0.134188i	0.997747	+	0.0670938i	$0.0213727\pi$
$107$	148.521i	0.134188i	−0.997747	+	0.0670938i	$0.978627\pi$
$108$	− 799.019i	− 0.711904i
$109$	703.114i	0.617854i	0.951086	+	0.308927i	$0.0999699\pi$
$109$	703.114i	0.617854i	−0.951086	+	0.308927i	$0.900030\pi$
$110$	309.254	0.268057
$111$	−1986.13	−1.69834
$112$	300.218i	0.253285i
$113$	1987.05i	1.65421i	0.562048	+	0.827104i	$0.310014\pi$
$113$	1987.05i	1.65421i	−0.562048	+	0.827104i	$0.689986\pi$
$114$	− 893.898i	− 0.734396i
$115$	2087.79	1.69293
$116$	6.96500i	0.00557487i
$117$	98.9012	0.0781489
$118$	−989.314	−0.771811
$119$	0	0
$120$	1537.35	1.16950
$121$	1147.88	0.862416
$122$	263.966i	0.195888i
$123$	919.996	0.674417
$124$	165.405i	0.119789i
$125$	596.402i	0.426750i
$126$	100.537i	0.0710837i
$127$	1561.88	1.09130	0.545649	−	0.838014i	$-0.316283\pi$
$127$	1561.88	1.09130	0.545649	+	0.838014i	$0.316283\pi$
$128$	−1157.03	−0.798970
$129$	− 675.838i	− 0.461273i
$130$	1045.88i	0.705613i
$131$	1317.70i	0.878839i	0.898282	+	0.439419i	$0.144816\pi$
$131$	1317.70i	0.878839i	−0.898282	+	0.439419i	$0.855184\pi$
$132$	−370.790	−0.244493
$133$	− 3334.72i	− 2.17411i
$134$	253.069	0.163148
$135$	−2099.66	−1.33859
$136$	0	0
$137$	485.214	0.302588	0.151294	−	0.988489i	$-0.451656\pi$
$137$	485.214	0.302588	0.151294	+	0.988489i	$0.451656\pi$
$138$	1139.29	0.702775
$139$	− 1016.66i	− 0.620376i	−0.950675	−	0.310188i	$-0.899608\pi$
$139$	− 1016.66i	− 0.620376i	0.950675	−	0.310188i	$-0.100392\pi$
$140$	2335.98	1.41019
$141$	1233.72i	0.736863i
$142$	30.7762i	0.0181879i
$143$	− 619.316i	− 0.362166i
$144$	22.0596	0.0127660
$145$	18.3026	0.0104824
$146$	531.559i	0.301316i
$147$	− 2601.44i	− 1.45961i
$148$	− 2190.93i	− 1.21685i
$149$	−1322.91	−0.727364	−0.363682	−	0.931523i	$-0.618481\pi$
$149$	−1322.91	−0.727364	−0.363682	+	0.931523i	$0.618481\pi$
$150$	− 660.008i	− 0.359263i
$151$	1467.32	0.790785	0.395392	−	0.918512i	$-0.370609\pi$
$151$	1467.32	0.790785	0.395392	+	0.918512i	$0.370609\pi$
$152$	2420.94	1.29187
$153$	0	0
$154$	629.559	0.329424
$155$	434.652	0.225239
$156$	− 1253.99i	− 0.643586i
$157$	−3403.23	−1.72998	−0.864991	−	0.501788i	$-0.832676\pi$
$157$	−3403.23	−1.72998	−0.864991	+	0.501788i	$0.832676\pi$
$158$	− 1695.97i	− 0.853949i
$159$	3165.66i	1.57895i
$160$	2701.00i	1.33458i
$161$	4250.17	2.08050
$162$	−1053.48	−0.510919
$163$	378.056i	0.181667i	0.995866	+	0.0908333i	$0.0289530\pi$
$163$	378.056i	0.181667i	−0.995866	+	0.0908333i	$0.971047\pi$
$164$	1014.86i	0.483216i
$165$	974.362i	0.459721i
$166$	−74.6540	−0.0349053
$167$	572.418i	0.265240i	0.991167	+	0.132620i	$0.0423390\pi$
$167$	572.418i	0.265240i	−0.991167	+	0.132620i	$0.957661\pi$
$168$	3129.63	1.43724
$169$	−102.512	−0.0466601
$170$	0	0
$171$	−245.031	−0.109579
$172$	745.528	0.330500
$173$	3321.64i	1.45977i	0.683571	+	0.729884i	$0.260426\pi$
$173$	3321.64i	1.45977i	−0.683571	+	0.729884i	$0.739574\pi$
$174$	9.98764	0.00435150
$175$	− 2462.18i	− 1.06356i
$176$	− 138.137i	− 0.0591616i
$177$	− 3117.01i	− 1.32367i
$178$	−990.726	−0.417180
$179$	2093.76	0.874273	0.437137	−	0.899395i	$-0.355993\pi$
$179$	2093.76	0.874273	0.437137	+	0.899395i	$0.355993\pi$
$180$	− 171.645i	− 0.0710758i
$181$	682.032i	0.280083i	0.990146	+	0.140042i	$0.0447236\pi$
$181$	682.032i	0.280083i	−0.990146	+	0.140042i	$0.955276\pi$
$182$	2129.13i	0.867152i
$183$	−831.673	−0.335951
$184$	3085.54i	1.23624i
$185$	−5757.33	−2.28804
$186$	237.187	0.0935021
$187$	0	0
$188$	−1360.93	−0.527958
$189$	−4274.35	−1.64504
$190$	− 2591.20i	− 0.989398i
$191$	1921.15	0.727799	0.363900	−	0.931438i	$-0.381445\pi$
$191$	1921.15	0.727799	0.363900	+	0.931438i	$0.381445\pi$
$192$	1066.92i	0.401032i
$193$	− 69.6893i	− 0.0259914i	−0.999916	−	0.0129957i	$-0.995863\pi$
$193$	− 69.6893i	− 0.0259914i	0.999916	−	0.0129957i	$-0.00413678\pi$
$194$	1094.70i	0.405130i
$195$	−3295.23	−1.21014
$196$	2869.68	1.04580
$197$	− 3043.10i	− 1.10057i	−0.834977	−	0.550285i	$-0.814519\pi$
$197$	− 3043.10i	− 1.10057i	0.834977	−	0.550285i	$-0.185481\pi$
$198$	− 46.2592i	− 0.0166035i
$199$	1255.72i	0.447315i	0.974668	+	0.223657i	$0.0717997\pi$
$199$	1255.72i	0.447315i	−0.974668	+	0.223657i	$0.928200\pi$
$200$	1787.49	0.631975
$201$	797.340i	0.279801i
$202$	−872.651	−0.303958
$203$	37.2593	0.0128822
$204$	0	0
$205$	2666.85	0.908591
$206$	−278.389	−0.0941566
$207$	− 312.297i	− 0.104861i
$208$	467.170	0.155733
$209$	1534.38i	0.507823i
$210$	− 3349.73i	− 1.10073i
$211$	− 1060.15i	− 0.345894i	−0.984931	−	0.172947i	$-0.944671\pi$
$211$	− 1060.15i	− 0.345894i	0.984931	−	0.172947i	$-0.0553289\pi$
$212$	−3492.09	−1.13131
$213$	−96.9658	−0.0311924
$214$	234.937i	0.0750464i
$215$	− 1959.10i	− 0.621439i
$216$	− 3103.09i	− 0.977494i
$217$	884.835	0.276804
$218$	1112.21i	0.345544i
$219$	−1674.77	−0.516761
$220$	−1074.83	−0.329387
$221$	0	0
$222$	−3141.74	−0.949819
$223$	1398.85	0.420064	0.210032	−	0.977695i	$-0.432643\pi$
$223$	1398.85	0.420064	0.210032	+	0.977695i	$0.432643\pi$
$224$	5498.50i	1.64011i
$225$	−180.918	−0.0536054
$226$	3143.19i	0.925141i
$227$	− 2436.65i	− 0.712451i	−0.934400	−	0.356226i	$-0.884063\pi$
$227$	− 2436.65i	− 0.712451i	0.934400	−	0.356226i	$-0.115937\pi$
$228$	3106.80i	0.902424i
$229$	−5027.28	−1.45071	−0.725354	−	0.688376i	$-0.758324\pi$
$229$	−5027.28	−1.45071	−0.725354	+	0.688376i	$0.758324\pi$
$230$	3302.54	0.946797
$231$	1983.54i	0.564967i
$232$	27.0495i	0.00765467i
$233$	− 891.120i	− 0.250555i	−0.992122	−	0.125277i	$-0.960018\pi$
$233$	− 891.120i	− 0.250555i	0.992122	−	0.125277i	$-0.0399821\pi$
$234$	156.446	0.0437059
$235$	3576.26i	0.992721i
$236$	3438.42	0.948399
$237$	5343.45	1.46453
$238$	0	0
$239$	−161.975	−0.0438382	−0.0219191	−	0.999760i	$-0.506978\pi$
$239$	−161.975	−0.0438382	−0.0219191	+	0.999760i	$0.506978\pi$
$240$	−734.992	−0.197681
$241$	3467.39i	0.926782i	0.886154	+	0.463391i	$0.153367\pi$
$241$	3467.39i	0.926782i	−0.886154	+	0.463391i	$0.846633\pi$
$242$	1815.75	0.482319
$243$	604.873i	0.159682i
$244$	− 917.432i	− 0.240707i
$245$	− 7540.96i	− 1.96642i
$246$	1455.29	0.377177
$247$	−5189.17	−1.33676
$248$	642.372i	0.164478i
$249$	− 235.211i	− 0.0598630i
$250$	943.412i	0.238666i
$251$	−4328.65	−1.08853	−0.544267	−	0.838912i	$-0.683192\pi$
$251$	−4328.65	−1.08853	−0.544267	+	0.838912i	$0.683192\pi$
$252$	− 349.423i	− 0.0873474i
$253$	−1955.60	−0.485958
$254$	2470.65	0.610324
$255$	0	0
$256$	−3542.83	−0.864950
$257$	4567.47	1.10860	0.554301	−	0.832316i	$-0.312986\pi$
$257$	4567.47	1.10860	0.554301	+	0.832316i	$0.312986\pi$
$258$	− 1069.07i	− 0.257974i
$259$	−11720.4	−2.81185
$260$	− 3635.02i	− 0.867055i
$261$	− 2.73777i	0 0.000649285i
$262$	2084.39i	0.491504i
$263$	−4711.02	−1.10454	−0.552270	−	0.833665i	$-0.686238\pi$
$263$	−4711.02	−1.10454	−0.552270	+	0.833665i	$0.686238\pi$
$264$	−1440.01	−0.335706
$265$	9176.52i	2.12721i
$266$	− 5274.99i	− 1.21590i
$267$	− 3121.46i	− 0.715469i
$268$	−879.558	−0.200476
$269$	− 3659.09i	− 0.829362i	−0.909967	−	0.414681i	$-0.863893\pi$
$269$	− 3659.09i	− 0.829362i	0.909967	−	0.414681i	$-0.136107\pi$
$270$	−3321.33	−0.748629
$271$	−7702.43	−1.72653	−0.863264	−	0.504753i	$-0.831584\pi$
$271$	−7702.43	−1.72653	−0.863264	+	0.504753i	$0.831584\pi$
$272$	0	0
$273$	−6708.21	−1.48718
$274$	767.530	0.169227
$275$	1132.90i	0.248424i
$276$	−3959.68	−0.863568
$277$	− 3252.61i	− 0.705526i	−0.935713	−	0.352763i	$-0.885242\pi$
$277$	− 3252.61i	− 0.705526i	0.935713	−	0.352763i	$-0.114758\pi$
$278$	− 1608.20i	− 0.346954i
$279$	− 65.0166i	− 0.0139514i
$280$	9072.06	1.93628
$281$	6839.31	1.45195	0.725977	−	0.687719i	$-0.241388\pi$
$281$	6839.31	1.45195	0.725977	+	0.687719i	$0.241388\pi$
$282$	1951.54i	0.412101i
$283$	− 3131.30i	− 0.657726i	−0.944378	−	0.328863i	$-0.893335\pi$
$283$	− 3131.30i	− 0.657726i	0.944378	−	0.328863i	$-0.106665\pi$
$284$	− 106.965i	− 0.0223492i
$285$	8164.05	1.69683
$286$	− 979.659i	− 0.202547i
$287$	5429.00	1.11660
$288$	404.023	0.0826642
$289$	0	0
$290$	28.9519	0.00586245
$291$	−3449.06	−0.694803
$292$	− 1847.47i	− 0.370256i
$293$	5396.03	1.07590	0.537951	−	0.842976i	$-0.319198\pi$
$293$	5396.03	1.07590	0.537951	+	0.842976i	$0.319198\pi$
$294$	− 4115.05i	− 0.816309i
$295$	− 9035.49i	− 1.78328i
$296$	− 8508.76i	− 1.67082i
$297$	1966.72	0.384245
$298$	−2092.64	−0.406789
$299$	− 6613.70i	− 1.27920i
$300$	2293.90i	0.441461i
$301$	− 3988.20i	− 0.763708i
$302$	2321.06	0.442258
$303$	− 2749.44i	− 0.521292i
$304$	−1157.43	−0.218365
$305$	−2410.83	−0.452602
$306$	0	0
$307$	2156.62	0.400927	0.200463	−	0.979701i	$-0.435755\pi$
$307$	2156.62	0.400927	0.200463	+	0.979701i	$0.435755\pi$
$308$	−2188.07	−0.404795
$309$	− 877.113i	− 0.161480i
$310$	687.550	0.125968
$311$	2729.42i	0.497656i	0.968548	+	0.248828i	$0.0800454\pi$
$311$	2729.42i	0.497656i	−0.968548	+	0.248828i	$0.919955\pi$
$312$	− 4870.02i	− 0.883687i
$313$	5014.77i	0.905595i	0.891613	+	0.452798i	$0.149574\pi$
$313$	5014.77i	0.905595i	−0.891613	+	0.452798i	$0.850426\pi$
$314$	−5383.36	−0.967518
$315$	−918.213	−0.164240
$316$	5894.44i	1.04933i
$317$	1164.53i	0.206329i	0.994664	+	0.103165i	$0.0328968\pi$
$317$	1164.53i	0.206329i	−0.994664	+	0.103165i	$0.967103\pi$
$318$	5007.57i	0.883052i
$319$	−17.1438	−0.00300899
$320$	3092.75i	0.540281i
$321$	−740.210	−0.128706
$322$	6723.09	1.16355
$323$	0	0
$324$	3661.42	0.627816
$325$	−3831.41	−0.653934
$326$	598.024i	0.101600i
$327$	−3504.23	−0.592612
$328$	3941.34i	0.663488i
$329$	7280.30i	1.21999i
$330$	1541.28i	0.257106i
$331$	−2144.03	−0.356032	−0.178016	−	0.984028i	$-0.556968\pi$
$331$	−2144.03	−0.356032	−0.178016	+	0.984028i	$0.556968\pi$
$332$	259.465	0.0428915
$333$	861.200i	0.141722i
$334$	905.474i	0.148339i
$335$	2311.30i	0.376955i
$336$	−1496.25	−0.242937
$337$	− 1955.09i	− 0.316025i	−0.987437	−	0.158012i	$-0.949491\pi$
$337$	− 1955.09i	− 0.316025i	0.987437	−	0.158012i	$-0.0505086\pi$
$338$	−162.158	−0.0260954
$339$	−9903.18	−1.58663
$340$	0	0
$341$	−407.132	−0.0646552
$342$	−387.600	−0.0612836
$343$	− 5263.61i	− 0.828595i
$344$	2895.35	0.453799
$345$	10405.3i	1.62377i
$346$	5254.31i	0.816397i
$347$	10906.7i	1.68732i	0.536875	+	0.843662i	$0.319605\pi$
$347$	10906.7i	1.68732i	−0.536875	+	0.843662i	$0.680395\pi$
$348$	−34.7127	−0.00534711
$349$	−9245.27	−1.41802	−0.709009	−	0.705200i	$-0.750857\pi$
$349$	−9245.27	−1.41802	−0.709009	+	0.705200i	$0.750857\pi$
$350$	− 3894.78i	− 0.594814i
$351$	6651.33i	1.01146i
$352$	− 2529.98i	− 0.383092i
$353$	−4026.13	−0.607051	−0.303526	−	0.952823i	$-0.598164\pi$
$353$	−4026.13	−0.607051	−0.303526	+	0.952823i	$0.598164\pi$
$354$	− 4930.61i	− 0.740280i
$355$	−281.081	−0.0420233
$356$	3443.33	0.512630
$357$	0	0
$358$	3311.99	0.488950
$359$	10382.3	1.52634	0.763171	−	0.646197i	$-0.223641\pi$
$359$	10382.3	1.52634	0.763171	+	0.646197i	$0.223641\pi$
$360$	− 666.604i	− 0.0975919i
$361$	5997.34	0.874375
$362$	1078.87i	0.156641i
$363$	5720.86i	0.827183i
$364$	− 7399.92i	− 1.06555i
$365$	−4854.78	−0.696194
$366$	−1315.57	−0.187886
$367$	− 12159.2i	− 1.72945i	−0.502249	−	0.864723i	$-0.667494\pi$
$367$	− 12159.2i	− 1.72945i	0.502249	−	0.864723i	$-0.332506\pi$
$368$	− 1475.17i	− 0.208963i
$369$	− 398.916i	− 0.0562785i
$370$	−9107.18	−1.27962
$371$	18680.9i	2.61419i
$372$	−824.358	−0.114895
$373$	11494.5	1.59561	0.797807	−	0.602913i	$-0.205993\pi$
$373$	11494.5	1.59561	0.797807	+	0.602913i	$0.205993\pi$
$374$	0	0
$375$	−2972.39	−0.409316
$376$	−5285.35	−0.724923
$377$	− 57.9793i	− 0.00792065i
$378$	−6761.34	−0.920016
$379$	− 1085.96i	− 0.147182i	−0.997289	−	0.0735908i	$-0.976554\pi$
$379$	− 1085.96i	− 0.147182i	0.997289	−	0.0735908i	$-0.0234459\pi$
$380$	9005.88i	1.21577i
$381$	7784.23i	1.04671i
$382$	3038.95	0.407032
$383$	8246.14	1.10015	0.550076	−	0.835115i	$-0.314599\pi$
$383$	8246.14	1.10015	0.550076	+	0.835115i	$0.314599\pi$
$384$	− 5766.50i	− 0.766329i
$385$	5749.82i	0.761138i
$386$	− 110.237i	− 0.0145361i
$387$	−293.048	−0.0384922
$388$	− 3804.72i	− 0.497823i
$389$	3801.18	0.495443	0.247722	−	0.968831i	$-0.420318\pi$
$389$	3801.18	0.495443	0.247722	+	0.968831i	$0.420318\pi$
$390$	−5212.53	−0.676786
$391$	0	0
$392$	11144.8	1.43596
$393$	−6567.24	−0.842935
$394$	− 4813.70i	− 0.615510i
$395$	15489.4	1.97306
$396$	160.777i	0.0204024i
$397$	8372.90i	1.05850i	0.848466	+	0.529249i	$0.177526\pi$
$397$	8372.90i	1.05850i	−0.848466	+	0.529249i	$0.822474\pi$
$398$	1986.35i	0.250167i
$399$	16619.8	2.08529
$400$	−854.586	−0.106823
$401$	2173.63i	0.270688i	0.990799	+	0.135344i	$0.0432139\pi$
$401$	2173.63i	0.270688i	−0.990799	+	0.135344i	$0.956786\pi$
$402$	1261.26i	0.156483i
$403$	− 1376.89i	− 0.170194i
$404$	3032.95	0.373503
$405$	− 9621.49i	− 1.18048i
$406$	58.9382	0.00720457
$407$	5392.80	0.656785
$408$	0	0
$409$	4538.13	0.548646	0.274323	−	0.961638i	$-0.411546\pi$
$409$	4538.13	0.548646	0.274323	+	0.961638i	$0.411546\pi$
$410$	4218.54	0.508143
$411$	2418.24i	0.290227i
$412$	967.557	0.115699
$413$	− 18393.8i	− 2.19153i
$414$	− 494.005i	− 0.0586449i
$415$	− 681.822i	− 0.0806490i
$416$	8556.23	1.00842
$417$	5066.91	0.595031
$418$	2427.14i	0.284008i
$419$	− 4239.67i	− 0.494323i	−0.968974	−	0.247161i	$-0.920502\pi$
$419$	− 4239.67i	− 0.494323i	0.968974	−	0.247161i	$-0.0794978\pi$
$420$	11642.2i	1.35257i
$421$	14001.6	1.62090	0.810448	−	0.585811i	$-0.199224\pi$
$421$	14001.6	1.62090	0.810448	+	0.585811i	$0.199224\pi$
$422$	− 1676.98i	− 0.193446i
$423$	534.948	0.0614895
$424$	−13562.0	−1.55337
$425$	0	0
$426$	−153.384	−0.0174448
$427$	−4907.80	−0.556218
$428$	− 816.537i	− 0.0922168i
$429$	3086.59	0.347371
$430$	− 3098.98i	− 0.347549i
$431$	− 6401.94i	− 0.715477i	−0.933822	−	0.357738i	$-0.883548\pi$
$431$	− 6401.94i	− 0.715477i	0.933822	−	0.357738i	$-0.116452\pi$
$432$	1483.56i	0.165227i
$433$	−13.2865	−0.00147462	−0.000737310	−	1.00000i	$-0.500235\pi$
$433$	−13.2865	−0.00147462	−0.000737310		1.00000i	$0.500235\pi$
$434$	1399.67	0.154807
$435$	91.2180i	0.0100542i
$436$	− 3865.57i	− 0.424603i
$437$	16385.7i	1.79367i
$438$	−2649.22	−0.289006
$439$	− 14294.8i	− 1.55410i	−0.629437	−	0.777052i	$-0.716714\pi$
$439$	− 14294.8i	− 1.55410i	0.629437	−	0.777052i	$-0.283286\pi$
$440$	−4174.25	−0.452272
$441$	−1128.00	−0.121801
$442$	0	0
$443$	16530.3	1.77286	0.886430	−	0.462862i	$-0.153177\pi$
$443$	16530.3	1.77286	0.886430	+	0.462862i	$0.153177\pi$
$444$	10919.3	1.16713
$445$	− 9048.39i	− 0.963898i
$446$	2212.76	0.234927
$447$	− 6593.22i	− 0.697648i
$448$	6296.01i	0.663970i
$449$	− 16571.9i	− 1.74182i	−0.491443	−	0.870910i	$-0.663530\pi$
$449$	− 16571.9i	− 1.74182i	0.491443	−	0.870910i	$-0.336470\pi$
$450$	−286.184	−0.0299796
$451$	−2498.00	−0.260812
$452$	− 10924.4i	− 1.13681i
$453$	7312.91i	0.758478i
$454$	− 3854.40i	− 0.398449i
$455$	−19445.5	−2.00356
$456$	12065.6i	1.23909i
$457$	14718.1	1.50653	0.753263	−	0.657719i	$-0.228479\pi$
$457$	14718.1	1.50653	0.753263	+	0.657719i	$0.228479\pi$
$458$	−7952.35	−0.811329
$459$	0	0
$460$	−11478.2	−1.16342
$461$	15330.5	1.54884	0.774419	−	0.632672i	$-0.218042\pi$
$461$	15330.5	1.54884	0.774419	+	0.632672i	$0.218042\pi$
$462$	3137.64i	0.315966i
$463$	11158.0	1.11999	0.559993	−	0.828497i	$-0.310804\pi$
$463$	11158.0	1.11999	0.559993	+	0.828497i	$0.310804\pi$
$464$	− 12.9321i	− 0.00129388i
$465$	2166.25i	0.216037i
$466$	− 1409.61i	− 0.140126i
$467$	14748.4	1.46140	0.730700	−	0.682699i	$-0.239194\pi$
$467$	14748.4	1.46140	0.730700	+	0.682699i	$0.239194\pi$
$468$	−543.737	−0.0537057
$469$	4705.19i	0.463253i
$470$	5657.07i	0.555194i
$471$	− 16961.2i	− 1.65931i
$472$	13353.5	1.30222
$473$	1835.06i	0.178385i
$474$	8452.48	0.819062
$475$	9492.45	0.916934
$476$	0	0
$477$	1372.65	0.131760
$478$	−256.219	−0.0245171
$479$	14914.1i	1.42264i	0.702868	+	0.711320i	$0.251902\pi$
$479$	14914.1i	1.42264i	−0.702868	+	0.711320i	$0.748098\pi$
$480$	−13461.4	−1.28006
$481$	18238.1i	1.72887i
$482$	5484.86i	0.518317i
$483$	21182.3i	1.99550i
$484$	−6310.77	−0.592672
$485$	−9998.04	−0.936057
$486$	956.813i	0.0893043i
$487$	− 3034.34i	− 0.282339i	−0.989985	−	0.141169i	$-0.954914\pi$
$487$	− 3034.34i	− 0.282339i	0.989985	−	0.141169i	$-0.0450862\pi$
$488$	− 3562.96i	− 0.330507i
$489$	−1884.18	−0.174245
$490$	− 11928.6i	− 1.09975i
$491$	−10032.0	−0.922073	−0.461036	−	0.887381i	$-0.652522\pi$
$491$	−10032.0	−0.922073	−0.461036	+	0.887381i	$0.652522\pi$
$492$	−5057.94	−0.463475
$493$	0	0
$494$	−8208.43	−0.747601
$495$	422.490	0.0383626
$496$	− 307.112i	− 0.0278019i
$497$	−572.207	−0.0516438
$498$	− 372.066i	− 0.0334793i
$499$	− 13921.2i	− 1.24890i	−0.781066	−	0.624449i	$-0.785324\pi$
$499$	− 13921.2i	− 1.24890i	0.781066	−	0.624449i	$-0.214676\pi$
$500$	− 3278.89i	− 0.293273i
$501$	−2852.86	−0.254404
$502$	−6847.23	−0.608779
$503$	10270.2i	0.910387i	0.890392	+	0.455194i	$0.150430\pi$
$503$	10270.2i	0.910387i	−0.890392	+	0.455194i	$0.849570\pi$
$504$	− 1357.03i	− 0.119934i
$505$	− 7970.00i	− 0.702298i
$506$	−3093.44	−0.271779
$507$	− 510.908i	− 0.0447539i
$508$	−8586.90	−0.749965
$509$	18020.5	1.56924	0.784621	−	0.619975i	$-0.212857\pi$
$509$	18020.5	1.56924	0.784621	+	0.619975i	$0.212857\pi$
$510$	0	0
$511$	−9883.03	−0.855576
$512$	3652.06	0.315234
$513$	− 16478.9i	− 1.41825i
$514$	7225.00	0.620002
$515$	− 2542.55i	− 0.217550i
$516$	3715.61i	0.316998i
$517$	− 3349.82i	− 0.284962i
$518$	−18539.8	−1.57257
$519$	−16554.6	−1.40013
$520$	− 14117.1i	− 1.19053i
$521$	18326.3i	1.54106i	0.637405	+	0.770529i	$0.280008\pi$
$521$	18326.3i	1.54106i	−0.637405	+	0.770529i	$0.719992\pi$
$522$	− 4.33071i	0 0.000363122i
$523$	−15369.6	−1.28502	−0.642512	−	0.766276i	$-0.722108\pi$
$523$	−15369.6	−1.28502	−0.642512	+	0.766276i	$0.722108\pi$
$524$	− 7244.42i	− 0.603958i
$525$	12271.2	1.02011
$526$	−7452.09	−0.617731
$527$	0	0
$528$	688.455	0.0567447
$529$	−8716.90	−0.716438
$530$	14515.8i	1.18967i
$531$	−1351.56	−0.110457
$532$	18333.6i	1.49410i
$533$	− 8448.08i	− 0.686542i
$534$	− 4937.65i	− 0.400137i
$535$	−2145.70	−0.173395
$536$	−3415.87	−0.275267
$537$	10435.0i	0.838556i
$538$	− 5788.09i	− 0.463833i
$539$	7063.50i	0.564464i
$540$	11543.5	0.919913
$541$	18606.9i	1.47870i	0.673323	+	0.739348i	$0.264866\pi$
$541$	18606.9i	1.47870i	−0.673323	+	0.739348i	$0.735134\pi$
$542$	−12184.0	−0.965586
$543$	−3399.16	−0.268641
$544$	0	0
$545$	−10157.9	−0.798382
$546$	−10611.3	−0.831725
$547$	− 3946.20i	− 0.308459i	−0.988035	−	0.154230i	$-0.950710\pi$
$547$	− 3946.20i	− 0.308459i	0.988035	−	0.154230i	$-0.0492896\pi$
$548$	−2667.60	−0.207946
$549$	360.619i	0.0280343i
$550$	1792.07i	0.138935i
$551$	143.646i	0.0111062i
$552$	−15377.9	−1.18574
$553$	31532.3	2.42476
$554$	− 5145.11i	− 0.394576i
$555$	− 28693.8i	− 2.19457i
$556$	5589.39i	0.426336i
$557$	14883.1	1.13217	0.566084	−	0.824347i	$-0.308458\pi$
$557$	14883.1	1.13217	0.566084	+	0.824347i	$0.308458\pi$
$558$	− 102.846i	− 0.00780253i
$559$	−6206.05	−0.469567
$560$	−4337.27	−0.327291
$561$	0	0
$562$	10818.7	0.812027
$563$	−25751.7	−1.92771	−0.963857	−	0.266420i	$-0.914159\pi$
$563$	−25751.7	−1.92771	−0.963857	+	0.266420i	$0.914159\pi$
$564$	− 6782.71i	− 0.506389i
$565$	−28707.0	−2.13755
$566$	− 4953.22i	− 0.367843i
$567$	− 19586.8i	− 1.45074i
$568$	− 415.410i	− 0.0306870i
$569$	2194.88	0.161712	0.0808561	−	0.996726i	$-0.474235\pi$
$569$	2194.88	0.161712	0.0808561	+	0.996726i	$0.474235\pi$
$570$	12914.2	0.948977
$571$	5227.87i	0.383151i	0.981478	+	0.191576i	$0.0613598\pi$
$571$	5227.87i	0.383151i	−0.981478	+	0.191576i	$0.938640\pi$
$572$	3404.86i	0.248889i
$573$	9574.77i	0.698066i
$574$	8587.81	0.624474
$575$	12098.3i	0.877453i
$576$	462.623	0.0334652
$577$	−6527.86	−0.470985	−0.235492	−	0.971876i	$-0.575670\pi$
$577$	−6527.86	−0.470985	−0.235492	+	0.971876i	$0.575670\pi$
$578$	0	0
$579$	347.323	0.0249296
$580$	−100.624	−0.00720376
$581$	− 1388.01i	− 0.0991122i
$582$	−5455.87	−0.388579
$583$	− 8595.50i	− 0.610617i
$584$	− 7174.87i	− 0.508387i
$585$	1428.83i	0.100983i
$586$	8535.65	0.601714
$587$	6648.24	0.467465	0.233733	−	0.972301i	$-0.424906\pi$
$587$	6648.24	0.467465	0.233733	+	0.972301i	$0.424906\pi$
$588$	14302.1i	1.00308i
$589$	3411.30i	0.238642i
$590$	− 14292.7i	− 0.997324i
$591$	15166.4	1.05561
$592$	4067.96i	0.282419i
$593$	1450.60	0.100453	0.0502267	−	0.998738i	$-0.484006\pi$
$593$	1450.60	0.100453	0.0502267	+	0.998738i	$0.484006\pi$
$594$	3111.04	0.214895
$595$	0	0
$596$	7273.09	0.499861
$597$	−6258.35	−0.429040
$598$	− 10461.8i	− 0.715411i
$599$	−10070.9	−0.686957	−0.343479	−	0.939161i	$-0.611605\pi$
$599$	−10070.9	−0.686957	−0.343479	+	0.939161i	$0.611605\pi$
$600$	8908.64i	0.606156i
$601$	− 9209.45i	− 0.625060i	−0.949908	−	0.312530i	$-0.898823\pi$
$601$	− 9209.45i	− 0.625060i	0.949908	−	0.312530i	$-0.101177\pi$
$602$	− 6308.69i	− 0.427115i
$603$	345.732	0.0233487
$604$	−8066.99	−0.543445
$605$	16583.5i	1.11440i
$606$	− 4349.18i	− 0.291540i
$607$	− 5813.89i	− 0.388762i	−0.980926	−	0.194381i	$-0.937730\pi$
$607$	− 5813.89i	− 0.388762i	0.980926	−	0.194381i	$-0.0622698\pi$
$608$	−21198.4	−1.41399
$609$	185.695i	0.0123559i
$610$	−3813.54	−0.253124
$611$	11328.9	0.750111
$612$	0	0
$613$	−23778.5	−1.56673	−0.783365	−	0.621562i	$-0.786498\pi$
$613$	−23778.5	−1.56673	−0.783365	+	0.621562i	$0.786498\pi$
$614$	3411.42	0.224224
$615$	13291.3i	0.871472i
$616$	−8497.65	−0.555812
$617$	− 19716.9i	− 1.28650i	−0.765655	−	0.643251i	$-0.777585\pi$
$617$	− 19716.9i	− 1.28650i	0.765655	−	0.643251i	$-0.222415\pi$
$618$	− 1387.45i	− 0.0903099i
$619$	6184.28i	0.401562i	0.979636	+	0.200781i	$0.0643480\pi$
$619$	6184.28i	0.401562i	−0.979636	+	0.200781i	$0.935652\pi$
$620$	−2389.62	−0.154790
$621$	21002.7	1.35718
$622$	4317.50i	0.278322i
$623$	− 18420.1i	− 1.18457i
$624$	2328.31i	0.149370i
$625$	−19081.0	−1.22119
$626$	7932.56i	0.506468i
$627$	−7647.13	−0.487077
$628$	18710.2	1.18888
$629$	0	0
$630$	−1452.47	−0.0918534
$631$	−14216.8	−0.896926	−0.448463	−	0.893801i	$-0.648028\pi$
$631$	−14216.8	−0.896926	−0.448463	+	0.893801i	$0.648028\pi$
$632$	22891.8i	1.44080i
$633$	5283.64	0.331763
$634$	1842.09i	0.115393i
$635$	22564.7i	1.41016i
$636$	− 17404.1i	− 1.08509i
$637$	−23888.3	−1.48585
$638$	−27.1187	−0.00168282
$639$	42.0450i	0.00260294i
$640$	− 16715.7i	− 1.03242i
$641$	− 14249.8i	− 0.878058i	−0.898473	−	0.439029i	$-0.855323\pi$
$641$	− 14249.8i	− 0.878058i	0.898473	−	0.439029i	$-0.144677\pi$
$642$	−1170.89	−0.0719805
$643$	− 1767.83i	− 0.108424i	−0.998529	−	0.0542118i	$-0.982735\pi$
$643$	− 1767.83i	− 0.108424i	0.998529	−	0.0542118i	$-0.0172646\pi$
$644$	−23366.5	−1.42977
$645$	9763.89	0.596051
$646$	0	0
$647$	−14274.5	−0.867369	−0.433685	−	0.901065i	$-0.642787\pi$
$647$	−14274.5	−0.867369	−0.433685	+	0.901065i	$0.642787\pi$
$648$	14219.6	0.862035
$649$	8463.40i	0.511891i
$650$	−6060.68	−0.365722
$651$	4409.90i	0.265496i
$652$	− 2078.47i	− 0.124845i
$653$	14516.1i	0.869924i	0.900449	+	0.434962i	$0.143238\pi$
$653$	14516.1i	0.869924i	−0.900449	+	0.434962i	$0.856762\pi$
$654$	−5543.13	−0.331427
$655$	−19036.9	−1.13562
$656$	− 1884.32i	− 0.112150i
$657$	726.193i	0.0431225i
$658$	11516.3i	0.682296i
$659$	23248.8	1.37427	0.687135	−	0.726530i	$-0.258868\pi$
$659$	23248.8	1.37427	0.687135	+	0.726530i	$0.258868\pi$
$660$	− 5356.83i	− 0.315931i
$661$	28695.7	1.68855	0.844277	−	0.535907i	$-0.180030\pi$
$661$	28695.7	1.68855	0.844277	+	0.535907i	$0.180030\pi$
$662$	−3391.51	−0.199116
$663$	0	0
$664$	1007.66	0.0588930
$665$	48177.0	2.80936
$666$	1362.28i	0.0792602i
$667$	−183.079	−0.0106280
$668$	− 3147.03i	− 0.182279i
$669$	6971.71i	0.402903i
$670$	3656.11i	0.210818i
$671$	2258.18	0.129920
$672$	−27403.8	−1.57310
$673$	− 6852.55i	− 0.392491i	−0.980555	−	0.196246i	$-0.937125\pi$
$673$	− 6852.55i	− 0.392491i	0.980555	−	0.196246i	$-0.0628750\pi$
$674$	− 3092.63i	− 0.176742i
$675$	− 12167.2i	− 0.693799i
$676$	563.590	0.0320659
$677$	2123.19i	0.120533i	0.998182	+	0.0602665i	$0.0191951\pi$
$677$	2123.19i	0.120533i	−0.998182	+	0.0602665i	$0.980805\pi$
$678$	−15665.2	−0.887345
$679$	−20353.3	−1.15035
$680$	0	0
$681$	12144.0	0.683345
$682$	−644.017	−0.0361594
$683$	4035.12i	0.226061i	0.993592	+	0.113030i	$0.0360558\pi$
$683$	4035.12i	0.226061i	−0.993592	+	0.113030i	$0.963944\pi$
$684$	1347.13	0.0753052
$685$	7009.92i	0.391001i
$686$	− 8326.18i	− 0.463404i
$687$	− 25055.3i	− 1.39144i
$688$	−1384.24	−0.0767060
$689$	29069.5	1.60734
$690$	16459.4i	0.908116i
$691$	20015.0i	1.10189i	0.834540	+	0.550947i	$0.185733\pi$
$691$	20015.0i	1.10189i	−0.834540	+	0.550947i	$0.814267\pi$
$692$	− 18261.7i	− 1.00319i
$693$	860.076	0.0471451
$694$	17252.6i	0.943661i
$695$	14687.8	0.801641
$696$	−134.811	−0.00734195
$697$	0	0
$698$	−14624.5	−0.793047
$699$	4441.23	0.240319
$700$	13536.6i	0.730905i
$701$	19025.4	1.02508	0.512539	−	0.858664i	$-0.328705\pi$
$701$	19025.4	1.02508	0.512539	+	0.858664i	$0.328705\pi$
$702$	10521.3i	0.565673i
$703$	− 45185.6i	− 2.42419i
$704$	− 2896.93i	− 0.155088i
$705$	−17823.6	−0.952164
$706$	−6368.69	−0.339502
$707$	− 16224.8i	− 0.863077i
$708$	17136.6i	0.909653i
$709$	11992.4i	0.635237i	0.948219	+	0.317619i	$0.102883\pi$
$709$	11992.4i	0.635237i	−0.948219	+	0.317619i	$0.897117\pi$
$710$	−444.626	−0.0235021
$711$	− 2316.96i	− 0.122212i
$712$	13372.6	0.703876
$713$	−4347.78	−0.228367
$714$	0	0
$715$	8947.31	0.467987
$716$	−11511.0	−0.600821
$717$	− 807.265i	− 0.0420472i
$718$	16423.1	0.853629
$719$	12005.7i	0.622721i	0.950292	+	0.311361i	$0.100785\pi$
$719$	12005.7i	0.622721i	−0.950292	+	0.311361i	$0.899215\pi$
$720$	318.697i	0.0164960i
$721$	− 5175.95i	− 0.267354i
$722$	9486.83	0.489007
$723$	−17281.0	−0.888919
$724$	− 3749.66i	− 0.192479i
$725$	106.060i	0.00543309i
$726$	9049.49i	0.462614i
$727$	−22403.9	−1.14294	−0.571468	−	0.820624i	$-0.693626\pi$
$727$	−22403.9	−1.14294	−0.571468	+	0.820624i	$0.693626\pi$
$728$	− 28738.5i	− 1.46308i
$729$	−20996.1	−1.06671
$730$	−7679.48	−0.389357
$731$	0	0
$732$	4572.36	0.230873
$733$	17400.5	0.876810	0.438405	−	0.898777i	$-0.355543\pi$
$733$	17400.5	0.876810	0.438405	+	0.898777i	$0.355543\pi$
$734$	− 19233.9i	− 0.967218i
$735$	37583.2	1.88609
$736$	− 27017.8i	− 1.35311i
$737$	− 2164.96i	− 0.108205i
$738$	− 631.022i	− 0.0314746i
$739$	7691.26	0.382852	0.191426	−	0.981507i	$-0.438689\pi$
$739$	7691.26	0.382852	0.191426	+	0.981507i	$0.438689\pi$
$740$	31652.6	1.57239
$741$	− 25862.1i	− 1.28214i
$742$	29550.3i	1.46203i
$743$	33377.5i	1.64805i	0.566551	+	0.824027i	$0.308277\pi$
$743$	33377.5i	1.64805i	−0.566551	+	0.824027i	$0.691723\pi$
$744$	−3201.50	−0.157759
$745$	− 19112.2i	− 0.939890i
$746$	18182.5	0.892371
$747$	−101.989	−0.00499543
$748$	0	0
$749$	−4368.06	−0.213092
$750$	−4701.84	−0.228916
$751$	30285.6i	1.47155i	0.677224	+	0.735777i	$0.263183\pi$
$751$	30285.6i	1.47155i	−0.677224	+	0.735777i	$0.736817\pi$
$752$	2526.88	0.122534
$753$	− 21573.4i	− 1.04406i
$754$	− 91.7139i	− 0.00442974i
$755$	21198.4i	1.02184i
$756$	23499.5	1.13051
$757$	13809.3	0.663023	0.331511	−	0.943451i	$-0.392441\pi$
$757$	13809.3	0.663023	0.331511	+	0.943451i	$0.392441\pi$
$758$	− 1717.81i	− 0.0823135i
$759$	− 9746.44i	− 0.466104i
$760$	34975.5i	1.66933i
$761$	33269.2	1.58477	0.792383	−	0.610023i	$-0.208840\pi$
$761$	33269.2	1.58477	0.792383	+	0.610023i	$0.208840\pi$
$762$	12313.4i	0.585390i
$763$	−20678.8	−0.981159
$764$	−10562.1	−0.500160
$765$	0	0
$766$	13044.1	0.615276
$767$	−28622.7	−1.34746
$768$	− 17657.0i	− 0.829613i
$769$	−22932.1	−1.07536	−0.537681	−	0.843148i	$-0.680700\pi$
$769$	−22932.1	−1.07536	−0.537681	+	0.843148i	$0.680700\pi$
$770$	9095.29i	0.425677i
$771$	22763.7i	1.06331i
$772$	383.137i	0.0178619i
$773$	−3142.46	−0.146218	−0.0731090	−	0.997324i	$-0.523292\pi$
$773$	−3142.46	−0.146218	−0.0731090	+	0.997324i	$0.523292\pi$
$774$	−463.555	−0.0215273
$775$	2518.73i	0.116742i
$776$	− 14776.1i	− 0.683545i
$777$	− 58412.9i	− 2.69698i
$778$	6012.85	0.277084
$779$	20930.4i	0.962657i
$780$	18116.5	0.831633
$781$	263.285	0.0120628
$782$	0	0
$783$	184.121	0.00840351
$784$	−5328.22	−0.242721
$785$	− 49166.7i	− 2.23546i
$786$	−10388.3	−0.471424
$787$	− 11712.3i	− 0.530492i	−0.964181	−	0.265246i	$-0.914547\pi$
$787$	− 11712.3i	− 0.530492i	0.964181	−	0.265246i	$-0.0854532\pi$
$788$	16730.3i	0.756337i
$789$	− 23479.1i	− 1.05942i
$790$	24501.8	1.10346
$791$	−58439.8	−2.62690
$792$	624.397i	0.0280139i
$793$	7637.04i	0.341991i
$794$	13244.6i	0.591981i
$795$	−45734.6	−2.04030
$796$	− 6903.68i	− 0.307405i
$797$	12691.2	0.564045	0.282022	−	0.959408i	$-0.408995\pi$
$797$	12691.2	0.564045	0.282022	+	0.959408i	$0.408995\pi$
$798$	26289.9	1.16623
$799$	0	0
$800$	−15651.8	−0.691717
$801$	−1353.49	−0.0597042
$802$	3438.33i	0.151386i
$803$	4547.39	0.199843
$804$	− 4383.60i	− 0.192286i
$805$	61402.6i	2.68839i
$806$	− 2178.03i	− 0.0951832i
$807$	18236.4	0.795480
$808$	11778.9	0.512845
$809$	− 22034.9i	− 0.957610i	−0.877921	−	0.478805i	$-0.841070\pi$
$809$	− 22034.9i	− 0.957610i	0.877921	−	0.478805i	$-0.158930\pi$
$810$	− 15219.7i	− 0.660203i
$811$	2987.60i	0.129357i	0.997906	+	0.0646787i	$0.0206023\pi$
$811$	2987.60i	0.129357i	−0.997906	+	0.0646787i	$0.979398\pi$
$812$	−204.843	−0.00885295
$813$	− 38387.9i	− 1.65599i
$814$	8530.55	0.367316
$815$	−5461.81	−0.234747
$816$	0	0
$817$	15375.7	0.658418
$818$	7178.59	0.306838
$819$	2908.72i	0.124101i
$820$	−14661.8	−0.624405
$821$	39918.9i	1.69693i	0.529251	+	0.848465i	$0.322473\pi$
$821$	39918.9i	1.69693i	−0.529251	+	0.848465i	$0.677527\pi$
$822$	3825.27i	0.162313i
$823$	15017.3i	0.636050i	0.948082	+	0.318025i	$0.103020\pi$
$823$	15017.3i	0.636050i	−0.948082	+	0.318025i	$0.896980\pi$
$824$	3757.63	0.158863
$825$	−5646.25	−0.238275
$826$	− 29096.1i	− 1.22564i
$827$	3898.57i	0.163926i	0.996635	+	0.0819628i	$0.0261189\pi$
$827$	3898.57i	0.163926i	−0.996635	+	0.0819628i	$0.973881\pi$
$828$	1716.94i	0.0720627i
$829$	−4410.66	−0.184787	−0.0923937	−	0.995723i	$-0.529452\pi$
$829$	−4410.66	−0.184787	−0.0923937	+	0.995723i	$0.529452\pi$
$830$	− 1078.53i	− 0.0451041i
$831$	16210.6	0.676702
$832$	9797.23	0.408243
$833$	0	0
$834$	8015.05	0.332780
$835$	−8269.77	−0.342739
$836$	− 8435.67i	− 0.348988i
$837$	4372.52	0.180569
$838$	− 6706.47i	− 0.276457i
$839$	29154.8i	1.19969i	0.800118	+	0.599843i	$0.204770\pi$
$839$	29154.8i	1.19969i	−0.800118	+	0.599843i	$0.795230\pi$
$840$	45214.0i	1.85718i
$841$	24387.4	0.999934
$842$	22148.3	0.906510
$843$	34086.2i	1.39264i
$844$	5828.46i	0.237706i
$845$	− 1481.00i	− 0.0602936i
$846$	846.202	0.0343889
$847$	33759.5i	1.36953i
$848$	6483.86	0.262567
$849$	15606.0	0.630856
$850$	0	0
$851$	57590.0	2.31981
$852$	533.097	0.0214362
$853$	− 24310.2i	− 0.975808i	−0.872897	−	0.487904i	$-0.837762\pi$
$853$	− 24310.2i	− 0.975808i	0.872897	−	0.487904i	$-0.162238\pi$
$854$	−7763.35	−0.311073
$855$	− 3539.99i	− 0.141596i
$856$	− 3171.12i	− 0.126620i
$857$	13081.1i	0.521403i	0.965419	+	0.260701i	$0.0839538\pi$
$857$	13081.1i	0.521403i	−0.965419	+	0.260701i	$0.916046\pi$
$858$	4882.49	0.194272
$859$	−1740.97	−0.0691514	−0.0345757	−	0.999402i	$-0.511008\pi$
$859$	−1740.97	−0.0691514	−0.0345757	+	0.999402i	$0.511008\pi$
$860$	10770.7i	0.427067i
$861$	27057.4i	1.07098i
$862$	− 10126.8i	− 0.400141i
$863$	−15561.9	−0.613826	−0.306913	−	0.951738i	$-0.599296\pi$
$863$	−15561.9	−0.613826	−0.306913	+	0.951738i	$0.599296\pi$
$864$	27171.5i	1.06990i
$865$	−47988.0	−1.88629
$866$	−21.0172	−0.000824703 0
$867$	0	0
$868$	−4864.63	−0.190226
$869$	−14508.7	−0.566368
$870$	144.292i	0.00562295i
$871$	7321.76	0.284832
$872$	− 15012.4i	− 0.583010i
$873$	1495.54i	0.0579797i
$874$	25919.5i	1.00314i
$875$	−17540.4	−0.677685
$876$	9207.54	0.355130
$877$	32043.6i	1.23379i	0.787045	+	0.616895i	$0.211610\pi$
$877$	32043.6i	1.23379i	−0.787045	+	0.616895i	$0.788390\pi$
$878$	− 22612.0i	− 0.869156i
$879$	26893.1i	1.03195i
$880$	1995.67	0.0764478
$881$	− 38673.0i	− 1.47892i	−0.673202	−	0.739458i	$-0.735082\pi$
$881$	− 38673.0i	− 1.47892i	0.673202	−	0.739458i	$-0.264918\pi$
$882$	−1784.31	−0.0681191
$883$	−45977.4	−1.75228	−0.876140	−	0.482056i	$-0.839890\pi$
$883$	−45977.4	−1.75228	−0.876140	+	0.482056i	$0.839890\pi$
$884$	0	0
$885$	45031.7	1.71042
$886$	26148.3	0.991499
$887$	− 5325.60i	− 0.201596i	−0.994907	−	0.100798i	$-0.967860\pi$
$887$	− 5325.60i	− 0.201596i	0.994907	−	0.100798i	$-0.0321396\pi$
$888$	42406.5	1.60256
$889$	45935.6i	1.73299i
$890$	− 14313.1i	− 0.539074i
$891$	9012.30i	0.338859i
$892$	−7690.60	−0.288677
$893$	−28067.7	−1.05179
$894$	− 10429.4i	− 0.390170i
$895$	30248.7i	1.12972i
$896$	− 34028.8i	− 1.26877i
$897$	32961.8	1.22694
$898$	− 26214.1i	− 0.974139i
$899$	−38.1150	−0.00141402
$900$	994.649	0.0368389
$901$	0	0
$902$	−3951.44	−0.145863
$903$	19876.7	0.732507
$904$	− 42426.1i	− 1.56092i
$905$	−9853.37	−0.361919
$906$	11567.9i	0.424190i
$907$	12188.4i	0.446207i	0.974795	+	0.223104i	$0.0716189\pi$
$907$	12188.4i	0.446207i	−0.974795	+	0.223104i	$0.928381\pi$
$908$	13396.2i	0.489613i
$909$	−1192.18	−0.0435006
$910$	−30759.7	−1.12052
$911$	− 24738.8i	− 0.899707i	−0.893102	−	0.449854i	$-0.851476\pi$
$911$	− 24738.8i	− 0.899707i	0.893102	−	0.449854i	$-0.148524\pi$
$912$	− 5768.48i	− 0.209444i
$913$	638.652i	0.0231504i
$914$	23281.6	0.842547
$915$	− 12015.3i	− 0.434111i
$916$	27638.9	0.996959
$917$	−38754.0	−1.39561
$918$	0	0
$919$	1584.70	0.0568820	0.0284410	−	0.999595i	$-0.490946\pi$
$919$	1584.70	0.0568820	0.0284410	+	0.999595i	$0.490946\pi$
$920$	−44577.0	−1.59746
$921$	10748.3i	0.384548i
$922$	24250.5	0.866211
$923$	890.412i	0.0317533i
$924$	− 10905.1i	− 0.388258i
$925$	− 33362.7i	− 1.18590i
$926$	17650.1	0.626369
$927$	−380.322	−0.0134751
$928$	− 236.852i	− 0.00837829i
$929$	7266.92i	0.256642i	0.991733	+	0.128321i	$0.0409587\pi$
$929$	7266.92i	0.256642i	−0.991733	+	0.128321i	$0.959041\pi$
$930$	3426.66i	0.120822i
$931$	59184.1	2.08344
$932$	4899.19i	0.172187i
$933$	−13603.1	−0.477325
$934$	23329.6	0.817310
$935$	0	0
$936$	−2111.67	−0.0737416
$937$	3617.62	0.126129	0.0630643	−	0.998009i	$-0.479913\pi$
$937$	3617.62	0.126129	0.0630643	+	0.998009i	$0.479913\pi$
$938$	7442.86i	0.259081i
$939$	−24992.9	−0.868598
$940$	− 19661.5i	− 0.682220i
$941$	− 35841.7i	− 1.24166i	−0.783944	−	0.620832i	$-0.786795\pi$
$941$	− 35841.7i	− 1.24166i	0.783944	−	0.620832i	$-0.213205\pi$
$942$	− 26830.0i	− 0.927991i
$943$	−26676.3	−0.921208
$944$	−6384.21	−0.220115
$945$	− 61751.9i	− 2.12570i
$946$	2902.77i	0.0997643i
$947$	33544.8i	1.15107i	0.817778	+	0.575534i	$0.195206\pi$
$947$	33544.8i	1.15107i	−0.817778	+	0.575534i	$0.804794\pi$
$948$	−29377.1	−1.00646
$949$	15379.0i	0.526052i
$950$	15015.5	0.512809
$951$	−5803.85	−0.197900
$952$	0	0
$953$	−51190.9	−1.74002	−0.870008	−	0.493038i	$-0.835886\pi$
$953$	−51190.9	−1.74002	−0.870008	+	0.493038i	$0.835886\pi$
$954$	2171.32	0.0736886
$955$	27755.0i	0.940452i
$956$	890.506	0.0301266
$957$	− 85.4425i	− 0.00288606i
$958$	23591.8i	0.795633i
$959$	14270.3i	0.480514i
$960$	−15413.9	−0.518209
$961$	28885.8	0.969616
$962$	28849.8i	0.966896i
$963$	320.960i	0.0107402i
$964$	− 19063.0i	− 0.636906i
$965$	1006.81	0.0335858
$966$	33507.0i	1.11602i
$967$	−11754.1	−0.390885	−0.195442	−	0.980715i	$-0.562614\pi$
$967$	−11754.1	−0.390885	−0.195442	+	0.980715i	$0.562614\pi$
$968$	−24508.7	−0.813779
$969$	0	0
$970$	−15815.3	−0.523503
$971$	16608.7	0.548919	0.274459	−	0.961599i	$-0.411501\pi$
$971$	16608.7	0.548919	0.274459	+	0.961599i	$0.411501\pi$
$972$	− 3325.46i	− 0.109737i
$973$	29900.4	0.985164
$974$	− 4799.84i	− 0.157902i
$975$	− 19095.3i	− 0.627218i
$976$	1703.42i	0.0558659i
$977$	47963.6	1.57062	0.785308	−	0.619105i	$-0.212504\pi$
$977$	47963.6	1.57062	0.785308	+	0.619105i	$0.212504\pi$
$978$	−2980.48	−0.0974490
$979$	8475.48i	0.276688i
$980$	41458.5i	1.35137i
$981$	1519.46i	0.0494521i
$982$	−15869.0	−0.515683
$983$	− 49858.2i	− 1.61773i	−0.587993	−	0.808866i	$-0.700082\pi$
$983$	− 49858.2i	− 1.61773i	0.587993	−	0.808866i	$-0.299918\pi$
$984$	−19643.1	−0.636382
$985$	43964.0	1.42214
$986$	0	0
$987$	−36284.1	−1.17015
$988$	28528.9	0.918649
$989$	19596.7i	0.630068i
$990$	668.311	0.0214549
$991$	− 50467.6i	− 1.61772i	−0.588005	−	0.808858i	$-0.700086\pi$
$991$	− 50467.6i	− 1.61772i	0.588005	−	0.808858i	$-0.299914\pi$
$992$	− 5624.78i	− 0.180027i
$993$	− 10685.6i	− 0.341487i
$994$	−905.139	−0.0288826
$995$	−18141.5	−0.578014
$996$	1293.14i	0.0411392i
$997$	− 40927.6i	− 1.30009i	−0.759895	−	0.650046i	$-0.774750\pi$
$997$	− 40927.6i	− 1.30009i	0.759895	−	0.650046i	$-0.225250\pi$
$998$	− 22021.2i	− 0.698465i
$999$	−57917.6	−1.83427

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.d.288.6		8
17.4	even	4		289.4.a.c.1.2	✓	4
17.13	even	4		289.4.a.d.1.2	yes	4
17.16	even	2	inner	289.4.b.d.288.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.c.1.2	✓	4	17.4	even	4
289.4.a.d.1.2	yes	4	17.13	even	4
289.4.b.d.288.5		8	17.16	even	2	inner
289.4.b.d.288.6		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	289.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.58184 q^{2} +4.98387i q^{3} -5.49778 q^{4} +14.4471i q^{5} +7.88368i q^{6} +29.4104i q^{7} -21.3513 q^{8} +2.16104 q^{9} +22.8530i q^{10} -13.5324i q^{11} -27.4002i q^{12} +45.7656 q^{13} +46.5225i q^{14} -72.0024 q^{15} +10.2079 q^{16} +3.41842 q^{18} -113.386 q^{19} -79.4269i q^{20} -146.578 q^{21} -21.4060i q^{22} -144.513i q^{23} -106.412i q^{24} -83.7182 q^{25} +72.3938 q^{26} +145.335i q^{27} -161.692i q^{28} -1.26688i q^{29} -113.896 q^{30} -30.0858i q^{31} +186.958 q^{32} +67.4435 q^{33} -424.894 q^{35} -11.8809 q^{36} +398.512i q^{37} -179.358 q^{38} +228.090i q^{39} -308.464i q^{40} -184.595i q^{41} -231.862 q^{42} -135.605 q^{43} +74.3979i q^{44} +31.2207i q^{45} -228.596i q^{46} +247.542 q^{47} +50.8748i q^{48} -521.971 q^{49} -132.429 q^{50} -251.609 q^{52} +635.182 q^{53} +229.896i q^{54} +195.503 q^{55} -627.951i q^{56} -565.100i q^{57} -2.00399i q^{58} -625.420 q^{59} +395.853 q^{60} +166.873i q^{61} -47.5909i q^{62} +63.5570i q^{63} +214.074 q^{64} +661.179i q^{65} +106.685 q^{66} +159.984 q^{67} +720.232 q^{69} -672.115 q^{70} +19.4559i q^{71} -46.1411 q^{72} +336.039i q^{73} +630.382i q^{74} -417.241i q^{75} +623.370 q^{76} +397.992 q^{77} +360.801i q^{78} -1072.15i q^{79} +147.474i q^{80} -665.982 q^{81} -291.999i q^{82} -47.1944 q^{83} +805.852 q^{84} -214.506 q^{86} +6.31394 q^{87} +288.934i q^{88} -626.312 q^{89} +49.3862i q^{90} +1345.98i q^{91} +794.499i q^{92} +149.944 q^{93} +391.572 q^{94} -1638.09i q^{95} +931.774i q^{96} +692.045i q^{97} -825.674 q^{98} -29.2439i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 22 q^{4} - 120 q^{8} - 12 q^{9} - 44 q^{13} - 108 q^{15} + 126 q^{16} - 668 q^{18} - 44 q^{19} - 704 q^{21} - 756 q^{25} + 896 q^{26} + 626 q^{30} - 662 q^{32} + 188 q^{33} - 484 q^{35}+ \cdots - 3754 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 22 * q^4 - 120 * q^8 - 12 * q^9 - 44 * q^13 - 108 * q^15 + 126 * q^16 - 668 * q^18 - 44 * q^19 - 704 * q^21 - 756 * q^25 + 896 * q^26 + 626 * q^30 - 662 * q^32 + 188 * q^33 - 484 * q^35 - 282 * q^36 - 1048 * q^38 - 2910 * q^42 + 228 * q^43 + 20 * q^47 - 2012 * q^49 + 1610 * q^50 - 3074 * q^52 - 100 * q^53 - 2632 * q^55 - 1992 * q^59 + 434 * q^60 - 300 * q^64 + 2180 * q^66 - 1736 * q^67 - 2256 * q^69 - 2104 * q^70 - 78 * q^72 + 1746 * q^76 + 1788 * q^77 + 2160 * q^81 - 1700 * q^83 + 886 * q^84 + 4822 * q^86 + 768 * q^87 + 1568 * q^89 + 3100 * q^93 - 2238 * q^94 - 3754 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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