LMFDB - Embedded newform 1617.2.c.b.538.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(538,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1617.538");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1617.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.9118100068$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			538.19
Character	$\chi$	$=$	1617.538
Dual form			1617.2.c.b.538.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.01496i q^{2} -1.00000i q^{3} +0.969853 q^{4} +1.37003i q^{5} -1.01496 q^{6} -3.01429i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.01496i q^{2} -1.00000i q^{3} +0.969853 q^{4} +1.37003i q^{5} -1.01496 q^{6} -3.01429i q^{8} -1.00000 q^{9} +1.39053 q^{10} +(1.32773 + 3.03927i) q^{11} -0.969853i q^{12} -0.625331 q^{13} +1.37003 q^{15} -1.11968 q^{16} -1.13716 q^{17} +1.01496i q^{18} +3.04466 q^{19} +1.32873i q^{20} +(3.08474 - 1.34759i) q^{22} +7.31454 q^{23} -3.01429 q^{24} +3.12302 q^{25} +0.634687i q^{26} +1.00000i q^{27} -5.55588i q^{29} -1.39053i q^{30} -2.26924i q^{31} -4.89214i q^{32} +(3.03927 - 1.32773i) q^{33} +1.15418i q^{34} -0.969853 q^{36} +4.09821 q^{37} -3.09022i q^{38} +0.625331i q^{39} +4.12966 q^{40} -7.17337 q^{41} -3.72728i q^{43} +(1.28770 + 2.94764i) q^{44} -1.37003i q^{45} -7.42398i q^{46} +3.00283i q^{47} +1.11968i q^{48} -3.16975i q^{50} +1.13716i q^{51} -0.606479 q^{52} +0.473245 q^{53} +1.01496 q^{54} +(-4.16388 + 1.81902i) q^{55} -3.04466i q^{57} -5.63900 q^{58} +3.07242i q^{59} +1.32873 q^{60} +13.7593 q^{61} -2.30319 q^{62} -7.20470 q^{64} -0.856721i q^{65} +(-1.34759 - 3.08474i) q^{66} +4.78603 q^{67} -1.10288 q^{68} -7.31454i q^{69} +3.67206 q^{71} +3.01429i q^{72} +13.2048 q^{73} -4.15953i q^{74} -3.12302i q^{75} +2.95288 q^{76} +0.634687 q^{78} +2.26033i q^{79} -1.53399i q^{80} +1.00000 q^{81} +7.28070i q^{82} -8.93208 q^{83} -1.55794i q^{85} -3.78304 q^{86} -5.55588 q^{87} +(9.16122 - 4.00215i) q^{88} +7.47349i q^{89} -1.39053 q^{90} +7.09403 q^{92} -2.26924 q^{93} +3.04776 q^{94} +4.17127i q^{95} -4.89214 q^{96} -14.5069i q^{97} +(-1.32773 - 3.03927i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) $ 48 * q - 64 * q^4 - 48 * q^9	$ 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) $ 48 * q - 64 * q^4 - 48 * q^9 - 16 * q^11 + 64 * q^16 + 16 * q^22 + 32 * q^23 - 80 * q^25 + 64 * q^36 - 96 * q^37 - 32 * q^44 + 64 * q^53 + 48 * q^58 - 48 * q^60 - 240 * q^64 + 96 * q^67 - 32 * q^71 + 48 * q^78 + 48 * q^81 - 96 * q^86 - 48 * q^88 - 32 * q^92 + 96 * q^93 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1617\mathbb{Z}\right)^\times$.

$n$	$199$	$442$	$1079$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	1.01496i		−	0.717686i	−0.933398	−	0.358843i	$-0.883171\pi$
$2$		−	1.01496i		−	0.717686i	0.933398	−	0.358843i	$-0.116829\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	0.969853			0.484926
$5$			1.37003i			0.612695i	0.951920	+	0.306348i	$0.0991070\pi$
$5$			1.37003i			0.612695i	−0.951920	+	0.306348i	$0.900893\pi$
$6$	−1.01496			−0.414356
$7$		0			0
$7$		0			0
$8$		−	3.01429i		−	1.06571i
$9$	−1.00000			−0.333333
$10$	1.39053			0.439723
$11$	1.32773	+	3.03927i	0.400324	+	0.916373i
$11$	1.32773	+	3.03927i	0.400324	+	0.916373i
$12$		−	0.969853i		−	0.279972i
$13$	−0.625331			−0.173436			−0.0867178	−	0.996233i	$-0.527638\pi$
$13$	−0.625331			−0.173436			−0.0867178	+	0.996233i	$0.527638\pi$
$14$		0			0
$15$	1.37003			0.353740
$16$	−1.11968			−0.279920
$17$	−1.13716			−0.275802			−0.137901	−	0.990446i	$-0.544036\pi$
$17$	−1.13716			−0.275802			−0.137901	+	0.990446i	$0.544036\pi$
$18$			1.01496i			0.239229i
$19$	3.04466			0.698494			0.349247	−	0.937031i	$-0.386437\pi$
$19$	3.04466			0.698494			0.349247	+	0.937031i	$0.386437\pi$
$20$			1.32873i			0.297112i
$21$		0			0
$22$	3.08474	−	1.34759i	0.657669	−	0.287307i
$23$	7.31454			1.52519			0.762594	−	0.646877i	$-0.223925\pi$
$23$	7.31454			1.52519			0.762594	+	0.646877i	$0.223925\pi$
$24$	−3.01429			−0.615289
$25$	3.12302			0.624605
$26$			0.634687i			0.124472i
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	5.55588i		−	1.03170i	−0.856679	−	0.515850i	$-0.827476\pi$
$29$		−	5.55588i		−	1.03170i	0.856679	−	0.515850i	$-0.172524\pi$
$30$		−	1.39053i		−	0.253874i
$31$		−	2.26924i		−	0.407568i	−0.979016	−	0.203784i	$-0.934676\pi$
$31$		−	2.26924i		−	0.407568i	0.979016	−	0.203784i	$-0.0653240\pi$
$32$		−	4.89214i		−	0.864817i
$33$	3.03927	−	1.32773i	0.529068	−	0.231127i
$34$			1.15418i			0.197940i
$35$		0			0
$36$	−0.969853			−0.161642
$37$	4.09821			0.673742			0.336871	−	0.941551i	$-0.390631\pi$
$37$	4.09821			0.673742			0.336871	+	0.941551i	$0.390631\pi$
$38$		−	3.09022i		−	0.501299i
$39$			0.625331i			0.100133i
$40$	4.12966			0.652956
$41$	−7.17337			−1.12029			−0.560146	−	0.828394i	$-0.689255\pi$
$41$	−7.17337			−1.12029			−0.560146	+	0.828394i	$0.689255\pi$
$42$		0			0
$43$		−	3.72728i		−	0.568405i	−0.958764	−	0.284202i	$-0.908271\pi$
$43$		−	3.72728i		−	0.568405i	0.958764	−	0.284202i	$-0.0917287\pi$
$44$	1.28770	+	2.94764i	0.194128	+	0.444374i
$45$		−	1.37003i		−	0.204232i
$46$		−	7.42398i		−	1.09461i
$47$			3.00283i			0.438008i	0.975724	+	0.219004i	$0.0702808\pi$
$47$			3.00283i			0.438008i	−0.975724	+	0.219004i	$0.929719\pi$
$48$			1.11968i			0.161612i
$49$		0			0
$50$		−	3.16975i		−	0.448270i
$51$			1.13716i			0.159235i
$52$	−0.606479			−0.0841035
$53$	0.473245			0.0650052			0.0325026	−	0.999472i	$-0.489652\pi$
$53$	0.473245			0.0650052			0.0325026	+	0.999472i	$0.489652\pi$
$54$	1.01496			0.138119
$55$	−4.16388	+	1.81902i	−0.561458	+	0.245277i
$56$		0			0
$57$		−	3.04466i		−	0.403276i
$58$	−5.63900			−0.740437
$59$			3.07242i			0.399996i	0.979796	+	0.199998i	$0.0640935\pi$
$59$			3.07242i			0.399996i	−0.979796	+	0.199998i	$0.935907\pi$
$60$	1.32873			0.171538
$61$	13.7593			1.76169			0.880847	−	0.473400i	$-0.156974\pi$
$61$	13.7593			1.76169			0.880847	+	0.473400i	$0.156974\pi$
$62$	−2.30319			−0.292506
$63$		0			0
$64$	−7.20470			−0.900587
$65$		−	0.856721i		−	0.106263i
$66$	−1.34759	−	3.08474i	−0.165877	−	0.379705i
$67$	4.78603			0.584707			0.292353	−	0.956310i	$-0.405562\pi$
$67$	4.78603			0.584707			0.292353	+	0.956310i	$0.405562\pi$
$68$	−1.10288			−0.133744
$69$		−	7.31454i		−	0.880568i
$70$		0			0
$71$	3.67206			0.435793			0.217897	−	0.975972i	$-0.430080\pi$
$71$	3.67206			0.435793			0.217897	+	0.975972i	$0.430080\pi$
$72$			3.01429i			0.355237i
$73$	13.2048			1.54550			0.772752	−	0.634708i	$-0.218880\pi$
$73$	13.2048			1.54550			0.772752	+	0.634708i	$0.218880\pi$
$74$		−	4.15953i		−	0.483535i
$75$		−	3.12302i		−	0.360616i
$76$	2.95288			0.338718
$77$		0			0
$78$	0.634687			0.0718642
$79$			2.26033i			0.254307i	0.991883	+	0.127153i	$0.0405840\pi$
$79$			2.26033i			0.254307i	−0.991883	+	0.127153i	$0.959416\pi$
$80$		−	1.53399i		−	0.171506i
$81$	1.00000			0.111111
$82$			7.28070i			0.804019i
$83$	−8.93208			−0.980423			−0.490212	−	0.871603i	$-0.663081\pi$
$83$	−8.93208			−0.980423			−0.490212	+	0.871603i	$0.663081\pi$
$84$		0			0
$85$		−	1.55794i		−	0.168983i
$86$	−3.78304			−0.407936
$87$	−5.55588			−0.595653
$88$	9.16122	−	4.00215i	0.976590	−	0.426630i
$89$			7.47349i			0.792189i	0.918210	+	0.396094i	$0.129635\pi$
$89$			7.47349i			0.792189i	−0.918210	+	0.396094i	$0.870365\pi$
$90$	−1.39053			−0.146574
$91$		0			0
$92$	7.09403			0.739604
$93$	−2.26924			−0.235309
$94$	3.04776			0.314352
$95$			4.17127i			0.427964i
$96$	−4.89214			−0.499302
$97$		−	14.5069i		−	1.47296i	−0.676461	−	0.736478i	$-0.736487\pi$
$97$		−	14.5069i		−	1.47296i	0.676461	−	0.736478i	$-0.263513\pi$
$98$		0			0
$99$	−1.32773	−	3.03927i	−0.133441	−	0.305458i
$100$	3.02887			0.302887
$101$	−14.1499			−1.40796			−0.703982	−	0.710218i	$-0.748596\pi$
$101$	−14.1499			−1.40796			−0.703982	+	0.710218i	$0.748596\pi$
$102$	1.15418			0.114280
$103$		−	4.26874i		−	0.420611i	−0.977636	−	0.210306i	$-0.932554\pi$
$103$		−	4.26874i		−	0.420611i	0.977636	−	0.210306i	$-0.0674459\pi$
$104$			1.88493i			0.184832i
$105$		0			0
$106$		−	0.480325i		−	0.0466533i
$107$		−	1.16689i		−	0.112808i	−0.998408	−	0.0564040i	$-0.982037\pi$
$107$		−	1.16689i		−	0.112808i	0.998408	−	0.0564040i	$-0.0179635\pi$
$108$			0.969853i			0.0933241i
$109$		−	18.8138i		−	1.80203i	−0.433784	−	0.901017i	$-0.642822\pi$
$109$		−	18.8138i		−	1.80203i	0.433784	−	0.901017i	$-0.357178\pi$
$110$	1.84624	+	4.22618i	0.176032	+	0.402950i
$111$		−	4.09821i		−	0.388985i
$112$		0			0
$113$	11.2307			1.05649			0.528247	−	0.849091i	$-0.322849\pi$
$113$	11.2307			1.05649			0.528247	+	0.849091i	$0.322849\pi$
$114$	−3.09022			−0.289425
$115$			10.0211i			0.934475i
$116$		−	5.38838i		−	0.500299i
$117$	0.625331			0.0578119
$118$	3.11839			0.287071
$119$		0			0
$120$		−	4.12966i		−	0.376984i
$121$	−7.47429	+	8.07063i	−0.679481	+	0.733693i
$122$		−	13.9651i		−	1.26434i
$123$			7.17337i			0.646801i
$124$		−	2.20083i		−	0.197640i
$125$			11.1288i			0.995387i
$126$		0			0
$127$			10.8122i			0.959426i	0.877425	+	0.479713i	$0.159259\pi$
$127$			10.8122i			0.959426i	−0.877425	+	0.479713i	$0.840741\pi$
$128$		−	2.47179i		−	0.218478i
$129$	−3.72728			−0.328169
$130$	−0.869539			−0.0762636
$131$	0.402389			0.0351569			0.0175784	−	0.999845i	$-0.494404\pi$
$131$	0.402389			0.0351569			0.0175784	+	0.999845i	$0.494404\pi$
$132$	2.94764	−	1.28770i	0.256559	−	0.112080i
$133$		0			0
$134$		−	4.85764i		−	0.419636i
$135$	−1.37003			−0.117913
$136$			3.42773i			0.293926i
$137$	−20.4546			−1.74755			−0.873777	−	0.486326i	$-0.838337\pi$
$137$	−20.4546			−1.74755			−0.873777	+	0.486326i	$0.838337\pi$
$138$	−7.42398			−0.631971
$139$	−19.9646			−1.69338			−0.846688	−	0.532090i	$-0.821407\pi$
$139$	−19.9646			−1.69338			−0.846688	+	0.532090i	$0.821407\pi$
$140$		0			0
$141$	3.00283			0.252884
$142$		−	3.72700i		−	0.312763i
$143$	−0.830269	−	1.90055i	−0.0694306	−	0.158932i
$144$	1.11968			0.0933066
$145$	7.61171			0.632118
$146$		−	13.4024i		−	1.10919i
$147$		0			0
$148$	3.97466			0.326715
$149$			16.0308i			1.31330i	0.754197	+	0.656648i	$0.228026\pi$
$149$			16.0308i			1.31330i	−0.754197	+	0.656648i	$0.771974\pi$
$150$	−3.16975			−0.258809
$151$			1.82731i			0.148704i	0.997232	+	0.0743522i	$0.0236889\pi$
$151$			1.82731i			0.148704i	−0.997232	+	0.0743522i	$0.976311\pi$
$152$		−	9.17749i		−	0.744393i
$153$	1.13716			0.0919341
$154$		0			0
$155$	3.10892			0.249715
$156$			0.606479i			0.0485572i
$157$			20.2714i			1.61784i	0.587921	+	0.808919i	$0.299947\pi$
$157$			20.2714i			1.61784i	−0.587921	+	0.808919i	$0.700053\pi$
$158$	2.29415			0.182512
$159$		−	0.473245i		−	0.0375308i
$160$	6.70237			0.529869
$161$		0			0
$162$		−	1.01496i		−	0.0797429i
$163$	3.85496			0.301944			0.150972	−	0.988538i	$-0.451760\pi$
$163$	3.85496			0.301944			0.150972	+	0.988538i	$0.451760\pi$
$164$	−6.95712			−0.543259
$165$	1.81902	+	4.16388i	0.141611	+	0.324158i
$166$			9.06572i			0.703636i
$167$	5.31934			0.411623			0.205811	−	0.978592i	$-0.434017\pi$
$167$	5.31934			0.411623			0.205811	+	0.978592i	$0.434017\pi$
$168$		0			0
$169$	−12.6090			−0.969920
$170$	−1.58125			−0.121277
$171$	−3.04466			−0.232831
$172$		−	3.61491i		−	0.275634i
$173$	−10.9221			−0.830395			−0.415197	−	0.909731i	$-0.636288\pi$
$173$	−10.9221			−0.830395			−0.415197	+	0.909731i	$0.636288\pi$
$174$			5.63900i			0.427492i
$175$		0			0
$176$	−1.48663	−	3.40301i	−0.112059	−	0.256511i
$177$	3.07242			0.230938
$178$	7.58531			0.568543
$179$	−17.2651			−1.29046			−0.645228	−	0.763990i	$-0.723238\pi$
$179$	−17.2651			−1.29046			−0.645228	+	0.763990i	$0.723238\pi$
$180$		−	1.32873i		−	0.0990374i
$181$		−	5.34619i		−	0.397379i	−0.980062	−	0.198690i	$-0.936331\pi$
$181$		−	5.34619i		−	0.397379i	0.980062	−	0.198690i	$-0.0636686\pi$
$182$		0			0
$183$		−	13.7593i		−	1.01712i
$184$		−	22.0481i		−	1.62541i
$185$			5.61467i			0.412799i
$186$			2.30319i			0.168878i
$187$	−1.50984	−	3.45614i	−0.110410	−	0.252738i
$188$			2.91230i			0.212402i
$189$		0			0
$190$	4.23368			0.307144
$191$	−9.12310			−0.660124			−0.330062	−	0.943959i	$-0.607070\pi$
$191$	−9.12310			−0.660124			−0.330062	+	0.943959i	$0.607070\pi$
$192$			7.20470i			0.519954i
$193$			0.275464i			0.0198283i	0.999951	+	0.00991417i	$0.00315583\pi$
$193$			0.275464i			0.0198283i	−0.999951	+	0.00991417i	$0.996844\pi$
$194$	−14.7240			−1.05712
$195$	−0.856721			−0.0613511
$196$		0			0
$197$			2.83392i			0.201909i	0.994891	+	0.100954i	$0.0321896\pi$
$197$			2.83392i			0.201909i	−0.994891	+	0.100954i	$0.967810\pi$
$198$	−3.08474	+	1.34759i	−0.219223	+	0.0957691i
$199$		−	9.87491i		−	0.700013i	−0.936747	−	0.350007i	$-0.886179\pi$
$199$		−	9.87491i		−	0.700013i	0.936747	−	0.350007i	$-0.113821\pi$
$200$		−	9.41369i		−	0.665648i
$201$		−	4.78603i		−	0.337581i
$202$			14.3616i			1.01048i
$203$		0			0
$204$			1.10288i			0.0772171i
$205$		−	9.82772i		−	0.686398i
$206$	−4.33261			−0.301867
$207$	−7.31454			−0.508396
$208$	0.700171			0.0485481
$209$	4.04248	+	9.25355i	0.279624	+	0.640081i
$210$		0			0
$211$			6.49539i			0.447161i	0.974685	+	0.223581i	$0.0717746\pi$
$211$			6.49539i			0.447161i	−0.974685	+	0.223581i	$0.928225\pi$
$212$	0.458978			0.0315227
$213$		−	3.67206i		−	0.251605i
$214$	−1.18435			−0.0809607
$215$	5.10648			0.348259
$216$	3.01429			0.205096
$217$		0			0
$218$	−19.0953			−1.29330
$219$		−	13.2048i		−	0.892297i
$220$	−4.03835	+	1.76418i	−0.272266	+	0.118941i
$221$	0.711103			0.0478340
$222$	−4.15953			−0.279169
$223$		−	18.6474i		−	1.24872i	−0.781135	−	0.624362i	$-0.785359\pi$
$223$		−	18.6474i		−	1.24872i	0.781135	−	0.624362i	$-0.214641\pi$
$224$		0			0
$225$	−3.12302			−0.208202
$226$		−	11.3987i		−	0.758232i
$227$	10.2301			0.678997			0.339499	−	0.940607i	$-0.389743\pi$
$227$	10.2301			0.678997			0.339499	+	0.940607i	$0.389743\pi$
$228$		−	2.95288i		−	0.195559i
$229$		−	0.808234i		−	0.0534096i	−0.999643	−	0.0267048i	$-0.991499\pi$
$229$		−	0.808234i		−	0.0534096i	0.999643	−	0.0267048i	$-0.00850141\pi$
$230$	10.1711			0.670660
$231$		0			0
$232$	−16.7470			−1.09950
$233$			6.93056i			0.454036i	0.973891	+	0.227018i	$0.0728976\pi$
$233$			6.93056i			0.454036i	−0.973891	+	0.227018i	$0.927102\pi$
$234$		−	0.634687i		−	0.0414908i
$235$	−4.11396			−0.268365
$236$			2.97980i			0.193968i
$237$	2.26033			0.146824
$238$		0			0
$239$		−	9.10113i		−	0.588703i	−0.955697	−	0.294352i	$-0.904896\pi$
$239$		−	9.10113i		−	0.588703i	0.955697	−	0.294352i	$-0.0951037\pi$
$240$	−1.53399			−0.0990188
$241$	21.8957			1.41043			0.705213	−	0.708995i	$-0.250851\pi$
$241$	21.8957			1.41043			0.705213	+	0.708995i	$0.250851\pi$
$242$	8.19138	+	7.58611i	0.526562	+	0.487654i
$243$		−	1.00000i		−	0.0641500i
$244$	13.3445			0.854292
$245$		0			0
$246$	7.28070			0.464200
$247$	−1.90392			−0.121144
$248$	−6.84014			−0.434349
$249$			8.93208i			0.566048i
$250$	11.2953			0.714376
$251$		−	14.9374i		−	0.942843i	−0.881908	−	0.471422i	$-0.843741\pi$
$251$		−	14.9374i		−	0.942843i	0.881908	−	0.471422i	$-0.156259\pi$
$252$		0			0
$253$	9.71171	+	22.2309i	0.610570	+	1.39764i
$254$	10.9740			0.688567
$255$	−1.55794			−0.0975623
$256$	−16.9182			−1.05739
$257$		−	7.38630i		−	0.460745i	−0.973103	−	0.230372i	$-0.926006\pi$
$257$		−	7.38630i		−	0.460745i	0.973103	−	0.230372i	$-0.0739944\pi$
$258$			3.78304i			0.235522i
$259$		0			0
$260$		−	0.830894i		−	0.0515298i
$261$			5.55588i			0.343900i
$262$		−	0.408409i		−	0.0252316i
$263$			30.5031i			1.88090i	0.339932	+	0.940450i	$0.389596\pi$
$263$			30.5031i			1.88090i	−0.339932	+	0.940450i	$0.610404\pi$
$264$	−4.00215	−	9.16122i	−0.246315	−	0.563834i
$265$			0.648359i			0.0398284i
$266$		0			0
$267$	7.47349			0.457370
$268$	4.64175			0.283540
$269$			6.22460i			0.379520i	0.981830	+	0.189760i	$0.0607710\pi$
$269$			6.22460i			0.379520i	−0.981830	+	0.189760i	$0.939229\pi$
$270$			1.39053i			0.0846247i
$271$	−15.4887			−0.940871			−0.470435	−	0.882434i	$-0.655903\pi$
$271$	−15.4887			−0.940871			−0.470435	+	0.882434i	$0.655903\pi$
$272$	1.27326			0.0772026
$273$		0			0
$274$			20.7606i			1.25420i
$275$	4.14652	+	9.49170i	0.250045	+	0.572371i
$276$		−	7.09403i		−	0.427011i
$277$			8.63624i			0.518901i	0.965756	+	0.259451i	$0.0835415\pi$
$277$			8.63624i			0.518901i	−0.965756	+	0.259451i	$0.916458\pi$
$278$			20.2633i			1.21531i
$279$			2.26924i			0.135856i
$280$		0			0
$281$			14.2958i			0.852817i	0.904531	+	0.426408i	$0.140221\pi$
$281$			14.2958i			0.852817i	−0.904531	+	0.426408i	$0.859779\pi$
$282$		−	3.04776i		−	0.181491i
$283$	31.6709			1.88264			0.941320	−	0.337515i	$-0.109586\pi$
$283$	31.6709			1.88264			0.941320	+	0.337515i	$0.109586\pi$
$284$	3.56136			0.211328
$285$	4.17127			0.247085
$286$	−1.92898	+	0.842691i	−0.114063	+	0.0498294i
$287$		0			0
$288$			4.89214i			0.288272i
$289$	−15.7069			−0.923933
$290$		−	7.72559i		−	0.453662i
$291$	−14.5069			−0.850412
$292$	12.8067			0.749456
$293$	17.2704			1.00895			0.504475	−	0.863426i	$-0.331686\pi$
$293$	17.2704			1.00895			0.504475	+	0.863426i	$0.331686\pi$
$294$		0			0
$295$	−4.20931			−0.245075
$296$		−	12.3532i		−	0.718015i
$297$	−3.03927	+	1.32773i	−0.176356	+	0.0770425i
$298$	16.2707			0.942535
$299$	−4.57401			−0.264522
$300$		−	3.02887i		−	0.174872i
$301$		0			0
$302$	1.85465			0.106723
$303$			14.1499i			0.812888i
$304$	−3.40905			−0.195522
$305$			18.8506i			1.07938i
$306$		−	1.15418i		−	0.0659799i
$307$	−7.76275			−0.443044			−0.221522	−	0.975155i	$-0.571102\pi$
$307$	−7.76275			−0.443044			−0.221522	+	0.975155i	$0.571102\pi$
$308$		0			0
$309$	−4.26874			−0.242840
$310$		−	3.15544i		−	0.179217i
$311$			18.7087i			1.06087i	0.847724	+	0.530437i	$0.177972\pi$
$311$			18.7087i			1.06087i	−0.847724	+	0.530437i	$0.822028\pi$
$312$	1.88493			0.106713
$313$			18.9781i			1.07271i	0.843993	+	0.536354i	$0.180199\pi$
$313$			18.9781i			1.07271i	−0.843993	+	0.536354i	$0.819801\pi$
$314$	20.5747			1.16110
$315$		0			0
$316$			2.19218i			0.123320i
$317$	4.08151			0.229240			0.114620	−	0.993409i	$-0.463435\pi$
$317$	4.08151			0.229240			0.114620	+	0.993409i	$0.463435\pi$
$318$	−0.480325			−0.0269353
$319$	16.8858	−	7.37668i	0.945423	−	0.413015i
$320$		−	9.87064i		−	0.551785i
$321$	−1.16689			−0.0651297
$322$		0			0
$323$	−3.46228			−0.192646
$324$	0.969853			0.0538807
$325$	−1.95292			−0.108329
$326$		−	3.91264i		−	0.216701i
$327$	−18.8138			−1.04040
$328$			21.6226i			1.19391i
$329$		0			0
$330$	4.22618	−	1.84624i	0.232644	−	0.101632i
$331$	27.4396			1.50822			0.754108	−	0.656750i	$-0.228069\pi$
$331$	27.4396			1.50822			0.754108	+	0.656750i	$0.228069\pi$
$332$	−8.66281			−0.475433
$333$	−4.09821			−0.224581
$334$		−	5.39893i		−	0.295416i
$335$			6.55700i			0.358247i
$336$		0			0
$337$			20.1887i			1.09975i	0.835247	+	0.549875i	$0.185325\pi$
$337$			20.1887i			1.09975i	−0.835247	+	0.549875i	$0.814675\pi$
$338$			12.7976i			0.696098i
$339$		−	11.2307i		−	0.609967i
$340$		−	1.51098i		−	0.0819442i
$341$	6.89683	−	3.01293i	0.373484	−	0.163159i
$342$			3.09022i			0.167100i
$343$		0			0
$344$	−11.2351			−0.605755
$345$	10.0211			0.539520
$346$			11.0856i			0.595963i
$347$			2.46328i			0.132236i	0.997812	+	0.0661179i	$0.0210613\pi$
$347$			2.46328i			0.132236i	−0.997812	+	0.0661179i	$0.978939\pi$
$348$	−5.38838			−0.288848
$349$	15.7590			0.843558			0.421779	−	0.906699i	$-0.361406\pi$
$349$	15.7590			0.843558			0.421779	+	0.906699i	$0.361406\pi$
$350$		0			0
$351$		−	0.625331i		−	0.0333777i
$352$	14.8685	−	6.49542i	0.792495	−	0.346207i
$353$		−	15.5815i		−	0.829318i	−0.909977	−	0.414659i	$-0.863901\pi$
$353$		−	15.5815i		−	0.829318i	0.909977	−	0.414659i	$-0.136099\pi$
$354$		−	3.11839i		−	0.165741i
$355$			5.03082i			0.267008i
$356$			7.24819i			0.384153i
$357$		0			0
$358$			17.5234i			0.926143i
$359$		−	28.9410i		−	1.52745i	−0.645543	−	0.763724i	$-0.723369\pi$
$359$		−	28.9410i		−	1.52745i	0.645543	−	0.763724i	$-0.276631\pi$
$360$	−4.12966			−0.217652
$361$	−9.73002			−0.512106
$362$	−5.42618			−0.285194
$363$	8.07063	+	7.47429i	0.423598	+	0.392298i
$364$		0			0
$365$			18.0909i			0.946923i
$366$	−13.9651			−0.729969
$367$			8.32913i			0.434777i	0.976085	+	0.217389i	$0.0697539\pi$
$367$			8.32913i			0.434777i	−0.976085	+	0.217389i	$0.930246\pi$
$368$	−8.18995			−0.426930
$369$	7.17337			0.373431
$370$	5.69867			0.296260
$371$		0			0
$372$	−2.20083			−0.114108
$373$			36.8876i			1.90997i	0.296656	+	0.954984i	$0.404129\pi$
$373$			36.8876i			1.90997i	−0.296656	+	0.954984i	$0.595871\pi$
$374$	−3.50785	+	1.53243i	−0.181387	+	0.0792401i
$375$	11.1288			0.574687
$376$	9.05140			0.466790
$377$			3.47426i			0.178934i
$378$		0			0
$379$	3.59264			0.184542			0.0922708	−	0.995734i	$-0.470587\pi$
$379$	3.59264			0.184542			0.0922708	+	0.995734i	$0.470587\pi$
$380$			4.04552i			0.207531i
$381$	10.8122			0.553925
$382$			9.25960i			0.473762i
$383$		−	21.5594i		−	1.10163i	−0.834626	−	0.550817i	$-0.814316\pi$
$383$		−	21.5594i		−	1.10163i	0.834626	−	0.550817i	$-0.185684\pi$
$384$	−2.47179			−0.126138
$385$		0			0
$386$	0.279586			0.0142305
$387$			3.72728i			0.189468i
$388$		−	14.0696i		−	0.714276i
$389$	35.4378			1.79677			0.898384	−	0.439211i	$-0.144742\pi$
$389$	35.4378			1.79677			0.898384	+	0.439211i	$0.144742\pi$
$390$			0.869539i			0.0440308i
$391$	−8.31782			−0.420650
$392$		0			0
$393$		−	0.402389i		−	0.0202978i
$394$	2.87632			0.144907
$395$	−3.09671			−0.155812
$396$	−1.28770	−	2.94764i	−0.0647093	−	0.148125i
$397$		−	26.0547i		−	1.30765i	−0.756646	−	0.653825i	$-0.773163\pi$
$397$		−	26.0547i		−	1.30765i	0.756646	−	0.653825i	$-0.226837\pi$
$398$	−10.0227			−0.502390
$399$		0			0
$400$	−3.49679			−0.174839
$401$	−29.8269			−1.48949			−0.744743	−	0.667351i	$-0.767428\pi$
$401$	−29.8269			−1.48949			−0.744743	+	0.667351i	$0.767428\pi$
$402$	−4.85764			−0.242277
$403$			1.41903i			0.0706868i
$404$	−13.7233			−0.682759
$405$			1.37003i			0.0680772i
$406$		0			0
$407$	5.44130	+	12.4556i	0.269715	+	0.617399i
$408$	3.42773			0.169698
$409$	10.0229			0.495600			0.247800	−	0.968811i	$-0.420292\pi$
$409$	10.0229			0.495600			0.247800	+	0.968811i	$0.420292\pi$
$410$	−9.97476			−0.492618
$411$			20.4546i			1.00895i
$412$		−	4.14005i		−	0.203966i
$413$		0			0
$414$			7.42398i			0.364869i
$415$		−	12.2372i		−	0.600701i
$416$			3.05921i			0.149990i
$417$			19.9646i			0.977671i
$418$	9.39199	−	4.10296i	0.459377	−	0.200682i
$419$			20.1178i			0.982818i	0.870929	+	0.491409i	$0.163518\pi$
$419$			20.1178i			0.982818i	−0.870929	+	0.491409i	$0.836482\pi$
$420$		0			0
$421$	−30.6522			−1.49390			−0.746948	−	0.664883i	$-0.768481\pi$
$421$	−30.6522			−1.49390			−0.746948	+	0.664883i	$0.768481\pi$
$422$	6.59258			0.320922
$423$		−	3.00283i		−	0.146003i
$424$		−	1.42650i		−	0.0692767i
$425$	−3.55138			−0.172267
$426$	−3.72700			−0.180574
$427$		0			0
$428$		−	1.13172i		−	0.0547036i
$429$	−1.90055	+	0.830269i	−0.0917594	+	0.0400857i
$430$		−	5.18288i		−	0.249941i
$431$			27.2150i			1.31090i	0.755239	+	0.655449i	$0.227521\pi$
$431$			27.2150i			1.31090i	−0.755239	+	0.655449i	$0.772479\pi$
$432$		−	1.11968i		−	0.0538706i
$433$			36.3504i			1.74689i	0.486923	+	0.873445i	$0.338119\pi$
$433$			36.3504i			1.74689i	−0.486923	+	0.873445i	$0.661881\pi$
$434$		0			0
$435$		−	7.61171i		−	0.364954i
$436$		−	18.2466i		−	0.873854i
$437$	22.2703			1.06533
$438$	−13.4024			−0.640389
$439$	−33.3393			−1.59120			−0.795600	−	0.605823i	$-0.792844\pi$
$439$	−33.3393			−1.59120			−0.795600	+	0.605823i	$0.792844\pi$
$440$	5.48305	+	12.5511i	0.261394	+	0.598352i
$441$		0			0
$442$		−	0.721742i		−	0.0343298i
$443$	−26.6169			−1.26461			−0.632304	−	0.774721i	$-0.717890\pi$
$443$	−26.6169			−1.26461			−0.632304	+	0.774721i	$0.717890\pi$
$444$		−	3.97466i		−	0.188629i
$445$	−10.2389			−0.485370
$446$	−18.9264			−0.896192
$447$	16.0308			0.758232
$448$		0			0
$449$	−16.3390			−0.771087			−0.385544	−	0.922690i	$-0.625986\pi$
$449$	−16.3390			−0.771087			−0.385544	+	0.922690i	$0.625986\pi$
$450$			3.16975i			0.149423i
$451$	−9.52427	−	21.8018i	−0.448480	−	1.02661i
$452$	10.8921			0.512322
$453$	1.82731			0.0858545
$454$		−	10.3832i		−	0.487307i
$455$		0			0
$456$	−9.17749			−0.429775
$457$			23.4115i			1.09514i	0.836758	+	0.547572i	$0.184448\pi$
$457$			23.4115i			1.09514i	−0.836758	+	0.547572i	$0.815552\pi$
$458$	−0.820326			−0.0383313
$459$		−	1.13716i		−	0.0530782i
$460$			9.71902i			0.453152i
$461$	−30.2464			−1.40872			−0.704358	−	0.709845i	$-0.748765\pi$
$461$	−30.2464			−1.40872			−0.704358	+	0.709845i	$0.748765\pi$
$462$		0			0
$463$	−9.36678			−0.435311			−0.217656	−	0.976026i	$-0.569841\pi$
$463$	−9.36678			−0.435311			−0.217656	+	0.976026i	$0.569841\pi$
$464$			6.22080i			0.288794i
$465$		−	3.10892i		−	0.144173i
$466$	7.03425			0.325855
$467$		−	33.4114i		−	1.54609i	−0.634349	−	0.773047i	$-0.718732\pi$
$467$		−	33.4114i		−	1.54609i	0.634349	−	0.773047i	$-0.281268\pi$
$468$	0.606479			0.0280345
$469$		0			0
$470$			4.17552i			0.192602i
$471$	20.2714			0.934059
$472$	9.26117			0.426280
$473$	11.3282	−	4.94881i	0.520871	−	0.227546i
$474$		−	2.29415i		−	0.105374i
$475$	9.50855			0.436282
$476$		0			0
$477$	−0.473245			−0.0216684
$478$	−9.23730			−0.422504
$479$	−11.7964			−0.538992			−0.269496	−	0.963002i	$-0.586857\pi$
$479$	−11.7964			−0.538992			−0.269496	+	0.963002i	$0.586857\pi$
$480$		−	6.70237i		−	0.305920i
$481$	−2.56274			−0.116851
$482$		−	22.2233i		−	1.01224i
$483$		0			0
$484$	−7.24896	+	7.82732i	−0.329498	+	0.355787i
$485$	19.8749			0.902474
$486$	−1.01496			−0.0460396
$487$	−31.5176			−1.42820			−0.714099	−	0.700044i	$-0.753164\pi$
$487$	−31.5176			−1.42820			−0.714099	+	0.700044i	$0.753164\pi$
$488$		−	41.4744i		−	1.87746i
$489$		−	3.85496i		−	0.174328i
$490$		0			0
$491$		−	34.3103i		−	1.54840i	−0.632939	−	0.774202i	$-0.718151\pi$
$491$		−	34.3103i		−	1.54840i	0.632939	−	0.774202i	$-0.281849\pi$
$492$			6.95712i			0.313651i
$493$			6.31794i			0.284545i
$494$			1.93241i			0.0869432i
$495$	4.16388	−	1.81902i	0.187153	−	0.0817590i
$496$			2.54082i			0.114086i
$497$		0			0
$498$	9.06572			0.406245
$499$	−8.42234			−0.377036			−0.188518	−	0.982070i	$-0.560368\pi$
$499$	−8.42234			−0.377036			−0.188518	+	0.982070i	$0.560368\pi$
$500$			10.7933i			0.482690i
$501$		−	5.31934i		−	0.237651i
$502$	−15.1609			−0.676665
$503$	−11.8253			−0.527265			−0.263633	−	0.964623i	$-0.584921\pi$
$503$	−11.8253			−0.527265			−0.263633	+	0.964623i	$0.584921\pi$
$504$		0			0
$505$		−	19.3857i		−	0.862652i
$506$	22.5635	−	9.85701i	1.00307	−	0.438198i
$507$			12.6090i			0.559984i
$508$			10.4862i			0.465251i
$509$			31.4785i			1.39526i	0.716459	+	0.697629i	$0.245762\pi$
$509$			31.4785i			1.39526i	−0.716459	+	0.697629i	$0.754238\pi$
$510$			1.58125i			0.0700191i
$511$		0			0
$512$			12.2277i			0.540393i
$513$			3.04466i			0.134425i
$514$	−7.49681			−0.330670
$515$	5.84829			0.257706
$516$	−3.61491			−0.159138
$517$	−9.12641	+	3.98694i	−0.401379	+	0.175345i
$518$		0			0
$519$			10.9221i			0.479429i
$520$	−2.58240			−0.113246
$521$		−	15.8332i		−	0.693667i	−0.937927	−	0.346833i	$-0.887257\pi$
$521$		−	15.8332i		−	0.693667i	0.937927	−	0.346833i	$-0.112743\pi$
$522$	5.63900			0.246812
$523$	−2.26835			−0.0991879			−0.0495939	−	0.998769i	$-0.515793\pi$
$523$	−2.26835			−0.0991879			−0.0495939	+	0.998769i	$0.515793\pi$
$524$	0.390258			0.0170485
$525$		0			0
$526$	30.9594			1.34990
$527$			2.58050i			0.112408i
$528$	−3.40301	+	1.48663i	−0.148097	+	0.0646972i
$529$	30.5026			1.32620
$530$	0.658059			0.0285843
$531$		−	3.07242i		−	0.133332i
$532$		0			0
$533$	4.48573			0.194299
$534$		−	7.58531i		−	0.328248i
$535$	1.59868			0.0691169
$536$		−	14.4265i		−	0.623129i
$537$			17.2651i			0.745045i
$538$	6.31773			0.272377
$539$		0			0
$540$	−1.32873			−0.0571793
$541$			40.7598i			1.75240i	0.481945	+	0.876201i	$0.339930\pi$
$541$			40.7598i			1.75240i	−0.481945	+	0.876201i	$0.660070\pi$
$542$			15.7204i			0.675250i
$543$	−5.34619			−0.229427
$544$			5.56316i			0.238518i
$545$	25.7754			1.10410
$546$		0			0
$547$		−	9.23593i		−	0.394900i	−0.980313	−	0.197450i	$-0.936734\pi$
$547$		−	9.23593i		−	0.394900i	0.980313	−	0.197450i	$-0.0632660\pi$
$548$	−19.8380			−0.847435
$549$	−13.7593			−0.587232
$550$	9.63371	−	4.20856i	0.410783	−	0.179454i
$551$		−	16.9158i		−	0.720636i
$552$	−22.0481			−0.938431
$553$		0			0
$554$	8.76545			0.372408
$555$	5.61467			0.238329
$556$	−19.3627			−0.821162
$557$		−	3.56623i		−	0.151106i	−0.997142	−	0.0755531i	$-0.975928\pi$
$557$		−	3.56623i		−	0.151106i	0.997142	−	0.0755531i	$-0.0240722\pi$
$558$	2.30319			0.0975019
$559$			2.33078i			0.0985817i
$560$		0			0
$561$	−3.45614	+	1.50984i	−0.145918	+	0.0637455i
$562$	14.5097			0.612055
$563$	31.2143			1.31552			0.657762	−	0.753226i	$-0.271503\pi$
$563$	31.2143			1.31552			0.657762	+	0.753226i	$0.271503\pi$
$564$	2.91230			0.122630
$565$			15.3864i			0.647309i
$566$		−	32.1448i		−	1.35115i
$567$		0			0
$568$		−	11.0686i		−	0.464430i
$569$		−	26.2526i		−	1.10057i	−0.834978	−	0.550283i	$-0.814520\pi$
$569$		−	26.2526i		−	1.10057i	0.834978	−	0.550283i	$-0.185480\pi$
$570$		−	4.23368i		−	0.177330i
$571$		−	3.62286i		−	0.151612i	−0.997123	−	0.0758060i	$-0.975847\pi$
$571$		−	3.62286i		−	0.151612i	0.997123	−	0.0758060i	$-0.0241530\pi$
$572$	−0.805238	−	1.84325i	−0.0336687	−	0.0770703i
$573$			9.12310i			0.381123i
$574$		0			0
$575$	22.8435			0.952639
$576$	7.20470			0.300196
$577$			6.72961i			0.280157i	0.990140	+	0.140079i	$0.0447356\pi$
$577$			6.72961i			0.280157i	−0.990140	+	0.140079i	$0.955264\pi$
$578$			15.9419i			0.663094i
$579$	0.275464			0.0114479
$580$	7.38224			0.306531
$581$		0			0
$582$			14.7240i			0.610329i
$583$	0.628339	+	1.43832i	0.0260232	+	0.0595690i
$584$		−	39.8030i		−	1.64706i
$585$			0.856721i			0.0354211i
$586$		−	17.5288i		−	0.724110i
$587$		−	39.9370i		−	1.64838i	−0.566317	−	0.824188i	$-0.691632\pi$
$587$		−	39.9370i		−	1.64838i	0.566317	−	0.824188i	$-0.308368\pi$
$588$		0			0
$589$		−	6.90907i		−	0.284683i
$590$			4.27229i			0.175887i
$591$	2.83392			0.116572
$592$	−4.58869			−0.188594
$593$	35.3214			1.45048			0.725238	−	0.688498i	$-0.241730\pi$
$593$	35.3214			1.45048			0.725238	+	0.688498i	$0.241730\pi$
$594$	1.34759	+	3.08474i	0.0552923	+	0.126568i
$595$		0			0
$596$			15.5475i			0.636852i
$597$	−9.87491			−0.404153
$598$			4.64245i			0.189844i
$599$	−14.4539			−0.590568			−0.295284	−	0.955409i	$-0.595414\pi$
$599$	−14.4539			−0.590568			−0.295284	+	0.955409i	$0.595414\pi$
$600$	−9.41369			−0.384312
$601$	−27.5005			−1.12177			−0.560885	−	0.827893i	$-0.689539\pi$
$601$	−27.5005			−1.12177			−0.560885	+	0.827893i	$0.689539\pi$
$602$		0			0
$603$	−4.78603			−0.194902
$604$			1.77222i			0.0721107i
$605$	−11.0570	−	10.2400i	−0.449530	−	0.416315i
$606$	14.3616			0.583399
$607$	−6.94840			−0.282027			−0.141013	−	0.990008i	$-0.545036\pi$
$607$	−6.94840			−0.282027			−0.141013	+	0.990008i	$0.545036\pi$
$608$		−	14.8949i		−	0.604069i
$609$		0			0
$610$	19.1326			0.774658
$611$		−	1.87776i		−	0.0759662i
$612$	1.10288			0.0445813
$613$		−	10.2741i		−	0.414968i	−0.978238	−	0.207484i	$-0.933473\pi$
$613$		−	10.2741i		−	0.414968i	0.978238	−	0.207484i	$-0.0665275\pi$
$614$			7.87890i			0.317966i
$615$	−9.82772			−0.396292
$616$		0			0
$617$	4.61580			0.185825			0.0929125	−	0.995674i	$-0.470382\pi$
$617$	4.61580			0.185825			0.0929125	+	0.995674i	$0.470382\pi$
$618$			4.33261i			0.174283i
$619$			33.7335i			1.35586i	0.735125	+	0.677932i	$0.237124\pi$
$619$			33.7335i			1.35586i	−0.735125	+	0.677932i	$0.762876\pi$
$620$	3.01520			0.121093
$621$			7.31454i			0.293523i
$622$	18.9886			0.761375
$623$		0			0
$624$		−	0.700171i		−	0.0280293i
$625$	0.368387			0.0147355
$626$	19.2621			0.769867
$627$	9.25355	−	4.04248i	0.369551	−	0.161441i
$628$			19.6603i			0.784532i
$629$	−4.66033			−0.185820
$630$		0			0
$631$	−3.29597			−0.131211			−0.0656053	−	0.997846i	$-0.520898\pi$
$631$	−3.29597			−0.131211			−0.0656053	+	0.997846i	$0.520898\pi$
$632$	6.81327			0.271017
$633$	6.49539			0.258169
$634$		−	4.14257i		−	0.164523i
$635$	−14.8130			−0.587836
$636$		−	0.458978i		−	0.0181997i
$637$		0			0
$638$	−7.48705	−	17.1384i	−0.296415	−	0.678517i
$639$	−3.67206			−0.145264
$640$	3.38643			0.133860
$641$	−31.6966			−1.25194			−0.625971	−	0.779847i	$-0.715297\pi$
$641$	−31.6966			−1.25194			−0.625971	+	0.779847i	$0.715297\pi$
$642$			1.18435i			0.0467427i
$643$		−	35.7474i		−	1.40974i	−0.709336	−	0.704870i	$-0.751005\pi$
$643$		−	35.7474i		−	1.40974i	0.709336	−	0.704870i	$-0.248995\pi$
$644$		0			0
$645$		−	5.10648i		−	0.201067i
$646$			3.51408i			0.138260i
$647$			11.3623i			0.446698i	0.974739	+	0.223349i	$0.0716989\pi$
$647$			11.3623i			0.446698i	−0.974739	+	0.223349i	$0.928301\pi$
$648$		−	3.01429i		−	0.118412i
$649$	−9.33792	+	4.07934i	−0.366545	+	0.160128i
$650$			1.98214i			0.0777460i
$651$		0			0
$652$	3.73875			0.146421
$653$	−18.2592			−0.714538			−0.357269	−	0.934002i	$-0.616292\pi$
$653$	−18.2592			−0.714538			−0.357269	+	0.934002i	$0.616292\pi$
$654$			19.0953i			0.746684i
$655$			0.551284i			0.0215405i
$656$	8.03188			0.313592
$657$	−13.2048			−0.515168
$658$		0			0
$659$		−	10.2117i		−	0.397793i	−0.980021	−	0.198896i	$-0.936264\pi$
$659$		−	10.2117i		−	0.397793i	0.980021	−	0.198896i	$-0.0637357\pi$
$660$	1.76418	+	4.03835i	0.0686708	+	0.157193i
$661$		−	17.6530i		−	0.686624i	−0.939221	−	0.343312i	$-0.888451\pi$
$661$		−	17.6530i		−	0.686624i	0.939221	−	0.343312i	$-0.111549\pi$
$662$		−	27.8501i		−	1.08243i
$663$		−	0.711103i		−	0.0276170i
$664$			26.9239i			1.04485i
$665$		0			0
$666$			4.15953i			0.161178i
$667$		−	40.6387i		−	1.57354i
$668$	5.15898			0.199607
$669$	−18.6474			−0.720951
$670$	6.65510			0.257109
$671$	18.2686	+	41.8181i	0.705250	+	1.61437i
$672$		0			0
$673$		−	14.1664i		−	0.546074i	−0.962004	−	0.273037i	$-0.911972\pi$
$673$		−	14.1664i		−	0.546074i	0.962004	−	0.273037i	$-0.0880282\pi$
$674$	20.4908			0.789275
$675$			3.12302i			0.120205i
$676$	−12.2288			−0.470340
$677$	26.4960			1.01832			0.509161	−	0.860671i	$-0.329956\pi$
$677$	26.4960			1.01832			0.509161	+	0.860671i	$0.329956\pi$
$678$	−11.3987			−0.437765
$679$		0			0
$680$	−4.69609			−0.180087
$681$		−	10.2301i		−	0.392019i
$682$	−3.05801	−	7.00002i	−0.117097	−	0.268044i
$683$	19.2646			0.737141			0.368571	−	0.929600i	$-0.379847\pi$
$683$	19.2646			0.737141			0.368571	+	0.929600i	$0.379847\pi$
$684$	−2.95288			−0.112906
$685$		−	28.0234i		−	1.07072i
$686$		0			0
$687$	−0.808234			−0.0308360
$688$			4.17336i			0.159108i
$689$	−0.295935			−0.0112742
$690$		−	10.1711i		−	0.387206i
$691$			31.9337i			1.21482i	0.794390	+	0.607408i	$0.207791\pi$
$691$			31.9337i			1.21482i	−0.794390	+	0.607408i	$0.792209\pi$
$692$	−10.5929			−0.402680
$693$		0			0
$694$	2.50013			0.0949038
$695$		−	27.3521i		−	1.03752i
$696$			16.7470i			0.634794i
$697$	8.15729			0.308979
$698$		−	15.9947i		−	0.605410i
$699$	6.93056			0.262138
$700$		0			0
$701$			11.9709i			0.452136i	0.974112	+	0.226068i	$0.0725871\pi$
$701$			11.9709i			0.452136i	−0.974112	+	0.226068i	$0.927413\pi$
$702$	−0.634687			−0.0239547
$703$	12.4777			0.470605
$704$	−9.56586	−	21.8970i	−0.360527	−	0.825274i
$705$			4.11396i			0.154941i
$706$	−15.8146			−0.595190
$707$		0			0
$708$	2.97980			0.111988
$709$	−28.6750			−1.07691			−0.538456	−	0.842654i	$-0.680992\pi$
$709$	−28.6750			−1.07691			−0.538456	+	0.842654i	$0.680992\pi$
$710$	5.10609			0.191628
$711$		−	2.26033i		−	0.0847689i
$712$	22.5273			0.844244
$713$		−	16.5985i		−	0.621617i
$714$		0			0
$715$	2.60381	−	1.13749i	0.0973768	−	0.0425398i
$716$	−16.7446			−0.625776
$717$	−9.10113			−0.339888
$718$	−29.3740			−1.09623
$719$			23.7570i			0.885988i	0.896525	+	0.442994i	$0.146084\pi$
$719$			23.7570i			0.885988i	−0.896525	+	0.442994i	$0.853916\pi$
$720$			1.53399i			0.0571685i
$721$		0			0
$722$			9.87560i			0.367532i
$723$		−	21.8957i		−	0.814310i
$724$		−	5.18502i		−	0.192700i
$725$		−	17.3511i		−	0.644405i
$726$	7.58611	−	8.19138i	0.281547	−	0.304011i
$727$			21.6385i			0.802529i	0.915962	+	0.401265i	$0.131429\pi$
$727$			21.6385i			0.802529i	−0.915962	+	0.401265i	$0.868571\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$	18.3616			0.679593
$731$			4.23852i			0.156767i
$732$		−	13.3445i		−	0.493226i
$733$	−5.71340			−0.211029			−0.105515	−	0.994418i	$-0.533649\pi$
$733$	−5.71340			−0.211029			−0.105515	+	0.994418i	$0.533649\pi$
$734$	8.45375			0.312034
$735$		0			0
$736$		−	35.7838i		−	1.31901i
$737$	6.35454	+	14.5460i	0.234073	+	0.535810i
$738$		−	7.28070i		−	0.268006i
$739$		−	22.7929i		−	0.838450i	−0.907882	−	0.419225i	$-0.862302\pi$
$739$		−	22.7929i		−	0.838450i	0.907882	−	0.419225i	$-0.137698\pi$
$740$			5.44540i			0.200177i
$741$			1.90392i			0.0699424i
$742$		0			0
$743$		−	48.3004i		−	1.77197i	−0.463714	−	0.885985i	$-0.653483\pi$
$743$		−	48.3004i		−	1.77197i	0.463714	−	0.885985i	$-0.346517\pi$
$744$			6.84014i			0.250772i
$745$	−21.9627			−0.804651
$746$	37.4395			1.37076
$747$	8.93208			0.326808
$748$	−1.46432	−	3.35195i	−0.0535409	−	0.122559i
$749$		0			0
$750$		−	11.2953i		−	0.412445i
$751$	−19.8899			−0.725793			−0.362897	−	0.931829i	$-0.618212\pi$
$751$	−19.8899			−0.725793			−0.362897	+	0.931829i	$0.618212\pi$
$752$		−	3.36221i		−	0.122607i
$753$	−14.9374			−0.544351
$754$	3.52624			0.128418
$755$	−2.50347			−0.0911104
$756$		0			0
$757$	34.0718			1.23836			0.619180	−	0.785249i	$-0.287465\pi$
$757$	34.0718			1.23836			0.619180	+	0.785249i	$0.287465\pi$
$758$		−	3.64639i		−	0.132443i
$759$	22.2309	−	9.71171i	0.806929	−	0.352513i
$760$	12.5734			0.456086
$761$	43.3617			1.57186			0.785930	−	0.618316i	$-0.212185\pi$
$761$	43.3617			1.57186			0.785930	+	0.618316i	$0.212185\pi$
$762$		−	10.9740i		−	0.397544i
$763$		0			0
$764$	−8.84807			−0.320112
$765$			1.55794i			0.0563276i
$766$	−21.8820			−0.790628
$767$		−	1.92128i		−	0.0693735i
$768$			16.9182i			0.610482i
$769$	36.3032			1.30913			0.654563	−	0.756008i	$-0.272853\pi$
$769$	36.3032			1.30913			0.654563	+	0.756008i	$0.272853\pi$
$770$		0			0
$771$	−7.38630			−0.266011
$772$			0.267160i			0.00961529i
$773$			40.2816i			1.44883i	0.689365	+	0.724414i	$0.257890\pi$
$773$			40.2816i			1.44883i	−0.689365	+	0.724414i	$0.742110\pi$
$774$	3.78304			0.135979
$775$		−	7.08689i		−	0.254569i
$776$	−43.7281			−1.56975
$777$		0			0
$778$		−	35.9680i		−	1.28952i
$779$	−21.8405			−0.782517
$780$	−0.830894			−0.0297508
$781$	4.87549	+	11.1604i	0.174459	+	0.399349i
$782$			8.44227i			0.301895i
$783$	5.55588			0.198551
$784$		0			0
$785$	−27.7724			−0.991241
$786$	−0.408409			−0.0145675
$787$	−49.1659			−1.75257			−0.876287	−	0.481789i	$-0.839987\pi$
$787$	−49.1659			−1.75257			−0.876287	+	0.481789i	$0.839987\pi$
$788$			2.74849i			0.0979108i
$789$	30.5031			1.08594
$790$			3.14304i			0.111824i
$791$		0			0
$792$	−9.16122	+	4.00215i	−0.325530	+	0.142210i
$793$	−8.60411			−0.305541
$794$	−26.4446			−0.938483
$795$	0.648359			0.0229949
$796$		−	9.57721i		−	0.339455i
$797$		−	27.7792i		−	0.983990i	−0.870598	−	0.491995i	$-0.836268\pi$
$797$		−	27.7792i		−	0.983990i	0.870598	−	0.491995i	$-0.163732\pi$
$798$		0			0
$799$		−	3.41471i		−	0.120804i
$800$		−	15.2783i		−	0.540168i
$801$		−	7.47349i		−	0.264063i
$802$			30.2732i			1.06898i
$803$	17.5323	+	40.1329i	0.618703	+	1.41626i
$804$		−	4.64175i		−	0.163702i
$805$		0			0
$806$	1.44026			0.0507309
$807$	6.22460			0.219116
$808$			42.6517i			1.50048i
$809$		−	37.4860i		−	1.31794i	−0.752170	−	0.658970i	$-0.770993\pi$
$809$		−	37.4860i		−	1.31794i	0.752170	−	0.658970i	$-0.229007\pi$
$810$	1.39053			0.0488581
$811$	−30.4875			−1.07056			−0.535280	−	0.844675i	$-0.679794\pi$
$811$	−30.4875			−1.07056			−0.535280	+	0.844675i	$0.679794\pi$
$812$		0			0
$813$			15.4887i			0.543212i
$814$	12.6419	−	5.52272i	0.443099	−	0.193571i
$815$			5.28141i			0.185000i
$816$		−	1.27326i		−	0.0445729i
$817$		−	11.3483i		−	0.397027i
$818$		−	10.1729i		−	0.355685i
$819$		0			0
$820$		−	9.53144i		−	0.332852i
$821$			14.1052i			0.492274i	0.969235	+	0.246137i	$0.0791612\pi$
$821$			14.1052i			0.492274i	−0.969235	+	0.246137i	$0.920839\pi$
$822$	20.7606			0.724110
$823$	−43.1239			−1.50321			−0.751603	−	0.659616i	$-0.770719\pi$
$823$	−43.1239			−1.50321			−0.751603	+	0.659616i	$0.770719\pi$
$824$	−12.8672			−0.448250
$825$	9.49170	−	4.14652i	0.330459	−	0.144363i
$826$		0			0
$827$		−	10.1597i		−	0.353287i	−0.984275	−	0.176644i	$-0.943476\pi$
$827$		−	10.1597i		−	0.353287i	0.984275	−	0.176644i	$-0.0565240\pi$
$828$	−7.09403			−0.246535
$829$		−	53.7652i		−	1.86734i	−0.358130	−	0.933672i	$-0.616585\pi$
$829$		−	53.7652i		−	1.86734i	0.358130	−	0.933672i	$-0.383415\pi$
$830$	−12.4203			−0.431115
$831$	8.63624			0.299588
$832$	4.50532			0.156194
$833$		0			0
$834$	20.2633			0.701661
$835$			7.28765i			0.252199i
$836$	3.92061	+	8.97458i	0.135597	+	0.310392i
$837$	2.26924			0.0784364
$838$	20.4188			0.705355
$839$		−	1.44926i		−	0.0500341i	−0.999687	−	0.0250170i	$-0.992036\pi$
$839$		−	1.44926i		−	0.0500341i	0.999687	−	0.0250170i	$-0.00796400\pi$
$840$		0			0
$841$	−1.86778			−0.0644062
$842$			31.1108i			1.07215i
$843$	14.2958			0.492374
$844$			6.29958i			0.216840i
$845$		−	17.2746i		−	0.594265i
$846$	−3.04776			−0.104784
$847$		0			0
$848$	−0.529883			−0.0181962
$849$		−	31.6709i		−	1.08694i
$850$			3.60452i			0.123634i
$851$	29.9766			1.02758
$852$		−	3.56136i		−	0.122010i
$853$	−38.4998			−1.31821			−0.659103	−	0.752052i	$-0.729064\pi$
$853$	−38.4998			−1.31821			−0.659103	+	0.752052i	$0.729064\pi$
$854$		0			0
$855$		−	4.17127i		−	0.142655i
$856$	−3.51736			−0.120221
$857$	−40.8008			−1.39373			−0.696864	−	0.717203i	$-0.745422\pi$
$857$	−40.8008			−1.39373			−0.696864	+	0.717203i	$0.745422\pi$
$858$	0.842691	+	1.92898i	0.0287690	+	0.0658544i
$859$		−	44.3273i		−	1.51243i	−0.654325	−	0.756214i	$-0.727047\pi$
$859$		−	44.3273i		−	1.51243i	0.654325	−	0.756214i	$-0.272953\pi$
$860$	4.95253			0.168880
$861$		0			0
$862$	27.6221			0.940814
$863$	−45.2853			−1.54153			−0.770765	−	0.637120i	$-0.780126\pi$
$863$	−45.2853			−1.54153			−0.770765	+	0.637120i	$0.780126\pi$
$864$	4.89214			0.166434
$865$		−	14.9636i		−	0.508779i
$866$	36.8943			1.25372
$867$			15.7069i			0.533433i
$868$		0			0
$869$	−6.86974	+	3.00110i	−0.233040	+	0.101805i
$870$	−7.72559			−0.261922
$871$	−2.99286			−0.101409
$872$	−56.7102			−1.92045
$873$			14.5069i			0.490986i
$874$		−	22.6035i		−	0.764576i
$875$		0			0
$876$		−	12.8067i		−	0.432698i
$877$			31.9168i			1.07775i	0.842385	+	0.538876i	$0.181151\pi$
$877$			31.9168i			1.07775i	−0.842385	+	0.538876i	$0.818849\pi$
$878$			33.8381i			1.14198i
$879$		−	17.2704i		−	0.582518i
$880$	4.66221	−	2.03672i	0.157163	−	0.0686579i
$881$			32.8806i			1.10778i	0.832592	+	0.553888i	$0.186856\pi$
$881$			32.8806i			1.10778i	−0.832592	+	0.553888i	$0.813144\pi$
$882$		0			0
$883$	21.6070			0.727132			0.363566	−	0.931568i	$-0.381559\pi$
$883$	21.6070			0.727132			0.363566	+	0.931568i	$0.381559\pi$
$884$	0.689665			0.0231960
$885$			4.20931i			0.141494i
$886$			27.0151i			0.907591i
$887$	5.52566			0.185533			0.0927667	−	0.995688i	$-0.470429\pi$
$887$	5.52566			0.185533			0.0927667	+	0.995688i	$0.470429\pi$
$888$	−12.3532			−0.414546
$889$		0			0
$890$			10.3921i			0.348344i
$891$	1.32773	+	3.03927i	0.0444805	+	0.101819i
$892$		−	18.0853i		−	0.605539i
$893$			9.14261i			0.305946i
$894$		−	16.2707i		−	0.544173i
$895$		−	23.6537i		−	0.790656i
$896$		0			0
$897$			4.57401i			0.152722i
$898$			16.5835i			0.553399i
$899$	−12.6076			−0.420488
$900$	−3.02887			−0.100962
$901$	−0.538156			−0.0179286
$902$	−22.1280	+	9.66677i	−0.736781	+	0.321868i
$903$		0			0
$904$		−	33.8525i		−	1.12592i
$905$	7.32443			0.243472
$906$		−	1.85465i		−	0.0616166i
$907$	9.63715			0.319996			0.159998	−	0.987117i	$-0.448851\pi$
$907$	9.63715			0.319996			0.159998	+	0.987117i	$0.448851\pi$
$908$	9.92172			0.329264
$909$	14.1499			0.469321
$910$		0			0
$911$	19.4818			0.645460			0.322730	−	0.946491i	$-0.395399\pi$
$911$	19.4818			0.645460			0.322730	+	0.946491i	$0.395399\pi$
$912$			3.40905i			0.112885i
$913$	−11.8594	−	27.1470i	−0.392488	−	0.898434i
$914$	23.7618			0.785970
$915$	18.8506			0.623182
$916$		−	0.783868i		−	0.0258997i
$917$		0			0
$918$	−1.15418			−0.0380935
$919$		−	19.0538i		−	0.628528i	−0.949336	−	0.314264i	$-0.898242\pi$
$919$		−	19.0538i		−	0.628528i	0.949336	−	0.314264i	$-0.101758\pi$
$920$	30.2066			0.995881
$921$			7.76275i			0.255791i
$922$			30.6989i			1.01102i
$923$	−2.29625			−0.0755821
$924$		0			0
$925$	12.7988			0.420822
$926$			9.50692i			0.312417i
$927$			4.26874i			0.140204i
$928$	−27.1801			−0.892232
$929$			38.5783i			1.26571i	0.774270	+	0.632856i	$0.218117\pi$
$929$			38.5783i			1.26571i	−0.774270	+	0.632856i	$0.781883\pi$
$930$	−3.15544			−0.103471
$931$		0			0
$932$			6.72162i			0.220174i
$933$	18.7087			0.612496
$934$	−33.9113			−1.10961
$935$	4.73501	−	2.06852i	0.154851	−	0.0676479i
$936$		−	1.88493i		−	0.0616108i
$937$	46.9089			1.53245			0.766223	−	0.642574i	$-0.222134\pi$
$937$	46.9089			1.53245			0.766223	+	0.642574i	$0.222134\pi$
$938$		0			0
$939$	18.9781			0.619328
$940$	−3.98994			−0.130137
$941$	−9.86719			−0.321661			−0.160831	−	0.986982i	$-0.551417\pi$
$941$	−9.86719			−0.321661			−0.160831	+	0.986982i	$0.551417\pi$
$942$		−	20.5747i		−	0.670361i
$943$	−52.4699			−1.70866
$944$		−	3.44013i		−	0.111967i
$945$		0			0
$946$	−5.02285	−	11.4977i	−0.163307	−	0.373822i
$947$	55.9144			1.81697			0.908487	−	0.417913i	$-0.137238\pi$
$947$	55.9144			1.81697			0.908487	+	0.417913i	$0.137238\pi$
$948$	2.19218			0.0711989
$949$	−8.25737			−0.268046
$950$		−	9.65082i		−	0.313114i
$951$		−	4.08151i		−	0.132352i
$952$		0			0
$953$			2.10965i			0.0683382i	0.999416	+	0.0341691i	$0.0108785\pi$
$953$			2.10965i			0.0683382i	−0.999416	+	0.0341691i	$0.989122\pi$
$954$			0.480325i			0.0155511i
$955$		−	12.4989i		−	0.404455i
$956$		−	8.82676i		−	0.285478i
$957$	−7.37668	−	16.8858i	−0.238454	−	0.545840i
$958$			11.9729i			0.386827i
$959$		0			0
$960$	−9.87064			−0.318573
$961$	25.8505			0.833889
$962$			2.60108i			0.0838623i
$963$			1.16689i			0.0376027i
$964$	21.2356			0.683953
$965$	−0.377394			−0.0121487
$966$		0			0
$967$		−	51.1660i		−	1.64539i	−0.568485	−	0.822694i	$-0.692470\pi$
$967$		−	51.1660i		−	1.64539i	0.568485	−	0.822694i	$-0.307530\pi$
$968$	24.3272	+	22.5296i	0.781905	+	0.724130i
$969$			3.46228i			0.111224i
$970$		−	20.1723i		−	0.647693i
$971$			50.6134i			1.62426i	0.583475	+	0.812131i	$0.301693\pi$
$971$			50.6134i			1.62426i	−0.583475	+	0.812131i	$0.698307\pi$
$972$		−	0.969853i		−	0.0311080i
$973$		0			0
$974$			31.9892i			1.02500i
$975$			1.95292i			0.0625436i
$976$	−15.4060			−0.493133
$977$	32.1719			1.02927			0.514635	−	0.857409i	$-0.327928\pi$
$977$	32.1719			1.02927			0.514635	+	0.857409i	$0.327928\pi$
$978$	−3.91264			−0.125112
$979$	−22.7139	+	9.92275i	−0.725941	+	0.317133i
$980$		0			0
$981$			18.8138i			0.600678i
$982$	−34.8237			−1.11127
$983$		−	4.36394i		−	0.139188i	−0.997575	−	0.0695941i	$-0.977830\pi$
$983$		−	4.36394i		−	0.139188i	0.997575	−	0.0695941i	$-0.0221704\pi$
$984$	21.6226			0.689303
$985$	−3.88255			−0.123708
$986$	6.41246			0.204214
$987$		0			0
$988$	−1.84653			−0.0587458
$989$		−	27.2633i		−	0.866924i
$990$	−1.84624	−	4.22618i	−0.0586773	−	0.134317i
$991$	33.0305			1.04925			0.524625	−	0.851333i	$-0.324206\pi$
$991$	33.0305			1.04925			0.524625	+	0.851333i	$0.324206\pi$
$992$	−11.1014			−0.352471
$993$		−	27.4396i		−	0.870769i
$994$		0			0
$995$	13.5289			0.428895
$996$			8.66281i			0.274492i
$997$	−34.8547			−1.10386			−0.551929	−	0.833891i	$-0.686108\pi$
$997$	−34.8547			−1.10386			−0.551929	+	0.833891i	$0.686108\pi$
$998$			8.54835i			0.270593i
$999$			4.09821i			0.129662i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.2.c.b.538.19	✓	48
7.6	odd	2	inner	1617.2.c.b.538.20	yes	48
11.10	odd	2	inner	1617.2.c.b.538.29	yes	48
77.76	even	2	inner	1617.2.c.b.538.30	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.c.b.538.19	✓	48	1.1	even	1	trivial
1617.2.c.b.538.20	yes	48	7.6	odd	2	inner
1617.2.c.b.538.29	yes	48	11.10	odd	2	inner
1617.2.c.b.538.30	yes	48	77.76	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1617.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.01496i q^{2} -1.00000i q^{3} +0.969853 q^{4} +1.37003i q^{5} -1.01496 q^{6} -3.01429i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.01496i q^{2} -1.00000i q^{3} +0.969853 q^{4} +1.37003i q^{5} -1.01496 q^{6} -3.01429i q^{8} -1.00000 q^{9} +1.39053 q^{10} +(1.32773 + 3.03927i) q^{11} -0.969853i q^{12} -0.625331 q^{13} +1.37003 q^{15} -1.11968 q^{16} -1.13716 q^{17} +1.01496i q^{18} +3.04466 q^{19} +1.32873i q^{20} +(3.08474 - 1.34759i) q^{22} +7.31454 q^{23} -3.01429 q^{24} +3.12302 q^{25} +0.634687i q^{26} +1.00000i q^{27} -5.55588i q^{29} -1.39053i q^{30} -2.26924i q^{31} -4.89214i q^{32} +(3.03927 - 1.32773i) q^{33} +1.15418i q^{34} -0.969853 q^{36} +4.09821 q^{37} -3.09022i q^{38} +0.625331i q^{39} +4.12966 q^{40} -7.17337 q^{41} -3.72728i q^{43} +(1.28770 + 2.94764i) q^{44} -1.37003i q^{45} -7.42398i q^{46} +3.00283i q^{47} +1.11968i q^{48} -3.16975i q^{50} +1.13716i q^{51} -0.606479 q^{52} +0.473245 q^{53} +1.01496 q^{54} +(-4.16388 + 1.81902i) q^{55} -3.04466i q^{57} -5.63900 q^{58} +3.07242i q^{59} +1.32873 q^{60} +13.7593 q^{61} -2.30319 q^{62} -7.20470 q^{64} -0.856721i q^{65} +(-1.34759 - 3.08474i) q^{66} +4.78603 q^{67} -1.10288 q^{68} -7.31454i q^{69} +3.67206 q^{71} +3.01429i q^{72} +13.2048 q^{73} -4.15953i q^{74} -3.12302i q^{75} +2.95288 q^{76} +0.634687 q^{78} +2.26033i q^{79} -1.53399i q^{80} +1.00000 q^{81} +7.28070i q^{82} -8.93208 q^{83} -1.55794i q^{85} -3.78304 q^{86} -5.55588 q^{87} +(9.16122 - 4.00215i) q^{88} +7.47349i q^{89} -1.39053 q^{90} +7.09403 q^{92} -2.26924 q^{93} +3.04776 q^{94} +4.17127i q^{95} -4.89214 q^{96} -14.5069i q^{97} +(-1.32773 - 3.03927i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 64 q^{4} - 48 q^{9}+O(q^{10}) \) 48 * q - 64 * q^4 - 48 * q^9	\( 48 q - 64 q^{4} - 48 q^{9} - 16 q^{11} + 64 q^{16} + 16 q^{22} + 32 q^{23} - 80 q^{25} + 64 q^{36} - 96 q^{37} - 32 q^{44} + 64 q^{53} + 48 q^{58} - 48 q^{60} - 240 q^{64} + 96 q^{67} - 32 q^{71} + 48 q^{78} + 48 q^{81} - 96 q^{86} - 48 q^{88} - 32 q^{92} + 96 q^{93} + 16 q^{99}+O(q^{100}) \) 48 * q - 64 * q^4 - 48 * q^9 - 16 * q^11 + 64 * q^16 + 16 * q^22 + 32 * q^23 - 80 * q^25 + 64 * q^36 - 96 * q^37 - 32 * q^44 + 64 * q^53 + 48 * q^58 - 48 * q^60 - 240 * q^64 + 96 * q^67 - 32 * q^71 + 48 * q^78 + 48 * q^81 - 96 * q^86 - 48 * q^88 - 32 * q^92 + 96 * q^93 + 16 * q^99

Introduction

L-functions

Modular forms

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