LMFDB - Embedded newform 246.2.d.b.163.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [246,2,Mod(163,246)] 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("246.163"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(246, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 246 = 2 \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	246.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.96431988972$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{57})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 196 $ x^4 + 29*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			163.1
Root			$-4.27492i$ of defining polynomial
Character	$\chi$	$=$	246.163
Dual form			246.2.d.b.163.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -3.27492 q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +3.27492 q^{10} +4.00000i q^{11} -1.00000i q^{12} +3.27492i q^{13} -2.00000i q^{14} +3.27492i q^{15} +1.00000 q^{16} -1.27492i q^{17} +1.00000 q^{18} +1.27492i q^{19} -3.27492 q^{20} +2.00000 q^{21} -4.00000i q^{22} +1.00000i q^{24} +5.72508 q^{25} -3.27492i q^{26} +1.00000i q^{27} +2.00000i q^{28} +8.54983i q^{29} -3.27492i q^{30} -9.27492 q^{31} -1.00000 q^{32} +4.00000 q^{33} +1.27492i q^{34} -6.54983i q^{35} -1.00000 q^{36} -6.00000 q^{37} -1.27492i q^{38} +3.27492 q^{39} +3.27492 q^{40} +(-6.27492 + 1.27492i) q^{41} -2.00000 q^{42} +2.54983 q^{43} +4.00000i q^{44} +3.27492 q^{45} -12.5498i q^{47} -1.00000i q^{48} +3.00000 q^{49} -5.72508 q^{50} -1.27492 q^{51} +3.27492i q^{52} +2.00000i q^{53} -1.00000i q^{54} -13.0997i q^{55} -2.00000i q^{56} +1.27492 q^{57} -8.54983i q^{58} +5.27492 q^{59} +3.27492i q^{60} +8.54983 q^{61} +9.27492 q^{62} -2.00000i q^{63} +1.00000 q^{64} -10.7251i q^{65} -4.00000 q^{66} -9.27492i q^{67} -1.27492i q^{68} +6.54983i q^{70} +11.2749i q^{71} +1.00000 q^{72} -7.27492 q^{73} +6.00000 q^{74} -5.72508i q^{75} +1.27492i q^{76} -8.00000 q^{77} -3.27492 q^{78} -8.54983i q^{79} -3.27492 q^{80} +1.00000 q^{81} +(6.27492 - 1.27492i) q^{82} +1.27492 q^{83} +2.00000 q^{84} +4.17525i q^{85} -2.54983 q^{86} +8.54983 q^{87} -4.00000i q^{88} -7.82475i q^{89} -3.27492 q^{90} -6.54983 q^{91} +9.27492i q^{93} +12.5498i q^{94} -4.17525i q^{95} +1.00000i q^{96} +14.5498i q^{97} -3.00000 q^{98} -4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{8} - 4 q^{9} - 2 q^{10} + 4 q^{16} + 4 q^{18} + 2 q^{20} + 8 q^{21} + 38 q^{25} - 22 q^{31} - 4 q^{32} + 16 q^{33} - 4 q^{36} - 24 q^{37} - 2 q^{39} - 2 q^{40} - 10 q^{41}+ \cdots - 12 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^8 - 4 * q^9 - 2 * q^10 + 4 * q^16 + 4 * q^18 + 2 * q^20 + 8 * q^21 + 38 * q^25 - 22 * q^31 - 4 * q^32 + 16 * q^33 - 4 * q^36 - 24 * q^37 - 2 * q^39 - 2 * q^40 - 10 * q^41 - 8 * q^42 - 20 * q^43 - 2 * q^45 + 12 * q^49 - 38 * q^50 + 10 * q^51 - 10 * q^57 + 6 * q^59 + 4 * q^61 + 22 * q^62 + 4 * q^64 - 16 * q^66 + 4 * q^72 - 14 * q^73 + 24 * q^74 - 32 * q^77 + 2 * q^78 + 2 * q^80 + 4 * q^81 + 10 * q^82 - 10 * q^83 + 8 * q^84 + 20 * q^86 + 4 * q^87 + 2 * q^90 + 4 * q^91 - 12 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/246\mathbb{Z}\right)^\times$.

$n$	$83$	$211$
$\chi(n)$	$1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	1.00000			0.500000
$5$	−3.27492			−1.46459			−0.732294	−	0.680989i	$-0.761550\pi$
$5$	−3.27492			−1.46459			−0.732294	+	0.680989i	$0.761550\pi$
$6$			1.00000i			0.408248i
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	−1.00000			−0.353553
$9$	−1.00000			−0.333333
$10$	3.27492			1.03562
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		−	1.00000i		−	0.288675i
$13$			3.27492i			0.908299i	0.890926	+	0.454149i	$0.150057\pi$
$13$			3.27492i			0.908299i	−0.890926	+	0.454149i	$0.849943\pi$
$14$		−	2.00000i		−	0.534522i
$15$			3.27492i			0.845580i
$16$	1.00000			0.250000
$17$		−	1.27492i		−	0.309213i	−0.987976	−	0.154606i	$-0.950589\pi$
$17$		−	1.27492i		−	0.309213i	0.987976	−	0.154606i	$-0.0494109\pi$
$18$	1.00000			0.235702
$19$			1.27492i			0.292486i	0.989249	+	0.146243i	$0.0467182\pi$
$19$			1.27492i			0.292486i	−0.989249	+	0.146243i	$0.953282\pi$
$20$	−3.27492			−0.732294
$21$	2.00000			0.436436
$22$		−	4.00000i		−	0.852803i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$			1.00000i			0.204124i
$25$	5.72508			1.14502
$26$		−	3.27492i		−	0.642264i
$27$			1.00000i			0.192450i
$28$			2.00000i			0.377964i
$29$			8.54983i			1.58766i	0.608137	+	0.793832i	$0.291917\pi$
$29$			8.54983i			1.58766i	−0.608137	+	0.793832i	$0.708083\pi$
$30$		−	3.27492i		−	0.597915i
$31$	−9.27492			−1.66582			−0.832912	−	0.553405i	$-0.813328\pi$
$31$	−9.27492			−1.66582			−0.832912	+	0.553405i	$0.813328\pi$
$32$	−1.00000			−0.176777
$33$	4.00000			0.696311
$34$			1.27492i			0.218646i
$35$		−	6.54983i		−	1.10712i
$36$	−1.00000			−0.166667
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		−	1.27492i		−	0.206819i
$39$	3.27492			0.524406
$40$	3.27492			0.517810
$41$	−6.27492	+	1.27492i	−0.979977	+	0.199109i
$41$	−6.27492	+	1.27492i	−0.979977	+	0.199109i
$42$	−2.00000			−0.308607
$43$	2.54983			0.388846			0.194423	−	0.980918i	$-0.437717\pi$
$43$	2.54983			0.388846			0.194423	+	0.980918i	$0.437717\pi$
$44$			4.00000i			0.603023i
$45$	3.27492			0.488196
$46$		0			0
$47$		−	12.5498i		−	1.83058i	−0.402794	−	0.915291i	$-0.631961\pi$
$47$		−	12.5498i		−	1.83058i	0.402794	−	0.915291i	$-0.368039\pi$
$48$		−	1.00000i		−	0.144338i
$49$	3.00000			0.428571
$50$	−5.72508			−0.809649
$51$	−1.27492			−0.178524
$52$			3.27492i			0.454149i
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		−	1.00000i		−	0.136083i
$55$		−	13.0997i		−	1.76636i
$56$		−	2.00000i		−	0.267261i
$57$	1.27492			0.168867
$58$		−	8.54983i		−	1.12265i
$59$	5.27492			0.686736			0.343368	−	0.939201i	$-0.388432\pi$
$59$	5.27492			0.686736			0.343368	+	0.939201i	$0.388432\pi$
$60$			3.27492i			0.422790i
$61$	8.54983			1.09469			0.547347	−	0.836906i	$-0.315638\pi$
$61$	8.54983			1.09469			0.547347	+	0.836906i	$0.315638\pi$
$62$	9.27492			1.17792
$63$		−	2.00000i		−	0.251976i
$64$	1.00000			0.125000
$65$		−	10.7251i		−	1.33028i
$66$	−4.00000			−0.492366
$67$		−	9.27492i		−	1.13311i	−0.824023	−	0.566556i	$-0.808276\pi$
$67$		−	9.27492i		−	1.13311i	0.824023	−	0.566556i	$-0.191724\pi$
$68$		−	1.27492i		−	0.154606i
$69$		0			0
$70$			6.54983i			0.782855i
$71$			11.2749i			1.33809i	0.743223	+	0.669043i	$0.233296\pi$
$71$			11.2749i			1.33809i	−0.743223	+	0.669043i	$0.766704\pi$
$72$	1.00000			0.117851
$73$	−7.27492			−0.851465			−0.425732	−	0.904849i	$-0.639983\pi$
$73$	−7.27492			−0.851465			−0.425732	+	0.904849i	$0.639983\pi$
$74$	6.00000			0.697486
$75$		−	5.72508i		−	0.661076i
$76$			1.27492i			0.146243i
$77$	−8.00000			−0.911685
$78$	−3.27492			−0.370811
$79$		−	8.54983i		−	0.961932i	−0.876739	−	0.480966i	$-0.840286\pi$
$79$		−	8.54983i		−	0.961932i	0.876739	−	0.480966i	$-0.159714\pi$
$80$	−3.27492			−0.366147
$81$	1.00000			0.111111
$82$	6.27492	−	1.27492i	0.692949	−	0.140791i
$83$	1.27492			0.139940			0.0699702	−	0.997549i	$-0.477710\pi$
$83$	1.27492			0.139940			0.0699702	+	0.997549i	$0.477710\pi$
$84$	2.00000			0.218218
$85$			4.17525i			0.452869i
$86$	−2.54983			−0.274956
$87$	8.54983			0.916638
$88$		−	4.00000i		−	0.426401i
$89$		−	7.82475i		−	0.829422i	−0.909953	−	0.414711i	$-0.863883\pi$
$89$		−	7.82475i		−	0.829422i	0.909953	−	0.414711i	$-0.136117\pi$
$90$	−3.27492			−0.345207
$91$	−6.54983			−0.686609
$92$		0			0
$93$			9.27492i			0.961764i
$94$			12.5498i			1.29442i
$95$		−	4.17525i		−	0.428371i
$96$			1.00000i			0.102062i
$97$			14.5498i			1.47731i	0.674083	+	0.738656i	$0.264539\pi$
$97$			14.5498i			1.47731i	−0.674083	+	0.738656i	$0.735461\pi$
$98$	−3.00000			−0.303046
$99$		−	4.00000i		−	0.402015i
$100$	5.72508			0.572508
$101$			16.5498i			1.64677i	0.567483	+	0.823385i	$0.307917\pi$
$101$			16.5498i			1.64677i	−0.567483	+	0.823385i	$0.692083\pi$
$102$	1.27492			0.126236
$103$	−11.8248			−1.16513			−0.582564	−	0.812785i	$-0.697950\pi$
$103$	−11.8248			−1.16513			−0.582564	+	0.812785i	$0.697950\pi$
$104$		−	3.27492i		−	0.321132i
$105$	−6.54983			−0.639198
$106$		−	2.00000i		−	0.194257i
$107$	11.8248			1.14314			0.571571	−	0.820553i	$-0.306334\pi$
$107$	11.8248			1.14314			0.571571	+	0.820553i	$0.306334\pi$
$108$			1.00000i			0.0962250i
$109$			19.0997i			1.82942i	0.404115	+	0.914708i	$0.367580\pi$
$109$			19.0997i			1.82942i	−0.404115	+	0.914708i	$0.632420\pi$
$110$			13.0997i			1.24900i
$111$			6.00000i			0.569495i
$112$			2.00000i			0.188982i
$113$	4.54983			0.428012			0.214006	−	0.976832i	$-0.431349\pi$
$113$	4.54983			0.428012			0.214006	+	0.976832i	$0.431349\pi$
$114$	−1.27492			−0.119407
$115$		0			0
$116$			8.54983i			0.793832i
$117$		−	3.27492i		−	0.302766i
$118$	−5.27492			−0.485595
$119$	2.54983			0.233743
$120$		−	3.27492i		−	0.298958i
$121$	−5.00000			−0.454545
$122$	−8.54983			−0.774066
$123$	1.27492	+	6.27492i	0.114955	+	0.565790i
$124$	−9.27492			−0.832912
$125$	−2.37459			−0.212389
$126$			2.00000i			0.178174i
$127$	1.27492			0.113131			0.0565653	−	0.998399i	$-0.481985\pi$
$127$	1.27492			0.113131			0.0565653	+	0.998399i	$0.481985\pi$
$128$	−1.00000			−0.0883883
$129$		−	2.54983i		−	0.224500i
$130$			10.7251i			0.940652i
$131$	16.0000			1.39793			0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000			1.39793			0.698963	+	0.715158i	$0.253645\pi$
$132$	4.00000			0.348155
$133$	−2.54983			−0.221099
$134$			9.27492i			0.801231i
$135$		−	3.27492i		−	0.281860i
$136$			1.27492i			0.109323i
$137$		−	10.5498i		−	0.901333i	−0.892692	−	0.450667i	$-0.851186\pi$
$137$		−	10.5498i		−	0.901333i	0.892692	−	0.450667i	$-0.148814\pi$
$138$		0			0
$139$	−10.5498			−0.894825			−0.447413	−	0.894328i	$-0.647654\pi$
$139$	−10.5498			−0.894825			−0.447413	+	0.894328i	$0.647654\pi$
$140$		−	6.54983i		−	0.553562i
$141$	−12.5498			−1.05689
$142$		−	11.2749i		−	0.946170i
$143$	−13.0997			−1.09545
$144$	−1.00000			−0.0833333
$145$		−	28.0000i		−	2.32527i
$146$	7.27492			0.602076
$147$		−	3.00000i		−	0.247436i
$148$	−6.00000			−0.493197
$149$			16.5498i			1.35582i	0.735147	+	0.677908i	$0.237113\pi$
$149$			16.5498i			1.35582i	−0.735147	+	0.677908i	$0.762887\pi$
$150$			5.72508i			0.467451i
$151$		−	2.00000i		−	0.162758i	−0.996683	−	0.0813788i	$-0.974068\pi$
$151$		−	2.00000i		−	0.162758i	0.996683	−	0.0813788i	$-0.0259324\pi$
$152$		−	1.27492i		−	0.103409i
$153$			1.27492i			0.103071i
$154$	8.00000			0.644658
$155$	30.3746			2.43975
$156$	3.27492			0.262203
$157$		−	14.0000i		−	1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$		−	14.0000i		−	1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$			8.54983i			0.680188i
$159$	2.00000			0.158610
$160$	3.27492			0.258905
$161$		0			0
$162$	−1.00000			−0.0785674
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	−6.27492	+	1.27492i	−0.489989	+	0.0995543i
$165$	−13.0997			−1.01981
$166$	−1.27492			−0.0989528
$167$			0.725083i			0.0561086i	0.999606	+	0.0280543i	$0.00893113\pi$
$167$			0.725083i			0.0561086i	−0.999606	+	0.0280543i	$0.991069\pi$
$168$	−2.00000			−0.154303
$169$	2.27492			0.174994
$170$		−	4.17525i		−	0.320227i
$171$		−	1.27492i		−	0.0974954i
$172$	2.54983			0.194423
$173$	23.0997			1.75624			0.878118	−	0.478445i	$-0.158799\pi$
$173$	23.0997			1.75624			0.878118	+	0.478445i	$0.158799\pi$
$174$	−8.54983			−0.648161
$175$			11.4502i			0.865551i
$176$			4.00000i			0.301511i
$177$		−	5.27492i		−	0.396487i
$178$			7.82475i			0.586490i
$179$			9.09967i			0.680141i	0.940400	+	0.340071i	$0.110451\pi$
$179$			9.09967i			0.680141i	−0.940400	+	0.340071i	$0.889549\pi$
$180$	3.27492			0.244098
$181$			11.2749i			0.838058i	0.907973	+	0.419029i	$0.137629\pi$
$181$			11.2749i			0.838058i	−0.907973	+	0.419029i	$0.862371\pi$
$182$	6.54983			0.485506
$183$		−	8.54983i		−	0.632022i
$184$		0			0
$185$	19.6495			1.44466
$186$		−	9.27492i		−	0.680070i
$187$	5.09967			0.372925
$188$		−	12.5498i		−	0.915291i
$189$	−2.00000			−0.145479
$190$			4.17525i			0.302904i
$191$			4.54983i			0.329214i	0.986359	+	0.164607i	$0.0526357\pi$
$191$			4.54983i			0.329214i	−0.986359	+	0.164607i	$0.947364\pi$
$192$		−	1.00000i		−	0.0721688i
$193$			20.0000i			1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$			20.0000i			1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$		−	14.5498i		−	1.04462i
$195$	−10.7251			−0.768039
$196$	3.00000			0.214286
$197$	−23.2749			−1.65827			−0.829135	−	0.559049i	$-0.811166\pi$
$197$	−23.2749			−1.65827			−0.829135	+	0.559049i	$0.811166\pi$
$198$			4.00000i			0.284268i
$199$			2.00000i			0.141776i	0.997484	+	0.0708881i	$0.0225833\pi$
$199$			2.00000i			0.141776i	−0.997484	+	0.0708881i	$0.977417\pi$
$200$	−5.72508			−0.404824
$201$	−9.27492			−0.654202
$202$		−	16.5498i		−	1.16444i
$203$	−17.0997			−1.20016
$204$	−1.27492			−0.0892621
$205$	20.5498	−	4.17525i	1.43526	−	0.291612i
$206$	11.8248			0.823869
$207$		0			0
$208$			3.27492i			0.227075i
$209$	−5.09967			−0.352751
$210$	6.54983			0.451982
$211$		−	9.27492i		−	0.638512i	−0.947669	−	0.319256i	$-0.896567\pi$
$211$		−	9.27492i		−	0.638512i	0.947669	−	0.319256i	$-0.103433\pi$
$212$			2.00000i			0.137361i
$213$	11.2749			0.772545
$214$	−11.8248			−0.808323
$215$	−8.35050			−0.569499
$216$		−	1.00000i		−	0.0680414i
$217$		−	18.5498i		−	1.25924i
$218$		−	19.0997i		−	1.29359i
$219$			7.27492i			0.491593i
$220$		−	13.0997i		−	0.883179i
$221$	4.17525			0.280858
$222$		−	6.00000i		−	0.402694i
$223$	25.2749			1.69253			0.846267	−	0.532759i	$-0.178845\pi$
$223$	25.2749			1.69253			0.846267	+	0.532759i	$0.178845\pi$
$224$		−	2.00000i		−	0.133631i
$225$	−5.72508			−0.381672
$226$	−4.54983			−0.302650
$227$		−	26.5498i		−	1.76217i	−0.472954	−	0.881087i	$-0.656812\pi$
$227$		−	26.5498i		−	1.76217i	0.472954	−	0.881087i	$-0.343188\pi$
$228$	1.27492			0.0844335
$229$		−	19.0997i		−	1.26214i	−0.775725	−	0.631071i	$-0.782616\pi$
$229$		−	19.0997i		−	1.26214i	0.775725	−	0.631071i	$-0.217384\pi$
$230$		0			0
$231$			8.00000i			0.526361i
$232$		−	8.54983i		−	0.561324i
$233$		−	7.82475i		−	0.512617i	−0.966595	−	0.256308i	$-0.917494\pi$
$233$		−	7.82475i		−	0.512617i	0.966595	−	0.256308i	$-0.0825062\pi$
$234$			3.27492i			0.214088i
$235$			41.0997i			2.68105i
$236$	5.27492			0.343368
$237$	−8.54983			−0.555371
$238$	−2.54983			−0.165281
$239$		−	15.2749i		−	0.988052i	−0.869447	−	0.494026i	$-0.835525\pi$
$239$		−	15.2749i		−	0.988052i	0.869447	−	0.494026i	$-0.164475\pi$
$240$			3.27492i			0.211395i
$241$	9.82475			0.632868			0.316434	−	0.948615i	$-0.397514\pi$
$241$	9.82475			0.632868			0.316434	+	0.948615i	$0.397514\pi$
$242$	5.00000			0.321412
$243$		−	1.00000i		−	0.0641500i
$244$	8.54983			0.547347
$245$	−9.82475			−0.627680
$246$	−1.27492	−	6.27492i	−0.0812858	−	0.400074i
$247$	−4.17525			−0.265665
$248$	9.27492			0.588958
$249$		−	1.27492i		−	0.0807946i
$250$	2.37459			0.150182
$251$	−29.0997			−1.83675			−0.918377	−	0.395706i	$-0.870500\pi$
$251$	−29.0997			−1.83675			−0.918377	+	0.395706i	$0.870500\pi$
$252$		−	2.00000i		−	0.125988i
$253$		0			0
$254$	−1.27492			−0.0799954
$255$	4.17525			0.261464
$256$	1.00000			0.0625000
$257$			1.45017i			0.0904588i	0.998977	+	0.0452294i	$0.0144019\pi$
$257$			1.45017i			0.0904588i	−0.998977	+	0.0452294i	$0.985598\pi$
$258$			2.54983i			0.158746i
$259$		−	12.0000i		−	0.745644i
$260$		−	10.7251i		−	0.665141i
$261$		−	8.54983i		−	0.529221i
$262$	−16.0000			−0.988483
$263$			22.9244i			1.41358i	0.707423	+	0.706790i	$0.249858\pi$
$263$			22.9244i			1.41358i	−0.707423	+	0.706790i	$0.750142\pi$
$264$	−4.00000			−0.246183
$265$		−	6.54983i		−	0.402353i
$266$	2.54983			0.156340
$267$	−7.82475			−0.478867
$268$		−	9.27492i		−	0.566556i
$269$	0.725083			0.0442091			0.0221045	−	0.999756i	$-0.492963\pi$
$269$	0.725083			0.0442091			0.0221045	+	0.999756i	$0.492963\pi$
$270$			3.27492i			0.199305i
$271$	−5.09967			−0.309783			−0.154891	−	0.987932i	$-0.549503\pi$
$271$	−5.09967			−0.309783			−0.154891	+	0.987932i	$0.549503\pi$
$272$		−	1.27492i		−	0.0773032i
$273$			6.54983i			0.396414i
$274$			10.5498i			0.637339i
$275$			22.9003i			1.38094i
$276$		0			0
$277$	15.4502			0.928310			0.464155	−	0.885754i	$-0.346358\pi$
$277$	15.4502			0.928310			0.464155	+	0.885754i	$0.346358\pi$
$278$	10.5498			0.632737
$279$	9.27492			0.555275
$280$			6.54983i			0.391427i
$281$			5.27492i			0.314675i	0.987545	+	0.157338i	$0.0502911\pi$
$281$			5.27492i			0.314675i	−0.987545	+	0.157338i	$0.949709\pi$
$282$	12.5498			0.747332
$283$	−23.6495			−1.40582			−0.702909	−	0.711280i	$-0.748116\pi$
$283$	−23.6495			−1.40582			−0.702909	+	0.711280i	$0.748116\pi$
$284$			11.2749i			0.669043i
$285$	−4.17525			−0.247320
$286$	13.0997			0.774600
$287$	−2.54983	−	12.5498i	−0.150512	−	0.740793i
$288$	1.00000			0.0589256
$289$	15.3746			0.904387
$290$			28.0000i			1.64422i
$291$	14.5498			0.852926
$292$	−7.27492			−0.425732
$293$			2.00000i			0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$			2.00000i			0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$			3.00000i			0.174964i
$295$	−17.2749			−1.00578
$296$	6.00000			0.348743
$297$	−4.00000			−0.232104
$298$		−	16.5498i		−	0.958706i
$299$		0			0
$300$		−	5.72508i		−	0.330538i
$301$			5.09967i			0.293940i
$302$			2.00000i			0.115087i
$303$	16.5498			0.950763
$304$			1.27492i			0.0731215i
$305$	−28.0000			−1.60328
$306$		−	1.27492i		−	0.0728822i
$307$	33.0997			1.88910			0.944549	−	0.328371i	$-0.106500\pi$
$307$	33.0997			1.88910			0.944549	+	0.328371i	$0.106500\pi$
$308$	−8.00000			−0.455842
$309$			11.8248i			0.672687i
$310$	−30.3746			−1.72516
$311$		−	20.7251i		−	1.17521i	−0.809147	−	0.587606i	$-0.800071\pi$
$311$		−	20.7251i		−	1.17521i	0.809147	−	0.587606i	$-0.199929\pi$
$312$	−3.27492			−0.185406
$313$			19.6495i			1.11066i	0.831632	+	0.555328i	$0.187407\pi$
$313$			19.6495i			1.11066i	−0.831632	+	0.555328i	$0.812593\pi$
$314$			14.0000i			0.790066i
$315$			6.54983i			0.369041i
$316$		−	8.54983i		−	0.480966i
$317$			15.4502i			0.867768i	0.900969	+	0.433884i	$0.142857\pi$
$317$			15.4502i			0.867768i	−0.900969	+	0.433884i	$0.857143\pi$
$318$	−2.00000			−0.112154
$319$	−34.1993			−1.91480
$320$	−3.27492			−0.183073
$321$		−	11.8248i		−	0.659993i
$322$		0			0
$323$	1.62541			0.0904404
$324$	1.00000			0.0555556
$325$			18.7492i			1.04002i
$326$	4.00000			0.221540
$327$	19.0997			1.05621
$328$	6.27492	−	1.27492i	0.346474	−	0.0703955i
$329$	25.0997			1.38379
$330$	13.0997			0.721113
$331$			29.0997i			1.59946i	0.600358	+	0.799731i	$0.295025\pi$
$331$			29.0997i			1.59946i	−0.600358	+	0.799731i	$0.704975\pi$
$332$	1.27492			0.0699702
$333$	6.00000			0.328798
$334$		−	0.725083i		−	0.0396748i
$335$			30.3746i			1.65954i
$336$	2.00000			0.109109
$337$	9.82475			0.535188			0.267594	−	0.963532i	$-0.413771\pi$
$337$	9.82475			0.535188			0.267594	+	0.963532i	$0.413771\pi$
$338$	−2.27492			−0.123739
$339$		−	4.54983i		−	0.247113i
$340$			4.17525i			0.226435i
$341$		−	37.0997i		−	2.00906i
$342$			1.27492i			0.0689396i
$343$			20.0000i			1.07990i
$344$	−2.54983			−0.137478
$345$		0			0
$346$	−23.0997			−1.24185
$347$		−	13.0997i		−	0.703227i	−0.936145	−	0.351614i	$-0.885633\pi$
$347$		−	13.0997i		−	0.703227i	0.936145	−	0.351614i	$-0.114367\pi$
$348$	8.54983			0.458319
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		−	11.4502i		−	0.612037i
$351$	−3.27492			−0.174802
$352$		−	4.00000i		−	0.213201i
$353$	−4.54983			−0.242163			−0.121082	−	0.992643i	$-0.538636\pi$
$353$	−4.54983			−0.242163			−0.121082	+	0.992643i	$0.538636\pi$
$354$			5.27492i			0.280359i
$355$		−	36.9244i		−	1.95974i
$356$		−	7.82475i		−	0.414711i
$357$		−	2.54983i		−	0.134952i
$358$		−	9.09967i		−	0.480932i
$359$	−13.0997			−0.691374			−0.345687	−	0.938350i	$-0.612354\pi$
$359$	−13.0997			−0.691374			−0.345687	+	0.938350i	$0.612354\pi$
$360$	−3.27492			−0.172603
$361$	17.3746			0.914452
$362$		−	11.2749i		−	0.592596i
$363$			5.00000i			0.262432i
$364$	−6.54983			−0.343305
$365$	23.8248			1.24704
$366$			8.54983i			0.446907i
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		0			0
$369$	6.27492	−	1.27492i	0.326659	−	0.0663695i
$370$	−19.6495			−1.02153
$371$	−4.00000			−0.207670
$372$			9.27492i			0.480882i
$373$	−23.0997			−1.19606			−0.598028	−	0.801475i	$-0.704049\pi$
$373$	−23.0997			−1.19606			−0.598028	+	0.801475i	$0.704049\pi$
$374$	−5.09967			−0.263698
$375$			2.37459i			0.122623i
$376$			12.5498i			0.647208i
$377$	−28.0000			−1.44207
$378$	2.00000			0.102869
$379$	13.4502			0.690889			0.345444	−	0.938439i	$-0.387728\pi$
$379$	13.4502			0.690889			0.345444	+	0.938439i	$0.387728\pi$
$380$		−	4.17525i		−	0.214186i
$381$		−	1.27492i		−	0.0653160i
$382$		−	4.54983i		−	0.232790i
$383$			21.8248i			1.11519i	0.830112	+	0.557596i	$0.188276\pi$
$383$			21.8248i			1.11519i	−0.830112	+	0.557596i	$0.811724\pi$
$384$			1.00000i			0.0510310i
$385$	26.1993			1.33524
$386$		−	20.0000i		−	1.01797i
$387$	−2.54983			−0.129615
$388$			14.5498i			0.738656i
$389$	−30.9244			−1.56793			−0.783965	−	0.620805i	$-0.786806\pi$
$389$	−30.9244			−1.56793			−0.783965	+	0.620805i	$0.786806\pi$
$390$	10.7251			0.543086
$391$		0			0
$392$	−3.00000			−0.151523
$393$		−	16.0000i		−	0.807093i
$394$	23.2749			1.17257
$395$			28.0000i			1.40883i
$396$		−	4.00000i		−	0.201008i
$397$		−	29.8248i		−	1.49686i	−0.663213	−	0.748431i	$-0.730808\pi$
$397$		−	29.8248i		−	1.49686i	0.663213	−	0.748431i	$-0.269192\pi$
$398$		−	2.00000i		−	0.100251i
$399$			2.54983i			0.127651i
$400$	5.72508			0.286254
$401$	7.09967			0.354541			0.177270	−	0.984162i	$-0.443273\pi$
$401$	7.09967			0.354541			0.177270	+	0.984162i	$0.443273\pi$
$402$	9.27492			0.462591
$403$		−	30.3746i		−	1.51307i
$404$			16.5498i			0.823385i
$405$	−3.27492			−0.162732
$406$	17.0997			0.848642
$407$		−	24.0000i		−	1.18964i
$408$	1.27492			0.0631178
$409$	−0.725083			−0.0358530			−0.0179265	−	0.999839i	$-0.505706\pi$
$409$	−0.725083			−0.0358530			−0.0179265	+	0.999839i	$0.505706\pi$
$410$	−20.5498	+	4.17525i	−1.01488	+	0.206201i
$411$	−10.5498			−0.520385
$412$	−11.8248			−0.582564
$413$			10.5498i			0.519123i
$414$		0			0
$415$	−4.17525			−0.204955
$416$		−	3.27492i		−	0.160566i
$417$			10.5498i			0.516628i
$418$	5.09967			0.249433
$419$	−26.7251			−1.30561			−0.652803	−	0.757528i	$-0.726407\pi$
$419$	−26.7251			−1.30561			−0.652803	+	0.757528i	$0.726407\pi$
$420$	−6.54983			−0.319599
$421$			25.8248i			1.25862i	0.777154	+	0.629311i	$0.216663\pi$
$421$			25.8248i			1.25862i	−0.777154	+	0.629311i	$0.783337\pi$
$422$			9.27492i			0.451496i
$423$			12.5498i			0.610194i
$424$		−	2.00000i		−	0.0971286i
$425$		−	7.29901i		−	0.354054i
$426$	−11.2749			−0.546272
$427$			17.0997i			0.827511i
$428$	11.8248			0.571571
$429$			13.0997i			0.632458i
$430$	8.35050			0.402697
$431$	34.5498			1.66421			0.832103	−	0.554620i	$-0.187137\pi$
$431$	34.5498			1.66421			0.832103	+	0.554620i	$0.187137\pi$
$432$			1.00000i			0.0481125i
$433$	−8.72508			−0.419301			−0.209650	−	0.977776i	$-0.567233\pi$
$433$	−8.72508			−0.419301			−0.209650	+	0.977776i	$0.567233\pi$
$434$			18.5498i			0.890421i
$435$	−28.0000			−1.34250
$436$			19.0997i			0.914708i
$437$		0			0
$438$		−	7.27492i		−	0.347609i
$439$			8.90033i			0.424790i	0.977184	+	0.212395i	$0.0681263\pi$
$439$			8.90033i			0.424790i	−0.977184	+	0.212395i	$0.931874\pi$
$440$			13.0997i			0.624502i
$441$	−3.00000			−0.142857
$442$	−4.17525			−0.198596
$443$	−14.3746			−0.682957			−0.341479	−	0.939890i	$-0.610928\pi$
$443$	−14.3746			−0.682957			−0.341479	+	0.939890i	$0.610928\pi$
$444$			6.00000i			0.284747i
$445$			25.6254i			1.21476i
$446$	−25.2749			−1.19680
$447$	16.5498			0.782780
$448$			2.00000i			0.0944911i
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$	5.72508			0.269883
$451$	−5.09967	−	25.0997i	−0.240134	−	1.18190i
$452$	4.54983			0.214006
$453$	−2.00000			−0.0939682
$454$			26.5498i			1.24605i
$455$	21.4502			1.00560
$456$	−1.27492			−0.0597035
$457$			1.09967i			0.0514403i	0.999669	+	0.0257202i	$0.00818789\pi$
$457$			1.09967i			0.0514403i	−0.999669	+	0.0257202i	$0.991812\pi$
$458$			19.0997i			0.892469i
$459$	1.27492			0.0595080
$460$		0			0
$461$	3.09967			0.144366			0.0721830	−	0.997391i	$-0.477003\pi$
$461$	3.09967			0.144366			0.0721830	+	0.997391i	$0.477003\pi$
$462$		−	8.00000i		−	0.372194i
$463$		−	11.4502i		−	0.532134i	−0.963954	−	0.266067i	$-0.914276\pi$
$463$		−	11.4502i		−	0.532134i	0.963954	−	0.266067i	$-0.0857243\pi$
$464$			8.54983i			0.396916i
$465$		−	30.3746i		−	1.40859i
$466$			7.82475i			0.362475i
$467$	−29.2749			−1.35468			−0.677341	−	0.735669i	$-0.736868\pi$
$467$	−29.2749			−1.35468			−0.677341	+	0.735669i	$0.736868\pi$
$468$		−	3.27492i		−	0.151383i
$469$	18.5498			0.856552
$470$		−	41.0997i		−	1.89579i
$471$	−14.0000			−0.645086
$472$	−5.27492			−0.242798
$473$			10.1993i			0.468966i
$474$	8.54983			0.392707
$475$			7.29901i			0.334901i
$476$	2.54983			0.116871
$477$		−	2.00000i		−	0.0915737i
$478$			15.2749i			0.698658i
$479$		−	18.9244i		−	0.864679i	−0.901711	−	0.432339i	$-0.857688\pi$
$479$		−	18.9244i		−	0.864679i	0.901711	−	0.432339i	$-0.142312\pi$
$480$		−	3.27492i		−	0.149479i
$481$		−	19.6495i		−	0.895940i
$482$	−9.82475			−0.447505
$483$		0			0
$484$	−5.00000			−0.227273
$485$		−	47.6495i		−	2.16365i
$486$			1.00000i			0.0453609i
$487$	35.8248			1.62337			0.811687	−	0.584092i	$-0.198549\pi$
$487$	35.8248			1.62337			0.811687	+	0.584092i	$0.198549\pi$
$488$	−8.54983			−0.387033
$489$			4.00000i			0.180886i
$490$	9.82475			0.443837
$491$	7.82475			0.353126			0.176563	−	0.984289i	$-0.443502\pi$
$491$	7.82475			0.353126			0.176563	+	0.984289i	$0.443502\pi$
$492$	1.27492	+	6.27492i	0.0574777	+	0.282895i
$493$	10.9003			0.490926
$494$	4.17525			0.187853
$495$			13.0997i			0.588786i
$496$	−9.27492			−0.416456
$497$	−22.5498			−1.01150
$498$			1.27492i			0.0571304i
$499$			14.3746i			0.643495i	0.946826	+	0.321747i	$0.104270\pi$
$499$			14.3746i			0.643495i	−0.946826	+	0.321747i	$0.895730\pi$
$500$	−2.37459			−0.106195
$501$	0.725083			0.0323943
$502$	29.0997			1.29878
$503$			12.7251i			0.567383i	0.958916	+	0.283692i	$0.0915592\pi$
$503$			12.7251i			0.567383i	−0.958916	+	0.283692i	$0.908441\pi$
$504$			2.00000i			0.0890871i
$505$		−	54.1993i		−	2.41184i
$506$		0			0
$507$		−	2.27492i		−	0.101033i
$508$	1.27492			0.0565653
$509$		−	37.6495i		−	1.66878i	−0.551171	−	0.834392i	$-0.685819\pi$
$509$		−	37.6495i		−	1.66878i	0.551171	−	0.834392i	$-0.314181\pi$
$510$	−4.17525			−0.184883
$511$		−	14.5498i		−	0.643647i
$512$	−1.00000			−0.0441942
$513$	−1.27492			−0.0562890
$514$		−	1.45017i		−	0.0639641i
$515$	38.7251			1.70643
$516$		−	2.54983i		−	0.112250i
$517$	50.1993			2.20776
$518$			12.0000i			0.527250i
$519$		−	23.0997i		−	1.01396i
$520$			10.7251i			0.470326i
$521$			25.2749i			1.10731i	0.832745	+	0.553657i	$0.186768\pi$
$521$			25.2749i			1.10731i	−0.832745	+	0.553657i	$0.813232\pi$
$522$			8.54983i			0.374216i
$523$	11.6495			0.509397			0.254699	−	0.967020i	$-0.418024\pi$
$523$	11.6495			0.509397			0.254699	+	0.967020i	$0.418024\pi$
$524$	16.0000			0.698963
$525$	11.4502			0.499726
$526$		−	22.9244i		−	0.999552i
$527$			11.8248i			0.515094i
$528$	4.00000			0.174078
$529$	−23.0000			−1.00000
$530$			6.54983i			0.284507i
$531$	−5.27492			−0.228912
$532$	−2.54983			−0.110549
$533$	−4.17525	−	20.5498i	−0.180850	−	0.890112i
$534$	7.82475			0.338610
$535$	−38.7251			−1.67423
$536$			9.27492i			0.400615i
$537$	9.09967			0.392680
$538$	−0.725083			−0.0312605
$539$			12.0000i			0.516877i
$540$		−	3.27492i		−	0.140930i
$541$	−34.7492			−1.49398			−0.746992	−	0.664833i	$-0.768503\pi$
$541$	−34.7492			−1.49398			−0.746992	+	0.664833i	$0.768503\pi$
$542$	5.09967			0.219050
$543$	11.2749			0.483853
$544$			1.27492i			0.0546616i
$545$		−	62.5498i		−	2.67934i
$546$		−	6.54983i		−	0.280307i
$547$		−	5.62541i		−	0.240525i	−0.992742	−	0.120263i	$-0.961626\pi$
$547$		−	5.62541i		−	0.240525i	0.992742	−	0.120263i	$-0.0383737\pi$
$548$		−	10.5498i		−	0.450667i
$549$	−8.54983			−0.364898
$550$		−	22.9003i		−	0.976473i
$551$	−10.9003			−0.464370
$552$		0			0
$553$	17.0997			0.727152
$554$	−15.4502			−0.656415
$555$		−	19.6495i		−	0.834075i
$556$	−10.5498			−0.447413
$557$			12.9003i			0.546605i	0.961928	+	0.273302i	$0.0881159\pi$
$557$			12.9003i			0.546605i	−0.961928	+	0.273302i	$0.911884\pi$
$558$	−9.27492			−0.392639
$559$			8.35050i			0.353188i
$560$		−	6.54983i		−	0.276781i
$561$		−	5.09967i		−	0.215308i
$562$		−	5.27492i		−	0.222509i
$563$			10.5498i			0.444623i	0.974976	+	0.222311i	$0.0713601\pi$
$563$			10.5498i			0.444623i	−0.974976	+	0.222311i	$0.928640\pi$
$564$	−12.5498			−0.528443
$565$	−14.9003			−0.626862
$566$	23.6495			0.994063
$567$			2.00000i			0.0839921i
$568$		−	11.2749i		−	0.473085i
$569$	−19.4502			−0.815393			−0.407697	−	0.913117i	$-0.633668\pi$
$569$	−19.4502			−0.815393			−0.407697	+	0.913117i	$0.633668\pi$
$570$	4.17525			0.174882
$571$			2.90033i			0.121375i	0.998157	+	0.0606875i	$0.0193293\pi$
$571$			2.90033i			0.121375i	−0.998157	+	0.0606875i	$0.980671\pi$
$572$	−13.0997			−0.547725
$573$	4.54983			0.190072
$574$	2.54983	+	12.5498i	0.106428	+	0.523820i
$575$		0			0
$576$	−1.00000			−0.0416667
$577$			13.0997i			0.545346i	0.962107	+	0.272673i	$0.0879078\pi$
$577$			13.0997i			0.545346i	−0.962107	+	0.272673i	$0.912092\pi$
$578$	−15.3746			−0.639498
$579$	20.0000			0.831172
$580$		−	28.0000i		−	1.16264i
$581$			2.54983i			0.105785i
$582$	−14.5498			−0.603110
$583$	−8.00000			−0.331326
$584$	7.27492			0.301038
$585$			10.7251i			0.443428i
$586$		−	2.00000i		−	0.0826192i
$587$			22.5498i			0.930731i	0.885119	+	0.465366i	$0.154077\pi$
$587$			22.5498i			0.930731i	−0.885119	+	0.465366i	$0.845923\pi$
$588$		−	3.00000i		−	0.123718i
$589$		−	11.8248i		−	0.487230i
$590$	17.2749			0.711197
$591$			23.2749i			0.957402i
$592$	−6.00000			−0.246598
$593$		−	8.17525i		−	0.335717i	−0.985811	−	0.167859i	$-0.946315\pi$
$593$		−	8.17525i		−	0.335717i	0.985811	−	0.167859i	$-0.0536852\pi$
$594$	4.00000			0.164122
$595$	−8.35050			−0.342337
$596$			16.5498i			0.677908i
$597$	2.00000			0.0818546
$598$		0			0
$599$	2.54983			0.104183			0.0520917	−	0.998642i	$-0.483411\pi$
$599$	2.54983			0.104183			0.0520917	+	0.998642i	$0.483411\pi$
$600$			5.72508i			0.233726i
$601$		−	26.5498i		−	1.08299i	−0.840704	−	0.541495i	$-0.817858\pi$
$601$		−	26.5498i		−	1.08299i	0.840704	−	0.541495i	$-0.182142\pi$
$602$		−	5.09967i		−	0.207847i
$603$			9.27492i			0.377704i
$604$		−	2.00000i		−	0.0813788i
$605$	16.3746			0.665722
$606$	−16.5498			−0.672291
$607$	−24.0000			−0.974130			−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000			−0.974130			−0.487065	+	0.873366i	$0.661933\pi$
$608$		−	1.27492i		−	0.0517047i
$609$			17.0997i			0.692914i
$610$	28.0000			1.13369
$611$	41.0997			1.66271
$612$			1.27492i			0.0515355i
$613$	−8.54983			−0.345325			−0.172662	−	0.984981i	$-0.555237\pi$
$613$	−8.54983			−0.345325			−0.172662	+	0.984981i	$0.555237\pi$
$614$	−33.0997			−1.33579
$615$	−4.17525	−	20.5498i	−0.168362	−	0.828649i
$616$	8.00000			0.322329
$617$	7.45017			0.299932			0.149966	−	0.988691i	$-0.452084\pi$
$617$	7.45017			0.299932			0.149966	+	0.988691i	$0.452084\pi$
$618$		−	11.8248i		−	0.475661i
$619$	13.0997			0.526520			0.263260	−	0.964725i	$-0.415202\pi$
$619$	13.0997			0.526520			0.263260	+	0.964725i	$0.415202\pi$
$620$	30.3746			1.21987
$621$		0			0
$622$			20.7251i			0.831000i
$623$	15.6495			0.626984
$624$	3.27492			0.131102
$625$	−20.8488			−0.833954
$626$		−	19.6495i		−	0.785352i
$627$			5.09967i			0.203661i
$628$		−	14.0000i		−	0.558661i
$629$			7.64950i			0.305006i
$630$		−	6.54983i		−	0.260952i
$631$	20.1752			0.803164			0.401582	−	0.915823i	$-0.368460\pi$
$631$	20.1752			0.803164			0.401582	+	0.915823i	$0.368460\pi$
$632$			8.54983i			0.340094i
$633$	−9.27492			−0.368645
$634$		−	15.4502i		−	0.613604i
$635$	−4.17525			−0.165690
$636$	2.00000			0.0793052
$637$			9.82475i			0.389271i
$638$	34.1993			1.35396
$639$		−	11.2749i		−	0.446029i
$640$	3.27492			0.129452
$641$		−	5.45017i		−	0.215269i	−0.994191	−	0.107634i	$-0.965672\pi$
$641$		−	5.45017i		−	0.215269i	0.994191	−	0.107634i	$-0.0343276\pi$
$642$			11.8248i			0.466686i
$643$			41.2749i			1.62772i	0.581058	+	0.813862i	$0.302639\pi$
$643$			41.2749i			1.62772i	−0.581058	+	0.813862i	$0.697361\pi$
$644$		0			0
$645$			8.35050i			0.328800i
$646$	−1.62541			−0.0639511
$647$	10.9003			0.428536			0.214268	−	0.976775i	$-0.431263\pi$
$647$	10.9003			0.428536			0.214268	+	0.976775i	$0.431263\pi$
$648$	−1.00000			−0.0392837
$649$			21.0997i			0.828234i
$650$		−	18.7492i		−	0.735403i
$651$	−18.5498			−0.727025
$652$	−4.00000			−0.156652
$653$			37.6495i			1.47334i	0.676253	+	0.736669i	$0.263603\pi$
$653$			37.6495i			1.47334i	−0.676253	+	0.736669i	$0.736397\pi$
$654$	−19.0997			−0.746856
$655$	−52.3987			−2.04739
$656$	−6.27492	+	1.27492i	−0.244994	+	0.0497772i
$657$	7.27492			0.283822
$658$	−25.0997			−0.978487
$659$			2.54983i			0.0993274i	0.998766	+	0.0496637i	$0.0158150\pi$
$659$			2.54983i			0.0993274i	−0.998766	+	0.0496637i	$0.984185\pi$
$660$	−13.0997			−0.509904
$661$	28.1993			1.09683			0.548414	−	0.836207i	$-0.315232\pi$
$661$	28.1993			1.09683			0.548414	+	0.836207i	$0.315232\pi$
$662$		−	29.0997i		−	1.13099i
$663$		−	4.17525i		−	0.162153i
$664$	−1.27492			−0.0494764
$665$	8.35050			0.323818
$666$	−6.00000			−0.232495
$667$		0			0
$668$			0.725083i			0.0280543i
$669$		−	25.2749i		−	0.977185i
$670$		−	30.3746i		−	1.17347i
$671$			34.1993i			1.32025i
$672$	−2.00000			−0.0771517
$673$		−	22.1993i		−	0.855722i	−0.903845	−	0.427861i	$-0.859267\pi$
$673$		−	22.1993i		−	0.855722i	0.903845	−	0.427861i	$-0.140733\pi$
$674$	−9.82475			−0.378435
$675$			5.72508i			0.220359i
$676$	2.27492			0.0874968
$677$	−37.4743			−1.44025			−0.720126	−	0.693843i	$-0.755916\pi$
$677$	−37.4743			−1.44025			−0.720126	+	0.693843i	$0.755916\pi$
$678$			4.54983i			0.174735i
$679$	−29.0997			−1.11674
$680$		−	4.17525i		−	0.160113i
$681$	−26.5498			−1.01739
$682$			37.0997i			1.42062i
$683$		−	33.0997i		−	1.26652i	−0.773938	−	0.633262i	$-0.781716\pi$
$683$		−	33.0997i		−	1.26652i	0.773938	−	0.633262i	$-0.218284\pi$
$684$		−	1.27492i		−	0.0487477i
$685$			34.5498i			1.32008i
$686$		−	20.0000i		−	0.763604i
$687$	−19.0997			−0.728698
$688$	2.54983			0.0972115
$689$	−6.54983			−0.249529
$690$		0			0
$691$			17.0997i			0.650502i	0.945628	+	0.325251i	$0.105449\pi$
$691$			17.0997i			0.650502i	−0.945628	+	0.325251i	$0.894551\pi$
$692$	23.0997			0.878118
$693$	8.00000			0.303895
$694$			13.0997i			0.497257i
$695$	34.5498			1.31055
$696$	−8.54983			−0.324081
$697$	1.62541	+	8.00000i	0.0615669	+	0.303022i
$698$	6.00000			0.227103
$699$	−7.82475			−0.295959
$700$			11.4502i			0.432776i
$701$	32.1993			1.21615			0.608076	−	0.793879i	$-0.291942\pi$
$701$	32.1993			1.21615			0.608076	+	0.793879i	$0.291942\pi$
$702$	3.27492			0.123604
$703$		−	7.64950i		−	0.288506i
$704$			4.00000i			0.150756i
$705$	41.0997			1.54790
$706$	4.54983			0.171235
$707$	−33.0997			−1.24484
$708$		−	5.27492i		−	0.198244i
$709$		−	3.27492i		−	0.122992i	−0.998107	−	0.0614960i	$-0.980413\pi$
$709$		−	3.27492i		−	0.122992i	0.998107	−	0.0614960i	$-0.0195872\pi$
$710$			36.9244i			1.38575i
$711$			8.54983i			0.320644i
$712$			7.82475i			0.293245i
$713$		0			0
$714$			2.54983i			0.0954252i
$715$	42.9003			1.60438
$716$			9.09967i			0.340071i
$717$	−15.2749			−0.570452
$718$	13.0997			0.488875
$719$		−	9.82475i		−	0.366401i	−0.983076	−	0.183201i	$-0.941354\pi$
$719$		−	9.82475i		−	0.366401i	0.983076	−	0.183201i	$-0.0586458\pi$
$720$	3.27492			0.122049
$721$		−	23.6495i		−	0.880754i
$722$	−17.3746			−0.646615
$723$		−	9.82475i		−	0.365386i
$724$			11.2749i			0.419029i
$725$			48.9485i			1.81790i
$726$		−	5.00000i		−	0.185567i
$727$			16.5498i			0.613799i	0.951742	+	0.306900i	$0.0992916\pi$
$727$			16.5498i			0.613799i	−0.951742	+	0.306900i	$0.900708\pi$
$728$	6.54983			0.242753
$729$	−1.00000			−0.0370370
$730$	−23.8248			−0.881794
$731$		−	3.25083i		−	0.120236i
$732$		−	8.54983i		−	0.316011i
$733$	−17.6495			−0.651899			−0.325950	−	0.945387i	$-0.605684\pi$
$733$	−17.6495			−0.651899			−0.325950	+	0.945387i	$0.605684\pi$
$734$	−8.00000			−0.295285
$735$			9.82475i			0.362391i
$736$		0			0
$737$	37.0997			1.36658
$738$	−6.27492	+	1.27492i	−0.230983	+	0.0469304i
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	19.6495			0.722330
$741$			4.17525i			0.153382i
$742$	4.00000			0.146845
$743$	32.0000			1.17397			0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000			1.17397			0.586983	+	0.809599i	$0.300316\pi$
$744$		−	9.27492i		−	0.340035i
$745$		−	54.1993i		−	1.98571i
$746$	23.0997			0.845739
$747$	−1.27492			−0.0466468
$748$	5.09967			0.186462
$749$			23.6495i			0.864134i
$750$		−	2.37459i		−	0.0867076i
$751$			27.0997i			0.988881i	0.869211	+	0.494440i	$0.164627\pi$
$751$			27.0997i			0.988881i	−0.869211	+	0.494440i	$0.835373\pi$
$752$		−	12.5498i		−	0.457645i
$753$			29.0997i			1.06045i
$754$	28.0000			1.01970
$755$			6.54983i			0.238373i
$756$	−2.00000			−0.0727393
$757$			47.0997i			1.71187i	0.517086	+	0.855933i	$0.327017\pi$
$757$			47.0997i			1.71187i	−0.517086	+	0.855933i	$0.672983\pi$
$758$	−13.4502			−0.488532
$759$		0			0
$760$			4.17525i			0.151452i
$761$	28.5498			1.03493			0.517465	−	0.855704i	$-0.326876\pi$
$761$	28.5498			1.03493			0.517465	+	0.855704i	$0.326876\pi$
$762$			1.27492i			0.0461854i
$763$	−38.1993			−1.38291
$764$			4.54983i			0.164607i
$765$		−	4.17525i		−	0.150956i
$766$		−	21.8248i		−	0.788560i
$767$			17.2749i			0.623761i
$768$		−	1.00000i		−	0.0360844i
$769$	6.17525			0.222685			0.111343	−	0.993782i	$-0.464485\pi$
$769$	6.17525			0.222685			0.111343	+	0.993782i	$0.464485\pi$
$770$	−26.1993			−0.944159
$771$	1.45017			0.0522264
$772$			20.0000i			0.719816i
$773$			52.5498i			1.89009i	0.326946	+	0.945043i	$0.393981\pi$
$773$			52.5498i			1.89009i	−0.326946	+	0.945043i	$0.606019\pi$
$774$	2.54983			0.0916519
$775$	−53.0997			−1.90740
$776$		−	14.5498i		−	0.522309i
$777$	−12.0000			−0.430498
$778$	30.9244			1.10869
$779$	−1.62541	−	8.00000i	−0.0582365	−	0.286630i
$780$	−10.7251			−0.384020
$781$	−45.0997			−1.61379
$782$		0			0
$783$	−8.54983			−0.305546
$784$	3.00000			0.107143
$785$			45.8488i			1.63642i
$786$			16.0000i			0.570701i
$787$	11.6495			0.415260			0.207630	−	0.978207i	$-0.433425\pi$
$787$	11.6495			0.415260			0.207630	+	0.978207i	$0.433425\pi$
$788$	−23.2749			−0.829135
$789$	22.9244			0.816131
$790$		−	28.0000i		−	0.996195i
$791$			9.09967i			0.323547i
$792$			4.00000i			0.142134i
$793$			28.0000i			0.994309i
$794$			29.8248i			1.05844i
$795$	−6.54983			−0.232299
$796$			2.00000i			0.0708881i
$797$	−38.9244			−1.37877			−0.689387	−	0.724393i	$-0.742120\pi$
$797$	−38.9244			−1.37877			−0.689387	+	0.724393i	$0.742120\pi$
$798$		−	2.54983i		−	0.0902632i
$799$	−16.0000			−0.566039
$800$	−5.72508			−0.202412
$801$			7.82475i			0.276474i
$802$	−7.09967			−0.250698
$803$		−	29.0997i		−	1.02691i
$804$	−9.27492			−0.327101
$805$		0			0
$806$			30.3746i			1.06990i
$807$		−	0.725083i		−	0.0255241i
$808$		−	16.5498i		−	0.582221i
$809$			9.62541i			0.338412i	0.985581	+	0.169206i	$0.0541202\pi$
$809$			9.62541i			0.338412i	−0.985581	+	0.169206i	$0.945880\pi$
$810$	3.27492			0.115069
$811$	6.54983			0.229996			0.114998	−	0.993366i	$-0.463314\pi$
$811$	6.54983			0.229996			0.114998	+	0.993366i	$0.463314\pi$
$812$	−17.0997			−0.600081
$813$			5.09967i			0.178853i
$814$			24.0000i			0.841200i
$815$	13.0997			0.458861
$816$	−1.27492			−0.0446310
$817$			3.25083i			0.113732i
$818$	0.725083			0.0253519
$819$	6.54983			0.228870
$820$	20.5498	−	4.17525i	0.717631	−	0.145806i
$821$	−18.9244			−0.660467			−0.330233	−	0.943899i	$-0.607127\pi$
$821$	−18.9244			−0.660467			−0.330233	+	0.943899i	$0.607127\pi$
$822$	10.5498			0.367968
$823$		−	36.5498i		−	1.27405i	−0.770844	−	0.637024i	$-0.780165\pi$
$823$		−	36.5498i		−	1.27405i	0.770844	−	0.637024i	$-0.219835\pi$
$824$	11.8248			0.411935
$825$	22.9003			0.797287
$826$		−	10.5498i		−	0.367076i
$827$		−	10.5498i		−	0.366854i	−0.983033	−	0.183427i	$-0.941281\pi$
$827$		−	10.5498i		−	0.366854i	0.983033	−	0.183427i	$-0.0587190\pi$
$828$		0			0
$829$	−43.0997			−1.49691			−0.748457	−	0.663184i	$-0.769205\pi$
$829$	−43.0997			−1.49691			−0.748457	+	0.663184i	$0.769205\pi$
$830$	4.17525			0.144925
$831$		−	15.4502i		−	0.535960i
$832$			3.27492i			0.113537i
$833$		−	3.82475i		−	0.132520i
$834$		−	10.5498i		−	0.365311i
$835$		−	2.37459i		−	0.0821759i
$836$	−5.09967			−0.176376
$837$		−	9.27492i		−	0.320588i
$838$	26.7251			0.923203
$839$		−	10.9244i		−	0.377153i	−0.982059	−	0.188576i	$-0.939613\pi$
$839$		−	10.9244i		−	0.377153i	0.982059	−	0.188576i	$-0.0603873\pi$
$840$	6.54983			0.225991
$841$	−44.0997			−1.52068
$842$		−	25.8248i		−	0.889980i
$843$	5.27492			0.181678
$844$		−	9.27492i		−	0.319256i
$845$	−7.45017			−0.256293
$846$		−	12.5498i		−	0.431472i
$847$		−	10.0000i		−	0.343604i
$848$			2.00000i			0.0686803i
$849$			23.6495i			0.811649i
$850$			7.29901i			0.250354i
$851$		0			0
$852$	11.2749			0.386272
$853$	24.5498			0.840570			0.420285	−	0.907392i	$-0.361930\pi$
$853$	24.5498			0.840570			0.420285	+	0.907392i	$0.361930\pi$
$854$		−	17.0997i		−	0.585139i
$855$			4.17525i			0.142790i
$856$	−11.8248			−0.404162
$857$	6.35050			0.216929			0.108464	−	0.994100i	$-0.465407\pi$
$857$	6.35050			0.216929			0.108464	+	0.994100i	$0.465407\pi$
$858$		−	13.0997i		−	0.447215i
$859$	54.1993			1.84926			0.924629	−	0.380870i	$-0.124375\pi$
$859$	54.1993			1.84926			0.924629	+	0.380870i	$0.124375\pi$
$860$	−8.35050			−0.284750
$861$	−12.5498	+	2.54983i	−0.427697	+	0.0868981i
$862$	−34.5498			−1.17677
$863$	26.5498			0.903767			0.451883	−	0.892077i	$-0.350752\pi$
$863$	26.5498			0.903767			0.451883	+	0.892077i	$0.350752\pi$
$864$		−	1.00000i		−	0.0340207i
$865$	−75.6495			−2.57216
$866$	8.72508			0.296490
$867$		−	15.3746i		−	0.522148i
$868$		−	18.5498i		−	0.629622i
$869$	34.1993			1.16013
$870$	28.0000			0.949289
$871$	30.3746			1.02920
$872$		−	19.0997i		−	0.646796i
$873$		−	14.5498i		−	0.492437i
$874$		0			0
$875$		−	4.74917i		−	0.160551i
$876$			7.27492i			0.245797i
$877$	−25.6495			−0.866122			−0.433061	−	0.901365i	$-0.642567\pi$
$877$	−25.6495			−0.866122			−0.433061	+	0.901365i	$0.642567\pi$
$878$		−	8.90033i		−	0.300372i
$879$	2.00000			0.0674583
$880$		−	13.0997i		−	0.441590i
$881$	24.1993			0.815296			0.407648	−	0.913139i	$-0.366349\pi$
$881$	24.1993			0.815296			0.407648	+	0.913139i	$0.366349\pi$
$882$	3.00000			0.101015
$883$			5.62541i			0.189310i	0.995510	+	0.0946551i	$0.0301748\pi$
$883$			5.62541i			0.189310i	−0.995510	+	0.0946551i	$0.969825\pi$
$884$	4.17525			0.140429
$885$			17.2749i			0.580690i
$886$	14.3746			0.482924
$887$		−	26.9244i		−	0.904034i	−0.892009	−	0.452017i	$-0.850705\pi$
$887$		−	26.9244i		−	0.904034i	0.892009	−	0.452017i	$-0.149295\pi$
$888$		−	6.00000i		−	0.201347i
$889$			2.54983i			0.0855187i
$890$		−	25.6254i		−	0.858966i
$891$			4.00000i			0.134005i
$892$	25.2749			0.846267
$893$	16.0000			0.535420
$894$	−16.5498			−0.553509
$895$		−	29.8007i		−	0.996126i
$896$		−	2.00000i		−	0.0668153i
$897$		0			0
$898$	−10.0000			−0.333704
$899$		−	79.2990i		−	2.64477i
$900$	−5.72508			−0.190836
$901$	2.54983			0.0849473
$902$	5.09967	+	25.0997i	0.169800	+	0.835728i
$903$	5.09967			0.169706
$904$	−4.54983			−0.151325
$905$		−	36.9244i		−	1.22741i
$906$	2.00000			0.0664455
$907$	−32.7492			−1.08742			−0.543709	−	0.839274i	$-0.682981\pi$
$907$	−32.7492			−1.08742			−0.543709	+	0.839274i	$0.682981\pi$
$908$		−	26.5498i		−	0.881087i
$909$		−	16.5498i		−	0.548923i
$910$	−21.4502			−0.711066
$911$	13.4502			0.445624			0.222812	−	0.974861i	$-0.428476\pi$
$911$	13.4502			0.445624			0.222812	+	0.974861i	$0.428476\pi$
$912$	1.27492			0.0422167
$913$			5.09967i			0.168774i
$914$		−	1.09967i		−	0.0363738i
$915$			28.0000i			0.925651i
$916$		−	19.0997i		−	0.631071i
$917$			32.0000i			1.05673i
$918$	−1.27492			−0.0420785
$919$		−	52.5498i		−	1.73346i	−0.498778	−	0.866730i	$-0.666218\pi$
$919$		−	52.5498i		−	1.73346i	0.498778	−	0.866730i	$-0.333782\pi$
$920$		0			0
$921$		−	33.0997i		−	1.09067i
$922$	−3.09967			−0.102082
$923$	−36.9244			−1.21538
$924$			8.00000i			0.263181i
$925$	−34.3505			−1.12944
$926$			11.4502i			0.376276i
$927$	11.8248			0.388376
$928$		−	8.54983i		−	0.280662i
$929$		−	4.17525i		−	0.136985i	−0.997652	−	0.0684927i	$-0.978181\pi$
$929$		−	4.17525i		−	0.136985i	0.997652	−	0.0684927i	$-0.0218190\pi$
$930$			30.3746i			0.996022i
$931$			3.82475i			0.125351i
$932$		−	7.82475i		−	0.256308i
$933$	−20.7251			−0.678509
$934$	29.2749			0.957905
$935$	−16.7010			−0.546181
$936$			3.27492i			0.107044i
$937$			18.1993i			0.594546i	0.954792	+	0.297273i	$0.0960772\pi$
$937$			18.1993i			0.594546i	−0.954792	+	0.297273i	$0.903923\pi$
$938$	−18.5498			−0.605674
$939$	19.6495			0.641237
$940$			41.0997i			1.34052i
$941$	40.3746			1.31617			0.658087	−	0.752942i	$-0.271366\pi$
$941$	40.3746			1.31617			0.658087	+	0.752942i	$0.271366\pi$
$942$	14.0000			0.456145
$943$		0			0
$944$	5.27492			0.171684
$945$	6.54983			0.213066
$946$		−	10.1993i		−	0.331609i
$947$	−29.2749			−0.951307			−0.475653	−	0.879633i	$-0.657788\pi$
$947$	−29.2749			−0.951307			−0.475653	+	0.879633i	$0.657788\pi$
$948$	−8.54983			−0.277686
$949$		−	23.8248i		−	0.773384i
$950$		−	7.29901i		−	0.236811i
$951$	15.4502			0.501006
$952$	−2.54983			−0.0826406
$953$	−15.4502			−0.500480			−0.250240	−	0.968184i	$-0.580510\pi$
$953$	−15.4502			−0.500480			−0.250240	+	0.968184i	$0.580510\pi$
$954$			2.00000i			0.0647524i
$955$		−	14.9003i		−	0.482163i
$956$		−	15.2749i		−	0.494026i
$957$			34.1993i			1.10551i
$958$			18.9244i			0.611420i
$959$	21.0997			0.681344
$960$			3.27492i			0.105697i
$961$	55.0241			1.77497
$962$			19.6495i			0.633525i
$963$	−11.8248			−0.381047
$964$	9.82475			0.316434
$965$		−	65.4983i		−	2.10847i
$966$		0			0
$967$		−	32.5498i		−	1.04673i	−0.852108	−	0.523366i	$-0.824676\pi$
$967$		−	32.5498i		−	1.04673i	0.852108	−	0.523366i	$-0.175324\pi$
$968$	5.00000			0.160706
$969$		−	1.62541i		−	0.0522158i
$970$			47.6495i			1.52993i
$971$			11.6495i			0.373850i	0.982374	+	0.186925i	$0.0598522\pi$
$971$			11.6495i			0.373850i	−0.982374	+	0.186925i	$0.940148\pi$
$972$		−	1.00000i		−	0.0320750i
$973$		−	21.0997i		−	0.676424i
$974$	−35.8248			−1.14790
$975$	18.7492			0.600454
$976$	8.54983			0.273674
$977$			46.3746i			1.48365i	0.670591	+	0.741827i	$0.266041\pi$
$977$			46.3746i			1.48365i	−0.670591	+	0.741827i	$0.733959\pi$
$978$		−	4.00000i		−	0.127906i
$979$	31.2990			1.00032
$980$	−9.82475			−0.313840
$981$		−	19.0997i		−	0.609805i
$982$	−7.82475			−0.249698
$983$	−10.9003			−0.347667			−0.173833	−	0.984775i	$-0.555615\pi$
$983$	−10.9003			−0.347667			−0.173833	+	0.984775i	$0.555615\pi$
$984$	−1.27492	−	6.27492i	−0.0406429	−	0.200037i
$985$	76.2234			2.42868
$986$	−10.9003			−0.347137
$987$		−	25.0997i		−	0.798931i
$988$	−4.17525			−0.132832
$989$		0			0
$990$		−	13.0997i		−	0.416335i
$991$			16.9003i			0.536857i	0.963300	+	0.268428i	$0.0865043\pi$
$991$			16.9003i			0.536857i	−0.963300	+	0.268428i	$0.913496\pi$
$992$	9.27492			0.294479
$993$	29.0997			0.923450
$994$	22.5498			0.715237
$995$		−	6.54983i		−	0.207644i
$996$		−	1.27492i		−	0.0403973i
$997$			42.9244i			1.35943i	0.733476	+	0.679715i	$0.237897\pi$
$997$			42.9244i			1.35943i	−0.733476	+	0.679715i	$0.762103\pi$
$998$		−	14.3746i		−	0.455020i
$999$		−	6.00000i		−	0.189832i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	246.2.d.b.163.1	✓	4
3.2	odd	2		738.2.d.h.163.4		4
4.3	odd	2		1968.2.j.c.1393.3		4
41.40	even	2	inner	246.2.d.b.163.3	yes	4
123.122	odd	2		738.2.d.h.163.3		4
164.163	odd	2		1968.2.j.c.1393.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
246.2.d.b.163.1	✓	4	1.1	even	1	trivial
246.2.d.b.163.3	yes	4	41.40	even	2	inner
738.2.d.h.163.3		4	123.122	odd	2
738.2.d.h.163.4		4	3.2	odd	2
1968.2.j.c.1393.1		4	164.163	odd	2
1968.2.j.c.1393.3		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 246 = 2 \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	246.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -3.27492 q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +3.27492 q^{10} +4.00000i q^{11} -1.00000i q^{12} +3.27492i q^{13} -2.00000i q^{14} +3.27492i q^{15} +1.00000 q^{16} -1.27492i q^{17} +1.00000 q^{18} +1.27492i q^{19} -3.27492 q^{20} +2.00000 q^{21} -4.00000i q^{22} +1.00000i q^{24} +5.72508 q^{25} -3.27492i q^{26} +1.00000i q^{27} +2.00000i q^{28} +8.54983i q^{29} -3.27492i q^{30} -9.27492 q^{31} -1.00000 q^{32} +4.00000 q^{33} +1.27492i q^{34} -6.54983i q^{35} -1.00000 q^{36} -6.00000 q^{37} -1.27492i q^{38} +3.27492 q^{39} +3.27492 q^{40} +(-6.27492 + 1.27492i) q^{41} -2.00000 q^{42} +2.54983 q^{43} +4.00000i q^{44} +3.27492 q^{45} -12.5498i q^{47} -1.00000i q^{48} +3.00000 q^{49} -5.72508 q^{50} -1.27492 q^{51} +3.27492i q^{52} +2.00000i q^{53} -1.00000i q^{54} -13.0997i q^{55} -2.00000i q^{56} +1.27492 q^{57} -8.54983i q^{58} +5.27492 q^{59} +3.27492i q^{60} +8.54983 q^{61} +9.27492 q^{62} -2.00000i q^{63} +1.00000 q^{64} -10.7251i q^{65} -4.00000 q^{66} -9.27492i q^{67} -1.27492i q^{68} +6.54983i q^{70} +11.2749i q^{71} +1.00000 q^{72} -7.27492 q^{73} +6.00000 q^{74} -5.72508i q^{75} +1.27492i q^{76} -8.00000 q^{77} -3.27492 q^{78} -8.54983i q^{79} -3.27492 q^{80} +1.00000 q^{81} +(6.27492 - 1.27492i) q^{82} +1.27492 q^{83} +2.00000 q^{84} +4.17525i q^{85} -2.54983 q^{86} +8.54983 q^{87} -4.00000i q^{88} -7.82475i q^{89} -3.27492 q^{90} -6.54983 q^{91} +9.27492i q^{93} +12.5498i q^{94} -4.17525i q^{95} +1.00000i q^{96} +14.5498i q^{97} -3.00000 q^{98} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{8} - 4 q^{9} - 2 q^{10} + 4 q^{16} + 4 q^{18} + 2 q^{20} + 8 q^{21} + 38 q^{25} - 22 q^{31} - 4 q^{32} + 16 q^{33} - 4 q^{36} - 24 q^{37} - 2 q^{39} - 2 q^{40} - 10 q^{41}+ \cdots - 12 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^8 - 4 * q^9 - 2 * q^10 + 4 * q^16 + 4 * q^18 + 2 * q^20 + 8 * q^21 + 38 * q^25 - 22 * q^31 - 4 * q^32 + 16 * q^33 - 4 * q^36 - 24 * q^37 - 2 * q^39 - 2 * q^40 - 10 * q^41 - 8 * q^42 - 20 * q^43 - 2 * q^45 + 12 * q^49 - 38 * q^50 + 10 * q^51 - 10 * q^57 + 6 * q^59 + 4 * q^61 + 22 * q^62 + 4 * q^64 - 16 * q^66 + 4 * q^72 - 14 * q^73 + 24 * q^74 - 32 * q^77 + 2 * q^78 + 2 * q^80 + 4 * q^81 + 10 * q^82 - 10 * q^83 + 8 * q^84 + 20 * q^86 + 4 * q^87 + 2 * q^90 + 4 * q^91 - 12 * q^98

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