LMFDB - Embedded newform 1190.2.c.f.1121.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1190,2,Mod(1121,1190)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1190, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1190.1121"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1190 = 2 \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1190.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,4,0,4,0,0,0,4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$9.50219784053$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
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			1121.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	1190.1121
Dual form			1190.2.c.f.1121.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.56155i q^{3} +1.00000 q^{4} -1.00000i q^{5} +2.56155i q^{6} -1.00000i q^{7} +1.00000 q^{8} -3.56155 q^{9} -1.00000i q^{10} +2.00000i q^{11} +2.56155i q^{12} +6.56155 q^{13} -1.00000i q^{14} +2.56155 q^{15} +1.00000 q^{16} +4.12311i q^{17} -3.56155 q^{18} -2.56155 q^{19} -1.00000i q^{20} +2.56155 q^{21} +2.00000i q^{22} +1.12311i q^{23} +2.56155i q^{24} -1.00000 q^{25} +6.56155 q^{26} -1.43845i q^{27} -1.00000i q^{28} +10.5616i q^{29} +2.56155 q^{30} -2.56155i q^{31} +1.00000 q^{32} -5.12311 q^{33} +4.12311i q^{34} -1.00000 q^{35} -3.56155 q^{36} +2.00000i q^{37} -2.56155 q^{38} +16.8078i q^{39} -1.00000i q^{40} -11.1231i q^{41} +2.56155 q^{42} +10.2462 q^{43} +2.00000i q^{44} +3.56155i q^{45} +1.12311i q^{46} -4.56155 q^{47} +2.56155i q^{48} -1.00000 q^{49} -1.00000 q^{50} -10.5616 q^{51} +6.56155 q^{52} +1.68466 q^{53} -1.43845i q^{54} +2.00000 q^{55} -1.00000i q^{56} -6.56155i q^{57} +10.5616i q^{58} -6.56155 q^{59} +2.56155 q^{60} +5.68466i q^{61} -2.56155i q^{62} +3.56155i q^{63} +1.00000 q^{64} -6.56155i q^{65} -5.12311 q^{66} -1.12311 q^{67} +4.12311i q^{68} -2.87689 q^{69} -1.00000 q^{70} +4.56155i q^{71} -3.56155 q^{72} -10.8078i q^{73} +2.00000i q^{74} -2.56155i q^{75} -2.56155 q^{76} +2.00000 q^{77} +16.8078i q^{78} -17.3693i q^{79} -1.00000i q^{80} -7.00000 q^{81} -11.1231i q^{82} -12.2462 q^{83} +2.56155 q^{84} +4.12311 q^{85} +10.2462 q^{86} -27.0540 q^{87} +2.00000i q^{88} +16.5616 q^{89} +3.56155i q^{90} -6.56155i q^{91} +1.12311i q^{92} +6.56155 q^{93} -4.56155 q^{94} +2.56155i q^{95} +2.56155i q^{96} -9.68466i q^{97} -1.00000 q^{98} -7.12311i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 6 q^{9} + 18 q^{13} + 2 q^{15} + 4 q^{16} - 6 q^{18} - 2 q^{19} + 2 q^{21} - 4 q^{25} + 18 q^{26} + 2 q^{30} + 4 q^{32} - 4 q^{33} - 4 q^{35} - 6 q^{36} - 2 q^{38} + 2 q^{42}+ \cdots - 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 6 * q^9 + 18 * q^13 + 2 * q^15 + 4 * q^16 - 6 * q^18 - 2 * q^19 + 2 * q^21 - 4 * q^25 + 18 * q^26 + 2 * q^30 + 4 * q^32 - 4 * q^33 - 4 * q^35 - 6 * q^36 - 2 * q^38 + 2 * q^42 + 8 * q^43 - 10 * q^47 - 4 * q^49 - 4 * q^50 - 34 * q^51 + 18 * q^52 - 18 * q^53 + 8 * q^55 - 18 * q^59 + 2 * q^60 + 4 * q^64 - 4 * q^66 + 12 * q^67 - 28 * q^69 - 4 * q^70 - 6 * q^72 - 2 * q^76 + 8 * q^77 - 28 * q^81 - 16 * q^83 + 2 * q^84 + 8 * q^86 - 34 * q^87 + 58 * q^89 + 18 * q^93 - 10 * q^94 - 4 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1190\mathbb{Z}\right)^\times$.

$n$	$71$	$171$	$477$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	1.00000	0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	2.56155i	1.04575i
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.00000	0.353553
$9$	−3.56155	−1.18718
$10$	− 1.00000i	− 0.316228i
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	2.56155i	0.739457i
$13$	6.56155	1.81985	0.909924	−	0.414776i	$-0.136140\pi$
$13$	6.56155	1.81985	0.909924	+	0.414776i	$0.136140\pi$
$14$	− 1.00000i	− 0.267261i
$15$	2.56155	0.661390
$16$	1.00000	0.250000
$17$	4.12311i	1.00000i
$17$	4.12311i	1.00000i
$18$	−3.56155	−0.839466
$19$	−2.56155	−0.587661	−0.293830	−	0.955858i	$-0.594930\pi$
$19$	−2.56155	−0.587661	−0.293830	+	0.955858i	$0.594930\pi$
$20$	− 1.00000i	− 0.223607i
$21$	2.56155	0.558977
$22$	2.00000i	0.426401i
$23$	1.12311i	0.234184i	0.993121	+	0.117092i	$0.0373572\pi$
$23$	1.12311i	0.234184i	−0.993121	+	0.117092i	$0.962643\pi$
$24$	2.56155i	0.522875i
$25$	−1.00000	−0.200000
$26$	6.56155	1.28683
$27$	− 1.43845i	− 0.276829i
$28$	− 1.00000i	− 0.188982i
$29$	10.5616i	1.96123i	0.195942	+	0.980616i	$0.437224\pi$
$29$	10.5616i	1.96123i	−0.195942	+	0.980616i	$0.562776\pi$
$30$	2.56155	0.467673
$31$	− 2.56155i	− 0.460068i	−0.973183	−	0.230034i	$-0.926116\pi$
$31$	− 2.56155i	− 0.460068i	0.973183	−	0.230034i	$-0.0738838\pi$
$32$	1.00000	0.176777
$33$	−5.12311	−0.891818
$34$	4.12311i	0.707107i
$35$	−1.00000	−0.169031
$36$	−3.56155	−0.593592
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	−2.56155	−0.415539
$39$	16.8078i	2.69140i
$40$	− 1.00000i	− 0.158114i
$41$	− 11.1231i	− 1.73714i	−0.495569	−	0.868569i	$-0.665040\pi$
$41$	− 11.1231i	− 1.73714i	0.495569	−	0.868569i	$-0.334960\pi$
$42$	2.56155	0.395256
$43$	10.2462	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$43$	10.2462	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$44$	2.00000i	0.301511i
$45$	3.56155i	0.530925i
$46$	1.12311i	0.165593i
$47$	−4.56155	−0.665371	−0.332685	−	0.943038i	$-0.607955\pi$
$47$	−4.56155	−0.665371	−0.332685	+	0.943038i	$0.607955\pi$
$48$	2.56155i	0.369728i
$49$	−1.00000	−0.142857
$50$	−1.00000	−0.141421
$51$	−10.5616	−1.47891
$52$	6.56155	0.909924
$53$	1.68466	0.231406	0.115703	−	0.993284i	$-0.463088\pi$
$53$	1.68466	0.231406	0.115703	+	0.993284i	$0.463088\pi$
$54$	− 1.43845i	− 0.195748i
$55$	2.00000	0.269680
$56$	− 1.00000i	− 0.133631i
$57$	− 6.56155i	− 0.869099i
$58$	10.5616i	1.38680i
$59$	−6.56155	−0.854241	−0.427121	−	0.904195i	$-0.640472\pi$
$59$	−6.56155	−0.854241	−0.427121	+	0.904195i	$0.640472\pi$
$60$	2.56155	0.330695
$61$	5.68466i	0.727846i	0.931429	+	0.363923i	$0.118563\pi$
$61$	5.68466i	0.727846i	−0.931429	+	0.363923i	$0.881437\pi$
$62$	− 2.56155i	− 0.325318i
$63$	3.56155i	0.448713i
$64$	1.00000	0.125000
$65$	− 6.56155i	− 0.813860i
$66$	−5.12311	−0.630611
$67$	−1.12311	−0.137209	−0.0686046	−	0.997644i	$-0.521855\pi$
$67$	−1.12311	−0.137209	−0.0686046	+	0.997644i	$0.521855\pi$
$68$	4.12311i	0.500000i
$69$	−2.87689	−0.346337
$70$	−1.00000	−0.119523
$71$	4.56155i	0.541357i	0.962670	+	0.270678i	$0.0872480\pi$
$71$	4.56155i	0.541357i	−0.962670	+	0.270678i	$0.912752\pi$
$72$	−3.56155	−0.419733
$73$	− 10.8078i	− 1.26495i	−0.774580	−	0.632477i	$-0.782038\pi$
$73$	− 10.8078i	− 1.26495i	0.774580	−	0.632477i	$-0.217962\pi$
$74$	2.00000i	0.232495i
$75$	− 2.56155i	− 0.295783i
$76$	−2.56155	−0.293830
$77$	2.00000	0.227921
$78$	16.8078i	1.90310i
$79$	− 17.3693i	− 1.95420i	−0.212779	−	0.977100i	$-0.568251\pi$
$79$	− 17.3693i	− 1.95420i	0.212779	−	0.977100i	$-0.431749\pi$
$80$	− 1.00000i	− 0.111803i
$81$	−7.00000	−0.777778
$82$	− 11.1231i	− 1.22834i
$83$	−12.2462	−1.34420	−0.672098	−	0.740462i	$-0.734607\pi$
$83$	−12.2462	−1.34420	−0.672098	+	0.740462i	$0.734607\pi$
$84$	2.56155	0.279488
$85$	4.12311	0.447214
$86$	10.2462	1.10488
$87$	−27.0540	−2.90049
$88$	2.00000i	0.213201i
$89$	16.5616	1.75552	0.877761	−	0.479100i	$-0.159037\pi$
$89$	16.5616	1.75552	0.877761	+	0.479100i	$0.159037\pi$
$90$	3.56155i	0.375421i
$91$	− 6.56155i	− 0.687838i
$92$	1.12311i	0.117092i
$93$	6.56155	0.680401
$94$	−4.56155	−0.470488
$95$	2.56155i	0.262810i
$96$	2.56155i	0.261437i
$97$	− 9.68466i	− 0.983328i	−0.870785	−	0.491664i	$-0.836389\pi$
$97$	− 9.68466i	− 0.983328i	0.870785	−	0.491664i	$-0.163611\pi$
$98$	−1.00000	−0.101015
$99$	− 7.12311i	− 0.715899i
$100$	−1.00000	−0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−10.5616	−1.04575
$103$	−11.1231	−1.09599	−0.547996	−	0.836481i	$-0.684609\pi$
$103$	−11.1231	−1.09599	−0.547996	+	0.836481i	$0.684609\pi$
$104$	6.56155	0.643413
$105$	− 2.56155i	− 0.249982i
$106$	1.68466	0.163628
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	− 1.43845i	− 0.138415i
$109$	4.31534i	0.413335i	0.978411	+	0.206667i	$0.0662618\pi$
$109$	4.31534i	0.413335i	−0.978411	+	0.206667i	$0.933738\pi$
$110$	2.00000	0.190693
$111$	−5.12311	−0.486264
$112$	− 1.00000i	− 0.0944911i
$113$	7.43845i	0.699750i	0.936796	+	0.349875i	$0.113776\pi$
$113$	7.43845i	0.699750i	−0.936796	+	0.349875i	$0.886224\pi$
$114$	− 6.56155i	− 0.614546i
$115$	1.12311	0.104730
$116$	10.5616i	0.980616i
$117$	−23.3693	−2.16049
$118$	−6.56155	−0.604040
$119$	4.12311	0.377964
$120$	2.56155	0.233837
$121$	7.00000	0.636364
$122$	5.68466i	0.514665i
$123$	28.4924	2.56908
$124$	− 2.56155i	− 0.230034i
$125$	1.00000i	0.0894427i
$126$	3.56155i	0.317288i
$127$	−17.9309	−1.59111	−0.795554	−	0.605883i	$-0.792820\pi$
$127$	−17.9309	−1.59111	−0.795554	+	0.605883i	$0.792820\pi$
$128$	1.00000	0.0883883
$129$	26.2462i	2.31085i
$130$	− 6.56155i	− 0.575486i
$131$	4.00000i	0.349482i	0.984614	+	0.174741i	$0.0559088\pi$
$131$	4.00000i	0.349482i	−0.984614	+	0.174741i	$0.944091\pi$
$132$	−5.12311	−0.445909
$133$	2.56155i	0.222115i
$134$	−1.12311	−0.0970215
$135$	−1.43845	−0.123802
$136$	4.12311i	0.353553i
$137$	23.1231	1.97554	0.987770	−	0.155917i	$-0.0498333\pi$
$137$	23.1231	1.97554	0.987770	+	0.155917i	$0.0498333\pi$
$138$	−2.87689	−0.244898
$139$	15.3693i	1.30361i	0.758387	+	0.651804i	$0.225988\pi$
$139$	15.3693i	1.30361i	−0.758387	+	0.651804i	$0.774012\pi$
$140$	−1.00000	−0.0845154
$141$	− 11.6847i	− 0.984026i
$142$	4.56155i	0.382797i
$143$	13.1231i	1.09741i
$144$	−3.56155	−0.296796
$145$	10.5616	0.877089
$146$	− 10.8078i	− 0.894457i
$147$	− 2.56155i	− 0.211273i
$148$	2.00000i	0.164399i
$149$	−8.24621	−0.675556	−0.337778	−	0.941226i	$-0.609675\pi$
$149$	−8.24621	−0.675556	−0.337778	+	0.941226i	$0.609675\pi$
$150$	− 2.56155i	− 0.209150i
$151$	17.6155	1.43353	0.716766	−	0.697314i	$-0.245622\pi$
$151$	17.6155	1.43353	0.716766	+	0.697314i	$0.245622\pi$
$152$	−2.56155	−0.207769
$153$	− 14.6847i	− 1.18718i
$154$	2.00000	0.161165
$155$	−2.56155	−0.205749
$156$	16.8078i	1.34570i
$157$	−7.36932	−0.588136	−0.294068	−	0.955785i	$-0.595009\pi$
$157$	−7.36932	−0.588136	−0.294068	+	0.955785i	$0.595009\pi$
$158$	− 17.3693i	− 1.38183i
$159$	4.31534i	0.342229i
$160$	− 1.00000i	− 0.0790569i
$161$	1.12311	0.0885131
$162$	−7.00000	−0.549972
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	− 11.1231i	− 0.868569i
$165$	5.12311i	0.398833i
$166$	−12.2462	−0.950490
$167$	− 23.3693i	− 1.80837i	−0.427139	−	0.904186i	$-0.640478\pi$
$167$	− 23.3693i	− 1.80837i	0.427139	−	0.904186i	$-0.359522\pi$
$168$	2.56155	0.197628
$169$	30.0540	2.31184
$170$	4.12311	0.316228
$171$	9.12311	0.697661
$172$	10.2462	0.781266
$173$	− 16.2462i	− 1.23518i	−0.786502	−	0.617588i	$-0.788110\pi$
$173$	− 16.2462i	− 1.23518i	0.786502	−	0.617588i	$-0.211890\pi$
$174$	−27.0540	−2.05096
$175$	1.00000i	0.0755929i
$176$	2.00000i	0.150756i
$177$	− 16.8078i	− 1.26335i
$178$	16.5616	1.24134
$179$	−1.75379	−0.131084	−0.0655422	−	0.997850i	$-0.520878\pi$
$179$	−1.75379	−0.131084	−0.0655422	+	0.997850i	$0.520878\pi$
$180$	3.56155i	0.265462i
$181$	− 22.4924i	− 1.67185i	−0.548845	−	0.835924i	$-0.684932\pi$
$181$	− 22.4924i	− 1.67185i	0.548845	−	0.835924i	$-0.315068\pi$
$182$	− 6.56155i	− 0.486375i
$183$	−14.5616	−1.07642
$184$	1.12311i	0.0827964i
$185$	2.00000	0.147043
$186$	6.56155	0.481116
$187$	−8.24621	−0.603023
$188$	−4.56155	−0.332685
$189$	−1.43845	−0.104632
$190$	2.56155i	0.185835i
$191$	13.1231	0.949555	0.474777	−	0.880106i	$-0.342529\pi$
$191$	13.1231	0.949555	0.474777	+	0.880106i	$0.342529\pi$
$192$	2.56155i	0.184864i
$193$	− 22.4924i	− 1.61904i	−0.587092	−	0.809520i	$-0.699727\pi$
$193$	− 22.4924i	− 1.61904i	0.587092	−	0.809520i	$-0.300273\pi$
$194$	− 9.68466i	− 0.695318i
$195$	16.8078	1.20363
$196$	−1.00000	−0.0714286
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	− 7.12311i	− 0.506217i
$199$	− 1.43845i	− 0.101969i	−0.998699	−	0.0509844i	$-0.983764\pi$
$199$	− 1.43845i	− 0.101969i	0.998699	−	0.0509844i	$-0.0162359\pi$
$200$	−1.00000	−0.0707107
$201$	− 2.87689i	− 0.202920i
$202$	10.0000	0.703598
$203$	10.5616	0.741276
$204$	−10.5616	−0.739457
$205$	−11.1231	−0.776871
$206$	−11.1231	−0.774983
$207$	− 4.00000i	− 0.278019i
$208$	6.56155	0.454962
$209$	− 5.12311i	− 0.354373i
$210$	− 2.56155i	− 0.176764i
$211$	22.0000i	1.51454i	0.653101	+	0.757271i	$0.273468\pi$
$211$	22.0000i	1.51454i	−0.653101	+	0.757271i	$0.726532\pi$
$212$	1.68466	0.115703
$213$	−11.6847	−0.800620
$214$	− 8.00000i	− 0.546869i
$215$	− 10.2462i	− 0.698786i
$216$	− 1.43845i	− 0.0978739i
$217$	−2.56155	−0.173890
$218$	4.31534i	0.292272i
$219$	27.6847	1.87076
$220$	2.00000	0.134840
$221$	27.0540i	1.81985i
$222$	−5.12311	−0.343840
$223$	−2.31534	−0.155047	−0.0775234	−	0.996991i	$-0.524701\pi$
$223$	−2.31534	−0.155047	−0.0775234	+	0.996991i	$0.524701\pi$
$224$	− 1.00000i	− 0.0668153i
$225$	3.56155	0.237437
$226$	7.43845i	0.494798i
$227$	− 17.9309i	− 1.19011i	−0.803684	−	0.595057i	$-0.797130\pi$
$227$	− 17.9309i	− 1.19011i	0.803684	−	0.595057i	$-0.202870\pi$
$228$	− 6.56155i	− 0.434549i
$229$	−1.36932	−0.0904870	−0.0452435	−	0.998976i	$-0.514406\pi$
$229$	−1.36932	−0.0904870	−0.0452435	+	0.998976i	$0.514406\pi$
$230$	1.12311	0.0740554
$231$	5.12311i	0.337076i
$232$	10.5616i	0.693400i
$233$	2.80776i	0.183943i	0.995762	+	0.0919714i	$0.0293168\pi$
$233$	2.80776i	0.183943i	−0.995762	+	0.0919714i	$0.970683\pi$
$234$	−23.3693	−1.52770
$235$	4.56155i	0.297563i
$236$	−6.56155	−0.427121
$237$	44.4924	2.89009
$238$	4.12311	0.267261
$239$	−12.4924	−0.808068	−0.404034	−	0.914744i	$-0.632392\pi$
$239$	−12.4924	−0.808068	−0.404034	+	0.914744i	$0.632392\pi$
$240$	2.56155	0.165348
$241$	− 1.36932i	− 0.0882055i	−0.999027	−	0.0441027i	$-0.985957\pi$
$241$	− 1.36932i	− 0.0882055i	0.999027	−	0.0441027i	$-0.0140429\pi$
$242$	7.00000	0.449977
$243$	− 22.2462i	− 1.42710i
$244$	5.68466i	0.363923i
$245$	1.00000i	0.0638877i
$246$	28.4924	1.81661
$247$	−16.8078	−1.06945
$248$	− 2.56155i	− 0.162659i
$249$	− 31.3693i	− 1.98795i
$250$	1.00000i	0.0632456i
$251$	12.4924	0.788515	0.394257	−	0.919000i	$-0.371002\pi$
$251$	12.4924	0.788515	0.394257	+	0.919000i	$0.371002\pi$
$252$	3.56155i	0.224357i
$253$	−2.24621	−0.141218
$254$	−17.9309	−1.12508
$255$	10.5616i	0.661390i
$256$	1.00000	0.0625000
$257$	−18.2462	−1.13817	−0.569084	−	0.822280i	$-0.692702\pi$
$257$	−18.2462	−1.13817	−0.569084	+	0.822280i	$0.692702\pi$
$258$	26.2462i	1.63402i
$259$	2.00000	0.124274
$260$	− 6.56155i	− 0.406930i
$261$	− 37.6155i	− 2.32834i
$262$	4.00000i	0.247121i
$263$	11.6847	0.720507	0.360253	−	0.932854i	$-0.382690\pi$
$263$	11.6847	0.720507	0.360253	+	0.932854i	$0.382690\pi$
$264$	−5.12311	−0.315305
$265$	− 1.68466i	− 0.103488i
$266$	2.56155i	0.157059i
$267$	42.4233i	2.59626i
$268$	−1.12311	−0.0686046
$269$	− 5.68466i	− 0.346600i	−0.984869	−	0.173300i	$-0.944557\pi$
$269$	− 5.68466i	− 0.346600i	0.984869	−	0.173300i	$-0.0554430\pi$
$270$	−1.43845	−0.0875411
$271$	−14.2462	−0.865396	−0.432698	−	0.901539i	$-0.642438\pi$
$271$	−14.2462	−0.865396	−0.432698	+	0.901539i	$0.642438\pi$
$272$	4.12311i	0.250000i
$273$	16.8078	1.01725
$274$	23.1231	1.39692
$275$	− 2.00000i	− 0.120605i
$276$	−2.87689	−0.173169
$277$	3.75379i	0.225543i	0.993621	+	0.112772i	$0.0359729\pi$
$277$	3.75379i	0.225543i	−0.993621	+	0.112772i	$0.964027\pi$
$278$	15.3693i	0.921790i
$279$	9.12311i	0.546186i
$280$	−1.00000	−0.0597614
$281$	−6.31534	−0.376742	−0.188371	−	0.982098i	$-0.560321\pi$
$281$	−6.31534	−0.376742	−0.188371	+	0.982098i	$0.560321\pi$
$282$	− 11.6847i	− 0.695811i
$283$	− 7.68466i	− 0.456806i	−0.973567	−	0.228403i	$-0.926650\pi$
$283$	− 7.68466i	− 0.456806i	0.973567	−	0.228403i	$-0.0733503\pi$
$284$	4.56155i	0.270678i
$285$	−6.56155	−0.388673
$286$	13.1231i	0.775986i
$287$	−11.1231	−0.656576
$288$	−3.56155	−0.209867
$289$	−17.0000	−1.00000
$290$	10.5616	0.620196
$291$	24.8078	1.45426
$292$	− 10.8078i	− 0.632477i
$293$	9.93087	0.580168	0.290084	−	0.957001i	$-0.406317\pi$
$293$	9.93087	0.580168	0.290084	+	0.957001i	$0.406317\pi$
$294$	− 2.56155i	− 0.149393i
$295$	6.56155i	0.382028i
$296$	2.00000i	0.116248i
$297$	2.87689	0.166934
$298$	−8.24621	−0.477690
$299$	7.36932i	0.426179i
$300$	− 2.56155i	− 0.147891i
$301$	− 10.2462i	− 0.590582i
$302$	17.6155	1.01366
$303$	25.6155i	1.47157i
$304$	−2.56155	−0.146915
$305$	5.68466	0.325503
$306$	− 14.6847i	− 0.839466i
$307$	−22.4924	−1.28371	−0.641855	−	0.766826i	$-0.721835\pi$
$307$	−22.4924	−1.28371	−0.641855	+	0.766826i	$0.721835\pi$
$308$	2.00000	0.113961
$309$	− 28.4924i	− 1.62088i
$310$	−2.56155	−0.145486
$311$	22.2462i	1.26147i	0.776000	+	0.630733i	$0.217246\pi$
$311$	22.2462i	1.26147i	−0.776000	+	0.630733i	$0.782754\pi$
$312$	16.8078i	0.951552i
$313$	− 16.2462i	− 0.918290i	−0.888361	−	0.459145i	$-0.848156\pi$
$313$	− 16.2462i	− 0.918290i	0.888361	−	0.459145i	$-0.151844\pi$
$314$	−7.36932	−0.415875
$315$	3.56155	0.200671
$316$	− 17.3693i	− 0.977100i
$317$	− 1.36932i	− 0.0769085i	−0.999260	−	0.0384542i	$-0.987757\pi$
$317$	− 1.36932i	− 0.0769085i	0.999260	−	0.0384542i	$-0.0122434\pi$
$318$	4.31534i	0.241992i
$319$	−21.1231	−1.18267
$320$	− 1.00000i	− 0.0559017i
$321$	20.4924	1.14378
$322$	1.12311	0.0625882
$323$	− 10.5616i	− 0.587661i
$324$	−7.00000	−0.388889
$325$	−6.56155	−0.363969
$326$	− 4.00000i	− 0.221540i
$327$	−11.0540	−0.611286
$328$	− 11.1231i	− 0.614171i
$329$	4.56155i	0.251487i
$330$	5.12311i	0.282018i
$331$	18.5616	1.02024	0.510118	−	0.860105i	$-0.329602\pi$
$331$	18.5616	1.02024	0.510118	+	0.860105i	$0.329602\pi$
$332$	−12.2462	−0.672098
$333$	− 7.12311i	− 0.390344i
$334$	− 23.3693i	− 1.27871i
$335$	1.12311i	0.0613618i
$336$	2.56155	0.139744
$337$	− 24.5616i	− 1.33795i	−0.743284	−	0.668977i	$-0.766733\pi$
$337$	− 24.5616i	− 1.33795i	0.743284	−	0.668977i	$-0.233267\pi$
$338$	30.0540	1.63472
$339$	−19.0540	−1.03487
$340$	4.12311	0.223607
$341$	5.12311	0.277432
$342$	9.12311	0.493321
$343$	1.00000i	0.0539949i
$344$	10.2462	0.552439
$345$	2.87689i	0.154887i
$346$	− 16.2462i	− 0.873402i
$347$	− 14.5616i	− 0.781705i	−0.920453	−	0.390853i	$-0.872180\pi$
$347$	− 14.5616i	− 0.781705i	0.920453	−	0.390853i	$-0.127820\pi$
$348$	−27.0540	−1.45025
$349$	−0.876894	−0.0469391	−0.0234695	−	0.999725i	$-0.507471\pi$
$349$	−0.876894	−0.0469391	−0.0234695	+	0.999725i	$0.507471\pi$
$350$	1.00000i	0.0534522i
$351$	− 9.43845i	− 0.503787i
$352$	2.00000i	0.106600i
$353$	−13.1231	−0.698472	−0.349236	−	0.937035i	$-0.613559\pi$
$353$	−13.1231	−0.698472	−0.349236	+	0.937035i	$0.613559\pi$
$354$	− 16.8078i	− 0.893323i
$355$	4.56155	0.242102
$356$	16.5616	0.877761
$357$	10.5616i	0.558977i
$358$	−1.75379	−0.0926906
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	3.56155i	0.187710i
$361$	−12.4384	−0.654655
$362$	− 22.4924i	− 1.18218i
$363$	17.9309i	0.941127i
$364$	− 6.56155i	− 0.343919i
$365$	−10.8078	−0.565704
$366$	−14.5616	−0.761145
$367$	17.6155i	0.919523i	0.888042	+	0.459762i	$0.152065\pi$
$367$	17.6155i	0.919523i	−0.888042	+	0.459762i	$0.847935\pi$
$368$	1.12311i	0.0585459i
$369$	39.6155i	2.06230i
$370$	2.00000	0.103975
$371$	− 1.68466i	− 0.0874631i
$372$	6.56155	0.340201
$373$	−2.49242	−0.129053	−0.0645264	−	0.997916i	$-0.520554\pi$
$373$	−2.49242	−0.129053	−0.0645264	+	0.997916i	$0.520554\pi$
$374$	−8.24621	−0.426401
$375$	−2.56155	−0.132278
$376$	−4.56155	−0.235244
$377$	69.3002i	3.56914i
$378$	−1.43845	−0.0739857
$379$	− 15.6155i	− 0.802116i	−0.916053	−	0.401058i	$-0.868643\pi$
$379$	− 15.6155i	− 0.802116i	0.916053	−	0.401058i	$-0.131357\pi$
$380$	2.56155i	0.131405i
$381$	− 45.9309i	− 2.35311i
$382$	13.1231	0.671436
$383$	−30.8078	−1.57420	−0.787102	−	0.616823i	$-0.788419\pi$
$383$	−30.8078	−1.57420	−0.787102	+	0.616823i	$0.788419\pi$
$384$	2.56155i	0.130719i
$385$	− 2.00000i	− 0.101929i
$386$	− 22.4924i	− 1.14483i
$387$	−36.4924	−1.85501
$388$	− 9.68466i	− 0.491664i
$389$	−23.1231	−1.17239	−0.586194	−	0.810171i	$-0.699374\pi$
$389$	−23.1231	−1.17239	−0.586194	+	0.810171i	$0.699374\pi$
$390$	16.8078	0.851094
$391$	−4.63068	−0.234184
$392$	−1.00000	−0.0505076
$393$	−10.2462	−0.516853
$394$	10.0000i	0.503793i
$395$	−17.3693	−0.873945
$396$	− 7.12311i	− 0.357950i
$397$	2.63068i	0.132030i	0.997819	+	0.0660151i	$0.0210286\pi$
$397$	2.63068i	0.132030i	−0.997819	+	0.0660151i	$0.978971\pi$
$398$	− 1.43845i	− 0.0721028i
$399$	−6.56155	−0.328489
$400$	−1.00000	−0.0500000
$401$	26.7386i	1.33526i	0.744492	+	0.667632i	$0.232692\pi$
$401$	26.7386i	1.33526i	−0.744492	+	0.667632i	$0.767308\pi$
$402$	− 2.87689i	− 0.143486i
$403$	− 16.8078i	− 0.837254i
$404$	10.0000	0.497519
$405$	7.00000i	0.347833i
$406$	10.5616	0.524161
$407$	−4.00000	−0.198273
$408$	−10.5616	−0.522875
$409$	−4.56155	−0.225554	−0.112777	−	0.993620i	$-0.535975\pi$
$409$	−4.56155	−0.225554	−0.112777	+	0.993620i	$0.535975\pi$
$410$	−11.1231	−0.549331
$411$	59.2311i	2.92165i
$412$	−11.1231	−0.547996
$413$	6.56155i	0.322873i
$414$	− 4.00000i	− 0.196589i
$415$	12.2462i	0.601143i
$416$	6.56155	0.321707
$417$	−39.3693	−1.92792
$418$	− 5.12311i	− 0.250579i
$419$	− 6.87689i	− 0.335958i	−0.985791	−	0.167979i	$-0.946276\pi$
$419$	− 6.87689i	− 0.335958i	0.985791	−	0.167979i	$-0.0537241\pi$
$420$	− 2.56155i	− 0.124991i
$421$	20.7386	1.01074	0.505370	−	0.862903i	$-0.331356\pi$
$421$	20.7386	1.01074	0.505370	+	0.862903i	$0.331356\pi$
$422$	22.0000i	1.07094i
$423$	16.2462	0.789918
$424$	1.68466	0.0818142
$425$	− 4.12311i	− 0.200000i
$426$	−11.6847	−0.566124
$427$	5.68466	0.275100
$428$	− 8.00000i	− 0.386695i
$429$	−33.6155	−1.62297
$430$	− 10.2462i	− 0.494116i
$431$	9.36932i	0.451304i	0.974208	+	0.225652i	$0.0724512\pi$
$431$	9.36932i	0.451304i	−0.974208	+	0.225652i	$0.927549\pi$
$432$	− 1.43845i	− 0.0692073i
$433$	18.7386	0.900521	0.450261	−	0.892897i	$-0.351331\pi$
$433$	18.7386	0.900521	0.450261	+	0.892897i	$0.351331\pi$
$434$	−2.56155	−0.122958
$435$	27.0540i	1.29714i
$436$	4.31534i	0.206667i
$437$	− 2.87689i	− 0.137621i
$438$	27.6847	1.32282
$439$	− 24.0000i	− 1.14546i	−0.819745	−	0.572729i	$-0.805885\pi$
$439$	− 24.0000i	− 1.14546i	0.819745	−	0.572729i	$-0.194115\pi$
$440$	2.00000	0.0953463
$441$	3.56155	0.169598
$442$	27.0540i	1.28683i
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	−5.12311	−0.243132
$445$	− 16.5616i	− 0.785093i
$446$	−2.31534	−0.109635
$447$	− 21.1231i	− 0.999089i
$448$	− 1.00000i	− 0.0472456i
$449$	− 10.8769i	− 0.513312i	−0.966503	−	0.256656i	$-0.917379\pi$
$449$	− 10.8769i	− 0.513312i	0.966503	−	0.256656i	$-0.0826208\pi$
$450$	3.56155	0.167893
$451$	22.2462	1.04753
$452$	7.43845i	0.349875i
$453$	45.1231i	2.12007i
$454$	− 17.9309i	− 0.841537i
$455$	−6.56155	−0.307610
$456$	− 6.56155i	− 0.307273i
$457$	−40.1080	−1.87617	−0.938085	−	0.346404i	$-0.887403\pi$
$457$	−40.1080	−1.87617	−0.938085	+	0.346404i	$0.887403\pi$
$458$	−1.36932	−0.0639840
$459$	5.93087	0.276829
$460$	1.12311	0.0523651
$461$	−15.7538	−0.733727	−0.366864	−	0.930275i	$-0.619568\pi$
$461$	−15.7538	−0.733727	−0.366864	+	0.930275i	$0.619568\pi$
$462$	5.12311i	0.238348i
$463$	15.0540	0.699618	0.349809	−	0.936821i	$-0.386247\pi$
$463$	15.0540	0.699618	0.349809	+	0.936821i	$0.386247\pi$
$464$	10.5616i	0.490308i
$465$	− 6.56155i	− 0.304285i
$466$	2.80776i	0.130067i
$467$	14.0000	0.647843	0.323921	−	0.946084i	$-0.394999\pi$
$467$	14.0000	0.647843	0.323921	+	0.946084i	$0.394999\pi$
$468$	−23.3693	−1.08025
$469$	1.12311i	0.0518602i
$470$	4.56155i	0.210409i
$471$	− 18.8769i	− 0.869801i
$472$	−6.56155	−0.302020
$473$	20.4924i	0.942243i
$474$	44.4924	2.04360
$475$	2.56155	0.117532
$476$	4.12311	0.188982
$477$	−6.00000	−0.274721
$478$	−12.4924	−0.571390
$479$	− 7.19224i	− 0.328622i	−0.986409	−	0.164311i	$-0.947460\pi$
$479$	− 7.19224i	− 0.328622i	0.986409	−	0.164311i	$-0.0525400\pi$
$480$	2.56155	0.116918
$481$	13.1231i	0.598362i
$482$	− 1.36932i	− 0.0623707i
$483$	2.87689i	0.130903i
$484$	7.00000	0.318182
$485$	−9.68466	−0.439758
$486$	− 22.2462i	− 1.00911i
$487$	9.75379i	0.441986i	0.975275	+	0.220993i	$0.0709299\pi$
$487$	9.75379i	0.441986i	−0.975275	+	0.220993i	$0.929070\pi$
$488$	5.68466i	0.257332i
$489$	10.2462	0.463350
$490$	1.00000i	0.0451754i
$491$	−28.1771	−1.27161	−0.635807	−	0.771848i	$-0.719333\pi$
$491$	−28.1771	−1.27161	−0.635807	+	0.771848i	$0.719333\pi$
$492$	28.4924	1.28454
$493$	−43.5464	−1.96123
$494$	−16.8078	−0.756217
$495$	−7.12311	−0.320160
$496$	− 2.56155i	− 0.115017i
$497$	4.56155	0.204614
$498$	− 31.3693i	− 1.40569i
$499$	13.3693i	0.598493i	0.954176	+	0.299246i	$0.0967353\pi$
$499$	13.3693i	0.598493i	−0.954176	+	0.299246i	$0.903265\pi$
$500$	1.00000i	0.0447214i
$501$	59.8617	2.67443
$502$	12.4924	0.557564
$503$	22.7386i	1.01387i	0.861986	+	0.506933i	$0.169221\pi$
$503$	22.7386i	1.01387i	−0.861986	+	0.506933i	$0.830779\pi$
$504$	3.56155i	0.158644i
$505$	− 10.0000i	− 0.444994i
$506$	−2.24621	−0.0998563
$507$	76.9848i	3.41902i
$508$	−17.9309	−0.795554
$509$	28.8769	1.27995	0.639973	−	0.768397i	$-0.278946\pi$
$509$	28.8769	1.27995	0.639973	+	0.768397i	$0.278946\pi$
$510$	10.5616i	0.467673i
$511$	−10.8078	−0.478107
$512$	1.00000	0.0441942
$513$	3.68466i	0.162682i
$514$	−18.2462	−0.804806
$515$	11.1231i	0.490143i
$516$	26.2462i	1.15543i
$517$	− 9.12311i	− 0.401234i
$518$	2.00000	0.0878750
$519$	41.6155	1.82672
$520$	− 6.56155i	− 0.287743i
$521$	3.12311i	0.136826i	0.997657	+	0.0684129i	$0.0217935\pi$
$521$	3.12311i	0.136826i	−0.997657	+	0.0684129i	$0.978206\pi$
$522$	− 37.6155i	− 1.64639i
$523$	6.49242	0.283894	0.141947	−	0.989874i	$-0.454664\pi$
$523$	6.49242	0.283894	0.141947	+	0.989874i	$0.454664\pi$
$524$	4.00000i	0.174741i
$525$	−2.56155	−0.111795
$526$	11.6847	0.509475
$527$	10.5616	0.460068
$528$	−5.12311	−0.222955
$529$	21.7386	0.945158
$530$	− 1.68466i	− 0.0731769i
$531$	23.3693	1.01414
$532$	2.56155i	0.111057i
$533$	− 72.9848i	− 3.16132i
$534$	42.4233i	1.83584i
$535$	−8.00000	−0.345870
$536$	−1.12311	−0.0485108
$537$	− 4.49242i	− 0.193862i
$538$	− 5.68466i	− 0.245083i
$539$	− 2.00000i	− 0.0861461i
$540$	−1.43845	−0.0619009
$541$	− 31.8617i	− 1.36984i	−0.728617	−	0.684922i	$-0.759836\pi$
$541$	− 31.8617i	− 1.36984i	0.728617	−	0.684922i	$-0.240164\pi$
$542$	−14.2462	−0.611927
$543$	57.6155	2.47252
$544$	4.12311i	0.176777i
$545$	4.31534	0.184849
$546$	16.8078	0.719306
$547$	2.56155i	0.109524i	0.998499	+	0.0547620i	$0.0174400\pi$
$547$	2.56155i	0.109524i	−0.998499	+	0.0547620i	$0.982560\pi$
$548$	23.1231	0.987770
$549$	− 20.2462i	− 0.864087i
$550$	− 2.00000i	− 0.0852803i
$551$	− 27.0540i	− 1.15254i
$552$	−2.87689	−0.122449
$553$	−17.3693	−0.738618
$554$	3.75379i	0.159483i
$555$	5.12311i	0.217464i
$556$	15.3693i	0.651804i
$557$	7.93087	0.336042	0.168021	−	0.985783i	$-0.446262\pi$
$557$	7.93087	0.336042	0.168021	+	0.985783i	$0.446262\pi$
$558$	9.12311i	0.386212i
$559$	67.2311	2.84357
$560$	−1.00000	−0.0422577
$561$	− 21.1231i	− 0.891818i
$562$	−6.31534	−0.266397
$563$	30.4924	1.28510	0.642551	−	0.766243i	$-0.277876\pi$
$563$	30.4924	1.28510	0.642551	+	0.766243i	$0.277876\pi$
$564$	− 11.6847i	− 0.492013i
$565$	7.43845	0.312938
$566$	− 7.68466i	− 0.323010i
$567$	7.00000i	0.293972i
$568$	4.56155i	0.191399i
$569$	−33.0540	−1.38569	−0.692847	−	0.721084i	$-0.743644\pi$
$569$	−33.0540	−1.38569	−0.692847	+	0.721084i	$0.743644\pi$
$570$	−6.56155	−0.274833
$571$	− 7.12311i	− 0.298093i	−0.988830	−	0.149046i	$-0.952380\pi$
$571$	− 7.12311i	− 0.298093i	0.988830	−	0.149046i	$-0.0476204\pi$
$572$	13.1231i	0.548705i
$573$	33.6155i	1.40431i
$574$	−11.1231	−0.464269
$575$	− 1.12311i	− 0.0468367i
$576$	−3.56155	−0.148398
$577$	−10.7386	−0.447055	−0.223528	−	0.974698i	$-0.571757\pi$
$577$	−10.7386	−0.447055	−0.223528	+	0.974698i	$0.571757\pi$
$578$	−17.0000	−0.707107
$579$	57.6155	2.39442
$580$	10.5616	0.438545
$581$	12.2462i	0.508058i
$582$	24.8078	1.02831
$583$	3.36932i	0.139543i
$584$	− 10.8078i	− 0.447228i
$585$	23.3693i	0.966202i
$586$	9.93087	0.410240
$587$	11.7538	0.485131	0.242565	−	0.970135i	$-0.422011\pi$
$587$	11.7538	0.485131	0.242565	+	0.970135i	$0.422011\pi$
$588$	− 2.56155i	− 0.105637i
$589$	6.56155i	0.270364i
$590$	6.56155i	0.270135i
$591$	−25.6155	−1.05368
$592$	2.00000i	0.0821995i
$593$	−11.3693	−0.466882	−0.233441	−	0.972371i	$-0.574999\pi$
$593$	−11.3693	−0.466882	−0.233441	+	0.972371i	$0.574999\pi$
$594$	2.87689	0.118040
$595$	− 4.12311i	− 0.169031i
$596$	−8.24621	−0.337778
$597$	3.68466	0.150803
$598$	7.36932i	0.301354i
$599$	−23.3693	−0.954844	−0.477422	−	0.878674i	$-0.658429\pi$
$599$	−23.3693	−0.954844	−0.477422	+	0.878674i	$0.658429\pi$
$600$	− 2.56155i	− 0.104575i
$601$	16.8769i	0.688423i	0.938892	+	0.344211i	$0.111854\pi$
$601$	16.8769i	0.688423i	−0.938892	+	0.344211i	$0.888146\pi$
$602$	− 10.2462i	− 0.417604i
$603$	4.00000	0.162893
$604$	17.6155	0.716766
$605$	− 7.00000i	− 0.284590i
$606$	25.6155i	1.04056i
$607$	5.75379i	0.233539i	0.993159	+	0.116770i	$0.0372539\pi$
$607$	5.75379i	0.233539i	−0.993159	+	0.116770i	$0.962746\pi$
$608$	−2.56155	−0.103885
$609$	27.0540i	1.09628i
$610$	5.68466	0.230165
$611$	−29.9309	−1.21087
$612$	− 14.6847i	− 0.593592i
$613$	3.43845	0.138878	0.0694388	−	0.997586i	$-0.477879\pi$
$613$	3.43845	0.138878	0.0694388	+	0.997586i	$0.477879\pi$
$614$	−22.4924	−0.907720
$615$	− 28.4924i	− 1.14893i
$616$	2.00000	0.0805823
$617$	25.5464i	1.02846i	0.857653	+	0.514230i	$0.171922\pi$
$617$	25.5464i	1.02846i	−0.857653	+	0.514230i	$0.828078\pi$
$618$	− 28.4924i	− 1.14613i
$619$	− 21.6155i	− 0.868801i	−0.900720	−	0.434401i	$-0.856960\pi$
$619$	− 21.6155i	− 0.868801i	0.900720	−	0.434401i	$-0.143040\pi$
$620$	−2.56155	−0.102874
$621$	1.61553	0.0648289
$622$	22.2462i	0.891992i
$623$	− 16.5616i	− 0.663525i
$624$	16.8078i	0.672849i
$625$	1.00000	0.0400000
$626$	− 16.2462i	− 0.649329i
$627$	13.1231	0.524086
$628$	−7.36932	−0.294068
$629$	−8.24621	−0.328798
$630$	3.56155	0.141896
$631$	−13.7538	−0.547530	−0.273765	−	0.961797i	$-0.588269\pi$
$631$	−13.7538	−0.547530	−0.273765	+	0.961797i	$0.588269\pi$
$632$	− 17.3693i	− 0.690914i
$633$	−56.3542	−2.23988
$634$	− 1.36932i	− 0.0543825i
$635$	17.9309i	0.711565i
$636$	4.31534i	0.171114i
$637$	−6.56155	−0.259978
$638$	−21.1231	−0.836272
$639$	− 16.2462i	− 0.642690i
$640$	− 1.00000i	− 0.0395285i
$641$	9.61553i	0.379791i	0.981804	+	0.189895i	$0.0608148\pi$
$641$	9.61553i	0.379791i	−0.981804	+	0.189895i	$0.939185\pi$
$642$	20.4924	0.808771
$643$	8.49242i	0.334908i	0.985880	+	0.167454i	$0.0535546\pi$
$643$	8.49242i	0.334908i	−0.985880	+	0.167454i	$0.946445\pi$
$644$	1.12311	0.0442566
$645$	26.2462	1.03344
$646$	− 10.5616i	− 0.415539i
$647$	35.3002	1.38779	0.693897	−	0.720074i	$-0.255892\pi$
$647$	35.3002	1.38779	0.693897	+	0.720074i	$0.255892\pi$
$648$	−7.00000	−0.274986
$649$	− 13.1231i	− 0.515127i
$650$	−6.56155	−0.257365
$651$	− 6.56155i	− 0.257168i
$652$	− 4.00000i	− 0.156652i
$653$	24.8769i	0.973508i	0.873539	+	0.486754i	$0.161819\pi$
$653$	24.8769i	0.973508i	−0.873539	+	0.486754i	$0.838181\pi$
$654$	−11.0540	−0.432245
$655$	4.00000	0.156293
$656$	− 11.1231i	− 0.434284i
$657$	38.4924i	1.50173i
$658$	4.56155i	0.177828i
$659$	−15.0540	−0.586420	−0.293210	−	0.956048i	$-0.594723\pi$
$659$	−15.0540	−0.586420	−0.293210	+	0.956048i	$0.594723\pi$
$660$	5.12311i	0.199417i
$661$	31.1231	1.21055	0.605274	−	0.796017i	$-0.293063\pi$
$661$	31.1231	1.21055	0.605274	+	0.796017i	$0.293063\pi$
$662$	18.5616	0.721415
$663$	−69.3002	−2.69140
$664$	−12.2462	−0.475245
$665$	2.56155	0.0993328
$666$	− 7.12311i	− 0.276015i
$667$	−11.8617	−0.459288
$668$	− 23.3693i	− 0.904186i
$669$	− 5.93087i	− 0.229301i
$670$	1.12311i	0.0433894i
$671$	−11.3693	−0.438908
$672$	2.56155	0.0988140
$673$	6.80776i	0.262420i	0.991355	+	0.131210i	$0.0418862\pi$
$673$	6.80776i	0.262420i	−0.991355	+	0.131210i	$0.958114\pi$
$674$	− 24.5616i	− 0.946076i
$675$	1.43845i	0.0553659i
$676$	30.0540	1.15592
$677$	29.3693i	1.12875i	0.825517	+	0.564377i	$0.190884\pi$
$677$	29.3693i	1.12875i	−0.825517	+	0.564377i	$0.809116\pi$
$678$	−19.0540	−0.731764
$679$	−9.68466	−0.371663
$680$	4.12311	0.158114
$681$	45.9309	1.76007
$682$	5.12311	0.196174
$683$	− 3.19224i	− 0.122148i	−0.998133	−	0.0610738i	$-0.980548\pi$
$683$	− 3.19224i	− 0.122148i	0.998133	−	0.0610738i	$-0.0194525\pi$
$684$	9.12311	0.348831
$685$	− 23.1231i	− 0.883488i
$686$	1.00000i	0.0381802i
$687$	− 3.50758i	− 0.133822i
$688$	10.2462	0.390633
$689$	11.0540	0.421123
$690$	2.87689i	0.109521i
$691$	− 16.6307i	− 0.632661i	−0.948649	−	0.316331i	$-0.897549\pi$
$691$	− 16.6307i	− 0.632661i	0.948649	−	0.316331i	$-0.102451\pi$
$692$	− 16.2462i	− 0.617588i
$693$	−7.12311	−0.270584
$694$	− 14.5616i	− 0.552749i
$695$	15.3693	0.582991
$696$	−27.0540	−1.02548
$697$	45.8617	1.73714
$698$	−0.876894	−0.0331909
$699$	−7.19224	−0.272035
$700$	1.00000i	0.0377964i
$701$	43.6155	1.64734	0.823668	−	0.567073i	$-0.191924\pi$
$701$	43.6155	1.64734	0.823668	+	0.567073i	$0.191924\pi$
$702$	− 9.43845i	− 0.356231i
$703$	− 5.12311i	− 0.193222i
$704$	2.00000i	0.0753778i
$705$	−11.6847	−0.440070
$706$	−13.1231	−0.493895
$707$	− 10.0000i	− 0.376089i
$708$	− 16.8078i	− 0.631674i
$709$	12.8078i	0.481006i	0.970648	+	0.240503i	$0.0773123\pi$
$709$	12.8078i	0.481006i	−0.970648	+	0.240503i	$0.922688\pi$
$710$	4.56155	0.171192
$711$	61.8617i	2.32000i
$712$	16.5616	0.620670
$713$	2.87689	0.107741
$714$	10.5616i	0.395256i
$715$	13.1231	0.490776
$716$	−1.75379	−0.0655422
$717$	− 32.0000i	− 1.19506i
$718$	−8.00000	−0.298557
$719$	32.1771i	1.20000i	0.799999	+	0.600001i	$0.204833\pi$
$719$	32.1771i	1.20000i	−0.799999	+	0.600001i	$0.795167\pi$
$720$	3.56155i	0.132731i
$721$	11.1231i	0.414246i
$722$	−12.4384	−0.462911
$723$	3.50758	0.130448
$724$	− 22.4924i	− 0.835924i
$725$	− 10.5616i	− 0.392246i
$726$	17.9309i	0.665477i
$727$	16.4233	0.609106	0.304553	−	0.952495i	$-0.401493\pi$
$727$	16.4233	0.609106	0.304553	+	0.952495i	$0.401493\pi$
$728$	− 6.56155i	− 0.243187i
$729$	35.9848	1.33277
$730$	−10.8078	−0.400013
$731$	42.2462i	1.56253i
$732$	−14.5616	−0.538210
$733$	−33.6155	−1.24162	−0.620809	−	0.783962i	$-0.713196\pi$
$733$	−33.6155	−1.24162	−0.620809	+	0.783962i	$0.713196\pi$
$734$	17.6155i	0.650201i
$735$	−2.56155	−0.0944843
$736$	1.12311i	0.0413982i
$737$	− 2.24621i	− 0.0827403i
$738$	39.6155i	1.45827i
$739$	−9.93087	−0.365313	−0.182656	−	0.983177i	$-0.558470\pi$
$739$	−9.93087	−0.365313	−0.182656	+	0.983177i	$0.558470\pi$
$740$	2.00000	0.0735215
$741$	− 43.0540i	− 1.58163i
$742$	− 1.68466i	− 0.0618458i
$743$	42.2462i	1.54986i	0.632045	+	0.774932i	$0.282216\pi$
$743$	42.2462i	1.54986i	−0.632045	+	0.774932i	$0.717784\pi$
$744$	6.56155	0.240558
$745$	8.24621i	0.302118i
$746$	−2.49242	−0.0912541
$747$	43.6155	1.59581
$748$	−8.24621	−0.301511
$749$	−8.00000	−0.292314
$750$	−2.56155	−0.0935347
$751$	45.6847i	1.66706i	0.552477	+	0.833528i	$0.313683\pi$
$751$	45.6847i	1.66706i	−0.552477	+	0.833528i	$0.686317\pi$
$752$	−4.56155	−0.166343
$753$	32.0000i	1.16614i
$754$	69.3002i	2.52376i
$755$	− 17.6155i	− 0.641095i
$756$	−1.43845	−0.0523158
$757$	−44.4233	−1.61459	−0.807296	−	0.590147i	$-0.799070\pi$
$757$	−44.4233	−1.61459	−0.807296	+	0.590147i	$0.799070\pi$
$758$	− 15.6155i	− 0.567182i
$759$	− 5.75379i	− 0.208849i
$760$	2.56155i	0.0929173i
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	− 45.9309i	− 1.66390i
$763$	4.31534	0.156226
$764$	13.1231	0.474777
$765$	−14.6847	−0.530925
$766$	−30.8078	−1.11313
$767$	−43.0540	−1.55459
$768$	2.56155i	0.0924321i
$769$	38.6695	1.39446	0.697229	−	0.716848i	$-0.254416\pi$
$769$	38.6695	1.39446	0.697229	+	0.716848i	$0.254416\pi$
$770$	− 2.00000i	− 0.0720750i
$771$	− 46.7386i	− 1.68325i
$772$	− 22.4924i	− 0.809520i
$773$	38.1080	1.37065	0.685324	−	0.728238i	$-0.259661\pi$
$773$	38.1080	1.37065	0.685324	+	0.728238i	$0.259661\pi$
$774$	−36.4924	−1.31169
$775$	2.56155i	0.0920137i
$776$	− 9.68466i	− 0.347659i
$777$	5.12311i	0.183790i
$778$	−23.1231	−0.829004
$779$	28.4924i	1.02085i
$780$	16.8078	0.601814
$781$	−9.12311	−0.326450
$782$	−4.63068	−0.165593
$783$	15.1922	0.542926
$784$	−1.00000	−0.0357143
$785$	7.36932i	0.263022i
$786$	−10.2462	−0.365470
$787$	− 27.1922i	− 0.969299i	−0.874709	−	0.484649i	$-0.838947\pi$
$787$	− 27.1922i	− 0.969299i	0.874709	−	0.484649i	$-0.161053\pi$
$788$	10.0000i	0.356235i
$789$	29.9309i	1.06557i
$790$	−17.3693	−0.617973
$791$	7.43845	0.264481
$792$	− 7.12311i	− 0.253109i
$793$	37.3002i	1.32457i
$794$	2.63068i	0.0933595i
$795$	4.31534	0.153049
$796$	− 1.43845i	− 0.0509844i
$797$	−12.6307	−0.447402	−0.223701	−	0.974658i	$-0.571814\pi$
$797$	−12.6307	−0.447402	−0.223701	+	0.974658i	$0.571814\pi$
$798$	−6.56155	−0.232276
$799$	− 18.8078i	− 0.665371i
$800$	−1.00000	−0.0353553
$801$	−58.9848	−2.08413
$802$	26.7386i	0.944174i
$803$	21.6155	0.762795
$804$	− 2.87689i	− 0.101460i
$805$	− 1.12311i	− 0.0395843i
$806$	− 16.8078i	− 0.592028i
$807$	14.5616	0.512591
$808$	10.0000	0.351799
$809$	− 36.0000i	− 1.26569i	−0.774277	−	0.632846i	$-0.781886\pi$
$809$	− 36.0000i	− 1.26569i	0.774277	−	0.632846i	$-0.218114\pi$
$810$	7.00000i	0.245955i
$811$	44.9848i	1.57963i	0.613344	+	0.789816i	$0.289824\pi$
$811$	44.9848i	1.57963i	−0.613344	+	0.789816i	$0.710176\pi$
$812$	10.5616	0.370638
$813$	− 36.4924i	− 1.27985i
$814$	−4.00000	−0.140200
$815$	−4.00000	−0.140114
$816$	−10.5616	−0.369728
$817$	−26.2462	−0.918239
$818$	−4.56155	−0.159491
$819$	23.3693i	0.816590i
$820$	−11.1231	−0.388436
$821$	40.1771i	1.40219i	0.713068	+	0.701095i	$0.247305\pi$
$821$	40.1771i	1.40219i	−0.713068	+	0.701095i	$0.752695\pi$
$822$	59.2311i	2.06592i
$823$	0.630683i	0.0219842i	0.999940	+	0.0109921i	$0.00349897\pi$
$823$	0.630683i	0.0219842i	−0.999940	+	0.0109921i	$0.996501\pi$
$824$	−11.1231	−0.387492
$825$	5.12311	0.178364
$826$	6.56155i	0.228306i
$827$	28.9848i	1.00790i	0.863732	+	0.503951i	$0.168121\pi$
$827$	28.9848i	1.00790i	−0.863732	+	0.503951i	$0.831879\pi$
$828$	− 4.00000i	− 0.139010i
$829$	−51.6155	−1.79268	−0.896341	−	0.443366i	$-0.853784\pi$
$829$	−51.6155	−1.79268	−0.896341	+	0.443366i	$0.853784\pi$
$830$	12.2462i	0.425072i
$831$	−9.61553	−0.333559
$832$	6.56155	0.227481
$833$	− 4.12311i	− 0.142857i
$834$	−39.3693	−1.36325
$835$	−23.3693	−0.808729
$836$	− 5.12311i	− 0.177186i
$837$	−3.68466	−0.127360
$838$	− 6.87689i	− 0.237558i
$839$	− 14.5616i	− 0.502721i	−0.967894	−	0.251360i	$-0.919122\pi$
$839$	− 14.5616i	− 0.502721i	0.967894	−	0.251360i	$-0.0808779\pi$
$840$	− 2.56155i	− 0.0883820i
$841$	−82.5464	−2.84643
$842$	20.7386	0.714701
$843$	− 16.1771i	− 0.557168i
$844$	22.0000i	0.757271i
$845$	− 30.0540i	− 1.03389i
$846$	16.2462	0.558556
$847$	− 7.00000i	− 0.240523i
$848$	1.68466	0.0578514
$849$	19.6847	0.675576
$850$	− 4.12311i	− 0.141421i
$851$	−2.24621	−0.0769991
$852$	−11.6847	−0.400310
$853$	− 32.1080i	− 1.09936i	−0.835377	−	0.549678i	$-0.814751\pi$
$853$	− 32.1080i	− 1.09936i	0.835377	−	0.549678i	$-0.185249\pi$
$854$	5.68466	0.194525
$855$	− 9.12311i	− 0.312004i
$856$	− 8.00000i	− 0.273434i
$857$	34.6695i	1.18429i	0.805832	+	0.592144i	$0.201718\pi$
$857$	34.6695i	1.18429i	−0.805832	+	0.592144i	$0.798282\pi$
$858$	−33.6155	−1.14762
$859$	44.6695	1.52410	0.762052	−	0.647516i	$-0.224192\pi$
$859$	44.6695	1.52410	0.762052	+	0.647516i	$0.224192\pi$
$860$	− 10.2462i	− 0.349393i
$861$	− 28.4924i	− 0.971019i
$862$	9.36932i	0.319120i
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	− 1.43845i	− 0.0489370i
$865$	−16.2462	−0.552388
$866$	18.7386	0.636765
$867$	− 43.5464i	− 1.47891i
$868$	−2.56155	−0.0869448
$869$	34.7386	1.17843
$870$	27.0540i	0.917216i
$871$	−7.36932	−0.249700
$872$	4.31534i	0.146136i
$873$	34.4924i	1.16739i
$874$	− 2.87689i	− 0.0973124i
$875$	1.00000	0.0338062
$876$	27.6847	0.935378
$877$	− 10.9848i	− 0.370932i	−0.982651	−	0.185466i	$-0.940621\pi$
$877$	− 10.9848i	− 0.370932i	0.982651	−	0.185466i	$-0.0593794\pi$
$878$	− 24.0000i	− 0.809961i
$879$	25.4384i	0.858018i
$880$	2.00000	0.0674200
$881$	20.2462i	0.682112i	0.940043	+	0.341056i	$0.110785\pi$
$881$	20.2462i	0.682112i	−0.940043	+	0.341056i	$0.889215\pi$
$882$	3.56155	0.119924
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	27.0540i	0.909924i
$885$	−16.8078	−0.564987
$886$	8.00000	0.268765
$887$	− 32.0000i	− 1.07445i	−0.843437	−	0.537227i	$-0.819472\pi$
$887$	− 32.0000i	− 1.07445i	0.843437	−	0.537227i	$-0.180528\pi$
$888$	−5.12311	−0.171920
$889$	17.9309i	0.601382i
$890$	− 16.5616i	− 0.555145i
$891$	− 14.0000i	− 0.469018i
$892$	−2.31534	−0.0775234
$893$	11.6847	0.391012
$894$	− 21.1231i	− 0.706462i
$895$	1.75379i	0.0586227i
$896$	− 1.00000i	− 0.0334077i
$897$	−18.8769	−0.630281
$898$	− 10.8769i	− 0.362967i
$899$	27.0540	0.902301
$900$	3.56155	0.118718
$901$	6.94602i	0.231406i
$902$	22.2462	0.740718
$903$	26.2462	0.873419
$904$	7.43845i	0.247399i
$905$	−22.4924	−0.747673
$906$	45.1231i	1.49911i
$907$	− 45.9309i	− 1.52511i	−0.646924	−	0.762555i	$-0.723945\pi$
$907$	− 45.9309i	− 1.52511i	0.646924	−	0.762555i	$-0.276055\pi$
$908$	− 17.9309i	− 0.595057i
$909$	−35.6155	−1.18129
$910$	−6.56155	−0.217513
$911$	57.3693i	1.90073i	0.311137	+	0.950365i	$0.399290\pi$
$911$	57.3693i	1.90073i	−0.311137	+	0.950365i	$0.600710\pi$
$912$	− 6.56155i	− 0.217275i
$913$	− 24.4924i	− 0.810581i
$914$	−40.1080	−1.32665
$915$	14.5616i	0.481390i
$916$	−1.36932	−0.0452435
$917$	4.00000	0.132092
$918$	5.93087	0.195748
$919$	4.49242	0.148191	0.0740957	−	0.997251i	$-0.476393\pi$
$919$	4.49242	0.148191	0.0740957	+	0.997251i	$0.476393\pi$
$920$	1.12311	0.0370277
$921$	− 57.6155i	− 1.89850i
$922$	−15.7538	−0.518823
$923$	29.9309i	0.985187i
$924$	5.12311i	0.168538i
$925$	− 2.00000i	− 0.0657596i
$926$	15.0540	0.494704
$927$	39.6155	1.30114
$928$	10.5616i	0.346700i
$929$	33.8617i	1.11097i	0.831527	+	0.555484i	$0.187467\pi$
$929$	33.8617i	1.11097i	−0.831527	+	0.555484i	$0.812533\pi$
$930$	− 6.56155i	− 0.215162i
$931$	2.56155	0.0839515
$932$	2.80776i	0.0919714i
$933$	−56.9848	−1.86560
$934$	14.0000	0.458094
$935$	8.24621i	0.269680i
$936$	−23.3693	−0.763850
$937$	−31.3693	−1.02479	−0.512395	−	0.858750i	$-0.671242\pi$
$937$	−31.3693	−1.02479	−0.512395	+	0.858750i	$0.671242\pi$
$938$	1.12311i	0.0366707i
$939$	41.6155	1.35807
$940$	4.56155i	0.148781i
$941$	− 31.4384i	− 1.02486i	−0.858728	−	0.512432i	$-0.828745\pi$
$941$	− 31.4384i	− 1.02486i	0.858728	−	0.512432i	$-0.171255\pi$
$942$	− 18.8769i	− 0.615042i
$943$	12.4924	0.406809
$944$	−6.56155	−0.213560
$945$	1.43845i	0.0467927i
$946$	20.4924i	0.666266i
$947$	− 4.31534i	− 0.140230i	−0.997539	−	0.0701149i	$-0.977663\pi$
$947$	− 4.31534i	− 0.140230i	0.997539	−	0.0701149i	$-0.0223366\pi$
$948$	44.4924	1.44505
$949$	− 70.9157i	− 2.30202i
$950$	2.56155	0.0831077
$951$	3.50758	0.113741
$952$	4.12311	0.133631
$953$	48.2462	1.56285	0.781424	−	0.624000i	$-0.214494\pi$
$953$	48.2462	1.56285	0.781424	+	0.624000i	$0.214494\pi$
$954$	−6.00000	−0.194257
$955$	− 13.1231i	− 0.424654i
$956$	−12.4924	−0.404034
$957$	− 54.1080i	− 1.74906i
$958$	− 7.19224i	− 0.232371i
$959$	− 23.1231i	− 0.746684i
$960$	2.56155	0.0826738
$961$	24.4384	0.788337
$962$	13.1231i	0.423106i
$963$	28.4924i	0.918155i
$964$	− 1.36932i	− 0.0441027i
$965$	−22.4924	−0.724057
$966$	2.87689i	0.0925626i
$967$	17.7538	0.570923	0.285462	−	0.958390i	$-0.407853\pi$
$967$	17.7538	0.570923	0.285462	+	0.958390i	$0.407853\pi$
$968$	7.00000	0.224989
$969$	27.0540	0.869099
$970$	−9.68466	−0.310956
$971$	−61.7926	−1.98302	−0.991510	−	0.130034i	$-0.958492\pi$
$971$	−61.7926	−1.98302	−0.991510	+	0.130034i	$0.958492\pi$
$972$	− 22.2462i	− 0.713548i
$973$	15.3693	0.492718
$974$	9.75379i	0.312532i
$975$	− 16.8078i	− 0.538279i
$976$	5.68466i	0.181961i
$977$	0.246211	0.00787700	0.00393850	−	0.999992i	$-0.498746\pi$
$977$	0.246211	0.00787700	0.00393850	+	0.999992i	$0.498746\pi$
$978$	10.2462	0.327638
$979$	33.1231i	1.05862i
$980$	1.00000i	0.0319438i
$981$	− 15.3693i	− 0.490705i
$982$	−28.1771	−0.899167
$983$	− 26.2462i	− 0.837124i	−0.908188	−	0.418562i	$-0.862534\pi$
$983$	− 26.2462i	− 0.837124i	0.908188	−	0.418562i	$-0.137466\pi$
$984$	28.4924	0.908305
$985$	10.0000	0.318626
$986$	−43.5464	−1.38680
$987$	−11.6847	−0.371927
$988$	−16.8078	−0.534726
$989$	11.5076i	0.365920i
$990$	−7.12311	−0.226387
$991$	17.0540i	0.541737i	0.962616	+	0.270869i	$0.0873109\pi$
$991$	17.0540i	0.541737i	−0.962616	+	0.270869i	$0.912689\pi$
$992$	− 2.56155i	− 0.0813294i
$993$	47.5464i	1.50884i
$994$	4.56155	0.144684
$995$	−1.43845	−0.0456018
$996$	− 31.3693i	− 0.993975i
$997$	33.3693i	1.05682i	0.848990	+	0.528408i	$0.177211\pi$
$997$	33.3693i	1.05682i	−0.848990	+	0.528408i	$0.822789\pi$
$998$	13.3693i	0.423198i
$999$	2.87689	0.0910209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1190.2.c.f.1121.4	yes	4
17.16	even	2	inner	1190.2.c.f.1121.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.c.f.1121.1	✓	4	17.16	even	2	inner
1190.2.c.f.1121.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1190 = 2 \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1190.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.56155i q^{3} +1.00000 q^{4} -1.00000i q^{5} +2.56155i q^{6} -1.00000i q^{7} +1.00000 q^{8} -3.56155 q^{9} -1.00000i q^{10} +2.00000i q^{11} +2.56155i q^{12} +6.56155 q^{13} -1.00000i q^{14} +2.56155 q^{15} +1.00000 q^{16} +4.12311i q^{17} -3.56155 q^{18} -2.56155 q^{19} -1.00000i q^{20} +2.56155 q^{21} +2.00000i q^{22} +1.12311i q^{23} +2.56155i q^{24} -1.00000 q^{25} +6.56155 q^{26} -1.43845i q^{27} -1.00000i q^{28} +10.5616i q^{29} +2.56155 q^{30} -2.56155i q^{31} +1.00000 q^{32} -5.12311 q^{33} +4.12311i q^{34} -1.00000 q^{35} -3.56155 q^{36} +2.00000i q^{37} -2.56155 q^{38} +16.8078i q^{39} -1.00000i q^{40} -11.1231i q^{41} +2.56155 q^{42} +10.2462 q^{43} +2.00000i q^{44} +3.56155i q^{45} +1.12311i q^{46} -4.56155 q^{47} +2.56155i q^{48} -1.00000 q^{49} -1.00000 q^{50} -10.5616 q^{51} +6.56155 q^{52} +1.68466 q^{53} -1.43845i q^{54} +2.00000 q^{55} -1.00000i q^{56} -6.56155i q^{57} +10.5616i q^{58} -6.56155 q^{59} +2.56155 q^{60} +5.68466i q^{61} -2.56155i q^{62} +3.56155i q^{63} +1.00000 q^{64} -6.56155i q^{65} -5.12311 q^{66} -1.12311 q^{67} +4.12311i q^{68} -2.87689 q^{69} -1.00000 q^{70} +4.56155i q^{71} -3.56155 q^{72} -10.8078i q^{73} +2.00000i q^{74} -2.56155i q^{75} -2.56155 q^{76} +2.00000 q^{77} +16.8078i q^{78} -17.3693i q^{79} -1.00000i q^{80} -7.00000 q^{81} -11.1231i q^{82} -12.2462 q^{83} +2.56155 q^{84} +4.12311 q^{85} +10.2462 q^{86} -27.0540 q^{87} +2.00000i q^{88} +16.5616 q^{89} +3.56155i q^{90} -6.56155i q^{91} +1.12311i q^{92} +6.56155 q^{93} -4.56155 q^{94} +2.56155i q^{95} +2.56155i q^{96} -9.68466i q^{97} -1.00000 q^{98} -7.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 6 q^{9} + 18 q^{13} + 2 q^{15} + 4 q^{16} - 6 q^{18} - 2 q^{19} + 2 q^{21} - 4 q^{25} + 18 q^{26} + 2 q^{30} + 4 q^{32} - 4 q^{33} - 4 q^{35} - 6 q^{36} - 2 q^{38} + 2 q^{42}+ \cdots - 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 6 * q^9 + 18 * q^13 + 2 * q^15 + 4 * q^16 - 6 * q^18 - 2 * q^19 + 2 * q^21 - 4 * q^25 + 18 * q^26 + 2 * q^30 + 4 * q^32 - 4 * q^33 - 4 * q^35 - 6 * q^36 - 2 * q^38 + 2 * q^42 + 8 * q^43 - 10 * q^47 - 4 * q^49 - 4 * q^50 - 34 * q^51 + 18 * q^52 - 18 * q^53 + 8 * q^55 - 18 * q^59 + 2 * q^60 + 4 * q^64 - 4 * q^66 + 12 * q^67 - 28 * q^69 - 4 * q^70 - 6 * q^72 - 2 * q^76 + 8 * q^77 - 28 * q^81 - 16 * q^83 + 2 * q^84 + 8 * q^86 - 34 * q^87 + 58 * q^89 + 18 * q^93 - 10 * q^94 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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