LMFDB - Embedded newform 1690.2.d.j.1351.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1690,2,Mod(1351,1690)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("1690.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,2,-6,0,0,0,0,-8,6,0,-2,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.5
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.j.1351.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +0.445042 q^{3} -1.00000 q^{4} -1.00000i q^{5} +0.445042i q^{6} -3.24698i q^{7} -1.00000i q^{8} -2.80194 q^{9} +1.00000 q^{10} +1.60388i q^{11} -0.445042 q^{12} +3.24698 q^{14} -0.445042i q^{15} +1.00000 q^{16} -3.10992 q^{17} -2.80194i q^{18} +2.89008i q^{19} +1.00000i q^{20} -1.44504i q^{21} -1.60388 q^{22} -6.78986 q^{23} -0.445042i q^{24} -1.00000 q^{25} -2.58211 q^{27} +3.24698i q^{28} +4.04892 q^{29} +0.445042 q^{30} +8.31767i q^{31} +1.00000i q^{32} +0.713792i q^{33} -3.10992i q^{34} -3.24698 q^{35} +2.80194 q^{36} +3.20775i q^{37} -2.89008 q^{38} -1.00000 q^{40} +10.6746i q^{41} +1.44504 q^{42} +5.18598 q^{43} -1.60388i q^{44} +2.80194i q^{45} -6.78986i q^{46} -3.97823i q^{47} +0.445042 q^{48} -3.54288 q^{49} -1.00000i q^{50} -1.38404 q^{51} -11.0858 q^{53} -2.58211i q^{54} +1.60388 q^{55} -3.24698 q^{56} +1.28621i q^{57} +4.04892i q^{58} -5.28621i q^{59} +0.445042i q^{60} -13.1250 q^{61} -8.31767 q^{62} +9.09783i q^{63} -1.00000 q^{64} -0.713792 q^{66} +12.3937i q^{67} +3.10992 q^{68} -3.02177 q^{69} -3.24698i q^{70} +7.50604i q^{71} +2.80194i q^{72} +12.2741i q^{73} -3.20775 q^{74} -0.445042 q^{75} -2.89008i q^{76} +5.20775 q^{77} -11.3056 q^{79} -1.00000i q^{80} +7.25667 q^{81} -10.6746 q^{82} -10.4209i q^{83} +1.44504i q^{84} +3.10992i q^{85} +5.18598i q^{86} +1.80194 q^{87} +1.60388 q^{88} -11.3817i q^{89} -2.80194 q^{90} +6.78986 q^{92} +3.70171i q^{93} +3.97823 q^{94} +2.89008 q^{95} +0.445042i q^{96} -6.05429i q^{97} -3.54288i q^{98} -4.49396i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 6 q^{4} - 8 q^{9} + 6 q^{10} - 2 q^{12} + 10 q^{14} + 6 q^{16} - 20 q^{17} + 8 q^{22} + 6 q^{23} - 6 q^{25} - 4 q^{27} + 6 q^{29} + 2 q^{30} - 10 q^{35} + 8 q^{36} - 16 q^{38} - 6 q^{40}+ \cdots + 16 q^{95}+O(q^{100}) $ 6 * q + 2 * q^3 - 6 * q^4 - 8 * q^9 + 6 * q^10 - 2 * q^12 + 10 * q^14 + 6 * q^16 - 20 * q^17 + 8 * q^22 + 6 * q^23 - 6 * q^25 - 4 * q^27 + 6 * q^29 + 2 * q^30 - 10 * q^35 + 8 * q^36 - 16 * q^38 - 6 * q^40 + 8 * q^42 + 2 * q^43 + 2 * q^48 + 16 * q^49 + 12 * q^51 + 8 * q^53 - 8 * q^55 - 10 * q^56 - 30 * q^61 - 16 * q^62 - 6 * q^64 + 12 * q^66 + 20 * q^68 - 12 * q^69 + 16 * q^74 - 2 * q^75 - 4 * q^77 + 4 * q^79 - 10 * q^81 - 22 * q^82 + 2 * q^87 - 8 * q^88 - 8 * q^90 - 6 * q^92 + 30 * q^94 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0.445042	0.256945	0.128473	−	0.991713i	$-0.458993\pi$
$3$	0.445042	0.256945	0.128473	+	0.991713i	$0.458993\pi$
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0.445042i	0.181688i
$7$	− 3.24698i	− 1.22724i	−0.789600	−	0.613621i	$-0.789712\pi$
$7$	− 3.24698i	− 1.22724i	0.789600	−	0.613621i	$-0.210288\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−2.80194	−0.933979
$10$	1.00000	0.316228
$11$	1.60388i	0.483587i	0.970328	+	0.241793i	$0.0777356\pi$
$11$	1.60388i	0.483587i	−0.970328	+	0.241793i	$0.922264\pi$
$12$	−0.445042	−0.128473
$13$	0	0
$13$	0	0
$14$	3.24698	0.867792
$15$	− 0.445042i	− 0.114909i
$16$	1.00000	0.250000
$17$	−3.10992	−0.754265	−0.377133	−	0.926159i	$-0.623090\pi$
$17$	−3.10992	−0.754265	−0.377133	+	0.926159i	$0.623090\pi$
$18$	− 2.80194i	− 0.660423i
$19$	2.89008i	0.663031i	0.943450	+	0.331515i	$0.107560\pi$
$19$	2.89008i	0.663031i	−0.943450	+	0.331515i	$0.892440\pi$
$20$	1.00000i	0.223607i
$21$	− 1.44504i	− 0.315334i
$22$	−1.60388	−0.341947
$23$	−6.78986	−1.41578	−0.707891	−	0.706321i	$-0.750353\pi$
$23$	−6.78986	−1.41578	−0.707891	+	0.706321i	$0.750353\pi$
$24$	− 0.445042i	− 0.0908438i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−2.58211	−0.496926
$28$	3.24698i	0.613621i
$29$	4.04892	0.751865	0.375933	−	0.926647i	$-0.377322\pi$
$29$	4.04892	0.751865	0.375933	+	0.926647i	$0.377322\pi$
$30$	0.445042	0.0812532
$31$	8.31767i	1.49390i	0.664882	+	0.746949i	$0.268482\pi$
$31$	8.31767i	1.49390i	−0.664882	+	0.746949i	$0.731518\pi$
$32$	1.00000i	0.176777i
$33$	0.713792i	0.124255i
$34$	− 3.10992i	− 0.533346i
$35$	−3.24698	−0.548840
$36$	2.80194	0.466990
$37$	3.20775i	0.527351i	0.964611	+	0.263676i	$0.0849348\pi$
$37$	3.20775i	0.527351i	−0.964611	+	0.263676i	$0.915065\pi$
$38$	−2.89008	−0.468833
$39$	0	0
$40$	−1.00000	−0.158114
$41$	10.6746i	1.66709i	0.552454	+	0.833543i	$0.313691\pi$
$41$	10.6746i	1.66709i	−0.552454	+	0.833543i	$0.686309\pi$
$42$	1.44504	0.222975
$43$	5.18598	0.790855	0.395427	−	0.918497i	$-0.370597\pi$
$43$	5.18598	0.790855	0.395427	+	0.918497i	$0.370597\pi$
$44$	− 1.60388i	− 0.241793i
$45$	2.80194i	0.417688i
$46$	− 6.78986i	− 1.00111i
$47$	− 3.97823i	− 0.580284i	−0.956984	−	0.290142i	$-0.906297\pi$
$47$	− 3.97823i	− 0.580284i	0.956984	−	0.290142i	$-0.0937026\pi$
$48$	0.445042	0.0642363
$49$	−3.54288	−0.506125
$50$	− 1.00000i	− 0.141421i
$51$	−1.38404	−0.193805
$52$	0	0
$53$	−11.0858	−1.52275	−0.761373	−	0.648314i	$-0.775474\pi$
$53$	−11.0858	−1.52275	−0.761373	+	0.648314i	$0.775474\pi$
$54$	− 2.58211i	− 0.351380i
$55$	1.60388	0.216267
$56$	−3.24698	−0.433896
$57$	1.28621i	0.170362i
$58$	4.04892i	0.531649i
$59$	− 5.28621i	− 0.688206i	−0.938932	−	0.344103i	$-0.888183\pi$
$59$	− 5.28621i	− 0.688206i	0.938932	−	0.344103i	$-0.111817\pi$
$60$	0.445042i	0.0574547i
$61$	−13.1250	−1.68048	−0.840241	−	0.542213i	$-0.817586\pi$
$61$	−13.1250	−1.68048	−0.840241	+	0.542213i	$0.817586\pi$
$62$	−8.31767	−1.05634
$63$	9.09783i	1.14622i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−0.713792	−0.0878617
$67$	12.3937i	1.51414i	0.653336	+	0.757068i	$0.273369\pi$
$67$	12.3937i	1.51414i	−0.653336	+	0.757068i	$0.726631\pi$
$68$	3.10992	0.377133
$69$	−3.02177	−0.363778
$70$	− 3.24698i	− 0.388088i
$71$	7.50604i	0.890803i	0.895331	+	0.445402i	$0.146939\pi$
$71$	7.50604i	0.890803i	−0.895331	+	0.445402i	$0.853061\pi$
$72$	2.80194i	0.330212i
$73$	12.2741i	1.43658i	0.695745	+	0.718289i	$0.255074\pi$
$73$	12.2741i	1.43658i	−0.695745	+	0.718289i	$0.744926\pi$
$74$	−3.20775	−0.372893
$75$	−0.445042	−0.0513890
$76$	− 2.89008i	− 0.331515i
$77$	5.20775	0.593478
$78$	0	0
$79$	−11.3056	−1.27198	−0.635989	−	0.771698i	$-0.719408\pi$
$79$	−11.3056	−1.27198	−0.635989	+	0.771698i	$0.719408\pi$
$80$	− 1.00000i	− 0.111803i
$81$	7.25667	0.806296
$82$	−10.6746	−1.17881
$83$	− 10.4209i	− 1.14384i	−0.820309	−	0.571920i	$-0.806199\pi$
$83$	− 10.4209i	− 1.14384i	0.820309	−	0.571920i	$-0.193801\pi$
$84$	1.44504i	0.157667i
$85$	3.10992i	0.337318i
$86$	5.18598i	0.559219i
$87$	1.80194	0.193188
$88$	1.60388	0.170974
$89$	− 11.3817i	− 1.20645i	−0.797570	−	0.603226i	$-0.793882\pi$
$89$	− 11.3817i	− 1.20645i	0.797570	−	0.603226i	$-0.206118\pi$
$90$	−2.80194	−0.295350
$91$	0	0
$92$	6.78986	0.707891
$93$	3.70171i	0.383849i
$94$	3.97823	0.410323
$95$	2.89008	0.296516
$96$	0.445042i	0.0454219i
$97$	− 6.05429i	− 0.614720i	−0.951593	−	0.307360i	$-0.900554\pi$
$97$	− 6.05429i	− 0.614720i	0.951593	−	0.307360i	$-0.0994456\pi$
$98$	− 3.54288i	− 0.357885i
$99$	− 4.49396i	− 0.451660i
$100$	1.00000	0.100000
$101$	4.19806	0.417723	0.208861	−	0.977945i	$-0.433024\pi$
$101$	4.19806	0.417723	0.208861	+	0.977945i	$0.433024\pi$
$102$	− 1.38404i	− 0.137041i
$103$	−11.5211	−1.13521	−0.567604	−	0.823302i	$-0.692130\pi$
$103$	−11.5211	−1.13521	−0.567604	+	0.823302i	$0.692130\pi$
$104$	0	0
$105$	−1.44504	−0.141022
$106$	− 11.0858i	− 1.07674i
$107$	−4.43967	−0.429199	−0.214599	−	0.976702i	$-0.568845\pi$
$107$	−4.43967	−0.429199	−0.214599	+	0.976702i	$0.568845\pi$
$108$	2.58211	0.248463
$109$	− 10.5526i	− 1.01075i	−0.862899	−	0.505376i	$-0.831354\pi$
$109$	− 10.5526i	− 1.01075i	0.862899	−	0.505376i	$-0.168646\pi$
$110$	1.60388i	0.152924i
$111$	1.42758i	0.135500i
$112$	− 3.24698i	− 0.306811i
$113$	4.31767	0.406172	0.203086	−	0.979161i	$-0.434903\pi$
$113$	4.31767	0.406172	0.203086	+	0.979161i	$0.434903\pi$
$114$	−1.28621	−0.120464
$115$	6.78986i	0.633157i
$116$	−4.04892	−0.375933
$117$	0	0
$118$	5.28621	0.486635
$119$	10.0978i	0.925667i
$120$	−0.445042	−0.0406266
$121$	8.42758	0.766144
$122$	− 13.1250i	− 1.18828i
$123$	4.75063i	0.428350i
$124$	− 8.31767i	− 0.746949i
$125$	1.00000i	0.0894427i
$126$	−9.09783	−0.810500
$127$	2.80731	0.249109	0.124554	−	0.992213i	$-0.460250\pi$
$127$	2.80731	0.249109	0.124554	+	0.992213i	$0.460250\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	2.30798	0.203206
$130$	0	0
$131$	2.37196	0.207239	0.103620	−	0.994617i	$-0.466958\pi$
$131$	2.37196	0.207239	0.103620	+	0.994617i	$0.466958\pi$
$132$	− 0.713792i	− 0.0621276i
$133$	9.38404	0.813700
$134$	−12.3937	−1.07066
$135$	2.58211i	0.222232i
$136$	3.10992i	0.266673i
$137$	− 16.8552i	− 1.44003i	−0.693956	−	0.720017i	$-0.744134\pi$
$137$	− 16.8552i	− 1.44003i	0.693956	−	0.720017i	$-0.255866\pi$
$138$	− 3.02177i	− 0.257230i
$139$	5.97584	0.506864	0.253432	−	0.967353i	$-0.418441\pi$
$139$	5.97584	0.506864	0.253432	+	0.967353i	$0.418441\pi$
$140$	3.24698	0.274420
$141$	− 1.77048i	− 0.149101i
$142$	−7.50604	−0.629893
$143$	0	0
$144$	−2.80194	−0.233495
$145$	− 4.04892i	− 0.336244i
$146$	−12.2741	−1.01581
$147$	−1.57673	−0.130046
$148$	− 3.20775i	− 0.263676i
$149$	− 14.9148i	− 1.22187i	−0.791680	−	0.610936i	$-0.790793\pi$
$149$	− 14.9148i	− 1.22187i	0.791680	−	0.610936i	$-0.209207\pi$
$150$	− 0.445042i	− 0.0363375i
$151$	9.56033i	0.778009i	0.921236	+	0.389005i	$0.127181\pi$
$151$	9.56033i	0.778009i	−0.921236	+	0.389005i	$0.872819\pi$
$152$	2.89008	0.234417
$153$	8.71379	0.704468
$154$	5.20775i	0.419653i
$155$	8.31767	0.668091
$156$	0	0
$157$	−13.7560	−1.09785	−0.548924	−	0.835872i	$-0.684962\pi$
$157$	−13.7560	−1.09785	−0.548924	+	0.835872i	$0.684962\pi$
$158$	− 11.3056i	− 0.899424i
$159$	−4.93362	−0.391262
$160$	1.00000	0.0790569
$161$	22.0465i	1.73751i
$162$	7.25667i	0.570138i
$163$	− 13.5308i	− 1.05981i	−0.848056	−	0.529907i	$-0.822227\pi$
$163$	− 13.5308i	− 1.05981i	0.848056	−	0.529907i	$-0.177773\pi$
$164$	− 10.6746i	− 0.833543i
$165$	0.713792	0.0555686
$166$	10.4209	0.808817
$167$	− 15.1685i	− 1.17378i	−0.809668	−	0.586888i	$-0.800353\pi$
$167$	− 15.1685i	− 1.17378i	0.809668	−	0.586888i	$-0.199647\pi$
$168$	−1.44504	−0.111487
$169$	0	0
$170$	−3.10992	−0.238520
$171$	− 8.09783i	− 0.619257i
$172$	−5.18598	−0.395427
$173$	−4.27413	−0.324956	−0.162478	−	0.986712i	$-0.551949\pi$
$173$	−4.27413	−0.324956	−0.162478	+	0.986712i	$0.551949\pi$
$174$	1.80194i	0.136605i
$175$	3.24698i	0.245449i
$176$	1.60388i	0.120897i
$177$	− 2.35258i	− 0.176831i
$178$	11.3817	0.853091
$179$	−15.6233	−1.16774	−0.583868	−	0.811848i	$-0.698462\pi$
$179$	−15.6233	−1.16774	−0.583868	+	0.811848i	$0.698462\pi$
$180$	− 2.80194i	− 0.208844i
$181$	−16.1933	−1.20364	−0.601818	−	0.798633i	$-0.705557\pi$
$181$	−16.1933	−1.20364	−0.601818	+	0.798633i	$0.705557\pi$
$182$	0	0
$183$	−5.84117	−0.431791
$184$	6.78986i	0.500555i
$185$	3.20775	0.235839
$186$	−3.70171	−0.271423
$187$	− 4.98792i	− 0.364753i
$188$	3.97823i	0.290142i
$189$	8.38404i	0.609849i
$190$	2.89008i	0.209669i
$191$	−26.5676	−1.92237	−0.961183	−	0.275911i	$-0.911020\pi$
$191$	−26.5676	−1.92237	−0.961183	+	0.275911i	$0.911020\pi$
$192$	−0.445042	−0.0321181
$193$	13.6582i	0.983137i	0.870839	+	0.491568i	$0.163576\pi$
$193$	13.6582i	0.983137i	−0.870839	+	0.491568i	$0.836424\pi$
$194$	6.05429	0.434673
$195$	0	0
$196$	3.54288	0.253063
$197$	14.0978i	1.00443i	0.864743	+	0.502215i	$0.167481\pi$
$197$	14.0978i	1.00443i	−0.864743	+	0.502215i	$0.832519\pi$
$198$	4.49396	0.319372
$199$	21.1293	1.49782	0.748908	−	0.662674i	$-0.230578\pi$
$199$	21.1293	1.49782	0.748908	+	0.662674i	$0.230578\pi$
$200$	1.00000i	0.0707107i
$201$	5.51573i	0.389050i
$202$	4.19806i	0.295375i
$203$	− 13.1468i	− 0.922721i
$204$	1.38404	0.0969024
$205$	10.6746	0.745544
$206$	− 11.5211i	− 0.802714i
$207$	19.0248	1.32231
$208$	0	0
$209$	−4.63533	−0.320633
$210$	− 1.44504i	− 0.0997174i
$211$	−18.3720	−1.26478	−0.632389	−	0.774651i	$-0.717926\pi$
$211$	−18.3720	−1.26478	−0.632389	+	0.774651i	$0.717926\pi$
$212$	11.0858	0.761373
$213$	3.34050i	0.228887i
$214$	− 4.43967i	− 0.303489i
$215$	− 5.18598i	− 0.353681i
$216$	2.58211i	0.175690i
$217$	27.0073	1.83337
$218$	10.5526	0.714710
$219$	5.46250i	0.369122i
$220$	−1.60388	−0.108133
$221$	0	0
$222$	−1.42758	−0.0958131
$223$	− 14.2687i	− 0.955506i	−0.878494	−	0.477753i	$-0.841451\pi$
$223$	− 14.2687i	− 0.955506i	0.878494	−	0.477753i	$-0.158549\pi$
$224$	3.24698	0.216948
$225$	2.80194	0.186796
$226$	4.31767i	0.287207i
$227$	− 27.4959i	− 1.82497i	−0.409115	−	0.912483i	$-0.634163\pi$
$227$	− 27.4959i	− 1.82497i	0.409115	−	0.912483i	$-0.365837\pi$
$228$	− 1.28621i	− 0.0851812i
$229$	19.6963i	1.30157i	0.759262	+	0.650785i	$0.225560\pi$
$229$	19.6963i	1.30157i	−0.759262	+	0.650785i	$0.774440\pi$
$230$	−6.78986	−0.447710
$231$	2.31767	0.152491
$232$	− 4.04892i	− 0.265824i
$233$	23.5362	1.54191	0.770953	−	0.636892i	$-0.219780\pi$
$233$	23.5362	1.54191	0.770953	+	0.636892i	$0.219780\pi$
$234$	0	0
$235$	−3.97823	−0.259511
$236$	5.28621i	0.344103i
$237$	−5.03146	−0.326828
$238$	−10.0978	−0.654545
$239$	14.4155i	0.932461i	0.884663	+	0.466231i	$0.154388\pi$
$239$	14.4155i	0.932461i	−0.884663	+	0.466231i	$0.845612\pi$
$240$	− 0.445042i	− 0.0287273i
$241$	15.6692i	1.00934i	0.863312	+	0.504671i	$0.168386\pi$
$241$	15.6692i	1.00934i	−0.863312	+	0.504671i	$0.831614\pi$
$242$	8.42758i	0.541746i
$243$	10.9758	0.704100
$244$	13.1250	0.840241
$245$	3.54288i	0.226346i
$246$	−4.75063	−0.302889
$247$	0	0
$248$	8.31767	0.528172
$249$	− 4.63773i	− 0.293904i
$250$	−1.00000	−0.0632456
$251$	15.1535	0.956478	0.478239	−	0.878230i	$-0.341275\pi$
$251$	15.1535	0.956478	0.478239	+	0.878230i	$0.341275\pi$
$252$	− 9.09783i	− 0.573110i
$253$	− 10.8901i	− 0.684654i
$254$	2.80731i	0.176147i
$255$	1.38404i	0.0866721i
$256$	1.00000	0.0625000
$257$	−18.8552	−1.17615	−0.588076	−	0.808805i	$-0.700115\pi$
$257$	−18.8552	−1.17615	−0.588076	+	0.808805i	$0.700115\pi$
$258$	2.30798i	0.143688i
$259$	10.4155	0.647188
$260$	0	0
$261$	−11.3448	−0.702226
$262$	2.37196i	0.146540i
$263$	−2.18060	−0.134462	−0.0672309	−	0.997737i	$-0.521416\pi$
$263$	−2.18060	−0.134462	−0.0672309	+	0.997737i	$0.521416\pi$
$264$	0.713792	0.0439308
$265$	11.0858i	0.680992i
$266$	9.38404i	0.575373i
$267$	− 5.06531i	− 0.309992i
$268$	− 12.3937i	− 0.757068i
$269$	9.73125	0.593325	0.296662	−	0.954982i	$-0.404126\pi$
$269$	9.73125	0.593325	0.296662	+	0.954982i	$0.404126\pi$
$270$	−2.58211	−0.157142
$271$	4.05429i	0.246281i	0.992389	+	0.123140i	$0.0392966\pi$
$271$	4.05429i	0.246281i	−0.992389	+	0.123140i	$0.960703\pi$
$272$	−3.10992	−0.188566
$273$	0	0
$274$	16.8552	1.01826
$275$	− 1.60388i	− 0.0967173i
$276$	3.02177	0.181889
$277$	−30.0301	−1.80434	−0.902168	−	0.431385i	$-0.858025\pi$
$277$	−30.0301	−1.80434	−0.902168	+	0.431385i	$0.858025\pi$
$278$	5.97584i	0.358407i
$279$	− 23.3056i	− 1.39527i
$280$	3.24698i	0.194044i
$281$	22.2640i	1.32816i	0.747663	+	0.664078i	$0.231176\pi$
$281$	22.2640i	1.32816i	−0.747663	+	0.664078i	$0.768824\pi$
$282$	1.77048	0.105430
$283$	8.88471	0.528141	0.264071	−	0.964503i	$-0.414935\pi$
$283$	8.88471	0.528141	0.264071	+	0.964503i	$0.414935\pi$
$284$	− 7.50604i	− 0.445402i
$285$	1.28621	0.0761884
$286$	0	0
$287$	34.6601	2.04592
$288$	− 2.80194i	− 0.165106i
$289$	−7.32842	−0.431084
$290$	4.04892	0.237761
$291$	− 2.69441i	− 0.157949i
$292$	− 12.2741i	− 0.718289i
$293$	21.5362i	1.25816i	0.777342	+	0.629078i	$0.216568\pi$
$293$	21.5362i	1.25816i	−0.777342	+	0.629078i	$0.783432\pi$
$294$	− 1.57673i	− 0.0919567i
$295$	−5.28621	−0.307775
$296$	3.20775	0.186447
$297$	− 4.14138i	− 0.240307i
$298$	14.9148	0.863993
$299$	0	0
$300$	0.445042	0.0256945
$301$	− 16.8388i	− 0.970571i
$302$	−9.56033	−0.550135
$303$	1.86831	0.107332
$304$	2.89008i	0.165758i
$305$	13.1250i	0.751534i
$306$	8.71379i	0.498134i
$307$	− 10.3230i	− 0.589167i	−0.955626	−	0.294584i	$-0.904819\pi$
$307$	− 10.3230i	− 0.589167i	0.955626	−	0.294584i	$-0.0951810\pi$
$308$	−5.20775	−0.296739
$309$	−5.12737	−0.291686
$310$	8.31767i	0.472412i
$311$	−7.50125	−0.425357	−0.212679	−	0.977122i	$-0.568219\pi$
$311$	−7.50125	−0.425357	−0.212679	+	0.977122i	$0.568219\pi$
$312$	0	0
$313$	5.60388	0.316750	0.158375	−	0.987379i	$-0.449375\pi$
$313$	5.60388	0.316750	0.158375	+	0.987379i	$0.449375\pi$
$314$	− 13.7560i	− 0.776296i
$315$	9.09783	0.512605
$316$	11.3056	0.635989
$317$	11.7802i	0.661640i	0.943694	+	0.330820i	$0.107325\pi$
$317$	11.7802i	0.661640i	−0.943694	+	0.330820i	$0.892675\pi$
$318$	− 4.93362i	− 0.276664i
$319$	6.49396i	0.363592i
$320$	1.00000i	0.0559017i
$321$	−1.97584	−0.110280
$322$	−22.0465	−1.22860
$323$	− 8.98792i	− 0.500101i
$324$	−7.25667	−0.403148
$325$	0	0
$326$	13.5308	0.749401
$327$	− 4.69633i	− 0.259708i
$328$	10.6746	0.589404
$329$	−12.9172	−0.712150
$330$	0.713792i	0.0392929i
$331$	6.46980i	0.355612i	0.984066	+	0.177806i	$0.0569000\pi$
$331$	6.46980i	0.355612i	−0.984066	+	0.177806i	$0.943100\pi$
$332$	10.4209i	0.571920i
$333$	− 8.98792i	− 0.492535i
$334$	15.1685	0.829985
$335$	12.3937	0.677142
$336$	− 1.44504i	− 0.0788335i
$337$	5.35988	0.291971	0.145986	−	0.989287i	$-0.453365\pi$
$337$	5.35988	0.291971	0.145986	+	0.989287i	$0.453365\pi$
$338$	0	0
$339$	1.92154	0.104364
$340$	− 3.10992i	− 0.168659i
$341$	−13.3405	−0.722429
$342$	8.09783	0.437881
$343$	− 11.2252i	− 0.606104i
$344$	− 5.18598i	− 0.279609i
$345$	3.02177i	0.162687i
$346$	− 4.27413i	− 0.229778i
$347$	3.56571	0.191417	0.0957087	−	0.995409i	$-0.469488\pi$
$347$	3.56571	0.191417	0.0957087	+	0.995409i	$0.469488\pi$
$348$	−1.80194	−0.0965940
$349$	− 7.97584i	− 0.426937i	−0.976950	−	0.213468i	$-0.931524\pi$
$349$	− 7.97584i	− 0.426937i	0.976950	−	0.213468i	$-0.0684760\pi$
$350$	−3.24698	−0.173558
$351$	0	0
$352$	−1.60388	−0.0854868
$353$	1.60388i	0.0853657i	0.999089	+	0.0426828i	$0.0135905\pi$
$353$	1.60388i	0.0853657i	−0.999089	+	0.0426828i	$0.986410\pi$
$354$	2.35258	0.125038
$355$	7.50604	0.398379
$356$	11.3817i	0.603226i
$357$	4.49396i	0.237846i
$358$	− 15.6233i	− 0.825715i
$359$	− 8.60255i	− 0.454025i	−0.973892	−	0.227013i	$-0.927104\pi$
$359$	− 8.60255i	− 0.454025i	0.973892	−	0.227013i	$-0.0728958\pi$
$360$	2.80194	0.147675
$361$	10.6474	0.560390
$362$	− 16.1933i	− 0.851100i
$363$	3.75063	0.196857
$364$	0	0
$365$	12.2741	0.642457
$366$	− 5.84117i	− 0.305323i
$367$	7.41358	0.386986	0.193493	−	0.981102i	$-0.438018\pi$
$367$	7.41358	0.386986	0.193493	+	0.981102i	$0.438018\pi$
$368$	−6.78986	−0.353946
$369$	− 29.9095i	− 1.55702i
$370$	3.20775i	0.166763i
$371$	35.9952i	1.86878i
$372$	− 3.70171i	− 0.191925i
$373$	29.6039	1.53283	0.766415	−	0.642345i	$-0.222039\pi$
$373$	29.6039	1.53283	0.766415	+	0.642345i	$0.222039\pi$
$374$	4.98792	0.257919
$375$	0.445042i	0.0229819i
$376$	−3.97823	−0.205162
$377$	0	0
$378$	−8.38404	−0.431229
$379$	8.45042i	0.434069i	0.976164	+	0.217034i	$0.0696384\pi$
$379$	8.45042i	0.434069i	−0.976164	+	0.217034i	$0.930362\pi$
$380$	−2.89008	−0.148258
$381$	1.24937	0.0640073
$382$	− 26.5676i	− 1.35932i
$383$	− 10.4644i	− 0.534707i	−0.963599	−	0.267353i	$-0.913851\pi$
$383$	− 10.4644i	− 0.534707i	0.963599	−	0.267353i	$-0.0861491\pi$
$384$	− 0.445042i	− 0.0227109i
$385$	− 5.20775i	− 0.265412i
$386$	−13.6582	−0.695183
$387$	−14.5308	−0.738642
$388$	6.05429i	0.307360i
$389$	−29.2295	−1.48200	−0.740998	−	0.671507i	$-0.765647\pi$
$389$	−29.2295	−1.48200	−0.740998	+	0.671507i	$0.765647\pi$
$390$	0	0
$391$	21.1159	1.06788
$392$	3.54288i	0.178942i
$393$	1.05562	0.0532491
$394$	−14.0978	−0.710239
$395$	11.3056i	0.568846i
$396$	4.49396i	0.225830i
$397$	20.7439i	1.04111i	0.853829	+	0.520554i	$0.174274\pi$
$397$	20.7439i	1.04111i	−0.853829	+	0.520554i	$0.825726\pi$
$398$	21.1293i	1.05912i
$399$	4.17629	0.209076
$400$	−1.00000	−0.0500000
$401$	− 24.2892i	− 1.21294i	−0.795105	−	0.606472i	$-0.792584\pi$
$401$	− 24.2892i	− 1.21294i	0.795105	−	0.606472i	$-0.207416\pi$
$402$	−5.51573	−0.275100
$403$	0	0
$404$	−4.19806	−0.208861
$405$	− 7.25667i	− 0.360587i
$406$	13.1468	0.652462
$407$	−5.14483	−0.255020
$408$	1.38404i	0.0685203i
$409$	36.7676i	1.81804i	0.416751	+	0.909021i	$0.363169\pi$
$409$	36.7676i	1.81804i	−0.416751	+	0.909021i	$0.636831\pi$
$410$	10.6746i	0.527179i
$411$	− 7.50125i	− 0.370010i
$412$	11.5211	0.567604
$413$	−17.1642	−0.844596
$414$	19.0248i	0.935016i
$415$	−10.4209	−0.511541
$416$	0	0
$417$	2.65950	0.130236
$418$	− 4.63533i	− 0.226722i
$419$	−30.8418	−1.50672	−0.753359	−	0.657609i	$-0.771568\pi$
$419$	−30.8418	−1.50672	−0.753359	+	0.657609i	$0.771568\pi$
$420$	1.44504	0.0705108
$421$	31.7458i	1.54720i	0.633676	+	0.773599i	$0.281545\pi$
$421$	31.7458i	1.54720i	−0.633676	+	0.773599i	$0.718455\pi$
$422$	− 18.3720i	− 0.894333i
$423$	11.1468i	0.541974i
$424$	11.0858i	0.538372i
$425$	3.10992	0.150853
$426$	−3.34050	−0.161848
$427$	42.6165i	2.06236i
$428$	4.43967	0.214599
$429$	0	0
$430$	5.18598	0.250090
$431$	− 25.5120i	− 1.22887i	−0.788967	−	0.614435i	$-0.789384\pi$
$431$	− 25.5120i	− 1.22887i	0.788967	−	0.614435i	$-0.210616\pi$
$432$	−2.58211	−0.124232
$433$	31.8189	1.52912	0.764560	−	0.644553i	$-0.222956\pi$
$433$	31.8189	1.52912	0.764560	+	0.644553i	$0.222956\pi$
$434$	27.0073i	1.29639i
$435$	− 1.80194i	− 0.0863963i
$436$	10.5526i	0.505376i
$437$	− 19.6233i	− 0.938707i
$438$	−5.46250	−0.261008
$439$	−26.9095	−1.28432	−0.642159	−	0.766571i	$-0.721961\pi$
$439$	−26.9095	−1.28432	−0.642159	+	0.766571i	$0.721961\pi$
$440$	− 1.60388i	− 0.0764618i
$441$	9.92692	0.472710
$442$	0	0
$443$	12.7385	0.605227	0.302613	−	0.953113i	$-0.402141\pi$
$443$	12.7385	0.605227	0.302613	+	0.953113i	$0.402141\pi$
$444$	− 1.42758i	− 0.0677501i
$445$	−11.3817	−0.539542
$446$	14.2687	0.675645
$447$	− 6.63773i	− 0.313954i
$448$	3.24698i	0.153405i
$449$	7.75063i	0.365775i	0.983134	+	0.182887i	$0.0585444\pi$
$449$	7.75063i	0.365775i	−0.983134	+	0.182887i	$0.941456\pi$
$450$	2.80194i	0.132085i
$451$	−17.1207	−0.806181
$452$	−4.31767	−0.203086
$453$	4.25475i	0.199906i
$454$	27.4959	1.29045
$455$	0	0
$456$	1.28621	0.0602322
$457$	− 6.94438i	− 0.324844i	−0.986721	−	0.162422i	$-0.948069\pi$
$457$	− 6.94438i	− 0.324844i	0.986721	−	0.162422i	$-0.0519306\pi$
$458$	−19.6963	−0.920349
$459$	8.03013	0.374814
$460$	− 6.78986i	− 0.316579i
$461$	12.6377i	0.588598i	0.955713	+	0.294299i	$0.0950861\pi$
$461$	12.6377i	0.588598i	−0.955713	+	0.294299i	$0.904914\pi$
$462$	2.31767i	0.107828i
$463$	− 5.62325i	− 0.261335i	−0.991426	−	0.130667i	$-0.958288\pi$
$463$	− 5.62325i	− 0.261335i	0.991426	−	0.130667i	$-0.0417120\pi$
$464$	4.04892	0.187966
$465$	3.70171	0.171663
$466$	23.5362i	1.09029i
$467$	35.2054	1.62911	0.814555	−	0.580087i	$-0.196981\pi$
$467$	35.2054	1.62911	0.814555	+	0.580087i	$0.196981\pi$
$468$	0	0
$469$	40.2422	1.85821
$470$	− 3.97823i	− 0.183502i
$471$	−6.12200	−0.282087
$472$	−5.28621	−0.243317
$473$	8.31767i	0.382447i
$474$	− 5.03146i	− 0.231103i
$475$	− 2.89008i	− 0.132606i
$476$	− 10.0978i	− 0.462833i
$477$	31.0616	1.42221
$478$	−14.4155	−0.659350
$479$	7.16421i	0.327341i	0.986515	+	0.163671i	$0.0523334\pi$
$479$	7.16421i	0.327341i	−0.986515	+	0.163671i	$0.947667\pi$
$480$	0.445042	0.0203133
$481$	0	0
$482$	−15.6692	−0.713712
$483$	9.81163i	0.446444i
$484$	−8.42758	−0.383072
$485$	−6.05429	−0.274911
$486$	10.9758i	0.497874i
$487$	− 28.7006i	− 1.30055i	−0.759699	−	0.650275i	$-0.774654\pi$
$487$	− 28.7006i	− 1.30055i	0.759699	−	0.650275i	$-0.225346\pi$
$488$	13.1250i	0.594140i
$489$	− 6.02177i	− 0.272314i
$490$	−3.54288	−0.160051
$491$	42.0689	1.89854	0.949271	−	0.314459i	$-0.101823\pi$
$491$	42.0689	1.89854	0.949271	+	0.314459i	$0.101823\pi$
$492$	− 4.75063i	− 0.214175i
$493$	−12.5918	−0.567106
$494$	0	0
$495$	−4.49396	−0.201988
$496$	8.31767i	0.373474i
$497$	24.3720	1.09323
$498$	4.63773	0.207822
$499$	− 33.2379i	− 1.48793i	−0.668218	−	0.743966i	$-0.732942\pi$
$499$	− 33.2379i	− 1.48793i	0.668218	−	0.743966i	$-0.267058\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 6.75063i	− 0.301596i
$502$	15.1535i	0.676332i
$503$	−37.7797	−1.68451	−0.842257	−	0.539077i	$-0.818773\pi$
$503$	−37.7797	−1.68451	−0.842257	+	0.539077i	$0.818773\pi$
$504$	9.09783	0.405250
$505$	− 4.19806i	− 0.186811i
$506$	10.8901	0.484123
$507$	0	0
$508$	−2.80731	−0.124554
$509$	− 2.26875i	− 0.100561i	−0.998735	−	0.0502803i	$-0.983989\pi$
$509$	− 2.26875i	− 0.100561i	0.998735	−	0.0502803i	$-0.0160115\pi$
$510$	−1.38404	−0.0612865
$511$	39.8538	1.76303
$512$	1.00000i	0.0441942i
$513$	− 7.46250i	− 0.329477i
$514$	− 18.8552i	− 0.831666i
$515$	11.5211i	0.507681i
$516$	−2.30798	−0.101603
$517$	6.38059	0.280618
$518$	10.4155i	0.457631i
$519$	−1.90217	−0.0834958
$520$	0	0
$521$	−22.7899	−0.998442	−0.499221	−	0.866475i	$-0.666380\pi$
$521$	−22.7899	−0.998442	−0.499221	+	0.866475i	$0.666380\pi$
$522$	− 11.3448i	− 0.496549i
$523$	6.61356	0.289191	0.144595	−	0.989491i	$-0.453812\pi$
$523$	6.61356	0.289191	0.144595	+	0.989491i	$0.453812\pi$
$524$	−2.37196	−0.103620
$525$	1.44504i	0.0630668i
$526$	− 2.18060i	− 0.0950788i
$527$	− 25.8672i	− 1.12680i
$528$	0.713792i	0.0310638i
$529$	23.1021	1.00444
$530$	−11.0858	−0.481534
$531$	14.8116i	0.642770i
$532$	−9.38404	−0.406850
$533$	0	0
$534$	5.06531	0.219197
$535$	4.43967i	0.191943i
$536$	12.3937	0.535328
$537$	−6.95300	−0.300044
$538$	9.73125i	0.419544i
$539$	− 5.68233i	− 0.244755i
$540$	− 2.58211i	− 0.111116i
$541$	24.2553i	1.04282i	0.853307	+	0.521409i	$0.174593\pi$
$541$	24.2553i	1.04282i	−0.853307	+	0.521409i	$0.825407\pi$
$542$	−4.05429	−0.174147
$543$	−7.20669	−0.309268
$544$	− 3.10992i	− 0.133337i
$545$	−10.5526	−0.452022
$546$	0	0
$547$	11.8086	0.504901	0.252451	−	0.967610i	$-0.418763\pi$
$547$	11.8086	0.504901	0.252451	+	0.967610i	$0.418763\pi$
$548$	16.8552i	0.720017i
$549$	36.7754	1.56954
$550$	1.60388	0.0683895
$551$	11.7017i	0.498510i
$552$	3.02177i	0.128615i
$553$	36.7090i	1.56103i
$554$	− 30.0301i	− 1.27586i
$555$	1.42758	0.0605975
$556$	−5.97584	−0.253432
$557$	− 15.0180i	− 0.636335i	−0.948034	−	0.318168i	$-0.896933\pi$
$557$	− 15.0180i	− 0.636335i	0.948034	−	0.318168i	$-0.103067\pi$
$558$	23.3056	0.986604
$559$	0	0
$560$	−3.24698	−0.137210
$561$	− 2.21983i	− 0.0937214i
$562$	−22.2640	−0.939149
$563$	−10.4198	−0.439143	−0.219571	−	0.975596i	$-0.570466\pi$
$563$	−10.4198	−0.439143	−0.219571	+	0.975596i	$0.570466\pi$
$564$	1.77048i	0.0745506i
$565$	− 4.31767i	− 0.181646i
$566$	8.88471i	0.373452i
$567$	− 23.5623i	− 0.989522i
$568$	7.50604	0.314946
$569$	−12.3461	−0.517577	−0.258789	−	0.965934i	$-0.583323\pi$
$569$	−12.3461	−0.517577	−0.258789	+	0.965934i	$0.583323\pi$
$570$	1.28621i	0.0538733i
$571$	2.21983	0.0928971	0.0464486	−	0.998921i	$-0.485210\pi$
$571$	2.21983	0.0928971	0.0464486	+	0.998921i	$0.485210\pi$
$572$	0	0
$573$	−11.8237	−0.493942
$574$	34.6601i	1.44668i
$575$	6.78986	0.283157
$576$	2.80194	0.116747
$577$	21.3405i	0.888417i	0.895924	+	0.444208i	$0.146515\pi$
$577$	21.3405i	0.888417i	−0.895924	+	0.444208i	$0.853485\pi$
$578$	− 7.32842i	− 0.304822i
$579$	6.07846i	0.252612i
$580$	4.04892i	0.168122i
$581$	−33.8364	−1.40377
$582$	2.69441	0.111687
$583$	− 17.7802i	− 0.736379i
$584$	12.2741	0.507907
$585$	0	0
$586$	−21.5362	−0.889651
$587$	10.1371i	0.418401i	0.977873	+	0.209201i	$0.0670862\pi$
$587$	10.1371i	0.418401i	−0.977873	+	0.209201i	$0.932914\pi$
$588$	1.57673	0.0650232
$589$	−24.0388	−0.990500
$590$	− 5.28621i	− 0.217630i
$591$	6.27413i	0.258083i
$592$	3.20775i	0.131838i
$593$	17.6668i	0.725488i	0.931889	+	0.362744i	$0.118160\pi$
$593$	17.6668i	0.725488i	−0.931889	+	0.362744i	$0.881840\pi$
$594$	4.14138	0.169923
$595$	10.0978	0.413971
$596$	14.9148i	0.610936i
$597$	9.40342	0.384856
$598$	0	0
$599$	9.14005	0.373452	0.186726	−	0.982412i	$-0.440212\pi$
$599$	9.14005	0.373452	0.186726	+	0.982412i	$0.440212\pi$
$600$	0.445042i	0.0181688i
$601$	37.0116	1.50973	0.754867	−	0.655877i	$-0.227701\pi$
$601$	37.0116	1.50973	0.754867	+	0.655877i	$0.227701\pi$
$602$	16.8388	0.686297
$603$	− 34.7265i	− 1.41417i
$604$	− 9.56033i	− 0.389005i
$605$	− 8.42758i	− 0.342630i
$606$	1.86831i	0.0758950i
$607$	10.9420	0.444121	0.222061	−	0.975033i	$-0.428722\pi$
$607$	10.9420	0.444121	0.222061	+	0.975033i	$0.428722\pi$
$608$	−2.89008	−0.117208
$609$	− 5.85086i	− 0.237089i
$610$	−13.1250	−0.531415
$611$	0	0
$612$	−8.71379	−0.352234
$613$	− 10.9336i	− 0.441605i	−0.975319	−	0.220802i	$-0.929132\pi$
$613$	− 10.9336i	− 0.441605i	0.975319	−	0.220802i	$-0.0708676\pi$
$614$	10.3230	0.416604
$615$	4.75063	0.191564
$616$	− 5.20775i	− 0.209826i
$617$	− 6.33704i	− 0.255120i	−0.991831	−	0.127560i	$-0.959285\pi$
$617$	− 6.33704i	− 0.255120i	0.991831	−	0.127560i	$-0.0407145\pi$
$618$	− 5.12737i	− 0.206253i
$619$	12.8659i	0.517125i	0.965995	+	0.258563i	$0.0832488\pi$
$619$	12.8659i	0.517125i	−0.965995	+	0.258563i	$0.916751\pi$
$620$	−8.31767	−0.334046
$621$	17.5321	0.703540
$622$	− 7.50125i	− 0.300773i
$623$	−36.9560	−1.48061
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.60388i	0.223976i
$627$	−2.06292	−0.0823850
$628$	13.7560	0.548924
$629$	− 9.97584i	− 0.397763i
$630$	9.09783i	0.362466i
$631$	− 47.8491i	− 1.90484i	−0.304788	−	0.952420i	$-0.598585\pi$
$631$	− 47.8491i	− 1.90484i	0.304788	−	0.952420i	$-0.401415\pi$
$632$	11.3056i	0.449712i
$633$	−8.17629	−0.324978
$634$	−11.7802	−0.467850
$635$	− 2.80731i	− 0.111405i
$636$	4.93362	0.195631
$637$	0	0
$638$	−6.49396	−0.257098
$639$	− 21.0315i	− 0.831992i
$640$	−1.00000	−0.0395285
$641$	−45.7168	−1.80570	−0.902852	−	0.429951i	$-0.858531\pi$
$641$	−45.7168	−1.80570	−0.902852	+	0.429951i	$0.858531\pi$
$642$	− 1.97584i	− 0.0779801i
$643$	33.1836i	1.30863i	0.756221	+	0.654316i	$0.227044\pi$
$643$	33.1836i	1.30863i	−0.756221	+	0.654316i	$0.772956\pi$
$644$	− 22.0465i	− 0.868755i
$645$	− 2.30798i	− 0.0908766i
$646$	8.98792	0.353625
$647$	9.53617	0.374906	0.187453	−	0.982274i	$-0.439977\pi$
$647$	9.53617	0.374906	0.187453	+	0.982274i	$0.439977\pi$
$648$	− 7.25667i	− 0.285069i
$649$	8.47842	0.332807
$650$	0	0
$651$	12.0194	0.471077
$652$	13.5308i	0.529907i
$653$	−40.5918	−1.58848	−0.794240	−	0.607604i	$-0.792131\pi$
$653$	−40.5918	−1.58848	−0.794240	+	0.607604i	$0.792131\pi$
$654$	4.69633	0.183641
$655$	− 2.37196i	− 0.0926802i
$656$	10.6746i	0.416772i
$657$	− 34.3913i	− 1.34173i
$658$	− 12.9172i	− 0.503566i
$659$	−0.537500	−0.0209380	−0.0104690	−	0.999945i	$-0.503332\pi$
$659$	−0.537500	−0.0209380	−0.0104690	+	0.999945i	$0.503332\pi$
$660$	−0.713792	−0.0277843
$661$	23.9928i	0.933213i	0.884465	+	0.466606i	$0.154523\pi$
$661$	23.9928i	0.933213i	−0.884465	+	0.466606i	$0.845477\pi$
$662$	−6.46980	−0.251456
$663$	0	0
$664$	−10.4209	−0.404409
$665$	− 9.38404i	− 0.363898i
$666$	8.98792	0.348275
$667$	−27.4916	−1.06448
$668$	15.1685i	0.586888i
$669$	− 6.35019i	− 0.245513i
$670$	12.3937i	0.478812i
$671$	− 21.0508i	− 0.812659i
$672$	1.44504	0.0557437
$673$	−7.26683	−0.280116	−0.140058	−	0.990143i	$-0.544729\pi$
$673$	−7.26683	−0.280116	−0.140058	+	0.990143i	$0.544729\pi$
$674$	5.35988i	0.206455i
$675$	2.58211	0.0993853
$676$	0	0
$677$	5.11470	0.196574	0.0982870	−	0.995158i	$-0.468664\pi$
$677$	5.11470	0.196574	0.0982870	+	0.995158i	$0.468664\pi$
$678$	1.92154i	0.0737964i
$679$	−19.6582	−0.754411
$680$	3.10992	0.119260
$681$	− 12.2368i	− 0.468916i
$682$	− 13.3405i	− 0.510834i
$683$	− 23.0532i	− 0.882107i	−0.897481	−	0.441054i	$-0.854605\pi$
$683$	− 23.0532i	− 0.882107i	0.897481	−	0.441054i	$-0.145395\pi$
$684$	8.09783i	0.309628i
$685$	−16.8552	−0.644003
$686$	11.2252	0.428580
$687$	8.76569i	0.334432i
$688$	5.18598	0.197714
$689$	0	0
$690$	−3.02177	−0.115037
$691$	− 31.1051i	− 1.18329i	−0.806197	−	0.591647i	$-0.798478\pi$
$691$	− 31.1051i	− 1.18329i	0.806197	−	0.591647i	$-0.201522\pi$
$692$	4.27413	0.162478
$693$	−14.5918	−0.554296
$694$	3.56571i	0.135353i
$695$	− 5.97584i	− 0.226676i
$696$	− 1.80194i	− 0.0683023i
$697$	− 33.1970i	− 1.25743i
$698$	7.97584	0.301890
$699$	10.4746	0.396185
$700$	− 3.24698i	− 0.122724i
$701$	−7.07069	−0.267056	−0.133528	−	0.991045i	$-0.542631\pi$
$701$	−7.07069	−0.267056	−0.133528	+	0.991045i	$0.542631\pi$
$702$	0	0
$703$	−9.27067	−0.349650
$704$	− 1.60388i	− 0.0604483i
$705$	−1.77048	−0.0666801
$706$	−1.60388	−0.0603626
$707$	− 13.6310i	− 0.512647i
$708$	2.35258i	0.0884155i
$709$	8.65519i	0.325052i	0.986704	+	0.162526i	$0.0519642\pi$
$709$	8.65519i	0.325052i	−0.986704	+	0.162526i	$0.948036\pi$
$710$	7.50604i	0.281697i
$711$	31.6775	1.18800
$712$	−11.3817	−0.426545
$713$	− 56.4758i	− 2.11503i
$714$	−4.49396	−0.168182
$715$	0	0
$716$	15.6233	0.583868
$717$	6.41550i	0.239591i
$718$	8.60255	0.321044
$719$	8.48055	0.316271	0.158136	−	0.987417i	$-0.449452\pi$
$719$	8.48055	0.316271	0.158136	+	0.987417i	$0.449452\pi$
$720$	2.80194i	0.104422i
$721$	37.4088i	1.39318i
$722$	10.6474i	0.396256i
$723$	6.97344i	0.259345i
$724$	16.1933	0.601818
$725$	−4.04892	−0.150373
$726$	3.75063i	0.139199i
$727$	−31.2683	−1.15968	−0.579838	−	0.814732i	$-0.696884\pi$
$727$	−31.2683	−1.15968	−0.579838	+	0.814732i	$0.696884\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	12.2741i	0.454286i
$731$	−16.1280	−0.596514
$732$	5.84117	0.215896
$733$	− 6.69441i	− 0.247264i	−0.992328	−	0.123632i	$-0.960546\pi$
$733$	− 6.69441i	− 0.247264i	0.992328	−	0.123632i	$-0.0394542\pi$
$734$	7.41358i	0.273640i
$735$	1.57673i	0.0581585i
$736$	− 6.78986i	− 0.250277i
$737$	−19.8780	−0.732216
$738$	29.9095	1.10098
$739$	− 5.29696i	− 0.194852i	−0.995243	−	0.0974259i	$-0.968939\pi$
$739$	− 5.29696i	− 0.194852i	0.995243	−	0.0974259i	$-0.0310609\pi$
$740$	−3.20775	−0.117919
$741$	0	0
$742$	−35.9952	−1.32143
$743$	13.0755i	0.479693i	0.970811	+	0.239846i	$0.0770970\pi$
$743$	13.0755i	0.479693i	−0.970811	+	0.239846i	$0.922903\pi$
$744$	3.70171	0.135711
$745$	−14.9148	−0.546437
$746$	29.6039i	1.08387i
$747$	29.1987i	1.06832i
$748$	4.98792i	0.182376i
$749$	14.4155i	0.526731i
$750$	−0.445042	−0.0162506
$751$	26.6461	0.972330	0.486165	−	0.873867i	$-0.338395\pi$
$751$	26.6461	0.972330	0.486165	+	0.873867i	$0.338395\pi$
$752$	− 3.97823i	− 0.145071i
$753$	6.74392	0.245762
$754$	0	0
$755$	9.56033	0.347936
$756$	− 8.38404i	− 0.304925i
$757$	54.3236	1.97443	0.987213	−	0.159406i	$-0.0509580\pi$
$757$	54.3236	1.97443	0.987213	+	0.159406i	$0.0509580\pi$
$758$	−8.45042	−0.306933
$759$	− 4.84654i	− 0.175918i
$760$	− 2.89008i	− 0.104834i
$761$	− 8.55363i	− 0.310069i	−0.987909	−	0.155034i	$-0.950451\pi$
$761$	− 8.55363i	− 0.310069i	0.987909	−	0.155034i	$-0.0495488\pi$
$762$	1.24937i	0.0452600i
$763$	−34.2640	−1.24044
$764$	26.5676	0.961183
$765$	− 8.71379i	− 0.315048i
$766$	10.4644	0.378095
$767$	0	0
$768$	0.445042	0.0160591
$769$	20.6504i	0.744672i	0.928098	+	0.372336i	$0.121443\pi$
$769$	20.6504i	0.744672i	−0.928098	+	0.372336i	$0.878557\pi$
$770$	5.20775	0.187674
$771$	−8.39134	−0.302207
$772$	− 13.6582i	− 0.491568i
$773$	5.27545i	0.189745i	0.995489	+	0.0948725i	$0.0302443\pi$
$773$	5.27545i	0.189745i	−0.995489	+	0.0948725i	$0.969756\pi$
$774$	− 14.5308i	− 0.522299i
$775$	− 8.31767i	− 0.298779i
$776$	−6.05429	−0.217336
$777$	4.63533	0.166292
$778$	− 29.2295i	− 1.04793i
$779$	−30.8504	−1.10533
$780$	0	0
$781$	−12.0388	−0.430781
$782$	21.1159i	0.755102i
$783$	−10.4547	−0.373622
$784$	−3.54288	−0.126531
$785$	13.7560i	0.490973i
$786$	1.05562i	0.0376528i
$787$	− 39.7077i	− 1.41543i	−0.706500	−	0.707713i	$-0.749727\pi$
$787$	− 39.7077i	− 1.41543i	0.706500	−	0.707713i	$-0.250273\pi$
$788$	− 14.0978i	− 0.502215i
$789$	−0.970460	−0.0345493
$790$	−11.3056	−0.402235
$791$	− 14.0194i	− 0.498472i
$792$	−4.49396	−0.159686
$793$	0	0
$794$	−20.7439	−0.736174
$795$	4.93362i	0.174978i
$796$	−21.1293	−0.748908
$797$	−36.5676	−1.29529	−0.647646	−	0.761941i	$-0.724246\pi$
$797$	−36.5676	−1.29529	−0.647646	+	0.761941i	$0.724246\pi$
$798$	4.17629i	0.147839i
$799$	12.3720i	0.437689i
$800$	− 1.00000i	− 0.0353553i
$801$	31.8907i	1.12680i
$802$	24.2892	0.857681
$803$	−19.6862	−0.694710
$804$	− 5.51573i	− 0.194525i
$805$	22.0465	0.777038
$806$	0	0
$807$	4.33081	0.152452
$808$	− 4.19806i	− 0.147687i
$809$	−21.9694	−0.772403	−0.386201	−	0.922414i	$-0.626213\pi$
$809$	−21.9694	−0.772403	−0.386201	+	0.922414i	$0.626213\pi$
$810$	7.25667	0.254973
$811$	− 31.3250i	− 1.09997i	−0.835175	−	0.549984i	$-0.814634\pi$
$811$	− 31.3250i	− 1.09997i	0.835175	−	0.549984i	$-0.185366\pi$
$812$	13.1468i	0.461361i
$813$	1.80433i	0.0632806i
$814$	− 5.14483i	− 0.180326i
$815$	−13.5308	−0.473963
$816$	−1.38404	−0.0484512
$817$	14.9879i	0.524361i
$818$	−36.7676	−1.28555
$819$	0	0
$820$	−10.6746	−0.372772
$821$	13.9734i	0.487677i	0.969816	+	0.243838i	$0.0784066\pi$
$821$	13.9734i	0.487677i	−0.969816	+	0.243838i	$0.921593\pi$
$822$	7.50125	0.261636
$823$	−32.0871	−1.11849	−0.559243	−	0.829004i	$-0.688908\pi$
$823$	−32.0871	−1.11849	−0.559243	+	0.829004i	$0.688908\pi$
$824$	11.5211i	0.401357i
$825$	− 0.713792i	− 0.0248510i
$826$	− 17.1642i	− 0.597219i
$827$	1.82430i	0.0634371i	0.999497	+	0.0317185i	$0.0100980\pi$
$827$	1.82430i	0.0634371i	−0.999497	+	0.0317185i	$0.989902\pi$
$828$	−19.0248	−0.661156
$829$	32.1360	1.11613	0.558065	−	0.829797i	$-0.311544\pi$
$829$	32.1360	1.11613	0.558065	+	0.829797i	$0.311544\pi$
$830$	− 10.4209i	− 0.361714i
$831$	−13.3647	−0.463615
$832$	0	0
$833$	11.0180	0.381753
$834$	2.65950i	0.0920909i
$835$	−15.1685	−0.524928
$836$	4.63533	0.160316
$837$	− 21.4771i	− 0.742357i
$838$	− 30.8418i	− 1.06541i
$839$	− 8.30904i	− 0.286860i	−0.989660	−	0.143430i	$-0.954187\pi$
$839$	− 8.30904i	− 0.286860i	0.989660	−	0.143430i	$-0.0458132\pi$
$840$	1.44504i	0.0498587i
$841$	−12.6063	−0.434699
$842$	−31.7458	−1.09403
$843$	9.90840i	0.341263i
$844$	18.3720	0.632389
$845$	0	0
$846$	−11.1468	−0.383233
$847$	− 27.3642i	− 0.940245i
$848$	−11.0858	−0.380686
$849$	3.95407	0.135703
$850$	3.10992i	0.106669i
$851$	− 21.7802i	− 0.746615i
$852$	− 3.34050i	− 0.114444i
$853$	31.7995i	1.08880i	0.838827	+	0.544398i	$0.183242\pi$
$853$	31.7995i	1.08880i	−0.838827	+	0.544398i	$0.816758\pi$
$854$	−42.6165	−1.45831
$855$	−8.09783	−0.276940
$856$	4.43967i	0.151745i
$857$	9.32975	0.318698	0.159349	−	0.987222i	$-0.449060\pi$
$857$	9.32975	0.318698	0.159349	+	0.987222i	$0.449060\pi$
$858$	0	0
$859$	−58.3913	−1.99229	−0.996143	−	0.0877403i	$-0.972035\pi$
$859$	−58.3913	−1.99229	−0.996143	+	0.0877403i	$0.972035\pi$
$860$	5.18598i	0.176840i
$861$	15.4252	0.525689
$862$	25.5120	0.868942
$863$	− 8.79284i	− 0.299312i	−0.988738	−	0.149656i	$-0.952183\pi$
$863$	− 8.79284i	− 0.299312i	0.988738	−	0.149656i	$-0.0478166\pi$
$864$	− 2.58211i	− 0.0878450i
$865$	4.27413i	0.145325i
$866$	31.8189i	1.08125i
$867$	−3.26145	−0.110765
$868$	−27.0073	−0.916687
$869$	− 18.1328i	− 0.615111i
$870$	1.80194	0.0610914
$871$	0	0
$872$	−10.5526	−0.357355
$873$	16.9638i	0.574136i
$874$	19.6233	0.663766
$875$	3.24698	0.109768
$876$	− 5.46250i	− 0.184561i
$877$	33.6668i	1.13685i	0.822736	+	0.568423i	$0.192446\pi$
$877$	33.6668i	1.13685i	−0.822736	+	0.568423i	$0.807554\pi$
$878$	− 26.9095i	− 0.908150i
$879$	9.58450i	0.323277i
$880$	1.60388	0.0540666
$881$	−24.8068	−0.835764	−0.417882	−	0.908501i	$-0.637227\pi$
$881$	−24.8068	−0.835764	−0.417882	+	0.908501i	$0.637227\pi$
$882$	9.92692i	0.334257i
$883$	−15.5120	−0.522021	−0.261010	−	0.965336i	$-0.584056\pi$
$883$	−15.5120	−0.522021	−0.261010	+	0.965336i	$0.584056\pi$
$884$	0	0
$885$	−2.35258	−0.0790812
$886$	12.7385i	0.427960i
$887$	−12.9487	−0.434774	−0.217387	−	0.976085i	$-0.569753\pi$
$887$	−12.9487	−0.434774	−0.217387	+	0.976085i	$0.569753\pi$
$888$	1.42758	0.0479066
$889$	− 9.11529i	− 0.305717i
$890$	− 11.3817i	− 0.381514i
$891$	11.6388i	0.389914i
$892$	14.2687i	0.477753i
$893$	11.4974	0.384746
$894$	6.63773	0.221999
$895$	15.6233i	0.522228i
$896$	−3.24698	−0.108474
$897$	0	0
$898$	−7.75063	−0.258642
$899$	33.6775i	1.12321i
$900$	−2.80194	−0.0933979
$901$	34.4758	1.14855
$902$	− 17.1207i	− 0.570056i
$903$	− 7.49396i	− 0.249383i
$904$	− 4.31767i	− 0.143603i
$905$	16.1933i	0.538283i
$906$	−4.25475	−0.141355
$907$	26.4983	0.879861	0.439930	−	0.898032i	$-0.355003\pi$
$907$	26.4983	0.879861	0.439930	+	0.898032i	$0.355003\pi$
$908$	27.4959i	0.912483i
$909$	−11.7627	−0.390144
$910$	0	0
$911$	−15.5797	−0.516179	−0.258089	−	0.966121i	$-0.583093\pi$
$911$	−15.5797	−0.516179	−0.258089	+	0.966121i	$0.583093\pi$
$912$	1.28621i	0.0425906i
$913$	16.7138	0.553146
$914$	6.94438	0.229700
$915$	5.84117i	0.193103i
$916$	− 19.6963i	− 0.650785i
$917$	− 7.70171i	− 0.254333i
$918$	8.03013i	0.265034i
$919$	−44.5870	−1.47079	−0.735395	−	0.677639i	$-0.763003\pi$
$919$	−44.5870	−1.47079	−0.735395	+	0.677639i	$0.763003\pi$
$920$	6.78986	0.223855
$921$	− 4.59419i	− 0.151384i
$922$	−12.6377	−0.416201
$923$	0	0
$924$	−2.31767	−0.0762457
$925$	− 3.20775i	− 0.105470i
$926$	5.62325	0.184792
$927$	32.2814	1.06026
$928$	4.04892i	0.132912i
$929$	− 7.09890i	− 0.232907i	−0.993196	−	0.116454i	$-0.962847\pi$
$929$	− 7.09890i	− 0.232907i	0.993196	−	0.116454i	$-0.0371527\pi$
$930$	3.70171i	0.121384i
$931$	− 10.2392i	− 0.335577i
$932$	−23.5362	−0.770953
$933$	−3.33837	−0.109293
$934$	35.2054i	1.15195i
$935$	−4.98792	−0.163122
$936$	0	0
$937$	−11.3297	−0.370127	−0.185063	−	0.982727i	$-0.559249\pi$
$937$	−11.3297	−0.370127	−0.185063	+	0.982727i	$0.559249\pi$
$938$	40.2422i	1.31395i
$939$	2.49396	0.0813873
$940$	3.97823	0.129756
$941$	− 49.8689i	− 1.62568i	−0.582487	−	0.812840i	$-0.697920\pi$
$941$	− 49.8689i	− 1.62568i	0.582487	−	0.812840i	$-0.302080\pi$
$942$	− 6.12200i	− 0.199465i
$943$	− 72.4787i	− 2.36023i
$944$	− 5.28621i	− 0.172051i
$945$	8.38404	0.272733
$946$	−8.31767	−0.270431
$947$	− 22.2457i	− 0.722887i	−0.932394	−	0.361443i	$-0.882284\pi$
$947$	− 22.2457i	− 0.722887i	0.932394	−	0.361443i	$-0.117716\pi$
$948$	5.03146	0.163414
$949$	0	0
$950$	2.89008	0.0937667
$951$	5.24267i	0.170005i
$952$	10.0978	0.327273
$953$	−35.3900	−1.14639	−0.573197	−	0.819417i	$-0.694297\pi$
$953$	−35.3900	−1.14639	−0.573197	+	0.819417i	$0.694297\pi$
$954$	31.0616i	1.00566i
$955$	26.5676i	0.859708i
$956$	− 14.4155i	− 0.466231i
$957$	2.89008i	0.0934231i
$958$	−7.16421	−0.231465
$959$	−54.7284	−1.76727
$960$	0.445042i	0.0143637i
$961$	−38.1836	−1.23173
$962$	0	0
$963$	12.4397	0.400863
$964$	− 15.6692i	− 0.504671i
$965$	13.6582	0.439672
$966$	−9.81163	−0.315684
$967$	51.0723i	1.64238i	0.570658	+	0.821188i	$0.306688\pi$
$967$	51.0723i	1.64238i	−0.570658	+	0.821188i	$0.693312\pi$
$968$	− 8.42758i	− 0.270873i
$969$	− 4.00000i	− 0.128499i
$970$	− 6.05429i	− 0.194392i
$971$	37.8840	1.21575	0.607877	−	0.794031i	$-0.292021\pi$
$971$	37.8840	1.21575	0.607877	+	0.794031i	$0.292021\pi$
$972$	−10.9758	−0.352050
$973$	− 19.4034i	− 0.622045i
$974$	28.7006	0.919628
$975$	0	0
$976$	−13.1250	−0.420120
$977$	31.3685i	1.00357i	0.864993	+	0.501784i	$0.167323\pi$
$977$	31.3685i	1.00357i	−0.864993	+	0.501784i	$0.832677\pi$
$978$	6.02177	0.192555
$979$	18.2547	0.583424
$980$	− 3.54288i	− 0.113173i
$981$	29.5676i	0.944022i
$982$	42.0689i	1.34247i
$983$	− 39.3653i	− 1.25556i	−0.778392	−	0.627778i	$-0.783964\pi$
$983$	− 39.3653i	− 1.25556i	0.778392	−	0.627778i	$-0.216036\pi$
$984$	4.75063	0.151444
$985$	14.0978	0.449194
$986$	− 12.5918i	− 0.401004i
$987$	−5.74871	−0.182983
$988$	0	0
$989$	−35.2121	−1.11968
$990$	− 4.49396i	− 0.142827i
$991$	43.2922	1.37522	0.687611	−	0.726080i	$-0.258660\pi$
$991$	43.2922	1.37522	0.687611	+	0.726080i	$0.258660\pi$
$992$	−8.31767	−0.264086
$993$	2.87933i	0.0913728i
$994$	24.3720i	0.773032i
$995$	− 21.1293i	− 0.669844i
$996$	4.63773i	0.146952i
$997$	−4.64609	−0.147143	−0.0735715	−	0.997290i	$-0.523440\pi$
$997$	−4.64609	−0.147143	−0.0735715	+	0.997290i	$0.523440\pi$
$998$	33.2379	1.05213
$999$	− 8.28275i	− 0.262055i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.j.1351.5		6
13.2	odd	12		1690.2.e.o.191.2		6
13.3	even	3		1690.2.l.l.1161.2		12
13.4	even	6		1690.2.l.l.361.2		12
13.5	odd	4		1690.2.a.s.1.2	yes	3
13.6	odd	12		1690.2.e.o.991.2		6
13.7	odd	12		1690.2.e.q.991.2		6
13.8	odd	4		1690.2.a.q.1.2	✓	3
13.9	even	3		1690.2.l.l.361.5		12
13.10	even	6		1690.2.l.l.1161.5		12
13.11	odd	12		1690.2.e.q.191.2		6
13.12	even	2	inner	1690.2.d.j.1351.2		6
65.34	odd	4		8450.2.a.bz.1.2		3
65.44	odd	4		8450.2.a.bo.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.q.1.2	✓	3	13.8	odd	4
1690.2.a.s.1.2	yes	3	13.5	odd	4
1690.2.d.j.1351.2		6	13.12	even	2	inner
1690.2.d.j.1351.5		6	1.1	even	1	trivial
1690.2.e.o.191.2		6	13.2	odd	12
1690.2.e.o.991.2		6	13.6	odd	12
1690.2.e.q.191.2		6	13.11	odd	12
1690.2.e.q.991.2		6	13.7	odd	12
1690.2.l.l.361.2		12	13.4	even	6
1690.2.l.l.361.5		12	13.9	even	3
1690.2.l.l.1161.2		12	13.3	even	3
1690.2.l.l.1161.5		12	13.10	even	6
8450.2.a.bo.1.2		3	65.44	odd	4
8450.2.a.bz.1.2		3	65.34	odd	4

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:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +0.445042 q^{3} -1.00000 q^{4} -1.00000i q^{5} +0.445042i q^{6} -3.24698i q^{7} -1.00000i q^{8} -2.80194 q^{9} +1.00000 q^{10} +1.60388i q^{11} -0.445042 q^{12} +3.24698 q^{14} -0.445042i q^{15} +1.00000 q^{16} -3.10992 q^{17} -2.80194i q^{18} +2.89008i q^{19} +1.00000i q^{20} -1.44504i q^{21} -1.60388 q^{22} -6.78986 q^{23} -0.445042i q^{24} -1.00000 q^{25} -2.58211 q^{27} +3.24698i q^{28} +4.04892 q^{29} +0.445042 q^{30} +8.31767i q^{31} +1.00000i q^{32} +0.713792i q^{33} -3.10992i q^{34} -3.24698 q^{35} +2.80194 q^{36} +3.20775i q^{37} -2.89008 q^{38} -1.00000 q^{40} +10.6746i q^{41} +1.44504 q^{42} +5.18598 q^{43} -1.60388i q^{44} +2.80194i q^{45} -6.78986i q^{46} -3.97823i q^{47} +0.445042 q^{48} -3.54288 q^{49} -1.00000i q^{50} -1.38404 q^{51} -11.0858 q^{53} -2.58211i q^{54} +1.60388 q^{55} -3.24698 q^{56} +1.28621i q^{57} +4.04892i q^{58} -5.28621i q^{59} +0.445042i q^{60} -13.1250 q^{61} -8.31767 q^{62} +9.09783i q^{63} -1.00000 q^{64} -0.713792 q^{66} +12.3937i q^{67} +3.10992 q^{68} -3.02177 q^{69} -3.24698i q^{70} +7.50604i q^{71} +2.80194i q^{72} +12.2741i q^{73} -3.20775 q^{74} -0.445042 q^{75} -2.89008i q^{76} +5.20775 q^{77} -11.3056 q^{79} -1.00000i q^{80} +7.25667 q^{81} -10.6746 q^{82} -10.4209i q^{83} +1.44504i q^{84} +3.10992i q^{85} +5.18598i q^{86} +1.80194 q^{87} +1.60388 q^{88} -11.3817i q^{89} -2.80194 q^{90} +6.78986 q^{92} +3.70171i q^{93} +3.97823 q^{94} +2.89008 q^{95} +0.445042i q^{96} -6.05429i q^{97} -3.54288i q^{98} -4.49396i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 6 q^{4} - 8 q^{9} + 6 q^{10} - 2 q^{12} + 10 q^{14} + 6 q^{16} - 20 q^{17} + 8 q^{22} + 6 q^{23} - 6 q^{25} - 4 q^{27} + 6 q^{29} + 2 q^{30} - 10 q^{35} + 8 q^{36} - 16 q^{38} - 6 q^{40}+ \cdots + 16 q^{95}+O(q^{100}) \) 6 * q + 2 * q^3 - 6 * q^4 - 8 * q^9 + 6 * q^10 - 2 * q^12 + 10 * q^14 + 6 * q^16 - 20 * q^17 + 8 * q^22 + 6 * q^23 - 6 * q^25 - 4 * q^27 + 6 * q^29 + 2 * q^30 - 10 * q^35 + 8 * q^36 - 16 * q^38 - 6 * q^40 + 8 * q^42 + 2 * q^43 + 2 * q^48 + 16 * q^49 + 12 * q^51 + 8 * q^53 - 8 * q^55 - 10 * q^56 - 30 * q^61 - 16 * q^62 - 6 * q^64 + 12 * q^66 + 20 * q^68 - 12 * q^69 + 16 * q^74 - 2 * q^75 - 4 * q^77 + 4 * q^79 - 10 * q^81 - 22 * q^82 + 2 * q^87 - 8 * q^88 - 8 * q^90 - 6 * q^92 + 30 * q^94 + 16 * q^95

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