LMFDB - Embedded newform 8038.2.a.d.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.78655 q^{3} +1.00000 q^{4} +0.381834 q^{5} -1.78655 q^{6} +1.78538 q^{7} +1.00000 q^{8} +0.191778 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.78655 q^{3} +1.00000 q^{4} +0.381834 q^{5} -1.78655 q^{6} +1.78538 q^{7} +1.00000 q^{8} +0.191778 q^{9} +0.381834 q^{10} +0.177528 q^{11} -1.78655 q^{12} +6.05294 q^{13} +1.78538 q^{14} -0.682168 q^{15} +1.00000 q^{16} +8.00871 q^{17} +0.191778 q^{18} +7.58161 q^{19} +0.381834 q^{20} -3.18968 q^{21} +0.177528 q^{22} -0.594083 q^{23} -1.78655 q^{24} -4.85420 q^{25} +6.05294 q^{26} +5.01704 q^{27} +1.78538 q^{28} +1.81788 q^{29} -0.682168 q^{30} -2.47751 q^{31} +1.00000 q^{32} -0.317164 q^{33} +8.00871 q^{34} +0.681718 q^{35} +0.191778 q^{36} +6.87222 q^{37} +7.58161 q^{38} -10.8139 q^{39} +0.381834 q^{40} -7.34339 q^{41} -3.18968 q^{42} +11.2304 q^{43} +0.177528 q^{44} +0.0732273 q^{45} -0.594083 q^{46} -6.76844 q^{47} -1.78655 q^{48} -3.81243 q^{49} -4.85420 q^{50} -14.3080 q^{51} +6.05294 q^{52} +8.38571 q^{53} +5.01704 q^{54} +0.0677864 q^{55} +1.78538 q^{56} -13.5450 q^{57} +1.81788 q^{58} +14.8791 q^{59} -0.682168 q^{60} -10.5692 q^{61} -2.47751 q^{62} +0.342395 q^{63} +1.00000 q^{64} +2.31122 q^{65} -0.317164 q^{66} -14.7610 q^{67} +8.00871 q^{68} +1.06136 q^{69} +0.681718 q^{70} -5.43460 q^{71} +0.191778 q^{72} -0.343317 q^{73} +6.87222 q^{74} +8.67230 q^{75} +7.58161 q^{76} +0.316955 q^{77} -10.8139 q^{78} +8.73578 q^{79} +0.381834 q^{80} -9.53855 q^{81} -7.34339 q^{82} -6.08214 q^{83} -3.18968 q^{84} +3.05800 q^{85} +11.2304 q^{86} -3.24774 q^{87} +0.177528 q^{88} -7.71869 q^{89} +0.0732273 q^{90} +10.8068 q^{91} -0.594083 q^{92} +4.42621 q^{93} -6.76844 q^{94} +2.89492 q^{95} -1.78655 q^{96} -8.61967 q^{97} -3.81243 q^{98} +0.0340459 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) $ 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.78655	−1.03147	−0.515734	−	0.856749i	$-0.672481\pi$
$3$	−1.78655	−1.03147	−0.515734	+	0.856749i	$0.672481\pi$
$4$	1.00000	0.500000
$5$	0.381834	0.170761	0.0853807	−	0.996348i	$-0.472789\pi$
$5$	0.381834	0.170761	0.0853807	+	0.996348i	$0.472789\pi$
$6$	−1.78655	−0.729358
$7$	1.78538	0.674809	0.337405	−	0.941360i	$-0.390451\pi$
$7$	1.78538	0.674809	0.337405	+	0.941360i	$0.390451\pi$
$8$	1.00000	0.353553
$9$	0.191778	0.0639259
$10$	0.381834	0.120747
$11$	0.177528	0.0535268	0.0267634	−	0.999642i	$-0.491480\pi$
$11$	0.177528	0.0535268	0.0267634	+	0.999642i	$0.491480\pi$
$12$	−1.78655	−0.515734
$13$	6.05294	1.67878	0.839392	−	0.543527i	$-0.182911\pi$
$13$	6.05294	1.67878	0.839392	+	0.543527i	$0.182911\pi$
$14$	1.78538	0.477162
$15$	−0.682168	−0.176135
$16$	1.00000	0.250000
$17$	8.00871	1.94240	0.971199	−	0.238271i	$-0.0765806\pi$
$17$	8.00871	1.94240	0.971199	+	0.238271i	$0.0765806\pi$
$18$	0.191778	0.0452024
$19$	7.58161	1.73934	0.869670	−	0.493634i	$-0.164332\pi$
$19$	7.58161	1.73934	0.869670	+	0.493634i	$0.164332\pi$
$20$	0.381834	0.0853807
$21$	−3.18968	−0.696044
$22$	0.177528	0.0378491
$23$	−0.594083	−0.123875	−0.0619375	−	0.998080i	$-0.519728\pi$
$23$	−0.594083	−0.123875	−0.0619375	+	0.998080i	$0.519728\pi$
$24$	−1.78655	−0.364679
$25$	−4.85420	−0.970841
$26$	6.05294	1.18708
$27$	5.01704	0.965530
$28$	1.78538	0.337405
$29$	1.81788	0.337572	0.168786	−	0.985653i	$-0.446015\pi$
$29$	1.81788	0.337572	0.168786	+	0.985653i	$0.446015\pi$
$30$	−0.682168	−0.124546
$31$	−2.47751	−0.444974	−0.222487	−	0.974936i	$-0.571417\pi$
$31$	−2.47751	−0.444974	−0.222487	+	0.974936i	$0.571417\pi$
$32$	1.00000	0.176777
$33$	−0.317164	−0.0552112
$34$	8.00871	1.37348
$35$	0.681718	0.115231
$36$	0.191778	0.0319629
$37$	6.87222	1.12979	0.564893	−	0.825164i	$-0.308918\pi$
$37$	6.87222	1.12979	0.564893	+	0.825164i	$0.308918\pi$
$38$	7.58161	1.22990
$39$	−10.8139	−1.73161
$40$	0.381834	0.0603733
$41$	−7.34339	−1.14684	−0.573422	−	0.819260i	$-0.694384\pi$
$41$	−7.34339	−1.14684	−0.573422	+	0.819260i	$0.694384\pi$
$42$	−3.18968	−0.492178
$43$	11.2304	1.71261	0.856307	−	0.516466i	$-0.172753\pi$
$43$	11.2304	1.71261	0.856307	+	0.516466i	$0.172753\pi$
$44$	0.177528	0.0267634
$45$	0.0732273	0.0109161
$46$	−0.594083	−0.0875928
$47$	−6.76844	−0.987279	−0.493639	−	0.869667i	$-0.664334\pi$
$47$	−6.76844	−0.987279	−0.493639	+	0.869667i	$0.664334\pi$
$48$	−1.78655	−0.257867
$49$	−3.81243	−0.544632
$50$	−4.85420	−0.686488
$51$	−14.3080	−2.00352
$52$	6.05294	0.839392
$53$	8.38571	1.15187	0.575933	−	0.817497i	$-0.304639\pi$
$53$	8.38571	1.15187	0.575933	+	0.817497i	$0.304639\pi$
$54$	5.01704	0.682733
$55$	0.0677864	0.00914031
$56$	1.78538	0.238581
$57$	−13.5450	−1.79407
$58$	1.81788	0.238700
$59$	14.8791	1.93709	0.968547	−	0.248829i	$-0.0800459\pi$
$59$	14.8791	1.93709	0.968547	+	0.248829i	$0.0800459\pi$
$60$	−0.682168	−0.0880675
$61$	−10.5692	−1.35325	−0.676625	−	0.736327i	$-0.736558\pi$
$61$	−10.5692	−1.35325	−0.676625	+	0.736327i	$0.736558\pi$
$62$	−2.47751	−0.314644
$63$	0.342395	0.0431378
$64$	1.00000	0.125000
$65$	2.31122	0.286672
$66$	−0.317164	−0.0390402
$67$	−14.7610	−1.80334	−0.901670	−	0.432424i	$-0.857658\pi$
$67$	−14.7610	−1.80334	−0.901670	+	0.432424i	$0.857658\pi$
$68$	8.00871	0.971199
$69$	1.06136	0.127773
$70$	0.681718	0.0814809
$71$	−5.43460	−0.644969	−0.322484	−	0.946575i	$-0.604518\pi$
$71$	−5.43460	−0.644969	−0.322484	+	0.946575i	$0.604518\pi$
$72$	0.191778	0.0226012
$73$	−0.343317	−0.0401822	−0.0200911	−	0.999798i	$-0.506396\pi$
$73$	−0.343317	−0.0401822	−0.0200911	+	0.999798i	$0.506396\pi$
$74$	6.87222	0.798879
$75$	8.67230	1.00139
$76$	7.58161	0.869670
$77$	0.316955	0.0361204
$78$	−10.8139	−1.22443
$79$	8.73578	0.982852	0.491426	−	0.870919i	$-0.336476\pi$
$79$	8.73578	0.982852	0.491426	+	0.870919i	$0.336476\pi$
$80$	0.381834	0.0426904
$81$	−9.53855	−1.05984
$82$	−7.34339	−0.810941
$83$	−6.08214	−0.667602	−0.333801	−	0.942644i	$-0.608331\pi$
$83$	−6.08214	−0.667602	−0.333801	+	0.942644i	$0.608331\pi$
$84$	−3.18968	−0.348022
$85$	3.05800	0.331687
$86$	11.2304	1.21100
$87$	−3.24774	−0.348195
$88$	0.177528	0.0189246
$89$	−7.71869	−0.818180	−0.409090	−	0.912494i	$-0.634154\pi$
$89$	−7.71869	−0.818180	−0.409090	+	0.912494i	$0.634154\pi$
$90$	0.0732273	0.00771883
$91$	10.8068	1.13286
$92$	−0.594083	−0.0619375
$93$	4.42621	0.458976
$94$	−6.76844	−0.698112
$95$	2.89492	0.297012
$96$	−1.78655	−0.182339
$97$	−8.61967	−0.875195	−0.437598	−	0.899171i	$-0.644171\pi$
$97$	−8.61967	−0.875195	−0.437598	+	0.899171i	$0.644171\pi$
$98$	−3.81243	−0.385113
$99$	0.0340459	0.00342175
$100$	−4.85420	−0.485420
$101$	1.91610	0.190659	0.0953293	−	0.995446i	$-0.469610\pi$
$101$	1.91610	0.190659	0.0953293	+	0.995446i	$0.469610\pi$
$102$	−14.3080	−1.41670
$103$	−12.5056	−1.23221	−0.616107	−	0.787662i	$-0.711291\pi$
$103$	−12.5056	−1.23221	−0.616107	+	0.787662i	$0.711291\pi$
$104$	6.05294	0.593540
$105$	−1.21793	−0.118858
$106$	8.38571	0.814492
$107$	−6.14692	−0.594246	−0.297123	−	0.954839i	$-0.596027\pi$
$107$	−6.14692	−0.594246	−0.297123	+	0.954839i	$0.596027\pi$
$108$	5.01704	0.482765
$109$	11.9529	1.14488	0.572439	−	0.819948i	$-0.305997\pi$
$109$	11.9529	1.14488	0.572439	+	0.819948i	$0.305997\pi$
$110$	0.0677864	0.00646318
$111$	−12.2776	−1.16534
$112$	1.78538	0.168702
$113$	5.07011	0.476956	0.238478	−	0.971148i	$-0.423352\pi$
$113$	5.07011	0.476956	0.238478	+	0.971148i	$0.423352\pi$
$114$	−13.5450	−1.26860
$115$	−0.226841	−0.0211531
$116$	1.81788	0.168786
$117$	1.16082	0.107318
$118$	14.8791	1.36973
$119$	14.2986	1.31075
$120$	−0.682168	−0.0622731
$121$	−10.9685	−0.997135
$122$	−10.5692	−0.956893
$123$	13.1194	1.18293
$124$	−2.47751	−0.222487
$125$	−3.76267	−0.336544
$126$	0.342395	0.0305030
$127$	−13.2017	−1.17147	−0.585733	−	0.810504i	$-0.699193\pi$
$127$	−13.2017	−1.17147	−0.585733	+	0.810504i	$0.699193\pi$
$128$	1.00000	0.0883883
$129$	−20.0637	−1.76651
$130$	2.31122	0.202707
$131$	16.8969	1.47629	0.738146	−	0.674641i	$-0.235702\pi$
$131$	16.8969	1.47629	0.738146	+	0.674641i	$0.235702\pi$
$132$	−0.317164	−0.0276056
$133$	13.5360	1.17372
$134$	−14.7610	−1.27515
$135$	1.91568	0.164875
$136$	8.00871	0.686741
$137$	−10.5098	−0.897916	−0.448958	−	0.893553i	$-0.648205\pi$
$137$	−10.5098	−0.897916	−0.448958	+	0.893553i	$0.648205\pi$
$138$	1.06136	0.0903492
$139$	−4.01972	−0.340949	−0.170474	−	0.985362i	$-0.554530\pi$
$139$	−4.01972	−0.340949	−0.170474	+	0.985362i	$0.554530\pi$
$140$	0.681718	0.0576157
$141$	12.0922	1.01835
$142$	−5.43460	−0.456062
$143$	1.07457	0.0898599
$144$	0.191778	0.0159815
$145$	0.694129	0.0576443
$146$	−0.343317	−0.0284131
$147$	6.81111	0.561771
$148$	6.87222	0.564893
$149$	2.63776	0.216093	0.108047	−	0.994146i	$-0.465540\pi$
$149$	2.63776	0.216093	0.108047	+	0.994146i	$0.465540\pi$
$150$	8.67230	0.708090
$151$	20.2696	1.64952	0.824760	−	0.565482i	$-0.191310\pi$
$151$	20.2696	1.64952	0.824760	+	0.565482i	$0.191310\pi$
$152$	7.58161	0.614950
$153$	1.53589	0.124169
$154$	0.316955	0.0255410
$155$	−0.945998	−0.0759844
$156$	−10.8139	−0.865806
$157$	5.03885	0.402144	0.201072	−	0.979577i	$-0.435558\pi$
$157$	5.03885	0.402144	0.201072	+	0.979577i	$0.435558\pi$
$158$	8.73578	0.694981
$159$	−14.9815	−1.18811
$160$	0.381834	0.0301866
$161$	−1.06066	−0.0835920
$162$	−9.53855	−0.749420
$163$	15.2242	1.19245	0.596224	−	0.802818i	$-0.296667\pi$
$163$	15.2242	1.19245	0.596224	+	0.802818i	$0.296667\pi$
$164$	−7.34339	−0.573422
$165$	−0.121104	−0.00942794
$166$	−6.08214	−0.472066
$167$	8.90923	0.689417	0.344709	−	0.938710i	$-0.387978\pi$
$167$	8.90923	0.689417	0.344709	+	0.938710i	$0.387978\pi$
$168$	−3.18968	−0.246089
$169$	23.6381	1.81832
$170$	3.05800	0.234538
$171$	1.45398	0.111189
$172$	11.2304	0.856307
$173$	11.6538	0.886019	0.443010	−	0.896517i	$-0.353911\pi$
$173$	11.6538	0.886019	0.443010	+	0.896517i	$0.353911\pi$
$174$	−3.24774	−0.246211
$175$	−8.66659	−0.655132
$176$	0.177528	0.0133817
$177$	−26.5823	−1.99805
$178$	−7.71869	−0.578540
$179$	6.21953	0.464869	0.232435	−	0.972612i	$-0.425331\pi$
$179$	6.21953	0.464869	0.232435	+	0.972612i	$0.425331\pi$
$180$	0.0732273	0.00545804
$181$	0.450166	0.0334606	0.0167303	−	0.999860i	$-0.494674\pi$
$181$	0.450166	0.0334606	0.0167303	+	0.999860i	$0.494674\pi$
$182$	10.8068	0.801052
$183$	18.8825	1.39583
$184$	−0.594083	−0.0437964
$185$	2.62405	0.192924
$186$	4.42621	0.324545
$187$	1.42177	0.103970
$188$	−6.76844	−0.493639
$189$	8.95732	0.651549
$190$	2.89492	0.210019
$191$	25.4358	1.84047	0.920235	−	0.391366i	$-0.127997\pi$
$191$	25.4358	1.84047	0.920235	+	0.391366i	$0.127997\pi$
$192$	−1.78655	−0.128933
$193$	−22.7238	−1.63569	−0.817847	−	0.575436i	$-0.804832\pi$
$193$	−22.7238	−1.63569	−0.817847	+	0.575436i	$0.804832\pi$
$194$	−8.61967	−0.618857
$195$	−4.12912	−0.295693
$196$	−3.81243	−0.272316
$197$	−16.6832	−1.18863	−0.594316	−	0.804232i	$-0.702577\pi$
$197$	−16.6832	−1.18863	−0.594316	+	0.804232i	$0.702577\pi$
$198$	0.0340459	0.00241954
$199$	−21.5626	−1.52853	−0.764267	−	0.644900i	$-0.776899\pi$
$199$	−21.5626	−1.52853	−0.764267	+	0.644900i	$0.776899\pi$
$200$	−4.85420	−0.343244
$201$	26.3713	1.86009
$202$	1.91610	0.134816
$203$	3.24561	0.227797
$204$	−14.3080	−1.00176
$205$	−2.80396	−0.195837
$206$	−12.5056	−0.871307
$207$	−0.113932	−0.00791881
$208$	6.05294	0.419696
$209$	1.34595	0.0931013
$210$	−1.21793	−0.0840450
$211$	12.3845	0.852585	0.426292	−	0.904585i	$-0.359819\pi$
$211$	12.3845	0.852585	0.426292	+	0.904585i	$0.359819\pi$
$212$	8.38571	0.575933
$213$	9.70922	0.665265
$214$	−6.14692	−0.420195
$215$	4.28814	0.292449
$216$	5.01704	0.341367
$217$	−4.42329	−0.300273
$218$	11.9529	0.809551
$219$	0.613355	0.0414467
$220$	0.0677864	0.00457016
$221$	48.4762	3.26087
$222$	−12.2776	−0.824018
$223$	−15.2195	−1.01918	−0.509588	−	0.860419i	$-0.670202\pi$
$223$	−15.2195	−1.01918	−0.509588	+	0.860419i	$0.670202\pi$
$224$	1.78538	0.119291
$225$	−0.930927	−0.0620618
$226$	5.07011	0.337259
$227$	−24.8597	−1.65000	−0.824999	−	0.565135i	$-0.808824\pi$
$227$	−24.8597	−1.65000	−0.824999	+	0.565135i	$0.808824\pi$
$228$	−13.5450	−0.897037
$229$	−20.7757	−1.37290	−0.686449	−	0.727178i	$-0.740832\pi$
$229$	−20.7757	−1.37290	−0.686449	+	0.727178i	$0.740832\pi$
$230$	−0.226841	−0.0149575
$231$	−0.566257	−0.0372570
$232$	1.81788	0.119350
$233$	3.18832	0.208874	0.104437	−	0.994532i	$-0.466696\pi$
$233$	3.18832	0.208874	0.104437	+	0.994532i	$0.466696\pi$
$234$	1.16082	0.0758851
$235$	−2.58442	−0.168589
$236$	14.8791	0.968547
$237$	−15.6069	−1.01378
$238$	14.2986	0.926839
$239$	13.3018	0.860420	0.430210	−	0.902729i	$-0.358439\pi$
$239$	13.3018	0.860420	0.430210	+	0.902729i	$0.358439\pi$
$240$	−0.682168	−0.0440337
$241$	−4.41873	−0.284635	−0.142318	−	0.989821i	$-0.545455\pi$
$241$	−4.41873	−0.284635	−0.142318	+	0.989821i	$0.545455\pi$
$242$	−10.9685	−0.705081
$243$	1.99002	0.127660
$244$	−10.5692	−0.676625
$245$	−1.45571	−0.0930022
$246$	13.1194	0.836460
$247$	45.8910	2.91998
$248$	−2.47751	−0.157322
$249$	10.8661	0.688610
$250$	−3.76267	−0.237972
$251$	18.3763	1.15990	0.579950	−	0.814652i	$-0.303072\pi$
$251$	18.3763	1.15990	0.579950	+	0.814652i	$0.303072\pi$
$252$	0.342395	0.0215689
$253$	−0.105467	−0.00663063
$254$	−13.2017	−0.828351
$255$	−5.46328	−0.342124
$256$	1.00000	0.0625000
$257$	9.13578	0.569875	0.284937	−	0.958546i	$-0.408027\pi$
$257$	9.13578	0.569875	0.284937	+	0.958546i	$0.408027\pi$
$258$	−20.0637	−1.24911
$259$	12.2695	0.762390
$260$	2.31122	0.143336
$261$	0.348629	0.0215796
$262$	16.8969	1.04390
$263$	−2.75499	−0.169880	−0.0849399	−	0.996386i	$-0.527070\pi$
$263$	−2.75499	−0.169880	−0.0849399	+	0.996386i	$0.527070\pi$
$264$	−0.317164	−0.0195201
$265$	3.20195	0.196694
$266$	13.5360	0.829947
$267$	13.7899	0.843926
$268$	−14.7610	−0.901670
$269$	13.0334	0.794660	0.397330	−	0.917676i	$-0.369937\pi$
$269$	13.0334	0.794660	0.397330	+	0.917676i	$0.369937\pi$
$270$	1.91568	0.116585
$271$	−12.2055	−0.741429	−0.370715	−	0.928747i	$-0.620887\pi$
$271$	−12.2055	−0.741429	−0.370715	+	0.928747i	$0.620887\pi$
$272$	8.00871	0.485599
$273$	−19.3069	−1.16851
$274$	−10.5098	−0.634923
$275$	−0.861758	−0.0519660
$276$	1.06136	0.0638865
$277$	−12.5074	−0.751498	−0.375749	−	0.926721i	$-0.622615\pi$
$277$	−12.5074	−0.751498	−0.375749	+	0.926721i	$0.622615\pi$
$278$	−4.01972	−0.241087
$279$	−0.475131	−0.0284454
$280$	0.681718	0.0407405
$281$	−23.5779	−1.40654	−0.703271	−	0.710922i	$-0.748278\pi$
$281$	−23.5779	−1.40654	−0.703271	+	0.710922i	$0.748278\pi$
$282$	12.0922	0.720080
$283$	0.480125	0.0285405	0.0142702	−	0.999898i	$-0.495457\pi$
$283$	0.480125	0.0285405	0.0142702	+	0.999898i	$0.495457\pi$
$284$	−5.43460	−0.322484
$285$	−5.17193	−0.306359
$286$	1.07457	0.0635405
$287$	−13.1107	−0.773901
$288$	0.191778	0.0113006
$289$	47.1394	2.77291
$290$	0.694129	0.0407607
$291$	15.3995	0.902736
$292$	−0.343317	−0.0200911
$293$	−18.4877	−1.08006	−0.540031	−	0.841645i	$-0.681587\pi$
$293$	−18.4877	−1.08006	−0.540031	+	0.841645i	$0.681587\pi$
$294$	6.81111	0.397232
$295$	5.68135	0.330781
$296$	6.87222	0.399439
$297$	0.890667	0.0516817
$298$	2.63776	0.152801
$299$	−3.59595	−0.207959
$300$	8.67230	0.500695
$301$	20.0504	1.15569
$302$	20.2696	1.16639
$303$	−3.42321	−0.196658
$304$	7.58161	0.434835
$305$	−4.03569	−0.231083
$306$	1.53589	0.0878011
$307$	8.44770	0.482136	0.241068	−	0.970508i	$-0.422502\pi$
$307$	8.44770	0.482136	0.241068	+	0.970508i	$0.422502\pi$
$308$	0.316955	0.0180602
$309$	22.3420	1.27099
$310$	−0.945998	−0.0537291
$311$	−6.78631	−0.384816	−0.192408	−	0.981315i	$-0.561630\pi$
$311$	−6.78631	−0.384816	−0.192408	+	0.981315i	$0.561630\pi$
$312$	−10.8139	−0.612217
$313$	13.4593	0.760764	0.380382	−	0.924829i	$-0.375792\pi$
$313$	13.4593	0.760764	0.380382	+	0.924829i	$0.375792\pi$
$314$	5.03885	0.284358
$315$	0.130738	0.00736627
$316$	8.73578	0.491426
$317$	8.34851	0.468899	0.234449	−	0.972128i	$-0.424671\pi$
$317$	8.34851	0.468899	0.234449	+	0.972128i	$0.424671\pi$
$318$	−14.9815	−0.840122
$319$	0.322725	0.0180691
$320$	0.381834	0.0213452
$321$	10.9818	0.612945
$322$	−1.06066	−0.0591085
$323$	60.7189	3.37849
$324$	−9.53855	−0.529920
$325$	−29.3822	−1.62983
$326$	15.2242	0.843188
$327$	−21.3545	−1.18090
$328$	−7.34339	−0.405471
$329$	−12.0842	−0.666225
$330$	−0.121104	−0.00666656
$331$	−25.9952	−1.42883	−0.714413	−	0.699725i	$-0.753306\pi$
$331$	−25.9952	−1.42883	−0.714413	+	0.699725i	$0.753306\pi$
$332$	−6.08214	−0.333801
$333$	1.31794	0.0722225
$334$	8.90923	0.487491
$335$	−5.63625	−0.307941
$336$	−3.18968	−0.174011
$337$	−2.24485	−0.122285	−0.0611425	−	0.998129i	$-0.519474\pi$
$337$	−2.24485	−0.122285	−0.0611425	+	0.998129i	$0.519474\pi$
$338$	23.6381	1.28574
$339$	−9.05803	−0.491964
$340$	3.05800	0.165843
$341$	−0.439828	−0.0238180
$342$	1.45398	0.0786224
$343$	−19.3043	−1.04233
$344$	11.2304	0.605501
$345$	0.405265	0.0218187
$346$	11.6538	0.626510
$347$	25.6017	1.37437	0.687185	−	0.726482i	$-0.258846\pi$
$347$	25.6017	1.37437	0.687185	+	0.726482i	$0.258846\pi$
$348$	−3.24774	−0.174097
$349$	−30.2702	−1.62033	−0.810163	−	0.586204i	$-0.800622\pi$
$349$	−30.2702	−1.62033	−0.810163	+	0.586204i	$0.800622\pi$
$350$	−8.66659	−0.463248
$351$	30.3679	1.62092
$352$	0.177528	0.00946229
$353$	34.0676	1.81324	0.906619	−	0.421950i	$-0.138654\pi$
$353$	34.0676	1.81324	0.906619	+	0.421950i	$0.138654\pi$
$354$	−26.5823	−1.41284
$355$	−2.07512	−0.110136
$356$	−7.71869	−0.409090
$357$	−25.5452	−1.35199
$358$	6.21953	0.328712
$359$	3.35128	0.176874	0.0884370	−	0.996082i	$-0.471813\pi$
$359$	3.35128	0.176874	0.0884370	+	0.996082i	$0.471813\pi$
$360$	0.0732273	0.00385942
$361$	38.4808	2.02530
$362$	0.450166	0.0236602
$363$	19.5958	1.02851
$364$	10.8068	0.566430
$365$	−0.131090	−0.00686158
$366$	18.8825	0.987004
$367$	21.5647	1.12567	0.562834	−	0.826570i	$-0.309711\pi$
$367$	21.5647	1.12567	0.562834	+	0.826570i	$0.309711\pi$
$368$	−0.594083	−0.0309687
$369$	−1.40830	−0.0733130
$370$	2.62405	0.136418
$371$	14.9717	0.777290
$372$	4.42621	0.229488
$373$	−31.0397	−1.60717	−0.803587	−	0.595187i	$-0.797078\pi$
$373$	−31.0397	−1.60717	−0.803587	+	0.595187i	$0.797078\pi$
$374$	1.42177	0.0735181
$375$	6.72222	0.347134
$376$	−6.76844	−0.349056
$377$	11.0035	0.566711
$378$	8.95732	0.460715
$379$	33.9098	1.74183	0.870915	−	0.491433i	$-0.163527\pi$
$379$	33.9098	1.74183	0.870915	+	0.491433i	$0.163527\pi$
$380$	2.89492	0.148506
$381$	23.5856	1.20833
$382$	25.4358	1.30141
$383$	−12.5799	−0.642805	−0.321403	−	0.946943i	$-0.604154\pi$
$383$	−12.5799	−0.642805	−0.321403	+	0.946943i	$0.604154\pi$
$384$	−1.78655	−0.0911697
$385$	0.121024	0.00616797
$386$	−22.7238	−1.15661
$387$	2.15373	0.109480
$388$	−8.61967	−0.437598
$389$	14.2327	0.721624	0.360812	−	0.932638i	$-0.382500\pi$
$389$	14.2327	0.721624	0.360812	+	0.932638i	$0.382500\pi$
$390$	−4.12912	−0.209086
$391$	−4.75784	−0.240614
$392$	−3.81243	−0.192557
$393$	−30.1873	−1.52275
$394$	−16.6832	−0.840489
$395$	3.33562	0.167833
$396$	0.0340459	0.00171087
$397$	18.2081	0.913836	0.456918	−	0.889509i	$-0.348953\pi$
$397$	18.2081	0.913836	0.456918	+	0.889509i	$0.348953\pi$
$398$	−21.5626	−1.08084
$399$	−24.1829	−1.21066
$400$	−4.85420	−0.242710
$401$	−15.1199	−0.755051	−0.377525	−	0.925999i	$-0.623225\pi$
$401$	−15.1199	−0.755051	−0.377525	+	0.925999i	$0.623225\pi$
$402$	26.3713	1.31528
$403$	−14.9962	−0.747015
$404$	1.91610	0.0953293
$405$	−3.64215	−0.180980
$406$	3.24561	0.161077
$407$	1.22001	0.0604738
$408$	−14.3080	−0.708351
$409$	−29.4635	−1.45688	−0.728439	−	0.685111i	$-0.759754\pi$
$409$	−29.4635	−1.45688	−0.728439	+	0.685111i	$0.759754\pi$
$410$	−2.80396	−0.138478
$411$	18.7764	0.926172
$412$	−12.5056	−0.616107
$413$	26.5648	1.30717
$414$	−0.113932	−0.00559945
$415$	−2.32237	−0.114001
$416$	6.05294	0.296770
$417$	7.18146	0.351677
$418$	1.34595	0.0658325
$419$	−11.9549	−0.584033	−0.292017	−	0.956413i	$-0.594326\pi$
$419$	−11.9549	−0.584033	−0.292017	+	0.956413i	$0.594326\pi$
$420$	−1.21793	−0.0594288
$421$	18.1980	0.886915	0.443458	−	0.896295i	$-0.353752\pi$
$421$	18.1980	0.886915	0.443458	+	0.896295i	$0.353752\pi$
$422$	12.3845	0.602868
$423$	−1.29804	−0.0631127
$424$	8.38571	0.407246
$425$	−38.8759	−1.88576
$426$	9.70922	0.470413
$427$	−18.8701	−0.913186
$428$	−6.14692	−0.297123
$429$	−1.91977	−0.0926876
$430$	4.28814	0.206792
$431$	−24.6175	−1.18579	−0.592893	−	0.805281i	$-0.702014\pi$
$431$	−24.6175	−1.18579	−0.592893	+	0.805281i	$0.702014\pi$
$432$	5.01704	0.241383
$433$	11.7116	0.562825	0.281413	−	0.959587i	$-0.409197\pi$
$433$	11.7116	0.562825	0.281413	+	0.959587i	$0.409197\pi$
$434$	−4.42329	−0.212325
$435$	−1.24010	−0.0594583
$436$	11.9529	0.572439
$437$	−4.50411	−0.215461
$438$	0.613355	0.0293072
$439$	37.5221	1.79083	0.895415	−	0.445232i	$-0.146879\pi$
$439$	37.5221	1.79083	0.895415	+	0.445232i	$0.146879\pi$
$440$	0.0677864	0.00323159
$441$	−0.731138	−0.0348161
$442$	48.4762	2.30578
$443$	−19.5435	−0.928541	−0.464271	−	0.885693i	$-0.653684\pi$
$443$	−19.5435	−0.928541	−0.464271	+	0.885693i	$0.653684\pi$
$444$	−12.2776	−0.582669
$445$	−2.94726	−0.139714
$446$	−15.2195	−0.720666
$447$	−4.71250	−0.222893
$448$	1.78538	0.0843512
$449$	2.97063	0.140193	0.0700965	−	0.997540i	$-0.477669\pi$
$449$	2.97063	0.140193	0.0700965	+	0.997540i	$0.477669\pi$
$450$	−0.930927	−0.0438843
$451$	−1.30366	−0.0613869
$452$	5.07011	0.238478
$453$	−36.2128	−1.70143
$454$	−24.8597	−1.16672
$455$	4.12640	0.193449
$456$	−13.5450	−0.634301
$457$	−1.08729	−0.0508613	−0.0254307	−	0.999677i	$-0.508096\pi$
$457$	−1.08729	−0.0508613	−0.0254307	+	0.999677i	$0.508096\pi$
$458$	−20.7757	−0.970786
$459$	40.1800	1.87544
$460$	−0.226841	−0.0105765
$461$	−6.45272	−0.300533	−0.150267	−	0.988646i	$-0.548013\pi$
$461$	−6.45272	−0.300533	−0.150267	+	0.988646i	$0.548013\pi$
$462$	−0.566257	−0.0263447
$463$	−9.94521	−0.462193	−0.231096	−	0.972931i	$-0.574231\pi$
$463$	−9.94521	−0.462193	−0.231096	+	0.972931i	$0.574231\pi$
$464$	1.81788	0.0843930
$465$	1.69008	0.0783755
$466$	3.18832	0.147696
$467$	20.0304	0.926898	0.463449	−	0.886124i	$-0.346612\pi$
$467$	20.0304	0.926898	0.463449	+	0.886124i	$0.346612\pi$
$468$	1.16082	0.0536589
$469$	−26.3539	−1.21691
$470$	−2.58442	−0.119211
$471$	−9.00217	−0.414798
$472$	14.8791	0.684866
$473$	1.99371	0.0916708
$474$	−15.6069	−0.716851
$475$	−36.8027	−1.68862
$476$	14.2986	0.655374
$477$	1.60819	0.0736340
$478$	13.3018	0.608409
$479$	23.7586	1.08556	0.542780	−	0.839875i	$-0.317372\pi$
$479$	23.7586	1.08556	0.542780	+	0.839875i	$0.317372\pi$
$480$	−0.682168	−0.0311366
$481$	41.5971	1.89667
$482$	−4.41873	−0.201267
$483$	1.89493	0.0862224
$484$	−10.9685	−0.498567
$485$	−3.29129	−0.149450
$486$	1.99002	0.0902691
$487$	−31.9577	−1.44814	−0.724070	−	0.689726i	$-0.757731\pi$
$487$	−31.9577	−1.44814	−0.724070	+	0.689726i	$0.757731\pi$
$488$	−10.5692	−0.478446
$489$	−27.1988	−1.22997
$490$	−1.45571	−0.0657625
$491$	17.9899	0.811875	0.405937	−	0.913901i	$-0.366945\pi$
$491$	17.9899	0.811875	0.405937	+	0.913901i	$0.366945\pi$
$492$	13.1194	0.591466
$493$	14.5589	0.655699
$494$	45.8910	2.06473
$495$	0.0129999	0.000584302 0
$496$	−2.47751	−0.111244
$497$	−9.70282	−0.435231
$498$	10.8661	0.486920
$499$	−2.22670	−0.0996808	−0.0498404	−	0.998757i	$-0.515871\pi$
$499$	−2.22670	−0.0996808	−0.0498404	+	0.998757i	$0.515871\pi$
$500$	−3.76267	−0.168272
$501$	−15.9168	−0.711111
$502$	18.3763	0.820173
$503$	20.1458	0.898255	0.449127	−	0.893468i	$-0.351735\pi$
$503$	20.1458	0.898255	0.449127	+	0.893468i	$0.351735\pi$
$504$	0.342395	0.0152515
$505$	0.731631	0.0325572
$506$	−0.105467	−0.00468856
$507$	−42.2308	−1.87553
$508$	−13.2017	−0.585733
$509$	1.74143	0.0771877	0.0385939	−	0.999255i	$-0.487712\pi$
$509$	1.74143	0.0771877	0.0385939	+	0.999255i	$0.487712\pi$
$510$	−5.46328	−0.241918
$511$	−0.612951	−0.0271154
$512$	1.00000	0.0441942
$513$	38.0372	1.67939
$514$	9.13578	0.402962
$515$	−4.77507	−0.210415
$516$	−20.0637	−0.883254
$517$	−1.20159	−0.0528459
$518$	12.2695	0.539091
$519$	−20.8201	−0.913900
$520$	2.31122	0.101354
$521$	−26.7017	−1.16982	−0.584911	−	0.811097i	$-0.698871\pi$
$521$	−26.7017	−1.16982	−0.584911	+	0.811097i	$0.698871\pi$
$522$	0.348629	0.0152591
$523$	−12.8693	−0.562735	−0.281368	−	0.959600i	$-0.590788\pi$
$523$	−12.8693	−0.562735	−0.281368	+	0.959600i	$0.590788\pi$
$524$	16.8969	0.738146
$525$	15.4833	0.675748
$526$	−2.75499	−0.120123
$527$	−19.8417	−0.864316
$528$	−0.317164	−0.0138028
$529$	−22.6471	−0.984655
$530$	3.20195	0.139084
$531$	2.85348	0.123830
$532$	13.5360	0.586861
$533$	−44.4491	−1.92530
$534$	13.7899	0.596746
$535$	−2.34711	−0.101474
$536$	−14.7610	−0.637577
$537$	−11.1115	−0.479498
$538$	13.0334	0.561910
$539$	−0.676813	−0.0291524
$540$	1.91568	0.0824377
$541$	9.78938	0.420878	0.210439	−	0.977607i	$-0.432511\pi$
$541$	9.78938	0.420878	0.210439	+	0.977607i	$0.432511\pi$
$542$	−12.2055	−0.524270
$543$	−0.804246	−0.0345135
$544$	8.00871	0.343371
$545$	4.56402	0.195501
$546$	−19.3069	−0.826260
$547$	13.9666	0.597167	0.298584	−	0.954383i	$-0.403486\pi$
$547$	13.9666	0.597167	0.298584	+	0.954383i	$0.403486\pi$
$548$	−10.5098	−0.448958
$549$	−2.02694	−0.0865077
$550$	−0.861758	−0.0367455
$551$	13.7825	0.587153
$552$	1.06136	0.0451746
$553$	15.5967	0.663238
$554$	−12.5074	−0.531390
$555$	−4.68800	−0.198995
$556$	−4.01972	−0.170474
$557$	−28.4541	−1.20564	−0.602820	−	0.797877i	$-0.705956\pi$
$557$	−28.4541	−1.20564	−0.602820	+	0.797877i	$0.705956\pi$
$558$	−0.475131	−0.0201139
$559$	67.9767	2.87511
$560$	0.681718	0.0288079
$561$	−2.54007	−0.107242
$562$	−23.5779	−0.994575
$563$	25.3874	1.06995	0.534976	−	0.844867i	$-0.320320\pi$
$563$	25.3874	1.06995	0.534976	+	0.844867i	$0.320320\pi$
$564$	12.0922	0.509173
$565$	1.93594	0.0814457
$566$	0.480125	0.0201812
$567$	−17.0299	−0.715190
$568$	−5.43460	−0.228031
$569$	2.04961	0.0859241	0.0429620	−	0.999077i	$-0.486321\pi$
$569$	2.04961	0.0859241	0.0429620	+	0.999077i	$0.486321\pi$
$570$	−5.17193	−0.216628
$571$	4.64036	0.194193	0.0970964	−	0.995275i	$-0.469044\pi$
$571$	4.64036	0.194193	0.0970964	+	0.995275i	$0.469044\pi$
$572$	1.07457	0.0449299
$573$	−45.4425	−1.89839
$574$	−13.1107	−0.547231
$575$	2.88380	0.120263
$576$	0.191778	0.00799073
$577$	8.76097	0.364724	0.182362	−	0.983231i	$-0.441626\pi$
$577$	8.76097	0.364724	0.182362	+	0.983231i	$0.441626\pi$
$578$	47.1394	1.96074
$579$	40.5973	1.68716
$580$	0.694129	0.0288222
$581$	−10.8589	−0.450504
$582$	15.3995	0.638331
$583$	1.48870	0.0616556
$584$	−0.343317	−0.0142066
$585$	0.443240	0.0183257
$586$	−18.4877	−0.763719
$587$	−30.0216	−1.23912	−0.619562	−	0.784947i	$-0.712690\pi$
$587$	−30.0216	−1.23912	−0.619562	+	0.784947i	$0.712690\pi$
$588$	6.81111	0.280885
$589$	−18.7835	−0.773961
$590$	5.68135	0.233898
$591$	29.8055	1.22603
$592$	6.87222	0.282446
$593$	−3.35034	−0.137582	−0.0687910	−	0.997631i	$-0.521914\pi$
$593$	−3.35034	−0.137582	−0.0687910	+	0.997631i	$0.521914\pi$
$594$	0.890667	0.0365445
$595$	5.45969	0.223825
$596$	2.63776	0.108047
$597$	38.5228	1.57663
$598$	−3.59595	−0.147049
$599$	−15.1228	−0.617902	−0.308951	−	0.951078i	$-0.599978\pi$
$599$	−15.1228	−0.617902	−0.308951	+	0.951078i	$0.599978\pi$
$600$	8.67230	0.354045
$601$	−0.974215	−0.0397391	−0.0198695	−	0.999803i	$-0.506325\pi$
$601$	−0.974215	−0.0397391	−0.0198695	+	0.999803i	$0.506325\pi$
$602$	20.0504	0.817195
$603$	−2.83082	−0.115280
$604$	20.2696	0.824760
$605$	−4.18814	−0.170272
$606$	−3.42321	−0.139058
$607$	16.9850	0.689401	0.344700	−	0.938713i	$-0.387981\pi$
$607$	16.9850	0.689401	0.344700	+	0.938713i	$0.387981\pi$
$608$	7.58161	0.307475
$609$	−5.79845	−0.234965
$610$	−4.03569	−0.163400
$611$	−40.9690	−1.65743
$612$	1.53589	0.0620847
$613$	−21.5791	−0.871573	−0.435786	−	0.900050i	$-0.643530\pi$
$613$	−21.5791	−0.871573	−0.435786	+	0.900050i	$0.643530\pi$
$614$	8.44770	0.340921
$615$	5.00942	0.201999
$616$	0.316955	0.0127705
$617$	17.4323	0.701799	0.350899	−	0.936413i	$-0.385876\pi$
$617$	17.4323	0.701799	0.350899	+	0.936413i	$0.385876\pi$
$618$	22.3420	0.898726
$619$	−26.1657	−1.05169	−0.525844	−	0.850581i	$-0.676250\pi$
$619$	−26.1657	−1.05169	−0.525844	+	0.850581i	$0.676250\pi$
$620$	−0.945998	−0.0379922
$621$	−2.98054	−0.119605
$622$	−6.78631	−0.272106
$623$	−13.7808	−0.552115
$624$	−10.8139	−0.432903
$625$	22.8343	0.913372
$626$	13.4593	0.537942
$627$	−2.40461	−0.0960310
$628$	5.03885	0.201072
$629$	55.0376	2.19449
$630$	0.130738	0.00520874
$631$	−28.8680	−1.14922	−0.574608	−	0.818429i	$-0.694845\pi$
$631$	−28.8680	−1.14922	−0.574608	+	0.818429i	$0.694845\pi$
$632$	8.73578	0.347491
$633$	−22.1256	−0.879414
$634$	8.34851	0.331562
$635$	−5.04088	−0.200041
$636$	−14.9815	−0.594056
$637$	−23.0764	−0.914320
$638$	0.322725	0.0127768
$639$	−1.04224	−0.0412302
$640$	0.381834	0.0150933
$641$	15.0122	0.592947	0.296473	−	0.955041i	$-0.404189\pi$
$641$	15.0122	0.592947	0.296473	+	0.955041i	$0.404189\pi$
$642$	10.9818	0.433418
$643$	−29.0339	−1.14498	−0.572492	−	0.819910i	$-0.694023\pi$
$643$	−29.0339	−1.14498	−0.572492	+	0.819910i	$0.694023\pi$
$644$	−1.06066	−0.0417960
$645$	−7.66099	−0.301651
$646$	60.7189	2.38895
$647$	28.2800	1.11180	0.555901	−	0.831249i	$-0.312373\pi$
$647$	28.2800	1.11180	0.555901	+	0.831249i	$0.312373\pi$
$648$	−9.53855	−0.374710
$649$	2.64146	0.103686
$650$	−29.3822	−1.15246
$651$	7.90245	0.309722
$652$	15.2242	0.596224
$653$	−40.6538	−1.59090	−0.795452	−	0.606016i	$-0.792767\pi$
$653$	−40.6538	−1.59090	−0.795452	+	0.606016i	$0.792767\pi$
$654$	−21.3545	−0.835025
$655$	6.45183	0.252094
$656$	−7.34339	−0.286711
$657$	−0.0658406	−0.00256868
$658$	−12.0842	−0.471092
$659$	30.9082	1.20401	0.602005	−	0.798492i	$-0.294369\pi$
$659$	30.9082	1.20401	0.602005	+	0.798492i	$0.294369\pi$
$660$	−0.121104	−0.00471397
$661$	9.16536	0.356491	0.178246	−	0.983986i	$-0.442958\pi$
$661$	9.16536	0.356491	0.178246	+	0.983986i	$0.442958\pi$
$662$	−25.9952	−1.01033
$663$	−86.6055	−3.36348
$664$	−6.08214	−0.236033
$665$	5.16852	0.200427
$666$	1.31794	0.0510690
$667$	−1.07997	−0.0418167
$668$	8.90923	0.344709
$669$	27.1905	1.05125
$670$	−5.63625	−0.217747
$671$	−1.87634	−0.0724352
$672$	−3.18968	−0.123044
$673$	37.8836	1.46031	0.730154	−	0.683283i	$-0.239449\pi$
$673$	37.8836	1.46031	0.730154	+	0.683283i	$0.239449\pi$
$674$	−2.24485	−0.0864686
$675$	−24.3537	−0.937376
$676$	23.6381	0.909158
$677$	4.28341	0.164625	0.0823124	−	0.996607i	$-0.473769\pi$
$677$	4.28341	0.164625	0.0823124	+	0.996607i	$0.473769\pi$
$678$	−9.05803	−0.347871
$679$	−15.3894	−0.590590
$680$	3.05800	0.117269
$681$	44.4132	1.70192
$682$	−0.439828	−0.0168419
$683$	−17.5596	−0.671900	−0.335950	−	0.941880i	$-0.609057\pi$
$683$	−17.5596	−0.671900	−0.335950	+	0.941880i	$0.609057\pi$
$684$	1.45398	0.0555944
$685$	−4.01302	−0.153330
$686$	−19.3043	−0.737040
$687$	37.1170	1.41610
$688$	11.2304	0.428154
$689$	50.7582	1.93373
$690$	0.405265	0.0154282
$691$	45.1627	1.71807	0.859035	−	0.511917i	$-0.171065\pi$
$691$	45.1627	1.71807	0.859035	+	0.511917i	$0.171065\pi$
$692$	11.6538	0.443010
$693$	0.0607849	0.00230903
$694$	25.6017	0.971826
$695$	−1.53487	−0.0582209
$696$	−3.24774	−0.123105
$697$	−58.8111	−2.22763
$698$	−30.2702	−1.14574
$699$	−5.69610	−0.215446
$700$	−8.66659	−0.327566
$701$	34.8953	1.31798	0.658989	−	0.752152i	$-0.270984\pi$
$701$	34.8953	1.31798	0.658989	+	0.752152i	$0.270984\pi$
$702$	30.3679	1.14616
$703$	52.1024	1.96508
$704$	0.177528	0.00669085
$705$	4.61721	0.173894
$706$	34.0676	1.28215
$707$	3.42096	0.128658
$708$	−26.5823	−0.999025
$709$	−15.9208	−0.597917	−0.298959	−	0.954266i	$-0.596639\pi$
$709$	−15.9208	−0.597917	−0.298959	+	0.954266i	$0.596639\pi$
$710$	−2.07512	−0.0778778
$711$	1.67533	0.0628297
$712$	−7.71869	−0.289270
$713$	1.47185	0.0551211
$714$	−25.5452	−0.956004
$715$	0.410307	0.0153446
$716$	6.21953	0.232435
$717$	−23.7644	−0.887496
$718$	3.35128	0.125069
$719$	36.7474	1.37045	0.685223	−	0.728333i	$-0.259705\pi$
$719$	36.7474	1.37045	0.685223	+	0.728333i	$0.259705\pi$
$720$	0.0732273	0.00272902
$721$	−22.3272	−0.831510
$722$	38.4808	1.43211
$723$	7.89430	0.293592
$724$	0.450166	0.0167303
$725$	−8.82437	−0.327729
$726$	19.5958	0.727268
$727$	−12.4746	−0.462656	−0.231328	−	0.972876i	$-0.574307\pi$
$727$	−12.4746	−0.462656	−0.231328	+	0.972876i	$0.574307\pi$
$728$	10.8068	0.400526
$729$	25.0604	0.928162
$730$	−0.131090	−0.00485187
$731$	89.9407	3.32658
$732$	18.8825	0.697917
$733$	43.2123	1.59608	0.798042	−	0.602602i	$-0.205869\pi$
$733$	43.2123	1.59608	0.798042	+	0.602602i	$0.205869\pi$
$734$	21.5647	0.795968
$735$	2.60071	0.0959288
$736$	−0.594083	−0.0218982
$737$	−2.62049	−0.0965270
$738$	−1.40830	−0.0518401
$739$	4.94860	0.182037	0.0910186	−	0.995849i	$-0.470988\pi$
$739$	4.94860	0.182037	0.0910186	+	0.995849i	$0.470988\pi$
$740$	2.62405	0.0964619
$741$	−81.9868	−3.01186
$742$	14.9717	0.549627
$743$	31.9010	1.17033	0.585167	−	0.810913i	$-0.301029\pi$
$743$	31.9010	1.17033	0.585167	+	0.810913i	$0.301029\pi$
$744$	4.42621	0.162273
$745$	1.00719	0.0369004
$746$	−31.0397	−1.13644
$747$	−1.16642	−0.0426770
$748$	1.42177	0.0519851
$749$	−10.9746	−0.401002
$750$	6.72222	0.245461
$751$	−12.9443	−0.472343	−0.236171	−	0.971711i	$-0.575893\pi$
$751$	−12.9443	−0.472343	−0.236171	+	0.971711i	$0.575893\pi$
$752$	−6.76844	−0.246820
$753$	−32.8302	−1.19640
$754$	11.0035	0.400725
$755$	7.73965	0.281675
$756$	8.95732	0.325774
$757$	−45.8857	−1.66774	−0.833871	−	0.551959i	$-0.813881\pi$
$757$	−45.8857	−1.66774	−0.833871	+	0.551959i	$0.813881\pi$
$758$	33.9098	1.23166
$759$	0.188422	0.00683928
$760$	2.89492	0.105010
$761$	−8.17045	−0.296179	−0.148089	−	0.988974i	$-0.547312\pi$
$761$	−8.17045	−0.296179	−0.148089	+	0.988974i	$0.547312\pi$
$762$	23.5856	0.854418
$763$	21.3404	0.772574
$764$	25.4358	0.920235
$765$	0.586456	0.0212034
$766$	−12.5799	−0.454532
$767$	90.0624	3.25196
$768$	−1.78655	−0.0644667
$769$	−24.1164	−0.869659	−0.434830	−	0.900513i	$-0.643191\pi$
$769$	−24.1164	−0.869659	−0.434830	+	0.900513i	$0.643191\pi$
$770$	0.121024	0.00436141
$771$	−16.3216	−0.587807
$772$	−22.7238	−0.817847
$773$	6.74455	0.242585	0.121292	−	0.992617i	$-0.461296\pi$
$773$	6.74455	0.242585	0.121292	+	0.992617i	$0.461296\pi$
$774$	2.15373	0.0774143
$775$	12.0263	0.431999
$776$	−8.61967	−0.309428
$777$	−21.9201	−0.786381
$778$	14.2327	0.510266
$779$	−55.6747	−1.99475
$780$	−4.12912	−0.147846
$781$	−0.964796	−0.0345231
$782$	−4.75784	−0.170140
$783$	9.12039	0.325936
$784$	−3.81243	−0.136158
$785$	1.92400	0.0686706
$786$	−30.1873	−1.07674
$787$	−42.0483	−1.49886	−0.749430	−	0.662083i	$-0.769673\pi$
$787$	−42.0483	−1.49886	−0.749430	+	0.662083i	$0.769673\pi$
$788$	−16.6832	−0.594316
$789$	4.92194	0.175226
$790$	3.33562	0.118676
$791$	9.05206	0.321854
$792$	0.0340459	0.00120977
$793$	−63.9749	−2.27182
$794$	18.2081	0.646180
$795$	−5.72046	−0.202884
$796$	−21.5626	−0.764267
$797$	16.4060	0.581129	0.290564	−	0.956855i	$-0.406157\pi$
$797$	16.4060	0.581129	0.290564	+	0.956855i	$0.406157\pi$
$798$	−24.1829	−0.856064
$799$	−54.2065	−1.91769
$800$	−4.85420	−0.171622
$801$	−1.48027	−0.0523028
$802$	−15.1199	−0.533901
$803$	−0.0609485	−0.00215083
$804$	26.3713	0.930044
$805$	−0.404998	−0.0142743
$806$	−14.9962	−0.528219
$807$	−23.2849	−0.819666
$808$	1.91610	0.0674080
$809$	−48.8743	−1.71833	−0.859164	−	0.511701i	$-0.829016\pi$
$809$	−48.8743	−1.71833	−0.859164	+	0.511701i	$0.829016\pi$
$810$	−3.64215	−0.127972
$811$	7.84190	0.275366	0.137683	−	0.990476i	$-0.456034\pi$
$811$	7.84190	0.275366	0.137683	+	0.990476i	$0.456034\pi$
$812$	3.24561	0.113898
$813$	21.8057	0.764761
$814$	1.22001	0.0427614
$815$	5.81311	0.203624
$816$	−14.3080	−0.500880
$817$	85.1442	2.97882
$818$	−29.4635	−1.03017
$819$	2.07250	0.0724190
$820$	−2.80396	−0.0979184
$821$	33.1585	1.15724	0.578620	−	0.815597i	$-0.303591\pi$
$821$	33.1585	1.15724	0.578620	+	0.815597i	$0.303591\pi$
$822$	18.7764	0.654902
$823$	22.3918	0.780528	0.390264	−	0.920703i	$-0.372384\pi$
$823$	22.3918	0.780528	0.390264	+	0.920703i	$0.372384\pi$
$824$	−12.5056	−0.435654
$825$	1.53958	0.0536012
$826$	26.5648	0.924309
$827$	22.7098	0.789697	0.394848	−	0.918746i	$-0.370797\pi$
$827$	22.7098	0.789697	0.394848	+	0.918746i	$0.370797\pi$
$828$	−0.113932	−0.00395941
$829$	32.2294	1.11937	0.559687	−	0.828704i	$-0.310921\pi$
$829$	32.2294	1.11937	0.559687	+	0.828704i	$0.310921\pi$
$830$	−2.32237	−0.0806106
$831$	22.3452	0.775146
$832$	6.05294	0.209848
$833$	−30.5326	−1.05789
$834$	7.18146	0.248674
$835$	3.40185	0.117726
$836$	1.34595	0.0465506
$837$	−12.4298	−0.429636
$838$	−11.9549	−0.412974
$839$	49.9917	1.72590	0.862952	−	0.505286i	$-0.168613\pi$
$839$	49.9917	1.72590	0.862952	+	0.505286i	$0.168613\pi$
$840$	−1.21793	−0.0420225
$841$	−25.6953	−0.886045
$842$	18.1980	0.627144
$843$	42.1233	1.45080
$844$	12.3845	0.426292
$845$	9.02583	0.310498
$846$	−1.29804	−0.0446274
$847$	−19.5829	−0.672876
$848$	8.38571	0.287966
$849$	−0.857770	−0.0294386
$850$	−38.8759	−1.33343
$851$	−4.08267	−0.139952
$852$	9.70922	0.332632
$853$	−16.3046	−0.558257	−0.279129	−	0.960254i	$-0.590046\pi$
$853$	−16.3046	−0.558257	−0.279129	+	0.960254i	$0.590046\pi$
$854$	−18.8701	−0.645720
$855$	0.555180	0.0189868
$856$	−6.14692	−0.210098
$857$	7.49553	0.256042	0.128021	−	0.991771i	$-0.459137\pi$
$857$	7.49553	0.256042	0.128021	+	0.991771i	$0.459137\pi$
$858$	−1.91977	−0.0655400
$859$	−19.9287	−0.679959	−0.339980	−	0.940433i	$-0.610420\pi$
$859$	−19.9287	−0.679959	−0.339980	+	0.940433i	$0.610420\pi$
$860$	4.28814	0.146224
$861$	23.4230	0.798254
$862$	−24.6175	−0.838477
$863$	16.6928	0.568229	0.284114	−	0.958790i	$-0.408300\pi$
$863$	16.6928	0.568229	0.284114	+	0.958790i	$0.408300\pi$
$864$	5.01704	0.170683
$865$	4.44981	0.151298
$866$	11.7116	0.397978
$867$	−84.2172	−2.86017
$868$	−4.42329	−0.150136
$869$	1.55085	0.0526089
$870$	−1.24010	−0.0420433
$871$	−89.3473	−3.02742
$872$	11.9529	0.404775
$873$	−1.65306	−0.0559476
$874$	−4.50411	−0.152354
$875$	−6.71779	−0.227103
$876$	0.613355	0.0207233
$877$	40.0764	1.35328	0.676642	−	0.736312i	$-0.263434\pi$
$877$	40.0764	1.35328	0.676642	+	0.736312i	$0.263434\pi$
$878$	37.5221	1.26631
$879$	33.0292	1.11405
$880$	0.0677864	0.00228508
$881$	28.8506	0.972001	0.486001	−	0.873958i	$-0.338455\pi$
$881$	28.8506	0.972001	0.486001	+	0.873958i	$0.338455\pi$
$882$	−0.731138	−0.0246187
$883$	−38.0291	−1.27978	−0.639890	−	0.768466i	$-0.721020\pi$
$883$	−38.0291	−1.27978	−0.639890	+	0.768466i	$0.721020\pi$
$884$	48.4762	1.63043
$885$	−10.1500	−0.341190
$886$	−19.5435	−0.656578
$887$	−43.8855	−1.47353	−0.736765	−	0.676148i	$-0.763648\pi$
$887$	−43.8855	−1.47353	−0.736765	+	0.676148i	$0.763648\pi$
$888$	−12.2776	−0.412009
$889$	−23.5701	−0.790516
$890$	−2.94726	−0.0987924
$891$	−1.69336	−0.0567298
$892$	−15.2195	−0.509588
$893$	−51.3157	−1.71721
$894$	−4.71250	−0.157609
$895$	2.37483	0.0793818
$896$	1.78538	0.0596453
$897$	6.42436	0.214503
$898$	2.97063	0.0991314
$899$	−4.50382	−0.150211
$900$	−0.930927	−0.0310309
$901$	67.1587	2.23738
$902$	−1.30366	−0.0434071
$903$	−35.8212	−1.19206
$904$	5.07011	0.168629
$905$	0.171889	0.00571378
$906$	−36.2128	−1.20309
$907$	37.6801	1.25115	0.625574	−	0.780165i	$-0.284865\pi$
$907$	37.6801	1.25115	0.625574	+	0.780165i	$0.284865\pi$
$908$	−24.8597	−0.824999
$909$	0.367464	0.0121880
$910$	4.12640	0.136789
$911$	−55.0122	−1.82263	−0.911317	−	0.411705i	$-0.864933\pi$
$911$	−55.0122	−1.82263	−0.911317	+	0.411705i	$0.864933\pi$
$912$	−13.5450	−0.448518
$913$	−1.07975	−0.0357346
$914$	−1.08729	−0.0359644
$915$	7.20999	0.238355
$916$	−20.7757	−0.686449
$917$	30.1674	0.996215
$918$	40.1800	1.32614
$919$	0.549466	0.0181252	0.00906261	−	0.999959i	$-0.497115\pi$
$919$	0.549466	0.0181252	0.00906261	+	0.999959i	$0.497115\pi$
$920$	−0.226841	−0.00747874
$921$	−15.0923	−0.497308
$922$	−6.45272	−0.212509
$923$	−32.8953	−1.08276
$924$	−0.566257	−0.0186285
$925$	−33.3591	−1.09684
$926$	−9.94521	−0.326820
$927$	−2.39830	−0.0787704
$928$	1.81788	0.0596749
$929$	−17.4052	−0.571045	−0.285522	−	0.958372i	$-0.592167\pi$
$929$	−17.4052	−0.571045	−0.285522	+	0.958372i	$0.592167\pi$
$930$	1.69008	0.0554198
$931$	−28.9043	−0.947301
$932$	3.18832	0.104437
$933$	12.1241	0.396926
$934$	20.0304	0.655416
$935$	0.542881	0.0177541
$936$	1.16082	0.0379425
$937$	−29.8065	−0.973736	−0.486868	−	0.873476i	$-0.661861\pi$
$937$	−29.8065	−0.973736	−0.486868	+	0.873476i	$0.661861\pi$
$938$	−26.3539	−0.860486
$939$	−24.0458	−0.784704
$940$	−2.58442	−0.0842946
$941$	−40.7854	−1.32957	−0.664783	−	0.747036i	$-0.731476\pi$
$941$	−40.7854	−1.32957	−0.664783	+	0.747036i	$0.731476\pi$
$942$	−9.00217	−0.293307
$943$	4.36258	0.142065
$944$	14.8791	0.484274
$945$	3.42021	0.111259
$946$	1.99371	0.0648210
$947$	57.7658	1.87714	0.938568	−	0.345095i	$-0.112153\pi$
$947$	57.7658	1.87714	0.938568	+	0.345095i	$0.112153\pi$
$948$	−15.6069	−0.506890
$949$	−2.07808	−0.0674573
$950$	−36.8027	−1.19404
$951$	−14.9151	−0.483654
$952$	14.2986	0.463419
$953$	19.3541	0.626941	0.313470	−	0.949598i	$-0.398508\pi$
$953$	19.3541	0.626941	0.313470	+	0.949598i	$0.398508\pi$
$954$	1.60819	0.0520671
$955$	9.71226	0.314281
$956$	13.3018	0.430210
$957$	−0.576566	−0.0186377
$958$	23.7586	0.767606
$959$	−18.7640	−0.605922
$960$	−0.682168	−0.0220169
$961$	−24.8619	−0.801998
$962$	41.5971	1.34114
$963$	−1.17884	−0.0379877
$964$	−4.41873	−0.142318
$965$	−8.67672	−0.279313
$966$	1.89493	0.0609685
$967$	−32.6874	−1.05115	−0.525577	−	0.850746i	$-0.676151\pi$
$967$	−32.6874	−1.05115	−0.525577	+	0.850746i	$0.676151\pi$
$968$	−10.9685	−0.352540
$969$	−108.478	−3.48480
$970$	−3.29129	−0.105677
$971$	26.0694	0.836608	0.418304	−	0.908307i	$-0.362625\pi$
$971$	26.0694	0.836608	0.418304	+	0.908307i	$0.362625\pi$
$972$	1.99002	0.0638299
$973$	−7.17673	−0.230075
$974$	−31.9577	−1.02399
$975$	52.4929	1.68112
$976$	−10.5692	−0.338313
$977$	−48.7114	−1.55842	−0.779208	−	0.626766i	$-0.784378\pi$
$977$	−48.7114	−1.55842	−0.779208	+	0.626766i	$0.784378\pi$
$978$	−27.1988	−0.869722
$979$	−1.37029	−0.0437945
$980$	−1.45571	−0.0465011
$981$	2.29229	0.0731873
$982$	17.9899	0.574082
$983$	24.4149	0.778714	0.389357	−	0.921087i	$-0.372697\pi$
$983$	24.4149	0.778714	0.389357	+	0.921087i	$0.372697\pi$
$984$	13.1194	0.418230
$985$	−6.37023	−0.202972
$986$	14.5589	0.463649
$987$	21.5891	0.687190
$988$	45.8910	1.45999
$989$	−6.67177	−0.212150
$990$	0.0129999	0.000413164 0
$991$	17.7474	0.563766	0.281883	−	0.959449i	$-0.409041\pi$
$991$	17.7474	0.563766	0.281883	+	0.959449i	$0.409041\pi$
$992$	−2.47751	−0.0786610
$993$	46.4418	1.47379
$994$	−9.70282	−0.307755
$995$	−8.23335	−0.261015
$996$	10.8661	0.344305
$997$	22.9409	0.726545	0.363273	−	0.931683i	$-0.381659\pi$
$997$	22.9409	0.726545	0.363273	+	0.931683i	$0.381659\pi$
$998$	−2.22670	−0.0704849
$999$	34.4782	1.09084

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.d.1.19	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.d.1.19	✓	92	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.78655 q^{3} +1.00000 q^{4} +0.381834 q^{5} -1.78655 q^{6} +1.78538 q^{7} +1.00000 q^{8} +0.191778 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.78655 q^{3} +1.00000 q^{4} +0.381834 q^{5} -1.78655 q^{6} +1.78538 q^{7} +1.00000 q^{8} +0.191778 q^{9} +0.381834 q^{10} +0.177528 q^{11} -1.78655 q^{12} +6.05294 q^{13} +1.78538 q^{14} -0.682168 q^{15} +1.00000 q^{16} +8.00871 q^{17} +0.191778 q^{18} +7.58161 q^{19} +0.381834 q^{20} -3.18968 q^{21} +0.177528 q^{22} -0.594083 q^{23} -1.78655 q^{24} -4.85420 q^{25} +6.05294 q^{26} +5.01704 q^{27} +1.78538 q^{28} +1.81788 q^{29} -0.682168 q^{30} -2.47751 q^{31} +1.00000 q^{32} -0.317164 q^{33} +8.00871 q^{34} +0.681718 q^{35} +0.191778 q^{36} +6.87222 q^{37} +7.58161 q^{38} -10.8139 q^{39} +0.381834 q^{40} -7.34339 q^{41} -3.18968 q^{42} +11.2304 q^{43} +0.177528 q^{44} +0.0732273 q^{45} -0.594083 q^{46} -6.76844 q^{47} -1.78655 q^{48} -3.81243 q^{49} -4.85420 q^{50} -14.3080 q^{51} +6.05294 q^{52} +8.38571 q^{53} +5.01704 q^{54} +0.0677864 q^{55} +1.78538 q^{56} -13.5450 q^{57} +1.81788 q^{58} +14.8791 q^{59} -0.682168 q^{60} -10.5692 q^{61} -2.47751 q^{62} +0.342395 q^{63} +1.00000 q^{64} +2.31122 q^{65} -0.317164 q^{66} -14.7610 q^{67} +8.00871 q^{68} +1.06136 q^{69} +0.681718 q^{70} -5.43460 q^{71} +0.191778 q^{72} -0.343317 q^{73} +6.87222 q^{74} +8.67230 q^{75} +7.58161 q^{76} +0.316955 q^{77} -10.8139 q^{78} +8.73578 q^{79} +0.381834 q^{80} -9.53855 q^{81} -7.34339 q^{82} -6.08214 q^{83} -3.18968 q^{84} +3.05800 q^{85} +11.2304 q^{86} -3.24774 q^{87} +0.177528 q^{88} -7.71869 q^{89} +0.0732273 q^{90} +10.8068 q^{91} -0.594083 q^{92} +4.42621 q^{93} -6.76844 q^{94} +2.89492 q^{95} -1.78655 q^{96} -8.61967 q^{97} -3.81243 q^{98} +0.0340459 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

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