LMFDB - Embedded newform 8031.2.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$1$
Dimension:	$102$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38361 q^{2} -1.00000 q^{3} +3.68158 q^{4} +1.26527 q^{5} +2.38361 q^{6} -2.62916 q^{7} -4.00822 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.38361 q^{2} -1.00000 q^{3} +3.68158 q^{4} +1.26527 q^{5} +2.38361 q^{6} -2.62916 q^{7} -4.00822 q^{8} +1.00000 q^{9} -3.01590 q^{10} +0.984672 q^{11} -3.68158 q^{12} +0.212001 q^{13} +6.26689 q^{14} -1.26527 q^{15} +2.19087 q^{16} +5.90052 q^{17} -2.38361 q^{18} +1.74925 q^{19} +4.65818 q^{20} +2.62916 q^{21} -2.34707 q^{22} +0.0254742 q^{23} +4.00822 q^{24} -3.39910 q^{25} -0.505326 q^{26} -1.00000 q^{27} -9.67947 q^{28} -0.0447304 q^{29} +3.01590 q^{30} +2.23264 q^{31} +2.79428 q^{32} -0.984672 q^{33} -14.0645 q^{34} -3.32659 q^{35} +3.68158 q^{36} -3.75890 q^{37} -4.16953 q^{38} -0.212001 q^{39} -5.07147 q^{40} -4.82004 q^{41} -6.26689 q^{42} +7.36351 q^{43} +3.62515 q^{44} +1.26527 q^{45} -0.0607204 q^{46} -1.50657 q^{47} -2.19087 q^{48} -0.0874982 q^{49} +8.10211 q^{50} -5.90052 q^{51} +0.780498 q^{52} -7.98354 q^{53} +2.38361 q^{54} +1.24587 q^{55} +10.5383 q^{56} -1.74925 q^{57} +0.106620 q^{58} -3.00731 q^{59} -4.65818 q^{60} -4.45462 q^{61} -5.32173 q^{62} -2.62916 q^{63} -11.0422 q^{64} +0.268238 q^{65} +2.34707 q^{66} -8.16233 q^{67} +21.7232 q^{68} -0.0254742 q^{69} +7.92929 q^{70} +1.36050 q^{71} -4.00822 q^{72} +15.5054 q^{73} +8.95974 q^{74} +3.39910 q^{75} +6.44001 q^{76} -2.58887 q^{77} +0.505326 q^{78} -7.62687 q^{79} +2.77203 q^{80} +1.00000 q^{81} +11.4891 q^{82} -7.20430 q^{83} +9.67947 q^{84} +7.46574 q^{85} -17.5517 q^{86} +0.0447304 q^{87} -3.94679 q^{88} -4.18899 q^{89} -3.01590 q^{90} -0.557385 q^{91} +0.0937852 q^{92} -2.23264 q^{93} +3.59108 q^{94} +2.21327 q^{95} -2.79428 q^{96} +8.03021 q^{97} +0.208561 q^{98} +0.984672 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) $ 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * 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 $

$2$	−2.38361	−1.68546	−0.842732	−	0.538333i	$-0.819054\pi$
$2$	−2.38361	−1.68546	−0.842732	+	0.538333i	$0.819054\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.68158	1.84079
$5$	1.26527	0.565845	0.282922	−	0.959143i	$-0.408696\pi$
$5$	1.26527	0.565845	0.282922	+	0.959143i	$0.408696\pi$
$6$	2.38361	0.973103
$7$	−2.62916	−0.993730	−0.496865	−	0.867828i	$-0.665516\pi$
$7$	−2.62916	−0.993730	−0.496865	+	0.867828i	$0.665516\pi$
$8$	−4.00822	−1.41712
$9$	1.00000	0.333333
$10$	−3.01590	−0.953711
$11$	0.984672	0.296890	0.148445	−	0.988921i	$-0.452573\pi$
$11$	0.984672	0.296890	0.148445	+	0.988921i	$0.452573\pi$
$12$	−3.68158	−1.06278
$13$	0.212001	0.0587984	0.0293992	−	0.999568i	$-0.490641\pi$
$13$	0.212001	0.0587984	0.0293992	+	0.999568i	$0.490641\pi$
$14$	6.26689	1.67490
$15$	−1.26527	−0.326691
$16$	2.19087	0.547717
$17$	5.90052	1.43109	0.715544	−	0.698568i	$-0.246179\pi$
$17$	5.90052	1.43109	0.715544	+	0.698568i	$0.246179\pi$
$18$	−2.38361	−0.561821
$19$	1.74925	0.401306	0.200653	−	0.979662i	$-0.435694\pi$
$19$	1.74925	0.401306	0.200653	+	0.979662i	$0.435694\pi$
$20$	4.65818	1.04160
$21$	2.62916	0.573731
$22$	−2.34707	−0.500397
$23$	0.0254742	0.00531173	0.00265587	−	0.999996i	$-0.499155\pi$
$23$	0.0254742	0.00531173	0.00265587	+	0.999996i	$0.499155\pi$
$24$	4.00822	0.818175
$25$	−3.39910	−0.679820
$26$	−0.505326	−0.0991026
$27$	−1.00000	−0.192450
$28$	−9.67947	−1.82925
$29$	−0.0447304	−0.00830624	−0.00415312	−	0.999991i	$-0.501322\pi$
$29$	−0.0447304	−0.00830624	−0.00415312	+	0.999991i	$0.501322\pi$
$30$	3.01590	0.550625
$31$	2.23264	0.400993	0.200497	−	0.979694i	$-0.435744\pi$
$31$	2.23264	0.400993	0.200497	+	0.979694i	$0.435744\pi$
$32$	2.79428	0.493964
$33$	−0.984672	−0.171409
$34$	−14.0645	−2.41205
$35$	−3.32659	−0.562297
$36$	3.68158	0.613597
$37$	−3.75890	−0.617960	−0.308980	−	0.951069i	$-0.599988\pi$
$37$	−3.75890	−0.617960	−0.308980	+	0.951069i	$0.599988\pi$
$38$	−4.16953	−0.676387
$39$	−0.212001	−0.0339473
$40$	−5.07147	−0.801870
$41$	−4.82004	−0.752765	−0.376382	−	0.926464i	$-0.622832\pi$
$41$	−4.82004	−0.752765	−0.376382	+	0.926464i	$0.622832\pi$
$42$	−6.26689	−0.967002
$43$	7.36351	1.12293	0.561463	−	0.827502i	$-0.310239\pi$
$43$	7.36351	1.12293	0.561463	+	0.827502i	$0.310239\pi$
$44$	3.62515	0.546512
$45$	1.26527	0.188615
$46$	−0.0607204	−0.00895274
$47$	−1.50657	−0.219757	−0.109878	−	0.993945i	$-0.535046\pi$
$47$	−1.50657	−0.219757	−0.109878	+	0.993945i	$0.535046\pi$
$48$	−2.19087	−0.316225
$49$	−0.0874982	−0.0124997
$50$	8.10211	1.14581
$51$	−5.90052	−0.826239
$52$	0.780498	0.108236
$53$	−7.98354	−1.09662	−0.548312	−	0.836274i	$-0.684729\pi$
$53$	−7.98354	−1.09662	−0.548312	+	0.836274i	$0.684729\pi$
$54$	2.38361	0.324368
$55$	1.24587	0.167994
$56$	10.5383	1.40824
$57$	−1.74925	−0.231694
$58$	0.106620	0.0139999
$59$	−3.00731	−0.391518	−0.195759	−	0.980652i	$-0.562717\pi$
$59$	−3.00731	−0.391518	−0.195759	+	0.980652i	$0.562717\pi$
$60$	−4.65818	−0.601369
$61$	−4.45462	−0.570355	−0.285178	−	0.958475i	$-0.592053\pi$
$61$	−4.45462	−0.570355	−0.285178	+	0.958475i	$0.592053\pi$
$62$	−5.32173	−0.675860
$63$	−2.62916	−0.331243
$64$	−11.0422	−1.38027
$65$	0.268238	0.0332708
$66$	2.34707	0.288905
$67$	−8.16233	−0.997187	−0.498594	−	0.866836i	$-0.666150\pi$
$67$	−8.16233	−0.997187	−0.498594	+	0.866836i	$0.666150\pi$
$68$	21.7232	2.63433
$69$	−0.0254742	−0.00306673
$70$	7.92929	0.947732
$71$	1.36050	0.161462	0.0807309	−	0.996736i	$-0.474275\pi$
$71$	1.36050	0.161462	0.0807309	+	0.996736i	$0.474275\pi$
$72$	−4.00822	−0.472374
$73$	15.5054	1.81477	0.907385	−	0.420300i	$-0.138075\pi$
$73$	15.5054	1.81477	0.907385	+	0.420300i	$0.138075\pi$
$74$	8.95974	1.04155
$75$	3.39910	0.392494
$76$	6.44001	0.738720
$77$	−2.58887	−0.295029
$78$	0.505326	0.0572169
$79$	−7.62687	−0.858090	−0.429045	−	0.903283i	$-0.641150\pi$
$79$	−7.62687	−0.858090	−0.429045	+	0.903283i	$0.641150\pi$
$80$	2.77203	0.309923
$81$	1.00000	0.111111
$82$	11.4891	1.26876
$83$	−7.20430	−0.790775	−0.395387	−	0.918514i	$-0.629390\pi$
$83$	−7.20430	−0.790775	−0.395387	+	0.918514i	$0.629390\pi$
$84$	9.67947	1.05612
$85$	7.46574	0.809773
$86$	−17.5517	−1.89265
$87$	0.0447304	0.00479561
$88$	−3.94679	−0.420729
$89$	−4.18899	−0.444032	−0.222016	−	0.975043i	$-0.571264\pi$
$89$	−4.18899	−0.444032	−0.222016	+	0.975043i	$0.571264\pi$
$90$	−3.01590	−0.317904
$91$	−0.557385	−0.0584298
$92$	0.0937852	0.00977779
$93$	−2.23264	−0.231514
$94$	3.59108	0.370392
$95$	2.21327	0.227077
$96$	−2.79428	−0.285190
$97$	8.03021	0.815345	0.407672	−	0.913128i	$-0.366341\pi$
$97$	8.03021	0.815345	0.407672	+	0.913128i	$0.366341\pi$
$98$	0.208561	0.0210679
$99$	0.984672	0.0989633
$100$	−12.5141	−1.25141
$101$	−12.5602	−1.24978	−0.624891	−	0.780712i	$-0.714857\pi$
$101$	−12.5602	−1.24978	−0.624891	+	0.780712i	$0.714857\pi$
$102$	14.0645	1.39260
$103$	9.71341	0.957091	0.478546	−	0.878063i	$-0.341164\pi$
$103$	9.71341	0.957091	0.478546	+	0.878063i	$0.341164\pi$
$104$	−0.849746	−0.0833245
$105$	3.32659	0.324642
$106$	19.0296	1.84832
$107$	13.4775	1.30292	0.651459	−	0.758684i	$-0.274157\pi$
$107$	13.4775	1.30292	0.651459	+	0.758684i	$0.274157\pi$
$108$	−3.68158	−0.354260
$109$	7.29056	0.698309	0.349155	−	0.937065i	$-0.386469\pi$
$109$	7.29056	0.698309	0.349155	+	0.937065i	$0.386469\pi$
$110$	−2.96967	−0.283147
$111$	3.75890	0.356779
$112$	−5.76015	−0.544283
$113$	−4.84038	−0.455345	−0.227672	−	0.973738i	$-0.573112\pi$
$113$	−4.84038	−0.455345	−0.227672	+	0.973738i	$0.573112\pi$
$114$	4.16953	0.390512
$115$	0.0322316	0.00300562
$116$	−0.164679	−0.0152900
$117$	0.212001	0.0195995
$118$	7.16824	0.659890
$119$	−15.5134	−1.42211
$120$	5.07147	0.462960
$121$	−10.0304	−0.911856
$122$	10.6181	0.961313
$123$	4.82004	0.434609
$124$	8.21963	0.738144
$125$	−10.6271	−0.950517
$126$	6.26689	0.558299
$127$	15.5247	1.37759	0.688797	−	0.724954i	$-0.258139\pi$
$127$	15.5247	1.37759	0.688797	+	0.724954i	$0.258139\pi$
$128$	20.7317	1.83244
$129$	−7.36351	−0.648321
$130$	−0.639373	−0.0560767
$131$	5.68888	0.497039	0.248520	−	0.968627i	$-0.420056\pi$
$131$	5.68888	0.497039	0.248520	+	0.968627i	$0.420056\pi$
$132$	−3.62515	−0.315529
$133$	−4.59907	−0.398790
$134$	19.4558	1.68072
$135$	−1.26527	−0.108897
$136$	−23.6506	−2.02802
$137$	−11.4539	−0.978576	−0.489288	−	0.872122i	$-0.662743\pi$
$137$	−11.4539	−0.978576	−0.489288	+	0.872122i	$0.662743\pi$
$138$	0.0607204	0.00516887
$139$	1.32674	0.112533	0.0562663	−	0.998416i	$-0.482080\pi$
$139$	1.32674	0.112533	0.0562663	+	0.998416i	$0.482080\pi$
$140$	−12.2471	−1.03507
$141$	1.50657	0.126876
$142$	−3.24290	−0.272138
$143$	0.208751	0.0174567
$144$	2.19087	0.182572
$145$	−0.0565960	−0.00470004
$146$	−36.9588	−3.05873
$147$	0.0874982	0.00721673
$148$	−13.8387	−1.13753
$149$	−9.07685	−0.743605	−0.371802	−	0.928312i	$-0.621260\pi$
$149$	−9.07685	−0.743605	−0.371802	+	0.928312i	$0.621260\pi$
$150$	−8.10211	−0.661535
$151$	5.08868	0.414111	0.207055	−	0.978329i	$-0.433612\pi$
$151$	5.08868	0.414111	0.207055	+	0.978329i	$0.433612\pi$
$152$	−7.01139	−0.568699
$153$	5.90052	0.477029
$154$	6.17084	0.497260
$155$	2.82488	0.226900
$156$	−0.780498	−0.0624898
$157$	6.23578	0.497670	0.248835	−	0.968546i	$-0.419952\pi$
$157$	6.23578	0.497670	0.248835	+	0.968546i	$0.419952\pi$
$158$	18.1795	1.44628
$159$	7.98354	0.633136
$160$	3.53551	0.279507
$161$	−0.0669758	−0.00527843
$162$	−2.38361	−0.187274
$163$	2.36233	0.185032	0.0925162	−	0.995711i	$-0.470509\pi$
$163$	2.36233	0.185032	0.0925162	+	0.995711i	$0.470509\pi$
$164$	−17.7454	−1.38568
$165$	−1.24587	−0.0969911
$166$	17.1722	1.33282
$167$	−5.76677	−0.446246	−0.223123	−	0.974790i	$-0.571625\pi$
$167$	−5.76677	−0.446246	−0.223123	+	0.974790i	$0.571625\pi$
$168$	−10.5383	−0.813046
$169$	−12.9551	−0.996543
$170$	−17.7954	−1.36484
$171$	1.74925	0.133769
$172$	27.1094	2.06707
$173$	15.9398	1.21188	0.605942	−	0.795509i	$-0.292796\pi$
$173$	15.9398	1.21188	0.605942	+	0.795509i	$0.292796\pi$
$174$	−0.106620	−0.00808282
$175$	8.93679	0.675558
$176$	2.15729	0.162612
$177$	3.00731	0.226043
$178$	9.98490	0.748400
$179$	−15.9744	−1.19398	−0.596991	−	0.802248i	$-0.703637\pi$
$179$	−15.9744	−1.19398	−0.596991	+	0.802248i	$0.703637\pi$
$180$	4.65818	0.347200
$181$	−6.97484	−0.518436	−0.259218	−	0.965819i	$-0.583465\pi$
$181$	−6.97484	−0.518436	−0.259218	+	0.965819i	$0.583465\pi$
$182$	1.32859	0.0984813
$183$	4.45462	0.329295
$184$	−0.102106	−0.00752737
$185$	−4.75602	−0.349669
$186$	5.32173	0.390208
$187$	5.81008	0.424875
$188$	−5.54657	−0.404526
$189$	2.62916	0.191244
$190$	−5.27557	−0.382730
$191$	−19.9892	−1.44636	−0.723182	−	0.690657i	$-0.757321\pi$
$191$	−19.9892	−1.44636	−0.723182	+	0.690657i	$0.757321\pi$
$192$	11.0422	0.796902
$193$	−20.3810	−1.46705	−0.733527	−	0.679661i	$-0.762127\pi$
$193$	−20.3810	−1.46705	−0.733527	+	0.679661i	$0.762127\pi$
$194$	−19.1409	−1.37423
$195$	−0.268238	−0.0192089
$196$	−0.322131	−0.0230094
$197$	−10.1221	−0.721169	−0.360584	−	0.932727i	$-0.617423\pi$
$197$	−10.1221	−0.721169	−0.360584	+	0.932727i	$0.617423\pi$
$198$	−2.34707	−0.166799
$199$	4.06969	0.288492	0.144246	−	0.989542i	$-0.453924\pi$
$199$	4.06969	0.288492	0.144246	+	0.989542i	$0.453924\pi$
$200$	13.6243	0.963387
$201$	8.16233	0.575726
$202$	29.9385	2.10646
$203$	0.117604	0.00825416
$204$	−21.7232	−1.52093
$205$	−6.09864	−0.425948
$206$	−23.1530	−1.61314
$207$	0.0254742	0.00177058
$208$	0.464466	0.0322049
$209$	1.72244	0.119144
$210$	−7.92929	−0.547173
$211$	−20.0058	−1.37726	−0.688628	−	0.725114i	$-0.741787\pi$
$211$	−20.0058	−1.37726	−0.688628	+	0.725114i	$0.741787\pi$
$212$	−29.3920	−2.01865
$213$	−1.36050	−0.0932201
$214$	−32.1250	−2.19602
$215$	9.31681	0.635401
$216$	4.00822	0.272725
$217$	−5.86997	−0.398479
$218$	−17.3778	−1.17698
$219$	−15.5054	−1.04776
$220$	4.58678	0.309241
$221$	1.25092	0.0841457
$222$	−8.95974	−0.601339
$223$	−4.64241	−0.310879	−0.155439	−	0.987845i	$-0.549679\pi$
$223$	−4.64241	−0.310879	−0.155439	+	0.987845i	$0.549679\pi$
$224$	−7.34662	−0.490867
$225$	−3.39910	−0.226607
$226$	11.5376	0.767467
$227$	5.27383	0.350036	0.175018	−	0.984565i	$-0.444002\pi$
$227$	5.27383	0.350036	0.175018	+	0.984565i	$0.444002\pi$
$228$	−6.44001	−0.426500
$229$	25.7910	1.70431	0.852157	−	0.523286i	$-0.175294\pi$
$229$	25.7910	1.70431	0.852157	+	0.523286i	$0.175294\pi$
$230$	−0.0768276	−0.00506586
$231$	2.58887	0.170335
$232$	0.179290	0.0117709
$233$	4.35070	0.285024	0.142512	−	0.989793i	$-0.454482\pi$
$233$	4.35070	0.285024	0.142512	+	0.989793i	$0.454482\pi$
$234$	−0.505326	−0.0330342
$235$	−1.90622	−0.124348
$236$	−11.0716	−0.720703
$237$	7.62687	0.495418
$238$	36.9779	2.39692
$239$	−4.07814	−0.263793	−0.131897	−	0.991263i	$-0.542107\pi$
$239$	−4.07814	−0.263793	−0.131897	+	0.991263i	$0.542107\pi$
$240$	−2.77203	−0.178934
$241$	4.89818	0.315520	0.157760	−	0.987478i	$-0.449573\pi$
$241$	4.89818	0.315520	0.157760	+	0.987478i	$0.449573\pi$
$242$	23.9086	1.53690
$243$	−1.00000	−0.0641500
$244$	−16.4000	−1.04990
$245$	−0.110709	−0.00707291
$246$	−11.4891	−0.732518
$247$	0.370843	0.0235962
$248$	−8.94890	−0.568256
$249$	7.20430	0.456554
$250$	25.3308	1.60206
$251$	22.9342	1.44760	0.723798	−	0.690012i	$-0.242395\pi$
$251$	22.9342	1.44760	0.723798	+	0.690012i	$0.242395\pi$
$252$	−9.67947	−0.609750
$253$	0.0250837	0.00157700
$254$	−37.0048	−2.32188
$255$	−7.46574	−0.467523
$256$	−27.3318	−1.70824
$257$	5.68607	0.354688	0.177344	−	0.984149i	$-0.443250\pi$
$257$	5.68607	0.354688	0.177344	+	0.984149i	$0.443250\pi$
$258$	17.5517	1.09272
$259$	9.88277	0.614086
$260$	0.987538	0.0612445
$261$	−0.0447304	−0.00276875
$262$	−13.5600	−0.837742
$263$	−6.19409	−0.381944	−0.190972	−	0.981595i	$-0.561164\pi$
$263$	−6.19409	−0.381944	−0.190972	+	0.981595i	$0.561164\pi$
$264$	3.94679	0.242908
$265$	−10.1013	−0.620519
$266$	10.9624	0.672146
$267$	4.18899	0.256362
$268$	−30.0503	−1.83561
$269$	−9.75177	−0.594576	−0.297288	−	0.954788i	$-0.596082\pi$
$269$	−9.75177	−0.594576	−0.297288	+	0.954788i	$0.596082\pi$
$270$	3.01590	0.183542
$271$	−8.58601	−0.521563	−0.260781	−	0.965398i	$-0.583980\pi$
$271$	−8.58601	−0.521563	−0.260781	+	0.965398i	$0.583980\pi$
$272$	12.9273	0.783831
$273$	0.557385	0.0337345
$274$	27.3017	1.64936
$275$	−3.34700	−0.201832
$276$	−0.0937852	−0.00564521
$277$	−6.39646	−0.384326	−0.192163	−	0.981363i	$-0.561550\pi$
$277$	−6.39646	−0.384326	−0.192163	+	0.981363i	$0.561550\pi$
$278$	−3.16243	−0.189670
$279$	2.23264	0.133664
$280$	13.3337	0.796843
$281$	−28.5867	−1.70534	−0.852669	−	0.522452i	$-0.825018\pi$
$281$	−28.5867	−1.70534	−0.852669	+	0.522452i	$0.825018\pi$
$282$	−3.59108	−0.213846
$283$	8.29181	0.492897	0.246448	−	0.969156i	$-0.420736\pi$
$283$	8.29181	0.492897	0.246448	+	0.969156i	$0.420736\pi$
$284$	5.00880	0.297217
$285$	−2.21327	−0.131103
$286$	−0.497581	−0.0294226
$287$	12.6727	0.748045
$288$	2.79428	0.164655
$289$	17.8162	1.04801
$290$	0.134902	0.00792175
$291$	−8.03021	−0.470740
$292$	57.0844	3.34061
$293$	22.3971	1.30845	0.654226	−	0.756299i	$-0.272994\pi$
$293$	22.3971	1.30845	0.654226	+	0.756299i	$0.272994\pi$
$294$	−0.208561	−0.0121635
$295$	−3.80505	−0.221538
$296$	15.0665	0.875724
$297$	−0.984672	−0.0571365
$298$	21.6356	1.25332
$299$	0.00540055	0.000312322 0
$300$	12.5141	0.722499
$301$	−19.3599	−1.11589
$302$	−12.1294	−0.697969
$303$	12.5602	0.721562
$304$	3.83238	0.219802
$305$	−5.63628	−0.322732
$306$	−14.0645	−0.804015
$307$	33.0830	1.88814	0.944072	−	0.329739i	$-0.106961\pi$
$307$	33.0830	1.88814	0.944072	+	0.329739i	$0.106961\pi$
$308$	−9.53111	−0.543086
$309$	−9.71341	−0.552577
$310$	−6.73340	−0.382432
$311$	18.6706	1.05871	0.529356	−	0.848400i	$-0.322433\pi$
$311$	18.6706	1.05871	0.529356	+	0.848400i	$0.322433\pi$
$312$	0.849746	0.0481074
$313$	−8.71587	−0.492650	−0.246325	−	0.969187i	$-0.579223\pi$
$313$	−8.71587	−0.492650	−0.246325	+	0.969187i	$0.579223\pi$
$314$	−14.8637	−0.838805
$315$	−3.32659	−0.187432
$316$	−28.0789	−1.57956
$317$	0.441949	0.0248223	0.0124112	−	0.999923i	$-0.496049\pi$
$317$	0.441949	0.0248223	0.0124112	+	0.999923i	$0.496049\pi$
$318$	−19.0296	−1.06713
$319$	−0.0440448	−0.00246604
$320$	−13.9713	−0.781021
$321$	−13.4775	−0.752240
$322$	0.159644	0.00889661
$323$	10.3215	0.574304
$324$	3.68158	0.204532
$325$	−0.720612	−0.0399723
$326$	−5.63088	−0.311865
$327$	−7.29056	−0.403169
$328$	19.3198	1.06676
$329$	3.96103	0.218379
$330$	2.96967	0.163475
$331$	22.9258	1.26011	0.630057	−	0.776549i	$-0.283031\pi$
$331$	22.9258	1.26011	0.630057	+	0.776549i	$0.283031\pi$
$332$	−26.5232	−1.45565
$333$	−3.75890	−0.205987
$334$	13.7457	0.752132
$335$	−10.3275	−0.564253
$336$	5.76015	0.314242
$337$	0.703681	0.0383319	0.0191660	−	0.999816i	$-0.493899\pi$
$337$	0.703681	0.0383319	0.0191660	+	0.999816i	$0.493899\pi$
$338$	30.8798	1.67964
$339$	4.84038	0.262893
$340$	27.4857	1.49062
$341$	2.19842	0.119051
$342$	−4.16953	−0.225462
$343$	18.6342	1.00615
$344$	−29.5146	−1.59132
$345$	−0.0322316	−0.00173529
$346$	−37.9943	−2.04259
$347$	−31.7267	−1.70318	−0.851588	−	0.524211i	$-0.824360\pi$
$347$	−31.7267	−1.70318	−0.851588	+	0.524211i	$0.824360\pi$
$348$	0.164679	0.00882770
$349$	−16.4420	−0.880117	−0.440059	−	0.897969i	$-0.645042\pi$
$349$	−16.4420	−0.880117	−0.440059	+	0.897969i	$0.645042\pi$
$350$	−21.3018	−1.13863
$351$	−0.212001	−0.0113158
$352$	2.75145	0.146653
$353$	−15.4521	−0.822433	−0.411216	−	0.911538i	$-0.634896\pi$
$353$	−15.4521	−0.822433	−0.411216	+	0.911538i	$0.634896\pi$
$354$	−7.16824	−0.380988
$355$	1.72140	0.0913623
$356$	−15.4221	−0.817370
$357$	15.5134	0.821058
$358$	38.0767	2.01241
$359$	8.84563	0.466855	0.233427	−	0.972374i	$-0.425006\pi$
$359$	8.84563	0.466855	0.233427	+	0.972374i	$0.425006\pi$
$360$	−5.07147	−0.267290
$361$	−15.9401	−0.838954
$362$	16.6253	0.873805
$363$	10.0304	0.526461
$364$	−2.05206	−0.107557
$365$	19.6185	1.02688
$366$	−10.6181	−0.555014
$367$	28.3974	1.48233	0.741167	−	0.671320i	$-0.234273\pi$
$367$	28.3974	1.48233	0.741167	+	0.671320i	$0.234273\pi$
$368$	0.0558106	0.00290933
$369$	−4.82004	−0.250922
$370$	11.3365	0.589355
$371$	20.9900	1.08975
$372$	−8.21963	−0.426168
$373$	21.3029	1.10302	0.551510	−	0.834168i	$-0.314052\pi$
$373$	21.3029	1.10302	0.551510	+	0.834168i	$0.314052\pi$
$374$	−13.8490	−0.716112
$375$	10.6271	0.548781
$376$	6.03869	0.311422
$377$	−0.00948289	−0.000488394 0
$378$	−6.26689	−0.322334
$379$	19.8354	1.01888	0.509438	−	0.860507i	$-0.329853\pi$
$379$	19.8354	1.01888	0.509438	+	0.860507i	$0.329853\pi$
$380$	8.14833	0.418001
$381$	−15.5247	−0.795354
$382$	47.6463	2.43780
$383$	−27.0641	−1.38291	−0.691455	−	0.722419i	$-0.743030\pi$
$383$	−27.0641	−1.38291	−0.691455	+	0.722419i	$0.743030\pi$
$384$	−20.7317	−1.05796
$385$	−3.27561	−0.166940
$386$	48.5802	2.47267
$387$	7.36351	0.374308
$388$	29.5639	1.50088
$389$	14.8738	0.754133	0.377067	−	0.926186i	$-0.376933\pi$
$389$	14.8738	0.754133	0.377067	+	0.926186i	$0.376933\pi$
$390$	0.639373	0.0323759
$391$	0.150311	0.00760155
$392$	0.350712	0.0177136
$393$	−5.68888	−0.286966
$394$	24.1271	1.21550
$395$	−9.65003	−0.485546
$396$	3.62515	0.182171
$397$	−0.0189198	−0.000949555 0	−0.000474778	−	1.00000i	$-0.500151\pi$
$397$	−0.0189198	−0.000949555 0	−0.000474778		1.00000i	$0.500151\pi$
$398$	−9.70053	−0.486244
$399$	4.59907	0.230241
$400$	−7.44698	−0.372349
$401$	19.7575	0.986644	0.493322	−	0.869847i	$-0.335782\pi$
$401$	19.7575	0.986644	0.493322	+	0.869847i	$0.335782\pi$
$402$	−19.4558	−0.970366
$403$	0.473320	0.0235778
$404$	−46.2412	−2.30059
$405$	1.26527	0.0628716
$406$	−0.280321	−0.0139121
$407$	−3.70129	−0.183466
$408$	23.6506	1.17088
$409$	21.1202	1.04433	0.522164	−	0.852845i	$-0.325125\pi$
$409$	21.1202	1.04433	0.522164	+	0.852845i	$0.325125\pi$
$410$	14.5368	0.717920
$411$	11.4539	0.564981
$412$	35.7607	1.76180
$413$	7.90671	0.389064
$414$	−0.0607204	−0.00298425
$415$	−9.11536	−0.447456
$416$	0.592389	0.0290443
$417$	−1.32674	−0.0649708
$418$	−4.10562	−0.200812
$419$	−20.6732	−1.00995	−0.504976	−	0.863133i	$-0.668499\pi$
$419$	−20.6732	−1.00995	−0.504976	+	0.863133i	$0.668499\pi$
$420$	12.2471	0.597598
$421$	39.8598	1.94265	0.971323	−	0.237763i	$-0.0764142\pi$
$421$	39.8598	1.94265	0.971323	+	0.237763i	$0.0764142\pi$
$422$	47.6860	2.32132
$423$	−1.50657	−0.0732522
$424$	31.9998	1.55405
$425$	−20.0565	−0.972881
$426$	3.24290	0.157119
$427$	11.7119	0.566779
$428$	49.6185	2.39840
$429$	−0.208751	−0.0100786
$430$	−22.2076	−1.07095
$431$	19.0618	0.918177	0.459088	−	0.888391i	$-0.348176\pi$
$431$	19.0618	0.918177	0.459088	+	0.888391i	$0.348176\pi$
$432$	−2.19087	−0.105408
$433$	−8.44226	−0.405709	−0.202855	−	0.979209i	$-0.565022\pi$
$433$	−8.44226	−0.405709	−0.202855	+	0.979209i	$0.565022\pi$
$434$	13.9917	0.671622
$435$	0.0565960	0.00271357
$436$	26.8408	1.28544
$437$	0.0445608	0.00213163
$438$	36.9588	1.76596
$439$	−17.2540	−0.823489	−0.411744	−	0.911299i	$-0.635080\pi$
$439$	−17.2540	−0.823489	−0.411744	+	0.911299i	$0.635080\pi$
$440$	−4.99374	−0.238067
$441$	−0.0874982	−0.00416658
$442$	−2.98169	−0.141825
$443$	17.5045	0.831662	0.415831	−	0.909442i	$-0.363491\pi$
$443$	17.5045	0.831662	0.415831	+	0.909442i	$0.363491\pi$
$444$	13.8387	0.656756
$445$	−5.30019	−0.251253
$446$	11.0657	0.523975
$447$	9.07685	0.429321
$448$	29.0318	1.37162
$449$	−24.6950	−1.16543	−0.582716	−	0.812676i	$-0.698010\pi$
$449$	−24.6950	−1.16543	−0.582716	+	0.812676i	$0.698010\pi$
$450$	8.10211	0.381937
$451$	−4.74617	−0.223488
$452$	−17.8202	−0.838194
$453$	−5.08868	−0.239087
$454$	−12.5707	−0.589973
$455$	−0.705240	−0.0330622
$456$	7.01139	0.328338
$457$	−11.0391	−0.516386	−0.258193	−	0.966093i	$-0.583127\pi$
$457$	−11.0391	−0.516386	−0.258193	+	0.966093i	$0.583127\pi$
$458$	−61.4755	−2.87256
$459$	−5.90052	−0.275413
$460$	0.118663	0.00553271
$461$	−3.85376	−0.179488	−0.0897438	−	0.995965i	$-0.528605\pi$
$461$	−3.85376	−0.179488	−0.0897438	+	0.995965i	$0.528605\pi$
$462$	−6.17084	−0.287093
$463$	−5.57471	−0.259079	−0.129539	−	0.991574i	$-0.541350\pi$
$463$	−5.57471	−0.259079	−0.129539	+	0.991574i	$0.541350\pi$
$464$	−0.0979985	−0.00454947
$465$	−2.82488	−0.131001
$466$	−10.3704	−0.480398
$467$	8.06486	0.373197	0.186599	−	0.982436i	$-0.440254\pi$
$467$	8.06486	0.373197	0.186599	+	0.982436i	$0.440254\pi$
$468$	0.780498	0.0360785
$469$	21.4601	0.990936
$470$	4.54368	0.209584
$471$	−6.23578	−0.287330
$472$	12.0540	0.554829
$473$	7.25065	0.333385
$474$	−18.1795	−0.835010
$475$	−5.94588	−0.272816
$476$	−57.1140	−2.61781
$477$	−7.98354	−0.365541
$478$	9.72069	0.444614
$479$	17.7700	0.811932	0.405966	−	0.913888i	$-0.366935\pi$
$479$	17.7700	0.811932	0.405966	+	0.913888i	$0.366935\pi$
$480$	−3.53551	−0.161373
$481$	−0.796890	−0.0363351
$482$	−11.6753	−0.531797
$483$	0.0669758	0.00304750
$484$	−36.9278	−1.67854
$485$	10.1604	0.461358
$486$	2.38361	0.108123
$487$	−1.26263	−0.0572153	−0.0286077	−	0.999591i	$-0.509107\pi$
$487$	−1.26263	−0.0572153	−0.0286077	+	0.999591i	$0.509107\pi$
$488$	17.8551	0.808262
$489$	−2.36233	−0.106828
$490$	0.263886	0.0119211
$491$	−25.5279	−1.15206	−0.576030	−	0.817429i	$-0.695399\pi$
$491$	−25.5279	−1.15206	−0.576030	+	0.817429i	$0.695399\pi$
$492$	17.7454	0.800023
$493$	−0.263933	−0.0118869
$494$	−0.883943	−0.0397705
$495$	1.24587	0.0559979
$496$	4.89141	0.219631
$497$	−3.57698	−0.160450
$498$	−17.1722	−0.769506
$499$	−16.5547	−0.741092	−0.370546	−	0.928814i	$-0.620829\pi$
$499$	−16.5547	−0.741092	−0.370546	+	0.928814i	$0.620829\pi$
$500$	−39.1245	−1.74970
$501$	5.76677	0.257640
$502$	−54.6662	−2.43987
$503$	−11.7107	−0.522153	−0.261077	−	0.965318i	$-0.584078\pi$
$503$	−11.7107	−0.522153	−0.261077	+	0.965318i	$0.584078\pi$
$504$	10.5383	0.469412
$505$	−15.8920	−0.707183
$506$	−0.0597897	−0.00265798
$507$	12.9551	0.575354
$508$	57.1554	2.53586
$509$	−33.2163	−1.47229	−0.736144	−	0.676825i	$-0.763355\pi$
$509$	−33.2163	−1.47229	−0.736144	+	0.676825i	$0.763355\pi$
$510$	17.7954	0.787993
$511$	−40.7662	−1.80339
$512$	23.6849	1.04673
$513$	−1.74925	−0.0772314
$514$	−13.5534	−0.597813
$515$	12.2901	0.541565
$516$	−27.1094	−1.19342
$517$	−1.48348	−0.0652435
$518$	−23.5566	−1.03502
$519$	−15.9398	−0.699681
$520$	−1.07516	−0.0471487
$521$	34.7931	1.52431	0.762156	−	0.647394i	$-0.224141\pi$
$521$	34.7931	1.52431	0.762156	+	0.647394i	$0.224141\pi$
$522$	0.106620	0.00466662
$523$	−38.3539	−1.67710	−0.838550	−	0.544825i	$-0.816596\pi$
$523$	−38.3539	−1.67710	−0.838550	+	0.544825i	$0.816596\pi$
$524$	20.9440	0.914945
$525$	−8.93679	−0.390033
$526$	14.7643	0.643753
$527$	13.1737	0.573856
$528$	−2.15729	−0.0938839
$529$	−22.9994	−0.999972
$530$	24.0775	1.04586
$531$	−3.00731	−0.130506
$532$	−16.9318	−0.734088
$533$	−1.02185	−0.0442614
$534$	−9.98490	−0.432089
$535$	17.0526	0.737249
$536$	32.7164	1.41314
$537$	15.9744	0.689346
$538$	23.2444	1.00214
$539$	−0.0861570	−0.00371105
$540$	−4.65818	−0.200456
$541$	14.2892	0.614340	0.307170	−	0.951655i	$-0.400618\pi$
$541$	14.2892	0.614340	0.307170	+	0.951655i	$0.400618\pi$
$542$	20.4657	0.879075
$543$	6.97484	0.299319
$544$	16.4877	0.706905
$545$	9.22451	0.395134
$546$	−1.32859	−0.0568582
$547$	−33.9696	−1.45243	−0.726217	−	0.687466i	$-0.758723\pi$
$547$	−33.9696	−1.45243	−0.726217	+	0.687466i	$0.758723\pi$
$548$	−42.1686	−1.80135
$549$	−4.45462	−0.190118
$550$	7.97793	0.340180
$551$	−0.0782448	−0.00333334
$552$	0.102106	0.00434593
$553$	20.0523	0.852710
$554$	15.2466	0.647767
$555$	4.75602	0.201882
$556$	4.88450	0.207149
$557$	−24.9876	−1.05876	−0.529380	−	0.848385i	$-0.677575\pi$
$557$	−24.9876	−1.05876	−0.529380	+	0.848385i	$0.677575\pi$
$558$	−5.32173	−0.225287
$559$	1.56107	0.0660262
$560$	−7.28813	−0.307980
$561$	−5.81008	−0.245302
$562$	68.1394	2.87429
$563$	43.2661	1.82345	0.911724	−	0.410803i	$-0.134752\pi$
$563$	43.2661	1.82345	0.911724	+	0.410803i	$0.134752\pi$
$564$	5.54657	0.233553
$565$	−6.12438	−0.257654
$566$	−19.7644	−0.830760
$567$	−2.62916	−0.110414
$568$	−5.45320	−0.228811
$569$	−33.5116	−1.40488	−0.702439	−	0.711744i	$-0.747906\pi$
$569$	−33.5116	−1.40488	−0.702439	+	0.711744i	$0.747906\pi$
$570$	5.27557	0.220969
$571$	1.81791	0.0760773	0.0380386	−	0.999276i	$-0.487889\pi$
$571$	1.81791	0.0760773	0.0380386	+	0.999276i	$0.487889\pi$
$572$	0.768534	0.0321340
$573$	19.9892	0.835059
$574$	−30.2067	−1.26080
$575$	−0.0865893	−0.00361102
$576$	−11.0422	−0.460092
$577$	−11.0886	−0.461626	−0.230813	−	0.972998i	$-0.574139\pi$
$577$	−11.0886	−0.461626	−0.230813	+	0.972998i	$0.574139\pi$
$578$	−42.4668	−1.76638
$579$	20.3810	0.847003
$580$	−0.208363	−0.00865178
$581$	18.9413	0.785817
$582$	19.1409	0.793415
$583$	−7.86117	−0.325576
$584$	−62.1491	−2.57175
$585$	0.268238	0.0110903
$586$	−53.3859	−2.20535
$587$	−42.1517	−1.73979	−0.869894	−	0.493238i	$-0.835813\pi$
$587$	−42.1517	−1.73979	−0.869894	+	0.493238i	$0.835813\pi$
$588$	0.322131	0.0132845
$589$	3.90544	0.160921
$590$	9.06974	0.373395
$591$	10.1221	0.416367
$592$	−8.23526	−0.338467
$593$	36.2865	1.49011	0.745054	−	0.667004i	$-0.232424\pi$
$593$	36.2865	1.49011	0.745054	+	0.667004i	$0.232424\pi$
$594$	2.34707	0.0963015
$595$	−19.6286	−0.804696
$596$	−33.4172	−1.36882
$597$	−4.06969	−0.166561
$598$	−0.0128728	−0.000526407 0
$599$	23.3534	0.954194	0.477097	−	0.878851i	$-0.341689\pi$
$599$	23.3534	0.954194	0.477097	+	0.878851i	$0.341689\pi$
$600$	−13.6243	−0.556212
$601$	−35.4552	−1.44625	−0.723123	−	0.690719i	$-0.757294\pi$
$601$	−35.4552	−1.44625	−0.723123	+	0.690719i	$0.757294\pi$
$602$	46.1463	1.88078
$603$	−8.16233	−0.332396
$604$	18.7344	0.762291
$605$	−12.6912	−0.515969
$606$	−29.9385	−1.21617
$607$	−5.92184	−0.240360	−0.120180	−	0.992752i	$-0.538347\pi$
$607$	−5.92184	−0.240360	−0.120180	+	0.992752i	$0.538347\pi$
$608$	4.88790	0.198231
$609$	−0.117604	−0.00476554
$610$	13.4347	0.543954
$611$	−0.319395	−0.0129213
$612$	21.7232	0.878110
$613$	−24.3588	−0.983842	−0.491921	−	0.870640i	$-0.663705\pi$
$613$	−24.3588	−0.983842	−0.491921	+	0.870640i	$0.663705\pi$
$614$	−78.8568	−3.18240
$615$	6.09864	0.245921
$616$	10.3767	0.418091
$617$	−38.7685	−1.56076	−0.780381	−	0.625305i	$-0.784975\pi$
$617$	−38.7685	−1.56076	−0.780381	+	0.625305i	$0.784975\pi$
$618$	23.1530	0.931349
$619$	−38.3424	−1.54111	−0.770556	−	0.637372i	$-0.780021\pi$
$619$	−38.3424	−1.54111	−0.770556	+	0.637372i	$0.780021\pi$
$620$	10.4000	0.417675
$621$	−0.0254742	−0.00102224
$622$	−44.5034	−1.78442
$623$	11.0135	0.441248
$624$	−0.464466	−0.0185935
$625$	3.54937	0.141975
$626$	20.7752	0.830344
$627$	−1.72244	−0.0687876
$628$	22.9575	0.916105
$629$	−22.1795	−0.884354
$630$	7.92929	0.315911
$631$	13.3398	0.531048	0.265524	−	0.964104i	$-0.414455\pi$
$631$	13.3398	0.531048	0.265524	+	0.964104i	$0.414455\pi$
$632$	30.5702	1.21602
$633$	20.0058	0.795160
$634$	−1.05343	−0.0418372
$635$	19.6429	0.779504
$636$	29.3920	1.16547
$637$	−0.0185497	−0.000734965 0
$638$	0.104986	0.00415642
$639$	1.36050	0.0538206
$640$	26.2311	1.03688
$641$	13.7955	0.544890	0.272445	−	0.962171i	$-0.412168\pi$
$641$	13.7955	0.544890	0.272445	+	0.962171i	$0.412168\pi$
$642$	32.1250	1.26787
$643$	−9.12235	−0.359750	−0.179875	−	0.983689i	$-0.557569\pi$
$643$	−9.12235	−0.359750	−0.179875	+	0.983689i	$0.557569\pi$
$644$	−0.246577	−0.00971648
$645$	−9.31681	−0.366849
$646$	−24.6024	−0.967968
$647$	11.9529	0.469919	0.234959	−	0.972005i	$-0.424504\pi$
$647$	11.9529	0.469919	0.234959	+	0.972005i	$0.424504\pi$
$648$	−4.00822	−0.157458
$649$	−2.96121	−0.116238
$650$	1.71765	0.0673719
$651$	5.86997	0.230062
$652$	8.69712	0.340606
$653$	−14.6333	−0.572645	−0.286323	−	0.958133i	$-0.592433\pi$
$653$	−14.6333	−0.572645	−0.286323	+	0.958133i	$0.592433\pi$
$654$	17.3778	0.679527
$655$	7.19795	0.281247
$656$	−10.5601	−0.412302
$657$	15.5054	0.604923
$658$	−9.44154	−0.368070
$659$	−28.8840	−1.12516	−0.562581	−	0.826742i	$-0.690191\pi$
$659$	−28.8840	−1.12516	−0.562581	+	0.826742i	$0.690191\pi$
$660$	−4.58678	−0.178540
$661$	−32.5326	−1.26537	−0.632685	−	0.774409i	$-0.718047\pi$
$661$	−32.5326	−1.26537	−0.632685	+	0.774409i	$0.718047\pi$
$662$	−54.6460	−2.12388
$663$	−1.25092	−0.0485815
$664$	28.8764	1.12062
$665$	−5.81905	−0.225653
$666$	8.95974	0.347183
$667$	−0.00113947	−4.41205e−5 0
$668$	−21.2308	−0.821445
$669$	4.64241	0.179486
$670$	24.6168	0.951028
$671$	−4.38634	−0.169333
$672$	7.34662	0.283402
$673$	29.9047	1.15274	0.576371	−	0.817188i	$-0.304468\pi$
$673$	29.9047	1.15274	0.576371	+	0.817188i	$0.304468\pi$
$674$	−1.67730	−0.0646071
$675$	3.39910	0.130831
$676$	−47.6951	−1.83443
$677$	−25.8058	−0.991798	−0.495899	−	0.868380i	$-0.665161\pi$
$677$	−25.8058	−0.991798	−0.495899	+	0.868380i	$0.665161\pi$
$678$	−11.5376	−0.443098
$679$	−21.1127	−0.810233
$680$	−29.9243	−1.14755
$681$	−5.27383	−0.202093
$682$	−5.24016	−0.200656
$683$	−12.5277	−0.479360	−0.239680	−	0.970852i	$-0.577043\pi$
$683$	−12.5277	−0.479360	−0.239680	+	0.970852i	$0.577043\pi$
$684$	6.44001	0.246240
$685$	−14.4923	−0.553722
$686$	−44.4166	−1.69583
$687$	−25.7910	−0.983987
$688$	16.1325	0.615045
$689$	−1.69252	−0.0644797
$690$	0.0768276	0.00292477
$691$	28.6392	1.08949	0.544743	−	0.838603i	$-0.316627\pi$
$691$	28.6392	1.08949	0.544743	+	0.838603i	$0.316627\pi$
$692$	58.6838	2.23082
$693$	−2.58887	−0.0983429
$694$	75.6239	2.87064
$695$	1.67868	0.0636760
$696$	−0.179290	−0.00679595
$697$	−28.4408	−1.07727
$698$	39.1911	1.48341
$699$	−4.35070	−0.164559
$700$	32.9015	1.24356
$701$	−31.8172	−1.20172	−0.600859	−	0.799355i	$-0.705175\pi$
$701$	−31.8172	−1.20172	−0.600859	+	0.799355i	$0.705175\pi$
$702$	0.505326	0.0190723
$703$	−6.57527	−0.247991
$704$	−10.8729	−0.409790
$705$	1.90622	0.0717924
$706$	36.8317	1.38618
$707$	33.0227	1.24195
$708$	11.0716	0.416098
$709$	41.3799	1.55405	0.777027	−	0.629468i	$-0.216727\pi$
$709$	41.3799	1.55405	0.777027	+	0.629468i	$0.216727\pi$
$710$	−4.10314	−0.153988
$711$	−7.62687	−0.286030
$712$	16.7904	0.629247
$713$	0.0568746	0.00212997
$714$	−36.9779	−1.38386
$715$	0.264126	0.00987776
$716$	−58.8110	−2.19787
$717$	4.07814	0.152301
$718$	−21.0845	−0.786867
$719$	−16.8485	−0.628343	−0.314172	−	0.949366i	$-0.601727\pi$
$719$	−16.8485	−0.628343	−0.314172	+	0.949366i	$0.601727\pi$
$720$	2.77203	0.103308
$721$	−25.5382	−0.951091
$722$	37.9950	1.41403
$723$	−4.89818	−0.182165
$724$	−25.6784	−0.954331
$725$	0.152043	0.00564674
$726$	−23.9086	−0.887330
$727$	0.520620	0.0193087	0.00965436	−	0.999953i	$-0.496927\pi$
$727$	0.520620	0.0193087	0.00965436	+	0.999953i	$0.496927\pi$
$728$	2.23412	0.0828021
$729$	1.00000	0.0370370
$730$	−46.7627	−1.73077
$731$	43.4486	1.60700
$732$	16.4000	0.606162
$733$	−33.2525	−1.22821	−0.614104	−	0.789225i	$-0.710482\pi$
$733$	−33.2525	−1.22821	−0.614104	+	0.789225i	$0.710482\pi$
$734$	−67.6883	−2.49842
$735$	0.110709	0.00408355
$736$	0.0711820	0.00262380
$737$	−8.03722	−0.296055
$738$	11.4891	0.422919
$739$	14.9782	0.550983	0.275491	−	0.961304i	$-0.411159\pi$
$739$	14.9782	0.550983	0.275491	+	0.961304i	$0.411159\pi$
$740$	−17.5096	−0.643668
$741$	−0.370843	−0.0136232
$742$	−50.0320	−1.83673
$743$	−14.9671	−0.549091	−0.274545	−	0.961574i	$-0.588527\pi$
$743$	−14.9671	−0.549091	−0.274545	+	0.961574i	$0.588527\pi$
$744$	8.94890	0.328083
$745$	−11.4846	−0.420765
$746$	−50.7776	−1.85910
$747$	−7.20430	−0.263592
$748$	21.3903	0.782106
$749$	−35.4345	−1.29475
$750$	−25.3308	−0.924951
$751$	−12.9314	−0.471873	−0.235936	−	0.971769i	$-0.575816\pi$
$751$	−12.9314	−0.471873	−0.235936	+	0.971769i	$0.575816\pi$
$752$	−3.30071	−0.120364
$753$	−22.9342	−0.835770
$754$	0.0226035	0.000823170 0
$755$	6.43853	0.234322
$756$	9.67947	0.352039
$757$	5.22075	0.189752	0.0948758	−	0.995489i	$-0.469755\pi$
$757$	5.22075	0.189752	0.0948758	+	0.995489i	$0.469755\pi$
$758$	−47.2798	−1.71728
$759$	−0.0250837	−0.000910482 0
$760$	−8.87128	−0.321795
$761$	−47.2107	−1.71139	−0.855694	−	0.517483i	$-0.826869\pi$
$761$	−47.2107	−1.71139	−0.855694	+	0.517483i	$0.826869\pi$
$762$	37.0048	1.34054
$763$	−19.1681	−0.693931
$764$	−73.5917	−2.66245
$765$	7.46574	0.269924
$766$	64.5102	2.33085
$767$	−0.637552	−0.0230207
$768$	27.3318	0.986252
$769$	−15.7529	−0.568065	−0.284032	−	0.958815i	$-0.591672\pi$
$769$	−15.7529	−0.568065	−0.284032	+	0.958815i	$0.591672\pi$
$770$	7.80775	0.281372
$771$	−5.68607	−0.204779
$772$	−75.0341	−2.70054
$773$	7.49921	0.269728	0.134864	−	0.990864i	$-0.456940\pi$
$773$	7.49921	0.269728	0.134864	+	0.990864i	$0.456940\pi$
$774$	−17.5517	−0.630883
$775$	−7.58895	−0.272603
$776$	−32.1869	−1.15544
$777$	−9.88277	−0.354542
$778$	−35.4534	−1.27106
$779$	−8.43147	−0.302089
$780$	−0.987538	−0.0353595
$781$	1.33965	0.0479364
$782$	−0.358282	−0.0128121
$783$	0.0447304	0.00159854
$784$	−0.191697	−0.00684632
$785$	7.88993	0.281604
$786$	13.5600	0.483671
$787$	−4.09826	−0.146087	−0.0730436	−	0.997329i	$-0.523271\pi$
$787$	−4.09826	−0.146087	−0.0730436	+	0.997329i	$0.523271\pi$
$788$	−37.2653	−1.32752
$789$	6.19409	0.220516
$790$	23.0019	0.818370
$791$	12.7262	0.452490
$792$	−3.94679	−0.140243
$793$	−0.944382	−0.0335360
$794$	0.0450973	0.00160044
$795$	10.1013	0.358257
$796$	14.9829	0.531054
$797$	28.4906	1.00919	0.504595	−	0.863356i	$-0.331642\pi$
$797$	28.4906	1.00919	0.504595	+	0.863356i	$0.331642\pi$
$798$	−10.9624	−0.388064
$799$	−8.88958	−0.314491
$800$	−9.49804	−0.335806
$801$	−4.18899	−0.148011
$802$	−47.0942	−1.66295
$803$	15.2677	0.538787
$804$	30.0503	1.05979
$805$	−0.0847423	−0.00298677
$806$	−1.12821	−0.0397395
$807$	9.75177	0.343279
$808$	50.3439	1.77109
$809$	−19.2084	−0.675331	−0.337666	−	0.941266i	$-0.609637\pi$
$809$	−19.2084	−0.675331	−0.337666	+	0.941266i	$0.609637\pi$
$810$	−3.01590	−0.105968
$811$	1.38842	0.0487540	0.0243770	−	0.999703i	$-0.492240\pi$
$811$	1.38842	0.0487540	0.0243770	+	0.999703i	$0.492240\pi$
$812$	0.432967	0.0151942
$813$	8.58601	0.301124
$814$	8.82241	0.309225
$815$	2.98898	0.104700
$816$	−12.9273	−0.452545
$817$	12.8806	0.450636
$818$	−50.3423	−1.76018
$819$	−0.557385	−0.0194766
$820$	−22.4526	−0.784080
$821$	−46.1247	−1.60976	−0.804881	−	0.593436i	$-0.797771\pi$
$821$	−46.1247	−1.60976	−0.804881	+	0.593436i	$0.797771\pi$
$822$	−27.3017	−0.952256
$823$	21.9749	0.765997	0.382999	−	0.923749i	$-0.374891\pi$
$823$	21.9749	0.765997	0.382999	+	0.923749i	$0.374891\pi$
$824$	−38.9335	−1.35631
$825$	3.34700	0.116528
$826$	−18.8465	−0.655753
$827$	−32.8344	−1.14176	−0.570881	−	0.821033i	$-0.693398\pi$
$827$	−32.8344	−1.14176	−0.570881	+	0.821033i	$0.693398\pi$
$828$	0.0937852	0.00325926
$829$	−27.0479	−0.939411	−0.469705	−	0.882823i	$-0.655640\pi$
$829$	−27.0479	−0.939411	−0.469705	+	0.882823i	$0.655640\pi$
$830$	21.7274	0.754171
$831$	6.39646	0.221891
$832$	−2.34095	−0.0811580
$833$	−0.516285	−0.0178882
$834$	3.16243	0.109506
$835$	−7.29650	−0.252506
$836$	6.34130	0.219318
$837$	−2.23264	−0.0771712
$838$	49.2768	1.70224
$839$	54.3231	1.87544	0.937720	−	0.347392i	$-0.112933\pi$
$839$	54.3231	1.87544	0.937720	+	0.347392i	$0.112933\pi$
$840$	−13.3337	−0.460057
$841$	−28.9980	−0.999931
$842$	−95.0100	−3.27426
$843$	28.5867	0.984577
$844$	−73.6530	−2.53524
$845$	−16.3916	−0.563888
$846$	3.59108	0.123464
$847$	26.3716	0.906139
$848$	−17.4909	−0.600639
$849$	−8.29181	−0.284574
$850$	47.8067	1.63976
$851$	−0.0957550	−0.00328244
$852$	−5.00880	−0.171599
$853$	−8.70810	−0.298160	−0.149080	−	0.988825i	$-0.547631\pi$
$853$	−8.70810	−0.298160	−0.149080	+	0.988825i	$0.547631\pi$
$854$	−27.9166	−0.955286
$855$	2.21327	0.0756923
$856$	−54.0208	−1.84639
$857$	−39.8814	−1.36232	−0.681162	−	0.732133i	$-0.738525\pi$
$857$	−39.8814	−1.36232	−0.681162	+	0.732133i	$0.738525\pi$
$858$	0.497581	0.0169871
$859$	37.6509	1.28463	0.642317	−	0.766439i	$-0.277973\pi$
$859$	37.6509	1.28463	0.642317	+	0.766439i	$0.277973\pi$
$860$	34.3006	1.16964
$861$	−12.6727	−0.431884
$862$	−45.4359	−1.54755
$863$	38.0194	1.29420	0.647098	−	0.762407i	$-0.275982\pi$
$863$	38.0194	1.29420	0.647098	+	0.762407i	$0.275982\pi$
$864$	−2.79428	−0.0950633
$865$	20.1681	0.685738
$866$	20.1230	0.683809
$867$	−17.8162	−0.605069
$868$	−21.6107	−0.733516
$869$	−7.50997	−0.254758
$870$	−0.134902	−0.00457362
$871$	−1.73042	−0.0586331
$872$	−29.2222	−0.989589
$873$	8.03021	0.271782
$874$	−0.106215	−0.00359279
$875$	27.9404	0.944558
$876$	−57.0844	−1.92870
$877$	−44.7523	−1.51118	−0.755589	−	0.655046i	$-0.772649\pi$
$877$	−44.7523	−1.51118	−0.755589	+	0.655046i	$0.772649\pi$
$878$	41.1268	1.38796
$879$	−22.3971	−0.755435
$880$	2.72954	0.0920129
$881$	−14.0629	−0.473790	−0.236895	−	0.971535i	$-0.576130\pi$
$881$	−14.0629	−0.473790	−0.236895	+	0.971535i	$0.576130\pi$
$882$	0.208561	0.00702262
$883$	−7.95640	−0.267754	−0.133877	−	0.990998i	$-0.542743\pi$
$883$	−7.95640	−0.267754	−0.133877	+	0.990998i	$0.542743\pi$
$884$	4.60534	0.154894
$885$	3.80505	0.127905
$886$	−41.7237	−1.40174
$887$	18.9765	0.637167	0.318583	−	0.947895i	$-0.396793\pi$
$887$	18.9765	0.637167	0.318583	+	0.947895i	$0.396793\pi$
$888$	−15.0665	−0.505599
$889$	−40.8170	−1.36896
$890$	12.6336	0.423478
$891$	0.984672	0.0329878
$892$	−17.0914	−0.572262
$893$	−2.63538	−0.0881896
$894$	−21.6356	−0.723604
$895$	−20.2119	−0.675608
$896$	−54.5070	−1.82095
$897$	−0.00540055	−0.000180319 0
$898$	58.8633	1.96429
$899$	−0.0998668	−0.00333074
$900$	−12.5141	−0.417135
$901$	−47.1071	−1.56936
$902$	11.3130	0.376681
$903$	19.3599	0.644257
$904$	19.4013	0.645279
$905$	−8.82504	−0.293354
$906$	12.1294	0.402972
$907$	−8.84026	−0.293536	−0.146768	−	0.989171i	$-0.546887\pi$
$907$	−8.84026	−0.293536	−0.146768	+	0.989171i	$0.546887\pi$
$908$	19.4160	0.644343
$909$	−12.5602	−0.416594
$910$	1.68102	0.0557251
$911$	12.6043	0.417599	0.208799	−	0.977959i	$-0.433044\pi$
$911$	12.6043	0.417599	0.208799	+	0.977959i	$0.433044\pi$
$912$	−3.83238	−0.126903
$913$	−7.09388	−0.234773
$914$	26.3128	0.870350
$915$	5.63628	0.186330
$916$	94.9515	3.13728
$917$	−14.9570	−0.493923
$918$	14.0645	0.464199
$919$	39.7504	1.31125	0.655623	−	0.755088i	$-0.272406\pi$
$919$	39.7504	1.31125	0.655623	+	0.755088i	$0.272406\pi$
$920$	−0.129192	−0.00425932
$921$	−33.0830	−1.09012
$922$	9.18585	0.302520
$923$	0.288427	0.00949370
$924$	9.53111	0.313551
$925$	12.7769	0.420101
$926$	13.2879	0.436668
$927$	9.71341	0.319030
$928$	−0.124989	−0.00410298
$929$	−10.8237	−0.355113	−0.177557	−	0.984111i	$-0.556819\pi$
$929$	−10.8237	−0.355113	−0.177557	+	0.984111i	$0.556819\pi$
$930$	6.73340	0.220797
$931$	−0.153056	−0.00501622
$932$	16.0175	0.524669
$933$	−18.6706	−0.611248
$934$	−19.2235	−0.629011
$935$	7.35131	0.240413
$936$	−0.849746	−0.0277748
$937$	−21.5739	−0.704789	−0.352395	−	0.935851i	$-0.614633\pi$
$937$	−21.5739	−0.704789	−0.352395	+	0.935851i	$0.614633\pi$
$938$	−51.1524	−1.67019
$939$	8.71587	0.284432
$940$	−7.01790	−0.228899
$941$	−14.3589	−0.468086	−0.234043	−	0.972226i	$-0.575196\pi$
$941$	−14.3589	−0.468086	−0.234043	+	0.972226i	$0.575196\pi$
$942$	14.8637	0.484284
$943$	−0.122787	−0.00399849
$944$	−6.58861	−0.214441
$945$	3.32659	0.108214
$946$	−17.2827	−0.561909
$947$	−33.8730	−1.10073	−0.550363	−	0.834926i	$-0.685511\pi$
$947$	−33.8730	−1.10073	−0.550363	+	0.834926i	$0.685511\pi$
$948$	28.0789	0.911961
$949$	3.28716	0.106706
$950$	14.1726	0.459821
$951$	−0.441949	−0.0143312
$952$	62.1813	2.01531
$953$	9.78877	0.317089	0.158545	−	0.987352i	$-0.449320\pi$
$953$	9.78877	0.317089	0.158545	+	0.987352i	$0.449320\pi$
$954$	19.0296	0.616107
$955$	−25.2916	−0.818418
$956$	−15.0140	−0.485588
$957$	0.0440448	0.00142377
$958$	−42.3567	−1.36848
$959$	30.1143	0.972441
$960$	13.9713	0.450923
$961$	−26.0153	−0.839204
$962$	1.89947	0.0612415
$963$	13.4775	0.434306
$964$	18.0330	0.580805
$965$	−25.7873	−0.830124
$966$	−0.159644	−0.00513646
$967$	11.7274	0.377128	0.188564	−	0.982061i	$-0.439617\pi$
$967$	11.7274	0.377128	0.188564	+	0.982061i	$0.439617\pi$
$968$	40.2042	1.29221
$969$	−10.3215	−0.331574
$970$	−24.2183	−0.777603
$971$	−30.0917	−0.965689	−0.482844	−	0.875706i	$-0.660396\pi$
$971$	−30.0917	−0.965689	−0.482844	+	0.875706i	$0.660396\pi$
$972$	−3.68158	−0.118087
$973$	−3.48822	−0.111827
$974$	3.00962	0.0964344
$975$	0.720612	0.0230780
$976$	−9.75948	−0.312393
$977$	−0.564236	−0.0180515	−0.00902576	−	0.999959i	$-0.502873\pi$
$977$	−0.564236	−0.0180515	−0.00902576	+	0.999959i	$0.502873\pi$
$978$	5.63088	0.180056
$979$	−4.12478	−0.131829
$980$	−0.407582	−0.0130197
$981$	7.29056	0.232770
$982$	60.8486	1.94176
$983$	−54.0396	−1.72360	−0.861798	−	0.507252i	$-0.830661\pi$
$983$	−54.0396	−1.72360	−0.861798	+	0.507252i	$0.830661\pi$
$984$	−19.3198	−0.615893
$985$	−12.8071	−0.408070
$986$	0.629112	0.0200350
$987$	−3.96103	−0.126081
$988$	1.36529	0.0434356
$989$	0.187579	0.00596468
$990$	−2.96967	−0.0943824
$991$	6.32936	0.201059	0.100529	−	0.994934i	$-0.467946\pi$
$991$	6.32936	0.201059	0.100529	+	0.994934i	$0.467946\pi$
$992$	6.23861	0.198076
$993$	−22.9258	−0.727528
$994$	8.52612	0.270432
$995$	5.14924	0.163242
$996$	26.5232	0.840420
$997$	0.387472	0.0122714	0.00613568	−	0.999981i	$-0.498047\pi$
$997$	0.387472	0.0122714	0.00613568	+	0.999981i	$0.498047\pi$
$998$	39.4600	1.24908
$999$	3.75890	0.118926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.b.1.11	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.b.1.11	✓	102	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38361 q^{2} -1.00000 q^{3} +3.68158 q^{4} +1.26527 q^{5} +2.38361 q^{6} -2.62916 q^{7} -4.00822 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.38361 q^{2} -1.00000 q^{3} +3.68158 q^{4} +1.26527 q^{5} +2.38361 q^{6} -2.62916 q^{7} -4.00822 q^{8} +1.00000 q^{9} -3.01590 q^{10} +0.984672 q^{11} -3.68158 q^{12} +0.212001 q^{13} +6.26689 q^{14} -1.26527 q^{15} +2.19087 q^{16} +5.90052 q^{17} -2.38361 q^{18} +1.74925 q^{19} +4.65818 q^{20} +2.62916 q^{21} -2.34707 q^{22} +0.0254742 q^{23} +4.00822 q^{24} -3.39910 q^{25} -0.505326 q^{26} -1.00000 q^{27} -9.67947 q^{28} -0.0447304 q^{29} +3.01590 q^{30} +2.23264 q^{31} +2.79428 q^{32} -0.984672 q^{33} -14.0645 q^{34} -3.32659 q^{35} +3.68158 q^{36} -3.75890 q^{37} -4.16953 q^{38} -0.212001 q^{39} -5.07147 q^{40} -4.82004 q^{41} -6.26689 q^{42} +7.36351 q^{43} +3.62515 q^{44} +1.26527 q^{45} -0.0607204 q^{46} -1.50657 q^{47} -2.19087 q^{48} -0.0874982 q^{49} +8.10211 q^{50} -5.90052 q^{51} +0.780498 q^{52} -7.98354 q^{53} +2.38361 q^{54} +1.24587 q^{55} +10.5383 q^{56} -1.74925 q^{57} +0.106620 q^{58} -3.00731 q^{59} -4.65818 q^{60} -4.45462 q^{61} -5.32173 q^{62} -2.62916 q^{63} -11.0422 q^{64} +0.268238 q^{65} +2.34707 q^{66} -8.16233 q^{67} +21.7232 q^{68} -0.0254742 q^{69} +7.92929 q^{70} +1.36050 q^{71} -4.00822 q^{72} +15.5054 q^{73} +8.95974 q^{74} +3.39910 q^{75} +6.44001 q^{76} -2.58887 q^{77} +0.505326 q^{78} -7.62687 q^{79} +2.77203 q^{80} +1.00000 q^{81} +11.4891 q^{82} -7.20430 q^{83} +9.67947 q^{84} +7.46574 q^{85} -17.5517 q^{86} +0.0447304 q^{87} -3.94679 q^{88} -4.18899 q^{89} -3.01590 q^{90} -0.557385 q^{91} +0.0937852 q^{92} -2.23264 q^{93} +3.59108 q^{94} +2.21327 q^{95} -2.79428 q^{96} +8.03021 q^{97} +0.208561 q^{98} +0.984672 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * q^99

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