LMFDB - Embedded newform 8038.2.a.b.1.7

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:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.88068 q^{3} +1.00000 q^{4} -2.36782 q^{5} +2.88068 q^{6} +0.285889 q^{7} -1.00000 q^{8} +5.29829 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.88068 q^{3} +1.00000 q^{4} -2.36782 q^{5} +2.88068 q^{6} +0.285889 q^{7} -1.00000 q^{8} +5.29829 q^{9} +2.36782 q^{10} +1.31435 q^{11} -2.88068 q^{12} -0.884758 q^{13} -0.285889 q^{14} +6.82091 q^{15} +1.00000 q^{16} +1.95424 q^{17} -5.29829 q^{18} +2.65314 q^{19} -2.36782 q^{20} -0.823553 q^{21} -1.31435 q^{22} +4.22542 q^{23} +2.88068 q^{24} +0.606558 q^{25} +0.884758 q^{26} -6.62064 q^{27} +0.285889 q^{28} -0.754962 q^{29} -6.82091 q^{30} -4.60849 q^{31} -1.00000 q^{32} -3.78621 q^{33} -1.95424 q^{34} -0.676932 q^{35} +5.29829 q^{36} +8.01828 q^{37} -2.65314 q^{38} +2.54870 q^{39} +2.36782 q^{40} +9.49569 q^{41} +0.823553 q^{42} +5.90766 q^{43} +1.31435 q^{44} -12.5454 q^{45} -4.22542 q^{46} +7.86253 q^{47} -2.88068 q^{48} -6.91827 q^{49} -0.606558 q^{50} -5.62954 q^{51} -0.884758 q^{52} +0.481935 q^{53} +6.62064 q^{54} -3.11213 q^{55} -0.285889 q^{56} -7.64285 q^{57} +0.754962 q^{58} +11.5218 q^{59} +6.82091 q^{60} +8.51768 q^{61} +4.60849 q^{62} +1.51472 q^{63} +1.00000 q^{64} +2.09495 q^{65} +3.78621 q^{66} +15.7717 q^{67} +1.95424 q^{68} -12.1721 q^{69} +0.676932 q^{70} -2.45124 q^{71} -5.29829 q^{72} +5.66837 q^{73} -8.01828 q^{74} -1.74730 q^{75} +2.65314 q^{76} +0.375757 q^{77} -2.54870 q^{78} -8.65853 q^{79} -2.36782 q^{80} +3.17703 q^{81} -9.49569 q^{82} -3.89408 q^{83} -0.823553 q^{84} -4.62729 q^{85} -5.90766 q^{86} +2.17480 q^{87} -1.31435 q^{88} +2.08583 q^{89} +12.5454 q^{90} -0.252942 q^{91} +4.22542 q^{92} +13.2756 q^{93} -7.86253 q^{94} -6.28216 q^{95} +2.88068 q^{96} +12.6273 q^{97} +6.91827 q^{98} +6.96379 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * 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
$3$	−2.88068	−1.66316	−0.831579	−	0.555406i	$-0.812563\pi$
$3$	−2.88068	−1.66316	−0.831579	+	0.555406i	$0.812563\pi$
$4$	1.00000	0.500000
$5$	−2.36782	−1.05892	−0.529460	−	0.848335i	$-0.677605\pi$
$5$	−2.36782	−1.05892	−0.529460	+	0.848335i	$0.677605\pi$
$6$	2.88068	1.17603
$7$	0.285889	0.108056	0.0540279	−	0.998539i	$-0.482794\pi$
$7$	0.285889	0.108056	0.0540279	+	0.998539i	$0.482794\pi$
$8$	−1.00000	−0.353553
$9$	5.29829	1.76610
$10$	2.36782	0.748770
$11$	1.31435	0.396290	0.198145	−	0.980173i	$-0.436508\pi$
$11$	1.31435	0.396290	0.198145	+	0.980173i	$0.436508\pi$
$12$	−2.88068	−0.831579
$13$	−0.884758	−0.245388	−0.122694	−	0.992445i	$-0.539153\pi$
$13$	−0.884758	−0.245388	−0.122694	+	0.992445i	$0.539153\pi$
$14$	−0.285889	−0.0764070
$15$	6.82091	1.76115
$16$	1.00000	0.250000
$17$	1.95424	0.473974	0.236987	−	0.971513i	$-0.423840\pi$
$17$	1.95424	0.473974	0.236987	+	0.971513i	$0.423840\pi$
$18$	−5.29829	−1.24882
$19$	2.65314	0.608673	0.304336	−	0.952565i	$-0.401565\pi$
$19$	2.65314	0.608673	0.304336	+	0.952565i	$0.401565\pi$
$20$	−2.36782	−0.529460
$21$	−0.823553	−0.179714
$22$	−1.31435	−0.280220
$23$	4.22542	0.881060	0.440530	−	0.897738i	$-0.354790\pi$
$23$	4.22542	0.881060	0.440530	+	0.897738i	$0.354790\pi$
$24$	2.88068	0.588015
$25$	0.606558	0.121312
$26$	0.884758	0.173515
$27$	−6.62064	−1.27414
$28$	0.285889	0.0540279
$29$	−0.754962	−0.140193	−0.0700964	−	0.997540i	$-0.522331\pi$
$29$	−0.754962	−0.140193	−0.0700964	+	0.997540i	$0.522331\pi$
$30$	−6.82091	−1.24532
$31$	−4.60849	−0.827709	−0.413854	−	0.910343i	$-0.635818\pi$
$31$	−4.60849	−0.827709	−0.413854	+	0.910343i	$0.635818\pi$
$32$	−1.00000	−0.176777
$33$	−3.78621	−0.659094
$34$	−1.95424	−0.335150
$35$	−0.676932	−0.114422
$36$	5.29829	0.883049
$37$	8.01828	1.31820	0.659098	−	0.752057i	$-0.270938\pi$
$37$	8.01828	1.31820	0.659098	+	0.752057i	$0.270938\pi$
$38$	−2.65314	−0.430397
$39$	2.54870	0.408119
$40$	2.36782	0.374385
$41$	9.49569	1.48298	0.741489	−	0.670965i	$-0.234120\pi$
$41$	9.49569	1.48298	0.741489	+	0.670965i	$0.234120\pi$
$42$	0.823553	0.127077
$43$	5.90766	0.900910	0.450455	−	0.892799i	$-0.351262\pi$
$43$	5.90766	0.900910	0.450455	+	0.892799i	$0.351262\pi$
$44$	1.31435	0.198145
$45$	−12.5454	−1.87016
$46$	−4.22542	−0.623004
$47$	7.86253	1.14687	0.573434	−	0.819252i	$-0.305611\pi$
$47$	7.86253	1.14687	0.573434	+	0.819252i	$0.305611\pi$
$48$	−2.88068	−0.415790
$49$	−6.91827	−0.988324
$50$	−0.606558	−0.0857802
$51$	−5.62954	−0.788293
$52$	−0.884758	−0.122694
$53$	0.481935	0.0661989	0.0330994	−	0.999452i	$-0.489462\pi$
$53$	0.481935	0.0661989	0.0330994	+	0.999452i	$0.489462\pi$
$54$	6.62064	0.900955
$55$	−3.11213	−0.419640
$56$	−0.285889	−0.0382035
$57$	−7.64285	−1.01232
$58$	0.754962	0.0991313
$59$	11.5218	1.50002	0.750008	−	0.661429i	$-0.230050\pi$
$59$	11.5218	1.50002	0.750008	+	0.661429i	$0.230050\pi$
$60$	6.82091	0.880576
$61$	8.51768	1.09058	0.545288	−	0.838249i	$-0.316420\pi$
$61$	8.51768	1.09058	0.545288	+	0.838249i	$0.316420\pi$
$62$	4.60849	0.585278
$63$	1.51472	0.190837
$64$	1.00000	0.125000
$65$	2.09495	0.259846
$66$	3.78621	0.466050
$67$	15.7717	1.92682	0.963411	−	0.268029i	$-0.0863724\pi$
$67$	15.7717	1.92682	0.963411	+	0.268029i	$0.0863724\pi$
$68$	1.95424	0.236987
$69$	−12.1721	−1.46534
$70$	0.676932	0.0809089
$71$	−2.45124	−0.290909	−0.145454	−	0.989365i	$-0.546464\pi$
$71$	−2.45124	−0.290909	−0.145454	+	0.989365i	$0.546464\pi$
$72$	−5.29829	−0.624410
$73$	5.66837	0.663432	0.331716	−	0.943379i	$-0.392372\pi$
$73$	5.66837	0.663432	0.331716	+	0.943379i	$0.392372\pi$
$74$	−8.01828	−0.932106
$75$	−1.74730	−0.201760
$76$	2.65314	0.304336
$77$	0.375757	0.0428214
$78$	−2.54870	−0.288584
$79$	−8.65853	−0.974161	−0.487080	−	0.873357i	$-0.661938\pi$
$79$	−8.65853	−0.974161	−0.487080	+	0.873357i	$0.661938\pi$
$80$	−2.36782	−0.264730
$81$	3.17703	0.353004
$82$	−9.49569	−1.04862
$83$	−3.89408	−0.427431	−0.213715	−	0.976896i	$-0.568557\pi$
$83$	−3.89408	−0.427431	−0.213715	+	0.976896i	$0.568557\pi$
$84$	−0.823553	−0.0898569
$85$	−4.62729	−0.501900
$86$	−5.90766	−0.637040
$87$	2.17480	0.233163
$88$	−1.31435	−0.140110
$89$	2.08583	0.221098	0.110549	−	0.993871i	$-0.464739\pi$
$89$	2.08583	0.221098	0.110549	+	0.993871i	$0.464739\pi$
$90$	12.5454	1.32240
$91$	−0.252942	−0.0265156
$92$	4.22542	0.440530
$93$	13.2756	1.37661
$94$	−7.86253	−0.810958
$95$	−6.28216	−0.644536
$96$	2.88068	0.294008
$97$	12.6273	1.28211	0.641054	−	0.767496i	$-0.278498\pi$
$97$	12.6273	1.28211	0.641054	+	0.767496i	$0.278498\pi$
$98$	6.91827	0.698851
$99$	6.96379	0.699887
$100$	0.606558	0.0606558
$101$	−8.63750	−0.859463	−0.429732	−	0.902957i	$-0.641392\pi$
$101$	−8.63750	−0.859463	−0.429732	+	0.902957i	$0.641392\pi$
$102$	5.62954	0.557408
$103$	−13.5093	−1.33111	−0.665555	−	0.746349i	$-0.731805\pi$
$103$	−13.5093	−1.33111	−0.665555	+	0.746349i	$0.731805\pi$
$104$	0.884758	0.0867577
$105$	1.95002	0.190303
$106$	−0.481935	−0.0468097
$107$	−7.40626	−0.715991	−0.357995	−	0.933723i	$-0.616540\pi$
$107$	−7.40626	−0.715991	−0.357995	+	0.933723i	$0.616540\pi$
$108$	−6.62064	−0.637071
$109$	−0.899069	−0.0861152	−0.0430576	−	0.999073i	$-0.513710\pi$
$109$	−0.899069	−0.0861152	−0.0430576	+	0.999073i	$0.513710\pi$
$110$	3.11213	0.296730
$111$	−23.0981	−2.19237
$112$	0.285889	0.0270139
$113$	−10.1068	−0.950770	−0.475385	−	0.879778i	$-0.657691\pi$
$113$	−10.1068	−0.950770	−0.475385	+	0.879778i	$0.657691\pi$
$114$	7.64285	0.715818
$115$	−10.0050	−0.932972
$116$	−0.754962	−0.0700964
$117$	−4.68771	−0.433379
$118$	−11.5218	−1.06067
$119$	0.558696	0.0512156
$120$	−6.82091	−0.622661
$121$	−9.27249	−0.842954
$122$	−8.51768	−0.771154
$123$	−27.3540	−2.46643
$124$	−4.60849	−0.413854
$125$	10.4029	0.930461
$126$	−1.51472	−0.134942
$127$	−5.35292	−0.474995	−0.237497	−	0.971388i	$-0.576327\pi$
$127$	−5.35292	−0.474995	−0.237497	+	0.971388i	$0.576327\pi$
$128$	−1.00000	−0.0883883
$129$	−17.0181	−1.49836
$130$	−2.09495	−0.183739
$131$	8.21937	0.718130	0.359065	−	0.933313i	$-0.383096\pi$
$131$	8.21937	0.718130	0.359065	+	0.933313i	$0.383096\pi$
$132$	−3.78621	−0.329547
$133$	0.758504	0.0657706
$134$	−15.7717	−1.36247
$135$	15.6765	1.34921
$136$	−1.95424	−0.167575
$137$	0.338373	0.0289091	0.0144546	−	0.999896i	$-0.495399\pi$
$137$	0.338373	0.0289091	0.0144546	+	0.999896i	$0.495399\pi$
$138$	12.1721	1.03615
$139$	−8.45541	−0.717179	−0.358589	−	0.933495i	$-0.616742\pi$
$139$	−8.45541	−0.717179	−0.358589	+	0.933495i	$0.616742\pi$
$140$	−0.676932	−0.0572112
$141$	−22.6494	−1.90742
$142$	2.45124	0.205703
$143$	−1.16288	−0.0972448
$144$	5.29829	0.441524
$145$	1.78761	0.148453
$146$	−5.66837	−0.469118
$147$	19.9293	1.64374
$148$	8.01828	0.659098
$149$	3.92129	0.321244	0.160622	−	0.987016i	$-0.448650\pi$
$149$	3.92129	0.321244	0.160622	+	0.987016i	$0.448650\pi$
$150$	1.74730	0.142666
$151$	9.01609	0.733719	0.366859	−	0.930276i	$-0.380433\pi$
$151$	9.01609	0.733719	0.366859	+	0.930276i	$0.380433\pi$
$152$	−2.65314	−0.215198
$153$	10.3542	0.837084
$154$	−0.375757	−0.0302793
$155$	10.9121	0.876477
$156$	2.54870	0.204059
$157$	6.72082	0.536380	0.268190	−	0.963366i	$-0.413575\pi$
$157$	6.72082	0.536380	0.268190	+	0.963366i	$0.413575\pi$
$158$	8.65853	0.688836
$159$	−1.38830	−0.110099
$160$	2.36782	0.187192
$161$	1.20800	0.0952036
$162$	−3.17703	−0.249611
$163$	−0.593155	−0.0464595	−0.0232297	−	0.999730i	$-0.507395\pi$
$163$	−0.593155	−0.0464595	−0.0232297	+	0.999730i	$0.507395\pi$
$164$	9.49569	0.741489
$165$	8.96504	0.697928
$166$	3.89408	0.302239
$167$	12.7279	0.984917	0.492458	−	0.870336i	$-0.336098\pi$
$167$	12.7279	0.984917	0.492458	+	0.870336i	$0.336098\pi$
$168$	0.823553	0.0635385
$169$	−12.2172	−0.939785
$170$	4.62729	0.354897
$171$	14.0571	1.07498
$172$	5.90766	0.450455
$173$	2.31348	0.175890	0.0879451	−	0.996125i	$-0.471970\pi$
$173$	2.31348	0.175890	0.0879451	+	0.996125i	$0.471970\pi$
$174$	−2.17480	−0.164871
$175$	0.173408	0.0131084
$176$	1.31435	0.0990726
$177$	−33.1907	−2.49476
$178$	−2.08583	−0.156340
$179$	−25.2497	−1.88725	−0.943624	−	0.331019i	$-0.892607\pi$
$179$	−25.2497	−1.88725	−0.943624	+	0.331019i	$0.892607\pi$
$180$	−12.5454	−0.935078
$181$	4.20240	0.312362	0.156181	−	0.987728i	$-0.450082\pi$
$181$	4.20240	0.312362	0.156181	+	0.987728i	$0.450082\pi$
$182$	0.252942	0.0187493
$183$	−24.5367	−1.81380
$184$	−4.22542	−0.311502
$185$	−18.9858	−1.39586
$186$	−13.2756	−0.973411
$187$	2.56855	0.187831
$188$	7.86253	0.573434
$189$	−1.89277	−0.137678
$190$	6.28216	0.455756
$191$	2.07735	0.150312	0.0751558	−	0.997172i	$-0.476055\pi$
$191$	2.07735	0.150312	0.0751558	+	0.997172i	$0.476055\pi$
$192$	−2.88068	−0.207895
$193$	−6.57015	−0.472930	−0.236465	−	0.971640i	$-0.575989\pi$
$193$	−6.57015	−0.472930	−0.236465	+	0.971640i	$0.575989\pi$
$194$	−12.6273	−0.906587
$195$	−6.03486	−0.432165
$196$	−6.91827	−0.494162
$197$	−9.31395	−0.663592	−0.331796	−	0.943351i	$-0.607655\pi$
$197$	−9.31395	−0.663592	−0.331796	+	0.943351i	$0.607655\pi$
$198$	−6.96379	−0.494895
$199$	−1.90200	−0.134829	−0.0674145	−	0.997725i	$-0.521475\pi$
$199$	−1.90200	−0.134829	−0.0674145	+	0.997725i	$0.521475\pi$
$200$	−0.606558	−0.0428901
$201$	−45.4332	−3.20461
$202$	8.63750	0.607732
$203$	−0.215835	−0.0151486
$204$	−5.62954	−0.394147
$205$	−22.4841	−1.57035
$206$	13.5093	0.941237
$207$	22.3875	1.55604
$208$	−0.884758	−0.0613470
$209$	3.48715	0.241211
$210$	−1.95002	−0.134564
$211$	4.61759	0.317888	0.158944	−	0.987288i	$-0.449191\pi$
$211$	4.61759	0.317888	0.158944	+	0.987288i	$0.449191\pi$
$212$	0.481935	0.0330994
$213$	7.06122	0.483827
$214$	7.40626	0.506282
$215$	−13.9883	−0.953992
$216$	6.62064	0.450477
$217$	−1.31751	−0.0894387
$218$	0.899069	0.0608927
$219$	−16.3287	−1.10339
$220$	−3.11213	−0.209820
$221$	−1.72903	−0.116307
$222$	23.0981	1.55024
$223$	−3.41705	−0.228823	−0.114411	−	0.993433i	$-0.536498\pi$
$223$	−3.41705	−0.228823	−0.114411	+	0.993433i	$0.536498\pi$
$224$	−0.285889	−0.0191017
$225$	3.21372	0.214248
$226$	10.1068	0.672296
$227$	−21.9128	−1.45440	−0.727202	−	0.686424i	$-0.759180\pi$
$227$	−21.9128	−1.45440	−0.727202	+	0.686424i	$0.759180\pi$
$228$	−7.64285	−0.506160
$229$	−0.563369	−0.0372285	−0.0186142	−	0.999827i	$-0.505925\pi$
$229$	−0.563369	−0.0372285	−0.0186142	+	0.999827i	$0.505925\pi$
$230$	10.0050	0.659711
$231$	−1.08243	−0.0712189
$232$	0.754962	0.0495657
$233$	−0.385486	−0.0252540	−0.0126270	−	0.999920i	$-0.504019\pi$
$233$	−0.385486	−0.0252540	−0.0126270	+	0.999920i	$0.504019\pi$
$234$	4.68771	0.306445
$235$	−18.6170	−1.21444
$236$	11.5218	0.750008
$237$	24.9424	1.62018
$238$	−0.558696	−0.0362149
$239$	17.0457	1.10259	0.551297	−	0.834309i	$-0.314133\pi$
$239$	17.0457	1.10259	0.551297	+	0.834309i	$0.314133\pi$
$240$	6.82091	0.440288
$241$	−5.92057	−0.381377	−0.190689	−	0.981651i	$-0.561072\pi$
$241$	−5.92057	−0.381377	−0.190689	+	0.981651i	$0.561072\pi$
$242$	9.27249	0.596058
$243$	10.7099	0.687041
$244$	8.51768	0.545288
$245$	16.3812	1.04656
$246$	27.3540	1.74403
$247$	−2.34739	−0.149361
$248$	4.60849	0.292639
$249$	11.2176	0.710886
$250$	−10.4029	−0.657935
$251$	16.6095	1.04839	0.524193	−	0.851600i	$-0.324367\pi$
$251$	16.6095	1.04839	0.524193	+	0.851600i	$0.324367\pi$
$252$	1.51472	0.0954185
$253$	5.55366	0.349156
$254$	5.35292	0.335872
$255$	13.3297	0.834740
$256$	1.00000	0.0625000
$257$	13.1654	0.821234	0.410617	−	0.911808i	$-0.365313\pi$
$257$	13.1654	0.821234	0.410617	+	0.911808i	$0.365313\pi$
$258$	17.0181	1.05950
$259$	2.29233	0.142439
$260$	2.09495	0.129923
$261$	−4.00001	−0.247594
$262$	−8.21937	−0.507795
$263$	9.43812	0.581980	0.290990	−	0.956726i	$-0.406015\pi$
$263$	9.43812	0.581980	0.290990	+	0.956726i	$0.406015\pi$
$264$	3.78621	0.233025
$265$	−1.14113	−0.0700993
$266$	−0.758504	−0.0465068
$267$	−6.00861	−0.367721
$268$	15.7717	0.963411
$269$	23.9911	1.46276	0.731381	−	0.681969i	$-0.238876\pi$
$269$	23.9911	1.46276	0.731381	+	0.681969i	$0.238876\pi$
$270$	−15.6765	−0.954039
$271$	6.22596	0.378200	0.189100	−	0.981958i	$-0.439443\pi$
$271$	6.22596	0.378200	0.189100	+	0.981958i	$0.439443\pi$
$272$	1.95424	0.118493
$273$	0.728645	0.0440996
$274$	−0.338373	−0.0204418
$275$	0.797227	0.0480746
$276$	−12.1721	−0.732672
$277$	−7.84424	−0.471315	−0.235657	−	0.971836i	$-0.575724\pi$
$277$	−7.84424	−0.471315	−0.235657	+	0.971836i	$0.575724\pi$
$278$	8.45541	0.507122
$279$	−24.4171	−1.46181
$280$	0.676932	0.0404544
$281$	3.64061	0.217181	0.108590	−	0.994087i	$-0.465366\pi$
$281$	3.64061	0.217181	0.108590	+	0.994087i	$0.465366\pi$
$282$	22.6494	1.34875
$283$	−26.8384	−1.59538	−0.797688	−	0.603071i	$-0.793944\pi$
$283$	−26.8384	−1.59538	−0.797688	+	0.603071i	$0.793944\pi$
$284$	−2.45124	−0.145454
$285$	18.0969	1.07197
$286$	1.16288	0.0687625
$287$	2.71471	0.160244
$288$	−5.29829	−0.312205
$289$	−13.1809	−0.775349
$290$	−1.78761	−0.104972
$291$	−36.3751	−2.13235
$292$	5.66837	0.331716
$293$	20.3721	1.19015	0.595077	−	0.803669i	$-0.297122\pi$
$293$	20.3721	1.19015	0.595077	+	0.803669i	$0.297122\pi$
$294$	−19.9293	−1.16230
$295$	−27.2816	−1.58840
$296$	−8.01828	−0.466053
$297$	−8.70181	−0.504930
$298$	−3.92129	−0.227154
$299$	−3.73847	−0.216201
$300$	−1.74730	−0.100880
$301$	1.68893	0.0973485
$302$	−9.01609	−0.518818
$303$	24.8818	1.42942
$304$	2.65314	0.152168
$305$	−20.1683	−1.15483
$306$	−10.3542	−0.591908
$307$	−17.0214	−0.971461	−0.485730	−	0.874109i	$-0.661446\pi$
$307$	−17.0214	−0.971461	−0.485730	+	0.874109i	$0.661446\pi$
$308$	0.375757	0.0214107
$309$	38.9159	2.21385
$310$	−10.9121	−0.619763
$311$	−7.97955	−0.452479	−0.226239	−	0.974072i	$-0.572643\pi$
$311$	−7.97955	−0.452479	−0.226239	+	0.974072i	$0.572643\pi$
$312$	−2.54870	−0.144292
$313$	0.832174	0.0470372	0.0235186	−	0.999723i	$-0.492513\pi$
$313$	0.832174	0.0470372	0.0235186	+	0.999723i	$0.492513\pi$
$314$	−6.72082	−0.379278
$315$	−3.58658	−0.202081
$316$	−8.65853	−0.487080
$317$	23.7220	1.33236	0.666181	−	0.745790i	$-0.267928\pi$
$317$	23.7220	1.33236	0.666181	+	0.745790i	$0.267928\pi$
$318$	1.38830	0.0778519
$319$	−0.992281	−0.0555571
$320$	−2.36782	−0.132365
$321$	21.3350	1.19081
$322$	−1.20800	−0.0673191
$323$	5.18489	0.288495
$324$	3.17703	0.176502
$325$	−0.536657	−0.0297684
$326$	0.593155	0.0328518
$327$	2.58993	0.143223
$328$	−9.49569	−0.524312
$329$	2.24781	0.123926
$330$	−8.96504	−0.493509
$331$	−10.3114	−0.566768	−0.283384	−	0.959006i	$-0.591457\pi$
$331$	−10.3114	−0.566768	−0.283384	+	0.959006i	$0.591457\pi$
$332$	−3.89408	−0.213715
$333$	42.4832	2.32806
$334$	−12.7279	−0.696441
$335$	−37.3445	−2.04035
$336$	−0.823553	−0.0449285
$337$	16.1538	0.879955	0.439977	−	0.898009i	$-0.354987\pi$
$337$	16.1538	0.879955	0.439977	+	0.898009i	$0.354987\pi$
$338$	12.2172	0.664528
$339$	29.1145	1.58128
$340$	−4.62729	−0.250950
$341$	−6.05715	−0.328013
$342$	−14.0571	−0.760123
$343$	−3.97907	−0.214850
$344$	−5.90766	−0.318520
$345$	28.8212	1.55168
$346$	−2.31348	−0.124373
$347$	−9.38085	−0.503591	−0.251795	−	0.967781i	$-0.581021\pi$
$347$	−9.38085	−0.503591	−0.251795	+	0.967781i	$0.581021\pi$
$348$	2.17480	0.116582
$349$	−26.2923	−1.40740	−0.703698	−	0.710499i	$-0.748469\pi$
$349$	−26.2923	−1.40740	−0.703698	+	0.710499i	$0.748469\pi$
$350$	−0.173408	−0.00926905
$351$	5.85767	0.312659
$352$	−1.31435	−0.0700549
$353$	11.0995	0.590766	0.295383	−	0.955379i	$-0.404553\pi$
$353$	11.0995	0.590766	0.295383	+	0.955379i	$0.404553\pi$
$354$	33.1907	1.76406
$355$	5.80409	0.308049
$356$	2.08583	0.110549
$357$	−1.60942	−0.0851796
$358$	25.2497	1.33449
$359$	−24.2763	−1.28126	−0.640628	−	0.767851i	$-0.721326\pi$
$359$	−24.2763	−1.28126	−0.640628	+	0.767851i	$0.721326\pi$
$360$	12.5454	0.661200
$361$	−11.9608	−0.629517
$362$	−4.20240	−0.220873
$363$	26.7110	1.40197
$364$	−0.252942	−0.0132578
$365$	−13.4217	−0.702522
$366$	24.5367	1.28255
$367$	−17.1146	−0.893373	−0.446686	−	0.894691i	$-0.647396\pi$
$367$	−17.1146	−0.893373	−0.446686	+	0.894691i	$0.647396\pi$
$368$	4.22542	0.220265
$369$	50.3109	2.61908
$370$	18.9858	0.987025
$371$	0.137780	0.00715317
$372$	13.2756	0.688306
$373$	16.2865	0.843281	0.421641	−	0.906763i	$-0.361454\pi$
$373$	16.2865	0.843281	0.421641	+	0.906763i	$0.361454\pi$
$374$	−2.56855	−0.132817
$375$	−29.9673	−1.54750
$376$	−7.86253	−0.405479
$377$	0.667959	0.0344016
$378$	1.89277	0.0973533
$379$	1.65858	0.0851958	0.0425979	−	0.999092i	$-0.486437\pi$
$379$	1.65858	0.0851958	0.0425979	+	0.999092i	$0.486437\pi$
$380$	−6.28216	−0.322268
$381$	15.4200	0.789992
$382$	−2.07735	−0.106286
$383$	15.2918	0.781375	0.390687	−	0.920523i	$-0.372237\pi$
$383$	15.2918	0.781375	0.390687	+	0.920523i	$0.372237\pi$
$384$	2.88068	0.147004
$385$	−0.889723	−0.0453445
$386$	6.57015	0.334412
$387$	31.3005	1.59110
$388$	12.6273	0.641054
$389$	−10.6892	−0.541966	−0.270983	−	0.962584i	$-0.587349\pi$
$389$	−10.6892	−0.541966	−0.270983	+	0.962584i	$0.587349\pi$
$390$	6.03486	0.305587
$391$	8.25749	0.417599
$392$	6.91827	0.349425
$393$	−23.6774	−1.19436
$394$	9.31395	0.469230
$395$	20.5018	1.03156
$396$	6.96379	0.349944
$397$	12.4568	0.625188	0.312594	−	0.949887i	$-0.398802\pi$
$397$	12.4568	0.625188	0.312594	+	0.949887i	$0.398802\pi$
$398$	1.90200	0.0953385
$399$	−2.18500	−0.109387
$400$	0.606558	0.0303279
$401$	−12.1458	−0.606532	−0.303266	−	0.952906i	$-0.598077\pi$
$401$	−12.1458	−0.606532	−0.303266	+	0.952906i	$0.598077\pi$
$402$	45.4332	2.26600
$403$	4.07740	0.203110
$404$	−8.63750	−0.429732
$405$	−7.52263	−0.373803
$406$	0.215835	0.0107117
$407$	10.5388	0.522388
$408$	5.62954	0.278704
$409$	−28.7938	−1.42376	−0.711881	−	0.702300i	$-0.752156\pi$
$409$	−28.7938	−1.42376	−0.711881	+	0.702300i	$0.752156\pi$
$410$	22.4841	1.11041
$411$	−0.974742	−0.0480805
$412$	−13.5093	−0.665555
$413$	3.29396	0.162085
$414$	−22.3875	−1.10029
$415$	9.22047	0.452615
$416$	0.884758	0.0433788
$417$	24.3573	1.19278
$418$	−3.48715	−0.170562
$419$	−26.5794	−1.29849	−0.649245	−	0.760579i	$-0.724915\pi$
$419$	−26.5794	−1.29849	−0.649245	+	0.760579i	$0.724915\pi$
$420$	1.95002	0.0951513
$421$	25.5626	1.24585	0.622923	−	0.782283i	$-0.285945\pi$
$421$	25.5626	1.24585	0.622923	+	0.782283i	$0.285945\pi$
$422$	−4.61759	−0.224781
$423$	41.6580	2.02548
$424$	−0.481935	−0.0234048
$425$	1.18536	0.0574985
$426$	−7.06122	−0.342117
$427$	2.43511	0.117843
$428$	−7.40626	−0.357995
$429$	3.34988	0.161734
$430$	13.9883	0.674574
$431$	−6.38697	−0.307649	−0.153825	−	0.988098i	$-0.549159\pi$
$431$	−6.38697	−0.307649	−0.153825	+	0.988098i	$0.549159\pi$
$432$	−6.62064	−0.318536
$433$	24.7516	1.18949	0.594744	−	0.803915i	$-0.297254\pi$
$433$	24.7516	1.18949	0.594744	+	0.803915i	$0.297254\pi$
$434$	1.31751	0.0632427
$435$	−5.14953	−0.246901
$436$	−0.899069	−0.0430576
$437$	11.2106	0.536278
$438$	16.3287	0.780217
$439$	−21.2319	−1.01334	−0.506671	−	0.862139i	$-0.669124\pi$
$439$	−21.2319	−1.01334	−0.506671	+	0.862139i	$0.669124\pi$
$440$	3.11213	0.148365
$441$	−36.6550	−1.74548
$442$	1.72903	0.0822417
$443$	25.5554	1.21417	0.607087	−	0.794635i	$-0.292338\pi$
$443$	25.5554	1.21417	0.607087	+	0.794635i	$0.292338\pi$
$444$	−23.0981	−1.09619
$445$	−4.93887	−0.234125
$446$	3.41705	0.161802
$447$	−11.2960	−0.534280
$448$	0.285889	0.0135070
$449$	−9.10253	−0.429575	−0.214787	−	0.976661i	$-0.568906\pi$
$449$	−9.10253	−0.429575	−0.214787	+	0.976661i	$0.568906\pi$
$450$	−3.21372	−0.151496
$451$	12.4806	0.587690
$452$	−10.1068	−0.475385
$453$	−25.9724	−1.22029
$454$	21.9128	1.02842
$455$	0.598921	0.0280779
$456$	7.64285	0.357909
$457$	25.6935	1.20189	0.600945	−	0.799290i	$-0.294791\pi$
$457$	25.6935	1.20189	0.600945	+	0.799290i	$0.294791\pi$
$458$	0.563369	0.0263245
$459$	−12.9383	−0.603910
$460$	−10.0050	−0.466486
$461$	4.83378	0.225131	0.112566	−	0.993644i	$-0.464093\pi$
$461$	4.83378	0.225131	0.112566	+	0.993644i	$0.464093\pi$
$462$	1.08243	0.0503594
$463$	−22.1105	−1.02756	−0.513781	−	0.857921i	$-0.671756\pi$
$463$	−22.1105	−1.02756	−0.513781	+	0.857921i	$0.671756\pi$
$464$	−0.754962	−0.0350482
$465$	−31.4341	−1.45772
$466$	0.385486	0.0178573
$467$	40.7641	1.88634	0.943168	−	0.332316i	$-0.107830\pi$
$467$	40.7641	1.88634	0.943168	+	0.332316i	$0.107830\pi$
$468$	−4.68771	−0.216689
$469$	4.50895	0.208204
$470$	18.6170	0.858740
$471$	−19.3605	−0.892085
$472$	−11.5218	−0.530335
$473$	7.76471	0.357022
$474$	−24.9424	−1.14564
$475$	1.60929	0.0738391
$476$	0.558696	0.0256078
$477$	2.55343	0.116914
$478$	−17.0457	−0.779652
$479$	31.7932	1.45267	0.726333	−	0.687343i	$-0.241223\pi$
$479$	31.7932	1.45267	0.726333	+	0.687343i	$0.241223\pi$
$480$	−6.82091	−0.311331
$481$	−7.09424	−0.323469
$482$	5.92057	0.269675
$483$	−3.47985	−0.158339
$484$	−9.27249	−0.421477
$485$	−29.8991	−1.35765
$486$	−10.7099	−0.485812
$487$	−3.80011	−0.172199	−0.0860997	−	0.996287i	$-0.527440\pi$
$487$	−3.80011	−0.172199	−0.0860997	+	0.996287i	$0.527440\pi$
$488$	−8.51768	−0.385577
$489$	1.70869	0.0772695
$490$	−16.3812	−0.740027
$491$	33.7797	1.52446	0.762229	−	0.647307i	$-0.224105\pi$
$491$	33.7797	1.52446	0.762229	+	0.647307i	$0.224105\pi$
$492$	−27.3540	−1.23321
$493$	−1.47538	−0.0664477
$494$	2.34739	0.105614
$495$	−16.4890	−0.741125
$496$	−4.60849	−0.206927
$497$	−0.700781	−0.0314343
$498$	−11.2176	−0.502672
$499$	−11.6496	−0.521509	−0.260754	−	0.965405i	$-0.583971\pi$
$499$	−11.6496	−0.521509	−0.260754	+	0.965405i	$0.583971\pi$
$500$	10.4029	0.465230
$501$	−36.6650	−1.63807
$502$	−16.6095	−0.741320
$503$	18.8236	0.839302	0.419651	−	0.907686i	$-0.362153\pi$
$503$	18.8236	0.839302	0.419651	+	0.907686i	$0.362153\pi$
$504$	−1.51472	−0.0674711
$505$	20.4520	0.910103
$506$	−5.55366	−0.246890
$507$	35.1938	1.56301
$508$	−5.35292	−0.237497
$509$	32.9445	1.46024	0.730120	−	0.683319i	$-0.239464\pi$
$509$	32.9445	1.46024	0.730120	+	0.683319i	$0.239464\pi$
$510$	−13.3297	−0.590250
$511$	1.62052	0.0716877
$512$	−1.00000	−0.0441942
$513$	−17.5655	−0.775536
$514$	−13.1654	−0.580700
$515$	31.9875	1.40954
$516$	−17.0181	−0.749178
$517$	10.3341	0.454493
$518$	−2.29233	−0.100719
$519$	−6.66437	−0.292533
$520$	−2.09495	−0.0918695
$521$	32.4721	1.42263	0.711314	−	0.702874i	$-0.248100\pi$
$521$	32.4721	1.42263	0.711314	+	0.702874i	$0.248100\pi$
$522$	4.00001	0.175076
$523$	−0.484951	−0.0212054	−0.0106027	−	0.999944i	$-0.503375\pi$
$523$	−0.484951	−0.0212054	−0.0106027	+	0.999944i	$0.503375\pi$
$524$	8.21937	0.359065
$525$	−0.499532	−0.0218014
$526$	−9.43812	−0.411522
$527$	−9.00610	−0.392312
$528$	−3.78621	−0.164773
$529$	−5.14585	−0.223733
$530$	1.14113	0.0495677
$531$	61.0461	2.64917
$532$	0.758504	0.0328853
$533$	−8.40139	−0.363905
$534$	6.00861	0.260018
$535$	17.5367	0.758177
$536$	−15.7717	−0.681234
$537$	72.7361	3.13879
$538$	−23.9911	−1.03433
$539$	−9.09300	−0.391663
$540$	15.6765	0.674607
$541$	11.8091	0.507712	0.253856	−	0.967242i	$-0.418301\pi$
$541$	11.8091	0.507712	0.253856	+	0.967242i	$0.418301\pi$
$542$	−6.22596	−0.267428
$543$	−12.1058	−0.519508
$544$	−1.95424	−0.0837875
$545$	2.12883	0.0911891
$546$	−0.728645	−0.0311831
$547$	−32.0499	−1.37036	−0.685178	−	0.728376i	$-0.740276\pi$
$547$	−32.0499	−1.37036	−0.685178	+	0.728376i	$0.740276\pi$
$548$	0.338373	0.0144546
$549$	45.1291	1.92606
$550$	−0.797227	−0.0339939
$551$	−2.00302	−0.0853316
$552$	12.1721	0.518077
$553$	−2.47538	−0.105264
$554$	7.84424	0.333270
$555$	54.6920	2.32154
$556$	−8.45541	−0.358589
$557$	39.1838	1.66027	0.830136	−	0.557561i	$-0.188263\pi$
$557$	39.1838	1.66027	0.830136	+	0.557561i	$0.188263\pi$
$558$	24.4171	1.03366
$559$	−5.22685	−0.221072
$560$	−0.676932	−0.0286056
$561$	−7.39917	−0.312393
$562$	−3.64061	−0.153570
$563$	22.8929	0.964819	0.482409	−	0.875946i	$-0.339762\pi$
$563$	22.8929	0.964819	0.482409	+	0.875946i	$0.339762\pi$
$564$	−22.6494	−0.953712
$565$	23.9311	1.00679
$566$	26.8384	1.12810
$567$	0.908278	0.0381441
$568$	2.45124	0.102852
$569$	−42.7823	−1.79353	−0.896764	−	0.442509i	$-0.854089\pi$
$569$	−42.7823	−1.79353	−0.896764	+	0.442509i	$0.854089\pi$
$570$	−18.0969	−0.757994
$571$	−23.9131	−1.00073	−0.500366	−	0.865814i	$-0.666801\pi$
$571$	−23.9131	−1.00073	−0.500366	+	0.865814i	$0.666801\pi$
$572$	−1.16288	−0.0486224
$573$	−5.98416	−0.249992
$574$	−2.71471	−0.113310
$575$	2.56296	0.106883
$576$	5.29829	0.220762
$577$	32.0571	1.33455	0.667277	−	0.744809i	$-0.267460\pi$
$577$	32.0571	1.33455	0.667277	+	0.744809i	$0.267460\pi$
$578$	13.1809	0.548255
$579$	18.9265	0.786557
$580$	1.78761	0.0742265
$581$	−1.11327	−0.0461864
$582$	36.3751	1.50780
$583$	0.633429	0.0262340
$584$	−5.66837	−0.234559
$585$	11.0996	0.458914
$586$	−20.3721	−0.841566
$587$	38.5989	1.59315	0.796573	−	0.604542i	$-0.206644\pi$
$587$	38.5989	1.59315	0.796573	+	0.604542i	$0.206644\pi$
$588$	19.9293	0.821870
$589$	−12.2270	−0.503804
$590$	27.2816	1.12317
$591$	26.8305	1.10366
$592$	8.01828	0.329549
$593$	17.3078	0.710747	0.355374	−	0.934724i	$-0.384354\pi$
$593$	17.3078	0.710747	0.355374	+	0.934724i	$0.384354\pi$
$594$	8.70181	0.357040
$595$	−1.32289	−0.0542332
$596$	3.92129	0.160622
$597$	5.47904	0.224242
$598$	3.73847	0.152878
$599$	−26.8940	−1.09886	−0.549430	−	0.835540i	$-0.685155\pi$
$599$	−26.8940	−1.09886	−0.549430	+	0.835540i	$0.685155\pi$
$600$	1.74730	0.0713331
$601$	−22.8079	−0.930353	−0.465176	−	0.885218i	$-0.654009\pi$
$601$	−22.8079	−0.930353	−0.465176	+	0.885218i	$0.654009\pi$
$602$	−1.68893	−0.0688358
$603$	83.5632	3.40296
$604$	9.01609	0.366859
$605$	21.9556	0.892621
$606$	−24.8818	−1.01076
$607$	7.57699	0.307541	0.153770	−	0.988107i	$-0.450858\pi$
$607$	7.57699	0.307541	0.153770	+	0.988107i	$0.450858\pi$
$608$	−2.65314	−0.107599
$609$	0.621751	0.0251946
$610$	20.1683	0.816590
$611$	−6.95644	−0.281428
$612$	10.3542	0.418542
$613$	11.2674	0.455087	0.227543	−	0.973768i	$-0.426931\pi$
$613$	11.2674	0.455087	0.227543	+	0.973768i	$0.426931\pi$
$614$	17.0214	0.686927
$615$	64.7693	2.61175
$616$	−0.375757	−0.0151397
$617$	−5.92047	−0.238349	−0.119175	−	0.992873i	$-0.538025\pi$
$617$	−5.92047	−0.238349	−0.119175	+	0.992873i	$0.538025\pi$
$618$	−38.9159	−1.56543
$619$	18.9257	0.760690	0.380345	−	0.924845i	$-0.375805\pi$
$619$	18.9257	0.760690	0.380345	+	0.924845i	$0.375805\pi$
$620$	10.9121	0.438239
$621$	−27.9750	−1.12260
$622$	7.97955	0.319951
$623$	0.596316	0.0238909
$624$	2.54870	0.102030
$625$	−27.6649	−1.10660
$626$	−0.832174	−0.0332604
$627$	−10.0453	−0.401173
$628$	6.72082	0.268190
$629$	15.6697	0.624790
$630$	3.58658	0.142893
$631$	24.9855	0.994658	0.497329	−	0.867562i	$-0.334314\pi$
$631$	24.9855	0.994658	0.497329	+	0.867562i	$0.334314\pi$
$632$	8.65853	0.344418
$633$	−13.3018	−0.528698
$634$	−23.7220	−0.942123
$635$	12.6747	0.502981
$636$	−1.38830	−0.0550496
$637$	6.12100	0.242523
$638$	0.992281	0.0392848
$639$	−12.9874	−0.513773
$640$	2.36782	0.0935962
$641$	42.0637	1.66142	0.830709	−	0.556707i	$-0.187935\pi$
$641$	42.0637	1.66142	0.830709	+	0.556707i	$0.187935\pi$
$642$	−21.3350	−0.842027
$643$	−23.8510	−0.940592	−0.470296	−	0.882509i	$-0.655853\pi$
$643$	−23.8510	−0.940592	−0.470296	+	0.882509i	$0.655853\pi$
$644$	1.20800	0.0476018
$645$	40.2956	1.58664
$646$	−5.18489	−0.203997
$647$	−31.7271	−1.24732	−0.623660	−	0.781695i	$-0.714355\pi$
$647$	−31.7271	−1.24732	−0.623660	+	0.781695i	$0.714355\pi$
$648$	−3.17703	−0.124806
$649$	15.1437	0.594441
$650$	0.536657	0.0210494
$651$	3.79533	0.148751
$652$	−0.593155	−0.0232297
$653$	5.62774	0.220230	0.110115	−	0.993919i	$-0.464878\pi$
$653$	5.62774	0.220230	0.110115	+	0.993919i	$0.464878\pi$
$654$	−2.58993	−0.101274
$655$	−19.4620	−0.760442
$656$	9.49569	0.370744
$657$	30.0327	1.17169
$658$	−2.24781	−0.0876287
$659$	−2.80582	−0.109299	−0.0546497	−	0.998506i	$-0.517404\pi$
$659$	−2.80582	−0.109299	−0.0546497	+	0.998506i	$0.517404\pi$
$660$	8.96504	0.348964
$661$	−16.0801	−0.625444	−0.312722	−	0.949845i	$-0.601241\pi$
$661$	−16.0801	−0.625444	−0.312722	+	0.949845i	$0.601241\pi$
$662$	10.3114	0.400766
$663$	4.98078	0.193438
$664$	3.89408	0.151120
$665$	−1.79600	−0.0696458
$666$	−42.4832	−1.64619
$667$	−3.19003	−0.123518
$668$	12.7279	0.492458
$669$	9.84341	0.380568
$670$	37.3445	1.44275
$671$	11.1952	0.432185
$672$	0.823553	0.0317692
$673$	14.1177	0.544197	0.272099	−	0.962269i	$-0.412282\pi$
$673$	14.1177	0.544197	0.272099	+	0.962269i	$0.412282\pi$
$674$	−16.1538	−0.622222
$675$	−4.01580	−0.154568
$676$	−12.2172	−0.469892
$677$	27.7928	1.06816	0.534081	−	0.845433i	$-0.320658\pi$
$677$	27.7928	1.06816	0.534081	+	0.845433i	$0.320658\pi$
$678$	−29.1145	−1.11814
$679$	3.61000	0.138539
$680$	4.62729	0.177448
$681$	63.1237	2.41890
$682$	6.05715	0.231940
$683$	30.0958	1.15158	0.575792	−	0.817596i	$-0.304694\pi$
$683$	30.0958	1.15158	0.575792	+	0.817596i	$0.304694\pi$
$684$	14.0571	0.537488
$685$	−0.801205	−0.0306125
$686$	3.97907	0.151922
$687$	1.62288	0.0619169
$688$	5.90766	0.225227
$689$	−0.426396	−0.0162444
$690$	−28.8212	−1.09720
$691$	12.7883	0.486491	0.243245	−	0.969965i	$-0.421788\pi$
$691$	12.7883	0.486491	0.243245	+	0.969965i	$0.421788\pi$
$692$	2.31348	0.0879451
$693$	1.99087	0.0756269
$694$	9.38085	0.356092
$695$	20.0209	0.759435
$696$	−2.17480	−0.0824356
$697$	18.5569	0.702892
$698$	26.2923	0.995180
$699$	1.11046	0.0420015
$700$	0.173408	0.00655421
$701$	−7.62538	−0.288007	−0.144003	−	0.989577i	$-0.545998\pi$
$701$	−7.62538	−0.288007	−0.144003	+	0.989577i	$0.545998\pi$
$702$	−5.85767	−0.221083
$703$	21.2736	0.802350
$704$	1.31435	0.0495363
$705$	53.6297	2.01981
$706$	−11.0995	−0.417734
$707$	−2.46936	−0.0928699
$708$	−33.1907	−1.24738
$709$	40.4039	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$709$	40.4039	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$710$	−5.80409	−0.217823
$711$	−45.8754	−1.72046
$712$	−2.08583	−0.0781699
$713$	−19.4728	−0.729261
$714$	1.60942	0.0602311
$715$	2.75348	0.102974
$716$	−25.2497	−0.943624
$717$	−49.1031	−1.83379
$718$	24.2763	0.905985
$719$	−36.1900	−1.34966	−0.674830	−	0.737973i	$-0.735783\pi$
$719$	−36.1900	−1.34966	−0.674830	+	0.737973i	$0.735783\pi$
$720$	−12.5454	−0.467539
$721$	−3.86215	−0.143834
$722$	11.9608	0.445136
$723$	17.0552	0.634291
$724$	4.20240	0.156181
$725$	−0.457928	−0.0170070
$726$	−26.7110	−0.991340
$727$	−25.2586	−0.936791	−0.468395	−	0.883519i	$-0.655168\pi$
$727$	−25.2586	−0.936791	−0.468395	+	0.883519i	$0.655168\pi$
$728$	0.252942	0.00937467
$729$	−40.3829	−1.49566
$730$	13.4217	0.496758
$731$	11.5450	0.427008
$732$	−24.5367	−0.906901
$733$	4.17100	0.154060	0.0770298	−	0.997029i	$-0.475456\pi$
$733$	4.17100	0.154060	0.0770298	+	0.997029i	$0.475456\pi$
$734$	17.1146	0.631710
$735$	−47.1889	−1.74059
$736$	−4.22542	−0.155751
$737$	20.7295	0.763581
$738$	−50.3109	−1.85197
$739$	−37.4558	−1.37783	−0.688916	−	0.724841i	$-0.741913\pi$
$739$	−37.4558	−1.37783	−0.688916	+	0.724841i	$0.741913\pi$
$740$	−18.9858	−0.697932
$741$	6.76207	0.248411
$742$	−0.137780	−0.00505805
$743$	7.29283	0.267548	0.133774	−	0.991012i	$-0.457290\pi$
$743$	7.29283	0.267548	0.133774	+	0.991012i	$0.457290\pi$
$744$	−13.2756	−0.486706
$745$	−9.28489	−0.340172
$746$	−16.2865	−0.596290
$747$	−20.6320	−0.754885
$748$	2.56855	0.0939156
$749$	−2.11737	−0.0773669
$750$	29.9673	1.09425
$751$	19.1614	0.699210	0.349605	−	0.936897i	$-0.386316\pi$
$751$	19.1614	0.699210	0.349605	+	0.936897i	$0.386316\pi$
$752$	7.86253	0.286717
$753$	−47.8467	−1.74363
$754$	−0.667959	−0.0243256
$755$	−21.3484	−0.776949
$756$	−1.89277	−0.0688392
$757$	50.2151	1.82510	0.912549	−	0.408968i	$-0.134111\pi$
$757$	50.2151	1.82510	0.912549	+	0.408968i	$0.134111\pi$
$758$	−1.65858	−0.0602425
$759$	−15.9983	−0.580701
$760$	6.28216	0.227878
$761$	−8.64223	−0.313281	−0.156640	−	0.987656i	$-0.550066\pi$
$761$	−8.64223	−0.313281	−0.156640	+	0.987656i	$0.550066\pi$
$762$	−15.4200	−0.558609
$763$	−0.257034	−0.00930525
$764$	2.07735	0.0751558
$765$	−24.5167	−0.886405
$766$	−15.2918	−0.552515
$767$	−10.1940	−0.368085
$768$	−2.88068	−0.103947
$769$	−53.3059	−1.92226	−0.961129	−	0.276099i	$-0.910958\pi$
$769$	−53.3059	−1.92226	−0.961129	+	0.276099i	$0.910958\pi$
$770$	0.889723	0.0320634
$771$	−37.9252	−1.36584
$772$	−6.57015	−0.236465
$773$	−11.4778	−0.412828	−0.206414	−	0.978465i	$-0.566179\pi$
$773$	−11.4778	−0.412828	−0.206414	+	0.978465i	$0.566179\pi$
$774$	−31.3005	−1.12507
$775$	−2.79531	−0.100411
$776$	−12.6273	−0.453293
$777$	−6.60347	−0.236898
$778$	10.6892	0.383228
$779$	25.1934	0.902648
$780$	−6.03486	−0.216083
$781$	−3.22178	−0.115284
$782$	−8.25749	−0.295287
$783$	4.99833	0.178626
$784$	−6.91827	−0.247081
$785$	−15.9137	−0.567983
$786$	23.6774	0.844543
$787$	45.3639	1.61705	0.808524	−	0.588463i	$-0.200267\pi$
$787$	45.3639	1.61705	0.808524	+	0.588463i	$0.200267\pi$
$788$	−9.31395	−0.331796
$789$	−27.1882	−0.967924
$790$	−20.5018	−0.729422
$791$	−2.88943	−0.102736
$792$	−6.96379	−0.247448
$793$	−7.53608	−0.267614
$794$	−12.4568	−0.442074
$795$	3.28724	0.116586
$796$	−1.90200	−0.0674145
$797$	−35.6204	−1.26174	−0.630870	−	0.775889i	$-0.717302\pi$
$797$	−35.6204	−1.26174	−0.630870	+	0.775889i	$0.717302\pi$
$798$	2.18500	0.0773483
$799$	15.3653	0.543585
$800$	−0.606558	−0.0214451
$801$	11.0514	0.390481
$802$	12.1458	0.428883
$803$	7.45020	0.262912
$804$	−45.4332	−1.60231
$805$	−2.86032	−0.100813
$806$	−4.07740	−0.143620
$807$	−69.1105	−2.43281
$808$	8.63750	0.303866
$809$	7.39843	0.260115	0.130058	−	0.991506i	$-0.458484\pi$
$809$	7.39843	0.260115	0.130058	+	0.991506i	$0.458484\pi$
$810$	7.52263	0.264318
$811$	27.3932	0.961905	0.480952	−	0.876747i	$-0.340291\pi$
$811$	27.3932	0.961905	0.480952	+	0.876747i	$0.340291\pi$
$812$	−0.215835	−0.00757432
$813$	−17.9350	−0.629007
$814$	−10.5388	−0.369384
$815$	1.40448	0.0491969
$816$	−5.62954	−0.197073
$817$	15.6739	0.548360
$818$	28.7938	1.00675
$819$	−1.34016	−0.0468291
$820$	−22.4841	−0.785177
$821$	36.3729	1.26942	0.634711	−	0.772750i	$-0.281119\pi$
$821$	36.3729	1.26942	0.634711	+	0.772750i	$0.281119\pi$
$822$	0.974742	0.0339980
$823$	−26.3390	−0.918118	−0.459059	−	0.888406i	$-0.651813\pi$
$823$	−26.3390	−0.918118	−0.459059	+	0.888406i	$0.651813\pi$
$824$	13.5093	0.470619
$825$	−2.29655	−0.0799557
$826$	−3.29396	−0.114612
$827$	−45.3255	−1.57612	−0.788061	−	0.615597i	$-0.788915\pi$
$827$	−45.3255	−1.57612	−0.788061	+	0.615597i	$0.788915\pi$
$828$	22.3875	0.778019
$829$	30.0357	1.04318	0.521592	−	0.853195i	$-0.325338\pi$
$829$	30.0357	1.04318	0.521592	+	0.853195i	$0.325338\pi$
$830$	−9.22047	−0.320047
$831$	22.5967	0.783871
$832$	−0.884758	−0.0306735
$833$	−13.5200	−0.468439
$834$	−24.3573	−0.843424
$835$	−30.1374	−1.04295
$836$	3.48715	0.120606
$837$	30.5111	1.05462
$838$	26.5794	0.918171
$839$	38.0368	1.31318	0.656588	−	0.754250i	$-0.271999\pi$
$839$	38.0368	1.31318	0.656588	+	0.754250i	$0.271999\pi$
$840$	−1.95002	−0.0672821
$841$	−28.4300	−0.980346
$842$	−25.5626	−0.880947
$843$	−10.4874	−0.361206
$844$	4.61759	0.158944
$845$	28.9281	0.995157
$846$	−41.6580	−1.43223
$847$	−2.65090	−0.0910860
$848$	0.481935	0.0165497
$849$	77.3126	2.65336
$850$	−1.18536	−0.0406576
$851$	33.8806	1.16141
$852$	7.06122	0.241914
$853$	5.12532	0.175488	0.0877438	−	0.996143i	$-0.472034\pi$
$853$	5.12532	0.175488	0.0877438	+	0.996143i	$0.472034\pi$
$854$	−2.43511	−0.0833276
$855$	−33.2847	−1.13831
$856$	7.40626	0.253141
$857$	17.7268	0.605536	0.302768	−	0.953064i	$-0.402089\pi$
$857$	17.7268	0.605536	0.302768	+	0.953064i	$0.402089\pi$
$858$	−3.34988	−0.114363
$859$	−42.8294	−1.46132	−0.730659	−	0.682742i	$-0.760787\pi$
$859$	−42.8294	−1.46132	−0.730659	+	0.682742i	$0.760787\pi$
$860$	−13.9883	−0.476996
$861$	−7.82020	−0.266512
$862$	6.38697	0.217541
$863$	−18.5064	−0.629966	−0.314983	−	0.949097i	$-0.601999\pi$
$863$	−18.5064	−0.629966	−0.314983	+	0.949097i	$0.601999\pi$
$864$	6.62064	0.225239
$865$	−5.47789	−0.186254
$866$	−24.7516	−0.841095
$867$	37.9700	1.28953
$868$	−1.31751	−0.0447193
$869$	−11.3803	−0.386050
$870$	5.14953	0.174585
$871$	−13.9542	−0.472819
$872$	0.899069	0.0304463
$873$	66.9031	2.26433
$874$	−11.2106	−0.379206
$875$	2.97406	0.100542
$876$	−16.3287	−0.551697
$877$	49.0486	1.65625	0.828127	−	0.560540i	$-0.189407\pi$
$877$	49.0486	1.65625	0.828127	+	0.560540i	$0.189407\pi$
$878$	21.2319	0.716542
$879$	−58.6855	−1.97941
$880$	−3.11213	−0.104910
$881$	−29.3237	−0.987939	−0.493969	−	0.869479i	$-0.664455\pi$
$881$	−29.3237	−0.987939	−0.493969	+	0.869479i	$0.664455\pi$
$882$	36.6550	1.23424
$883$	−34.3894	−1.15730	−0.578648	−	0.815578i	$-0.696419\pi$
$883$	−34.3894	−1.15730	−0.578648	+	0.815578i	$0.696419\pi$
$884$	−1.72903	−0.0581537
$885$	78.5894	2.64176
$886$	−25.5554	−0.858551
$887$	−6.04239	−0.202883	−0.101442	−	0.994841i	$-0.532346\pi$
$887$	−6.04239	−0.202883	−0.101442	+	0.994841i	$0.532346\pi$
$888$	23.0981	0.775120
$889$	−1.53034	−0.0513259
$890$	4.93887	0.165551
$891$	4.17572	0.139892
$892$	−3.41705	−0.114411
$893$	20.8604	0.698068
$894$	11.2960	0.377793
$895$	59.7866	1.99844
$896$	−0.285889	−0.00955087
$897$	10.7693	0.359577
$898$	9.10253	0.303755
$899$	3.47923	0.116039
$900$	3.21372	0.107124
$901$	0.941818	0.0313765
$902$	−12.4806	−0.415559
$903$	−4.86527	−0.161906
$904$	10.1068	0.336148
$905$	−9.95052	−0.330767
$906$	25.9724	0.862876
$907$	21.3959	0.710440	0.355220	−	0.934783i	$-0.384406\pi$
$907$	21.3959	0.710440	0.355220	+	0.934783i	$0.384406\pi$
$908$	−21.9128	−0.727202
$909$	−45.7640	−1.51790
$910$	−0.598921	−0.0198540
$911$	23.8485	0.790136	0.395068	−	0.918652i	$-0.370721\pi$
$911$	23.8485	0.790136	0.395068	+	0.918652i	$0.370721\pi$
$912$	−7.64285	−0.253080
$913$	−5.11817	−0.169387
$914$	−25.6935	−0.849865
$915$	58.0983	1.92067
$916$	−0.563369	−0.0186142
$917$	2.34983	0.0775981
$918$	12.9383	0.427029
$919$	16.3138	0.538142	0.269071	−	0.963120i	$-0.413283\pi$
$919$	16.3138	0.538142	0.269071	+	0.963120i	$0.413283\pi$
$920$	10.0050	0.329856
$921$	49.0330	1.61569
$922$	−4.83378	−0.159192
$923$	2.16875	0.0713854
$924$	−1.08243	−0.0356094
$925$	4.86355	0.159912
$926$	22.1105	0.726596
$927$	−71.5762	−2.35087
$928$	0.754962	0.0247828
$929$	36.2392	1.18897	0.594485	−	0.804107i	$-0.297356\pi$
$929$	36.2392	1.18897	0.594485	+	0.804107i	$0.297356\pi$
$930$	31.4341	1.03076
$931$	−18.3552	−0.601566
$932$	−0.385486	−0.0126270
$933$	22.9865	0.752544
$934$	−40.7641	−1.33384
$935$	−6.08186	−0.198898
$936$	4.68771	0.153223
$937$	−28.4073	−0.928026	−0.464013	−	0.885828i	$-0.653591\pi$
$937$	−28.4073	−0.928026	−0.464013	+	0.885828i	$0.653591\pi$
$938$	−4.50895	−0.147223
$939$	−2.39722	−0.0782304
$940$	−18.6170	−0.607221
$941$	9.93338	0.323819	0.161909	−	0.986806i	$-0.448235\pi$
$941$	9.93338	0.323819	0.161909	+	0.986806i	$0.448235\pi$
$942$	19.3605	0.630799
$943$	40.1232	1.30659
$944$	11.5218	0.375004
$945$	4.48172	0.145790
$946$	−7.76471	−0.252453
$947$	−46.5242	−1.51183	−0.755916	−	0.654669i	$-0.772808\pi$
$947$	−46.5242	−1.51183	−0.755916	+	0.654669i	$0.772808\pi$
$948$	24.9424	0.810092
$949$	−5.01514	−0.162798
$950$	−1.60929	−0.0522121
$951$	−68.3355	−2.21593
$952$	−0.558696	−0.0181074
$953$	−23.9121	−0.774590	−0.387295	−	0.921956i	$-0.626591\pi$
$953$	−23.9121	−0.774590	−0.387295	+	0.921956i	$0.626591\pi$
$954$	−2.55343	−0.0826704
$955$	−4.91878	−0.159168
$956$	17.0457	0.551297
$957$	2.85844	0.0924003
$958$	−31.7932	−1.02719
$959$	0.0967369	0.00312380
$960$	6.82091	0.220144
$961$	−9.76185	−0.314898
$962$	7.09424	0.228727
$963$	−39.2406	−1.26451
$964$	−5.92057	−0.190689
$965$	15.5569	0.500795
$966$	3.47985	0.111962
$967$	−0.775673	−0.0249440	−0.0124720	−	0.999922i	$-0.503970\pi$
$967$	−0.775673	−0.0249440	−0.0124720	+	0.999922i	$0.503970\pi$
$968$	9.27249	0.298029
$969$	−14.9360	−0.479813
$970$	29.8991	0.960003
$971$	−11.8290	−0.379611	−0.189806	−	0.981822i	$-0.560786\pi$
$971$	−11.8290	−0.379611	−0.189806	+	0.981822i	$0.560786\pi$
$972$	10.7099	0.343521
$973$	−2.41731	−0.0774953
$974$	3.80011	0.121763
$975$	1.54594	0.0495096
$976$	8.51768	0.272644
$977$	6.13673	0.196331	0.0981656	−	0.995170i	$-0.468703\pi$
$977$	6.13673	0.196331	0.0981656	+	0.995170i	$0.468703\pi$
$978$	−1.70869	−0.0546378
$979$	2.74151	0.0876190
$980$	16.3812	0.523278
$981$	−4.76353	−0.152088
$982$	−33.7797	−1.07795
$983$	4.78798	0.152713	0.0763564	−	0.997081i	$-0.475671\pi$
$983$	4.78798	0.152713	0.0763564	+	0.997081i	$0.475671\pi$
$984$	27.3540	0.872014
$985$	22.0537	0.702691
$986$	1.47538	0.0469856
$987$	−6.47521	−0.206108
$988$	−2.34739	−0.0746805
$989$	24.9623	0.793756
$990$	16.4890	0.524054
$991$	−14.2124	−0.451473	−0.225737	−	0.974188i	$-0.572479\pi$
$991$	−14.2124	−0.451473	−0.225737	+	0.974188i	$0.572479\pi$
$992$	4.60849	0.146320
$993$	29.7039	0.942626
$994$	0.700781	0.0222274
$995$	4.50358	0.142773
$996$	11.2176	0.355443
$997$	4.58595	0.145238	0.0726192	−	0.997360i	$-0.476864\pi$
$997$	4.58595	0.145238	0.0726192	+	0.997360i	$0.476864\pi$
$998$	11.6496	0.368762
$999$	−53.0861	−1.67957

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.7	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.7	✓	83	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.88068 q^{3} +1.00000 q^{4} -2.36782 q^{5} +2.88068 q^{6} +0.285889 q^{7} -1.00000 q^{8} +5.29829 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.88068 q^{3} +1.00000 q^{4} -2.36782 q^{5} +2.88068 q^{6} +0.285889 q^{7} -1.00000 q^{8} +5.29829 q^{9} +2.36782 q^{10} +1.31435 q^{11} -2.88068 q^{12} -0.884758 q^{13} -0.285889 q^{14} +6.82091 q^{15} +1.00000 q^{16} +1.95424 q^{17} -5.29829 q^{18} +2.65314 q^{19} -2.36782 q^{20} -0.823553 q^{21} -1.31435 q^{22} +4.22542 q^{23} +2.88068 q^{24} +0.606558 q^{25} +0.884758 q^{26} -6.62064 q^{27} +0.285889 q^{28} -0.754962 q^{29} -6.82091 q^{30} -4.60849 q^{31} -1.00000 q^{32} -3.78621 q^{33} -1.95424 q^{34} -0.676932 q^{35} +5.29829 q^{36} +8.01828 q^{37} -2.65314 q^{38} +2.54870 q^{39} +2.36782 q^{40} +9.49569 q^{41} +0.823553 q^{42} +5.90766 q^{43} +1.31435 q^{44} -12.5454 q^{45} -4.22542 q^{46} +7.86253 q^{47} -2.88068 q^{48} -6.91827 q^{49} -0.606558 q^{50} -5.62954 q^{51} -0.884758 q^{52} +0.481935 q^{53} +6.62064 q^{54} -3.11213 q^{55} -0.285889 q^{56} -7.64285 q^{57} +0.754962 q^{58} +11.5218 q^{59} +6.82091 q^{60} +8.51768 q^{61} +4.60849 q^{62} +1.51472 q^{63} +1.00000 q^{64} +2.09495 q^{65} +3.78621 q^{66} +15.7717 q^{67} +1.95424 q^{68} -12.1721 q^{69} +0.676932 q^{70} -2.45124 q^{71} -5.29829 q^{72} +5.66837 q^{73} -8.01828 q^{74} -1.74730 q^{75} +2.65314 q^{76} +0.375757 q^{77} -2.54870 q^{78} -8.65853 q^{79} -2.36782 q^{80} +3.17703 q^{81} -9.49569 q^{82} -3.89408 q^{83} -0.823553 q^{84} -4.62729 q^{85} -5.90766 q^{86} +2.17480 q^{87} -1.31435 q^{88} +2.08583 q^{89} +12.5454 q^{90} -0.252942 q^{91} +4.22542 q^{92} +13.2756 q^{93} -7.86253 q^{94} -6.28216 q^{95} +2.88068 q^{96} +12.6273 q^{97} +6.91827 q^{98} +6.96379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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