LMFDB - Embedded newform 8038.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.82708 q^{3} +1.00000 q^{4} -2.48763 q^{5} -2.82708 q^{6} -1.73566 q^{7} +1.00000 q^{8} +4.99240 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.82708 q^{3} +1.00000 q^{4} -2.48763 q^{5} -2.82708 q^{6} -1.73566 q^{7} +1.00000 q^{8} +4.99240 q^{9} -2.48763 q^{10} -2.30324 q^{11} -2.82708 q^{12} +1.43380 q^{13} -1.73566 q^{14} +7.03274 q^{15} +1.00000 q^{16} -3.43809 q^{17} +4.99240 q^{18} -5.30744 q^{19} -2.48763 q^{20} +4.90685 q^{21} -2.30324 q^{22} +2.54790 q^{23} -2.82708 q^{24} +1.18831 q^{25} +1.43380 q^{26} -5.63269 q^{27} -1.73566 q^{28} +0.912086 q^{29} +7.03274 q^{30} +4.78725 q^{31} +1.00000 q^{32} +6.51145 q^{33} -3.43809 q^{34} +4.31768 q^{35} +4.99240 q^{36} -10.9078 q^{37} -5.30744 q^{38} -4.05346 q^{39} -2.48763 q^{40} -2.72920 q^{41} +4.90685 q^{42} -4.12637 q^{43} -2.30324 q^{44} -12.4193 q^{45} +2.54790 q^{46} +8.74972 q^{47} -2.82708 q^{48} -3.98749 q^{49} +1.18831 q^{50} +9.71977 q^{51} +1.43380 q^{52} -11.9314 q^{53} -5.63269 q^{54} +5.72961 q^{55} -1.73566 q^{56} +15.0046 q^{57} +0.912086 q^{58} -12.0205 q^{59} +7.03274 q^{60} -10.0543 q^{61} +4.78725 q^{62} -8.66510 q^{63} +1.00000 q^{64} -3.56676 q^{65} +6.51145 q^{66} -12.1910 q^{67} -3.43809 q^{68} -7.20313 q^{69} +4.31768 q^{70} -13.0548 q^{71} +4.99240 q^{72} +11.6007 q^{73} -10.9078 q^{74} -3.35946 q^{75} -5.30744 q^{76} +3.99763 q^{77} -4.05346 q^{78} +7.56475 q^{79} -2.48763 q^{80} +0.946874 q^{81} -2.72920 q^{82} -5.38644 q^{83} +4.90685 q^{84} +8.55270 q^{85} -4.12637 q^{86} -2.57854 q^{87} -2.30324 q^{88} +10.6707 q^{89} -12.4193 q^{90} -2.48858 q^{91} +2.54790 q^{92} -13.5340 q^{93} +8.74972 q^{94} +13.2029 q^{95} -2.82708 q^{96} +7.82462 q^{97} -3.98749 q^{98} -11.4987 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) $ 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.82708	−1.63222	−0.816109	−	0.577898i	$-0.803873\pi$
$3$	−2.82708	−1.63222	−0.816109	+	0.577898i	$0.803873\pi$
$4$	1.00000	0.500000
$5$	−2.48763	−1.11250	−0.556251	−	0.831014i	$-0.687761\pi$
$5$	−2.48763	−1.11250	−0.556251	+	0.831014i	$0.687761\pi$
$6$	−2.82708	−1.15415
$7$	−1.73566	−0.656017	−0.328009	−	0.944675i	$-0.606377\pi$
$7$	−1.73566	−0.656017	−0.328009	+	0.944675i	$0.606377\pi$
$8$	1.00000	0.353553
$9$	4.99240	1.66413
$10$	−2.48763	−0.786658
$11$	−2.30324	−0.694452	−0.347226	−	0.937781i	$-0.612876\pi$
$11$	−2.30324	−0.694452	−0.347226	+	0.937781i	$0.612876\pi$
$12$	−2.82708	−0.816109
$13$	1.43380	0.397664	0.198832	−	0.980034i	$-0.436285\pi$
$13$	1.43380	0.397664	0.198832	+	0.980034i	$0.436285\pi$
$14$	−1.73566	−0.463874
$15$	7.03274	1.81585
$16$	1.00000	0.250000
$17$	−3.43809	−0.833859	−0.416930	−	0.908939i	$-0.636894\pi$
$17$	−3.43809	−0.833859	−0.416930	+	0.908939i	$0.636894\pi$
$18$	4.99240	1.17672
$19$	−5.30744	−1.21761	−0.608805	−	0.793320i	$-0.708351\pi$
$19$	−5.30744	−1.21761	−0.608805	+	0.793320i	$0.708351\pi$
$20$	−2.48763	−0.556251
$21$	4.90685	1.07076
$22$	−2.30324	−0.491052
$23$	2.54790	0.531274	0.265637	−	0.964073i	$-0.414418\pi$
$23$	2.54790	0.531274	0.265637	+	0.964073i	$0.414418\pi$
$24$	−2.82708	−0.577076
$25$	1.18831	0.237662
$26$	1.43380	0.281191
$27$	−5.63269	−1.08401
$28$	−1.73566	−0.328009
$29$	0.912086	0.169370	0.0846851	−	0.996408i	$-0.473012\pi$
$29$	0.912086	0.169370	0.0846851	+	0.996408i	$0.473012\pi$
$30$	7.03274	1.28400
$31$	4.78725	0.859816	0.429908	−	0.902873i	$-0.358546\pi$
$31$	4.78725	0.859816	0.429908	+	0.902873i	$0.358546\pi$
$32$	1.00000	0.176777
$33$	6.51145	1.13350
$34$	−3.43809	−0.589628
$35$	4.31768	0.729821
$36$	4.99240	0.832067
$37$	−10.9078	−1.79323	−0.896616	−	0.442809i	$-0.853982\pi$
$37$	−10.9078	−1.79323	−0.896616	+	0.442809i	$0.853982\pi$
$38$	−5.30744	−0.860980
$39$	−4.05346	−0.649073
$40$	−2.48763	−0.393329
$41$	−2.72920	−0.426229	−0.213115	−	0.977027i	$-0.568361\pi$
$41$	−2.72920	−0.426229	−0.213115	+	0.977027i	$0.568361\pi$
$42$	4.90685	0.757143
$43$	−4.12637	−0.629265	−0.314633	−	0.949213i	$-0.601881\pi$
$43$	−4.12637	−0.629265	−0.314633	+	0.949213i	$0.601881\pi$
$44$	−2.30324	−0.347226
$45$	−12.4193	−1.85135
$46$	2.54790	0.375668
$47$	8.74972	1.27628	0.638139	−	0.769921i	$-0.279704\pi$
$47$	8.74972	1.27628	0.638139	+	0.769921i	$0.279704\pi$
$48$	−2.82708	−0.408054
$49$	−3.98749	−0.569642
$50$	1.18831	0.168053
$51$	9.71977	1.36104
$52$	1.43380	0.198832
$53$	−11.9314	−1.63890	−0.819452	−	0.573148i	$-0.805722\pi$
$53$	−11.9314	−1.63890	−0.819452	+	0.573148i	$0.805722\pi$
$54$	−5.63269	−0.766512
$55$	5.72961	0.772580
$56$	−1.73566	−0.231937
$57$	15.0046	1.98740
$58$	0.912086	0.119763
$59$	−12.0205	−1.56493	−0.782467	−	0.622692i	$-0.786039\pi$
$59$	−12.0205	−1.56493	−0.782467	+	0.622692i	$0.786039\pi$
$60$	7.03274	0.907923
$61$	−10.0543	−1.28732	−0.643659	−	0.765312i	$-0.722584\pi$
$61$	−10.0543	−1.28732	−0.643659	+	0.765312i	$0.722584\pi$
$62$	4.78725	0.607982
$63$	−8.66510	−1.09170
$64$	1.00000	0.125000
$65$	−3.56676	−0.442402
$66$	6.51145	0.801504
$67$	−12.1910	−1.48936	−0.744682	−	0.667419i	$-0.767399\pi$
$67$	−12.1910	−1.48936	−0.744682	+	0.667419i	$0.767399\pi$
$68$	−3.43809	−0.416930
$69$	−7.20313	−0.867155
$70$	4.31768	0.516061
$71$	−13.0548	−1.54932	−0.774658	−	0.632381i	$-0.782078\pi$
$71$	−13.0548	−1.54932	−0.774658	+	0.632381i	$0.782078\pi$
$72$	4.99240	0.588360
$73$	11.6007	1.35775	0.678877	−	0.734252i	$-0.262467\pi$
$73$	11.6007	1.35775	0.678877	+	0.734252i	$0.262467\pi$
$74$	−10.9078	−1.26801
$75$	−3.35946	−0.387916
$76$	−5.30744	−0.608805
$77$	3.99763	0.455573
$78$	−4.05346	−0.458964
$79$	7.56475	0.851100	0.425550	−	0.904935i	$-0.360081\pi$
$79$	7.56475	0.851100	0.425550	+	0.904935i	$0.360081\pi$
$80$	−2.48763	−0.278126
$81$	0.946874	0.105208
$82$	−2.72920	−0.301390
$83$	−5.38644	−0.591239	−0.295619	−	0.955306i	$-0.595526\pi$
$83$	−5.38644	−0.591239	−0.295619	+	0.955306i	$0.595526\pi$
$84$	4.90685	0.535381
$85$	8.55270	0.927671
$86$	−4.12637	−0.444958
$87$	−2.57854	−0.276449
$88$	−2.30324	−0.245526
$89$	10.6707	1.13109	0.565545	−	0.824717i	$-0.308666\pi$
$89$	10.6707	1.13109	0.565545	+	0.824717i	$0.308666\pi$
$90$	−12.4193	−1.30910
$91$	−2.48858	−0.260874
$92$	2.54790	0.265637
$93$	−13.5340	−1.40341
$94$	8.74972	0.902465
$95$	13.2029	1.35459
$96$	−2.82708	−0.288538
$97$	7.82462	0.794470	0.397235	−	0.917717i	$-0.369970\pi$
$97$	7.82462	0.794470	0.397235	+	0.917717i	$0.369970\pi$
$98$	−3.98749	−0.402797
$99$	−11.4987	−1.15566
$100$	1.18831	0.118831
$101$	8.61317	0.857042	0.428521	−	0.903532i	$-0.359035\pi$
$101$	8.61317	0.857042	0.428521	+	0.903532i	$0.359035\pi$
$102$	9.71977	0.962400
$103$	−11.9013	−1.17267	−0.586335	−	0.810069i	$-0.699430\pi$
$103$	−11.9013	−1.17267	−0.586335	+	0.810069i	$0.699430\pi$
$104$	1.43380	0.140595
$105$	−12.2064	−1.19123
$106$	−11.9314	−1.15888
$107$	4.54159	0.439052	0.219526	−	0.975607i	$-0.429549\pi$
$107$	4.54159	0.439052	0.219526	+	0.975607i	$0.429549\pi$
$108$	−5.63269	−0.542006
$109$	−10.1215	−0.969463	−0.484731	−	0.874663i	$-0.661083\pi$
$109$	−10.1215	−0.969463	−0.484731	+	0.874663i	$0.661083\pi$
$110$	5.72961	0.546297
$111$	30.8373	2.92694
$112$	−1.73566	−0.164004
$113$	−6.70099	−0.630376	−0.315188	−	0.949029i	$-0.602068\pi$
$113$	−6.70099	−0.630376	−0.315188	+	0.949029i	$0.602068\pi$
$114$	15.0046	1.40531
$115$	−6.33824	−0.591044
$116$	0.912086	0.0846851
$117$	7.15809	0.661766
$118$	−12.0205	−1.10658
$119$	5.96735	0.547026
$120$	7.03274	0.641999
$121$	−5.69510	−0.517736
$122$	−10.0543	−0.910271
$123$	7.71567	0.695699
$124$	4.78725	0.429908
$125$	9.48208	0.848103
$126$	−8.66510	−0.771949
$127$	18.0575	1.60234	0.801170	−	0.598437i	$-0.204211\pi$
$127$	18.0575	1.60234	0.801170	+	0.598437i	$0.204211\pi$
$128$	1.00000	0.0883883
$129$	11.6656	1.02710
$130$	−3.56676	−0.312825
$131$	14.9932	1.30996	0.654979	−	0.755647i	$-0.272677\pi$
$131$	14.9932	1.30996	0.654979	+	0.755647i	$0.272677\pi$
$132$	6.51145	0.566749
$133$	9.21189	0.798772
$134$	−12.1910	−1.05314
$135$	14.0121	1.20597
$136$	−3.43809	−0.294814
$137$	−18.5808	−1.58746	−0.793732	−	0.608268i	$-0.791865\pi$
$137$	−18.5808	−1.58746	−0.793732	+	0.608268i	$0.791865\pi$
$138$	−7.20313	−0.613171
$139$	7.78293	0.660140	0.330070	−	0.943956i	$-0.392928\pi$
$139$	7.78293	0.660140	0.330070	+	0.943956i	$0.392928\pi$
$140$	4.31768	0.364910
$141$	−24.7362	−2.08316
$142$	−13.0548	−1.09553
$143$	−3.30237	−0.276158
$144$	4.99240	0.416034
$145$	−2.26893	−0.188425
$146$	11.6007	0.960077
$147$	11.2730	0.929779
$148$	−10.9078	−0.896616
$149$	−3.57877	−0.293184	−0.146592	−	0.989197i	$-0.546830\pi$
$149$	−3.57877	−0.293184	−0.146592	+	0.989197i	$0.546830\pi$
$150$	−3.35946	−0.274298
$151$	−1.57752	−0.128377	−0.0641883	−	0.997938i	$-0.520446\pi$
$151$	−1.57752	−0.128377	−0.0641883	+	0.997938i	$0.520446\pi$
$152$	−5.30744	−0.430490
$153$	−17.1643	−1.38765
$154$	3.99763	0.322138
$155$	−11.9089	−0.956548
$156$	−4.05346	−0.324537
$157$	11.0417	0.881221	0.440611	−	0.897698i	$-0.354762\pi$
$157$	11.0417	0.881221	0.440611	+	0.897698i	$0.354762\pi$
$158$	7.56475	0.601819
$159$	33.7311	2.67505
$160$	−2.48763	−0.196665
$161$	−4.42229	−0.348525
$162$	0.946874	0.0743934
$163$	11.8206	0.925862	0.462931	−	0.886394i	$-0.346798\pi$
$163$	11.8206	0.925862	0.462931	+	0.886394i	$0.346798\pi$
$164$	−2.72920	−0.213115
$165$	−16.1981	−1.26102
$166$	−5.38644	−0.418069
$167$	0.713744	0.0552311	0.0276156	−	0.999619i	$-0.491209\pi$
$167$	0.713744	0.0552311	0.0276156	+	0.999619i	$0.491209\pi$
$168$	4.90685	0.378572
$169$	−10.9442	−0.841864
$170$	8.55270	0.655962
$171$	−26.4969	−2.02626
$172$	−4.12637	−0.314633
$173$	−7.78446	−0.591841	−0.295921	−	0.955213i	$-0.595626\pi$
$173$	−7.78446	−0.591841	−0.295921	+	0.955213i	$0.595626\pi$
$174$	−2.57854	−0.195479
$175$	−2.06250	−0.155910
$176$	−2.30324	−0.173613
$177$	33.9829	2.55431
$178$	10.6707	0.799802
$179$	−8.66924	−0.647969	−0.323985	−	0.946062i	$-0.605023\pi$
$179$	−8.66924	−0.647969	−0.323985	+	0.946062i	$0.605023\pi$
$180$	−12.4193	−0.925677
$181$	−4.07244	−0.302702	−0.151351	−	0.988480i	$-0.548362\pi$
$181$	−4.07244	−0.302702	−0.151351	+	0.988480i	$0.548362\pi$
$182$	−2.48858	−0.184466
$183$	28.4243	2.10118
$184$	2.54790	0.187834
$185$	27.1346	1.99498
$186$	−13.5340	−0.992359
$187$	7.91874	0.579075
$188$	8.74972	0.638139
$189$	9.77642	0.711130
$190$	13.2029	0.957842
$191$	−7.71008	−0.557882	−0.278941	−	0.960308i	$-0.589983\pi$
$191$	−7.71008	−0.557882	−0.278941	+	0.960308i	$0.589983\pi$
$192$	−2.82708	−0.204027
$193$	−0.609783	−0.0438931	−0.0219466	−	0.999759i	$-0.506986\pi$
$193$	−0.609783	−0.0438931	−0.0219466	+	0.999759i	$0.506986\pi$
$194$	7.82462	0.561775
$195$	10.0835	0.722096
$196$	−3.98749	−0.284821
$197$	−5.92525	−0.422157	−0.211078	−	0.977469i	$-0.567698\pi$
$197$	−5.92525	−0.422157	−0.211078	+	0.977469i	$0.567698\pi$
$198$	−11.4987	−0.817176
$199$	−6.17206	−0.437525	−0.218763	−	0.975778i	$-0.570202\pi$
$199$	−6.17206	−0.437525	−0.218763	+	0.975778i	$0.570202\pi$
$200$	1.18831	0.0840263
$201$	34.4649	2.43097
$202$	8.61317	0.606020
$203$	−1.58307	−0.111110
$204$	9.71977	0.680520
$205$	6.78924	0.474181
$206$	−11.9013	−0.829203
$207$	12.7202	0.884112
$208$	1.43380	0.0994159
$209$	12.2243	0.845571
$210$	−12.2064	−0.842324
$211$	18.0500	1.24262	0.621308	−	0.783566i	$-0.286601\pi$
$211$	18.0500	1.24262	0.621308	+	0.783566i	$0.286601\pi$
$212$	−11.9314	−0.819452
$213$	36.9069	2.52882
$214$	4.54159	0.310457
$215$	10.2649	0.700060
$216$	−5.63269	−0.383256
$217$	−8.30903	−0.564054
$218$	−10.1215	−0.685514
$219$	−32.7960	−2.21615
$220$	5.72961	0.386290
$221$	−4.92952	−0.331595
$222$	30.8373	2.06966
$223$	−5.28259	−0.353748	−0.176874	−	0.984233i	$-0.556599\pi$
$223$	−5.28259	−0.353748	−0.176874	+	0.984233i	$0.556599\pi$
$224$	−1.73566	−0.115969
$225$	5.93253	0.395502
$226$	−6.70099	−0.445743
$227$	0.192854	0.0128002	0.00640008	−	0.999980i	$-0.497963\pi$
$227$	0.192854	0.0128002	0.00640008	+	0.999980i	$0.497963\pi$
$228$	15.0046	0.993701
$229$	−8.11208	−0.536062	−0.268031	−	0.963410i	$-0.586373\pi$
$229$	−8.11208	−0.536062	−0.268031	+	0.963410i	$0.586373\pi$
$230$	−6.33824	−0.417931
$231$	−11.3016	−0.743593
$232$	0.912086	0.0598814
$233$	−16.6850	−1.09307	−0.546535	−	0.837436i	$-0.684054\pi$
$233$	−16.6850	−1.09307	−0.546535	+	0.837436i	$0.684054\pi$
$234$	7.15809	0.467939
$235$	−21.7661	−1.41986
$236$	−12.0205	−0.782467
$237$	−21.3862	−1.38918
$238$	5.96735	0.386806
$239$	24.9767	1.61561	0.807804	−	0.589452i	$-0.200656\pi$
$239$	24.9767	1.61561	0.807804	+	0.589452i	$0.200656\pi$
$240$	7.03274	0.453962
$241$	−18.2581	−1.17611	−0.588054	−	0.808822i	$-0.700106\pi$
$241$	−18.2581	−1.17611	−0.588054	+	0.808822i	$0.700106\pi$
$242$	−5.69510	−0.366095
$243$	14.2212	0.912289
$244$	−10.0543	−0.643659
$245$	9.91941	0.633728
$246$	7.71567	0.491933
$247$	−7.60978	−0.484199
$248$	4.78725	0.303991
$249$	15.2279	0.965030
$250$	9.48208	0.599699
$251$	−5.05076	−0.318801	−0.159401	−	0.987214i	$-0.550956\pi$
$251$	−5.05076	−0.318801	−0.159401	+	0.987214i	$0.550956\pi$
$252$	−8.66510	−0.545850
$253$	−5.86842	−0.368945
$254$	18.0575	1.13303
$255$	−24.1792	−1.51416
$256$	1.00000	0.0625000
$257$	27.1948	1.69636	0.848181	−	0.529706i	$-0.177698\pi$
$257$	27.1948	1.69636	0.848181	+	0.529706i	$0.177698\pi$
$258$	11.6656	0.726268
$259$	18.9322	1.17639
$260$	−3.56676	−0.221201
$261$	4.55350	0.281855
$262$	14.9932	0.926281
$263$	−7.98979	−0.492671	−0.246336	−	0.969185i	$-0.579227\pi$
$263$	−7.98979	−0.492671	−0.246336	+	0.969185i	$0.579227\pi$
$264$	6.51145	0.400752
$265$	29.6809	1.82328
$266$	9.21189	0.564817
$267$	−30.1669	−1.84619
$268$	−12.1910	−0.744682
$269$	−17.7286	−1.08093	−0.540464	−	0.841367i	$-0.681751\pi$
$269$	−17.7286	−1.08093	−0.540464	+	0.841367i	$0.681751\pi$
$270$	14.0121	0.852746
$271$	13.2693	0.806051	0.403026	−	0.915189i	$-0.367959\pi$
$271$	13.2693	0.806051	0.403026	+	0.915189i	$0.367959\pi$
$272$	−3.43809	−0.208465
$273$	7.03543	0.425803
$274$	−18.5808	−1.12251
$275$	−2.73696	−0.165045
$276$	−7.20313	−0.433578
$277$	6.88400	0.413620	0.206810	−	0.978381i	$-0.433692\pi$
$277$	6.88400	0.413620	0.206810	+	0.978381i	$0.433692\pi$
$278$	7.78293	0.466789
$279$	23.8999	1.43085
$280$	4.31768	0.258031
$281$	−17.6905	−1.05533	−0.527663	−	0.849454i	$-0.676932\pi$
$281$	−17.6905	−1.05533	−0.527663	+	0.849454i	$0.676932\pi$
$282$	−24.7362	−1.47302
$283$	7.81975	0.464836	0.232418	−	0.972616i	$-0.425336\pi$
$283$	7.81975	0.464836	0.232418	+	0.972616i	$0.425336\pi$
$284$	−13.0548	−0.774658
$285$	−37.3258	−2.21099
$286$	−3.30237	−0.195273
$287$	4.73695	0.279614
$288$	4.99240	0.294180
$289$	−5.17954	−0.304679
$290$	−2.26893	−0.133236
$291$	−22.1209	−1.29675
$292$	11.6007	0.678877
$293$	24.6718	1.44134	0.720671	−	0.693277i	$-0.243834\pi$
$293$	24.6718	1.44134	0.720671	+	0.693277i	$0.243834\pi$
$294$	11.2730	0.657453
$295$	29.9026	1.74099
$296$	−10.9078	−0.634003
$297$	12.9734	0.752794
$298$	−3.57877	−0.207312
$299$	3.65317	0.211268
$300$	−3.35946	−0.193958
$301$	7.16197	0.412809
$302$	−1.57752	−0.0907759
$303$	−24.3501	−1.39888
$304$	−5.30744	−0.304402
$305$	25.0113	1.43214
$306$	−17.1643	−0.981219
$307$	3.73552	0.213197	0.106599	−	0.994302i	$-0.466004\pi$
$307$	3.73552	0.213197	0.106599	+	0.994302i	$0.466004\pi$
$308$	3.99763	0.227786
$309$	33.6460	1.91405
$310$	−11.9089	−0.676381
$311$	6.49530	0.368315	0.184157	−	0.982897i	$-0.441044\pi$
$311$	6.49530	0.368315	0.184157	+	0.982897i	$0.441044\pi$
$312$	−4.05346	−0.229482
$313$	16.2586	0.918989	0.459495	−	0.888181i	$-0.348031\pi$
$313$	16.2586	0.918989	0.459495	+	0.888181i	$0.348031\pi$
$314$	11.0417	0.623117
$315$	21.5556	1.21452
$316$	7.56475	0.425550
$317$	33.7229	1.89407	0.947035	−	0.321131i	$-0.104063\pi$
$317$	33.7229	1.89407	0.947035	+	0.321131i	$0.104063\pi$
$318$	33.7311	1.89154
$319$	−2.10075	−0.117620
$320$	−2.48763	−0.139063
$321$	−12.8395	−0.716629
$322$	−4.42229	−0.246444
$323$	18.2474	1.01531
$324$	0.946874	0.0526041
$325$	1.70380	0.0945096
$326$	11.8206	0.654683
$327$	28.6143	1.58237
$328$	−2.72920	−0.150695
$329$	−15.1865	−0.837260
$330$	−16.1981	−0.891675
$331$	−1.05776	−0.0581400	−0.0290700	−	0.999577i	$-0.509255\pi$
$331$	−1.05776	−0.0581400	−0.0290700	+	0.999577i	$0.509255\pi$
$332$	−5.38644	−0.295619
$333$	−54.4561	−2.98418
$334$	0.713744	0.0390543
$335$	30.3266	1.65692
$336$	4.90685	0.267691
$337$	−10.8341	−0.590171	−0.295085	−	0.955471i	$-0.595348\pi$
$337$	−10.8341	−0.590171	−0.295085	+	0.955471i	$0.595348\pi$
$338$	−10.9442	−0.595288
$339$	18.9442	1.02891
$340$	8.55270	0.463835
$341$	−11.0262	−0.597101
$342$	−26.4969	−1.43279
$343$	19.0705	1.02971
$344$	−4.12637	−0.222479
$345$	17.9187	0.964713
$346$	−7.78446	−0.418495
$347$	20.7601	1.11446	0.557232	−	0.830357i	$-0.311864\pi$
$347$	20.7601	1.11446	0.557232	+	0.830357i	$0.311864\pi$
$348$	−2.57854	−0.138224
$349$	−17.5460	−0.939215	−0.469608	−	0.882875i	$-0.655605\pi$
$349$	−17.5460	−0.939215	−0.469608	+	0.882875i	$0.655605\pi$
$350$	−2.06250	−0.110245
$351$	−8.07613	−0.431072
$352$	−2.30324	−0.122763
$353$	−0.359014	−0.0191084	−0.00955421	−	0.999954i	$-0.503041\pi$
$353$	−0.359014	−0.0191084	−0.00955421	+	0.999954i	$0.503041\pi$
$354$	33.9829	1.80617
$355$	32.4754	1.72362
$356$	10.6707	0.565545
$357$	−16.8702	−0.892865
$358$	−8.66924	−0.458183
$359$	1.26279	0.0666474	0.0333237	−	0.999445i	$-0.489391\pi$
$359$	1.26279	0.0666474	0.0333237	+	0.999445i	$0.489391\pi$
$360$	−12.4193	−0.654552
$361$	9.16887	0.482572
$362$	−4.07244	−0.214043
$363$	16.1005	0.845058
$364$	−2.48858	−0.130437
$365$	−28.8581	−1.51050
$366$	28.4243	1.48576
$367$	7.30969	0.381563	0.190781	−	0.981633i	$-0.438898\pi$
$367$	7.30969	0.381563	0.190781	+	0.981633i	$0.438898\pi$
$368$	2.54790	0.132819
$369$	−13.6253	−0.709302
$370$	27.1346	1.41066
$371$	20.7088	1.07515
$372$	−13.5340	−0.701703
$373$	−0.115843	−0.00599814	−0.00299907	−	0.999996i	$-0.500955\pi$
$373$	−0.115843	−0.00599814	−0.00299907	+	0.999996i	$0.500955\pi$
$374$	7.91874	0.409468
$375$	−26.8066	−1.38429
$376$	8.74972	0.451232
$377$	1.30775	0.0673524
$378$	9.77642	0.502845
$379$	36.2890	1.86404	0.932022	−	0.362403i	$-0.118044\pi$
$379$	36.2890	1.86404	0.932022	+	0.362403i	$0.118044\pi$
$380$	13.2029	0.677297
$381$	−51.0499	−2.61537
$382$	−7.71008	−0.394482
$383$	10.4069	0.531768	0.265884	−	0.964005i	$-0.414336\pi$
$383$	10.4069	0.531768	0.265884	+	0.964005i	$0.414336\pi$
$384$	−2.82708	−0.144269
$385$	−9.94464	−0.506826
$386$	−0.609783	−0.0310371
$387$	−20.6005	−1.04718
$388$	7.82462	0.397235
$389$	7.80537	0.395748	0.197874	−	0.980227i	$-0.436596\pi$
$389$	7.80537	0.395748	0.197874	+	0.980227i	$0.436596\pi$
$390$	10.0835	0.510599
$391$	−8.75992	−0.443008
$392$	−3.98749	−0.201399
$393$	−42.3869	−2.13814
$394$	−5.92525	−0.298510
$395$	−18.8183	−0.946852
$396$	−11.4987	−0.577831
$397$	−29.2989	−1.47047	−0.735234	−	0.677814i	$-0.762928\pi$
$397$	−29.2989	−1.47047	−0.735234	+	0.677814i	$0.762928\pi$
$398$	−6.17206	−0.309377
$399$	−26.0428	−1.30377
$400$	1.18831	0.0594156
$401$	−8.27570	−0.413269	−0.206634	−	0.978418i	$-0.566251\pi$
$401$	−8.27570	−0.413269	−0.206634	+	0.978418i	$0.566251\pi$
$402$	34.4649	1.71895
$403$	6.86395	0.341918
$404$	8.61317	0.428521
$405$	−2.35547	−0.117044
$406$	−1.58307	−0.0785664
$407$	25.1233	1.24531
$408$	9.71977	0.481200
$409$	32.9009	1.62684	0.813422	−	0.581675i	$-0.197602\pi$
$409$	32.9009	1.62684	0.813422	+	0.581675i	$0.197602\pi$
$410$	6.78924	0.335297
$411$	52.5294	2.59109
$412$	−11.9013	−0.586335
$413$	20.8635	1.02662
$414$	12.7202	0.625161
$415$	13.3995	0.657755
$416$	1.43380	0.0702977
$417$	−22.0030	−1.07749
$418$	12.2243	0.597909
$419$	12.0703	0.589672	0.294836	−	0.955548i	$-0.404735\pi$
$419$	12.0703	0.589672	0.294836	+	0.955548i	$0.404735\pi$
$420$	−12.2064	−0.595613
$421$	32.0863	1.56379	0.781896	−	0.623409i	$-0.214253\pi$
$421$	32.0863	1.56379	0.781896	+	0.623409i	$0.214253\pi$
$422$	18.0500	0.878663
$423$	43.6821	2.12390
$424$	−11.9314	−0.579440
$425$	−4.08552	−0.198177
$426$	36.9069	1.78815
$427$	17.4508	0.844503
$428$	4.54159	0.219526
$429$	9.33609	0.450751
$430$	10.2649	0.495017
$431$	−0.189885	−0.00914645	−0.00457322	−	0.999990i	$-0.501456\pi$
$431$	−0.189885	−0.00914645	−0.00457322	+	0.999990i	$0.501456\pi$
$432$	−5.63269	−0.271003
$433$	−24.7173	−1.18784	−0.593919	−	0.804525i	$-0.702420\pi$
$433$	−24.7173	−1.18784	−0.593919	+	0.804525i	$0.702420\pi$
$434$	−8.30903	−0.398846
$435$	6.41447	0.307550
$436$	−10.1215	−0.484731
$437$	−13.5228	−0.646885
$438$	−32.7960	−1.56705
$439$	20.4842	0.977655	0.488828	−	0.872380i	$-0.337425\pi$
$439$	20.4842	0.977655	0.488828	+	0.872380i	$0.337425\pi$
$440$	5.72961	0.273148
$441$	−19.9072	−0.947960
$442$	−4.92952	−0.234473
$443$	14.8504	0.705564	0.352782	−	0.935705i	$-0.385236\pi$
$443$	14.8504	0.705564	0.352782	+	0.935705i	$0.385236\pi$
$444$	30.8373	1.46347
$445$	−26.5447	−1.25834
$446$	−5.28259	−0.250138
$447$	10.1175	0.478540
$448$	−1.73566	−0.0820021
$449$	35.5553	1.67796	0.838980	−	0.544162i	$-0.183152\pi$
$449$	35.5553	1.67796	0.838980	+	0.544162i	$0.183152\pi$
$450$	5.93253	0.279662
$451$	6.28599	0.295996
$452$	−6.70099	−0.315188
$453$	4.45977	0.209538
$454$	0.192854	0.00905109
$455$	6.19067	0.290223
$456$	15.0046	0.702653
$457$	17.0580	0.797942	0.398971	−	0.916964i	$-0.369367\pi$
$457$	17.0580	0.797942	0.398971	+	0.916964i	$0.369367\pi$
$458$	−8.11208	−0.379053
$459$	19.3657	0.903913
$460$	−6.33824	−0.295522
$461$	42.5547	1.98197	0.990986	−	0.133967i	$-0.0427715\pi$
$461$	42.5547	1.98197	0.990986	+	0.133967i	$0.0427715\pi$
$462$	−11.3016	−0.525800
$463$	42.0928	1.95622	0.978109	−	0.208094i	$-0.0667260\pi$
$463$	42.0928	1.95622	0.978109	+	0.208094i	$0.0667260\pi$
$464$	0.912086	0.0423425
$465$	33.6675	1.56129
$466$	−16.6850	−0.772918
$467$	−15.1037	−0.698915	−0.349458	−	0.936952i	$-0.613634\pi$
$467$	−15.1037	−0.698915	−0.349458	+	0.936952i	$0.613634\pi$
$468$	7.15809	0.330883
$469$	21.1594	0.977048
$470$	−21.7661	−1.00399
$471$	−31.2157	−1.43834
$472$	−12.0205	−0.553288
$473$	9.50401	0.436995
$474$	−21.3862	−0.982299
$475$	−6.30688	−0.289380
$476$	5.96735	0.273513
$477$	−59.5663	−2.72735
$478$	24.9767	1.14241
$479$	3.29900	0.150735	0.0753675	−	0.997156i	$-0.475987\pi$
$479$	3.29900	0.150735	0.0753675	+	0.997156i	$0.475987\pi$
$480$	7.03274	0.320999
$481$	−15.6396	−0.713103
$482$	−18.2581	−0.831633
$483$	12.5022	0.568869
$484$	−5.69510	−0.258868
$485$	−19.4648	−0.883850
$486$	14.2212	0.645085
$487$	−29.1985	−1.32311	−0.661556	−	0.749896i	$-0.730103\pi$
$487$	−29.1985	−1.32311	−0.661556	+	0.749896i	$0.730103\pi$
$488$	−10.0543	−0.455136
$489$	−33.4179	−1.51121
$490$	9.91941	0.448113
$491$	−26.0672	−1.17640	−0.588198	−	0.808717i	$-0.700162\pi$
$491$	−26.0672	−1.17640	−0.588198	+	0.808717i	$0.700162\pi$
$492$	7.71567	0.347849
$493$	−3.13584	−0.141231
$494$	−7.60978	−0.342380
$495$	28.6045	1.28568
$496$	4.78725	0.214954
$497$	22.6586	1.01638
$498$	15.2279	0.682379
$499$	−31.9291	−1.42934	−0.714671	−	0.699461i	$-0.753423\pi$
$499$	−31.9291	−1.42934	−0.714671	+	0.699461i	$0.753423\pi$
$500$	9.48208	0.424051
$501$	−2.01781	−0.0901492
$502$	−5.05076	−0.225426
$503$	41.3496	1.84369	0.921844	−	0.387561i	$-0.126682\pi$
$503$	41.3496	1.84369	0.921844	+	0.387561i	$0.126682\pi$
$504$	−8.66510	−0.385974
$505$	−21.4264	−0.953462
$506$	−5.86842	−0.260883
$507$	30.9402	1.37410
$508$	18.0575	0.801170
$509$	−14.7019	−0.651650	−0.325825	−	0.945430i	$-0.605642\pi$
$509$	−14.7019	−0.651650	−0.325825	+	0.945430i	$0.605642\pi$
$510$	−24.1792	−1.07067
$511$	−20.1348	−0.890709
$512$	1.00000	0.0441942
$513$	29.8951	1.31990
$514$	27.1948	1.19951
$515$	29.6061	1.30460
$516$	11.6656	0.513549
$517$	−20.1527	−0.886314
$518$	18.9322	0.831834
$519$	22.0073	0.966014
$520$	−3.56676	−0.156413
$521$	32.3118	1.41561	0.707803	−	0.706410i	$-0.249686\pi$
$521$	32.3118	1.41561	0.707803	+	0.706410i	$0.249686\pi$
$522$	4.55350	0.199301
$523$	−29.4182	−1.28637	−0.643184	−	0.765712i	$-0.722387\pi$
$523$	−29.4182	−1.28637	−0.643184	+	0.765712i	$0.722387\pi$
$524$	14.9932	0.654979
$525$	5.83087	0.254480
$526$	−7.98979	−0.348371
$527$	−16.4590	−0.716966
$528$	6.51145	0.283374
$529$	−16.5082	−0.717747
$530$	29.6809	1.28926
$531$	−60.0111	−2.60426
$532$	9.21189	0.399386
$533$	−3.91311	−0.169496
$534$	−30.1669	−1.30545
$535$	−11.2978	−0.488447
$536$	−12.1910	−0.526570
$537$	24.5087	1.05763
$538$	−17.7286	−0.764332
$539$	9.18414	0.395589
$540$	14.0121	0.602983
$541$	−9.19095	−0.395150	−0.197575	−	0.980288i	$-0.563307\pi$
$541$	−9.19095	−0.395150	−0.197575	+	0.980288i	$0.563307\pi$
$542$	13.2693	0.569964
$543$	11.5131	0.494076
$544$	−3.43809	−0.147407
$545$	25.1785	1.07853
$546$	7.03543	0.301088
$547$	−38.4373	−1.64346	−0.821730	−	0.569878i	$-0.806991\pi$
$547$	−38.4373	−1.64346	−0.821730	+	0.569878i	$0.806991\pi$
$548$	−18.5808	−0.793732
$549$	−50.1950	−2.14227
$550$	−2.73696	−0.116704
$551$	−4.84084	−0.206227
$552$	−7.20313	−0.306586
$553$	−13.1298	−0.558336
$554$	6.88400	0.292473
$555$	−76.7118	−3.25623
$556$	7.78293	0.330070
$557$	−9.27188	−0.392862	−0.196431	−	0.980518i	$-0.562935\pi$
$557$	−9.27188	−0.392862	−0.196431	+	0.980518i	$0.562935\pi$
$558$	23.8999	1.01176
$559$	−5.91637	−0.250236
$560$	4.31768	0.182455
$561$	−22.3869	−0.945177
$562$	−17.6905	−0.746229
$563$	39.1250	1.64892	0.824462	−	0.565918i	$-0.191478\pi$
$563$	39.1250	1.64892	0.824462	+	0.565918i	$0.191478\pi$
$564$	−24.7362	−1.04158
$565$	16.6696	0.701295
$566$	7.81975	0.328689
$567$	−1.64345	−0.0690184
$568$	−13.0548	−0.547766
$569$	14.3148	0.600109	0.300055	−	0.953922i	$-0.402995\pi$
$569$	14.3148	0.600109	0.300055	+	0.953922i	$0.402995\pi$
$570$	−37.3258	−1.56341
$571$	27.7544	1.16149	0.580743	−	0.814087i	$-0.302762\pi$
$571$	27.7544	1.16149	0.580743	+	0.814087i	$0.302762\pi$
$572$	−3.30237	−0.138079
$573$	21.7971	0.910585
$574$	4.73695	0.197717
$575$	3.02770	0.126264
$576$	4.99240	0.208017
$577$	−40.6248	−1.69123	−0.845616	−	0.533792i	$-0.820767\pi$
$577$	−40.6248	−1.69123	−0.845616	+	0.533792i	$0.820767\pi$
$578$	−5.17954	−0.215440
$579$	1.72391	0.0716431
$580$	−2.26893	−0.0942124
$581$	9.34902	0.387863
$582$	−22.1209	−0.916939
$583$	27.4808	1.13814
$584$	11.6007	0.480038
$585$	−17.8067	−0.736216
$586$	24.6718	1.01918
$587$	−24.9408	−1.02942	−0.514708	−	0.857366i	$-0.672100\pi$
$587$	−24.9408	−1.02942	−0.514708	+	0.857366i	$0.672100\pi$
$588$	11.2730	0.464890
$589$	−25.4080	−1.04692
$590$	29.9026	1.23107
$591$	16.7512	0.689051
$592$	−10.9078	−0.448308
$593$	27.0740	1.11180	0.555898	−	0.831251i	$-0.312375\pi$
$593$	27.0740	1.11180	0.555898	+	0.831251i	$0.312375\pi$
$594$	12.9734	0.532306
$595$	−14.8446	−0.608568
$596$	−3.57877	−0.146592
$597$	17.4489	0.714137
$598$	3.65317	0.149389
$599$	−5.48581	−0.224144	−0.112072	−	0.993700i	$-0.535749\pi$
$599$	−5.48581	−0.224144	−0.112072	+	0.993700i	$0.535749\pi$
$600$	−3.35946	−0.137149
$601$	−36.2133	−1.47717	−0.738585	−	0.674160i	$-0.764506\pi$
$601$	−36.2133	−1.47717	−0.738585	+	0.674160i	$0.764506\pi$
$602$	7.16197	0.291900
$603$	−60.8622	−2.47850
$604$	−1.57752	−0.0641883
$605$	14.1673	0.575983
$606$	−24.3501	−0.989157
$607$	42.8629	1.73975	0.869875	−	0.493272i	$-0.164199\pi$
$607$	42.8629	1.73975	0.869875	+	0.493272i	$0.164199\pi$
$608$	−5.30744	−0.215245
$609$	4.47547	0.181355
$610$	25.0113	1.01268
$611$	12.5453	0.507529
$612$	−17.1643	−0.693827
$613$	−7.60748	−0.307263	−0.153632	−	0.988128i	$-0.549097\pi$
$613$	−7.60748	−0.307263	−0.153632	+	0.988128i	$0.549097\pi$
$614$	3.73552	0.150753
$615$	−19.1937	−0.773967
$616$	3.99763	0.161069
$617$	−26.2641	−1.05735	−0.528676	−	0.848824i	$-0.677311\pi$
$617$	−26.2641	−1.05735	−0.528676	+	0.848824i	$0.677311\pi$
$618$	33.6460	1.35344
$619$	34.5724	1.38958	0.694791	−	0.719212i	$-0.255497\pi$
$619$	34.5724	1.38958	0.694791	+	0.719212i	$0.255497\pi$
$620$	−11.9089	−0.478274
$621$	−14.3515	−0.575907
$622$	6.49530	0.260438
$623$	−18.5207	−0.742015
$624$	−4.05346	−0.162268
$625$	−29.5295	−1.18118
$626$	16.2586	0.649823
$627$	−34.5591	−1.38016
$628$	11.0417	0.440611
$629$	37.5020	1.49530
$630$	21.5556	0.858795
$631$	−27.3602	−1.08919	−0.544596	−	0.838699i	$-0.683317\pi$
$631$	−27.3602	−1.08919	−0.544596	+	0.838699i	$0.683317\pi$
$632$	7.56475	0.300909
$633$	−51.0290	−2.02822
$634$	33.7229	1.33931
$635$	−44.9203	−1.78261
$636$	33.7311	1.33752
$637$	−5.71725	−0.226526
$638$	−2.10075	−0.0831696
$639$	−65.1746	−2.57827
$640$	−2.48763	−0.0983323
$641$	−32.9032	−1.29960	−0.649799	−	0.760106i	$-0.725147\pi$
$641$	−32.9032	−1.29960	−0.649799	+	0.760106i	$0.725147\pi$
$642$	−12.8395	−0.506733
$643$	−28.5144	−1.12450	−0.562250	−	0.826968i	$-0.690064\pi$
$643$	−28.5144	−1.12450	−0.562250	+	0.826968i	$0.690064\pi$
$644$	−4.42229	−0.174263
$645$	−29.0197	−1.14265
$646$	18.2474	0.717936
$647$	36.4304	1.43223	0.716113	−	0.697985i	$-0.245920\pi$
$647$	36.4304	1.43223	0.716113	+	0.697985i	$0.245920\pi$
$648$	0.946874	0.0371967
$649$	27.6861	1.08677
$650$	1.70380	0.0668284
$651$	23.4903	0.920659
$652$	11.8206	0.462931
$653$	−7.29452	−0.285457	−0.142728	−	0.989762i	$-0.545588\pi$
$653$	−7.29452	−0.285457	−0.142728	+	0.989762i	$0.545588\pi$
$654$	28.6143	1.11891
$655$	−37.2975	−1.45733
$656$	−2.72920	−0.106557
$657$	57.9151	2.25948
$658$	−15.1865	−0.592032
$659$	−14.5928	−0.568457	−0.284228	−	0.958757i	$-0.591737\pi$
$659$	−14.5928	−0.568457	−0.284228	+	0.958757i	$0.591737\pi$
$660$	−16.1981	−0.630509
$661$	2.63619	0.102536	0.0512680	−	0.998685i	$-0.483674\pi$
$661$	2.63619	0.102536	0.0512680	+	0.998685i	$0.483674\pi$
$662$	−1.05776	−0.0411112
$663$	13.9362	0.541236
$664$	−5.38644	−0.209034
$665$	−22.9158	−0.888636
$666$	−54.4561	−2.11013
$667$	2.32391	0.0899820
$668$	0.713744	0.0276156
$669$	14.9343	0.577394
$670$	30.3266	1.17162
$671$	23.1574	0.893981
$672$	4.90685	0.189286
$673$	50.2848	1.93834	0.969169	−	0.246396i	$-0.0792464\pi$
$673$	50.2848	1.93834	0.969169	+	0.246396i	$0.0792464\pi$
$674$	−10.8341	−0.417314
$675$	−6.69339	−0.257629
$676$	−10.9442	−0.420932
$677$	35.2887	1.35625	0.678127	−	0.734945i	$-0.262792\pi$
$677$	35.2887	1.35625	0.678127	+	0.734945i	$0.262792\pi$
$678$	18.9442	0.727549
$679$	−13.5809	−0.521186
$680$	8.55270	0.327981
$681$	−0.545214	−0.0208927
$682$	−11.0262	−0.422214
$683$	−15.0529	−0.575982	−0.287991	−	0.957633i	$-0.592987\pi$
$683$	−15.0529	−0.575982	−0.287991	+	0.957633i	$0.592987\pi$
$684$	−26.4969	−1.01313
$685$	46.2221	1.76606
$686$	19.0705	0.728116
$687$	22.9335	0.874969
$688$	−4.12637	−0.157316
$689$	−17.1072	−0.651732
$690$	17.9187	0.682155
$691$	−43.3698	−1.64986	−0.824932	−	0.565232i	$-0.808787\pi$
$691$	−43.3698	−1.64986	−0.824932	+	0.565232i	$0.808787\pi$
$692$	−7.78446	−0.295921
$693$	19.9578	0.758134
$694$	20.7601	0.788044
$695$	−19.3611	−0.734407
$696$	−2.57854	−0.0977395
$697$	9.38323	0.355415
$698$	−17.5460	−0.664126
$699$	47.1699	1.78413
$700$	−2.06250	−0.0779552
$701$	32.4023	1.22382	0.611908	−	0.790929i	$-0.290402\pi$
$701$	32.4023	1.22382	0.611908	+	0.790929i	$0.290402\pi$
$702$	−8.07613	−0.304814
$703$	57.8925	2.18346
$704$	−2.30324	−0.0868065
$705$	61.5345	2.31753
$706$	−0.359014	−0.0135117
$707$	−14.9495	−0.562234
$708$	33.9829	1.27716
$709$	9.12246	0.342601	0.171301	−	0.985219i	$-0.445203\pi$
$709$	9.12246	0.342601	0.171301	+	0.985219i	$0.445203\pi$
$710$	32.4754	1.21878
$711$	37.7663	1.41635
$712$	10.6707	0.399901
$713$	12.1975	0.456798
$714$	−16.8702	−0.631351
$715$	8.21509	0.307227
$716$	−8.66924	−0.323985
$717$	−70.6112	−2.63702
$718$	1.26279	0.0471269
$719$	15.8016	0.589302	0.294651	−	0.955605i	$-0.404797\pi$
$719$	15.8016	0.589302	0.294651	+	0.955605i	$0.404797\pi$
$720$	−12.4193	−0.462838
$721$	20.6566	0.769292
$722$	9.16887	0.341230
$723$	51.6172	1.91966
$724$	−4.07244	−0.151351
$725$	1.08384	0.0402529
$726$	16.1005	0.597546
$727$	−24.9364	−0.924838	−0.462419	−	0.886661i	$-0.653018\pi$
$727$	−24.9364	−0.924838	−0.462419	+	0.886661i	$0.653018\pi$
$728$	−2.48858	−0.0922329
$729$	−43.0451	−1.59426
$730$	−28.8581	−1.06809
$731$	14.1868	0.524719
$732$	28.4243	1.05059
$733$	−0.738122	−0.0272632	−0.0136316	−	0.999907i	$-0.504339\pi$
$733$	−0.738122	−0.0272632	−0.0136316	+	0.999907i	$0.504339\pi$
$734$	7.30969	0.269806
$735$	−28.0430	−1.03438
$736$	2.54790	0.0939169
$737$	28.0787	1.03429
$738$	−13.6253	−0.501553
$739$	−30.8473	−1.13474	−0.567368	−	0.823465i	$-0.692038\pi$
$739$	−30.8473	−1.13474	−0.567368	+	0.823465i	$0.692038\pi$
$740$	27.1346	0.997488
$741$	21.5135	0.790318
$742$	20.7088	0.760245
$743$	35.3320	1.29621	0.648104	−	0.761552i	$-0.275562\pi$
$743$	35.3320	1.29621	0.648104	+	0.761552i	$0.275562\pi$
$744$	−13.5340	−0.496179
$745$	8.90265	0.326168
$746$	−0.115843	−0.00424132
$747$	−26.8913	−0.983900
$748$	7.91874	0.289538
$749$	−7.88265	−0.288026
$750$	−26.8066	−0.978840
$751$	−29.2383	−1.06692	−0.533461	−	0.845825i	$-0.679109\pi$
$751$	−29.2383	−1.06692	−0.533461	+	0.845825i	$0.679109\pi$
$752$	8.74972	0.319070
$753$	14.2789	0.520353
$754$	1.30775	0.0476253
$755$	3.92428	0.142819
$756$	9.77642	0.355565
$757$	−12.0172	−0.436774	−0.218387	−	0.975862i	$-0.570079\pi$
$757$	−12.0172	−0.436774	−0.218387	+	0.975862i	$0.570079\pi$
$758$	36.2890	1.31808
$759$	16.5905	0.602198
$760$	13.2029	0.478921
$761$	−29.3741	−1.06481	−0.532405	−	0.846490i	$-0.678711\pi$
$761$	−29.3741	−1.06481	−0.532405	+	0.846490i	$0.678711\pi$
$762$	−51.0499	−1.84934
$763$	17.5674	0.635984
$764$	−7.71008	−0.278941
$765$	42.6985	1.54377
$766$	10.4069	0.376017
$767$	−17.2349	−0.622318
$768$	−2.82708	−0.102014
$769$	−23.8490	−0.860018	−0.430009	−	0.902825i	$-0.641490\pi$
$769$	−23.8490	−0.860018	−0.430009	+	0.902825i	$0.641490\pi$
$770$	−9.94464	−0.358380
$771$	−76.8819	−2.76883
$772$	−0.609783	−0.0219466
$773$	1.16966	0.0420699	0.0210349	−	0.999779i	$-0.493304\pi$
$773$	1.16966	0.0420699	0.0210349	+	0.999779i	$0.493304\pi$
$774$	−20.6005	−0.740470
$775$	5.68875	0.204346
$776$	7.82462	0.280888
$777$	−53.5230	−1.92013
$778$	7.80537	0.279836
$779$	14.4850	0.518980
$780$	10.0835	0.361048
$781$	30.0682	1.07593
$782$	−8.75992	−0.313254
$783$	−5.13750	−0.183599
$784$	−3.98749	−0.142410
$785$	−27.4676	−0.980361
$786$	−42.3869	−1.51189
$787$	−15.5403	−0.553950	−0.276975	−	0.960877i	$-0.589332\pi$
$787$	−15.5403	−0.553950	−0.276975	+	0.960877i	$0.589332\pi$
$788$	−5.92525	−0.211078
$789$	22.5878	0.804147
$790$	−18.8183	−0.669525
$791$	11.6306	0.413537
$792$	−11.4987	−0.408588
$793$	−14.4158	−0.511920
$794$	−29.2989	−1.03978
$795$	−83.9104	−2.97600
$796$	−6.17206	−0.218763
$797$	26.8390	0.950688	0.475344	−	0.879800i	$-0.342324\pi$
$797$	26.8390	0.950688	0.475344	+	0.879800i	$0.342324\pi$
$798$	−26.0428	−0.921905
$799$	−30.0823	−1.06424
$800$	1.18831	0.0420131
$801$	53.2724	1.88229
$802$	−8.27570	−0.292225
$803$	−26.7191	−0.942895
$804$	34.4649	1.21548
$805$	11.0010	0.387735
$806$	6.86395	0.241772
$807$	50.1201	1.76431
$808$	8.61317	0.303010
$809$	−12.1541	−0.427315	−0.213658	−	0.976909i	$-0.568538\pi$
$809$	−12.1541	−0.427315	−0.213658	+	0.976909i	$0.568538\pi$
$810$	−2.35547	−0.0827629
$811$	−9.77229	−0.343151	−0.171576	−	0.985171i	$-0.554886\pi$
$811$	−9.77229	−0.343151	−0.171576	+	0.985171i	$0.554886\pi$
$812$	−1.58307	−0.0555549
$813$	−37.5134	−1.31565
$814$	25.1233	0.880570
$815$	−29.4053	−1.03002
$816$	9.71977	0.340260
$817$	21.9004	0.766199
$818$	32.9009	1.15035
$819$	−12.4240	−0.434129
$820$	6.78924	0.237091
$821$	27.9350	0.974938	0.487469	−	0.873140i	$-0.337920\pi$
$821$	27.9350	0.974938	0.487469	+	0.873140i	$0.337920\pi$
$822$	52.5294	1.83217
$823$	5.05898	0.176345	0.0881725	−	0.996105i	$-0.471897\pi$
$823$	5.05898	0.176345	0.0881725	+	0.996105i	$0.471897\pi$
$824$	−11.9013	−0.414602
$825$	7.73762	0.269389
$826$	20.8635	0.725933
$827$	11.1621	0.388146	0.194073	−	0.980987i	$-0.437830\pi$
$827$	11.1621	0.388146	0.194073	+	0.980987i	$0.437830\pi$
$828$	12.7202	0.442056
$829$	−49.3456	−1.71384	−0.856921	−	0.515448i	$-0.827626\pi$
$829$	−49.3456	−1.71384	−0.856921	+	0.515448i	$0.827626\pi$
$830$	13.3995	0.465103
$831$	−19.4617	−0.675117
$832$	1.43380	0.0497079
$833$	13.7094	0.475001
$834$	−22.0030	−0.761902
$835$	−1.77553	−0.0614448
$836$	12.2243	0.422786
$837$	−26.9651	−0.932050
$838$	12.0703	0.416961
$839$	−33.8882	−1.16995	−0.584975	−	0.811051i	$-0.698896\pi$
$839$	−33.8882	−1.16995	−0.584975	+	0.811051i	$0.698896\pi$
$840$	−12.2064	−0.421162
$841$	−28.1681	−0.971314
$842$	32.0863	1.10577
$843$	50.0125	1.72252
$844$	18.0500	0.621308
$845$	27.2252	0.936576
$846$	43.6821	1.50182
$847$	9.88474	0.339644
$848$	−11.9314	−0.409726
$849$	−22.1071	−0.758713
$850$	−4.08552	−0.140132
$851$	−27.7920	−0.952698
$852$	36.9069	1.26441
$853$	−35.9338	−1.23035	−0.615176	−	0.788390i	$-0.710915\pi$
$853$	−35.9338	−1.23035	−0.615176	+	0.788390i	$0.710915\pi$
$854$	17.4508	0.597154
$855$	65.9144	2.25423
$856$	4.54159	0.155228
$857$	28.8007	0.983813	0.491907	−	0.870648i	$-0.336300\pi$
$857$	28.8007	0.983813	0.491907	+	0.870648i	$0.336300\pi$
$858$	9.33609	0.318729
$859$	−45.8265	−1.56358	−0.781790	−	0.623542i	$-0.785693\pi$
$859$	−45.8265	−1.56358	−0.781790	+	0.623542i	$0.785693\pi$
$860$	10.2649	0.350030
$861$	−13.3918	−0.456390
$862$	−0.189885	−0.00646752
$863$	−17.5237	−0.596513	−0.298256	−	0.954486i	$-0.596405\pi$
$863$	−17.5237	−0.596513	−0.298256	+	0.954486i	$0.596405\pi$
$864$	−5.63269	−0.191628
$865$	19.3649	0.658425
$866$	−24.7173	−0.839928
$867$	14.6430	0.497302
$868$	−8.30903	−0.282027
$869$	−17.4234	−0.591049
$870$	6.41447	0.217471
$871$	−17.4794	−0.592266
$872$	−10.1215	−0.342757
$873$	39.0637	1.32210
$874$	−13.5228	−0.457416
$875$	−16.4576	−0.556370
$876$	−32.7960	−1.10807
$877$	25.8594	0.873210	0.436605	−	0.899653i	$-0.356181\pi$
$877$	25.8594	0.873210	0.436605	+	0.899653i	$0.356181\pi$
$878$	20.4842	0.691307
$879$	−69.7493	−2.35258
$880$	5.72961	0.193145
$881$	−39.3553	−1.32591	−0.662956	−	0.748658i	$-0.730698\pi$
$881$	−39.3553	−1.32591	−0.662956	+	0.748658i	$0.730698\pi$
$882$	−19.9072	−0.670309
$883$	0.964036	0.0324424	0.0162212	−	0.999868i	$-0.494836\pi$
$883$	0.964036	0.0324424	0.0162212	+	0.999868i	$0.494836\pi$
$884$	−4.92952	−0.165798
$885$	−84.5370	−2.84168
$886$	14.8504	0.498909
$887$	−41.0959	−1.37986	−0.689932	−	0.723874i	$-0.742360\pi$
$887$	−41.0959	−1.37986	−0.689932	+	0.723874i	$0.742360\pi$
$888$	30.8373	1.03483
$889$	−31.3416	−1.05116
$890$	−26.5447	−0.889782
$891$	−2.18088	−0.0730621
$892$	−5.28259	−0.176874
$893$	−46.4386	−1.55401
$894$	10.1175	0.338379
$895$	21.5659	0.720868
$896$	−1.73566	−0.0579843
$897$	−10.3278	−0.344836
$898$	35.5553	1.18650
$899$	4.36639	0.145627
$900$	5.93253	0.197751
$901$	41.0212	1.36661
$902$	6.28599	0.209301
$903$	−20.2475	−0.673794
$904$	−6.70099	−0.222871
$905$	10.1307	0.336757
$906$	4.45977	0.148166
$907$	50.0210	1.66092	0.830460	−	0.557078i	$-0.188078\pi$
$907$	50.0210	1.66092	0.830460	+	0.557078i	$0.188078\pi$
$908$	0.192854	0.00640008
$909$	43.0004	1.42623
$910$	6.19067	0.205219
$911$	20.1196	0.666592	0.333296	−	0.942822i	$-0.391839\pi$
$911$	20.1196	0.666592	0.333296	+	0.942822i	$0.391839\pi$
$912$	15.0046	0.496851
$913$	12.4063	0.410587
$914$	17.0580	0.564230
$915$	−70.7091	−2.33757
$916$	−8.11208	−0.268031
$917$	−26.0230	−0.859355
$918$	19.3657	0.639163
$919$	7.49463	0.247225	0.123613	−	0.992331i	$-0.460552\pi$
$919$	7.49463	0.247225	0.123613	+	0.992331i	$0.460552\pi$
$920$	−6.33824	−0.208966
$921$	−10.5606	−0.347985
$922$	42.5547	1.40147
$923$	−18.7179	−0.616106
$924$	−11.3016	−0.371797
$925$	−12.9619	−0.426183
$926$	42.0928	1.38325
$927$	−59.4161	−1.95148
$928$	0.912086	0.0299407
$929$	−24.9066	−0.817158	−0.408579	−	0.912723i	$-0.633976\pi$
$929$	−24.9066	−0.817158	−0.408579	+	0.912723i	$0.633976\pi$
$930$	33.6675	1.10400
$931$	21.1634	0.693601
$932$	−16.6850	−0.546535
$933$	−18.3628	−0.601170
$934$	−15.1037	−0.494208
$935$	−19.6989	−0.644223
$936$	7.15809	0.233969
$937$	−36.8694	−1.20447	−0.602235	−	0.798319i	$-0.705723\pi$
$937$	−36.8694	−1.20447	−0.602235	+	0.798319i	$0.705723\pi$
$938$	21.1594	0.690878
$939$	−45.9644	−1.49999
$940$	−21.7661	−0.709931
$941$	−7.20409	−0.234847	−0.117423	−	0.993082i	$-0.537463\pi$
$941$	−7.20409	−0.234847	−0.117423	+	0.993082i	$0.537463\pi$
$942$	−31.2157	−1.01706
$943$	−6.95373	−0.226445
$944$	−12.0205	−0.391234
$945$	−24.3201	−0.791134
$946$	9.50401	0.309002
$947$	−0.316763	−0.0102934	−0.00514671	−	0.999987i	$-0.501638\pi$
$947$	−0.316763	−0.0102934	−0.00514671	+	0.999987i	$0.501638\pi$
$948$	−21.3862	−0.694591
$949$	16.6330	0.539929
$950$	−6.30688	−0.204622
$951$	−95.3376	−3.09153
$952$	5.96735	0.193403
$953$	−39.0460	−1.26482	−0.632412	−	0.774632i	$-0.717935\pi$
$953$	−39.0460	−1.26482	−0.632412	+	0.774632i	$0.717935\pi$
$954$	−59.5663	−1.92853
$955$	19.1798	0.620645
$956$	24.9767	0.807804
$957$	5.93900	0.191981
$958$	3.29900	0.106586
$959$	32.2499	1.04140
$960$	7.03274	0.226981
$961$	−8.08221	−0.260716
$962$	−15.6396	−0.504240
$963$	22.6735	0.730642
$964$	−18.2581	−0.588054
$965$	1.51692	0.0488312
$966$	12.5022	0.402251
$967$	3.46233	0.111341	0.0556705	−	0.998449i	$-0.482270\pi$
$967$	3.46233	0.111341	0.0556705	+	0.998449i	$0.482270\pi$
$968$	−5.69510	−0.183047
$969$	−51.5870	−1.65721
$970$	−19.4648	−0.624976
$971$	−22.4578	−0.720706	−0.360353	−	0.932816i	$-0.617344\pi$
$971$	−22.4578	−0.720706	−0.360353	+	0.932816i	$0.617344\pi$
$972$	14.2212	0.456144
$973$	−13.5085	−0.433063
$974$	−29.1985	−0.935581
$975$	−4.81677	−0.154260
$976$	−10.0543	−0.321830
$977$	−16.5066	−0.528092	−0.264046	−	0.964510i	$-0.585057\pi$
$977$	−16.5066	−0.528092	−0.264046	+	0.964510i	$0.585057\pi$
$978$	−33.4179	−1.06859
$979$	−24.5771	−0.785488
$980$	9.91941	0.316864
$981$	−50.5305	−1.61332
$982$	−26.0672	−0.831837
$983$	−50.7893	−1.61993	−0.809964	−	0.586479i	$-0.800513\pi$
$983$	−50.7893	−1.61993	−0.809964	+	0.586479i	$0.800513\pi$
$984$	7.71567	0.245967
$985$	14.7398	0.469650
$986$	−3.13584	−0.0998653
$987$	42.9336	1.36659
$988$	−7.60978	−0.242099
$989$	−10.5136	−0.334313
$990$	28.6045	0.909111
$991$	28.8941	0.917851	0.458926	−	0.888475i	$-0.348234\pi$
$991$	28.8941	0.917851	0.458926	+	0.888475i	$0.348234\pi$
$992$	4.78725	0.151995
$993$	2.99039	0.0948971
$994$	22.6586	0.718687
$995$	15.3538	0.486748
$996$	15.2279	0.482515
$997$	25.0491	0.793312	0.396656	−	0.917967i	$-0.370171\pi$
$997$	25.0491	0.793312	0.396656	+	0.917967i	$0.370171\pi$
$998$	−31.9291	−1.01070
$999$	61.4403	1.94388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.d.1.4	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.d.1.4	✓	92	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.82708 q^{3} +1.00000 q^{4} -2.48763 q^{5} -2.82708 q^{6} -1.73566 q^{7} +1.00000 q^{8} +4.99240 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.82708 q^{3} +1.00000 q^{4} -2.48763 q^{5} -2.82708 q^{6} -1.73566 q^{7} +1.00000 q^{8} +4.99240 q^{9} -2.48763 q^{10} -2.30324 q^{11} -2.82708 q^{12} +1.43380 q^{13} -1.73566 q^{14} +7.03274 q^{15} +1.00000 q^{16} -3.43809 q^{17} +4.99240 q^{18} -5.30744 q^{19} -2.48763 q^{20} +4.90685 q^{21} -2.30324 q^{22} +2.54790 q^{23} -2.82708 q^{24} +1.18831 q^{25} +1.43380 q^{26} -5.63269 q^{27} -1.73566 q^{28} +0.912086 q^{29} +7.03274 q^{30} +4.78725 q^{31} +1.00000 q^{32} +6.51145 q^{33} -3.43809 q^{34} +4.31768 q^{35} +4.99240 q^{36} -10.9078 q^{37} -5.30744 q^{38} -4.05346 q^{39} -2.48763 q^{40} -2.72920 q^{41} +4.90685 q^{42} -4.12637 q^{43} -2.30324 q^{44} -12.4193 q^{45} +2.54790 q^{46} +8.74972 q^{47} -2.82708 q^{48} -3.98749 q^{49} +1.18831 q^{50} +9.71977 q^{51} +1.43380 q^{52} -11.9314 q^{53} -5.63269 q^{54} +5.72961 q^{55} -1.73566 q^{56} +15.0046 q^{57} +0.912086 q^{58} -12.0205 q^{59} +7.03274 q^{60} -10.0543 q^{61} +4.78725 q^{62} -8.66510 q^{63} +1.00000 q^{64} -3.56676 q^{65} +6.51145 q^{66} -12.1910 q^{67} -3.43809 q^{68} -7.20313 q^{69} +4.31768 q^{70} -13.0548 q^{71} +4.99240 q^{72} +11.6007 q^{73} -10.9078 q^{74} -3.35946 q^{75} -5.30744 q^{76} +3.99763 q^{77} -4.05346 q^{78} +7.56475 q^{79} -2.48763 q^{80} +0.946874 q^{81} -2.72920 q^{82} -5.38644 q^{83} +4.90685 q^{84} +8.55270 q^{85} -4.12637 q^{86} -2.57854 q^{87} -2.30324 q^{88} +10.6707 q^{89} -12.4193 q^{90} -2.48858 q^{91} +2.54790 q^{92} -13.5340 q^{93} +8.74972 q^{94} +13.2029 q^{95} -2.82708 q^{96} +7.82462 q^{97} -3.98749 q^{98} -11.4987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

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