LMFDB - Embedded newform 8042.2.a.c.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.02152 q^{3} +1.00000 q^{4} +3.49725 q^{5} +2.02152 q^{6} -2.15386 q^{7} -1.00000 q^{8} +1.08655 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.02152 q^{3} +1.00000 q^{4} +3.49725 q^{5} +2.02152 q^{6} -2.15386 q^{7} -1.00000 q^{8} +1.08655 q^{9} -3.49725 q^{10} +0.642902 q^{11} -2.02152 q^{12} -2.94242 q^{13} +2.15386 q^{14} -7.06976 q^{15} +1.00000 q^{16} +1.73141 q^{17} -1.08655 q^{18} +2.22670 q^{19} +3.49725 q^{20} +4.35408 q^{21} -0.642902 q^{22} +7.43269 q^{23} +2.02152 q^{24} +7.23075 q^{25} +2.94242 q^{26} +3.86809 q^{27} -2.15386 q^{28} -4.41638 q^{29} +7.06976 q^{30} -1.19527 q^{31} -1.00000 q^{32} -1.29964 q^{33} -1.73141 q^{34} -7.53260 q^{35} +1.08655 q^{36} +0.585059 q^{37} -2.22670 q^{38} +5.94816 q^{39} -3.49725 q^{40} +9.41374 q^{41} -4.35408 q^{42} +0.549585 q^{43} +0.642902 q^{44} +3.79992 q^{45} -7.43269 q^{46} +4.47659 q^{47} -2.02152 q^{48} -2.36087 q^{49} -7.23075 q^{50} -3.50008 q^{51} -2.94242 q^{52} -0.998661 q^{53} -3.86809 q^{54} +2.24839 q^{55} +2.15386 q^{56} -4.50133 q^{57} +4.41638 q^{58} +1.72800 q^{59} -7.06976 q^{60} -2.52227 q^{61} +1.19527 q^{62} -2.34027 q^{63} +1.00000 q^{64} -10.2904 q^{65} +1.29964 q^{66} -7.72287 q^{67} +1.73141 q^{68} -15.0253 q^{69} +7.53260 q^{70} +1.70003 q^{71} -1.08655 q^{72} -10.7477 q^{73} -0.585059 q^{74} -14.6171 q^{75} +2.22670 q^{76} -1.38472 q^{77} -5.94816 q^{78} -4.59092 q^{79} +3.49725 q^{80} -11.0791 q^{81} -9.41374 q^{82} +4.17517 q^{83} +4.35408 q^{84} +6.05518 q^{85} -0.549585 q^{86} +8.92781 q^{87} -0.642902 q^{88} -12.7294 q^{89} -3.79992 q^{90} +6.33757 q^{91} +7.43269 q^{92} +2.41627 q^{93} -4.47659 q^{94} +7.78734 q^{95} +2.02152 q^{96} -8.89877 q^{97} +2.36087 q^{98} +0.698543 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * 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
$3$	−2.02152	−1.16713	−0.583563	−	0.812068i	$-0.698342\pi$
$3$	−2.02152	−1.16713	−0.583563	+	0.812068i	$0.698342\pi$
$4$	1.00000	0.500000
$5$	3.49725	1.56402	0.782009	−	0.623267i	$-0.214195\pi$
$5$	3.49725	1.56402	0.782009	+	0.623267i	$0.214195\pi$
$6$	2.02152	0.825282
$7$	−2.15386	−0.814084	−0.407042	−	0.913409i	$-0.633440\pi$
$7$	−2.15386	−0.814084	−0.407042	+	0.913409i	$0.633440\pi$
$8$	−1.00000	−0.353553
$9$	1.08655	0.362182
$10$	−3.49725	−1.10593
$11$	0.642902	0.193842	0.0969211	−	0.995292i	$-0.469101\pi$
$11$	0.642902	0.193842	0.0969211	+	0.995292i	$0.469101\pi$
$12$	−2.02152	−0.583563
$13$	−2.94242	−0.816081	−0.408040	−	0.912964i	$-0.633788\pi$
$13$	−2.94242	−0.816081	−0.408040	+	0.912964i	$0.633788\pi$
$14$	2.15386	0.575644
$15$	−7.06976	−1.82540
$16$	1.00000	0.250000
$17$	1.73141	0.419929	0.209964	−	0.977709i	$-0.432665\pi$
$17$	1.73141	0.419929	0.209964	+	0.977709i	$0.432665\pi$
$18$	−1.08655	−0.256102
$19$	2.22670	0.510841	0.255420	−	0.966830i	$-0.417786\pi$
$19$	2.22670	0.510841	0.255420	+	0.966830i	$0.417786\pi$
$20$	3.49725	0.782009
$21$	4.35408	0.950138
$22$	−0.642902	−0.137067
$23$	7.43269	1.54982	0.774911	−	0.632070i	$-0.217795\pi$
$23$	7.43269	1.54982	0.774911	+	0.632070i	$0.217795\pi$
$24$	2.02152	0.412641
$25$	7.23075	1.44615
$26$	2.94242	0.577056
$27$	3.86809	0.744413
$28$	−2.15386	−0.407042
$29$	−4.41638	−0.820101	−0.410051	−	0.912063i	$-0.634489\pi$
$29$	−4.41638	−0.820101	−0.410051	+	0.912063i	$0.634489\pi$
$30$	7.06976	1.29076
$31$	−1.19527	−0.214677	−0.107339	−	0.994223i	$-0.534233\pi$
$31$	−1.19527	−0.214677	−0.107339	+	0.994223i	$0.534233\pi$
$32$	−1.00000	−0.176777
$33$	−1.29964	−0.226238
$34$	−1.73141	−0.296935
$35$	−7.53260	−1.27324
$36$	1.08655	0.181091
$37$	0.585059	0.0961831	0.0480916	−	0.998843i	$-0.484686\pi$
$37$	0.585059	0.0961831	0.0480916	+	0.998843i	$0.484686\pi$
$38$	−2.22670	−0.361219
$39$	5.94816	0.952469
$40$	−3.49725	−0.552964
$41$	9.41374	1.47018	0.735089	−	0.677970i	$-0.237140\pi$
$41$	9.41374	1.47018	0.735089	+	0.677970i	$0.237140\pi$
$42$	−4.35408	−0.671849
$43$	0.549585	0.0838109	0.0419054	−	0.999122i	$-0.486657\pi$
$43$	0.549585	0.0838109	0.0419054	+	0.999122i	$0.486657\pi$
$44$	0.642902	0.0969211
$45$	3.79992	0.566459
$46$	−7.43269	−1.09589
$47$	4.47659	0.652978	0.326489	−	0.945201i	$-0.394134\pi$
$47$	4.47659	0.652978	0.326489	+	0.945201i	$0.394134\pi$
$48$	−2.02152	−0.291781
$49$	−2.36087	−0.337267
$50$	−7.23075	−1.02258
$51$	−3.50008	−0.490110
$52$	−2.94242	−0.408040
$53$	−0.998661	−0.137177	−0.0685883	−	0.997645i	$-0.521849\pi$
$53$	−0.998661	−0.137177	−0.0685883	+	0.997645i	$0.521849\pi$
$54$	−3.86809	−0.526380
$55$	2.24839	0.303173
$56$	2.15386	0.287822
$57$	−4.50133	−0.596216
$58$	4.41638	0.579899
$59$	1.72800	0.224966	0.112483	−	0.993654i	$-0.464120\pi$
$59$	1.72800	0.224966	0.112483	+	0.993654i	$0.464120\pi$
$60$	−7.06976	−0.912702
$61$	−2.52227	−0.322944	−0.161472	−	0.986877i	$-0.551624\pi$
$61$	−2.52227	−0.322944	−0.161472	+	0.986877i	$0.551624\pi$
$62$	1.19527	0.151800
$63$	−2.34027	−0.294847
$64$	1.00000	0.125000
$65$	−10.2904	−1.27636
$66$	1.29964	0.159975
$67$	−7.72287	−0.943499	−0.471750	−	0.881733i	$-0.656377\pi$
$67$	−7.72287	−0.943499	−0.471750	+	0.881733i	$0.656377\pi$
$68$	1.73141	0.209964
$69$	−15.0253	−1.80884
$70$	7.53260	0.900318
$71$	1.70003	0.201757	0.100879	−	0.994899i	$-0.467835\pi$
$71$	1.70003	0.201757	0.100879	+	0.994899i	$0.467835\pi$
$72$	−1.08655	−0.128051
$73$	−10.7477	−1.25793	−0.628964	−	0.777434i	$-0.716521\pi$
$73$	−10.7477	−1.25793	−0.628964	+	0.777434i	$0.716521\pi$
$74$	−0.585059	−0.0680118
$75$	−14.6171	−1.68784
$76$	2.22670	0.255420
$77$	−1.38472	−0.157804
$78$	−5.94816	−0.673497
$79$	−4.59092	−0.516518	−0.258259	−	0.966076i	$-0.583149\pi$
$79$	−4.59092	−0.516518	−0.258259	+	0.966076i	$0.583149\pi$
$80$	3.49725	0.391004
$81$	−11.0791	−1.23101
$82$	−9.41374	−1.03957
$83$	4.17517	0.458284	0.229142	−	0.973393i	$-0.426408\pi$
$83$	4.17517	0.458284	0.229142	+	0.973393i	$0.426408\pi$
$84$	4.35408	0.475069
$85$	6.05518	0.656776
$86$	−0.549585	−0.0592632
$87$	8.92781	0.957161
$88$	−0.642902	−0.0685336
$89$	−12.7294	−1.34932	−0.674658	−	0.738130i	$-0.735709\pi$
$89$	−12.7294	−1.34932	−0.674658	+	0.738130i	$0.735709\pi$
$90$	−3.79992	−0.400547
$91$	6.33757	0.664358
$92$	7.43269	0.774911
$93$	2.41627	0.250555
$94$	−4.47659	−0.461725
$95$	7.78734	0.798964
$96$	2.02152	0.206321
$97$	−8.89877	−0.903534	−0.451767	−	0.892136i	$-0.649206\pi$
$97$	−8.89877	−0.903534	−0.451767	+	0.892136i	$0.649206\pi$
$98$	2.36087	0.238484
$99$	0.698543	0.0702062
$100$	7.23075	0.723075
$101$	19.9193	1.98204	0.991020	−	0.133716i	$-0.0426909\pi$
$101$	19.9193	1.98204	0.991020	+	0.133716i	$0.0426909\pi$
$102$	3.50008	0.346560
$103$	11.6970	1.15254	0.576272	−	0.817258i	$-0.304507\pi$
$103$	11.6970	1.15254	0.576272	+	0.817258i	$0.304507\pi$
$104$	2.94242	0.288528
$105$	15.2273	1.48603
$106$	0.998661	0.0969985
$107$	11.7161	1.13264	0.566320	−	0.824186i	$-0.308367\pi$
$107$	11.7161	1.13264	0.566320	+	0.824186i	$0.308367\pi$
$108$	3.86809	0.372207
$109$	−6.28371	−0.601870	−0.300935	−	0.953645i	$-0.597299\pi$
$109$	−6.28371	−0.601870	−0.300935	+	0.953645i	$0.597299\pi$
$110$	−2.24839	−0.214375
$111$	−1.18271	−0.112258
$112$	−2.15386	−0.203521
$113$	10.5949	0.996688	0.498344	−	0.866980i	$-0.333942\pi$
$113$	10.5949	0.996688	0.498344	+	0.866980i	$0.333942\pi$
$114$	4.50133	0.421588
$115$	25.9940	2.42395
$116$	−4.41638	−0.410051
$117$	−3.19708	−0.295570
$118$	−1.72800	−0.159075
$119$	−3.72922	−0.341857
$120$	7.06976	0.645378
$121$	−10.5867	−0.962425
$122$	2.52227	0.228356
$123$	−19.0301	−1.71588
$124$	−1.19527	−0.107339
$125$	7.80151	0.697788
$126$	2.34027	0.208488
$127$	12.9073	1.14534	0.572668	−	0.819787i	$-0.305908\pi$
$127$	12.9073	1.14534	0.572668	+	0.819787i	$0.305908\pi$
$128$	−1.00000	−0.0883883
$129$	−1.11100	−0.0978178
$130$	10.2904	0.902526
$131$	−2.00531	−0.175204	−0.0876022	−	0.996156i	$-0.527920\pi$
$131$	−2.00531	−0.175204	−0.0876022	+	0.996156i	$0.527920\pi$
$132$	−1.29964	−0.113119
$133$	−4.79602	−0.415867
$134$	7.72287	0.667155
$135$	13.5277	1.16428
$136$	−1.73141	−0.148467
$137$	−6.16772	−0.526944	−0.263472	−	0.964667i	$-0.584868\pi$
$137$	−6.16772	−0.526944	−0.263472	+	0.964667i	$0.584868\pi$
$138$	15.0253	1.27904
$139$	−8.87306	−0.752603	−0.376301	−	0.926497i	$-0.622804\pi$
$139$	−8.87306	−0.752603	−0.376301	+	0.926497i	$0.622804\pi$
$140$	−7.53260	−0.636621
$141$	−9.04953	−0.762108
$142$	−1.70003	−0.142664
$143$	−1.89169	−0.158191
$144$	1.08655	0.0905456
$145$	−15.4452	−1.28265
$146$	10.7477	0.889490
$147$	4.77255	0.393633
$148$	0.585059	0.0480916
$149$	−1.59975	−0.131057	−0.0655284	−	0.997851i	$-0.520873\pi$
$149$	−1.59975	−0.131057	−0.0655284	+	0.997851i	$0.520873\pi$
$150$	14.6171	1.19348
$151$	15.3028	1.24533	0.622664	−	0.782490i	$-0.286051\pi$
$151$	15.3028	1.24533	0.622664	+	0.782490i	$0.286051\pi$
$152$	−2.22670	−0.180610
$153$	1.88126	0.152091
$154$	1.38472	0.111584
$155$	−4.18016	−0.335759
$156$	5.94816	0.476234
$157$	−3.45824	−0.275998	−0.137999	−	0.990432i	$-0.544067\pi$
$157$	−3.45824	−0.275998	−0.137999	+	0.990432i	$0.544067\pi$
$158$	4.59092	0.365234
$159$	2.01881	0.160102
$160$	−3.49725	−0.276482
$161$	−16.0090	−1.26169
$162$	11.0791	0.870453
$163$	6.03012	0.472315	0.236158	−	0.971715i	$-0.424112\pi$
$163$	6.03012	0.472315	0.236158	+	0.971715i	$0.424112\pi$
$164$	9.41374	0.735089
$165$	−4.54517	−0.353841
$166$	−4.17517	−0.324056
$167$	12.0460	0.932151	0.466075	−	0.884745i	$-0.345668\pi$
$167$	12.0460	0.932151	0.466075	+	0.884745i	$0.345668\pi$
$168$	−4.35408	−0.335925
$169$	−4.34216	−0.334012
$170$	−6.05518	−0.464411
$171$	2.41942	0.185018
$172$	0.549585	0.0419054
$173$	23.1769	1.76211	0.881054	−	0.473015i	$-0.156834\pi$
$173$	23.1769	1.76211	0.881054	+	0.473015i	$0.156834\pi$
$174$	−8.92781	−0.676815
$175$	−15.5741	−1.17729
$176$	0.642902	0.0484606
$177$	−3.49318	−0.262564
$178$	12.7294	0.954111
$179$	−15.5092	−1.15921	−0.579607	−	0.814896i	$-0.696794\pi$
$179$	−15.5092	−1.15921	−0.579607	+	0.814896i	$0.696794\pi$
$180$	3.79992	0.283230
$181$	−20.2179	−1.50278	−0.751392	−	0.659856i	$-0.770617\pi$
$181$	−20.2179	−1.50278	−0.751392	+	0.659856i	$0.770617\pi$
$182$	−6.33757	−0.469772
$183$	5.09882	0.376916
$184$	−7.43269	−0.547945
$185$	2.04610	0.150432
$186$	−2.41627	−0.177169
$187$	1.11313	0.0814000
$188$	4.47659	0.326489
$189$	−8.33133	−0.606015
$190$	−7.78734	−0.564953
$191$	1.46415	0.105942	0.0529709	−	0.998596i	$-0.483131\pi$
$191$	1.46415	0.105942	0.0529709	+	0.998596i	$0.483131\pi$
$192$	−2.02152	−0.145891
$193$	16.3124	1.17419	0.587097	−	0.809516i	$-0.300271\pi$
$193$	16.3124	1.17419	0.587097	+	0.809516i	$0.300271\pi$
$194$	8.89877	0.638895
$195$	20.8022	1.48968
$196$	−2.36087	−0.168634
$197$	7.21408	0.513982	0.256991	−	0.966414i	$-0.417269\pi$
$197$	7.21408	0.513982	0.256991	+	0.966414i	$0.417269\pi$
$198$	−0.698543	−0.0496433
$199$	3.36486	0.238529	0.119264	−	0.992863i	$-0.461946\pi$
$199$	3.36486	0.238529	0.119264	+	0.992863i	$0.461946\pi$
$200$	−7.23075	−0.511292
$201$	15.6119	1.10118
$202$	−19.9193	−1.40151
$203$	9.51228	0.667631
$204$	−3.50008	−0.245055
$205$	32.9222	2.29939
$206$	−11.6970	−0.814971
$207$	8.07596	0.561318
$208$	−2.94242	−0.204020
$209$	1.43155	0.0990226
$210$	−15.2273	−1.05078
$211$	27.3162	1.88053	0.940263	−	0.340448i	$-0.110579\pi$
$211$	27.3162	1.88053	0.940263	+	0.340448i	$0.110579\pi$
$212$	−0.998661	−0.0685883
$213$	−3.43666	−0.235476
$214$	−11.7161	−0.800897
$215$	1.92203	0.131082
$216$	−3.86809	−0.263190
$217$	2.57445	0.174765
$218$	6.28371	0.425587
$219$	21.7268	1.46816
$220$	2.24839	0.151586
$221$	−5.09454	−0.342696
$222$	1.18271	0.0793783
$223$	16.7863	1.12410	0.562048	−	0.827105i	$-0.310014\pi$
$223$	16.7863	1.12410	0.562048	+	0.827105i	$0.310014\pi$
$224$	2.15386	0.143911
$225$	7.85655	0.523770
$226$	−10.5949	−0.704765
$227$	24.7395	1.64202	0.821008	−	0.570916i	$-0.193412\pi$
$227$	24.7395	1.64202	0.821008	+	0.570916i	$0.193412\pi$
$228$	−4.50133	−0.298108
$229$	4.21112	0.278279	0.139139	−	0.990273i	$-0.455566\pi$
$229$	4.21112	0.278279	0.139139	+	0.990273i	$0.455566\pi$
$230$	−25.9940	−1.71399
$231$	2.79925	0.184177
$232$	4.41638	0.289950
$233$	−15.6618	−1.02604	−0.513018	−	0.858378i	$-0.671473\pi$
$233$	−15.6618	−1.02604	−0.513018	+	0.858378i	$0.671473\pi$
$234$	3.19708	0.208999
$235$	15.6558	1.02127
$236$	1.72800	0.112483
$237$	9.28063	0.602842
$238$	3.72922	0.241730
$239$	−19.5755	−1.26623	−0.633117	−	0.774056i	$-0.718225\pi$
$239$	−19.5755	−1.26623	−0.633117	+	0.774056i	$0.718225\pi$
$240$	−7.06976	−0.456351
$241$	−24.5795	−1.58331	−0.791653	−	0.610971i	$-0.790779\pi$
$241$	−24.5795	−1.58331	−0.791653	+	0.610971i	$0.790779\pi$
$242$	10.5867	0.680537
$243$	10.7923	0.692325
$244$	−2.52227	−0.161472
$245$	−8.25656	−0.527492
$246$	19.0301	1.21331
$247$	−6.55190	−0.416887
$248$	1.19527	0.0758998
$249$	−8.44019	−0.534876
$250$	−7.80151	−0.493411
$251$	−15.1881	−0.958667	−0.479333	−	0.877633i	$-0.659122\pi$
$251$	−15.1881	−0.958667	−0.479333	+	0.877633i	$0.659122\pi$
$252$	−2.34027	−0.147423
$253$	4.77849	0.300421
$254$	−12.9073	−0.809875
$255$	−12.2407	−0.766540
$256$	1.00000	0.0625000
$257$	5.16747	0.322338	0.161169	−	0.986927i	$-0.448474\pi$
$257$	5.16747	0.322338	0.161169	+	0.986927i	$0.448474\pi$
$258$	1.11100	0.0691676
$259$	−1.26014	−0.0783012
$260$	−10.2904	−0.638182
$261$	−4.79860	−0.297026
$262$	2.00531	0.123888
$263$	2.16179	0.133301	0.0666507	−	0.997776i	$-0.478769\pi$
$263$	2.16179	0.133301	0.0666507	+	0.997776i	$0.478769\pi$
$264$	1.29964	0.0799873
$265$	−3.49257	−0.214547
$266$	4.79602	0.294063
$267$	25.7328	1.57482
$268$	−7.72287	−0.471750
$269$	30.1665	1.83928	0.919642	−	0.392757i	$-0.128478\pi$
$269$	30.1665	1.83928	0.919642	+	0.392757i	$0.128478\pi$
$270$	−13.5277	−0.823267
$271$	−19.3988	−1.17839	−0.589196	−	0.807990i	$-0.700555\pi$
$271$	−19.3988	−1.17839	−0.589196	+	0.807990i	$0.700555\pi$
$272$	1.73141	0.104982
$273$	−12.8115	−0.775389
$274$	6.16772	0.372606
$275$	4.64867	0.280325
$276$	−15.0253	−0.904419
$277$	−16.0873	−0.966590	−0.483295	−	0.875458i	$-0.660560\pi$
$277$	−16.0873	−0.966590	−0.483295	+	0.875458i	$0.660560\pi$
$278$	8.87306	0.532171
$279$	−1.29872	−0.0777522
$280$	7.53260	0.450159
$281$	19.9018	1.18724	0.593620	−	0.804746i	$-0.297698\pi$
$281$	19.9018	1.18724	0.593620	+	0.804746i	$0.297698\pi$
$282$	9.04953	0.538892
$283$	−11.3099	−0.672302	−0.336151	−	0.941808i	$-0.609125\pi$
$283$	−11.3099	−0.672302	−0.336151	+	0.941808i	$0.609125\pi$
$284$	1.70003	0.100879
$285$	−15.7423	−0.932492
$286$	1.89169	0.111858
$287$	−20.2759	−1.19685
$288$	−1.08655	−0.0640254
$289$	−14.0022	−0.823660
$290$	15.4452	0.906972
$291$	17.9891	1.05454
$292$	−10.7477	−0.628964
$293$	5.95670	0.347994	0.173997	−	0.984746i	$-0.444332\pi$
$293$	5.95670	0.347994	0.173997	+	0.984746i	$0.444332\pi$
$294$	−4.77255	−0.278341
$295$	6.04324	0.351851
$296$	−0.585059	−0.0340059
$297$	2.48680	0.144299
$298$	1.59975	0.0926712
$299$	−21.8701	−1.26478
$300$	−14.6171	−0.843920
$301$	−1.18373	−0.0682291
$302$	−15.3028	−0.880579
$303$	−40.2672	−2.31329
$304$	2.22670	0.127710
$305$	−8.82101	−0.505090
$306$	−1.88126	−0.107544
$307$	−15.4762	−0.883273	−0.441636	−	0.897194i	$-0.645602\pi$
$307$	−15.4762	−0.883273	−0.441636	+	0.897194i	$0.645602\pi$
$308$	−1.38472	−0.0789019
$309$	−23.6458	−1.34516
$310$	4.18016	0.237417
$311$	25.7236	1.45865	0.729326	−	0.684166i	$-0.239834\pi$
$311$	25.7236	1.45865	0.729326	+	0.684166i	$0.239834\pi$
$312$	−5.94816	−0.336749
$313$	−8.70889	−0.492256	−0.246128	−	0.969237i	$-0.579158\pi$
$313$	−8.70889	−0.492256	−0.246128	+	0.969237i	$0.579158\pi$
$314$	3.45824	0.195160
$315$	−8.18452	−0.461145
$316$	−4.59092	−0.258259
$317$	−2.20990	−0.124120	−0.0620601	−	0.998072i	$-0.519767\pi$
$317$	−2.20990	−0.124120	−0.0620601	+	0.998072i	$0.519767\pi$
$318$	−2.01881	−0.113209
$319$	−2.83930	−0.158970
$320$	3.49725	0.195502
$321$	−23.6844	−1.32193
$322$	16.0090	0.892146
$323$	3.85534	0.214517
$324$	−11.0791	−0.615503
$325$	−21.2759	−1.18018
$326$	−6.03012	−0.333977
$327$	12.7027	0.702458
$328$	−9.41374	−0.519787
$329$	−9.64197	−0.531579
$330$	4.54517	0.250203
$331$	−24.1651	−1.32824	−0.664118	−	0.747628i	$-0.731193\pi$
$331$	−24.1651	−1.32824	−0.664118	+	0.747628i	$0.731193\pi$
$332$	4.17517	0.229142
$333$	0.635694	0.0348358
$334$	−12.0460	−0.659130
$335$	−27.0088	−1.47565
$336$	4.35408	0.237535
$337$	−25.3401	−1.38036	−0.690181	−	0.723637i	$-0.742469\pi$
$337$	−25.3401	−1.38036	−0.690181	+	0.723637i	$0.742469\pi$
$338$	4.34216	0.236182
$339$	−21.4179	−1.16326
$340$	6.05518	0.328388
$341$	−0.768442	−0.0416135
$342$	−2.41942	−0.130827
$343$	20.1620	1.08865
$344$	−0.549585	−0.0296316
$345$	−52.5473	−2.82905
$346$	−23.1769	−1.24600
$347$	13.7829	0.739903	0.369951	−	0.929051i	$-0.379374\pi$
$347$	13.7829	0.739903	0.369951	+	0.929051i	$0.379374\pi$
$348$	8.92781	0.478581
$349$	2.28502	0.122314	0.0611572	−	0.998128i	$-0.480521\pi$
$349$	2.28502	0.122314	0.0611572	+	0.998128i	$0.480521\pi$
$350$	15.5741	0.832469
$351$	−11.3815	−0.607501
$352$	−0.642902	−0.0342668
$353$	3.99082	0.212410	0.106205	−	0.994344i	$-0.466130\pi$
$353$	3.99082	0.212410	0.106205	+	0.994344i	$0.466130\pi$
$354$	3.49318	0.185660
$355$	5.94545	0.315551
$356$	−12.7294	−0.674658
$357$	7.53871	0.398991
$358$	15.5092	0.819688
$359$	12.2982	0.649072	0.324536	−	0.945873i	$-0.394792\pi$
$359$	12.2982	0.649072	0.324536	+	0.945873i	$0.394792\pi$
$360$	−3.79992	−0.200274
$361$	−14.0418	−0.739041
$362$	20.2179	1.06263
$363$	21.4012	1.12327
$364$	6.33757	0.332179
$365$	−37.5876	−1.96742
$366$	−5.09882	−0.266520
$367$	−13.9872	−0.730128	−0.365064	−	0.930982i	$-0.618953\pi$
$367$	−13.9872	−0.730128	−0.365064	+	0.930982i	$0.618953\pi$
$368$	7.43269	0.387456
$369$	10.2285	0.532473
$370$	−2.04610	−0.106372
$371$	2.15098	0.111673
$372$	2.41627	0.125278
$373$	29.8800	1.54713	0.773563	−	0.633719i	$-0.218473\pi$
$373$	29.8800	1.54713	0.773563	+	0.633719i	$0.218473\pi$
$374$	−1.11313	−0.0575585
$375$	−15.7709	−0.814406
$376$	−4.47659	−0.230863
$377$	12.9948	0.669269
$378$	8.33133	0.428517
$379$	4.37390	0.224672	0.112336	−	0.993670i	$-0.464167\pi$
$379$	4.37390	0.224672	0.112336	+	0.993670i	$0.464167\pi$
$380$	7.78734	0.399482
$381$	−26.0923	−1.33675
$382$	−1.46415	−0.0749122
$383$	32.4114	1.65615	0.828074	−	0.560619i	$-0.189437\pi$
$383$	32.4114	1.65615	0.828074	+	0.560619i	$0.189437\pi$
$384$	2.02152	0.103160
$385$	−4.84272	−0.246808
$386$	−16.3124	−0.830281
$387$	0.597149	0.0303548
$388$	−8.89877	−0.451767
$389$	29.9822	1.52016	0.760078	−	0.649832i	$-0.225161\pi$
$389$	29.9822	1.52016	0.760078	+	0.649832i	$0.225161\pi$
$390$	−20.8022	−1.05336
$391$	12.8690	0.650815
$392$	2.36087	0.119242
$393$	4.05377	0.204486
$394$	−7.21408	−0.363440
$395$	−16.0556	−0.807844
$396$	0.698543	0.0351031
$397$	29.9500	1.50315	0.751574	−	0.659649i	$-0.229295\pi$
$397$	29.9500	1.50315	0.751574	+	0.659649i	$0.229295\pi$
$398$	−3.36486	−0.168665
$399$	9.69525	0.485370
$400$	7.23075	0.361538
$401$	15.5388	0.775969	0.387985	−	0.921666i	$-0.373171\pi$
$401$	15.5388	0.775969	0.387985	+	0.921666i	$0.373171\pi$
$402$	−15.6119	−0.778653
$403$	3.51699	0.175194
$404$	19.9193	0.991020
$405$	−38.7462	−1.92532
$406$	−9.51228	−0.472087
$407$	0.376136	0.0186444
$408$	3.50008	0.173280
$409$	13.7670	0.680733	0.340367	−	0.940293i	$-0.389449\pi$
$409$	13.7670	0.680733	0.340367	+	0.940293i	$0.389449\pi$
$410$	−32.9222	−1.62591
$411$	12.4682	0.615010
$412$	11.6970	0.576272
$413$	−3.72187	−0.183141
$414$	−8.07596	−0.396912
$415$	14.6016	0.716765
$416$	2.94242	0.144264
$417$	17.9371	0.878382
$418$	−1.43155	−0.0700195
$419$	27.8301	1.35959	0.679794	−	0.733404i	$-0.262069\pi$
$419$	27.8301	1.35959	0.679794	+	0.733404i	$0.262069\pi$
$420$	15.2273	0.743016
$421$	24.6876	1.20320	0.601600	−	0.798798i	$-0.294530\pi$
$421$	24.6876	1.20320	0.601600	+	0.798798i	$0.294530\pi$
$422$	−27.3162	−1.32973
$423$	4.86403	0.236497
$424$	0.998661	0.0484993
$425$	12.5194	0.607281
$426$	3.43666	0.166507
$427$	5.43263	0.262903
$428$	11.7161	0.566320
$429$	3.82409	0.184629
$430$	−1.92203	−0.0926887
$431$	4.81017	0.231698	0.115849	−	0.993267i	$-0.463041\pi$
$431$	4.81017	0.231698	0.115849	+	0.993267i	$0.463041\pi$
$432$	3.86809	0.186103
$433$	19.9556	0.959004	0.479502	−	0.877541i	$-0.340817\pi$
$433$	19.9556	0.959004	0.479502	+	0.877541i	$0.340817\pi$
$434$	−2.57445	−0.123578
$435$	31.2228	1.49702
$436$	−6.28371	−0.300935
$437$	16.5504	0.791713
$438$	−21.7268	−1.03815
$439$	−21.6464	−1.03313	−0.516563	−	0.856249i	$-0.672789\pi$
$439$	−21.6464	−1.03313	−0.516563	+	0.856249i	$0.672789\pi$
$440$	−2.24839	−0.107188
$441$	−2.56520	−0.122152
$442$	5.09454	0.242323
$443$	−31.3307	−1.48857	−0.744284	−	0.667863i	$-0.767209\pi$
$443$	−31.3307	−1.48857	−0.744284	+	0.667863i	$0.767209\pi$
$444$	−1.18271	−0.0561289
$445$	−44.5180	−2.11035
$446$	−16.7863	−0.794856
$447$	3.23393	0.152960
$448$	−2.15386	−0.101760
$449$	1.16615	0.0550341	0.0275171	−	0.999621i	$-0.491240\pi$
$449$	1.16615	0.0550341	0.0275171	+	0.999621i	$0.491240\pi$
$450$	−7.85655	−0.370361
$451$	6.05211	0.284983
$452$	10.5949	0.498344
$453$	−30.9350	−1.45345
$454$	−24.7395	−1.16108
$455$	22.1641	1.03907
$456$	4.50133	0.210794
$457$	4.73128	0.221320	0.110660	−	0.993858i	$-0.464704\pi$
$457$	4.73128	0.221320	0.110660	+	0.993858i	$0.464704\pi$
$458$	−4.21112	−0.196773
$459$	6.69725	0.312601
$460$	25.9940	1.21197
$461$	−8.73944	−0.407036	−0.203518	−	0.979071i	$-0.565238\pi$
$461$	−8.73944	−0.407036	−0.203518	+	0.979071i	$0.565238\pi$
$462$	−2.79925	−0.130233
$463$	−41.3562	−1.92199	−0.960993	−	0.276572i	$-0.910802\pi$
$463$	−41.3562	−1.92199	−0.960993	+	0.276572i	$0.910802\pi$
$464$	−4.41638	−0.205025
$465$	8.45028	0.391873
$466$	15.6618	0.725517
$467$	−19.6447	−0.909047	−0.454524	−	0.890735i	$-0.650191\pi$
$467$	−19.6447	−0.909047	−0.454524	+	0.890735i	$0.650191\pi$
$468$	−3.19708	−0.147785
$469$	16.6340	0.768087
$470$	−15.6558	−0.722147
$471$	6.99091	0.322124
$472$	−1.72800	−0.0795375
$473$	0.353329	0.0162461
$474$	−9.28063	−0.426274
$475$	16.1008	0.738753
$476$	−3.72922	−0.170929
$477$	−1.08509	−0.0496829
$478$	19.5755	0.895363
$479$	28.5291	1.30353	0.651763	−	0.758422i	$-0.274030\pi$
$479$	28.5291	1.30353	0.651763	+	0.758422i	$0.274030\pi$
$480$	7.06976	0.322689
$481$	−1.72149	−0.0784932
$482$	24.5795	1.11957
$483$	32.3625	1.47255
$484$	−10.5867	−0.481213
$485$	−31.1212	−1.41314
$486$	−10.7923	−0.489548
$487$	7.51487	0.340531	0.170266	−	0.985398i	$-0.445537\pi$
$487$	7.51487	0.340531	0.170266	+	0.985398i	$0.445537\pi$
$488$	2.52227	0.114178
$489$	−12.1900	−0.551251
$490$	8.25656	0.372993
$491$	12.2338	0.552106	0.276053	−	0.961142i	$-0.410974\pi$
$491$	12.2338	0.552106	0.276053	+	0.961142i	$0.410974\pi$
$492$	−19.0301	−0.857942
$493$	−7.64657	−0.344384
$494$	6.55190	0.294784
$495$	2.44298	0.109804
$496$	−1.19527	−0.0536693
$497$	−3.66164	−0.164247
$498$	8.44019	0.378214
$499$	−27.7120	−1.24056	−0.620279	−	0.784381i	$-0.712981\pi$
$499$	−27.7120	−1.24056	−0.620279	+	0.784381i	$0.712981\pi$
$500$	7.80151	0.348894
$501$	−24.3513	−1.08794
$502$	15.1881	0.677880
$503$	16.9979	0.757898	0.378949	−	0.925417i	$-0.376285\pi$
$503$	16.9979	0.757898	0.378949	+	0.925417i	$0.376285\pi$
$504$	2.34027	0.104244
$505$	69.6626	3.09994
$506$	−4.77849	−0.212430
$507$	8.77777	0.389834
$508$	12.9073	0.572668
$509$	−6.84593	−0.303440	−0.151720	−	0.988423i	$-0.548481\pi$
$509$	−6.84593	−0.303440	−0.151720	+	0.988423i	$0.548481\pi$
$510$	12.2407	0.542026
$511$	23.1492	1.02406
$512$	−1.00000	−0.0441942
$513$	8.61308	0.380277
$514$	−5.16747	−0.227927
$515$	40.9075	1.80260
$516$	−1.11100	−0.0489089
$517$	2.87801	0.126575
$518$	1.26014	0.0553673
$519$	−46.8526	−2.05660
$520$	10.2904	0.451263
$521$	36.6421	1.60532	0.802659	−	0.596438i	$-0.203418\pi$
$521$	36.6421	1.60532	0.802659	+	0.596438i	$0.203418\pi$
$522$	4.79860	0.210029
$523$	−6.38212	−0.279071	−0.139535	−	0.990217i	$-0.544561\pi$
$523$	−6.38212	−0.279071	−0.139535	+	0.990217i	$0.544561\pi$
$524$	−2.00531	−0.0876022
$525$	31.4833	1.37404
$526$	−2.16179	−0.0942584
$527$	−2.06951	−0.0901491
$528$	−1.29964	−0.0565596
$529$	32.2448	1.40195
$530$	3.49257	0.151707
$531$	1.87755	0.0814787
$532$	−4.79602	−0.207934
$533$	−27.6992	−1.19978
$534$	−25.7328	−1.11357
$535$	40.9742	1.77147
$536$	7.72287	0.333577
$537$	31.3522	1.35295
$538$	−30.1665	−1.30057
$539$	−1.51781	−0.0653767
$540$	13.5277	0.582138
$541$	1.38998	0.0597599	0.0298800	−	0.999553i	$-0.490487\pi$
$541$	1.38998	0.0597599	0.0298800	+	0.999553i	$0.490487\pi$
$542$	19.3988	0.833248
$543$	40.8709	1.75394
$544$	−1.73141	−0.0742337
$545$	−21.9757	−0.941336
$546$	12.8115	0.548283
$547$	8.78084	0.375441	0.187721	−	0.982222i	$-0.439890\pi$
$547$	8.78084	0.375441	0.187721	+	0.982222i	$0.439890\pi$
$548$	−6.16772	−0.263472
$549$	−2.74056	−0.116964
$550$	−4.64867	−0.198220
$551$	−9.83397	−0.418941
$552$	15.0253	0.639520
$553$	9.88821	0.420489
$554$	16.0873	0.683482
$555$	−4.13623	−0.175573
$556$	−8.87306	−0.376301
$557$	−11.1523	−0.472538	−0.236269	−	0.971688i	$-0.575925\pi$
$557$	−11.1523	−0.472538	−0.236269	+	0.971688i	$0.575925\pi$
$558$	1.29872	0.0549791
$559$	−1.61711	−0.0683964
$560$	−7.53260	−0.318310
$561$	−2.25021	−0.0950040
$562$	−19.9018	−0.839505
$563$	32.2555	1.35941	0.679704	−	0.733487i	$-0.262108\pi$
$563$	32.2555	1.35941	0.679704	+	0.733487i	$0.262108\pi$
$564$	−9.04953	−0.381054
$565$	37.0531	1.55884
$566$	11.3099	0.475390
$567$	23.8628	1.00214
$568$	−1.70003	−0.0713319
$569$	−0.577039	−0.0241907	−0.0120954	−	0.999927i	$-0.503850\pi$
$569$	−0.577039	−0.0241907	−0.0120954	+	0.999927i	$0.503850\pi$
$570$	15.7423	0.659371
$571$	27.0027	1.13003	0.565015	−	0.825081i	$-0.308870\pi$
$571$	27.0027	1.13003	0.565015	+	0.825081i	$0.308870\pi$
$572$	−1.89169	−0.0790955
$573$	−2.95980	−0.123647
$574$	20.2759	0.846300
$575$	53.7439	2.24128
$576$	1.08655	0.0452728
$577$	−0.178303	−0.00742286	−0.00371143	−	0.999993i	$-0.501181\pi$
$577$	−0.178303	−0.00742286	−0.00371143	+	0.999993i	$0.501181\pi$
$578$	14.0022	0.582415
$579$	−32.9759	−1.37043
$580$	−15.4452	−0.641326
$581$	−8.99275	−0.373082
$582$	−17.9891	−0.745670
$583$	−0.642041	−0.0265906
$584$	10.7477	0.444745
$585$	−11.1810	−0.462277
$586$	−5.95670	−0.246069
$587$	6.00843	0.247994	0.123997	−	0.992283i	$-0.460429\pi$
$587$	6.00843	0.247994	0.123997	+	0.992283i	$0.460429\pi$
$588$	4.77255	0.196817
$589$	−2.66152	−0.109666
$590$	−6.04324	−0.248796
$591$	−14.5834	−0.599882
$592$	0.585059	0.0240458
$593$	12.2715	0.503929	0.251965	−	0.967736i	$-0.418923\pi$
$593$	12.2715	0.503929	0.251965	+	0.967736i	$0.418923\pi$
$594$	−2.48680	−0.102035
$595$	−13.0420	−0.534671
$596$	−1.59975	−0.0655284
$597$	−6.80214	−0.278393
$598$	21.8701	0.894334
$599$	10.8323	0.442597	0.221298	−	0.975206i	$-0.428971\pi$
$599$	10.8323	0.442597	0.221298	+	0.975206i	$0.428971\pi$
$600$	14.6171	0.596741
$601$	4.39264	0.179179	0.0895897	−	0.995979i	$-0.471444\pi$
$601$	4.39264	0.179179	0.0895897	+	0.995979i	$0.471444\pi$
$602$	1.18373	0.0482452
$603$	−8.39126	−0.341719
$604$	15.3028	0.622664
$605$	−37.0243	−1.50525
$606$	40.2672	1.63574
$607$	16.0262	0.650482	0.325241	−	0.945631i	$-0.394555\pi$
$607$	16.0262	0.650482	0.325241	+	0.945631i	$0.394555\pi$
$608$	−2.22670	−0.0903048
$609$	−19.2293	−0.779210
$610$	8.82101	0.357152
$611$	−13.1720	−0.532883
$612$	1.88126	0.0760454
$613$	31.4899	1.27186	0.635932	−	0.771745i	$-0.280616\pi$
$613$	31.4899	1.27186	0.635932	+	0.771745i	$0.280616\pi$
$614$	15.4762	0.624568
$615$	−66.5529	−2.68367
$616$	1.38472	0.0557921
$617$	47.1116	1.89664	0.948321	−	0.317313i	$-0.102781\pi$
$617$	47.1116	1.89664	0.948321	+	0.317313i	$0.102781\pi$
$618$	23.6458	0.951174
$619$	18.2981	0.735464	0.367732	−	0.929932i	$-0.380134\pi$
$619$	18.2981	0.735464	0.367732	+	0.929932i	$0.380134\pi$
$620$	−4.18016	−0.167879
$621$	28.7503	1.15371
$622$	−25.7236	−1.03142
$623$	27.4174	1.09846
$624$	5.94816	0.238117
$625$	−8.86996	−0.354798
$626$	8.70889	0.348077
$627$	−2.89391	−0.115572
$628$	−3.45824	−0.137999
$629$	1.01298	0.0403901
$630$	8.18452	0.326079
$631$	11.3852	0.453238	0.226619	−	0.973983i	$-0.427233\pi$
$631$	11.3852	0.453238	0.226619	+	0.973983i	$0.427233\pi$
$632$	4.59092	0.182617
$633$	−55.2203	−2.19481
$634$	2.20990	0.0877662
$635$	45.1400	1.79133
$636$	2.01881	0.0800512
$637$	6.94668	0.275237
$638$	2.83930	0.112409
$639$	1.84717	0.0730728
$640$	−3.49725	−0.138241
$641$	41.5729	1.64203	0.821016	−	0.570906i	$-0.193408\pi$
$641$	41.5729	1.64203	0.821016	+	0.570906i	$0.193408\pi$
$642$	23.6844	0.934747
$643$	27.1837	1.07202	0.536011	−	0.844211i	$-0.319931\pi$
$643$	27.1837	1.07202	0.536011	+	0.844211i	$0.319931\pi$
$644$	−16.0090	−0.630843
$645$	−3.88543	−0.152989
$646$	−3.85534	−0.151686
$647$	−47.4397	−1.86505	−0.932523	−	0.361110i	$-0.882398\pi$
$647$	−47.4397	−1.86505	−0.932523	+	0.361110i	$0.882398\pi$
$648$	11.0791	0.435226
$649$	1.11093	0.0436079
$650$	21.2759	0.834510
$651$	−5.20431	−0.203973
$652$	6.03012	0.236158
$653$	−34.6478	−1.35587	−0.677936	−	0.735121i	$-0.737126\pi$
$653$	−34.6478	−1.35587	−0.677936	+	0.735121i	$0.737126\pi$
$654$	−12.7027	−0.496713
$655$	−7.01306	−0.274023
$656$	9.41374	0.367545
$657$	−11.6779	−0.455599
$658$	9.64197	0.375883
$659$	33.6518	1.31089	0.655444	−	0.755244i	$-0.272482\pi$
$659$	33.6518	1.31089	0.655444	+	0.755244i	$0.272482\pi$
$660$	−4.54517	−0.176920
$661$	−43.4029	−1.68818	−0.844089	−	0.536203i	$-0.819858\pi$
$661$	−43.4029	−1.68818	−0.844089	+	0.536203i	$0.819858\pi$
$662$	24.1651	0.939204
$663$	10.2987	0.399969
$664$	−4.17517	−0.162028
$665$	−16.7729	−0.650424
$666$	−0.635694	−0.0246326
$667$	−32.8256	−1.27101
$668$	12.0460	0.466075
$669$	−33.9339	−1.31196
$670$	27.0088	1.04344
$671$	−1.62157	−0.0626001
$672$	−4.35408	−0.167962
$673$	12.9572	0.499464	0.249732	−	0.968315i	$-0.419657\pi$
$673$	12.9572	0.499464	0.249732	+	0.968315i	$0.419657\pi$
$674$	25.3401	0.976063
$675$	27.9692	1.07653
$676$	−4.34216	−0.167006
$677$	−2.48900	−0.0956600	−0.0478300	−	0.998855i	$-0.515231\pi$
$677$	−2.48900	−0.0956600	−0.0478300	+	0.998855i	$0.515231\pi$
$678$	21.4179	0.822549
$679$	19.1667	0.735552
$680$	−6.05518	−0.232205
$681$	−50.0114	−1.91644
$682$	0.768442	0.0294252
$683$	21.3476	0.816843	0.408421	−	0.912794i	$-0.366079\pi$
$683$	21.3476	0.816843	0.408421	+	0.912794i	$0.366079\pi$
$684$	2.41942	0.0925088
$685$	−21.5701	−0.824150
$686$	−20.1620	−0.769790
$687$	−8.51286	−0.324786
$688$	0.549585	0.0209527
$689$	2.93848	0.111947
$690$	52.5473	2.00044
$691$	11.7815	0.448188	0.224094	−	0.974568i	$-0.428058\pi$
$691$	11.7815	0.448188	0.224094	+	0.974568i	$0.428058\pi$
$692$	23.1769	0.881054
$693$	−1.50457	−0.0571538
$694$	−13.7829	−0.523190
$695$	−31.0313	−1.17708
$696$	−8.92781	−0.338408
$697$	16.2991	0.617371
$698$	−2.28502	−0.0864894
$699$	31.6606	1.19751
$700$	−15.5741	−0.588644
$701$	−4.72292	−0.178382	−0.0891911	−	0.996015i	$-0.528428\pi$
$701$	−4.72292	−0.178382	−0.0891911	+	0.996015i	$0.528428\pi$
$702$	11.3815	0.429568
$703$	1.30275	0.0491343
$704$	0.642902	0.0242303
$705$	−31.6485	−1.19195
$706$	−3.99082	−0.150196
$707$	−42.9034	−1.61355
$708$	−3.49318	−0.131282
$709$	26.8042	1.00665	0.503326	−	0.864097i	$-0.332110\pi$
$709$	26.8042	1.00665	0.503326	+	0.864097i	$0.332110\pi$
$710$	−5.94545	−0.223129
$711$	−4.98824	−0.187074
$712$	12.7294	0.477055
$713$	−8.88408	−0.332711
$714$	−7.53871	−0.282129
$715$	−6.61571	−0.247413
$716$	−15.5092	−0.579607
$717$	39.5723	1.47785
$718$	−12.2982	−0.458964
$719$	−21.7152	−0.809841	−0.404920	−	0.914352i	$-0.632701\pi$
$719$	−21.7152	−0.809841	−0.404920	+	0.914352i	$0.632701\pi$
$720$	3.79992	0.141615
$721$	−25.1938	−0.938267
$722$	14.0418	0.522581
$723$	49.6880	1.84792
$724$	−20.2179	−0.751392
$725$	−31.9338	−1.18599
$726$	−21.4012	−0.794273
$727$	28.7190	1.06513	0.532564	−	0.846390i	$-0.321228\pi$
$727$	28.7190	1.06513	0.532564	+	0.846390i	$0.321228\pi$
$728$	−6.33757	−0.234886
$729$	11.4203	0.422975
$730$	37.5876	1.39118
$731$	0.951557	0.0351946
$732$	5.09882	0.188458
$733$	26.4461	0.976810	0.488405	−	0.872617i	$-0.337579\pi$
$733$	26.4461	0.976810	0.488405	+	0.872617i	$0.337579\pi$
$734$	13.9872	0.516278
$735$	16.6908	0.615649
$736$	−7.43269	−0.273972
$737$	−4.96505	−0.182890
$738$	−10.2285	−0.376515
$739$	−16.1899	−0.595555	−0.297777	−	0.954635i	$-0.596245\pi$
$739$	−16.1899	−0.595555	−0.297777	+	0.954635i	$0.596245\pi$
$740$	2.04610	0.0752161
$741$	13.2448	0.486560
$742$	−2.15098	−0.0789649
$743$	−23.8998	−0.876799	−0.438399	−	0.898780i	$-0.644454\pi$
$743$	−23.8998	−0.876799	−0.438399	+	0.898780i	$0.644454\pi$
$744$	−2.41627	−0.0885846
$745$	−5.59474	−0.204975
$746$	−29.8800	−1.09398
$747$	4.53652	0.165982
$748$	1.11313	0.0407000
$749$	−25.2349	−0.922064
$750$	15.7709	0.575872
$751$	−15.6871	−0.572431	−0.286215	−	0.958165i	$-0.592397\pi$
$751$	−15.6871	−0.572431	−0.286215	+	0.958165i	$0.592397\pi$
$752$	4.47659	0.163245
$753$	30.7031	1.11888
$754$	−12.9948	−0.473244
$755$	53.5178	1.94771
$756$	−8.33133	−0.303008
$757$	1.45948	0.0530456	0.0265228	−	0.999648i	$-0.491557\pi$
$757$	1.45948	0.0530456	0.0265228	+	0.999648i	$0.491557\pi$
$758$	−4.37390	−0.158867
$759$	−9.65982	−0.350629
$760$	−7.78734	−0.282477
$761$	26.5007	0.960649	0.480324	−	0.877091i	$-0.340519\pi$
$761$	26.5007	0.960649	0.480324	+	0.877091i	$0.340519\pi$
$762$	26.0923	0.945226
$763$	13.5343	0.489973
$764$	1.46415	0.0529709
$765$	6.57923	0.237873
$766$	−32.4114	−1.17107
$767$	−5.08449	−0.183590
$768$	−2.02152	−0.0729454
$769$	−6.55064	−0.236222	−0.118111	−	0.993000i	$-0.537684\pi$
$769$	−6.55064	−0.236222	−0.118111	+	0.993000i	$0.537684\pi$
$770$	4.84272	0.174520
$771$	−10.4461	−0.376209
$772$	16.3124	0.587097
$773$	−25.9131	−0.932028	−0.466014	−	0.884777i	$-0.654310\pi$
$773$	−25.9131	−0.932028	−0.466014	+	0.884777i	$0.654310\pi$
$774$	−0.597149	−0.0214641
$775$	−8.64271	−0.310455
$776$	8.89877	0.319447
$777$	2.54740	0.0913873
$778$	−29.9822	−1.07491
$779$	20.9616	0.751028
$780$	20.8022	0.744839
$781$	1.09296	0.0391090
$782$	−12.8690	−0.460196
$783$	−17.0829	−0.610494
$784$	−2.36087	−0.0843168
$785$	−12.0943	−0.431665
$786$	−4.05377	−0.144593
$787$	−10.4513	−0.372548	−0.186274	−	0.982498i	$-0.559641\pi$
$787$	−10.4513	−0.372548	−0.186274	+	0.982498i	$0.559641\pi$
$788$	7.21408	0.256991
$789$	−4.37010	−0.155580
$790$	16.0556	0.571232
$791$	−22.8200	−0.811387
$792$	−0.698543	−0.0248216
$793$	7.42158	0.263548
$794$	−29.9500	−1.06289
$795$	7.06029	0.250403
$796$	3.36486	0.119264
$797$	23.5546	0.834347	0.417174	−	0.908827i	$-0.363021\pi$
$797$	23.5546	0.834347	0.417174	+	0.908827i	$0.363021\pi$
$798$	−9.69525	−0.343208
$799$	7.75083	0.274205
$800$	−7.23075	−0.255646
$801$	−13.8311	−0.488698
$802$	−15.5388	−0.548693
$803$	−6.90975	−0.243840
$804$	15.6119	0.550591
$805$	−55.9874	−1.97330
$806$	−3.51699	−0.123881
$807$	−60.9822	−2.14668
$808$	−19.9193	−0.700757
$809$	−16.7190	−0.587810	−0.293905	−	0.955835i	$-0.594955\pi$
$809$	−16.7190	−0.587810	−0.293905	+	0.955835i	$0.594955\pi$
$810$	38.7462	1.36140
$811$	41.2526	1.44857	0.724287	−	0.689499i	$-0.242169\pi$
$811$	41.2526	1.44857	0.724287	+	0.689499i	$0.242169\pi$
$812$	9.51228	0.333816
$813$	39.2150	1.37533
$814$	−0.376136	−0.0131836
$815$	21.0888	0.738709
$816$	−3.50008	−0.122527
$817$	1.22376	0.0428140
$818$	−13.7670	−0.481351
$819$	6.88607	0.240619
$820$	32.9222	1.14969
$821$	−41.9822	−1.46519	−0.732595	−	0.680665i	$-0.761691\pi$
$821$	−41.9822	−1.46519	−0.732595	+	0.680665i	$0.761691\pi$
$822$	−12.4682	−0.434878
$823$	27.0241	0.942000	0.471000	−	0.882133i	$-0.343893\pi$
$823$	27.0241	0.942000	0.471000	+	0.882133i	$0.343893\pi$
$824$	−11.6970	−0.407486
$825$	−9.39738	−0.327175
$826$	3.72187	0.129500
$827$	56.1442	1.95233	0.976163	−	0.217040i	$-0.0696403\pi$
$827$	56.1442	1.95233	0.976163	+	0.217040i	$0.0696403\pi$
$828$	8.07596	0.280659
$829$	44.9217	1.56020	0.780098	−	0.625657i	$-0.215169\pi$
$829$	44.9217	1.56020	0.780098	+	0.625657i	$0.215169\pi$
$830$	−14.6016	−0.506829
$831$	32.5207	1.12813
$832$	−2.94242	−0.102010
$833$	−4.08764	−0.141628
$834$	−17.9371	−0.621110
$835$	42.1280	1.45790
$836$	1.43155	0.0495113
$837$	−4.62341	−0.159808
$838$	−27.8301	−0.961373
$839$	55.8999	1.92988	0.964940	−	0.262472i	$-0.0845378\pi$
$839$	55.8999	1.92988	0.964940	+	0.262472i	$0.0845378\pi$
$840$	−15.2273	−0.525392
$841$	−9.49558	−0.327434
$842$	−24.6876	−0.850790
$843$	−40.2318	−1.38566
$844$	27.3162	0.940263
$845$	−15.1856	−0.522401
$846$	−4.86403	−0.167229
$847$	22.8023	0.783495
$848$	−0.998661	−0.0342942
$849$	22.8632	0.784661
$850$	−12.5194	−0.429412
$851$	4.34856	0.149067
$852$	−3.43666	−0.117738
$853$	−18.9295	−0.648133	−0.324067	−	0.946034i	$-0.605050\pi$
$853$	−18.9295	−0.648133	−0.324067	+	0.946034i	$0.605050\pi$
$854$	−5.43263	−0.185901
$855$	8.46131	0.289371
$856$	−11.7161	−0.400449
$857$	4.72972	0.161564	0.0807821	−	0.996732i	$-0.474258\pi$
$857$	4.72972	0.161564	0.0807821	+	0.996732i	$0.474258\pi$
$858$	−3.82409	−0.130552
$859$	−29.1923	−0.996027	−0.498014	−	0.867169i	$-0.665937\pi$
$859$	−29.1923	−0.996027	−0.498014	+	0.867169i	$0.665937\pi$
$860$	1.92203	0.0655408
$861$	40.9882	1.39687
$862$	−4.81017	−0.163835
$863$	2.90882	0.0990173	0.0495087	−	0.998774i	$-0.484234\pi$
$863$	2.90882	0.0990173	0.0495087	+	0.998774i	$0.484234\pi$
$864$	−3.86809	−0.131595
$865$	81.0555	2.75597
$866$	−19.9556	−0.678119
$867$	28.3058	0.961314
$868$	2.57445	0.0873826
$869$	−2.95151	−0.100123
$870$	−31.2228	−1.05855
$871$	22.7239	0.769971
$872$	6.28371	0.212793
$873$	−9.66893	−0.327244
$874$	−16.5504	−0.559825
$875$	−16.8034	−0.568058
$876$	21.7268	0.734080
$877$	−5.43104	−0.183393	−0.0916966	−	0.995787i	$-0.529229\pi$
$877$	−5.43104	−0.183393	−0.0916966	+	0.995787i	$0.529229\pi$
$878$	21.6464	0.730530
$879$	−12.0416	−0.406153
$880$	2.24839	0.0757932
$881$	−3.91856	−0.132020	−0.0660099	−	0.997819i	$-0.521027\pi$
$881$	−3.91856	−0.132020	−0.0660099	+	0.997819i	$0.521027\pi$
$882$	2.56520	0.0863747
$883$	23.4133	0.787919	0.393960	−	0.919128i	$-0.371105\pi$
$883$	23.4133	0.787919	0.393960	+	0.919128i	$0.371105\pi$
$884$	−5.09454	−0.171348
$885$	−12.2165	−0.410654
$886$	31.3307	1.05258
$887$	−9.36886	−0.314575	−0.157288	−	0.987553i	$-0.550275\pi$
$887$	−9.36886	−0.314575	−0.157288	+	0.987553i	$0.550275\pi$
$888$	1.18271	0.0396891
$889$	−27.8005	−0.932400
$890$	44.5180	1.49225
$891$	−7.12275	−0.238621
$892$	16.7863	0.562048
$893$	9.96805	0.333568
$894$	−3.23393	−0.108159
$895$	−54.2396	−1.81303
$896$	2.15386	0.0719555
$897$	44.2108	1.47616
$898$	−1.16615	−0.0389150
$899$	5.27877	0.176057
$900$	7.85655	0.261885
$901$	−1.72909	−0.0576044
$902$	−6.05211	−0.201513
$903$	2.39294	0.0796319
$904$	−10.5949	−0.352382
$905$	−70.7070	−2.35038
$906$	30.9350	1.02775
$907$	−2.74371	−0.0911036	−0.0455518	−	0.998962i	$-0.514505\pi$
$907$	−2.74371	−0.0911036	−0.0455518	+	0.998962i	$0.514505\pi$
$908$	24.7395	0.821008
$909$	21.6432	0.717859
$910$	−22.1641	−0.734732
$911$	−6.33790	−0.209984	−0.104992	−	0.994473i	$-0.533482\pi$
$911$	−6.33790	−0.209984	−0.104992	+	0.994473i	$0.533482\pi$
$912$	−4.50133	−0.149054
$913$	2.68423	0.0888349
$914$	−4.73128	−0.156497
$915$	17.8319	0.589503
$916$	4.21112	0.139139
$917$	4.31916	0.142631
$918$	−6.69725	−0.221042
$919$	31.5485	1.04069	0.520345	−	0.853956i	$-0.325803\pi$
$919$	31.5485	1.04069	0.520345	+	0.853956i	$0.325803\pi$
$920$	−25.9940	−0.856995
$921$	31.2854	1.03089
$922$	8.73944	0.287818
$923$	−5.00222	−0.164650
$924$	2.79925	0.0920885
$925$	4.23042	0.139095
$926$	41.3562	1.35905
$927$	12.7094	0.417431
$928$	4.41638	0.144975
$929$	−31.1375	−1.02159	−0.510795	−	0.859703i	$-0.670649\pi$
$929$	−31.1375	−1.02159	−0.510795	+	0.859703i	$0.670649\pi$
$930$	−8.45028	−0.277096
$931$	−5.25696	−0.172290
$932$	−15.6618	−0.513018
$933$	−52.0008	−1.70243
$934$	19.6447	0.642794
$935$	3.89289	0.127311
$936$	3.19708	0.104500
$937$	14.5876	0.476555	0.238278	−	0.971197i	$-0.423417\pi$
$937$	14.5876	0.476555	0.238278	+	0.971197i	$0.423417\pi$
$938$	−16.6340	−0.543120
$939$	17.6052	0.574524
$940$	15.6558	0.510635
$941$	14.6477	0.477501	0.238750	−	0.971081i	$-0.423262\pi$
$941$	14.6477	0.477501	0.238750	+	0.971081i	$0.423262\pi$
$942$	−6.99091	−0.227776
$943$	69.9694	2.27852
$944$	1.72800	0.0562415
$945$	−29.1367	−0.947818
$946$	−0.353329	−0.0114877
$947$	−24.6723	−0.801742	−0.400871	−	0.916135i	$-0.631292\pi$
$947$	−24.6723	−0.801742	−0.400871	+	0.916135i	$0.631292\pi$
$948$	9.28063	0.301421
$949$	31.6244	1.02657
$950$	−16.1008	−0.522377
$951$	4.46735	0.144864
$952$	3.72922	0.120865
$953$	−21.4461	−0.694706	−0.347353	−	0.937734i	$-0.612919\pi$
$953$	−21.4461	−0.694706	−0.347353	+	0.937734i	$0.612919\pi$
$954$	1.08509	0.0351311
$955$	5.12048	0.165695
$956$	−19.5755	−0.633117
$957$	5.73970	0.185538
$958$	−28.5291	−0.921733
$959$	13.2844	0.428977
$960$	−7.06976	−0.228176
$961$	−29.5713	−0.953914
$962$	1.72149	0.0555031
$963$	12.7301	0.410222
$964$	−24.5795	−0.791653
$965$	57.0486	1.83646
$966$	−32.3625	−1.04125
$967$	21.3306	0.685947	0.342973	−	0.939345i	$-0.388566\pi$
$967$	21.3306	0.685947	0.342973	+	0.939345i	$0.388566\pi$
$968$	10.5867	0.340269
$969$	−7.79365	−0.250368
$970$	31.1212	0.999243
$971$	1.34944	0.0433054	0.0216527	−	0.999766i	$-0.493107\pi$
$971$	1.34944	0.0433054	0.0216527	+	0.999766i	$0.493107\pi$
$972$	10.7923	0.346163
$973$	19.1114	0.612682
$974$	−7.51487	−0.240792
$975$	43.0097	1.37741
$976$	−2.52227	−0.0807359
$977$	41.0832	1.31437	0.657184	−	0.753730i	$-0.271747\pi$
$977$	41.0832	1.31437	0.657184	+	0.753730i	$0.271747\pi$
$978$	12.1900	0.389794
$979$	−8.18377	−0.261555
$980$	−8.25656	−0.263746
$981$	−6.82754	−0.217987
$982$	−12.2338	−0.390398
$983$	−57.3111	−1.82794	−0.913970	−	0.405781i	$-0.866999\pi$
$983$	−57.3111	−1.82794	−0.913970	+	0.405781i	$0.866999\pi$
$984$	19.0301	0.606656
$985$	25.2294	0.803877
$986$	7.64657	0.243516
$987$	19.4914	0.620420
$988$	−6.55190	−0.208444
$989$	4.08489	0.129892
$990$	−2.44298	−0.0776430
$991$	12.5766	0.399508	0.199754	−	0.979846i	$-0.435986\pi$
$991$	12.5766	0.399508	0.199754	+	0.979846i	$0.435986\pi$
$992$	1.19527	0.0379499
$993$	48.8503	1.55022
$994$	3.66164	0.116140
$995$	11.7678	0.373063
$996$	−8.44019	−0.267438
$997$	−7.11949	−0.225477	−0.112738	−	0.993625i	$-0.535962\pi$
$997$	−7.11949	−0.225477	−0.112738	+	0.993625i	$0.535962\pi$
$998$	27.7120	0.877207
$999$	2.26306	0.0716000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.17	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.17	✓	86	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.02152 q^{3} +1.00000 q^{4} +3.49725 q^{5} +2.02152 q^{6} -2.15386 q^{7} -1.00000 q^{8} +1.08655 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.02152 q^{3} +1.00000 q^{4} +3.49725 q^{5} +2.02152 q^{6} -2.15386 q^{7} -1.00000 q^{8} +1.08655 q^{9} -3.49725 q^{10} +0.642902 q^{11} -2.02152 q^{12} -2.94242 q^{13} +2.15386 q^{14} -7.06976 q^{15} +1.00000 q^{16} +1.73141 q^{17} -1.08655 q^{18} +2.22670 q^{19} +3.49725 q^{20} +4.35408 q^{21} -0.642902 q^{22} +7.43269 q^{23} +2.02152 q^{24} +7.23075 q^{25} +2.94242 q^{26} +3.86809 q^{27} -2.15386 q^{28} -4.41638 q^{29} +7.06976 q^{30} -1.19527 q^{31} -1.00000 q^{32} -1.29964 q^{33} -1.73141 q^{34} -7.53260 q^{35} +1.08655 q^{36} +0.585059 q^{37} -2.22670 q^{38} +5.94816 q^{39} -3.49725 q^{40} +9.41374 q^{41} -4.35408 q^{42} +0.549585 q^{43} +0.642902 q^{44} +3.79992 q^{45} -7.43269 q^{46} +4.47659 q^{47} -2.02152 q^{48} -2.36087 q^{49} -7.23075 q^{50} -3.50008 q^{51} -2.94242 q^{52} -0.998661 q^{53} -3.86809 q^{54} +2.24839 q^{55} +2.15386 q^{56} -4.50133 q^{57} +4.41638 q^{58} +1.72800 q^{59} -7.06976 q^{60} -2.52227 q^{61} +1.19527 q^{62} -2.34027 q^{63} +1.00000 q^{64} -10.2904 q^{65} +1.29964 q^{66} -7.72287 q^{67} +1.73141 q^{68} -15.0253 q^{69} +7.53260 q^{70} +1.70003 q^{71} -1.08655 q^{72} -10.7477 q^{73} -0.585059 q^{74} -14.6171 q^{75} +2.22670 q^{76} -1.38472 q^{77} -5.94816 q^{78} -4.59092 q^{79} +3.49725 q^{80} -11.0791 q^{81} -9.41374 q^{82} +4.17517 q^{83} +4.35408 q^{84} +6.05518 q^{85} -0.549585 q^{86} +8.92781 q^{87} -0.642902 q^{88} -12.7294 q^{89} -3.79992 q^{90} +6.33757 q^{91} +7.43269 q^{92} +2.41627 q^{93} -4.47659 q^{94} +7.78734 q^{95} +2.02152 q^{96} -8.89877 q^{97} +2.36087 q^{98} +0.698543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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