LMFDB - Embedded newform 8042.2.a.c.1.20

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			1.20
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.75423 q^{3} +1.00000 q^{4} -3.88795 q^{5} +1.75423 q^{6} +3.77690 q^{7} -1.00000 q^{8} +0.0773276 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.75423 q^{3} +1.00000 q^{4} -3.88795 q^{5} +1.75423 q^{6} +3.77690 q^{7} -1.00000 q^{8} +0.0773276 q^{9} +3.88795 q^{10} -0.891621 q^{11} -1.75423 q^{12} +0.270119 q^{13} -3.77690 q^{14} +6.82037 q^{15} +1.00000 q^{16} +3.78696 q^{17} -0.0773276 q^{18} +6.97343 q^{19} -3.88795 q^{20} -6.62556 q^{21} +0.891621 q^{22} +1.59417 q^{23} +1.75423 q^{24} +10.1162 q^{25} -0.270119 q^{26} +5.12704 q^{27} +3.77690 q^{28} +5.23837 q^{29} -6.82037 q^{30} +5.29808 q^{31} -1.00000 q^{32} +1.56411 q^{33} -3.78696 q^{34} -14.6844 q^{35} +0.0773276 q^{36} +2.71579 q^{37} -6.97343 q^{38} -0.473851 q^{39} +3.88795 q^{40} -6.00482 q^{41} +6.62556 q^{42} +5.68254 q^{43} -0.891621 q^{44} -0.300646 q^{45} -1.59417 q^{46} -3.50723 q^{47} -1.75423 q^{48} +7.26499 q^{49} -10.1162 q^{50} -6.64321 q^{51} +0.270119 q^{52} +7.83966 q^{53} -5.12704 q^{54} +3.46658 q^{55} -3.77690 q^{56} -12.2330 q^{57} -5.23837 q^{58} +10.0165 q^{59} +6.82037 q^{60} +2.93760 q^{61} -5.29808 q^{62} +0.292059 q^{63} +1.00000 q^{64} -1.05021 q^{65} -1.56411 q^{66} +6.65517 q^{67} +3.78696 q^{68} -2.79654 q^{69} +14.6844 q^{70} -7.44458 q^{71} -0.0773276 q^{72} -0.533104 q^{73} -2.71579 q^{74} -17.7461 q^{75} +6.97343 q^{76} -3.36757 q^{77} +0.473851 q^{78} +3.23891 q^{79} -3.88795 q^{80} -9.22600 q^{81} +6.00482 q^{82} -9.69483 q^{83} -6.62556 q^{84} -14.7235 q^{85} -5.68254 q^{86} -9.18932 q^{87} +0.891621 q^{88} +15.0329 q^{89} +0.300646 q^{90} +1.02021 q^{91} +1.59417 q^{92} -9.29407 q^{93} +3.50723 q^{94} -27.1124 q^{95} +1.75423 q^{96} +4.42694 q^{97} -7.26499 q^{98} -0.0689469 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.75423	−1.01281	−0.506403	−	0.862297i	$-0.669025\pi$
$3$	−1.75423	−1.01281	−0.506403	+	0.862297i	$0.669025\pi$
$4$	1.00000	0.500000
$5$	−3.88795	−1.73875	−0.869373	−	0.494156i	$-0.835477\pi$
$5$	−3.88795	−1.73875	−0.869373	+	0.494156i	$0.835477\pi$
$6$	1.75423	0.716162
$7$	3.77690	1.42753	0.713767	−	0.700383i	$-0.246987\pi$
$7$	3.77690	1.42753	0.713767	+	0.700383i	$0.246987\pi$
$8$	−1.00000	−0.353553
$9$	0.0773276	0.0257759
$10$	3.88795	1.22948
$11$	−0.891621	−0.268834	−0.134417	−	0.990925i	$-0.542916\pi$
$11$	−0.891621	−0.268834	−0.134417	+	0.990925i	$0.542916\pi$
$12$	−1.75423	−0.506403
$13$	0.270119	0.0749175	0.0374588	−	0.999298i	$-0.488074\pi$
$13$	0.270119	0.0749175	0.0374588	+	0.999298i	$0.488074\pi$
$14$	−3.77690	−1.00942
$15$	6.82037	1.76101
$16$	1.00000	0.250000
$17$	3.78696	0.918474	0.459237	−	0.888314i	$-0.348123\pi$
$17$	3.78696	0.918474	0.459237	+	0.888314i	$0.348123\pi$
$18$	−0.0773276	−0.0182263
$19$	6.97343	1.59982	0.799908	−	0.600123i	$-0.204882\pi$
$19$	6.97343	1.59982	0.799908	+	0.600123i	$0.204882\pi$
$20$	−3.88795	−0.869373
$21$	−6.62556	−1.44582
$22$	0.891621	0.190094
$23$	1.59417	0.332407	0.166203	−	0.986091i	$-0.446849\pi$
$23$	1.59417	0.332407	0.166203	+	0.986091i	$0.446849\pi$
$24$	1.75423	0.358081
$25$	10.1162	2.02324
$26$	−0.270119	−0.0529747
$27$	5.12704	0.986700
$28$	3.77690	0.713767
$29$	5.23837	0.972741	0.486371	−	0.873753i	$-0.338320\pi$
$29$	5.23837	0.972741	0.486371	+	0.873753i	$0.338320\pi$
$30$	−6.82037	−1.24522
$31$	5.29808	0.951564	0.475782	−	0.879563i	$-0.342165\pi$
$31$	5.29808	0.951564	0.475782	+	0.879563i	$0.342165\pi$
$32$	−1.00000	−0.176777
$33$	1.56411	0.272277
$34$	−3.78696	−0.649459
$35$	−14.6844	−2.48212
$36$	0.0773276	0.0128879
$37$	2.71579	0.446474	0.223237	−	0.974764i	$-0.428338\pi$
$37$	2.71579	0.446474	0.223237	+	0.974764i	$0.428338\pi$
$38$	−6.97343	−1.13124
$39$	−0.473851	−0.0758769
$40$	3.88795	0.614740
$41$	−6.00482	−0.937796	−0.468898	−	0.883252i	$-0.655349\pi$
$41$	−6.00482	−0.937796	−0.468898	+	0.883252i	$0.655349\pi$
$42$	6.62556	1.02235
$43$	5.68254	0.866579	0.433289	−	0.901255i	$-0.357353\pi$
$43$	5.68254	0.866579	0.433289	+	0.901255i	$0.357353\pi$
$44$	−0.891621	−0.134417
$45$	−0.300646	−0.0448177
$46$	−1.59417	−0.235047
$47$	−3.50723	−0.511582	−0.255791	−	0.966732i	$-0.582336\pi$
$47$	−3.50723	−0.511582	−0.255791	+	0.966732i	$0.582336\pi$
$48$	−1.75423	−0.253201
$49$	7.26499	1.03786
$50$	−10.1162	−1.43064
$51$	−6.64321	−0.930235
$52$	0.270119	0.0374588
$53$	7.83966	1.07686	0.538430	−	0.842670i	$-0.319017\pi$
$53$	7.83966	1.07686	0.538430	+	0.842670i	$0.319017\pi$
$54$	−5.12704	−0.697702
$55$	3.46658	0.467434
$56$	−3.77690	−0.504710
$57$	−12.2330	−1.62030
$58$	−5.23837	−0.687832
$59$	10.0165	1.30403	0.652017	−	0.758205i	$-0.273923\pi$
$59$	10.0165	1.30403	0.652017	+	0.758205i	$0.273923\pi$
$60$	6.82037	0.880506
$61$	2.93760	0.376121	0.188061	−	0.982157i	$-0.439780\pi$
$61$	2.93760	0.376121	0.188061	+	0.982157i	$0.439780\pi$
$62$	−5.29808	−0.672857
$63$	0.292059	0.0367959
$64$	1.00000	0.125000
$65$	−1.05021	−0.130263
$66$	−1.56411	−0.192529
$67$	6.65517	0.813059	0.406529	−	0.913638i	$-0.366739\pi$
$67$	6.65517	0.813059	0.406529	+	0.913638i	$0.366739\pi$
$68$	3.78696	0.459237
$69$	−2.79654	−0.336664
$70$	14.6844	1.75512
$71$	−7.44458	−0.883509	−0.441755	−	0.897136i	$-0.645644\pi$
$71$	−7.44458	−0.883509	−0.441755	+	0.897136i	$0.645644\pi$
$72$	−0.0773276	−0.00911314
$73$	−0.533104	−0.0623951	−0.0311975	−	0.999513i	$-0.509932\pi$
$73$	−0.533104	−0.0623951	−0.0311975	+	0.999513i	$0.509932\pi$
$74$	−2.71579	−0.315705
$75$	−17.7461	−2.04915
$76$	6.97343	0.799908
$77$	−3.36757	−0.383770
$78$	0.473851	0.0536531
$79$	3.23891	0.364406	0.182203	−	0.983261i	$-0.441677\pi$
$79$	3.23891	0.364406	0.182203	+	0.983261i	$0.441677\pi$
$80$	−3.88795	−0.434686
$81$	−9.22600	−1.02511
$82$	6.00482	0.663122
$83$	−9.69483	−1.06415	−0.532073	−	0.846698i	$-0.678587\pi$
$83$	−9.69483	−1.06415	−0.532073	+	0.846698i	$0.678587\pi$
$84$	−6.62556	−0.722908
$85$	−14.7235	−1.59699
$86$	−5.68254	−0.612764
$87$	−9.18932	−0.985198
$88$	0.891621	0.0950471
$89$	15.0329	1.59349	0.796743	−	0.604319i	$-0.206555\pi$
$89$	15.0329	1.59349	0.796743	+	0.604319i	$0.206555\pi$
$90$	0.300646	0.0316909
$91$	1.02021	0.106947
$92$	1.59417	0.166203
$93$	−9.29407	−0.963750
$94$	3.50723	0.361743
$95$	−27.1124	−2.78167
$96$	1.75423	0.179040
$97$	4.42694	0.449488	0.224744	−	0.974418i	$-0.427845\pi$
$97$	4.42694	0.449488	0.224744	+	0.974418i	$0.427845\pi$
$98$	−7.26499	−0.733875
$99$	−0.0689469	−0.00692942
$100$	10.1162	1.01162
$101$	4.94998	0.492541	0.246271	−	0.969201i	$-0.420795\pi$
$101$	4.94998	0.492541	0.246271	+	0.969201i	$0.420795\pi$
$102$	6.64321	0.657776
$103$	4.25054	0.418818	0.209409	−	0.977828i	$-0.432846\pi$
$103$	4.25054	0.418818	0.209409	+	0.977828i	$0.432846\pi$
$104$	−0.270119	−0.0264873
$105$	25.7599	2.51391
$106$	−7.83966	−0.761456
$107$	−8.47450	−0.819261	−0.409630	−	0.912252i	$-0.634342\pi$
$107$	−8.47450	−0.819261	−0.409630	+	0.912252i	$0.634342\pi$
$108$	5.12704	0.493350
$109$	−4.94777	−0.473910	−0.236955	−	0.971521i	$-0.576149\pi$
$109$	−4.94777	−0.473910	−0.236955	+	0.971521i	$0.576149\pi$
$110$	−3.46658	−0.330526
$111$	−4.76413	−0.452191
$112$	3.77690	0.356884
$113$	−12.8103	−1.20509	−0.602544	−	0.798085i	$-0.705846\pi$
$113$	−12.8103	−1.20509	−0.602544	+	0.798085i	$0.705846\pi$
$114$	12.2330	1.14573
$115$	−6.19805	−0.577971
$116$	5.23837	0.486371
$117$	0.0208877	0.00193106
$118$	−10.0165	−0.922091
$119$	14.3030	1.31115
$120$	−6.82037	−0.622612
$121$	−10.2050	−0.927728
$122$	−2.93760	−0.265958
$123$	10.5338	0.949805
$124$	5.29808	0.475782
$125$	−19.8915	−1.77915
$126$	−0.292059	−0.0260187
$127$	0.461728	0.0409718	0.0204859	−	0.999790i	$-0.493479\pi$
$127$	0.461728	0.0409718	0.0204859	+	0.999790i	$0.493479\pi$
$128$	−1.00000	−0.0883883
$129$	−9.96849	−0.877676
$130$	1.05021	0.0921095
$131$	−6.70343	−0.585681	−0.292840	−	0.956161i	$-0.594600\pi$
$131$	−6.70343	−0.585681	−0.292840	+	0.956161i	$0.594600\pi$
$132$	1.56411	0.136138
$133$	26.3380	2.28379
$134$	−6.65517	−0.574919
$135$	−19.9337	−1.71562
$136$	−3.78696	−0.324729
$137$	12.1320	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$137$	12.1320	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$138$	2.79654	0.238057
$139$	17.9890	1.52580	0.762902	−	0.646514i	$-0.223774\pi$
$139$	17.9890	1.52580	0.762902	+	0.646514i	$0.223774\pi$
$140$	−14.6844	−1.24106
$141$	6.15249	0.518133
$142$	7.44458	0.624735
$143$	−0.240844	−0.0201404
$144$	0.0773276	0.00644397
$145$	−20.3666	−1.69135
$146$	0.533104	0.0441200
$147$	−12.7445	−1.05115
$148$	2.71579	0.223237
$149$	14.2877	1.17049	0.585245	−	0.810856i	$-0.300998\pi$
$149$	14.2877	1.17049	0.585245	+	0.810856i	$0.300998\pi$
$150$	17.7461	1.44897
$151$	−15.1888	−1.23605	−0.618023	−	0.786160i	$-0.712066\pi$
$151$	−15.1888	−1.23605	−0.618023	+	0.786160i	$0.712066\pi$
$152$	−6.97343	−0.565620
$153$	0.292837	0.0236744
$154$	3.36757	0.271366
$155$	−20.5987	−1.65453
$156$	−0.473851	−0.0379385
$157$	12.2721	0.979423	0.489711	−	0.871885i	$-0.337102\pi$
$157$	12.2721	0.979423	0.489711	+	0.871885i	$0.337102\pi$
$158$	−3.23891	−0.257674
$159$	−13.7526	−1.09065
$160$	3.88795	0.307370
$161$	6.02101	0.474522
$162$	9.22600	0.724863
$163$	−9.37968	−0.734673	−0.367336	−	0.930088i	$-0.619730\pi$
$163$	−9.37968	−0.734673	−0.367336	+	0.930088i	$0.619730\pi$
$164$	−6.00482	−0.468898
$165$	−6.08119	−0.473420
$166$	9.69483	0.752465
$167$	−12.3620	−0.956598	−0.478299	−	0.878197i	$-0.658747\pi$
$167$	−12.3620	−0.956598	−0.478299	+	0.878197i	$0.658747\pi$
$168$	6.62556	0.511173
$169$	−12.9270	−0.994387
$170$	14.7235	1.12924
$171$	0.539239	0.0412366
$172$	5.68254	0.433289
$173$	−3.83557	−0.291613	−0.145807	−	0.989313i	$-0.546578\pi$
$173$	−3.83557	−0.291613	−0.145807	+	0.989313i	$0.546578\pi$
$174$	9.18932	0.696640
$175$	38.2078	2.88824
$176$	−0.891621	−0.0672085
$177$	−17.5712	−1.32073
$178$	−15.0329	−1.12676
$179$	18.6789	1.39613	0.698065	−	0.716035i	$-0.254045\pi$
$179$	18.6789	1.39613	0.698065	+	0.716035i	$0.254045\pi$
$180$	−0.300646	−0.0224088
$181$	−5.94129	−0.441612	−0.220806	−	0.975318i	$-0.570869\pi$
$181$	−5.94129	−0.441612	−0.220806	+	0.975318i	$0.570869\pi$
$182$	−1.02021	−0.0756232
$183$	−5.15323	−0.380938
$184$	−1.59417	−0.117524
$185$	−10.5589	−0.776304
$186$	9.29407	0.681474
$187$	−3.37654	−0.246917
$188$	−3.50723	−0.255791
$189$	19.3643	1.40855
$190$	27.1124	1.96694
$191$	12.2179	0.884053	0.442027	−	0.897002i	$-0.354260\pi$
$191$	12.2179	0.884053	0.442027	+	0.897002i	$0.354260\pi$
$192$	−1.75423	−0.126601
$193$	10.0841	0.725871	0.362935	−	0.931814i	$-0.381775\pi$
$193$	10.0841	0.725871	0.362935	+	0.931814i	$0.381775\pi$
$194$	−4.42694	−0.317836
$195$	1.84231	0.131931
$196$	7.26499	0.518928
$197$	12.2085	0.869818	0.434909	−	0.900474i	$-0.356781\pi$
$197$	12.2085	0.869818	0.434909	+	0.900474i	$0.356781\pi$
$198$	0.0689469	0.00489984
$199$	−21.4321	−1.51928	−0.759640	−	0.650344i	$-0.774625\pi$
$199$	−21.4321	−1.51928	−0.759640	+	0.650344i	$0.774625\pi$
$200$	−10.1162	−0.715322
$201$	−11.6747	−0.823471
$202$	−4.94998	−0.348279
$203$	19.7848	1.38862
$204$	−6.64321	−0.465118
$205$	23.3465	1.63059
$206$	−4.25054	−0.296149
$207$	0.123273	0.00856807
$208$	0.270119	0.0187294
$209$	−6.21766	−0.430084
$210$	−25.7599	−1.77760
$211$	28.4055	1.95551	0.977756	−	0.209745i	$-0.0672634\pi$
$211$	28.4055	1.95551	0.977756	+	0.209745i	$0.0672634\pi$
$212$	7.83966	0.538430
$213$	13.0595	0.894823
$214$	8.47450	0.579305
$215$	−22.0934	−1.50676
$216$	−5.12704	−0.348851
$217$	20.0103	1.35839
$218$	4.94777	0.335105
$219$	0.935188	0.0631941
$220$	3.46658	0.233717
$221$	1.02293	0.0688098
$222$	4.76413	0.319747
$223$	−27.9759	−1.87340	−0.936701	−	0.350130i	$-0.886137\pi$
$223$	−27.9759	−1.87340	−0.936701	+	0.350130i	$0.886137\pi$
$224$	−3.77690	−0.252355
$225$	0.782260	0.0521507
$226$	12.8103	0.852126
$227$	−19.6923	−1.30703	−0.653513	−	0.756915i	$-0.726705\pi$
$227$	−19.6923	−1.30703	−0.653513	+	0.756915i	$0.726705\pi$
$228$	−12.2330	−0.810151
$229$	14.5783	0.963362	0.481681	−	0.876347i	$-0.340026\pi$
$229$	14.5783	0.963362	0.481681	+	0.876347i	$0.340026\pi$
$230$	6.19805	0.408687
$231$	5.90749	0.388684
$232$	−5.23837	−0.343916
$233$	12.2580	0.803049	0.401524	−	0.915848i	$-0.368480\pi$
$233$	12.2580	0.803049	0.401524	+	0.915848i	$0.368480\pi$
$234$	−0.0208877	−0.00136547
$235$	13.6359	0.889511
$236$	10.0165	0.652017
$237$	−5.68179	−0.369072
$238$	−14.3030	−0.927125
$239$	−4.81615	−0.311531	−0.155766	−	0.987794i	$-0.549784\pi$
$239$	−4.81615	−0.311531	−0.155766	+	0.987794i	$0.549784\pi$
$240$	6.82037	0.440253
$241$	−21.1441	−1.36201	−0.681005	−	0.732279i	$-0.738457\pi$
$241$	−21.1441	−1.36201	−0.681005	+	0.732279i	$0.738457\pi$
$242$	10.2050	0.656003
$243$	0.803413	0.0515390
$244$	2.93760	0.188061
$245$	−28.2459	−1.80457
$246$	−10.5338	−0.671614
$247$	1.88366	0.119854
$248$	−5.29808	−0.336429
$249$	17.0070	1.07777
$250$	19.8915	1.25805
$251$	−10.3012	−0.650205	−0.325102	−	0.945679i	$-0.605399\pi$
$251$	−10.3012	−0.650205	−0.325102	+	0.945679i	$0.605399\pi$
$252$	0.292059	0.0183980
$253$	−1.42139	−0.0893622
$254$	−0.461728	−0.0289714
$255$	25.8285	1.61744
$256$	1.00000	0.0625000
$257$	9.11865	0.568806	0.284403	−	0.958705i	$-0.408205\pi$
$257$	9.11865	0.568806	0.284403	+	0.958705i	$0.408205\pi$
$258$	9.96849	0.620611
$259$	10.2573	0.637357
$260$	−1.05021	−0.0651313
$261$	0.405071	0.0250732
$262$	6.70343	0.414139
$263$	0.0355164	0.00219004	0.00109502	−	0.999999i	$-0.499651\pi$
$263$	0.0355164	0.00219004	0.00109502	+	0.999999i	$0.499651\pi$
$264$	−1.56411	−0.0962643
$265$	−30.4803	−1.87239
$266$	−26.3380	−1.61488
$267$	−26.3712	−1.61389
$268$	6.65517	0.406529
$269$	−1.82249	−0.111119	−0.0555596	−	0.998455i	$-0.517694\pi$
$269$	−1.82249	−0.111119	−0.0555596	+	0.998455i	$0.517694\pi$
$270$	19.9337	1.21313
$271$	−16.9264	−1.02821	−0.514104	−	0.857728i	$-0.671875\pi$
$271$	−16.9264	−1.02821	−0.514104	+	0.857728i	$0.671875\pi$
$272$	3.78696	0.229618
$273$	−1.78969	−0.108317
$274$	−12.1320	−0.732921
$275$	−9.01981	−0.543915
$276$	−2.79654	−0.168332
$277$	−22.5995	−1.35787	−0.678937	−	0.734196i	$-0.737559\pi$
$277$	−22.5995	−1.35787	−0.678937	+	0.734196i	$0.737559\pi$
$278$	−17.9890	−1.07891
$279$	0.409688	0.0245274
$280$	14.6844	0.877562
$281$	−7.27928	−0.434245	−0.217123	−	0.976144i	$-0.569667\pi$
$281$	−7.27928	−0.434245	−0.217123	+	0.976144i	$0.569667\pi$
$282$	−6.15249	−0.366375
$283$	−11.9303	−0.709186	−0.354593	−	0.935021i	$-0.615380\pi$
$283$	−11.9303	−0.709186	−0.354593	+	0.935021i	$0.615380\pi$
$284$	−7.44458	−0.441755
$285$	47.5614	2.81729
$286$	0.240844	0.0142414
$287$	−22.6796	−1.33874
$288$	−0.0773276	−0.00455657
$289$	−2.65891	−0.156406
$290$	20.3666	1.19597
$291$	−7.76588	−0.455244
$292$	−0.533104	−0.0311975
$293$	7.81829	0.456749	0.228375	−	0.973573i	$-0.426659\pi$
$293$	7.81829	0.456749	0.228375	+	0.973573i	$0.426659\pi$
$294$	12.7445	0.743273
$295$	−38.9436	−2.26738
$296$	−2.71579	−0.157852
$297$	−4.57138	−0.265258
$298$	−14.2877	−0.827662
$299$	0.430615	0.0249031
$300$	−17.7461	−1.02457
$301$	21.4624	1.23707
$302$	15.1888	0.874016
$303$	−8.68341	−0.498849
$304$	6.97343	0.399954
$305$	−11.4213	−0.653979
$306$	−0.292837	−0.0167404
$307$	33.1253	1.89056	0.945280	−	0.326261i	$-0.105789\pi$
$307$	33.1253	1.89056	0.945280	+	0.326261i	$0.105789\pi$
$308$	−3.36757	−0.191885
$309$	−7.45644	−0.424182
$310$	20.5987	1.16993
$311$	−17.1099	−0.970214	−0.485107	−	0.874455i	$-0.661219\pi$
$311$	−17.1099	−0.970214	−0.485107	+	0.874455i	$0.661219\pi$
$312$	0.473851	0.0268265
$313$	2.05251	0.116015	0.0580074	−	0.998316i	$-0.481525\pi$
$313$	2.05251	0.116015	0.0580074	+	0.998316i	$0.481525\pi$
$314$	−12.2721	−0.692557
$315$	−1.13551	−0.0639788
$316$	3.23891	0.182203
$317$	15.0429	0.844892	0.422446	−	0.906388i	$-0.361172\pi$
$317$	15.0429	0.844892	0.422446	+	0.906388i	$0.361172\pi$
$318$	13.7526	0.771207
$319$	−4.67064	−0.261506
$320$	−3.88795	−0.217343
$321$	14.8662	0.829752
$322$	−6.02101	−0.335538
$323$	26.4081	1.46939
$324$	−9.22600	−0.512556
$325$	2.73257	0.151576
$326$	9.37968	0.519492
$327$	8.67953	0.479979
$328$	6.00482	0.331561
$329$	−13.2465	−0.730301
$330$	6.08119	0.334758
$331$	−20.6862	−1.13702	−0.568509	−	0.822677i	$-0.692480\pi$
$331$	−20.6862	−1.13702	−0.568509	+	0.822677i	$0.692480\pi$
$332$	−9.69483	−0.532073
$333$	0.210006	0.0115082
$334$	12.3620	0.676417
$335$	−25.8750	−1.41370
$336$	−6.62556	−0.361454
$337$	−24.8191	−1.35198	−0.675992	−	0.736909i	$-0.736284\pi$
$337$	−24.8191	−1.35198	−0.675992	+	0.736909i	$0.736284\pi$
$338$	12.9270	0.703138
$339$	22.4722	1.22052
$340$	−14.7235	−0.798496
$341$	−4.72388	−0.255813
$342$	−0.539239	−0.0291587
$343$	1.00084	0.0540400
$344$	−5.68254	−0.306382
$345$	10.8728	0.585373
$346$	3.83557	0.206202
$347$	29.5347	1.58551	0.792753	−	0.609543i	$-0.208647\pi$
$347$	29.5347	1.58551	0.792753	+	0.609543i	$0.208647\pi$
$348$	−9.18932	−0.492599
$349$	8.95922	0.479576	0.239788	−	0.970825i	$-0.422922\pi$
$349$	8.95922	0.479576	0.239788	+	0.970825i	$0.422922\pi$
$350$	−38.2078	−2.04230
$351$	1.38491	0.0739211
$352$	0.891621	0.0475236
$353$	−4.05140	−0.215634	−0.107817	−	0.994171i	$-0.534386\pi$
$353$	−4.05140	−0.215634	−0.107817	+	0.994171i	$0.534386\pi$
$354$	17.5712	0.933899
$355$	28.9442	1.53620
$356$	15.0329	0.796743
$357$	−25.0908	−1.32794
$358$	−18.6789	−0.987213
$359$	15.6541	0.826193	0.413096	−	0.910687i	$-0.364447\pi$
$359$	15.6541	0.826193	0.413096	+	0.910687i	$0.364447\pi$
$360$	0.300646	0.0158454
$361$	29.6288	1.55941
$362$	5.94129	0.312267
$363$	17.9020	0.939609
$364$	1.02021	0.0534737
$365$	2.07268	0.108489
$366$	5.15323	0.269364
$367$	26.1847	1.36683	0.683416	−	0.730029i	$-0.260494\pi$
$367$	26.1847	1.36683	0.683416	+	0.730029i	$0.260494\pi$
$368$	1.59417	0.0831017
$369$	−0.464338	−0.0241725
$370$	10.5589	0.548930
$371$	29.6096	1.53726
$372$	−9.29407	−0.481875
$373$	−14.0151	−0.725676	−0.362838	−	0.931852i	$-0.618192\pi$
$373$	−14.0151	−0.725676	−0.362838	+	0.931852i	$0.618192\pi$
$374$	3.37654	0.174597
$375$	34.8943	1.80193
$376$	3.50723	0.180871
$377$	1.41498	0.0728754
$378$	−19.3643	−0.995994
$379$	−5.24233	−0.269280	−0.134640	−	0.990895i	$-0.542988\pi$
$379$	−5.24233	−0.269280	−0.134640	+	0.990895i	$0.542988\pi$
$380$	−27.1124	−1.39084
$381$	−0.809978	−0.0414964
$382$	−12.2179	−0.625120
$383$	9.80693	0.501111	0.250555	−	0.968102i	$-0.419387\pi$
$383$	9.80693	0.501111	0.250555	+	0.968102i	$0.419387\pi$
$384$	1.75423	0.0895202
$385$	13.0929	0.667278
$386$	−10.0841	−0.513268
$387$	0.439417	0.0223368
$388$	4.42694	0.224744
$389$	−7.00766	−0.355303	−0.177651	−	0.984093i	$-0.556850\pi$
$389$	−7.00766	−0.355303	−0.177651	+	0.984093i	$0.556850\pi$
$390$	−1.84231	−0.0932891
$391$	6.03705	0.305307
$392$	−7.26499	−0.366937
$393$	11.7594	0.593181
$394$	−12.2085	−0.615054
$395$	−12.5927	−0.633609
$396$	−0.0689469	−0.00346471
$397$	17.3426	0.870398	0.435199	−	0.900334i	$-0.356678\pi$
$397$	17.3426	0.870398	0.435199	+	0.900334i	$0.356678\pi$
$398$	21.4321	1.07429
$399$	−46.2029	−2.31304
$400$	10.1162	0.505809
$401$	27.8421	1.39037	0.695185	−	0.718831i	$-0.255322\pi$
$401$	27.8421	1.39037	0.695185	+	0.718831i	$0.255322\pi$
$402$	11.6747	0.582282
$403$	1.43111	0.0712888
$404$	4.94998	0.246271
$405$	35.8703	1.78241
$406$	−19.7848	−0.981904
$407$	−2.42146	−0.120027
$408$	6.64321	0.328888
$409$	25.1732	1.24473	0.622367	−	0.782725i	$-0.286171\pi$
$409$	25.1732	1.24473	0.622367	+	0.782725i	$0.286171\pi$
$410$	−23.3465	−1.15300
$411$	−21.2823	−1.04978
$412$	4.25054	0.209409
$413$	37.8312	1.86155
$414$	−0.123273	−0.00605854
$415$	37.6931	1.85028
$416$	−0.270119	−0.0132437
$417$	−31.5568	−1.54534
$418$	6.21766	0.304116
$419$	−8.82035	−0.430902	−0.215451	−	0.976515i	$-0.569122\pi$
$419$	−8.82035	−0.430902	−0.215451	+	0.976515i	$0.569122\pi$
$420$	25.7599	1.25695
$421$	10.0679	0.490680	0.245340	−	0.969437i	$-0.421100\pi$
$421$	10.0679	0.490680	0.245340	+	0.969437i	$0.421100\pi$
$422$	−28.4055	−1.38276
$423$	−0.271205	−0.0131865
$424$	−7.83966	−0.380728
$425$	38.3096	1.85829
$426$	−13.0595	−0.632736
$427$	11.0950	0.536926
$428$	−8.47450	−0.409630
$429$	0.422496	0.0203983
$430$	22.0934	1.06544
$431$	−7.32818	−0.352986	−0.176493	−	0.984302i	$-0.556475\pi$
$431$	−7.32818	−0.352986	−0.176493	+	0.984302i	$0.556475\pi$
$432$	5.12704	0.246675
$433$	−19.4082	−0.932699	−0.466350	−	0.884600i	$-0.654431\pi$
$433$	−19.4082	−0.932699	−0.466350	+	0.884600i	$0.654431\pi$
$434$	−20.0103	−0.960527
$435$	35.7276	1.71301
$436$	−4.94777	−0.236955
$437$	11.1168	0.531790
$438$	−0.935188	−0.0446850
$439$	−9.96479	−0.475594	−0.237797	−	0.971315i	$-0.576425\pi$
$439$	−9.96479	−0.475594	−0.237797	+	0.971315i	$0.576425\pi$
$440$	−3.46658	−0.165263
$441$	0.561784	0.0267516
$442$	−1.02293	−0.0486559
$443$	−10.2194	−0.485538	−0.242769	−	0.970084i	$-0.578056\pi$
$443$	−10.2194	−0.485538	−0.242769	+	0.970084i	$0.578056\pi$
$444$	−4.76413	−0.226096
$445$	−58.4473	−2.77067
$446$	27.9759	1.32470
$447$	−25.0639	−1.18548
$448$	3.77690	0.178442
$449$	25.9723	1.22571	0.612855	−	0.790195i	$-0.290021\pi$
$449$	25.9723	1.22571	0.612855	+	0.790195i	$0.290021\pi$
$450$	−0.782260	−0.0368761
$451$	5.35403	0.252111
$452$	−12.8103	−0.602544
$453$	26.6446	1.25187
$454$	19.6923	0.924207
$455$	−3.96654	−0.185954
$456$	12.2330	0.572863
$457$	−9.99144	−0.467380	−0.233690	−	0.972311i	$-0.575080\pi$
$457$	−9.99144	−0.467380	−0.233690	+	0.972311i	$0.575080\pi$
$458$	−14.5783	−0.681200
$459$	19.4159	0.906258
$460$	−6.19805	−0.288986
$461$	31.9429	1.48773	0.743863	−	0.668332i	$-0.232991\pi$
$461$	31.9429	1.48773	0.743863	+	0.668332i	$0.232991\pi$
$462$	−5.90749	−0.274841
$463$	26.7749	1.24433	0.622167	−	0.782885i	$-0.286253\pi$
$463$	26.7749	1.24433	0.622167	+	0.782885i	$0.286253\pi$
$464$	5.23837	0.243185
$465$	36.1349	1.67572
$466$	−12.2580	−0.567841
$467$	−31.9630	−1.47907	−0.739535	−	0.673118i	$-0.764954\pi$
$467$	−31.9630	−1.47907	−0.739535	+	0.673118i	$0.764954\pi$
$468$	0.0208877	0.000965532 0
$469$	25.1359	1.16067
$470$	−13.6359	−0.628979
$471$	−21.5282	−0.991965
$472$	−10.0165	−0.461045
$473$	−5.06667	−0.232966
$474$	5.68179	0.260973
$475$	70.5446	3.23681
$476$	14.3030	0.655576
$477$	0.606222	0.0277570
$478$	4.81615	0.220286
$479$	14.5226	0.663553	0.331777	−	0.943358i	$-0.392352\pi$
$479$	14.5226	0.663553	0.331777	+	0.943358i	$0.392352\pi$
$480$	−6.82037	−0.311306
$481$	0.733587	0.0334487
$482$	21.1441	0.963086
$483$	−10.5623	−0.480599
$484$	−10.2050	−0.463864
$485$	−17.2117	−0.781545
$486$	−0.803413	−0.0364436
$487$	−11.8883	−0.538712	−0.269356	−	0.963041i	$-0.586811\pi$
$487$	−11.8883	−0.538712	−0.269356	+	0.963041i	$0.586811\pi$
$488$	−2.93760	−0.132979
$489$	16.4541	0.744081
$490$	28.2459	1.27602
$491$	−42.3600	−1.91168	−0.955841	−	0.293886i	$-0.905052\pi$
$491$	−42.3600	−1.91168	−0.955841	+	0.293886i	$0.905052\pi$
$492$	10.5338	0.474903
$493$	19.8375	0.893437
$494$	−1.88366	−0.0847497
$495$	0.268062	0.0120485
$496$	5.29808	0.237891
$497$	−28.1174	−1.26124
$498$	−17.0070	−0.762101
$499$	−2.53918	−0.113669	−0.0568347	−	0.998384i	$-0.518101\pi$
$499$	−2.53918	−0.113669	−0.0568347	+	0.998384i	$0.518101\pi$
$500$	−19.8915	−0.889575
$501$	21.6858	0.968848
$502$	10.3012	0.459764
$503$	17.5942	0.784484	0.392242	−	0.919862i	$-0.371700\pi$
$503$	17.5942	0.784484	0.392242	+	0.919862i	$0.371700\pi$
$504$	−0.292059	−0.0130093
$505$	−19.2453	−0.856404
$506$	1.42139	0.0631886
$507$	22.6770	1.00712
$508$	0.461728	0.0204859
$509$	3.83278	0.169885	0.0849425	−	0.996386i	$-0.472929\pi$
$509$	3.83278	0.169885	0.0849425	+	0.996386i	$0.472929\pi$
$510$	−25.8285	−1.14371
$511$	−2.01348	−0.0890712
$512$	−1.00000	−0.0441942
$513$	35.7531	1.57854
$514$	−9.11865	−0.402207
$515$	−16.5259	−0.728219
$516$	−9.96849	−0.438838
$517$	3.12712	0.137530
$518$	−10.2573	−0.450679
$519$	6.72848	0.295348
$520$	1.05021	0.0460548
$521$	−32.5902	−1.42780	−0.713901	−	0.700247i	$-0.753073\pi$
$521$	−32.5902	−1.42780	−0.713901	+	0.700247i	$0.753073\pi$
$522$	−0.405071	−0.0177295
$523$	9.55255	0.417704	0.208852	−	0.977947i	$-0.433027\pi$
$523$	9.55255	0.417704	0.208852	+	0.977947i	$0.433027\pi$
$524$	−6.70343	−0.292840
$525$	−67.0254	−2.92523
$526$	−0.0355164	−0.00154859
$527$	20.0637	0.873986
$528$	1.56411	0.0680691
$529$	−20.4586	−0.889506
$530$	30.4803	1.32398
$531$	0.774550	0.0336126
$532$	26.3380	1.14190
$533$	−1.62202	−0.0702573
$534$	26.3712	1.14119
$535$	32.9485	1.42449
$536$	−6.65517	−0.287460
$537$	−32.7672	−1.41401
$538$	1.82249	0.0785732
$539$	−6.47762	−0.279011
$540$	−19.9337	−0.857810
$541$	15.0495	0.647027	0.323514	−	0.946224i	$-0.395136\pi$
$541$	15.0495	0.647027	0.323514	+	0.946224i	$0.395136\pi$
$542$	16.9264	0.727053
$543$	10.4224	0.447268
$544$	−3.78696	−0.162365
$545$	19.2367	0.824009
$546$	1.78969	0.0765916
$547$	−20.7359	−0.886601	−0.443301	−	0.896373i	$-0.646193\pi$
$547$	−20.7359	−0.886601	−0.443301	+	0.896373i	$0.646193\pi$
$548$	12.1320	0.518254
$549$	0.227158	0.00969485
$550$	9.01981	0.384606
$551$	36.5294	1.55621
$552$	2.79654	0.119029
$553$	12.2330	0.520202
$554$	22.5995	0.960162
$555$	18.5227	0.786245
$556$	17.9890	0.762902
$557$	20.9110	0.886027	0.443014	−	0.896515i	$-0.353909\pi$
$557$	20.9110	0.886027	0.443014	+	0.896515i	$0.353909\pi$
$558$	−0.409688	−0.0173435
$559$	1.53496	0.0649219
$560$	−14.6844	−0.620530
$561$	5.92323	0.250079
$562$	7.27928	0.307058
$563$	18.2332	0.768437	0.384218	−	0.923242i	$-0.374471\pi$
$563$	18.2332	0.768437	0.384218	+	0.923242i	$0.374471\pi$
$564$	6.15249	0.259066
$565$	49.8057	2.09534
$566$	11.9303	0.501470
$567$	−34.8457	−1.46338
$568$	7.44458	0.312368
$569$	40.4829	1.69713	0.848565	−	0.529092i	$-0.177467\pi$
$569$	40.4829	1.69713	0.848565	+	0.529092i	$0.177467\pi$
$570$	−47.5614	−1.99213
$571$	43.5712	1.82340	0.911698	−	0.410860i	$-0.134771\pi$
$571$	43.5712	1.82340	0.911698	+	0.410860i	$0.134771\pi$
$572$	−0.240844	−0.0100702
$573$	−21.4330	−0.895375
$574$	22.6796	0.946629
$575$	16.1269	0.672538
$576$	0.0773276	0.00322198
$577$	16.9182	0.704314	0.352157	−	0.935941i	$-0.385448\pi$
$577$	16.9182	0.704314	0.352157	+	0.935941i	$0.385448\pi$
$578$	2.65891	0.110596
$579$	−17.6899	−0.735166
$580$	−20.3666	−0.845675
$581$	−36.6164	−1.51911
$582$	7.76588	0.321906
$583$	−6.99001	−0.289497
$584$	0.533104	0.0220600
$585$	−0.0812102	−0.00335763
$586$	−7.81829	−0.322970
$587$	40.5944	1.67551	0.837756	−	0.546045i	$-0.183867\pi$
$587$	40.5944	1.67551	0.837756	+	0.546045i	$0.183867\pi$
$588$	−12.7445	−0.525573
$589$	36.9458	1.52233
$590$	38.9436	1.60328
$591$	−21.4165	−0.880957
$592$	2.71579	0.111618
$593$	6.74010	0.276783	0.138391	−	0.990378i	$-0.455807\pi$
$593$	6.74010	0.276783	0.138391	+	0.990378i	$0.455807\pi$
$594$	4.57138	0.187566
$595$	−55.6094	−2.27976
$596$	14.2877	0.585245
$597$	37.5968	1.53874
$598$	−0.430615	−0.0176092
$599$	−25.1112	−1.02602	−0.513008	−	0.858384i	$-0.671469\pi$
$599$	−25.1112	−1.02602	−0.513008	+	0.858384i	$0.671469\pi$
$600$	17.7461	0.724483
$601$	28.4148	1.15906	0.579531	−	0.814950i	$-0.303236\pi$
$601$	28.4148	1.15906	0.579531	+	0.814950i	$0.303236\pi$
$602$	−21.4624	−0.874742
$603$	0.514628	0.0209573
$604$	−15.1888	−0.618023
$605$	39.6766	1.61308
$606$	8.68341	0.352739
$607$	0.920037	0.0373431	0.0186716	−	0.999826i	$-0.494056\pi$
$607$	0.920037	0.0373431	0.0186716	+	0.999826i	$0.494056\pi$
$608$	−6.97343	−0.282810
$609$	−34.7072	−1.40640
$610$	11.4213	0.462433
$611$	−0.947369	−0.0383264
$612$	0.292837	0.0118372
$613$	10.5382	0.425636	0.212818	−	0.977092i	$-0.431736\pi$
$613$	10.5382	0.425636	0.212818	+	0.977092i	$0.431736\pi$
$614$	−33.1253	−1.33683
$615$	−40.9551	−1.65147
$616$	3.36757	0.135683
$617$	17.8182	0.717332	0.358666	−	0.933466i	$-0.383232\pi$
$617$	17.8182	0.717332	0.358666	+	0.933466i	$0.383232\pi$
$618$	7.45644	0.299942
$619$	2.87281	0.115468	0.0577340	−	0.998332i	$-0.481612\pi$
$619$	2.87281	0.115468	0.0577340	+	0.998332i	$0.481612\pi$
$620$	−20.5987	−0.827264
$621$	8.17337	0.327986
$622$	17.1099	0.686045
$623$	56.7778	2.27476
$624$	−0.473851	−0.0189692
$625$	26.7563	1.07025
$626$	−2.05251	−0.0820348
$627$	10.9072	0.435592
$628$	12.2721	0.489711
$629$	10.2846	0.410074
$630$	1.13551	0.0452398
$631$	−6.33127	−0.252044	−0.126022	−	0.992027i	$-0.540221\pi$
$631$	−6.33127	−0.252044	−0.126022	+	0.992027i	$0.540221\pi$
$632$	−3.23891	−0.128837
$633$	−49.8297	−1.98055
$634$	−15.0429	−0.597429
$635$	−1.79518	−0.0712395
$636$	−13.7526	−0.545325
$637$	1.96241	0.0777536
$638$	4.67064	0.184913
$639$	−0.575671	−0.0227732
$640$	3.88795	0.153685
$641$	4.81431	0.190154	0.0950769	−	0.995470i	$-0.469690\pi$
$641$	4.81431	0.190154	0.0950769	+	0.995470i	$0.469690\pi$
$642$	−14.8662	−0.586723
$643$	19.0899	0.752833	0.376417	−	0.926451i	$-0.377156\pi$
$643$	19.0899	0.752833	0.376417	+	0.926451i	$0.377156\pi$
$644$	6.02101	0.237261
$645$	38.7570	1.52606
$646$	−26.4081	−1.03901
$647$	−13.3705	−0.525647	−0.262824	−	0.964844i	$-0.584654\pi$
$647$	−13.3705	−0.525647	−0.262824	+	0.964844i	$0.584654\pi$
$648$	9.22600	0.362432
$649$	−8.93090	−0.350568
$650$	−2.73257	−0.107180
$651$	−35.1028	−1.37579
$652$	−9.37968	−0.367336
$653$	2.41770	0.0946120	0.0473060	−	0.998880i	$-0.484936\pi$
$653$	2.41770	0.0946120	0.0473060	+	0.998880i	$0.484936\pi$
$654$	−8.67953	−0.339396
$655$	26.0626	1.01835
$656$	−6.00482	−0.234449
$657$	−0.0412236	−0.00160829
$658$	13.2465	0.516400
$659$	−44.4967	−1.73334	−0.866672	−	0.498878i	$-0.833745\pi$
$659$	−44.4967	−1.73334	−0.866672	+	0.498878i	$0.833745\pi$
$660$	−6.08119	−0.236710
$661$	−15.4372	−0.600438	−0.300219	−	0.953870i	$-0.597060\pi$
$661$	−15.4372	−0.600438	−0.300219	+	0.953870i	$0.597060\pi$
$662$	20.6862	0.803994
$663$	−1.79446	−0.0696909
$664$	9.69483	0.376232
$665$	−102.401	−3.97093
$666$	−0.210006	−0.00813756
$667$	8.35084	0.323346
$668$	−12.3620	−0.478299
$669$	49.0761	1.89739
$670$	25.8750	0.999639
$671$	−2.61923	−0.101114
$672$	6.62556	0.255587
$673$	−35.3144	−1.36127	−0.680635	−	0.732622i	$-0.738296\pi$
$673$	−35.3144	−1.36127	−0.680635	+	0.732622i	$0.738296\pi$
$674$	24.8191	0.955996
$675$	51.8661	1.99633
$676$	−12.9270	−0.497194
$677$	−5.70475	−0.219251	−0.109626	−	0.993973i	$-0.534965\pi$
$677$	−5.70475	−0.219251	−0.109626	+	0.993973i	$0.534965\pi$
$678$	−22.4722	−0.863039
$679$	16.7201	0.641660
$680$	14.7235	0.564622
$681$	34.5449	1.32376
$682$	4.72388	0.180887
$683$	2.39655	0.0917014	0.0458507	−	0.998948i	$-0.485400\pi$
$683$	2.39655	0.0917014	0.0458507	+	0.998948i	$0.485400\pi$
$684$	0.539239	0.0206183
$685$	−47.1687	−1.80222
$686$	−1.00084	−0.0382121
$687$	−25.5737	−0.975699
$688$	5.68254	0.216645
$689$	2.11764	0.0806757
$690$	−10.8728	−0.413921
$691$	9.08309	0.345537	0.172768	−	0.984962i	$-0.444729\pi$
$691$	9.08309	0.345537	0.172768	+	0.984962i	$0.444729\pi$
$692$	−3.83557	−0.145807
$693$	−0.260406	−0.00989199
$694$	−29.5347	−1.12112
$695$	−69.9403	−2.65299
$696$	9.18932	0.348320
$697$	−22.7400	−0.861341
$698$	−8.95922	−0.339111
$699$	−21.5034	−0.813333
$700$	38.2078	1.44412
$701$	−28.3666	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$701$	−28.3666	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$702$	−1.38491	−0.0522701
$703$	18.9384	0.714275
$704$	−0.891621	−0.0336042
$705$	−23.9206	−0.900902
$706$	4.05140	0.152477
$707$	18.6956	0.703120
$708$	−17.5712	−0.660366
$709$	−36.9497	−1.38768	−0.693838	−	0.720131i	$-0.744082\pi$
$709$	−36.9497	−1.38768	−0.693838	+	0.720131i	$0.744082\pi$
$710$	−28.9442	−1.08626
$711$	0.250457	0.00939287
$712$	−15.0329	−0.563382
$713$	8.44603	0.316306
$714$	25.0908	0.938998
$715$	0.936390	0.0350190
$716$	18.6789	0.698065
$717$	8.44865	0.315521
$718$	−15.6541	−0.584206
$719$	−30.3703	−1.13262	−0.566310	−	0.824192i	$-0.691629\pi$
$719$	−30.3703	−1.13262	−0.566310	+	0.824192i	$0.691629\pi$
$720$	−0.300646	−0.0112044
$721$	16.0539	0.597878
$722$	−29.6288	−1.10267
$723$	37.0916	1.37945
$724$	−5.94129	−0.220806
$725$	52.9924	1.96809
$726$	−17.9020	−0.664404
$727$	21.1943	0.786054	0.393027	−	0.919527i	$-0.371428\pi$
$727$	21.1943	0.786054	0.393027	+	0.919527i	$0.371428\pi$
$728$	−1.02021	−0.0378116
$729$	26.2686	0.972912
$730$	−2.07268	−0.0767135
$731$	21.5196	0.795930
$732$	−5.15323	−0.190469
$733$	−36.8248	−1.36016	−0.680078	−	0.733140i	$-0.738054\pi$
$733$	−36.8248	−1.36016	−0.680078	+	0.733140i	$0.738054\pi$
$734$	−26.1847	−0.966496
$735$	49.5499	1.82768
$736$	−1.59417	−0.0587618
$737$	−5.93389	−0.218578
$738$	0.464338	0.0170925
$739$	8.36301	0.307638	0.153819	−	0.988099i	$-0.450843\pi$
$739$	8.36301	0.307638	0.153819	+	0.988099i	$0.450843\pi$
$740$	−10.5589	−0.388152
$741$	−3.30437	−0.121389
$742$	−29.6096	−1.08700
$743$	43.8790	1.60976	0.804882	−	0.593435i	$-0.202229\pi$
$743$	43.8790	1.60976	0.804882	+	0.593435i	$0.202229\pi$
$744$	9.29407	0.340737
$745$	−55.5497	−2.03519
$746$	14.0151	0.513131
$747$	−0.749678	−0.0274293
$748$	−3.37654	−0.123458
$749$	−32.0073	−1.16952
$750$	−34.8943	−1.27416
$751$	49.2440	1.79694	0.898470	−	0.439034i	$-0.144679\pi$
$751$	49.2440	1.79694	0.898470	+	0.439034i	$0.144679\pi$
$752$	−3.50723	−0.127895
$753$	18.0707	0.658531
$754$	−1.41498	−0.0515307
$755$	59.0533	2.14917
$756$	19.3643	0.704274
$757$	−3.85726	−0.140195	−0.0700973	−	0.997540i	$-0.522331\pi$
$757$	−3.85726	−0.140195	−0.0700973	+	0.997540i	$0.522331\pi$
$758$	5.24233	0.190410
$759$	2.49345	0.0905066
$760$	27.1124	0.983470
$761$	17.2533	0.625432	0.312716	−	0.949847i	$-0.398761\pi$
$761$	17.2533	0.625432	0.312716	+	0.949847i	$0.398761\pi$
$762$	0.809978	0.0293424
$763$	−18.6872	−0.676523
$764$	12.2179	0.442027
$765$	−1.13854	−0.0411639
$766$	−9.80693	−0.354339
$767$	2.70564	0.0976950
$768$	−1.75423	−0.0633004
$769$	17.2644	0.622569	0.311285	−	0.950317i	$-0.399241\pi$
$769$	17.2644	0.622569	0.311285	+	0.950317i	$0.399241\pi$
$770$	−13.0929	−0.471837
$771$	−15.9962	−0.576090
$772$	10.0841	0.362935
$773$	−46.3937	−1.66866	−0.834332	−	0.551262i	$-0.814146\pi$
$773$	−46.3937	−1.66866	−0.834332	+	0.551262i	$0.814146\pi$
$774$	−0.439417	−0.0157945
$775$	53.5964	1.92524
$776$	−4.42694	−0.158918
$777$	−17.9936	−0.645519
$778$	7.00766	0.251237
$779$	−41.8742	−1.50030
$780$	1.84231	0.0659653
$781$	6.63774	0.237517
$782$	−6.03705	−0.215885
$783$	26.8574	0.959804
$784$	7.26499	0.259464
$785$	−47.7135	−1.70297
$786$	−11.7594	−0.419442
$787$	25.6917	0.915809	0.457905	−	0.889001i	$-0.348600\pi$
$787$	25.6917	0.915809	0.457905	+	0.889001i	$0.348600\pi$
$788$	12.2085	0.434909
$789$	−0.0623040	−0.00221808
$790$	12.5927	0.448029
$791$	−48.3831	−1.72031
$792$	0.0689469	0.00244992
$793$	0.793502	0.0281781
$794$	−17.3426	−0.615464
$795$	53.4694	1.89636
$796$	−21.4321	−0.759640
$797$	−17.1929	−0.609003	−0.304502	−	0.952512i	$-0.598490\pi$
$797$	−17.1929	−0.609003	−0.304502	+	0.952512i	$0.598490\pi$
$798$	46.2029	1.63556
$799$	−13.2817	−0.469874
$800$	−10.1162	−0.357661
$801$	1.16246	0.0410735
$802$	−27.8421	−0.983140
$803$	0.475327	0.0167739
$804$	−11.6747	−0.411735
$805$	−23.4094	−0.825074
$806$	−1.43111	−0.0504088
$807$	3.19707	0.112542
$808$	−4.94998	−0.174140
$809$	16.3961	0.576456	0.288228	−	0.957562i	$-0.406934\pi$
$809$	16.3961	0.576456	0.288228	+	0.957562i	$0.406934\pi$
$810$	−35.8703	−1.26035
$811$	−15.8166	−0.555397	−0.277698	−	0.960668i	$-0.589572\pi$
$811$	−15.8166	−0.555397	−0.277698	+	0.960668i	$0.589572\pi$
$812$	19.7848	0.694311
$813$	29.6929	1.04138
$814$	2.42146	0.0848721
$815$	36.4678	1.27741
$816$	−6.64321	−0.232559
$817$	39.6268	1.38637
$818$	−25.1732	−0.880161
$819$	0.0788906	0.00275666
$820$	23.3465	0.815294
$821$	22.8069	0.795968	0.397984	−	0.917392i	$-0.369710\pi$
$821$	22.8069	0.795968	0.397984	+	0.917392i	$0.369710\pi$
$822$	21.2823	0.742307
$823$	17.2997	0.603031	0.301516	−	0.953461i	$-0.402507\pi$
$823$	17.2997	0.603031	0.301516	+	0.953461i	$0.402507\pi$
$824$	−4.25054	−0.148075
$825$	15.8228	0.550880
$826$	−37.8312	−1.31632
$827$	−13.5049	−0.469611	−0.234806	−	0.972042i	$-0.575445\pi$
$827$	−13.5049	−0.469611	−0.234806	+	0.972042i	$0.575445\pi$
$828$	0.123273	0.00428404
$829$	0.0781965	0.00271588	0.00135794	−	0.999999i	$-0.499568\pi$
$829$	0.0781965	0.00271588	0.00135794	+	0.999999i	$0.499568\pi$
$830$	−37.6931	−1.30835
$831$	39.6448	1.37526
$832$	0.270119	0.00936469
$833$	27.5122	0.953243
$834$	31.5568	1.09272
$835$	48.0628	1.66328
$836$	−6.21766	−0.215042
$837$	27.1635	0.938908
$838$	8.82035	0.304694
$839$	7.63850	0.263710	0.131855	−	0.991269i	$-0.457907\pi$
$839$	7.63850	0.263710	0.131855	+	0.991269i	$0.457907\pi$
$840$	−25.7599	−0.888800
$841$	−1.55945	−0.0537742
$842$	−10.0679	−0.346963
$843$	12.7695	0.439806
$844$	28.4055	0.977756
$845$	50.2597	1.72899
$846$	0.271205	0.00932423
$847$	−38.5433	−1.32436
$848$	7.83966	0.269215
$849$	20.9286	0.718267
$850$	−38.3096	−1.31401
$851$	4.32943	0.148411
$852$	13.0595	0.447412
$853$	−19.5560	−0.669586	−0.334793	−	0.942292i	$-0.608666\pi$
$853$	−19.5560	−0.669586	−0.334793	+	0.942292i	$0.608666\pi$
$854$	−11.0950	−0.379664
$855$	−2.09654	−0.0717000
$856$	8.47450	0.289652
$857$	−23.7702	−0.811974	−0.405987	−	0.913879i	$-0.633072\pi$
$857$	−23.7702	−0.811974	−0.405987	+	0.913879i	$0.633072\pi$
$858$	−0.422496	−0.0144238
$859$	11.5432	0.393849	0.196925	−	0.980419i	$-0.436905\pi$
$859$	11.5432	0.393849	0.196925	+	0.980419i	$0.436905\pi$
$860$	−22.0934	−0.753380
$861$	39.7853	1.35588
$862$	7.32818	0.249599
$863$	45.2584	1.54061	0.770307	−	0.637674i	$-0.220103\pi$
$863$	45.2584	1.54061	0.770307	+	0.637674i	$0.220103\pi$
$864$	−5.12704	−0.174426
$865$	14.9125	0.507041
$866$	19.4082	0.659518
$867$	4.66434	0.158409
$868$	20.0103	0.679195
$869$	−2.88788	−0.0979646
$870$	−35.7276	−1.21128
$871$	1.79769	0.0609123
$872$	4.94777	0.167553
$873$	0.342325	0.0115859
$874$	−11.1168	−0.376032
$875$	−75.1283	−2.53980
$876$	0.935188	0.0315971
$877$	17.3178	0.584780	0.292390	−	0.956299i	$-0.405549\pi$
$877$	17.3178	0.584780	0.292390	+	0.956299i	$0.405549\pi$
$878$	9.96479	0.336296
$879$	−13.7151	−0.462598
$880$	3.46658	0.116858
$881$	35.0018	1.17924	0.589621	−	0.807680i	$-0.299277\pi$
$881$	35.0018	1.17924	0.589621	+	0.807680i	$0.299277\pi$
$882$	−0.561784	−0.0189163
$883$	41.5003	1.39660	0.698298	−	0.715807i	$-0.253941\pi$
$883$	41.5003	1.39660	0.698298	+	0.715807i	$0.253941\pi$
$884$	1.02293	0.0344049
$885$	68.3160	2.29642
$886$	10.2194	0.343327
$887$	35.5633	1.19410	0.597049	−	0.802205i	$-0.296340\pi$
$887$	35.5633	1.19410	0.597049	+	0.802205i	$0.296340\pi$
$888$	4.76413	0.159874
$889$	1.74390	0.0584886
$890$	58.4473	1.95916
$891$	8.22610	0.275585
$892$	−27.9759	−0.936701
$893$	−24.4574	−0.818436
$894$	25.0639	0.838261
$895$	−72.6229	−2.42751
$896$	−3.77690	−0.126177
$897$	−0.755398	−0.0252220
$898$	−25.9723	−0.866708
$899$	27.7533	0.925626
$900$	0.782260	0.0260753
$901$	29.6885	0.989068
$902$	−5.35403	−0.178270
$903$	−37.6500	−1.25291
$904$	12.8103	0.426063
$905$	23.0995	0.767852
$906$	−26.6446	−0.885209
$907$	1.01620	0.0337422	0.0168711	−	0.999858i	$-0.494630\pi$
$907$	1.01620	0.0337422	0.0168711	+	0.999858i	$0.494630\pi$
$908$	−19.6923	−0.653513
$909$	0.382770	0.0126957
$910$	3.96654	0.131490
$911$	−37.6952	−1.24890	−0.624449	−	0.781066i	$-0.714676\pi$
$911$	−37.6952	−1.24890	−0.624449	+	0.781066i	$0.714676\pi$
$912$	−12.2330	−0.405076
$913$	8.64412	0.286079
$914$	9.99144	0.330488
$915$	20.0355	0.662354
$916$	14.5783	0.481681
$917$	−25.3182	−0.836080
$918$	−19.4159	−0.640821
$919$	20.3518	0.671343	0.335672	−	0.941979i	$-0.391037\pi$
$919$	20.3518	0.671343	0.335672	+	0.941979i	$0.391037\pi$
$920$	6.19805	0.204344
$921$	−58.1094	−1.91477
$922$	−31.9429	−1.05198
$923$	−2.01092	−0.0661903
$924$	5.90749	0.194342
$925$	27.4735	0.903322
$926$	−26.7749	−0.879877
$927$	0.328684	0.0107954
$928$	−5.23837	−0.171958
$929$	−44.9593	−1.47507	−0.737533	−	0.675311i	$-0.764009\pi$
$929$	−44.9593	−1.47507	−0.737533	+	0.675311i	$0.764009\pi$
$930$	−36.1349	−1.18491
$931$	50.6619	1.66038
$932$	12.2580	0.401524
$933$	30.0147	0.982639
$934$	31.9630	1.04586
$935$	13.1278	0.429326
$936$	−0.0208877	−0.000682734 0
$937$	3.61097	0.117965	0.0589827	−	0.998259i	$-0.481214\pi$
$937$	3.61097	0.117965	0.0589827	+	0.998259i	$0.481214\pi$
$938$	−25.1359	−0.820717
$939$	−3.60058	−0.117500
$940$	13.6359	0.444755
$941$	−35.0560	−1.14279	−0.571397	−	0.820674i	$-0.693598\pi$
$941$	−35.0560	−1.14279	−0.571397	+	0.820674i	$0.693598\pi$
$942$	21.5282	0.701425
$943$	−9.57269	−0.311730
$944$	10.0165	0.326008
$945$	−75.2877	−2.44911
$946$	5.06667	0.164732
$947$	−16.0441	−0.521364	−0.260682	−	0.965425i	$-0.583947\pi$
$947$	−16.0441	−0.521364	−0.260682	+	0.965425i	$0.583947\pi$
$948$	−5.68179	−0.184536
$949$	−0.144002	−0.00467449
$950$	−70.5446	−2.28877
$951$	−26.3887	−0.855711
$952$	−14.3030	−0.463563
$953$	44.1901	1.43146	0.715729	−	0.698378i	$-0.246095\pi$
$953$	44.1901	1.43146	0.715729	+	0.698378i	$0.246095\pi$
$954$	−0.606222	−0.0196272
$955$	−47.5025	−1.53714
$956$	−4.81615	−0.155766
$957$	8.19339	0.264855
$958$	−14.5226	−0.469203
$959$	45.8214	1.47965
$960$	6.82037	0.220127
$961$	−2.93030	−0.0945258
$962$	−0.733587	−0.0236518
$963$	−0.655312	−0.0211171
$964$	−21.1441	−0.681005
$965$	−39.2066	−1.26211
$966$	10.5623	0.339835
$967$	−11.4345	−0.367710	−0.183855	−	0.982953i	$-0.558858\pi$
$967$	−11.4345	−0.367710	−0.183855	+	0.982953i	$0.558858\pi$
$968$	10.2050	0.328002
$969$	−46.3260	−1.48820
$970$	17.2117	0.552636
$971$	35.2985	1.13278	0.566391	−	0.824136i	$-0.308339\pi$
$971$	35.2985	1.13278	0.566391	+	0.824136i	$0.308339\pi$
$972$	0.803413	0.0257695
$973$	67.9426	2.17814
$974$	11.8883	0.380927
$975$	−4.79357	−0.153517
$976$	2.93760	0.0940303
$977$	−32.2080	−1.03043	−0.515213	−	0.857062i	$-0.672287\pi$
$977$	−32.2080	−1.03043	−0.515213	+	0.857062i	$0.672287\pi$
$978$	−16.4541	−0.526145
$979$	−13.4037	−0.428383
$980$	−28.2459	−0.902284
$981$	−0.382599	−0.0122154
$982$	42.3600	1.35176
$983$	−29.2237	−0.932092	−0.466046	−	0.884760i	$-0.654322\pi$
$983$	−29.2237	−0.932092	−0.466046	+	0.884760i	$0.654322\pi$
$984$	−10.5338	−0.335807
$985$	−47.4660	−1.51239
$986$	−19.8375	−0.631756
$987$	23.2373	0.739653
$988$	1.88366	0.0599271
$989$	9.05892	0.288057
$990$	−0.268062	−0.00851958
$991$	−24.7241	−0.785386	−0.392693	−	0.919670i	$-0.628457\pi$
$991$	−24.7241	−0.785386	−0.392693	+	0.919670i	$0.628457\pi$
$992$	−5.29808	−0.168214
$993$	36.2885	1.15158
$994$	28.1174	0.891831
$995$	83.3270	2.64164
$996$	17.0070	0.538887
$997$	−53.0189	−1.67912	−0.839562	−	0.543263i	$-0.817188\pi$
$997$	−53.0189	−1.67912	−0.839562	+	0.543263i	$0.817188\pi$
$998$	2.53918	0.0803764
$999$	13.9240	0.440535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.20	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.20	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.75423 q^{3} +1.00000 q^{4} -3.88795 q^{5} +1.75423 q^{6} +3.77690 q^{7} -1.00000 q^{8} +0.0773276 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.75423 q^{3} +1.00000 q^{4} -3.88795 q^{5} +1.75423 q^{6} +3.77690 q^{7} -1.00000 q^{8} +0.0773276 q^{9} +3.88795 q^{10} -0.891621 q^{11} -1.75423 q^{12} +0.270119 q^{13} -3.77690 q^{14} +6.82037 q^{15} +1.00000 q^{16} +3.78696 q^{17} -0.0773276 q^{18} +6.97343 q^{19} -3.88795 q^{20} -6.62556 q^{21} +0.891621 q^{22} +1.59417 q^{23} +1.75423 q^{24} +10.1162 q^{25} -0.270119 q^{26} +5.12704 q^{27} +3.77690 q^{28} +5.23837 q^{29} -6.82037 q^{30} +5.29808 q^{31} -1.00000 q^{32} +1.56411 q^{33} -3.78696 q^{34} -14.6844 q^{35} +0.0773276 q^{36} +2.71579 q^{37} -6.97343 q^{38} -0.473851 q^{39} +3.88795 q^{40} -6.00482 q^{41} +6.62556 q^{42} +5.68254 q^{43} -0.891621 q^{44} -0.300646 q^{45} -1.59417 q^{46} -3.50723 q^{47} -1.75423 q^{48} +7.26499 q^{49} -10.1162 q^{50} -6.64321 q^{51} +0.270119 q^{52} +7.83966 q^{53} -5.12704 q^{54} +3.46658 q^{55} -3.77690 q^{56} -12.2330 q^{57} -5.23837 q^{58} +10.0165 q^{59} +6.82037 q^{60} +2.93760 q^{61} -5.29808 q^{62} +0.292059 q^{63} +1.00000 q^{64} -1.05021 q^{65} -1.56411 q^{66} +6.65517 q^{67} +3.78696 q^{68} -2.79654 q^{69} +14.6844 q^{70} -7.44458 q^{71} -0.0773276 q^{72} -0.533104 q^{73} -2.71579 q^{74} -17.7461 q^{75} +6.97343 q^{76} -3.36757 q^{77} +0.473851 q^{78} +3.23891 q^{79} -3.88795 q^{80} -9.22600 q^{81} +6.00482 q^{82} -9.69483 q^{83} -6.62556 q^{84} -14.7235 q^{85} -5.68254 q^{86} -9.18932 q^{87} +0.891621 q^{88} +15.0329 q^{89} +0.300646 q^{90} +1.02021 q^{91} +1.59417 q^{92} -9.29407 q^{93} +3.50723 q^{94} -27.1124 q^{95} +1.75423 q^{96} +4.42694 q^{97} -7.26499 q^{98} -0.0689469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 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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