LMFDB - Embedded newform 8042.2.a.a.1.37

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:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.37
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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.176865 q^{3} +1.00000 q^{4} -4.18491 q^{5} +0.176865 q^{6} -4.23014 q^{7} +1.00000 q^{8} -2.96872 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.176865 q^{3} +1.00000 q^{4} -4.18491 q^{5} +0.176865 q^{6} -4.23014 q^{7} +1.00000 q^{8} -2.96872 q^{9} -4.18491 q^{10} +3.02812 q^{11} +0.176865 q^{12} +0.660935 q^{13} -4.23014 q^{14} -0.740165 q^{15} +1.00000 q^{16} +0.251301 q^{17} -2.96872 q^{18} +4.59953 q^{19} -4.18491 q^{20} -0.748164 q^{21} +3.02812 q^{22} +6.65567 q^{23} +0.176865 q^{24} +12.5135 q^{25} +0.660935 q^{26} -1.05566 q^{27} -4.23014 q^{28} +4.32449 q^{29} -0.740165 q^{30} +2.59280 q^{31} +1.00000 q^{32} +0.535568 q^{33} +0.251301 q^{34} +17.7028 q^{35} -2.96872 q^{36} -8.50000 q^{37} +4.59953 q^{38} +0.116896 q^{39} -4.18491 q^{40} +2.36142 q^{41} -0.748164 q^{42} +0.881347 q^{43} +3.02812 q^{44} +12.4238 q^{45} +6.65567 q^{46} -8.59990 q^{47} +0.176865 q^{48} +10.8941 q^{49} +12.5135 q^{50} +0.0444463 q^{51} +0.660935 q^{52} -9.09907 q^{53} -1.05566 q^{54} -12.6724 q^{55} -4.23014 q^{56} +0.813495 q^{57} +4.32449 q^{58} +2.05038 q^{59} -0.740165 q^{60} -2.20500 q^{61} +2.59280 q^{62} +12.5581 q^{63} +1.00000 q^{64} -2.76595 q^{65} +0.535568 q^{66} -2.78495 q^{67} +0.251301 q^{68} +1.17715 q^{69} +17.7028 q^{70} -10.8740 q^{71} -2.96872 q^{72} -3.99573 q^{73} -8.50000 q^{74} +2.21320 q^{75} +4.59953 q^{76} -12.8094 q^{77} +0.116896 q^{78} -3.04731 q^{79} -4.18491 q^{80} +8.71945 q^{81} +2.36142 q^{82} +7.29547 q^{83} -0.748164 q^{84} -1.05167 q^{85} +0.881347 q^{86} +0.764850 q^{87} +3.02812 q^{88} +10.3098 q^{89} +12.4238 q^{90} -2.79585 q^{91} +6.65567 q^{92} +0.458575 q^{93} -8.59990 q^{94} -19.2486 q^{95} +0.176865 q^{96} +6.10127 q^{97} +10.8941 q^{98} -8.98964 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	0.176865	0.102113	0.0510565	−	0.998696i	$-0.483741\pi$
$3$	0.176865	0.102113	0.0510565	+	0.998696i	$0.483741\pi$
$4$	1.00000	0.500000
$5$	−4.18491	−1.87155	−0.935775	−	0.352598i	$-0.885298\pi$
$5$	−4.18491	−1.87155	−0.935775	+	0.352598i	$0.885298\pi$
$6$	0.176865	0.0722048
$7$	−4.23014	−1.59884	−0.799422	−	0.600770i	$-0.794861\pi$
$7$	−4.23014	−1.59884	−0.799422	+	0.600770i	$0.794861\pi$
$8$	1.00000	0.353553
$9$	−2.96872	−0.989573
$10$	−4.18491	−1.32339
$11$	3.02812	0.913012	0.456506	−	0.889720i	$-0.349101\pi$
$11$	3.02812	0.913012	0.456506	+	0.889720i	$0.349101\pi$
$12$	0.176865	0.0510565
$13$	0.660935	0.183310	0.0916552	−	0.995791i	$-0.470784\pi$
$13$	0.660935	0.183310	0.0916552	+	0.995791i	$0.470784\pi$
$14$	−4.23014	−1.13055
$15$	−0.740165	−0.191110
$16$	1.00000	0.250000
$17$	0.251301	0.0609494	0.0304747	−	0.999536i	$-0.490298\pi$
$17$	0.251301	0.0609494	0.0304747	+	0.999536i	$0.490298\pi$
$18$	−2.96872	−0.699734
$19$	4.59953	1.05520	0.527602	−	0.849492i	$-0.323091\pi$
$19$	4.59953	1.05520	0.527602	+	0.849492i	$0.323091\pi$
$20$	−4.18491	−0.935775
$21$	−0.748164	−0.163263
$22$	3.02812	0.645597
$23$	6.65567	1.38780	0.693902	−	0.720070i	$-0.255890\pi$
$23$	6.65567	1.38780	0.693902	+	0.720070i	$0.255890\pi$
$24$	0.176865	0.0361024
$25$	12.5135	2.50270
$26$	0.660935	0.129620
$27$	−1.05566	−0.203161
$28$	−4.23014	−0.799422
$29$	4.32449	0.803037	0.401518	−	0.915851i	$-0.368483\pi$
$29$	4.32449	0.803037	0.401518	+	0.915851i	$0.368483\pi$
$30$	−0.740165	−0.135135
$31$	2.59280	0.465681	0.232840	−	0.972515i	$-0.425198\pi$
$31$	2.59280	0.465681	0.232840	+	0.972515i	$0.425198\pi$
$32$	1.00000	0.176777
$33$	0.535568	0.0932305
$34$	0.251301	0.0430978
$35$	17.7028	2.99232
$36$	−2.96872	−0.494786
$37$	−8.50000	−1.39739	−0.698696	−	0.715419i	$-0.746236\pi$
$37$	−8.50000	−1.39739	−0.698696	+	0.715419i	$0.746236\pi$
$38$	4.59953	0.746142
$39$	0.116896	0.0187184
$40$	−4.18491	−0.661693
$41$	2.36142	0.368793	0.184396	−	0.982852i	$-0.440967\pi$
$41$	2.36142	0.368793	0.184396	+	0.982852i	$0.440967\pi$
$42$	−0.748164	−0.115444
$43$	0.881347	0.134404	0.0672021	−	0.997739i	$-0.478593\pi$
$43$	0.881347	0.134404	0.0672021	+	0.997739i	$0.478593\pi$
$44$	3.02812	0.456506
$45$	12.4238	1.85204
$46$	6.65567	0.981325
$47$	−8.59990	−1.25442	−0.627212	−	0.778848i	$-0.715804\pi$
$47$	−8.59990	−1.25442	−0.627212	+	0.778848i	$0.715804\pi$
$48$	0.176865	0.0255283
$49$	10.8941	1.55630
$50$	12.5135	1.76968
$51$	0.0444463	0.00622373
$52$	0.660935	0.0916552
$53$	−9.09907	−1.24985	−0.624927	−	0.780684i	$-0.714871\pi$
$53$	−9.09907	−1.24985	−0.624927	+	0.780684i	$0.714871\pi$
$54$	−1.05566	−0.143657
$55$	−12.6724	−1.70875
$56$	−4.23014	−0.565277
$57$	0.813495	0.107750
$58$	4.32449	0.567833
$59$	2.05038	0.266937	0.133468	−	0.991053i	$-0.457389\pi$
$59$	2.05038	0.266937	0.133468	+	0.991053i	$0.457389\pi$
$60$	−0.740165	−0.0955548
$61$	−2.20500	−0.282321	−0.141160	−	0.989987i	$-0.545083\pi$
$61$	−2.20500	−0.282321	−0.141160	+	0.989987i	$0.545083\pi$
$62$	2.59280	0.329286
$63$	12.5581	1.58217
$64$	1.00000	0.125000
$65$	−2.76595	−0.343074
$66$	0.535568	0.0659239
$67$	−2.78495	−0.340236	−0.170118	−	0.985424i	$-0.554415\pi$
$67$	−2.78495	−0.340236	−0.170118	+	0.985424i	$0.554415\pi$
$68$	0.251301	0.0304747
$69$	1.17715	0.141713
$70$	17.7028	2.11589
$71$	−10.8740	−1.29050	−0.645252	−	0.763970i	$-0.723247\pi$
$71$	−10.8740	−1.29050	−0.645252	+	0.763970i	$0.723247\pi$
$72$	−2.96872	−0.349867
$73$	−3.99573	−0.467665	−0.233832	−	0.972277i	$-0.575127\pi$
$73$	−3.99573	−0.467665	−0.233832	+	0.972277i	$0.575127\pi$
$74$	−8.50000	−0.988105
$75$	2.21320	0.255558
$76$	4.59953	0.527602
$77$	−12.8094	−1.45976
$78$	0.116896	0.0132359
$79$	−3.04731	−0.342849	−0.171425	−	0.985197i	$-0.554837\pi$
$79$	−3.04731	−0.342849	−0.171425	+	0.985197i	$0.554837\pi$
$80$	−4.18491	−0.467887
$81$	8.71945	0.968828
$82$	2.36142	0.260776
$83$	7.29547	0.800782	0.400391	−	0.916344i	$-0.368874\pi$
$83$	7.29547	0.800782	0.400391	+	0.916344i	$0.368874\pi$
$84$	−0.748164	−0.0816314
$85$	−1.05167	−0.114070
$86$	0.881347	0.0950381
$87$	0.764850	0.0820005
$88$	3.02812	0.322799
$89$	10.3098	1.09283	0.546416	−	0.837514i	$-0.315992\pi$
$89$	10.3098	1.09283	0.546416	+	0.837514i	$0.315992\pi$
$90$	12.4238	1.30959
$91$	−2.79585	−0.293085
$92$	6.65567	0.693902
$93$	0.458575	0.0475521
$94$	−8.59990	−0.887012
$95$	−19.2486	−1.97487
$96$	0.176865	0.0180512
$97$	6.10127	0.619491	0.309745	−	0.950820i	$-0.399756\pi$
$97$	6.10127	0.619491	0.309745	+	0.950820i	$0.399756\pi$
$98$	10.8941	1.10047
$99$	−8.98964	−0.903492
$100$	12.5135	1.25135
$101$	1.65699	0.164877	0.0824383	−	0.996596i	$-0.473729\pi$
$101$	1.65699	0.164877	0.0824383	+	0.996596i	$0.473729\pi$
$102$	0.0444463	0.00440084
$103$	−9.71423	−0.957171	−0.478586	−	0.878041i	$-0.658850\pi$
$103$	−9.71423	−0.957171	−0.478586	+	0.878041i	$0.658850\pi$
$104$	0.660935	0.0648100
$105$	3.13100	0.305555
$106$	−9.09907	−0.883780
$107$	−0.806426	−0.0779601	−0.0389801	−	0.999240i	$-0.512411\pi$
$107$	−0.806426	−0.0779601	−0.0389801	+	0.999240i	$0.512411\pi$
$108$	−1.05566	−0.101581
$109$	−3.46489	−0.331877	−0.165938	−	0.986136i	$-0.553065\pi$
$109$	−3.46489	−0.331877	−0.165938	+	0.986136i	$0.553065\pi$
$110$	−12.6724	−1.20827
$111$	−1.50335	−0.142692
$112$	−4.23014	−0.399711
$113$	−3.52210	−0.331332	−0.165666	−	0.986182i	$-0.552977\pi$
$113$	−3.52210	−0.331332	−0.165666	+	0.986182i	$0.552977\pi$
$114$	0.813495	0.0761908
$115$	−27.8534	−2.59734
$116$	4.32449	0.401518
$117$	−1.96213	−0.181399
$118$	2.05038	0.188753
$119$	−1.06304	−0.0974487
$120$	−0.740165	−0.0675675
$121$	−1.83049	−0.166408
$122$	−2.20500	−0.199631
$123$	0.417653	0.0376585
$124$	2.59280	0.232840
$125$	−31.4433	−2.81238
$126$	12.5581	1.11877
$127$	2.39167	0.212226	0.106113	−	0.994354i	$-0.466159\pi$
$127$	2.39167	0.212226	0.106113	+	0.994354i	$0.466159\pi$
$128$	1.00000	0.0883883
$129$	0.155879	0.0137244
$130$	−2.76595	−0.242590
$131$	−13.0431	−1.13958	−0.569789	−	0.821791i	$-0.692975\pi$
$131$	−13.0431	−1.13958	−0.569789	+	0.821791i	$0.692975\pi$
$132$	0.535568	0.0466152
$133$	−19.4567	−1.68711
$134$	−2.78495	−0.240583
$135$	4.41783	0.380227
$136$	0.251301	0.0215489
$137$	−17.9457	−1.53321	−0.766604	−	0.642120i	$-0.778055\pi$
$137$	−17.9457	−1.53321	−0.766604	+	0.642120i	$0.778055\pi$
$138$	1.17715	0.100206
$139$	17.4439	1.47958	0.739788	−	0.672840i	$-0.234926\pi$
$139$	17.4439	1.47958	0.739788	+	0.672840i	$0.234926\pi$
$140$	17.7028	1.49616
$141$	−1.52102	−0.128093
$142$	−10.8740	−0.912524
$143$	2.00139	0.167365
$144$	−2.96872	−0.247393
$145$	−18.0976	−1.50292
$146$	−3.99573	−0.330689
$147$	1.92679	0.158919
$148$	−8.50000	−0.698696
$149$	4.18178	0.342584	0.171292	−	0.985220i	$-0.445206\pi$
$149$	4.18178	0.342584	0.171292	+	0.985220i	$0.445206\pi$
$150$	2.21320	0.180707
$151$	−2.21703	−0.180419	−0.0902096	−	0.995923i	$-0.528754\pi$
$151$	−2.21703	−0.180419	−0.0902096	+	0.995923i	$0.528754\pi$
$152$	4.59953	0.373071
$153$	−0.746042	−0.0603139
$154$	−12.8094	−1.03221
$155$	−10.8506	−0.871544
$156$	0.116896	0.00935919
$157$	4.86830	0.388533	0.194266	−	0.980949i	$-0.437767\pi$
$157$	4.86830	0.388533	0.194266	+	0.980949i	$0.437767\pi$
$158$	−3.04731	−0.242431
$159$	−1.60931	−0.127626
$160$	−4.18491	−0.330846
$161$	−28.1544	−2.21888
$162$	8.71945	0.685064
$163$	−0.907923	−0.0711140	−0.0355570	−	0.999368i	$-0.511321\pi$
$163$	−0.907923	−0.0711140	−0.0355570	+	0.999368i	$0.511321\pi$
$164$	2.36142	0.184396
$165$	−2.24131	−0.174485
$166$	7.29547	0.566238
$167$	−0.999769	−0.0773644	−0.0386822	−	0.999252i	$-0.512316\pi$
$167$	−0.999769	−0.0773644	−0.0386822	+	0.999252i	$0.512316\pi$
$168$	−0.748164	−0.0577221
$169$	−12.5632	−0.966397
$170$	−1.05167	−0.0806596
$171$	−13.6547	−1.04420
$172$	0.881347	0.0672021
$173$	−7.84449	−0.596406	−0.298203	−	0.954503i	$-0.596387\pi$
$173$	−7.84449	−0.596406	−0.298203	+	0.954503i	$0.596387\pi$
$174$	0.764850	0.0579831
$175$	−52.9339	−4.00143
$176$	3.02812	0.228253
$177$	0.362640	0.0272577
$178$	10.3098	0.772749
$179$	−8.13521	−0.608054	−0.304027	−	0.952663i	$-0.598331\pi$
$179$	−8.13521	−0.608054	−0.304027	+	0.952663i	$0.598331\pi$
$180$	12.4238	0.926018
$181$	25.0495	1.86192	0.930958	−	0.365126i	$-0.118974\pi$
$181$	25.0495	1.86192	0.930958	+	0.365126i	$0.118974\pi$
$182$	−2.79585	−0.207242
$183$	−0.389987	−0.0288286
$184$	6.65567	0.490663
$185$	35.5718	2.61529
$186$	0.458575	0.0336244
$187$	0.760969	0.0556476
$188$	−8.59990	−0.627212
$189$	4.46558	0.324823
$190$	−19.2486	−1.39644
$191$	−0.299812	−0.0216937	−0.0108468	−	0.999941i	$-0.503453\pi$
$191$	−0.299812	−0.0216937	−0.0108468	+	0.999941i	$0.503453\pi$
$192$	0.176865	0.0127641
$193$	25.3583	1.82533	0.912666	−	0.408706i	$-0.134020\pi$
$193$	25.3583	1.82533	0.912666	+	0.408706i	$0.134020\pi$
$194$	6.10127	0.438046
$195$	−0.489200	−0.0350324
$196$	10.8941	0.778152
$197$	−4.24548	−0.302478	−0.151239	−	0.988497i	$-0.548326\pi$
$197$	−4.24548	−0.302478	−0.151239	+	0.988497i	$0.548326\pi$
$198$	−8.98964	−0.638866
$199$	−2.46681	−0.174867	−0.0874337	−	0.996170i	$-0.527867\pi$
$199$	−2.46681	−0.174867	−0.0874337	+	0.996170i	$0.527867\pi$
$200$	12.5135	0.884838
$201$	−0.492561	−0.0347426
$202$	1.65699	0.116585
$203$	−18.2932	−1.28393
$204$	0.0444463	0.00311187
$205$	−9.88236	−0.690214
$206$	−9.71423	−0.676822
$207$	−19.7588	−1.37333
$208$	0.660935	0.0458276
$209$	13.9279	0.963414
$210$	3.13100	0.216060
$211$	−19.8304	−1.36518	−0.682591	−	0.730801i	$-0.739147\pi$
$211$	−19.8304	−1.36518	−0.682591	+	0.730801i	$0.739147\pi$
$212$	−9.09907	−0.624927
$213$	−1.92323	−0.131777
$214$	−0.806426	−0.0551261
$215$	−3.68836	−0.251544
$216$	−1.05566	−0.0718284
$217$	−10.9679	−0.744551
$218$	−3.46489	−0.234672
$219$	−0.706705	−0.0477547
$220$	−12.6724	−0.854374
$221$	0.166094	0.0111727
$222$	−1.50335	−0.100898
$223$	−7.49716	−0.502047	−0.251024	−	0.967981i	$-0.580767\pi$
$223$	−7.49716	−0.502047	−0.251024	+	0.967981i	$0.580767\pi$
$224$	−4.23014	−0.282638
$225$	−37.1491	−2.47660
$226$	−3.52210	−0.234287
$227$	−3.66273	−0.243104	−0.121552	−	0.992585i	$-0.538787\pi$
$227$	−3.66273	−0.243104	−0.121552	+	0.992585i	$0.538787\pi$
$228$	0.813495	0.0538750
$229$	27.9709	1.84837	0.924183	−	0.381950i	$-0.124747\pi$
$229$	27.9709	1.84837	0.924183	+	0.381950i	$0.124747\pi$
$230$	−27.8534	−1.83660
$231$	−2.26553	−0.149061
$232$	4.32449	0.283916
$233$	8.82322	0.578028	0.289014	−	0.957325i	$-0.406673\pi$
$233$	8.82322	0.578028	0.289014	+	0.957325i	$0.406673\pi$
$234$	−1.96213	−0.128268
$235$	35.9898	2.34772
$236$	2.05038	0.133468
$237$	−0.538963	−0.0350094
$238$	−1.06304	−0.0689066
$239$	6.10029	0.394595	0.197297	−	0.980344i	$-0.436784\pi$
$239$	6.10029	0.394595	0.197297	+	0.980344i	$0.436784\pi$
$240$	−0.740165	−0.0477774
$241$	−19.1964	−1.23655	−0.618274	−	0.785963i	$-0.712168\pi$
$241$	−19.1964	−1.23655	−0.618274	+	0.785963i	$0.712168\pi$
$242$	−1.83049	−0.117668
$243$	4.70914	0.302091
$244$	−2.20500	−0.141160
$245$	−45.5910	−2.91270
$246$	0.417653	0.0266286
$247$	3.03999	0.193430
$248$	2.59280	0.164643
$249$	1.29031	0.0817703
$250$	−31.4433	−1.98865
$251$	11.0350	0.696525	0.348262	−	0.937397i	$-0.386772\pi$
$251$	11.0350	0.696525	0.348262	+	0.937397i	$0.386772\pi$
$252$	12.5581	0.791087
$253$	20.1542	1.26708
$254$	2.39167	0.150067
$255$	−0.186004	−0.0116480
$256$	1.00000	0.0625000
$257$	−11.6428	−0.726260	−0.363130	−	0.931738i	$-0.618292\pi$
$257$	−11.6428	−0.726260	−0.363130	+	0.931738i	$0.618292\pi$
$258$	0.155879	0.00970463
$259$	35.9562	2.23421
$260$	−2.76595	−0.171537
$261$	−12.8382	−0.794663
$262$	−13.0431	−0.805803
$263$	−24.9696	−1.53969	−0.769846	−	0.638229i	$-0.779667\pi$
$263$	−24.9696	−1.53969	−0.769846	+	0.638229i	$0.779667\pi$
$264$	0.535568	0.0329620
$265$	38.0788	2.33916
$266$	−19.4567	−1.19296
$267$	1.82344	0.111592
$268$	−2.78495	−0.170118
$269$	−17.2110	−1.04937	−0.524685	−	0.851296i	$-0.675817\pi$
$269$	−17.2110	−1.04937	−0.524685	+	0.851296i	$0.675817\pi$
$270$	4.41783	0.268861
$271$	−30.4720	−1.85104	−0.925522	−	0.378695i	$-0.876373\pi$
$271$	−30.4720	−1.85104	−0.925522	+	0.378695i	$0.876373\pi$
$272$	0.251301	0.0152374
$273$	−0.494488	−0.0299278
$274$	−17.9457	−1.08414
$275$	37.8924	2.28500
$276$	1.17715	0.0708564
$277$	−10.6525	−0.640047	−0.320024	−	0.947410i	$-0.603691\pi$
$277$	−10.6525	−0.640047	−0.320024	+	0.947410i	$0.603691\pi$
$278$	17.4439	1.04622
$279$	−7.69729	−0.460825
$280$	17.7028	1.05794
$281$	22.8489	1.36305	0.681524	−	0.731796i	$-0.261317\pi$
$281$	22.8489	1.36305	0.681524	+	0.731796i	$0.261317\pi$
$282$	−1.52102	−0.0905755
$283$	−3.98736	−0.237024	−0.118512	−	0.992953i	$-0.537812\pi$
$283$	−3.98736	−0.237024	−0.118512	+	0.992953i	$0.537812\pi$
$284$	−10.8740	−0.645252
$285$	−3.40441	−0.201660
$286$	2.00139	0.118345
$287$	−9.98917	−0.589642
$288$	−2.96872	−0.174933
$289$	−16.9368	−0.996285
$290$	−18.0976	−1.06273
$291$	1.07910	0.0632581
$292$	−3.99573	−0.233832
$293$	−0.241692	−0.0141198	−0.00705989	−	0.999975i	$-0.502247\pi$
$293$	−0.241692	−0.0141198	−0.00705989	+	0.999975i	$0.502247\pi$
$294$	1.92679	0.112373
$295$	−8.58066	−0.499585
$296$	−8.50000	−0.494052
$297$	−3.19666	−0.185489
$298$	4.18178	0.242244
$299$	4.39896	0.254399
$300$	2.21320	0.127779
$301$	−3.72823	−0.214891
$302$	−2.21703	−0.127576
$303$	0.293063	0.0168360
$304$	4.59953	0.263801
$305$	9.22772	0.528378
$306$	−0.746042	−0.0426484
$307$	−15.9563	−0.910674	−0.455337	−	0.890319i	$-0.650481\pi$
$307$	−15.9563	−0.910674	−0.455337	+	0.890319i	$0.650481\pi$
$308$	−12.8094	−0.729882
$309$	−1.71811	−0.0977397
$310$	−10.8506	−0.616275
$311$	24.9863	1.41685	0.708423	−	0.705788i	$-0.249407\pi$
$311$	24.9863	1.41685	0.708423	+	0.705788i	$0.249407\pi$
$312$	0.116896	0.00661794
$313$	4.05538	0.229224	0.114612	−	0.993410i	$-0.463438\pi$
$313$	4.05538	0.229224	0.114612	+	0.993410i	$0.463438\pi$
$314$	4.86830	0.274734
$315$	−52.5546	−2.96112
$316$	−3.04731	−0.171425
$317$	−2.26851	−0.127412	−0.0637062	−	0.997969i	$-0.520292\pi$
$317$	−2.26851	−0.127412	−0.0637062	+	0.997969i	$0.520292\pi$
$318$	−1.60931	−0.0902454
$319$	13.0951	0.733183
$320$	−4.18491	−0.233944
$321$	−0.142628	−0.00796074
$322$	−28.1544	−1.56899
$323$	1.15587	0.0643141
$324$	8.71945	0.484414
$325$	8.27060	0.458771
$326$	−0.907923	−0.0502852
$327$	−0.612818	−0.0338889
$328$	2.36142	0.130388
$329$	36.3788	2.00563
$330$	−2.24131	−0.123380
$331$	−19.5247	−1.07317	−0.536587	−	0.843845i	$-0.680287\pi$
$331$	−19.5247	−1.07317	−0.536587	+	0.843845i	$0.680287\pi$
$332$	7.29547	0.400391
$333$	25.2341	1.38282
$334$	−0.999769	−0.0547049
$335$	11.6548	0.636769
$336$	−0.748164	−0.0408157
$337$	21.8666	1.19115	0.595576	−	0.803299i	$-0.296924\pi$
$337$	21.8666	1.19115	0.595576	+	0.803299i	$0.296924\pi$
$338$	−12.5632	−0.683346
$339$	−0.622937	−0.0338333
$340$	−1.05167	−0.0570350
$341$	7.85131	0.425172
$342$	−13.6547	−0.738362
$343$	−16.4727	−0.889443
$344$	0.881347	0.0475191
$345$	−4.92629	−0.265223
$346$	−7.84449	−0.421723
$347$	4.42743	0.237677	0.118838	−	0.992914i	$-0.462083\pi$
$347$	4.42743	0.237677	0.118838	+	0.992914i	$0.462083\pi$
$348$	0.764850	0.0410003
$349$	−0.814950	−0.0436233	−0.0218116	−	0.999762i	$-0.506943\pi$
$349$	−0.814950	−0.0436233	−0.0218116	+	0.999762i	$0.506943\pi$
$350$	−52.9339	−2.82944
$351$	−0.697721	−0.0372416
$352$	3.02812	0.161399
$353$	4.93782	0.262814	0.131407	−	0.991329i	$-0.458051\pi$
$353$	4.93782	0.262814	0.131407	+	0.991329i	$0.458051\pi$
$354$	0.362640	0.0192741
$355$	45.5066	2.41524
$356$	10.3098	0.546416
$357$	−0.188014	−0.00995078
$358$	−8.13521	−0.429959
$359$	9.61016	0.507205	0.253602	−	0.967309i	$-0.418385\pi$
$359$	9.61016	0.507205	0.253602	+	0.967309i	$0.418385\pi$
$360$	12.4238	0.654793
$361$	2.15564	0.113455
$362$	25.0495	1.31657
$363$	−0.323750	−0.0169925
$364$	−2.79585	−0.146542
$365$	16.7218	0.875258
$366$	−0.389987	−0.0203849
$367$	−15.1615	−0.791425	−0.395713	−	0.918374i	$-0.629502\pi$
$367$	−15.1615	−0.791425	−0.395713	+	0.918374i	$0.629502\pi$
$368$	6.65567	0.346951
$369$	−7.01041	−0.364947
$370$	35.5718	1.84929
$371$	38.4904	1.99832
$372$	0.458575	0.0237760
$373$	−33.7901	−1.74959	−0.874793	−	0.484497i	$-0.839003\pi$
$373$	−33.7901	−1.74959	−0.874793	+	0.484497i	$0.839003\pi$
$374$	0.760969	0.0393488
$375$	−5.56122	−0.287180
$376$	−8.59990	−0.443506
$377$	2.85820	0.147205
$378$	4.46558	0.229685
$379$	−4.08736	−0.209954	−0.104977	−	0.994475i	$-0.533477\pi$
$379$	−4.08736	−0.209954	−0.104977	+	0.994475i	$0.533477\pi$
$380$	−19.2486	−0.987433
$381$	0.423002	0.0216711
$382$	−0.299812	−0.0153397
$383$	−18.9861	−0.970146	−0.485073	−	0.874474i	$-0.661207\pi$
$383$	−18.9861	−0.970146	−0.485073	+	0.874474i	$0.661207\pi$
$384$	0.176865	0.00902560
$385$	53.6062	2.73202
$386$	25.3583	1.29070
$387$	−2.61647	−0.133003
$388$	6.10127	0.309745
$389$	−25.7824	−1.30722	−0.653611	−	0.756831i	$-0.726747\pi$
$389$	−25.7824	−1.30722	−0.653611	+	0.756831i	$0.726747\pi$
$390$	−0.489200	−0.0247716
$391$	1.67258	0.0845858
$392$	10.8941	0.550236
$393$	−2.30686	−0.116366
$394$	−4.24548	−0.213884
$395$	12.7527	0.641660
$396$	−8.98964	−0.451746
$397$	−6.21692	−0.312018	−0.156009	−	0.987756i	$-0.549863\pi$
$397$	−6.21692	−0.312018	−0.156009	+	0.987756i	$0.549863\pi$
$398$	−2.46681	−0.123650
$399$	−3.44120	−0.172276
$400$	12.5135	0.625675
$401$	−1.22902	−0.0613745	−0.0306873	−	0.999529i	$-0.509770\pi$
$401$	−1.22902	−0.0613745	−0.0306873	+	0.999529i	$0.509770\pi$
$402$	−0.492561	−0.0245667
$403$	1.71367	0.0853640
$404$	1.65699	0.0824383
$405$	−36.4901	−1.81321
$406$	−18.2932	−0.907876
$407$	−25.7390	−1.27584
$408$	0.0444463	0.00220042
$409$	−10.0258	−0.495744	−0.247872	−	0.968793i	$-0.579731\pi$
$409$	−10.0258	−0.495744	−0.247872	+	0.968793i	$0.579731\pi$
$410$	−9.88236	−0.488055
$411$	−3.17397	−0.156561
$412$	−9.71423	−0.478586
$413$	−8.67340	−0.426790
$414$	−19.7588	−0.971093
$415$	−30.5309	−1.49870
$416$	0.660935	0.0324050
$417$	3.08522	0.151084
$418$	13.9279	0.681237
$419$	−3.56896	−0.174355	−0.0871776	−	0.996193i	$-0.527785\pi$
$419$	−3.56896	−0.174355	−0.0871776	+	0.996193i	$0.527785\pi$
$420$	3.13100	0.152777
$421$	38.2338	1.86340	0.931700	−	0.363229i	$-0.118326\pi$
$421$	38.2338	1.86340	0.931700	+	0.363229i	$0.118326\pi$
$422$	−19.8304	−0.965330
$423$	25.5307	1.24134
$424$	−9.09907	−0.441890
$425$	3.14465	0.152538
$426$	−1.92323	−0.0931806
$427$	9.32745	0.451387
$428$	−0.806426	−0.0389801
$429$	0.353976	0.0170901
$430$	−3.68836	−0.177869
$431$	18.5853	0.895223	0.447611	−	0.894228i	$-0.352275\pi$
$431$	18.5853	0.895223	0.447611	+	0.894228i	$0.352275\pi$
$432$	−1.05566	−0.0507903
$433$	−19.5022	−0.937215	−0.468607	−	0.883407i	$-0.655244\pi$
$433$	−19.5022	−0.937215	−0.468607	+	0.883407i	$0.655244\pi$
$434$	−10.9679	−0.526477
$435$	−3.20083	−0.153468
$436$	−3.46489	−0.165938
$437$	30.6129	1.46441
$438$	−0.706705	−0.0337677
$439$	20.2213	0.965110	0.482555	−	0.875866i	$-0.339709\pi$
$439$	20.2213	0.965110	0.482555	+	0.875866i	$0.339709\pi$
$440$	−12.6724	−0.604134
$441$	−32.3416	−1.54008
$442$	0.166094	0.00790026
$443$	−13.5862	−0.645501	−0.322751	−	0.946484i	$-0.604608\pi$
$443$	−13.5862	−0.645501	−0.322751	+	0.946484i	$0.604608\pi$
$444$	−1.50335	−0.0713459
$445$	−43.1455	−2.04529
$446$	−7.49716	−0.355001
$447$	0.739610	0.0349823
$448$	−4.23014	−0.199856
$449$	4.49497	0.212131	0.106065	−	0.994359i	$-0.466175\pi$
$449$	4.49497	0.212131	0.106065	+	0.994359i	$0.466175\pi$
$450$	−37.1491	−1.75122
$451$	7.15068	0.336712
$452$	−3.52210	−0.165666
$453$	−0.392115	−0.0184232
$454$	−3.66273	−0.171900
$455$	11.7004	0.548523
$456$	0.813495	0.0380954
$457$	2.97830	0.139319	0.0696594	−	0.997571i	$-0.477809\pi$
$457$	2.97830	0.139319	0.0696594	+	0.997571i	$0.477809\pi$
$458$	27.9709	1.30699
$459$	−0.265288	−0.0123826
$460$	−27.8534	−1.29867
$461$	−42.4597	−1.97755	−0.988773	−	0.149423i	$-0.952258\pi$
$461$	−42.4597	−1.97755	−0.988773	+	0.149423i	$0.952258\pi$
$462$	−2.26553	−0.105402
$463$	−24.3917	−1.13358	−0.566789	−	0.823863i	$-0.691815\pi$
$463$	−24.3917	−1.13358	−0.566789	+	0.823863i	$0.691815\pi$
$464$	4.32449	0.200759
$465$	−1.91910	−0.0889960
$466$	8.82322	0.408728
$467$	−28.2253	−1.30611	−0.653055	−	0.757311i	$-0.726513\pi$
$467$	−28.2253	−1.30611	−0.653055	+	0.757311i	$0.726513\pi$
$468$	−1.96213	−0.0906995
$469$	11.7808	0.543985
$470$	35.9898	1.66009
$471$	0.861031	0.0396742
$472$	2.05038	0.0943763
$473$	2.66883	0.122713
$474$	−0.538963	−0.0247554
$475$	57.5561	2.64086
$476$	−1.06304	−0.0487243
$477$	27.0126	1.23682
$478$	6.10029	0.279021
$479$	31.6261	1.44503	0.722516	−	0.691354i	$-0.242985\pi$
$479$	31.6261	1.44503	0.722516	+	0.691354i	$0.242985\pi$
$480$	−0.740165	−0.0337837
$481$	−5.61795	−0.256156
$482$	−19.1964	−0.874371
$483$	−4.97954	−0.226577
$484$	−1.83049	−0.0832042
$485$	−25.5333	−1.15941
$486$	4.70914	0.213611
$487$	−37.7648	−1.71129	−0.855644	−	0.517565i	$-0.826838\pi$
$487$	−37.7648	−1.71129	−0.855644	+	0.517565i	$0.826838\pi$
$488$	−2.20500	−0.0998155
$489$	−0.160580	−0.00726166
$490$	−45.5910	−2.05959
$491$	−9.86419	−0.445165	−0.222582	−	0.974914i	$-0.571449\pi$
$491$	−9.86419	−0.445165	−0.222582	+	0.974914i	$0.571449\pi$
$492$	0.417653	0.0188293
$493$	1.08675	0.0489446
$494$	3.03999	0.136775
$495$	37.6208	1.69093
$496$	2.59280	0.116420
$497$	45.9985	2.06331
$498$	1.29031	0.0578203
$499$	−25.7528	−1.15285	−0.576426	−	0.817150i	$-0.695553\pi$
$499$	−25.7528	−1.15285	−0.576426	+	0.817150i	$0.695553\pi$
$500$	−31.4433	−1.40619
$501$	−0.176824	−0.00789992
$502$	11.0350	0.492517
$503$	38.9453	1.73648	0.868242	−	0.496141i	$-0.165250\pi$
$503$	38.9453	1.73648	0.868242	+	0.496141i	$0.165250\pi$
$504$	12.5581	0.559383
$505$	−6.93436	−0.308575
$506$	20.1542	0.895962
$507$	−2.22198	−0.0986818
$508$	2.39167	0.106113
$509$	0.889041	0.0394060	0.0197030	−	0.999806i	$-0.493728\pi$
$509$	0.889041	0.0394060	0.0197030	+	0.999806i	$0.493728\pi$
$510$	−0.186004	−0.00823640
$511$	16.9025	0.747723
$512$	1.00000	0.0441942
$513$	−4.85552	−0.214377
$514$	−11.6428	−0.513543
$515$	40.6532	1.79139
$516$	0.155879	0.00686221
$517$	−26.0415	−1.14531
$518$	35.9562	1.57983
$519$	−1.38742	−0.0609008
$520$	−2.76595	−0.121295
$521$	34.6875	1.51969	0.759843	−	0.650107i	$-0.225276\pi$
$521$	34.6875	1.51969	0.759843	+	0.650107i	$0.225276\pi$
$522$	−12.8382	−0.561912
$523$	11.1693	0.488399	0.244200	−	0.969725i	$-0.421475\pi$
$523$	11.1693	0.488399	0.244200	+	0.969725i	$0.421475\pi$
$524$	−13.0431	−0.569789
$525$	−9.36215	−0.408598
$526$	−24.9696	−1.08873
$527$	0.651573	0.0283830
$528$	0.535568	0.0233076
$529$	21.2979	0.925998
$530$	38.0788	1.65404
$531$	−6.08700	−0.264153
$532$	−19.4567	−0.843553
$533$	1.56075	0.0676035
$534$	1.82344	0.0789078
$535$	3.37482	0.145906
$536$	−2.78495	−0.120292
$537$	−1.43883	−0.0620902
$538$	−17.2110	−0.742017
$539$	32.9887	1.42092
$540$	4.41783	0.190113
$541$	−35.7187	−1.53567	−0.767833	−	0.640650i	$-0.778665\pi$
$541$	−35.7187	−1.53567	−0.767833	+	0.640650i	$0.778665\pi$
$542$	−30.4720	−1.30889
$543$	4.43038	0.190126
$544$	0.251301	0.0107744
$545$	14.5003	0.621124
$546$	−0.494488	−0.0211621
$547$	9.84820	0.421079	0.210539	−	0.977585i	$-0.432478\pi$
$547$	9.84820	0.421079	0.210539	+	0.977585i	$0.432478\pi$
$548$	−17.9457	−0.766604
$549$	6.54602	0.279377
$550$	37.8924	1.61574
$551$	19.8906	0.847367
$552$	1.17715	0.0501030
$553$	12.8906	0.548163
$554$	−10.6525	−0.452582
$555$	6.29140	0.267055
$556$	17.4439	0.739788
$557$	9.77874	0.414338	0.207169	−	0.978305i	$-0.433575\pi$
$557$	9.77874	0.414338	0.207169	+	0.978305i	$0.433575\pi$
$558$	−7.69729	−0.325852
$559$	0.582513	0.0246377
$560$	17.7028	0.748079
$561$	0.134589	0.00568234
$562$	22.8489	0.963821
$563$	−30.6650	−1.29238	−0.646189	−	0.763178i	$-0.723638\pi$
$563$	−30.6650	−1.29238	−0.646189	+	0.763178i	$0.723638\pi$
$564$	−1.52102	−0.0640466
$565$	14.7397	0.620104
$566$	−3.98736	−0.167601
$567$	−36.8845	−1.54900
$568$	−10.8740	−0.456262
$569$	−25.8555	−1.08392	−0.541959	−	0.840405i	$-0.682317\pi$
$569$	−25.8555	−1.08392	−0.541959	+	0.840405i	$0.682317\pi$
$570$	−3.40441	−0.142595
$571$	−34.3351	−1.43688	−0.718440	−	0.695589i	$-0.755144\pi$
$571$	−34.3351	−1.43688	−0.718440	+	0.695589i	$0.755144\pi$
$572$	2.00139	0.0836823
$573$	−0.0530263	−0.00221521
$574$	−9.98917	−0.416940
$575$	83.2857	3.47325
$576$	−2.96872	−0.123697
$577$	−35.8636	−1.49302	−0.746511	−	0.665373i	$-0.768273\pi$
$577$	−35.8636	−1.49302	−0.746511	+	0.665373i	$0.768273\pi$
$578$	−16.9368	−0.704480
$579$	4.48500	0.186390
$580$	−18.0976	−0.751462
$581$	−30.8609	−1.28033
$582$	1.07910	0.0447302
$583$	−27.5531	−1.14113
$584$	−3.99573	−0.165344
$585$	8.21134	0.339497
$586$	−0.241692	−0.00998420
$587$	23.9426	0.988217	0.494109	−	0.869400i	$-0.335495\pi$
$587$	23.9426	0.988217	0.494109	+	0.869400i	$0.335495\pi$
$588$	1.92679	0.0794594
$589$	11.9256	0.491388
$590$	−8.58066	−0.353260
$591$	−0.750876	−0.0308869
$592$	−8.50000	−0.349348
$593$	−21.9046	−0.899513	−0.449757	−	0.893151i	$-0.648489\pi$
$593$	−21.9046	−0.899513	−0.449757	+	0.893151i	$0.648489\pi$
$594$	−3.19666	−0.131160
$595$	4.44873	0.182380
$596$	4.18178	0.171292
$597$	−0.436292	−0.0178562
$598$	4.39896	0.179887
$599$	6.93520	0.283364	0.141682	−	0.989912i	$-0.454749\pi$
$599$	6.93520	0.283364	0.141682	+	0.989912i	$0.454749\pi$
$600$	2.21320	0.0903535
$601$	42.9065	1.75019	0.875096	−	0.483950i	$-0.160798\pi$
$601$	42.9065	1.75019	0.875096	+	0.483950i	$0.160798\pi$
$602$	−3.72823	−0.151951
$603$	8.26775	0.336689
$604$	−2.21703	−0.0902096
$605$	7.66045	0.311441
$606$	0.293063	0.0119049
$607$	−2.16910	−0.0880412	−0.0440206	−	0.999031i	$-0.514017\pi$
$607$	−2.16910	−0.0880412	−0.0440206	+	0.999031i	$0.514017\pi$
$608$	4.59953	0.186535
$609$	−3.23543	−0.131106
$610$	9.22772	0.373619
$611$	−5.68397	−0.229949
$612$	−0.746042	−0.0301570
$613$	−2.98927	−0.120735	−0.0603677	−	0.998176i	$-0.519227\pi$
$613$	−2.98927	−0.120735	−0.0603677	+	0.998176i	$0.519227\pi$
$614$	−15.9563	−0.643944
$615$	−1.74784	−0.0704798
$616$	−12.8094	−0.516105
$617$	11.3932	0.458675	0.229337	−	0.973347i	$-0.426344\pi$
$617$	11.3932	0.458675	0.229337	+	0.973347i	$0.426344\pi$
$618$	−1.71811	−0.0691124
$619$	11.9153	0.478915	0.239457	−	0.970907i	$-0.423030\pi$
$619$	11.9153	0.478915	0.239457	+	0.970907i	$0.423030\pi$
$620$	−10.8506	−0.435772
$621$	−7.02611	−0.281948
$622$	24.9863	1.00186
$623$	−43.6118	−1.74727
$624$	0.116896	0.00467959
$625$	69.0201	2.76080
$626$	4.05538	0.162086
$627$	2.46336	0.0983771
$628$	4.86830	0.194266
$629$	−2.13606	−0.0851702
$630$	−52.5546	−2.09383
$631$	−19.4483	−0.774223	−0.387111	−	0.922033i	$-0.626527\pi$
$631$	−19.4483	−0.774223	−0.387111	+	0.922033i	$0.626527\pi$
$632$	−3.04731	−0.121216
$633$	−3.50731	−0.139403
$634$	−2.26851	−0.0900941
$635$	−10.0089	−0.397192
$636$	−1.60931	−0.0638131
$637$	7.20030	0.285286
$638$	13.0951	0.518438
$639$	32.2818	1.27705
$640$	−4.18491	−0.165423
$641$	−23.9149	−0.944584	−0.472292	−	0.881442i	$-0.656573\pi$
$641$	−23.9149	−0.944584	−0.472292	+	0.881442i	$0.656573\pi$
$642$	−0.142628	−0.00562910
$643$	−23.0357	−0.908438	−0.454219	−	0.890890i	$-0.650082\pi$
$643$	−23.0357	−0.908438	−0.454219	+	0.890890i	$0.650082\pi$
$644$	−28.1544	−1.10944
$645$	−0.652342	−0.0256859
$646$	1.15587	0.0454769
$647$	42.8822	1.68587	0.842936	−	0.538014i	$-0.180825\pi$
$647$	42.8822	1.68587	0.842936	+	0.538014i	$0.180825\pi$
$648$	8.71945	0.342532
$649$	6.20879	0.243716
$650$	8.27060	0.324400
$651$	−1.93984	−0.0760283
$652$	−0.907923	−0.0355570
$653$	−43.2299	−1.69172	−0.845858	−	0.533409i	$-0.820911\pi$
$653$	−43.2299	−1.69172	−0.845858	+	0.533409i	$0.820911\pi$
$654$	−0.612818	−0.0239631
$655$	54.5841	2.13278
$656$	2.36142	0.0921981
$657$	11.8622	0.462788
$658$	36.3788	1.41819
$659$	25.0584	0.976138	0.488069	−	0.872805i	$-0.337701\pi$
$659$	25.0584	0.976138	0.488069	+	0.872805i	$0.337701\pi$
$660$	−2.24131	−0.0872427
$661$	25.5284	0.992941	0.496470	−	0.868054i	$-0.334629\pi$
$661$	25.5284	0.992941	0.496470	+	0.868054i	$0.334629\pi$
$662$	−19.5247	−0.758849
$663$	0.0293761	0.00114087
$664$	7.29547	0.283119
$665$	81.4244	3.15750
$666$	25.2341	0.977802
$667$	28.7823	1.11446
$668$	−0.999769	−0.0386822
$669$	−1.32599	−0.0512655
$670$	11.6548	0.450264
$671$	−6.67699	−0.257762
$672$	−0.748164	−0.0288611
$673$	37.2580	1.43619	0.718095	−	0.695945i	$-0.245014\pi$
$673$	37.2580	1.43619	0.718095	+	0.695945i	$0.245014\pi$
$674$	21.8666	0.842271
$675$	−13.2100	−0.508452
$676$	−12.5632	−0.483199
$677$	4.98835	0.191718	0.0958590	−	0.995395i	$-0.469440\pi$
$677$	4.98835	0.191718	0.0958590	+	0.995395i	$0.469440\pi$
$678$	−0.622937	−0.0239237
$679$	−25.8093	−0.990469
$680$	−1.05167	−0.0403298
$681$	−0.647809	−0.0248241
$682$	7.85131	0.300642
$683$	−10.1399	−0.387992	−0.193996	−	0.981002i	$-0.562145\pi$
$683$	−10.1399	−0.387992	−0.193996	+	0.981002i	$0.562145\pi$
$684$	−13.6547	−0.522100
$685$	75.1014	2.86948
$686$	−16.4727	−0.628931
$687$	4.94706	0.188742
$688$	0.881347	0.0336011
$689$	−6.01389	−0.229111
$690$	−4.92629	−0.187541
$691$	9.03457	0.343691	0.171846	−	0.985124i	$-0.445027\pi$
$691$	9.03457	0.343691	0.171846	+	0.985124i	$0.445027\pi$
$692$	−7.84449	−0.298203
$693$	38.0275	1.44454
$694$	4.42743	0.168063
$695$	−73.0014	−2.76910
$696$	0.764850	0.0289916
$697$	0.593428	0.0224777
$698$	−0.814950	−0.0308463
$699$	1.56052	0.0590242
$700$	−52.9339	−2.00071
$701$	−0.281630	−0.0106370	−0.00531851	−	0.999986i	$-0.501693\pi$
$701$	−0.281630	−0.0106370	−0.00531851	+	0.999986i	$0.501693\pi$
$702$	−0.697721	−0.0263338
$703$	−39.0960	−1.47453
$704$	3.02812	0.114127
$705$	6.36534	0.239733
$706$	4.93782	0.185837
$707$	−7.00930	−0.263612
$708$	0.362640	0.0136289
$709$	27.8421	1.04563	0.522816	−	0.852446i	$-0.324882\pi$
$709$	27.8421	1.04563	0.522816	+	0.852446i	$0.324882\pi$
$710$	45.5066	1.70783
$711$	9.04661	0.339274
$712$	10.3098	0.386375
$713$	17.2568	0.646273
$714$	−0.188014	−0.00703626
$715$	−8.37564	−0.313231
$716$	−8.13521	−0.304027
$717$	1.07893	0.0402933
$718$	9.61016	0.358648
$719$	17.6708	0.659009	0.329505	−	0.944154i	$-0.393118\pi$
$719$	17.6708	0.659009	0.329505	+	0.944154i	$0.393118\pi$
$720$	12.4238	0.463009
$721$	41.0926	1.53037
$722$	2.15564	0.0802245
$723$	−3.39517	−0.126268
$724$	25.0495	0.930958
$725$	54.1144	2.00976
$726$	−0.323750	−0.0120155
$727$	−0.657499	−0.0243853	−0.0121926	−	0.999926i	$-0.503881\pi$
$727$	−0.657499	−0.0243853	−0.0121926	+	0.999926i	$0.503881\pi$
$728$	−2.79585	−0.103621
$729$	−25.3255	−0.937980
$730$	16.7218	0.618901
$731$	0.221483	0.00819186
$732$	−0.389987	−0.0144143
$733$	−40.3214	−1.48930	−0.744652	−	0.667453i	$-0.767385\pi$
$733$	−40.3214	−1.48930	−0.744652	+	0.667453i	$0.767385\pi$
$734$	−15.1615	−0.559622
$735$	−8.06344	−0.297425
$736$	6.65567	0.245331
$737$	−8.43317	−0.310640
$738$	−7.01041	−0.258057
$739$	2.02482	0.0744843	0.0372422	−	0.999306i	$-0.488143\pi$
$739$	2.02482	0.0744843	0.0372422	+	0.999306i	$0.488143\pi$
$740$	35.5718	1.30764
$741$	0.537667	0.0197517
$742$	38.4904	1.41303
$743$	9.20988	0.337878	0.168939	−	0.985627i	$-0.445966\pi$
$743$	9.20988	0.337878	0.168939	+	0.985627i	$0.445966\pi$
$744$	0.458575	0.0168122
$745$	−17.5004	−0.641164
$746$	−33.7901	−1.23714
$747$	−21.6582	−0.792432
$748$	0.760969	0.0278238
$749$	3.41130	0.124646
$750$	−5.56122	−0.203067
$751$	20.1740	0.736158	0.368079	−	0.929795i	$-0.380016\pi$
$751$	20.1740	0.736158	0.368079	+	0.929795i	$0.380016\pi$
$752$	−8.59990	−0.313606
$753$	1.95171	0.0711242
$754$	2.85820	0.104090
$755$	9.27807	0.337664
$756$	4.46558	0.162412
$757$	−16.4395	−0.597504	−0.298752	−	0.954331i	$-0.596570\pi$
$757$	−16.4395	−0.597504	−0.298752	+	0.954331i	$0.596570\pi$
$758$	−4.08736	−0.148460
$759$	3.56457	0.129386
$760$	−19.2486	−0.698221
$761$	−42.1295	−1.52719	−0.763597	−	0.645693i	$-0.776569\pi$
$761$	−42.1295	−1.52719	−0.763597	+	0.645693i	$0.776569\pi$
$762$	0.423002	0.0153238
$763$	14.6570	0.530619
$764$	−0.299812	−0.0108468
$765$	3.12212	0.112880
$766$	−18.9861	−0.685997
$767$	1.35517	0.0489322
$768$	0.176865	0.00638207
$769$	−12.4942	−0.450551	−0.225276	−	0.974295i	$-0.572328\pi$
$769$	−12.4942	−0.450551	−0.225276	+	0.974295i	$0.572328\pi$
$770$	53.6062	1.93183
$771$	−2.05921	−0.0741606
$772$	25.3583	0.912666
$773$	34.1227	1.22731	0.613655	−	0.789574i	$-0.289699\pi$
$773$	34.1227	1.22731	0.613655	+	0.789574i	$0.289699\pi$
$774$	−2.61647	−0.0940472
$775$	32.4450	1.16546
$776$	6.10127	0.219023
$777$	6.35940	0.228142
$778$	−25.7824	−0.924345
$779$	10.8614	0.389151
$780$	−0.489200	−0.0175162
$781$	−32.9277	−1.17825
$782$	1.67258	0.0598112
$783$	−4.56517	−0.163146
$784$	10.8941	0.389076
$785$	−20.3734	−0.727158
$786$	−2.30686	−0.0822830
$787$	28.4127	1.01280	0.506401	−	0.862298i	$-0.330976\pi$
$787$	28.4127	1.01280	0.506401	+	0.862298i	$0.330976\pi$
$788$	−4.24548	−0.151239
$789$	−4.41625	−0.157223
$790$	12.7527	0.453722
$791$	14.8990	0.529748
$792$	−8.98964	−0.319433
$793$	−1.45736	−0.0517523
$794$	−6.21692	−0.220630
$795$	6.73481	0.238859
$796$	−2.46681	−0.0874337
$797$	47.8043	1.69331	0.846657	−	0.532138i	$-0.178611\pi$
$797$	47.8043	1.69331	0.846657	+	0.532138i	$0.178611\pi$
$798$	−3.44120	−0.121817
$799$	−2.16116	−0.0764565
$800$	12.5135	0.442419
$801$	−30.6068	−1.08144
$802$	−1.22902	−0.0433983
$803$	−12.0995	−0.426984
$804$	−0.492561	−0.0173713
$805$	117.824	4.15275
$806$	1.71367	0.0603615
$807$	−3.04402	−0.107154
$808$	1.65699	0.0582927
$809$	29.8698	1.05017	0.525084	−	0.851050i	$-0.324034\pi$
$809$	29.8698	1.05017	0.525084	+	0.851050i	$0.324034\pi$
$810$	−36.4901	−1.28213
$811$	−30.6321	−1.07564	−0.537819	−	0.843060i	$-0.680752\pi$
$811$	−30.6321	−1.07564	−0.537819	+	0.843060i	$0.680752\pi$
$812$	−18.2932	−0.641965
$813$	−5.38943	−0.189016
$814$	−25.7390	−0.902152
$815$	3.79958	0.133093
$816$	0.0444463	0.00155593
$817$	4.05378	0.141824
$818$	−10.0258	−0.350544
$819$	8.30009	0.290029
$820$	−9.88236	−0.345107
$821$	−9.86648	−0.344343	−0.172171	−	0.985067i	$-0.555078\pi$
$821$	−9.86648	−0.344343	−0.172171	+	0.985067i	$0.555078\pi$
$822$	−3.17397	−0.110705
$823$	−45.0359	−1.56985	−0.784926	−	0.619589i	$-0.787299\pi$
$823$	−45.0359	−1.56985	−0.784926	+	0.619589i	$0.787299\pi$
$824$	−9.71423	−0.338411
$825$	6.70183	0.233328
$826$	−8.67340	−0.301786
$827$	3.60771	0.125452	0.0627262	−	0.998031i	$-0.480021\pi$
$827$	3.60771	0.125452	0.0627262	+	0.998031i	$0.480021\pi$
$828$	−19.7588	−0.686666
$829$	51.7480	1.79728	0.898641	−	0.438686i	$-0.144556\pi$
$829$	51.7480	1.79728	0.898641	+	0.438686i	$0.144556\pi$
$830$	−30.5309	−1.05974
$831$	−1.88406	−0.0653572
$832$	0.660935	0.0229138
$833$	2.73770	0.0948558
$834$	3.08522	0.106833
$835$	4.18394	0.144791
$836$	13.9279	0.481707
$837$	−2.73711	−0.0946083
$838$	−3.56896	−0.123288
$839$	−14.3264	−0.494601	−0.247301	−	0.968939i	$-0.579544\pi$
$839$	−14.3264	−0.494601	−0.247301	+	0.968939i	$0.579544\pi$
$840$	3.13100	0.108030
$841$	−10.2988	−0.355132
$842$	38.2338	1.31762
$843$	4.04116	0.139185
$844$	−19.8304	−0.682591
$845$	52.5758	1.80866
$846$	25.5307	0.877763
$847$	7.74324	0.266061
$848$	−9.09907	−0.312463
$849$	−0.705224	−0.0242032
$850$	3.14465	0.107861
$851$	−56.5732	−1.93930
$852$	−1.92323	−0.0658886
$853$	−16.3169	−0.558679	−0.279340	−	0.960192i	$-0.590116\pi$
$853$	−16.3169	−0.558679	−0.279340	+	0.960192i	$0.590116\pi$
$854$	9.32745	0.319179
$855$	57.1437	1.95427
$856$	−0.806426	−0.0275631
$857$	31.8792	1.08897	0.544486	−	0.838770i	$-0.316725\pi$
$857$	31.8792	1.08897	0.544486	+	0.838770i	$0.316725\pi$
$858$	0.353976	0.0120845
$859$	−46.5714	−1.58899	−0.794497	−	0.607268i	$-0.792265\pi$
$859$	−46.5714	−1.58899	−0.794497	+	0.607268i	$0.792265\pi$
$860$	−3.68836	−0.125772
$861$	−1.76673	−0.0602101
$862$	18.5853	0.633018
$863$	−0.502746	−0.0171137	−0.00855683	−	0.999963i	$-0.502724\pi$
$863$	−0.502746	−0.0171137	−0.00855683	+	0.999963i	$0.502724\pi$
$864$	−1.05566	−0.0359142
$865$	32.8285	1.11620
$866$	−19.5022	−0.662711
$867$	−2.99554	−0.101734
$868$	−10.9679	−0.372275
$869$	−9.22762	−0.313026
$870$	−3.20083	−0.108518
$871$	−1.84067	−0.0623688
$872$	−3.46489	−0.117336
$873$	−18.1130	−0.613031
$874$	30.6129	1.03550
$875$	133.010	4.49655
$876$	−0.706705	−0.0238773
$877$	−36.9267	−1.24693	−0.623463	−	0.781853i	$-0.714275\pi$
$877$	−36.9267	−1.24693	−0.623463	+	0.781853i	$0.714275\pi$
$878$	20.2213	0.682436
$879$	−0.0427468	−0.00144181
$880$	−12.6724	−0.427187
$881$	−1.75023	−0.0589666	−0.0294833	−	0.999565i	$-0.509386\pi$
$881$	−1.75023	−0.0589666	−0.0294833	+	0.999565i	$0.509386\pi$
$882$	−32.3416	−1.08900
$883$	−30.4212	−1.02376	−0.511878	−	0.859058i	$-0.671050\pi$
$883$	−30.4212	−1.02376	−0.511878	+	0.859058i	$0.671050\pi$
$884$	0.166094	0.00558633
$885$	−1.51762	−0.0510142
$886$	−13.5862	−0.456438
$887$	36.4394	1.22352	0.611758	−	0.791045i	$-0.290463\pi$
$887$	36.4394	1.22352	0.611758	+	0.791045i	$0.290463\pi$
$888$	−1.50335	−0.0504492
$889$	−10.1171	−0.339317
$890$	−43.1455	−1.44624
$891$	26.4035	0.884552
$892$	−7.49716	−0.251024
$893$	−39.5555	−1.32367
$894$	0.739610	0.0247363
$895$	34.0451	1.13800
$896$	−4.23014	−0.141319
$897$	0.778023	0.0259774
$898$	4.49497	0.149999
$899$	11.2125	0.373959
$900$	−37.1491	−1.23830
$901$	−2.28660	−0.0761778
$902$	7.15068	0.238091
$903$	−0.659393	−0.0219432
$904$	−3.52210	−0.117143
$905$	−104.830	−3.48467
$906$	−0.392115	−0.0130271
$907$	−25.2710	−0.839111	−0.419556	−	0.907730i	$-0.637814\pi$
$907$	−25.2710	−0.839111	−0.419556	+	0.907730i	$0.637814\pi$
$908$	−3.66273	−0.121552
$909$	−4.91913	−0.163157
$910$	11.7004	0.387864
$911$	−56.2236	−1.86277	−0.931386	−	0.364034i	$-0.881399\pi$
$911$	−56.2236	−1.86277	−0.931386	+	0.364034i	$0.881399\pi$
$912$	0.813495	0.0269375
$913$	22.0916	0.731124
$914$	2.97830	0.0985133
$915$	1.63206	0.0539542
$916$	27.9709	0.924183
$917$	55.1741	1.82201
$918$	−0.265288	−0.00875580
$919$	15.1275	0.499010	0.249505	−	0.968373i	$-0.419732\pi$
$919$	15.1275	0.499010	0.249505	+	0.968373i	$0.419732\pi$
$920$	−27.8534	−0.918299
$921$	−2.82211	−0.0929917
$922$	−42.4597	−1.39834
$923$	−7.18699	−0.236563
$924$	−2.26553	−0.0745305
$925$	−106.365	−3.49725
$926$	−24.3917	−0.801561
$927$	28.8388	0.947191
$928$	4.32449	0.141958
$929$	−59.5532	−1.95388	−0.976939	−	0.213518i	$-0.931508\pi$
$929$	−59.5532	−1.95388	−0.976939	+	0.213518i	$0.931508\pi$
$930$	−1.91910	−0.0629297
$931$	50.1078	1.64222
$932$	8.82322	0.289014
$933$	4.41921	0.144678
$934$	−28.2253	−0.923559
$935$	−3.18459	−0.104147
$936$	−1.96213	−0.0641342
$937$	13.7877	0.450426	0.225213	−	0.974310i	$-0.427692\pi$
$937$	13.7877	0.450426	0.225213	+	0.974310i	$0.427692\pi$
$938$	11.7808	0.384655
$939$	0.717254	0.0234067
$940$	35.9898	1.17386
$941$	−26.3958	−0.860477	−0.430238	−	0.902715i	$-0.641571\pi$
$941$	−26.3958	−0.860477	−0.430238	+	0.902715i	$0.641571\pi$
$942$	0.861031	0.0280539
$943$	15.7169	0.511811
$944$	2.05038	0.0667341
$945$	−18.6881	−0.607923
$946$	2.66883	0.0867710
$947$	−19.0045	−0.617564	−0.308782	−	0.951133i	$-0.599921\pi$
$947$	−19.0045	−0.617564	−0.308782	+	0.951133i	$0.599921\pi$
$948$	−0.538963	−0.0175047
$949$	−2.64092	−0.0857278
$950$	57.5561	1.86737
$951$	−0.401220	−0.0130105
$952$	−1.06304	−0.0344533
$953$	36.5056	1.18253	0.591266	−	0.806477i	$-0.298628\pi$
$953$	36.5056	1.18253	0.591266	+	0.806477i	$0.298628\pi$
$954$	27.0126	0.874564
$955$	1.25469	0.0406008
$956$	6.10029	0.197297
$957$	2.31606	0.0748675
$958$	31.6261	1.02179
$959$	75.9131	2.45136
$960$	−0.740165	−0.0238887
$961$	−24.2774	−0.783142
$962$	−5.61795	−0.181130
$963$	2.39405	0.0771472
$964$	−19.1964	−0.618274
$965$	−106.122	−3.41620
$966$	−4.97954	−0.160214
$967$	−42.1128	−1.35426	−0.677129	−	0.735865i	$-0.736776\pi$
$967$	−42.1128	−1.35426	−0.677129	+	0.735865i	$0.736776\pi$
$968$	−1.83049	−0.0588342
$969$	0.204432	0.00656730
$970$	−25.5333	−0.819825
$971$	−54.7788	−1.75793	−0.878967	−	0.476883i	$-0.841767\pi$
$971$	−54.7788	−1.75793	−0.878967	+	0.476883i	$0.841767\pi$
$972$	4.70914	0.151046
$973$	−73.7904	−2.36561
$974$	−37.7648	−1.21006
$975$	1.46278	0.0468465
$976$	−2.20500	−0.0705802
$977$	−25.2522	−0.807888	−0.403944	−	0.914784i	$-0.632361\pi$
$977$	−25.2522	−0.807888	−0.403944	+	0.914784i	$0.632361\pi$
$978$	−0.160580	−0.00513477
$979$	31.2192	0.997770
$980$	−45.5910	−1.45635
$981$	10.2863	0.328416
$982$	−9.86419	−0.314779
$983$	−14.0553	−0.448296	−0.224148	−	0.974555i	$-0.571960\pi$
$983$	−14.0553	−0.448296	−0.224148	+	0.974555i	$0.571960\pi$
$984$	0.417653	0.0133143
$985$	17.7669	0.566102
$986$	1.08675	0.0346091
$987$	6.43414	0.204801
$988$	3.03999	0.0967148
$989$	5.86596	0.186527
$990$	37.6208	1.19567
$991$	13.6964	0.435082	0.217541	−	0.976051i	$-0.430196\pi$
$991$	13.6964	0.435082	0.217541	+	0.976051i	$0.430196\pi$
$992$	2.59280	0.0823215
$993$	−3.45323	−0.109585
$994$	45.9985	1.45898
$995$	10.3234	0.327273
$996$	1.29031	0.0408851
$997$	−47.4167	−1.50170	−0.750850	−	0.660472i	$-0.770356\pi$
$997$	−47.4167	−1.50170	−0.750850	+	0.660472i	$0.770356\pi$
$998$	−25.7528	−0.815189
$999$	8.97309	0.283896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.37	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.37	✓	67	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.176865 q^{3} +1.00000 q^{4} -4.18491 q^{5} +0.176865 q^{6} -4.23014 q^{7} +1.00000 q^{8} -2.96872 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.176865 q^{3} +1.00000 q^{4} -4.18491 q^{5} +0.176865 q^{6} -4.23014 q^{7} +1.00000 q^{8} -2.96872 q^{9} -4.18491 q^{10} +3.02812 q^{11} +0.176865 q^{12} +0.660935 q^{13} -4.23014 q^{14} -0.740165 q^{15} +1.00000 q^{16} +0.251301 q^{17} -2.96872 q^{18} +4.59953 q^{19} -4.18491 q^{20} -0.748164 q^{21} +3.02812 q^{22} +6.65567 q^{23} +0.176865 q^{24} +12.5135 q^{25} +0.660935 q^{26} -1.05566 q^{27} -4.23014 q^{28} +4.32449 q^{29} -0.740165 q^{30} +2.59280 q^{31} +1.00000 q^{32} +0.535568 q^{33} +0.251301 q^{34} +17.7028 q^{35} -2.96872 q^{36} -8.50000 q^{37} +4.59953 q^{38} +0.116896 q^{39} -4.18491 q^{40} +2.36142 q^{41} -0.748164 q^{42} +0.881347 q^{43} +3.02812 q^{44} +12.4238 q^{45} +6.65567 q^{46} -8.59990 q^{47} +0.176865 q^{48} +10.8941 q^{49} +12.5135 q^{50} +0.0444463 q^{51} +0.660935 q^{52} -9.09907 q^{53} -1.05566 q^{54} -12.6724 q^{55} -4.23014 q^{56} +0.813495 q^{57} +4.32449 q^{58} +2.05038 q^{59} -0.740165 q^{60} -2.20500 q^{61} +2.59280 q^{62} +12.5581 q^{63} +1.00000 q^{64} -2.76595 q^{65} +0.535568 q^{66} -2.78495 q^{67} +0.251301 q^{68} +1.17715 q^{69} +17.7028 q^{70} -10.8740 q^{71} -2.96872 q^{72} -3.99573 q^{73} -8.50000 q^{74} +2.21320 q^{75} +4.59953 q^{76} -12.8094 q^{77} +0.116896 q^{78} -3.04731 q^{79} -4.18491 q^{80} +8.71945 q^{81} +2.36142 q^{82} +7.29547 q^{83} -0.748164 q^{84} -1.05167 q^{85} +0.881347 q^{86} +0.764850 q^{87} +3.02812 q^{88} +10.3098 q^{89} +12.4238 q^{90} -2.79585 q^{91} +6.65567 q^{92} +0.458575 q^{93} -8.59990 q^{94} -19.2486 q^{95} +0.176865 q^{96} +6.10127 q^{97} +10.8941 q^{98} -8.98964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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