LMFDB - Embedded newform 8047.2.a.c.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$151$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58760 q^{2} -0.469727 q^{3} +4.69569 q^{4} -2.90773 q^{5} +1.21547 q^{6} +0.483013 q^{7} -6.97536 q^{8} -2.77936 q^{9} +O(q^{10})$	\(q-2.58760 q^{2} -0.469727 q^{3} +4.69569 q^{4} -2.90773 q^{5} +1.21547 q^{6} +0.483013 q^{7} -6.97536 q^{8} -2.77936 q^{9} +7.52406 q^{10} -4.39727 q^{11} -2.20569 q^{12} -1.00000 q^{13} -1.24985 q^{14} +1.36584 q^{15} +8.65810 q^{16} +7.71914 q^{17} +7.19187 q^{18} +2.87108 q^{19} -13.6538 q^{20} -0.226884 q^{21} +11.3784 q^{22} -0.0901109 q^{23} +3.27652 q^{24} +3.45492 q^{25} +2.58760 q^{26} +2.71472 q^{27} +2.26808 q^{28} -5.06275 q^{29} -3.53426 q^{30} -4.25059 q^{31} -8.45299 q^{32} +2.06552 q^{33} -19.9741 q^{34} -1.40447 q^{35} -13.0510 q^{36} +2.95659 q^{37} -7.42921 q^{38} +0.469727 q^{39} +20.2825 q^{40} -2.72527 q^{41} +0.587087 q^{42} +7.77014 q^{43} -20.6482 q^{44} +8.08163 q^{45} +0.233171 q^{46} -9.74746 q^{47} -4.06694 q^{48} -6.76670 q^{49} -8.93996 q^{50} -3.62589 q^{51} -4.69569 q^{52} -4.59219 q^{53} -7.02462 q^{54} +12.7861 q^{55} -3.36919 q^{56} -1.34862 q^{57} +13.1004 q^{58} -10.3065 q^{59} +6.41357 q^{60} +0.709858 q^{61} +10.9988 q^{62} -1.34247 q^{63} +4.55677 q^{64} +2.90773 q^{65} -5.34473 q^{66} -4.47804 q^{67} +36.2466 q^{68} +0.0423275 q^{69} +3.63422 q^{70} +12.0115 q^{71} +19.3870 q^{72} +4.09917 q^{73} -7.65047 q^{74} -1.62287 q^{75} +13.4817 q^{76} -2.12394 q^{77} -1.21547 q^{78} -0.208737 q^{79} -25.1755 q^{80} +7.06289 q^{81} +7.05192 q^{82} +3.66352 q^{83} -1.06538 q^{84} -22.4452 q^{85} -20.1060 q^{86} +2.37811 q^{87} +30.6725 q^{88} +0.366173 q^{89} -20.9120 q^{90} -0.483013 q^{91} -0.423132 q^{92} +1.99662 q^{93} +25.2226 q^{94} -8.34834 q^{95} +3.97060 q^{96} +7.61160 q^{97} +17.5095 q^{98} +12.2216 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) $ 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * 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 $

$2$	−2.58760	−1.82971	−0.914856	−	0.403781i	$-0.867696\pi$
$2$	−2.58760	−1.82971	−0.914856	+	0.403781i	$0.867696\pi$
$3$	−0.469727	−0.271197	−0.135599	−	0.990764i	$-0.543296\pi$
$3$	−0.469727	−0.271197	−0.135599	+	0.990764i	$0.543296\pi$
$4$	4.69569	2.34784
$5$	−2.90773	−1.30038	−0.650189	−	0.759772i	$-0.725310\pi$
$5$	−2.90773	−1.30038	−0.650189	+	0.759772i	$0.725310\pi$
$6$	1.21547	0.496212
$7$	0.483013	0.182562	0.0912809	−	0.995825i	$-0.470904\pi$
$7$	0.483013	0.182562	0.0912809	+	0.995825i	$0.470904\pi$
$8$	−6.97536	−2.46616
$9$	−2.77936	−0.926452
$10$	7.52406	2.37932
$11$	−4.39727	−1.32583	−0.662913	−	0.748696i	$-0.730680\pi$
$11$	−4.39727	−1.32583	−0.662913	+	0.748696i	$0.730680\pi$
$12$	−2.20569	−0.636728
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.24985	−0.334035
$15$	1.36584	0.352659
$16$	8.65810	2.16452
$17$	7.71914	1.87217	0.936083	−	0.351780i	$-0.114424\pi$
$17$	7.71914	1.87217	0.936083	+	0.351780i	$0.114424\pi$
$18$	7.19187	1.69514
$19$	2.87108	0.658671	0.329335	−	0.944213i	$-0.393175\pi$
$19$	2.87108	0.658671	0.329335	+	0.944213i	$0.393175\pi$
$20$	−13.6538	−3.05308
$21$	−0.226884	−0.0495102
$22$	11.3784	2.42588
$23$	−0.0901109	−0.0187894	−0.00939471	−	0.999956i	$-0.502990\pi$
$23$	−0.0901109	−0.0187894	−0.00939471	+	0.999956i	$0.502990\pi$
$24$	3.27652	0.668816
$25$	3.45492	0.690984
$26$	2.58760	0.507471
$27$	2.71472	0.522448
$28$	2.26808	0.428626
$29$	−5.06275	−0.940129	−0.470065	−	0.882632i	$-0.655769\pi$
$29$	−5.06275	−0.940129	−0.470065	+	0.882632i	$0.655769\pi$
$30$	−3.53426	−0.645264
$31$	−4.25059	−0.763429	−0.381714	−	0.924280i	$-0.624666\pi$
$31$	−4.25059	−0.763429	−0.381714	+	0.924280i	$0.624666\pi$
$32$	−8.45299	−1.49429
$33$	2.06552	0.359560
$34$	−19.9741	−3.42552
$35$	−1.40447	−0.237399
$36$	−13.0510	−2.17516
$37$	2.95659	0.486060	0.243030	−	0.970019i	$-0.421859\pi$
$37$	2.95659	0.486060	0.243030	+	0.970019i	$0.421859\pi$
$38$	−7.42921	−1.20518
$39$	0.469727	0.0752165
$40$	20.2825	3.20695
$41$	−2.72527	−0.425616	−0.212808	−	0.977094i	$-0.568261\pi$
$41$	−2.72527	−0.425616	−0.212808	+	0.977094i	$0.568261\pi$
$42$	0.587087	0.0905894
$43$	7.77014	1.18494	0.592468	−	0.805594i	$-0.298154\pi$
$43$	7.77014	1.18494	0.592468	+	0.805594i	$0.298154\pi$
$44$	−20.6482	−3.11283
$45$	8.08163	1.20474
$46$	0.233171	0.0343792
$47$	−9.74746	−1.42181	−0.710907	−	0.703286i	$-0.751715\pi$
$47$	−9.74746	−1.42181	−0.710907	+	0.703286i	$0.751715\pi$
$48$	−4.06694	−0.587013
$49$	−6.76670	−0.966671
$50$	−8.93996	−1.26430
$51$	−3.62589	−0.507726
$52$	−4.69569	−0.651175
$53$	−4.59219	−0.630786	−0.315393	−	0.948961i	$-0.602136\pi$
$53$	−4.59219	−0.630786	−0.315393	+	0.948961i	$0.602136\pi$
$54$	−7.02462	−0.955929
$55$	12.7861	1.72408
$56$	−3.36919	−0.450227
$57$	−1.34862	−0.178630
$58$	13.1004	1.72016
$59$	−10.3065	−1.34180	−0.670899	−	0.741549i	$-0.734091\pi$
$59$	−10.3065	−1.34180	−0.670899	+	0.741549i	$0.734091\pi$
$60$	6.41357	0.827988
$61$	0.709858	0.0908880	0.0454440	−	0.998967i	$-0.485530\pi$
$61$	0.709858	0.0908880	0.0454440	+	0.998967i	$0.485530\pi$
$62$	10.9988	1.39685
$63$	−1.34247	−0.169135
$64$	4.55677	0.569596
$65$	2.90773	0.360660
$66$	−5.34473	−0.657891
$67$	−4.47804	−0.547080	−0.273540	−	0.961861i	$-0.588195\pi$
$67$	−4.47804	−0.547080	−0.273540	+	0.961861i	$0.588195\pi$
$68$	36.2466	4.39555
$69$	0.0423275	0.00509564
$70$	3.63422	0.434372
$71$	12.0115	1.42550	0.712750	−	0.701418i	$-0.247449\pi$
$71$	12.0115	1.42550	0.712750	+	0.701418i	$0.247449\pi$
$72$	19.3870	2.28478
$73$	4.09917	0.479772	0.239886	−	0.970801i	$-0.422890\pi$
$73$	4.09917	0.479772	0.239886	+	0.970801i	$0.422890\pi$
$74$	−7.65047	−0.889350
$75$	−1.62287	−0.187393
$76$	13.4817	1.54646
$77$	−2.12394	−0.242045
$78$	−1.21547	−0.137625
$79$	−0.208737	−0.0234848	−0.0117424	−	0.999931i	$-0.503738\pi$
$79$	−0.208737	−0.0234848	−0.0117424	+	0.999931i	$0.503738\pi$
$80$	−25.1755	−2.81470
$81$	7.06289	0.784766
$82$	7.05192	0.778754
$83$	3.66352	0.402123	0.201062	−	0.979579i	$-0.435561\pi$
$83$	3.66352	0.402123	0.201062	+	0.979579i	$0.435561\pi$
$84$	−1.06538	−0.116242
$85$	−22.4452	−2.43452
$86$	−20.1060	−2.16809
$87$	2.37811	0.254960
$88$	30.6725	3.26970
$89$	0.366173	0.0388143	0.0194071	−	0.999812i	$-0.493822\pi$
$89$	0.366173	0.0388143	0.0194071	+	0.999812i	$0.493822\pi$
$90$	−20.9120	−2.20432
$91$	−0.483013	−0.0506335
$92$	−0.423132	−0.0441146
$93$	1.99662	0.207040
$94$	25.2226	2.60151
$95$	−8.34834	−0.856521
$96$	3.97060	0.405247
$97$	7.61160	0.772841	0.386420	−	0.922323i	$-0.373711\pi$
$97$	7.61160	0.772841	0.386420	+	0.922323i	$0.373711\pi$
$98$	17.5095	1.76873
$99$	12.2216	1.22831
$100$	16.2232	1.62232
$101$	−0.504925	−0.0502419	−0.0251210	−	0.999684i	$-0.507997\pi$
$101$	−0.504925	−0.0502419	−0.0251210	+	0.999684i	$0.507997\pi$
$102$	9.38236	0.928992
$103$	15.0667	1.48456	0.742281	−	0.670089i	$-0.233744\pi$
$103$	15.0667	1.48456	0.742281	+	0.670089i	$0.233744\pi$
$104$	6.97536	0.683991
$105$	0.659720	0.0643820
$106$	11.8828	1.15416
$107$	1.59772	0.154457	0.0772287	−	0.997013i	$-0.475393\pi$
$107$	1.59772	0.154457	0.0772287	+	0.997013i	$0.475393\pi$
$108$	12.7475	1.22663
$109$	18.5354	1.77537	0.887686	−	0.460450i	$-0.152312\pi$
$109$	18.5354	1.77537	0.887686	+	0.460450i	$0.152312\pi$
$110$	−33.0853	−3.15456
$111$	−1.38879	−0.131818
$112$	4.18197	0.395159
$113$	9.21879	0.867231	0.433616	−	0.901098i	$-0.357238\pi$
$113$	9.21879	0.867231	0.433616	+	0.901098i	$0.357238\pi$
$114$	3.48970	0.326841
$115$	0.262019	0.0244334
$116$	−23.7731	−2.20728
$117$	2.77936	0.256952
$118$	26.6692	2.45510
$119$	3.72844	0.341786
$120$	−9.52725	−0.869715
$121$	8.33596	0.757815
$122$	−1.83683	−0.166299
$123$	1.28013	0.115426
$124$	−19.9594	−1.79241
$125$	4.49268	0.401837
$126$	3.47377	0.309468
$127$	1.87932	0.166763	0.0833815	−	0.996518i	$-0.473428\pi$
$127$	1.87932	0.166763	0.0833815	+	0.996518i	$0.473428\pi$
$128$	5.11486	0.452094
$129$	−3.64985	−0.321351
$130$	−7.52406	−0.659904
$131$	−1.34287	−0.117327	−0.0586634	−	0.998278i	$-0.518684\pi$
$131$	−1.34287	−0.117327	−0.0586634	+	0.998278i	$0.518684\pi$
$132$	9.69902	0.844191
$133$	1.38677	0.120248
$134$	11.5874	1.00100
$135$	−7.89369	−0.679380
$136$	−53.8438	−4.61707
$137$	−2.41162	−0.206038	−0.103019	−	0.994679i	$-0.532850\pi$
$137$	−2.41162	−0.206038	−0.103019	+	0.994679i	$0.532850\pi$
$138$	−0.109527	−0.00932354
$139$	17.7974	1.50956	0.754779	−	0.655979i	$-0.227744\pi$
$139$	17.7974	1.50956	0.754779	+	0.655979i	$0.227744\pi$
$140$	−6.59497	−0.557377
$141$	4.57865	0.385592
$142$	−31.0809	−2.60826
$143$	4.39727	0.367718
$144$	−24.0639	−2.00533
$145$	14.7211	1.22252
$146$	−10.6070	−0.877844
$147$	3.17850	0.262158
$148$	13.8832	1.14119
$149$	−6.09384	−0.499227	−0.249613	−	0.968346i	$-0.580304\pi$
$149$	−6.09384	−0.499227	−0.249613	+	0.968346i	$0.580304\pi$
$150$	4.19934	0.342875
$151$	−18.2832	−1.48787	−0.743933	−	0.668254i	$-0.767042\pi$
$151$	−18.2832	−1.48787	−0.743933	+	0.668254i	$0.767042\pi$
$152$	−20.0268	−1.62439
$153$	−21.4542	−1.73447
$154$	5.49591	0.442873
$155$	12.3596	0.992746
$156$	2.20569	0.176597
$157$	3.55338	0.283591	0.141795	−	0.989896i	$-0.454713\pi$
$157$	3.55338	0.283591	0.141795	+	0.989896i	$0.454713\pi$
$158$	0.540129	0.0429704
$159$	2.15708	0.171067
$160$	24.5790	1.94314
$161$	−0.0435247	−0.00343023
$162$	−18.2760	−1.43589
$163$	0.814745	0.0638157	0.0319079	−	0.999491i	$-0.489842\pi$
$163$	0.814745	0.0638157	0.0319079	+	0.999491i	$0.489842\pi$
$164$	−12.7970	−0.999280
$165$	−6.00597	−0.467564
$166$	−9.47973	−0.735769
$167$	−9.03079	−0.698823	−0.349412	−	0.936969i	$-0.613619\pi$
$167$	−9.03079	−0.698823	−0.349412	+	0.936969i	$0.613619\pi$
$168$	1.58260	0.122100
$169$	1.00000	0.0769231
$170$	58.0793	4.45448
$171$	−7.97975	−0.610227
$172$	36.4862	2.78204
$173$	−2.87636	−0.218686	−0.109343	−	0.994004i	$-0.534875\pi$
$173$	−2.87636	−0.218686	−0.109343	+	0.994004i	$0.534875\pi$
$174$	−6.15361	−0.466504
$175$	1.66877	0.126147
$176$	−38.0720	−2.86978
$177$	4.84126	0.363892
$178$	−0.947511	−0.0710189
$179$	7.74006	0.578519	0.289259	−	0.957251i	$-0.406591\pi$
$179$	7.74006	0.578519	0.289259	+	0.957251i	$0.406591\pi$
$180$	37.9488	2.82854
$181$	−2.09918	−0.156031	−0.0780154	−	0.996952i	$-0.524858\pi$
$181$	−2.09918	−0.156031	−0.0780154	+	0.996952i	$0.524858\pi$
$182$	1.24985	0.0926447
$183$	−0.333440	−0.0246486
$184$	0.628556	0.0463378
$185$	−8.59697	−0.632062
$186$	−5.16645	−0.378823
$187$	−33.9431	−2.48217
$188$	−45.7710	−3.33820
$189$	1.31125	0.0953791
$190$	21.6022	1.56719
$191$	24.7615	1.79168	0.895841	−	0.444375i	$-0.146574\pi$
$191$	24.7615	1.79168	0.895841	+	0.444375i	$0.146574\pi$
$192$	−2.14044	−0.154473
$193$	−13.2341	−0.952613	−0.476306	−	0.879279i	$-0.658025\pi$
$193$	−13.2341	−0.952613	−0.476306	+	0.879279i	$0.658025\pi$
$194$	−19.6958	−1.41407
$195$	−1.36584	−0.0978100
$196$	−31.7743	−2.26959
$197$	14.7632	1.05183	0.525916	−	0.850536i	$-0.323723\pi$
$197$	14.7632	1.05183	0.525916	+	0.850536i	$0.323723\pi$
$198$	−31.6246	−2.24746
$199$	26.6893	1.89196	0.945978	−	0.324232i	$-0.105106\pi$
$199$	26.6893	1.89196	0.945978	+	0.324232i	$0.105106\pi$
$200$	−24.0993	−1.70408
$201$	2.10346	0.148367
$202$	1.30655	0.0919282
$203$	−2.44537	−0.171632
$204$	−17.0260	−1.19206
$205$	7.92437	0.553462
$206$	−38.9865	−2.71632
$207$	0.250450	0.0174075
$208$	−8.65810	−0.600331
$209$	−12.6249	−0.873283
$210$	−1.70709	−0.117801
$211$	9.50872	0.654607	0.327304	−	0.944919i	$-0.393860\pi$
$211$	9.50872	0.654607	0.327304	+	0.944919i	$0.393860\pi$
$212$	−21.5635	−1.48099
$213$	−5.64212	−0.386592
$214$	−4.13426	−0.282613
$215$	−22.5935	−1.54086
$216$	−18.9362	−1.28844
$217$	−2.05309	−0.139373
$218$	−47.9623	−3.24842
$219$	−1.92549	−0.130113
$220$	60.0395	4.04786
$221$	−7.71914	−0.519245
$222$	3.59364	0.241189
$223$	6.31524	0.422900	0.211450	−	0.977389i	$-0.432181\pi$
$223$	6.31524	0.422900	0.211450	+	0.977389i	$0.432181\pi$
$224$	−4.08290	−0.272800
$225$	−9.60246	−0.640164
$226$	−23.8546	−1.58678
$227$	18.7721	1.24595	0.622974	−	0.782243i	$-0.285924\pi$
$227$	18.7721	1.24595	0.622974	+	0.782243i	$0.285924\pi$
$228$	−6.33272	−0.419394
$229$	1.60988	0.106384	0.0531918	−	0.998584i	$-0.483061\pi$
$229$	1.60988	0.106384	0.0531918	+	0.998584i	$0.483061\pi$
$230$	−0.678000	−0.0447060
$231$	0.997671	0.0656420
$232$	35.3145	2.31851
$233$	−19.9854	−1.30929	−0.654644	−	0.755937i	$-0.727182\pi$
$233$	−19.9854	−1.30929	−0.654644	+	0.755937i	$0.727182\pi$
$234$	−7.19187	−0.470147
$235$	28.3430	1.84890
$236$	−48.3963	−3.15033
$237$	0.0980496	0.00636901
$238$	−9.64773	−0.625369
$239$	−25.1457	−1.62654	−0.813271	−	0.581885i	$-0.802315\pi$
$239$	−25.1457	−1.62654	−0.813271	+	0.581885i	$0.802315\pi$
$240$	11.8256	0.763339
$241$	−2.07034	−0.133363	−0.0666813	−	0.997774i	$-0.521241\pi$
$241$	−2.07034	−0.133363	−0.0666813	+	0.997774i	$0.521241\pi$
$242$	−21.5702	−1.38658
$243$	−11.4618	−0.735274
$244$	3.33327	0.213391
$245$	19.6758	1.25704
$246$	−3.31248	−0.211196
$247$	−2.87108	−0.182682
$248$	29.6494	1.88274
$249$	−1.72085	−0.109055
$250$	−11.6253	−0.735246
$251$	17.2960	1.09172	0.545858	−	0.837878i	$-0.316204\pi$
$251$	17.2960	1.09172	0.545858	+	0.837878i	$0.316204\pi$
$252$	−6.30380	−0.397102
$253$	0.396242	0.0249115
$254$	−4.86294	−0.305128
$255$	10.5431	0.660236
$256$	−22.3488	−1.39680
$257$	−8.77405	−0.547310	−0.273655	−	0.961828i	$-0.588233\pi$
$257$	−8.77405	−0.547310	−0.273655	+	0.961828i	$0.588233\pi$
$258$	9.44435	0.587980
$259$	1.42807	0.0887360
$260$	13.6538	0.846773
$261$	14.0712	0.870985
$262$	3.47480	0.214674
$263$	−1.50233	−0.0926378	−0.0463189	−	0.998927i	$-0.514749\pi$
$263$	−1.50233	−0.0926378	−0.0463189	+	0.998927i	$0.514749\pi$
$264$	−14.4077	−0.886734
$265$	13.3529	0.820261
$266$	−3.58841	−0.220019
$267$	−0.172001	−0.0105263
$268$	−21.0275	−1.28446
$269$	11.6399	0.709697	0.354848	−	0.934924i	$-0.384532\pi$
$269$	11.6399	0.709697	0.354848	+	0.934924i	$0.384532\pi$
$270$	20.4257	1.24307
$271$	−20.8395	−1.26591	−0.632956	−	0.774188i	$-0.718158\pi$
$271$	−20.8395	−1.26591	−0.632956	+	0.774188i	$0.718158\pi$
$272$	66.8330	4.05235
$273$	0.226884	0.0137317
$274$	6.24030	0.376991
$275$	−15.1922	−0.916125
$276$	0.198757	0.0119638
$277$	7.43546	0.446753	0.223377	−	0.974732i	$-0.428292\pi$
$277$	7.43546	0.446753	0.223377	+	0.974732i	$0.428292\pi$
$278$	−46.0527	−2.76205
$279$	11.8139	0.707280
$280$	9.79672	0.585466
$281$	9.45601	0.564098	0.282049	−	0.959400i	$-0.408986\pi$
$281$	9.45601	0.564098	0.282049	+	0.959400i	$0.408986\pi$
$282$	−11.8477	−0.705521
$283$	11.5453	0.686296	0.343148	−	0.939281i	$-0.388507\pi$
$283$	11.5453	0.686296	0.343148	+	0.939281i	$0.388507\pi$
$284$	56.4022	3.34685
$285$	3.92144	0.232286
$286$	−11.3784	−0.672818
$287$	−1.31634	−0.0777012
$288$	23.4939	1.38439
$289$	42.5851	2.50500
$290$	−38.0924	−2.23687
$291$	−3.57537	−0.209592
$292$	19.2484	1.12643
$293$	12.2035	0.712938	0.356469	−	0.934307i	$-0.383981\pi$
$293$	12.2035	0.712938	0.356469	+	0.934307i	$0.383981\pi$
$294$	−8.22470	−0.479674
$295$	29.9687	1.74484
$296$	−20.6233	−1.19870
$297$	−11.9374	−0.692675
$298$	15.7684	0.913441
$299$	0.0901109	0.00521125
$300$	−7.62049	−0.439969
$301$	3.75308	0.216324
$302$	47.3097	2.72236
$303$	0.237177	0.0136255
$304$	24.8581	1.42571
$305$	−2.06408	−0.118189
$306$	55.5150	3.17358
$307$	−23.6864	−1.35185	−0.675927	−	0.736969i	$-0.736256\pi$
$307$	−23.6864	−1.35185	−0.675927	+	0.736969i	$0.736256\pi$
$308$	−9.97335	−0.568284
$309$	−7.07722	−0.402609
$310$	−31.9817	−1.81644
$311$	−10.8643	−0.616059	−0.308030	−	0.951377i	$-0.599670\pi$
$311$	−10.8643	−0.616059	−0.308030	+	0.951377i	$0.599670\pi$
$312$	−3.27652	−0.185496
$313$	−27.3695	−1.54702	−0.773508	−	0.633787i	$-0.781500\pi$
$313$	−27.3695	−1.54702	−0.773508	+	0.633787i	$0.781500\pi$
$314$	−9.19473	−0.518889
$315$	3.90353	0.219939
$316$	−0.980165	−0.0551386
$317$	13.0829	0.734807	0.367403	−	0.930062i	$-0.380247\pi$
$317$	13.0829	0.734807	0.367403	+	0.930062i	$0.380247\pi$
$318$	−5.58166	−0.313004
$319$	22.2623	1.24645
$320$	−13.2499	−0.740691
$321$	−0.750493	−0.0418884
$322$	0.112625	0.00627633
$323$	22.1623	1.23314
$324$	33.1651	1.84251
$325$	−3.45492	−0.191645
$326$	−2.10824	−0.116764
$327$	−8.70659	−0.481476
$328$	19.0098	1.04964
$329$	−4.70815	−0.259569
$330$	15.5411	0.855508
$331$	24.4614	1.34452	0.672260	−	0.740315i	$-0.265323\pi$
$331$	24.4614	1.34452	0.672260	+	0.740315i	$0.265323\pi$
$332$	17.2027	0.944122
$333$	−8.21741	−0.450311
$334$	23.3681	1.27864
$335$	13.0210	0.711411
$336$	−1.96439	−0.107166
$337$	−25.7029	−1.40013	−0.700064	−	0.714080i	$-0.746845\pi$
$337$	−25.7029	−1.40013	−0.700064	+	0.714080i	$0.746845\pi$
$338$	−2.58760	−0.140747
$339$	−4.33032	−0.235191
$340$	−105.396	−5.71588
$341$	18.6910	1.01217
$342$	20.6484	1.11654
$343$	−6.64950	−0.359039
$344$	−54.1996	−2.92225
$345$	−0.123077	−0.00662626
$346$	7.44288	0.400131
$347$	−11.0628	−0.593881	−0.296941	−	0.954896i	$-0.595966\pi$
$347$	−11.0628	−0.593881	−0.296941	+	0.954896i	$0.595966\pi$
$348$	11.1669	0.598607
$349$	15.2024	0.813767	0.406883	−	0.913480i	$-0.366616\pi$
$349$	15.2024	0.813767	0.406883	+	0.913480i	$0.366616\pi$
$350$	−4.31812	−0.230813
$351$	−2.71472	−0.144901
$352$	37.1700	1.98117
$353$	−30.7480	−1.63655	−0.818277	−	0.574825i	$-0.805070\pi$
$353$	−30.7480	−1.63655	−0.818277	+	0.574825i	$0.805070\pi$
$354$	−12.5273	−0.665816
$355$	−34.9262	−1.85369
$356$	1.71943	0.0911298
$357$	−1.75135	−0.0926913
$358$	−20.0282	−1.05852
$359$	−14.1122	−0.744813	−0.372407	−	0.928070i	$-0.621467\pi$
$359$	−14.1122	−0.744813	−0.372407	+	0.928070i	$0.621467\pi$
$360$	−56.3723	−2.97108
$361$	−10.7569	−0.566153
$362$	5.43184	0.285491
$363$	−3.91563	−0.205517
$364$	−2.26808	−0.118880
$365$	−11.9193	−0.623885
$366$	0.862809	0.0450998
$367$	−9.34518	−0.487815	−0.243907	−	0.969799i	$-0.578429\pi$
$367$	−9.34518	−0.487815	−0.243907	+	0.969799i	$0.578429\pi$
$368$	−0.780189	−0.0406702
$369$	7.57450	0.394313
$370$	22.2456	1.15649
$371$	−2.21809	−0.115157
$372$	9.37549	0.486097
$373$	−15.6089	−0.808200	−0.404100	−	0.914715i	$-0.632415\pi$
$373$	−15.6089	−0.808200	−0.404100	+	0.914715i	$0.632415\pi$
$374$	87.8313	4.54165
$375$	−2.11033	−0.108977
$376$	67.9921	3.50643
$377$	5.06275	0.260745
$378$	−3.39298	−0.174516
$379$	−13.3503	−0.685761	−0.342880	−	0.939379i	$-0.611403\pi$
$379$	−13.3503	−0.685761	−0.342880	+	0.939379i	$0.611403\pi$
$380$	−39.2012	−2.01098
$381$	−0.882769	−0.0452256
$382$	−64.0730	−3.27826
$383$	−26.0157	−1.32934	−0.664670	−	0.747137i	$-0.731428\pi$
$383$	−26.0157	−1.32934	−0.664670	+	0.747137i	$0.731428\pi$
$384$	−2.40259	−0.122607
$385$	6.17585	0.314750
$386$	34.2446	1.74301
$387$	−21.5960	−1.09779
$388$	35.7417	1.81451
$389$	25.6729	1.30167	0.650833	−	0.759221i	$-0.274420\pi$
$389$	25.6729	1.30167	0.650833	+	0.759221i	$0.274420\pi$
$390$	3.53426	0.178964
$391$	−0.695578	−0.0351769
$392$	47.2002	2.38397
$393$	0.630781	0.0318187
$394$	−38.2012	−1.92455
$395$	0.606953	0.0305391
$396$	57.3887	2.88389
$397$	3.40391	0.170838	0.0854188	−	0.996345i	$-0.472777\pi$
$397$	3.40391	0.170838	0.0854188	+	0.996345i	$0.472777\pi$
$398$	−69.0613	−3.46173
$399$	−0.651403	−0.0326109
$400$	29.9130	1.49565
$401$	7.81312	0.390169	0.195084	−	0.980786i	$-0.437502\pi$
$401$	7.81312	0.390169	0.195084	+	0.980786i	$0.437502\pi$
$402$	−5.44291	−0.271468
$403$	4.25059	0.211737
$404$	−2.37097	−0.117960
$405$	−20.5370	−1.02049
$406$	6.32766	0.314036
$407$	−13.0009	−0.644431
$408$	25.2919	1.25214
$409$	−1.13488	−0.0561163	−0.0280582	−	0.999606i	$-0.508932\pi$
$409$	−1.13488	−0.0561163	−0.0280582	+	0.999606i	$0.508932\pi$
$410$	−20.5051	−1.01268
$411$	1.13280	0.0558770
$412$	70.7483	3.48552
$413$	−4.97819	−0.244961
$414$	−0.648066	−0.0318507
$415$	−10.6525	−0.522912
$416$	8.45299	0.414442
$417$	−8.35993	−0.409388
$418$	32.6682	1.59786
$419$	4.68149	0.228706	0.114353	−	0.993440i	$-0.463521\pi$
$419$	4.68149	0.228706	0.114353	+	0.993440i	$0.463521\pi$
$420$	3.09784	0.151159
$421$	22.6049	1.10169	0.550847	−	0.834606i	$-0.314305\pi$
$421$	22.6049	1.10169	0.550847	+	0.834606i	$0.314305\pi$
$422$	−24.6048	−1.19774
$423$	27.0917	1.31724
$424$	32.0322	1.55562
$425$	26.6690	1.29364
$426$	14.5996	0.707351
$427$	0.342871	0.0165927
$428$	7.50239	0.362642
$429$	−2.06552	−0.0997241
$430$	58.4630	2.81934
$431$	−14.4160	−0.694395	−0.347197	−	0.937792i	$-0.612867\pi$
$431$	−14.4160	−0.694395	−0.347197	+	0.937792i	$0.612867\pi$
$432$	23.5043	1.13085
$433$	−6.47112	−0.310982	−0.155491	−	0.987837i	$-0.549696\pi$
$433$	−6.47112	−0.310982	−0.155491	+	0.987837i	$0.549696\pi$
$434$	5.31258	0.255012
$435$	−6.91492	−0.331545
$436$	87.0365	4.16829
$437$	−0.258716	−0.0123760
$438$	4.98241	0.238069
$439$	−4.12577	−0.196912	−0.0984562	−	0.995141i	$-0.531390\pi$
$439$	−4.12577	−0.196912	−0.0984562	+	0.995141i	$0.531390\pi$
$440$	−89.1876	−4.25185
$441$	18.8071	0.895575
$442$	19.9741	0.950069
$443$	−34.4462	−1.63659	−0.818295	−	0.574798i	$-0.805081\pi$
$443$	−34.4462	−1.63659	−0.818295	+	0.574798i	$0.805081\pi$
$444$	−6.52132	−0.309488
$445$	−1.06473	−0.0504733
$446$	−16.3413	−0.773784
$447$	2.86244	0.135389
$448$	2.20098	0.103987
$449$	−16.8718	−0.796227	−0.398114	−	0.917336i	$-0.630335\pi$
$449$	−16.8718	−0.796227	−0.398114	+	0.917336i	$0.630335\pi$
$450$	24.8473	1.17132
$451$	11.9838	0.564293
$452$	43.2886	2.03612
$453$	8.58812	0.403505
$454$	−48.5747	−2.27973
$455$	1.40447	0.0658428
$456$	9.40714	0.440530
$457$	−24.4064	−1.14169	−0.570843	−	0.821059i	$-0.693383\pi$
$457$	−24.4064	−1.14169	−0.570843	+	0.821059i	$0.693383\pi$
$458$	−4.16572	−0.194651
$459$	20.9553	0.978110
$460$	1.23036	0.0573657
$461$	0.0682256	0.00317759	0.00158879	−	0.999999i	$-0.499494\pi$
$461$	0.0682256	0.00317759	0.00158879	+	0.999999i	$0.499494\pi$
$462$	−2.58158	−0.120106
$463$	33.9916	1.57972	0.789861	−	0.613285i	$-0.210152\pi$
$463$	33.9916	1.57972	0.789861	+	0.613285i	$0.210152\pi$
$464$	−43.8338	−2.03493
$465$	−5.80563	−0.269230
$466$	51.7143	2.39562
$467$	6.38791	0.295597	0.147799	−	0.989017i	$-0.452781\pi$
$467$	6.38791	0.295597	0.147799	+	0.989017i	$0.452781\pi$
$468$	13.0510	0.603282
$469$	−2.16295	−0.0998759
$470$	−73.3405	−3.38295
$471$	−1.66912	−0.0769089
$472$	71.8919	3.30909
$473$	−34.1674	−1.57102
$474$	−0.253713	−0.0116534
$475$	9.91935	0.455131
$476$	17.5076	0.802460
$477$	12.7633	0.584393
$478$	65.0672	2.97610
$479$	14.8015	0.676299	0.338150	−	0.941092i	$-0.390199\pi$
$479$	14.8015	0.676299	0.338150	+	0.941092i	$0.390199\pi$
$480$	−11.5454	−0.526975
$481$	−2.95659	−0.134809
$482$	5.35723	0.244015
$483$	0.0204448	0.000930268 0
$484$	39.1431	1.77923
$485$	−22.1325	−1.00499
$486$	29.6586	1.34534
$487$	38.1989	1.73096	0.865479	−	0.500945i	$-0.167014\pi$
$487$	38.1989	1.73096	0.865479	+	0.500945i	$0.167014\pi$
$488$	−4.95152	−0.224145
$489$	−0.382708	−0.0173066
$490$	−50.9131	−2.30002
$491$	−3.16854	−0.142994	−0.0714972	−	0.997441i	$-0.522778\pi$
$491$	−3.16854	−0.142994	−0.0714972	+	0.997441i	$0.522778\pi$
$492$	6.01111	0.271002
$493$	−39.0801	−1.76008
$494$	7.42921	0.334256
$495$	−35.5371	−1.59727
$496$	−36.8020	−1.65246
$497$	5.80170	0.260242
$498$	4.45289	0.199539
$499$	−1.01618	−0.0454904	−0.0227452	−	0.999741i	$-0.507241\pi$
$499$	−1.01618	−0.0454904	−0.0227452	+	0.999741i	$0.507241\pi$
$500$	21.0962	0.943451
$501$	4.24201	0.189519
$502$	−44.7553	−1.99753
$503$	12.8638	0.573567	0.286784	−	0.957995i	$-0.407414\pi$
$503$	12.8638	0.573567	0.286784	+	0.957995i	$0.407414\pi$
$504$	9.36419	0.417114
$505$	1.46819	0.0653335
$506$	−1.02532	−0.0455809
$507$	−0.469727	−0.0208613
$508$	8.82471	0.391533
$509$	−0.785030	−0.0347959	−0.0173979	−	0.999849i	$-0.505538\pi$
$509$	−0.785030	−0.0347959	−0.0173979	+	0.999849i	$0.505538\pi$
$510$	−27.2814	−1.20804
$511$	1.97995	0.0875880
$512$	47.6000	2.10364
$513$	7.79418	0.344121
$514$	22.7037	1.00142
$515$	−43.8098	−1.93049
$516$	−17.1385	−0.754482
$517$	42.8622	1.88508
$518$	−3.69528	−0.162361
$519$	1.35110	0.0593069
$520$	−20.2825	−0.889447
$521$	3.20695	0.140499	0.0702496	−	0.997529i	$-0.477620\pi$
$521$	3.20695	0.140499	0.0702496	+	0.997529i	$0.477620\pi$
$522$	−36.4106	−1.59365
$523$	−20.0752	−0.877825	−0.438913	−	0.898530i	$-0.644636\pi$
$523$	−20.0752	−0.877825	−0.438913	+	0.898530i	$0.644636\pi$
$524$	−6.30568	−0.275465
$525$	−0.783868	−0.0342108
$526$	3.88744	0.169500
$527$	−32.8109	−1.42926
$528$	17.8834	0.778277
$529$	−22.9919	−0.999647
$530$	−34.5519	−1.50084
$531$	28.6455	1.24311
$532$	6.51183	0.282324
$533$	2.72527	0.118045
$534$	0.445071	0.0192601
$535$	−4.64575	−0.200853
$536$	31.2360	1.34919
$537$	−3.63571	−0.156893
$538$	−30.1194	−1.29854
$539$	29.7550	1.28164
$540$	−37.0663	−1.59508
$541$	−14.0805	−0.605368	−0.302684	−	0.953091i	$-0.597883\pi$
$541$	−14.0805	−0.605368	−0.302684	+	0.953091i	$0.597883\pi$
$542$	53.9244	2.31625
$543$	0.986042	0.0423151
$544$	−65.2497	−2.79756
$545$	−53.8961	−2.30866
$546$	−0.587087	−0.0251250
$547$	26.8357	1.14741	0.573705	−	0.819062i	$-0.305506\pi$
$547$	26.8357	1.14741	0.573705	+	0.819062i	$0.305506\pi$
$548$	−11.3242	−0.483746
$549$	−1.97295	−0.0842034
$550$	39.3114	1.67624
$551$	−14.5356	−0.619236
$552$	−0.295250	−0.0125667
$553$	−0.100823	−0.00428743
$554$	−19.2400	−0.817429
$555$	4.03823	0.171413
$556$	83.5711	3.54420
$557$	26.7078	1.13164	0.565822	−	0.824528i	$-0.308559\pi$
$557$	26.7078	1.13164	0.565822	+	0.824528i	$0.308559\pi$
$558$	−30.5697	−1.29412
$559$	−7.77014	−0.328642
$560$	−12.1601	−0.513857
$561$	15.9440	0.673156
$562$	−24.4684	−1.03214
$563$	36.1967	1.52551	0.762754	−	0.646689i	$-0.223847\pi$
$563$	36.1967	1.52551	0.762754	+	0.646689i	$0.223847\pi$
$564$	21.4999	0.905309
$565$	−26.8058	−1.12773
$566$	−29.8746	−1.25572
$567$	3.41147	0.143268
$568$	−83.7845	−3.51552
$569$	−29.0976	−1.21983	−0.609916	−	0.792466i	$-0.708797\pi$
$569$	−29.0976	−1.21983	−0.609916	+	0.792466i	$0.708797\pi$
$570$	−10.1471	−0.425017
$571$	−38.4596	−1.60948	−0.804742	−	0.593625i	$-0.797696\pi$
$571$	−38.4596	−1.60948	−0.804742	+	0.593625i	$0.797696\pi$
$572$	20.6482	0.863344
$573$	−11.6312	−0.485899
$574$	3.40617	0.142171
$575$	−0.311326	−0.0129832
$576$	−12.6649	−0.527704
$577$	33.2198	1.38296	0.691479	−	0.722397i	$-0.256959\pi$
$577$	33.2198	1.38296	0.691479	+	0.722397i	$0.256959\pi$
$578$	−110.193	−4.58343
$579$	6.21642	0.258346
$580$	69.1258	2.87029
$581$	1.76953	0.0734123
$582$	9.25165	0.383493
$583$	20.1931	0.836312
$584$	−28.5932	−1.18320
$585$	−8.08163	−0.334134
$586$	−31.5779	−1.30447
$587$	−31.0981	−1.28356	−0.641779	−	0.766890i	$-0.721803\pi$
$587$	−31.0981	−1.28356	−0.641779	+	0.766890i	$0.721803\pi$
$588$	14.9252	0.615507
$589$	−12.2038	−0.502848
$590$	−77.5470	−3.19256
$591$	−6.93466	−0.285254
$592$	25.5984	1.05209
$593$	7.99640	0.328373	0.164186	−	0.986429i	$-0.447500\pi$
$593$	7.99640	0.328373	0.164186	+	0.986429i	$0.447500\pi$
$594$	30.8891	1.26740
$595$	−10.8413	−0.444451
$596$	−28.6147	−1.17211
$597$	−12.5367	−0.513093
$598$	−0.233171	−0.00953508
$599$	−5.69124	−0.232538	−0.116269	−	0.993218i	$-0.537093\pi$
$599$	−5.69124	−0.232538	−0.116269	+	0.993218i	$0.537093\pi$
$600$	11.3201	0.462142
$601$	−10.0483	−0.409878	−0.204939	−	0.978775i	$-0.565700\pi$
$601$	−10.0483	−0.409878	−0.204939	+	0.978775i	$0.565700\pi$
$602$	−9.71148	−0.395810
$603$	12.4461	0.506843
$604$	−85.8522	−3.49328
$605$	−24.2388	−0.985446
$606$	−0.613720	−0.0249307
$607$	2.52178	0.102356	0.0511780	−	0.998690i	$-0.483702\pi$
$607$	2.52178	0.102356	0.0511780	+	0.998690i	$0.483702\pi$
$608$	−24.2692	−0.984246
$609$	1.14866	0.0465460
$610$	5.34102	0.216251
$611$	9.74746	0.394340
$612$	−100.742	−4.07227
$613$	−5.38800	−0.217619	−0.108810	−	0.994063i	$-0.534704\pi$
$613$	−5.38800	−0.217619	−0.108810	+	0.994063i	$0.534704\pi$
$614$	61.2910	2.47350
$615$	−3.72229	−0.150097
$616$	14.8152	0.596923
$617$	−32.6165	−1.31309	−0.656546	−	0.754286i	$-0.727983\pi$
$617$	−32.6165	−1.31309	−0.656546	+	0.754286i	$0.727983\pi$
$618$	18.3130	0.736658
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	58.0368	2.33081
$621$	−0.244626	−0.00981650
$622$	28.1125	1.12721
$623$	0.176866	0.00708600
$624$	4.06694	0.162808
$625$	−30.3381	−1.21352
$626$	70.8214	2.83059
$627$	5.93026	0.236832
$628$	16.6856	0.665826
$629$	22.8223	0.909985
$630$	−10.1008	−0.402425
$631$	−16.2160	−0.645548	−0.322774	−	0.946476i	$-0.604615\pi$
$631$	−16.2160	−0.645548	−0.322774	+	0.946476i	$0.604615\pi$
$632$	1.45602	0.0579173
$633$	−4.46651	−0.177528
$634$	−33.8532	−1.34448
$635$	−5.46457	−0.216855
$636$	10.1290	0.401639
$637$	6.76670	0.268106
$638$	−57.6059	−2.28064
$639$	−33.3842	−1.32066
$640$	−14.8727	−0.587893
$641$	0.0509036	0.00201057	0.00100528	−	0.999999i	$-0.499680\pi$
$641$	0.0509036	0.00201057	0.00100528	+	0.999999i	$0.499680\pi$
$642$	1.94198	0.0766437
$643$	−29.5152	−1.16397	−0.581983	−	0.813201i	$-0.697723\pi$
$643$	−29.5152	−1.16397	−0.581983	+	0.813201i	$0.697723\pi$
$644$	−0.204379	−0.00805364
$645$	10.6128	0.417878
$646$	−57.3471	−2.25629
$647$	−0.159198	−0.00625872	−0.00312936	−	0.999995i	$-0.500996\pi$
$647$	−0.159198	−0.00625872	−0.00312936	+	0.999995i	$0.500996\pi$
$648$	−49.2662	−1.93536
$649$	45.3206	1.77899
$650$	8.93996	0.350654
$651$	0.964393	0.0377975
$652$	3.82578	0.149829
$653$	38.1520	1.49300	0.746501	−	0.665385i	$-0.231732\pi$
$653$	38.1520	1.49300	0.746501	+	0.665385i	$0.231732\pi$
$654$	22.5292	0.880961
$655$	3.90470	0.152569
$656$	−23.5957	−0.921256
$657$	−11.3931	−0.444486
$658$	12.1828	0.474936
$659$	−8.03317	−0.312928	−0.156464	−	0.987684i	$-0.550010\pi$
$659$	−8.03317	−0.312928	−0.156464	+	0.987684i	$0.550010\pi$
$660$	−28.2022	−1.09777
$661$	−18.5184	−0.720281	−0.360141	−	0.932898i	$-0.617271\pi$
$661$	−18.5184	−0.720281	−0.360141	+	0.932898i	$0.617271\pi$
$662$	−63.2964	−2.46009
$663$	3.62589	0.140818
$664$	−25.5544	−0.991702
$665$	−4.03236	−0.156368
$666$	21.2634	0.823940
$667$	0.456209	0.0176645
$668$	−42.4057	−1.64073
$669$	−2.96644	−0.114689
$670$	−33.6931	−1.30168
$671$	−3.12144	−0.120502
$672$	1.91785	0.0739827
$673$	3.82120	0.147296	0.0736482	−	0.997284i	$-0.476536\pi$
$673$	3.82120	0.147296	0.0736482	+	0.997284i	$0.476536\pi$
$674$	66.5090	2.56183
$675$	9.37915	0.361004
$676$	4.69569	0.180603
$677$	15.9280	0.612164	0.306082	−	0.952005i	$-0.400982\pi$
$677$	15.9280	0.612164	0.306082	+	0.952005i	$0.400982\pi$
$678$	11.2051	0.430331
$679$	3.67650	0.141091
$680$	156.563	6.00393
$681$	−8.81777	−0.337898
$682$	−48.3648	−1.85199
$683$	10.8502	0.415172	0.207586	−	0.978217i	$-0.433439\pi$
$683$	10.8502	0.415172	0.207586	+	0.978217i	$0.433439\pi$
$684$	−37.4704	−1.43272
$685$	7.01234	0.267928
$686$	17.2063	0.656938
$687$	−0.756202	−0.0288509
$688$	67.2747	2.56482
$689$	4.59219	0.174949
$690$	0.318475	0.0121241
$691$	−19.3472	−0.736002	−0.368001	−	0.929825i	$-0.619958\pi$
$691$	−19.3472	−0.736002	−0.368001	+	0.929825i	$0.619958\pi$
$692$	−13.5065	−0.513439
$693$	5.90318	0.224243
$694$	28.6261	1.08663
$695$	−51.7502	−1.96300
$696$	−16.5882	−0.628774
$697$	−21.0367	−0.796824
$698$	−39.3378	−1.48896
$699$	9.38770	0.355075
$700$	7.83603	0.296174
$701$	−26.8595	−1.01447	−0.507234	−	0.861808i	$-0.669332\pi$
$701$	−26.8595	−1.01447	−0.507234	+	0.861808i	$0.669332\pi$
$702$	7.02462	0.265127
$703$	8.48860	0.320154
$704$	−20.0373	−0.755186
$705$	−13.3135	−0.501415
$706$	79.5637	2.99442
$707$	−0.243885	−0.00917226
$708$	22.7330	0.854360
$709$	38.8582	1.45935	0.729676	−	0.683793i	$-0.239671\pi$
$709$	38.8582	1.45935	0.729676	+	0.683793i	$0.239671\pi$
$710$	90.3751	3.39172
$711$	0.580156	0.0217575
$712$	−2.55419	−0.0957224
$713$	0.383024	0.0143444
$714$	4.53180	0.169598
$715$	−12.7861	−0.478173
$716$	36.3449	1.35827
$717$	11.8116	0.441114
$718$	36.5168	1.36279
$719$	−34.1342	−1.27299	−0.636495	−	0.771281i	$-0.719617\pi$
$719$	−34.1342	−1.27299	−0.636495	+	0.771281i	$0.719617\pi$
$720$	69.9716	2.60769
$721$	7.27739	0.271024
$722$	27.8346	1.03590
$723$	0.972497	0.0361675
$724$	−9.85709	−0.366336
$725$	−17.4914	−0.649615
$726$	10.1321	0.376037
$727$	−17.4605	−0.647574	−0.323787	−	0.946130i	$-0.604956\pi$
$727$	−17.4605	−0.647574	−0.323787	+	0.946130i	$0.604956\pi$
$728$	3.36919	0.124871
$729$	−15.8048	−0.585361
$730$	30.8424	1.14153
$731$	59.9788	2.21840
$732$	−1.56573	−0.0578710
$733$	43.6603	1.61263	0.806315	−	0.591486i	$-0.201459\pi$
$733$	43.6603	1.61263	0.806315	+	0.591486i	$0.201459\pi$
$734$	24.1816	0.892560
$735$	−9.24224	−0.340905
$736$	0.761706	0.0280769
$737$	19.6912	0.725333
$738$	−19.5998	−0.721479
$739$	24.5510	0.903124	0.451562	−	0.892240i	$-0.350867\pi$
$739$	24.5510	0.903124	0.451562	+	0.892240i	$0.350867\pi$
$740$	−40.3687	−1.48398
$741$	1.34862	0.0495429
$742$	5.73953	0.210705
$743$	−6.80145	−0.249521	−0.124761	−	0.992187i	$-0.539816\pi$
$743$	−6.80145	−0.249521	−0.124761	+	0.992187i	$0.539816\pi$
$744$	−13.9271	−0.510594
$745$	17.7193	0.649184
$746$	40.3897	1.47877
$747$	−10.1822	−0.372548
$748$	−159.386	−5.82774
$749$	0.771720	0.0281980
$750$	5.46070	0.199397
$751$	−23.6604	−0.863381	−0.431691	−	0.902022i	$-0.642083\pi$
$751$	−23.6604	−0.863381	−0.431691	+	0.902022i	$0.642083\pi$
$752$	−84.3945	−3.07755
$753$	−8.12442	−0.296070
$754$	−13.1004	−0.477088
$755$	53.1627	1.93479
$756$	6.15720	0.223935
$757$	−33.8974	−1.23202	−0.616011	−	0.787737i	$-0.711252\pi$
$757$	−33.8974	−1.23202	−0.616011	+	0.787737i	$0.711252\pi$
$758$	34.5454	1.25474
$759$	−0.186125	−0.00675593
$760$	58.2327	2.11232
$761$	1.24601	0.0451678	0.0225839	−	0.999745i	$-0.492811\pi$
$761$	1.24601	0.0451678	0.0225839	+	0.999745i	$0.492811\pi$
$762$	2.28426	0.0827498
$763$	8.95285	0.324115
$764$	116.272	4.20659
$765$	62.3832	2.25547
$766$	67.3183	2.43231
$767$	10.3065	0.372148
$768$	10.4978	0.378808
$769$	−18.0574	−0.651167	−0.325584	−	0.945513i	$-0.605561\pi$
$769$	−18.0574	−0.651167	−0.325584	+	0.945513i	$0.605561\pi$
$770$	−15.9806	−0.575902
$771$	4.12141	0.148429
$772$	−62.1433	−2.23658
$773$	−47.4620	−1.70709	−0.853544	−	0.521021i	$-0.825551\pi$
$773$	−47.4620	−1.70709	−0.853544	+	0.521021i	$0.825551\pi$
$774$	55.8819	2.00863
$775$	−14.6855	−0.527517
$776$	−53.0937	−1.90595
$777$	−0.670804	−0.0240649
$778$	−66.4312	−2.38167
$779$	−7.82447	−0.280341
$780$	−6.41357	−0.229642
$781$	−52.8177	−1.88997
$782$	1.79988	0.0643636
$783$	−13.7440	−0.491169
$784$	−58.5867	−2.09238
$785$	−10.3323	−0.368775
$786$	−1.63221	−0.0582190
$787$	−27.2384	−0.970943	−0.485471	−	0.874253i	$-0.661352\pi$
$787$	−27.2384	−0.970943	−0.485471	+	0.874253i	$0.661352\pi$
$788$	69.3232	2.46954
$789$	0.705686	0.0251231
$790$	−1.57055	−0.0558778
$791$	4.45280	0.158323
$792$	−85.2499	−3.02922
$793$	−0.709858	−0.0252078
$794$	−8.80798	−0.312583
$795$	−6.27221	−0.222452
$796$	125.325	4.44201
$797$	19.8486	0.703074	0.351537	−	0.936174i	$-0.385659\pi$
$797$	19.8486	0.703074	0.351537	+	0.936174i	$0.385659\pi$
$798$	1.68557	0.0596686
$799$	−75.2420	−2.66187
$800$	−29.2044	−1.03253
$801$	−1.01773	−0.0359596
$802$	−20.2173	−0.713896
$803$	−18.0252	−0.636094
$804$	9.87718	0.348341
$805$	0.126558	0.00446060
$806$	−10.9988	−0.387418
$807$	−5.46757	−0.192468
$808$	3.52204	0.123905
$809$	−15.1412	−0.532336	−0.266168	−	0.963927i	$-0.585758\pi$
$809$	−15.1412	−0.532336	−0.266168	+	0.963927i	$0.585758\pi$
$810$	53.1416	1.86721
$811$	4.99345	0.175344	0.0876719	−	0.996149i	$-0.472057\pi$
$811$	4.99345	0.175344	0.0876719	+	0.996149i	$0.472057\pi$
$812$	−11.4827	−0.402964
$813$	9.78889	0.343311
$814$	33.6412	1.17912
$815$	−2.36906	−0.0829846
$816$	−31.3933	−1.09898
$817$	22.3087	0.780483
$818$	2.93663	0.102677
$819$	1.34247	0.0469095
$820$	37.2104	1.29944
$821$	−46.1274	−1.60986	−0.804929	−	0.593371i	$-0.797797\pi$
$821$	−46.1274	−1.60986	−0.804929	+	0.593371i	$0.797797\pi$
$822$	−2.93124	−0.102239
$823$	11.7570	0.409824	0.204912	−	0.978780i	$-0.434309\pi$
$823$	11.7570	0.409824	0.204912	+	0.978780i	$0.434309\pi$
$824$	−105.095	−3.66117
$825$	7.13620	0.248450
$826$	12.8816	0.448208
$827$	−20.8370	−0.724573	−0.362286	−	0.932067i	$-0.618004\pi$
$827$	−20.8370	−0.724573	−0.362286	+	0.932067i	$0.618004\pi$
$828$	1.17604	0.0408701
$829$	43.5810	1.51363	0.756815	−	0.653629i	$-0.226754\pi$
$829$	43.5810	1.51363	0.756815	+	0.653629i	$0.226754\pi$
$830$	27.5645	0.956779
$831$	−3.49264	−0.121158
$832$	−4.55677	−0.157978
$833$	−52.2331	−1.80977
$834$	21.6322	0.749061
$835$	26.2591	0.908735
$836$	−59.2826	−2.05033
$837$	−11.5392	−0.398852
$838$	−12.1138	−0.418465
$839$	−25.0486	−0.864774	−0.432387	−	0.901688i	$-0.642329\pi$
$839$	−25.0486	−0.864774	−0.432387	+	0.901688i	$0.642329\pi$
$840$	−4.60178	−0.158777
$841$	−3.36855	−0.116157
$842$	−58.4925	−2.01578
$843$	−4.44174	−0.152982
$844$	44.6500	1.53692
$845$	−2.90773	−0.100029
$846$	−70.1025	−2.41017
$847$	4.02638	0.138348
$848$	−39.7596	−1.36535
$849$	−5.42313	−0.186121
$850$	−69.0088	−2.36698
$851$	−0.266421	−0.00913279
$852$	−26.4936	−0.907657
$853$	−27.0775	−0.927117	−0.463558	−	0.886066i	$-0.653428\pi$
$853$	−27.0775	−0.927117	−0.463558	+	0.886066i	$0.653428\pi$
$854$	−0.887213	−0.0303598
$855$	23.2030	0.793526
$856$	−11.1447	−0.380917
$857$	2.17353	0.0742463	0.0371232	−	0.999311i	$-0.488181\pi$
$857$	2.17353	0.0742463	0.0371232	+	0.999311i	$0.488181\pi$
$858$	5.34473	0.182466
$859$	−14.4654	−0.493552	−0.246776	−	0.969073i	$-0.579371\pi$
$859$	−14.4654	−0.493552	−0.246776	+	0.969073i	$0.579371\pi$
$860$	−106.092	−3.61771
$861$	0.618322	0.0210723
$862$	37.3029	1.27054
$863$	−20.4663	−0.696682	−0.348341	−	0.937368i	$-0.613255\pi$
$863$	−20.4663	−0.696682	−0.348341	+	0.937368i	$0.613255\pi$
$864$	−22.9475	−0.780690
$865$	8.36369	0.284374
$866$	16.7447	0.569007
$867$	−20.0034	−0.679350
$868$	−9.64067	−0.327226
$869$	0.917874	0.0311367
$870$	17.8931	0.606631
$871$	4.47804	0.151733
$872$	−129.291	−4.37836
$873$	−21.1553	−0.716000
$874$	0.669453	0.0226446
$875$	2.17002	0.0733602
$876$	−9.04151	−0.305484
$877$	−11.0121	−0.371852	−0.185926	−	0.982564i	$-0.559528\pi$
$877$	−11.0121	−0.371852	−0.185926	+	0.982564i	$0.559528\pi$
$878$	10.6759	0.360293
$879$	−5.73233	−0.193347
$880$	110.703	3.73180
$881$	55.6911	1.87628	0.938141	−	0.346253i	$-0.112546\pi$
$881$	55.6911	1.87628	0.938141	+	0.346253i	$0.112546\pi$
$882$	−48.6652	−1.63864
$883$	−34.3647	−1.15647	−0.578233	−	0.815872i	$-0.696257\pi$
$883$	−34.3647	−1.15647	−0.578233	+	0.815872i	$0.696257\pi$
$884$	−36.2466	−1.21911
$885$	−14.0771	−0.473197
$886$	89.1332	2.99449
$887$	−23.1335	−0.776748	−0.388374	−	0.921502i	$-0.626963\pi$
$887$	−23.1335	−0.776748	−0.388374	+	0.921502i	$0.626963\pi$
$888$	9.68732	0.325085
$889$	0.907738	0.0304445
$890$	2.75511	0.0923515
$891$	−31.0574	−1.04046
$892$	29.6544	0.992902
$893$	−27.9857	−0.936507
$894$	−7.40686	−0.247722
$895$	−22.5060	−0.752294
$896$	2.47054	0.0825351
$897$	−0.0423275	−0.00141328
$898$	43.6574	1.45687
$899$	21.5197	0.717721
$900$	−45.0901	−1.50300
$901$	−35.4477	−1.18094
$902$	−31.0092	−1.03249
$903$	−1.76292	−0.0586664
$904$	−64.3045	−2.13873
$905$	6.10386	0.202899
$906$	−22.2226	−0.738297
$907$	−16.0588	−0.533223	−0.266612	−	0.963804i	$-0.585904\pi$
$907$	−16.0588	−0.533223	−0.266612	+	0.963804i	$0.585904\pi$
$908$	88.1479	2.92529
$909$	1.40337	0.0465467
$910$	−3.63422	−0.120473
$911$	20.6380	0.683769	0.341885	−	0.939742i	$-0.388935\pi$
$911$	20.6380	0.683769	0.341885	+	0.939742i	$0.388935\pi$
$912$	−11.6765	−0.386648
$913$	−16.1095	−0.533145
$914$	63.1542	2.08895
$915$	0.969554	0.0320525
$916$	7.55947	0.249772
$917$	−0.648622	−0.0214194
$918$	−54.2240	−1.78966
$919$	−20.6074	−0.679776	−0.339888	−	0.940466i	$-0.610389\pi$
$919$	−20.6074	−0.679776	−0.339888	+	0.940466i	$0.610389\pi$
$920$	−1.82767	−0.0602567
$921$	11.1261	0.366619
$922$	−0.176541	−0.00581406
$923$	−12.0115	−0.395363
$924$	4.68475	0.154117
$925$	10.2148	0.335860
$926$	−87.9567	−2.89044
$927$	−41.8756	−1.37538
$928$	42.7954	1.40483
$929$	50.6257	1.66097	0.830487	−	0.557038i	$-0.188062\pi$
$929$	50.6257	1.66097	0.830487	+	0.557038i	$0.188062\pi$
$930$	15.0227	0.492613
$931$	−19.4277	−0.636718
$932$	−93.8453	−3.07401
$933$	5.10327	0.167073
$934$	−16.5294	−0.540858
$935$	98.6975	3.22775
$936$	−19.3870	−0.633685
$937$	8.35208	0.272851	0.136425	−	0.990650i	$-0.456439\pi$
$937$	8.35208	0.272851	0.136425	+	0.990650i	$0.456439\pi$
$938$	5.59686	0.182744
$939$	12.8562	0.419546
$940$	133.090	4.34092
$941$	25.4540	0.829776	0.414888	−	0.909872i	$-0.363821\pi$
$941$	25.4540	0.829776	0.414888	+	0.909872i	$0.363821\pi$
$942$	4.31902	0.140721
$943$	0.245577	0.00799708
$944$	−89.2350	−2.90435
$945$	−3.81275	−0.124029
$946$	88.4116	2.87451
$947$	−41.4985	−1.34852	−0.674260	−	0.738494i	$-0.735537\pi$
$947$	−41.4985	−1.34852	−0.674260	+	0.738494i	$0.735537\pi$
$948$	0.460410	0.0149534
$949$	−4.09917	−0.133065
$950$	−25.6673	−0.832759
$951$	−6.14538	−0.199277
$952$	−26.0073	−0.842900
$953$	−14.9231	−0.483405	−0.241703	−	0.970350i	$-0.577706\pi$
$953$	−14.9231	−0.483405	−0.241703	+	0.970350i	$0.577706\pi$
$954$	−33.0264	−1.06927
$955$	−72.0000	−2.32986
$956$	−118.076	−3.81887
$957$	−10.4572	−0.338033
$958$	−38.3005	−1.23743
$959$	−1.16484	−0.0376147
$960$	6.22383	0.200873
$961$	−12.9325	−0.417177
$962$	7.65047	0.246661
$963$	−4.44063	−0.143097
$964$	−9.72169	−0.313114
$965$	38.4813	1.23876
$966$	−0.0529029	−0.00170212
$967$	60.7250	1.95278	0.976392	−	0.216007i	$-0.0693036\pi$
$967$	60.7250	1.95278	0.976392	+	0.216007i	$0.0693036\pi$
$968$	−58.1464	−1.86889
$969$	−10.4102	−0.334424
$970$	57.2701	1.83883
$971$	19.4645	0.624644	0.312322	−	0.949976i	$-0.398893\pi$
$971$	19.4645	0.624644	0.312322	+	0.949976i	$0.398893\pi$
$972$	−53.8210	−1.72631
$973$	8.59639	0.275588
$974$	−98.8436	−3.16715
$975$	1.62287	0.0519735
$976$	6.14602	0.196729
$977$	4.52923	0.144903	0.0724515	−	0.997372i	$-0.476918\pi$
$977$	4.52923	0.144903	0.0724515	+	0.997372i	$0.476918\pi$
$978$	0.990295	0.0316662
$979$	−1.61016	−0.0514610
$980$	92.3912	2.95133
$981$	−51.5165	−1.64480
$982$	8.19893	0.261638
$983$	26.6749	0.850796	0.425398	−	0.905006i	$-0.360134\pi$
$983$	26.6749	0.850796	0.425398	+	0.905006i	$0.360134\pi$
$984$	−8.92940	−0.284659
$985$	−42.9274	−1.36778
$986$	101.124	3.22043
$987$	2.21155	0.0703943
$988$	−13.4817	−0.428910
$989$	−0.700174	−0.0222643
$990$	91.9559	2.92255
$991$	34.2155	1.08689	0.543446	−	0.839444i	$-0.317119\pi$
$991$	34.2155	1.08689	0.543446	+	0.839444i	$0.317119\pi$
$992$	35.9302	1.14078
$993$	−11.4902	−0.364630
$994$	−15.0125	−0.476168
$995$	−77.6054	−2.46026
$996$	−8.08059	−0.256043
$997$	−9.90247	−0.313614	−0.156807	−	0.987629i	$-0.550120\pi$
$997$	−9.90247	−0.313614	−0.156807	+	0.987629i	$0.550120\pi$
$998$	2.62946	0.0832342
$999$	8.02631	0.253941

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.c.1.12	✓	151

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.c.1.12	✓	151	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58760 q^{2} -0.469727 q^{3} +4.69569 q^{4} -2.90773 q^{5} +1.21547 q^{6} +0.483013 q^{7} -6.97536 q^{8} -2.77936 q^{9} +O(q^{10})\)	\(q-2.58760 q^{2} -0.469727 q^{3} +4.69569 q^{4} -2.90773 q^{5} +1.21547 q^{6} +0.483013 q^{7} -6.97536 q^{8} -2.77936 q^{9} +7.52406 q^{10} -4.39727 q^{11} -2.20569 q^{12} -1.00000 q^{13} -1.24985 q^{14} +1.36584 q^{15} +8.65810 q^{16} +7.71914 q^{17} +7.19187 q^{18} +2.87108 q^{19} -13.6538 q^{20} -0.226884 q^{21} +11.3784 q^{22} -0.0901109 q^{23} +3.27652 q^{24} +3.45492 q^{25} +2.58760 q^{26} +2.71472 q^{27} +2.26808 q^{28} -5.06275 q^{29} -3.53426 q^{30} -4.25059 q^{31} -8.45299 q^{32} +2.06552 q^{33} -19.9741 q^{34} -1.40447 q^{35} -13.0510 q^{36} +2.95659 q^{37} -7.42921 q^{38} +0.469727 q^{39} +20.2825 q^{40} -2.72527 q^{41} +0.587087 q^{42} +7.77014 q^{43} -20.6482 q^{44} +8.08163 q^{45} +0.233171 q^{46} -9.74746 q^{47} -4.06694 q^{48} -6.76670 q^{49} -8.93996 q^{50} -3.62589 q^{51} -4.69569 q^{52} -4.59219 q^{53} -7.02462 q^{54} +12.7861 q^{55} -3.36919 q^{56} -1.34862 q^{57} +13.1004 q^{58} -10.3065 q^{59} +6.41357 q^{60} +0.709858 q^{61} +10.9988 q^{62} -1.34247 q^{63} +4.55677 q^{64} +2.90773 q^{65} -5.34473 q^{66} -4.47804 q^{67} +36.2466 q^{68} +0.0423275 q^{69} +3.63422 q^{70} +12.0115 q^{71} +19.3870 q^{72} +4.09917 q^{73} -7.65047 q^{74} -1.62287 q^{75} +13.4817 q^{76} -2.12394 q^{77} -1.21547 q^{78} -0.208737 q^{79} -25.1755 q^{80} +7.06289 q^{81} +7.05192 q^{82} +3.66352 q^{83} -1.06538 q^{84} -22.4452 q^{85} -20.1060 q^{86} +2.37811 q^{87} +30.6725 q^{88} +0.366173 q^{89} -20.9120 q^{90} -0.483013 q^{91} -0.423132 q^{92} +1.99662 q^{93} +25.2226 q^{94} -8.34834 q^{95} +3.97060 q^{96} +7.61160 q^{97} +17.5095 q^{98} +12.2216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * q^99

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