LMFDB - Embedded newform 8047.2.a.e.1.8

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:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-2.56793 q^{2} +0.140851 q^{3} +4.59428 q^{4} -0.0428559 q^{5} -0.361697 q^{6} -5.13541 q^{7} -6.66194 q^{8} -2.98016 q^{9} +O(q^{10})$	\(q-2.56793 q^{2} +0.140851 q^{3} +4.59428 q^{4} -0.0428559 q^{5} -0.361697 q^{6} -5.13541 q^{7} -6.66194 q^{8} -2.98016 q^{9} +0.110051 q^{10} +1.99789 q^{11} +0.647111 q^{12} +1.00000 q^{13} +13.1874 q^{14} -0.00603632 q^{15} +7.91887 q^{16} +3.74455 q^{17} +7.65285 q^{18} +2.97791 q^{19} -0.196892 q^{20} -0.723329 q^{21} -5.13044 q^{22} -6.20818 q^{23} -0.938344 q^{24} -4.99816 q^{25} -2.56793 q^{26} -0.842314 q^{27} -23.5935 q^{28} -8.91123 q^{29} +0.0155009 q^{30} -8.69920 q^{31} -7.01123 q^{32} +0.281405 q^{33} -9.61577 q^{34} +0.220083 q^{35} -13.6917 q^{36} -3.68037 q^{37} -7.64707 q^{38} +0.140851 q^{39} +0.285504 q^{40} +0.721146 q^{41} +1.85746 q^{42} +2.39562 q^{43} +9.17886 q^{44} +0.127718 q^{45} +15.9422 q^{46} -5.97305 q^{47} +1.11538 q^{48} +19.3724 q^{49} +12.8350 q^{50} +0.527426 q^{51} +4.59428 q^{52} +0.683044 q^{53} +2.16301 q^{54} -0.0856214 q^{55} +34.2118 q^{56} +0.419443 q^{57} +22.8834 q^{58} -7.51937 q^{59} -0.0277326 q^{60} -4.57599 q^{61} +22.3390 q^{62} +15.3043 q^{63} +2.16665 q^{64} -0.0428559 q^{65} -0.722630 q^{66} +1.34504 q^{67} +17.2035 q^{68} -0.874430 q^{69} -0.565158 q^{70} -7.72515 q^{71} +19.8537 q^{72} +11.9895 q^{73} +9.45094 q^{74} -0.703998 q^{75} +13.6814 q^{76} -10.2600 q^{77} -0.361697 q^{78} +1.77408 q^{79} -0.339370 q^{80} +8.82184 q^{81} -1.85185 q^{82} -2.25580 q^{83} -3.32318 q^{84} -0.160476 q^{85} -6.15179 q^{86} -1.25516 q^{87} -13.3098 q^{88} -9.22886 q^{89} -0.327970 q^{90} -5.13541 q^{91} -28.5221 q^{92} -1.22529 q^{93} +15.3384 q^{94} -0.127621 q^{95} -0.987542 q^{96} +0.637632 q^{97} -49.7471 q^{98} -5.95403 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * 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 $

$2$	−2.56793	−1.81580	−0.907902	−	0.419183i	$-0.862316\pi$
$2$	−2.56793	−1.81580	−0.907902	+	0.419183i	$0.862316\pi$
$3$	0.140851	0.0813206	0.0406603	−	0.999173i	$-0.487054\pi$
$3$	0.140851	0.0813206	0.0406603	+	0.999173i	$0.487054\pi$
$4$	4.59428	2.29714
$5$	−0.0428559	−0.0191658	−0.00958288	−	0.999954i	$-0.503050\pi$
$5$	−0.0428559	−0.0191658	−0.00958288	+	0.999954i	$0.503050\pi$
$6$	−0.361697	−0.147662
$7$	−5.13541	−1.94100	−0.970501	−	0.241097i	$-0.922493\pi$
$7$	−5.13541	−1.94100	−0.970501	+	0.241097i	$0.922493\pi$
$8$	−6.66194	−2.35535
$9$	−2.98016	−0.993387
$10$	0.110051	0.0348012
$11$	1.99789	0.602386	0.301193	−	0.953563i	$-0.402615\pi$
$11$	1.99789	0.602386	0.301193	+	0.953563i	$0.402615\pi$
$12$	0.647111	0.186805
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	13.1874	3.52448
$15$	−0.00603632	−0.00155857
$16$	7.91887	1.97972
$17$	3.74455	0.908188	0.454094	−	0.890954i	$-0.349963\pi$
$17$	3.74455	0.908188	0.454094	+	0.890954i	$0.349963\pi$
$18$	7.65285	1.80380
$19$	2.97791	0.683179	0.341590	−	0.939849i	$-0.389035\pi$
$19$	2.97791	0.683179	0.341590	+	0.939849i	$0.389035\pi$
$20$	−0.196892	−0.0440265
$21$	−0.723329	−0.157843
$22$	−5.13044	−1.09381
$23$	−6.20818	−1.29449	−0.647247	−	0.762280i	$-0.724080\pi$
$23$	−6.20818	−1.29449	−0.647247	+	0.762280i	$0.724080\pi$
$24$	−0.938344	−0.191539
$25$	−4.99816	−0.999633
$26$	−2.56793	−0.503613
$27$	−0.842314	−0.162103
$28$	−23.5935	−4.45876
$29$	−8.91123	−1.65477	−0.827387	−	0.561633i	$-0.810173\pi$
$29$	−8.91123	−1.65477	−0.827387	+	0.561633i	$0.810173\pi$
$30$	0.0155009	0.00283006
$31$	−8.69920	−1.56242	−0.781211	−	0.624267i	$-0.785398\pi$
$31$	−8.69920	−1.56242	−0.781211	+	0.624267i	$0.785398\pi$
$32$	−7.01123	−1.23942
$33$	0.281405	0.0489864
$34$	−9.61577	−1.64909
$35$	0.220083	0.0372008
$36$	−13.6917	−2.28195
$37$	−3.68037	−0.605048	−0.302524	−	0.953142i	$-0.597829\pi$
$37$	−3.68037	−0.605048	−0.302524	+	0.953142i	$0.597829\pi$
$38$	−7.64707	−1.24052
$39$	0.140851	0.0225543
$40$	0.285504	0.0451421
$41$	0.721146	0.112624	0.0563120	−	0.998413i	$-0.482066\pi$
$41$	0.721146	0.112624	0.0563120	+	0.998413i	$0.482066\pi$
$42$	1.85746	0.286613
$43$	2.39562	0.365328	0.182664	−	0.983175i	$-0.441528\pi$
$43$	2.39562	0.365328	0.182664	+	0.983175i	$0.441528\pi$
$44$	9.17886	1.38377
$45$	0.127718	0.0190390
$46$	15.9422	2.35055
$47$	−5.97305	−0.871258	−0.435629	−	0.900126i	$-0.643474\pi$
$47$	−5.97305	−0.871258	−0.435629	+	0.900126i	$0.643474\pi$
$48$	1.11538	0.160992
$49$	19.3724	2.76749
$50$	12.8350	1.81514
$51$	0.527426	0.0738544
$52$	4.59428	0.637112
$53$	0.683044	0.0938234	0.0469117	−	0.998899i	$-0.485062\pi$
$53$	0.683044	0.0938234	0.0469117	+	0.998899i	$0.485062\pi$
$54$	2.16301	0.294348
$55$	−0.0856214	−0.0115452
$56$	34.2118	4.57175
$57$	0.419443	0.0555565
$58$	22.8834	3.00474
$59$	−7.51937	−0.978939	−0.489469	−	0.872021i	$-0.662809\pi$
$59$	−7.51937	−0.978939	−0.489469	+	0.872021i	$0.662809\pi$
$60$	−0.0277326	−0.00358026
$61$	−4.57599	−0.585895	−0.292948	−	0.956128i	$-0.594636\pi$
$61$	−4.57599	−0.585895	−0.292948	+	0.956128i	$0.594636\pi$
$62$	22.3390	2.83705
$63$	15.3043	1.92817
$64$	2.16665	0.270831
$65$	−0.0428559	−0.00531563
$66$	−0.722630	−0.0889496
$67$	1.34504	0.164323	0.0821613	−	0.996619i	$-0.473818\pi$
$67$	1.34504	0.164323	0.0821613	+	0.996619i	$0.473818\pi$
$68$	17.2035	2.08624
$69$	−0.874430	−0.105269
$70$	−0.565158	−0.0675493
$71$	−7.72515	−0.916806	−0.458403	−	0.888744i	$-0.651578\pi$
$71$	−7.72515	−0.916806	−0.458403	+	0.888744i	$0.651578\pi$
$72$	19.8537	2.33978
$73$	11.9895	1.40326	0.701632	−	0.712540i	$-0.252455\pi$
$73$	11.9895	1.40326	0.701632	+	0.712540i	$0.252455\pi$
$74$	9.45094	1.09865
$75$	−0.703998	−0.0812907
$76$	13.6814	1.56936
$77$	−10.2600	−1.16923
$78$	−0.361697	−0.0409541
$79$	1.77408	0.199599	0.0997997	−	0.995008i	$-0.468180\pi$
$79$	1.77408	0.199599	0.0997997	+	0.995008i	$0.468180\pi$
$80$	−0.339370	−0.0379428
$81$	8.82184	0.980205
$82$	−1.85185	−0.204503
$83$	−2.25580	−0.247607	−0.123803	−	0.992307i	$-0.539509\pi$
$83$	−2.25580	−0.247607	−0.123803	+	0.992307i	$0.539509\pi$
$84$	−3.32318	−0.362589
$85$	−0.160476	−0.0174061
$86$	−6.15179	−0.663364
$87$	−1.25516	−0.134567
$88$	−13.3098	−1.41883
$89$	−9.22886	−0.978258	−0.489129	−	0.872212i	$-0.662685\pi$
$89$	−9.22886	−0.978258	−0.489129	+	0.872212i	$0.662685\pi$
$90$	−0.327970	−0.0345711
$91$	−5.13541	−0.538337
$92$	−28.5221	−2.97364
$93$	−1.22529	−0.127057
$94$	15.3384	1.58203
$95$	−0.127621	−0.0130937
$96$	−0.987542	−0.100791
$97$	0.637632	0.0647417	0.0323709	−	0.999476i	$-0.489694\pi$
$97$	0.637632	0.0647417	0.0323709	+	0.999476i	$0.489694\pi$
$98$	−49.7471	−5.02521
$99$	−5.95403	−0.598402
$100$	−22.9630	−2.29630
$101$	−3.24877	−0.323264	−0.161632	−	0.986851i	$-0.551676\pi$
$101$	−3.24877	−0.323264	−0.161632	+	0.986851i	$0.551676\pi$
$102$	−1.35439	−0.134105
$103$	2.23322	0.220046	0.110023	−	0.993929i	$-0.464908\pi$
$103$	2.23322	0.220046	0.110023	+	0.993929i	$0.464908\pi$
$104$	−6.66194	−0.653257
$105$	0.0309990	0.00302519
$106$	−1.75401	−0.170365
$107$	7.31144	0.706824	0.353412	−	0.935468i	$-0.385021\pi$
$107$	7.31144	0.706824	0.353412	+	0.935468i	$0.385021\pi$
$108$	−3.86983	−0.372374
$109$	−16.0578	−1.53806	−0.769031	−	0.639211i	$-0.779261\pi$
$109$	−16.0578	−1.53806	−0.769031	+	0.639211i	$0.779261\pi$
$110$	0.219870	0.0209638
$111$	−0.518385	−0.0492029
$112$	−40.6666	−3.84263
$113$	11.6606	1.09694	0.548470	−	0.836170i	$-0.315210\pi$
$113$	11.6606	1.09694	0.548470	+	0.836170i	$0.315210\pi$
$114$	−1.07710	−0.100880
$115$	0.266057	0.0248100
$116$	−40.9407	−3.80125
$117$	−2.98016	−0.275516
$118$	19.3092	1.77756
$119$	−19.2298	−1.76279
$120$	0.0402136	0.00367098
$121$	−7.00844	−0.637131
$122$	11.7508	1.06387
$123$	0.101574	0.00915865
$124$	−39.9666	−3.58911
$125$	0.428481	0.0383245
$126$	−39.3005	−3.50117
$127$	14.0236	1.24439	0.622195	−	0.782862i	$-0.286241\pi$
$127$	14.0236	1.24439	0.622195	+	0.782862i	$0.286241\pi$
$128$	8.45866	0.747647
$129$	0.337426	0.0297087
$130$	0.110051	0.00965213
$131$	8.35041	0.729579	0.364789	−	0.931090i	$-0.381141\pi$
$131$	8.35041	0.729579	0.364789	+	0.931090i	$0.381141\pi$
$132$	1.29286	0.112529
$133$	−15.2928	−1.32605
$134$	−3.45397	−0.298378
$135$	0.0360982	0.00310683
$136$	−24.9460	−2.13910
$137$	1.48199	0.126615	0.0633073	−	0.997994i	$-0.479835\pi$
$137$	1.48199	0.126615	0.0633073	+	0.997994i	$0.479835\pi$
$138$	2.24548	0.191148
$139$	−18.8961	−1.60275	−0.801374	−	0.598163i	$-0.795897\pi$
$139$	−18.8961	−1.60275	−0.801374	+	0.598163i	$0.795897\pi$
$140$	1.01112	0.0854554
$141$	−0.841312	−0.0708512
$142$	19.8377	1.66474
$143$	1.99789	0.167072
$144$	−23.5995	−1.96662
$145$	0.381899	0.0317150
$146$	−30.7882	−2.54805
$147$	2.72863	0.225054
$148$	−16.9086	−1.38988
$149$	−12.5766	−1.03031	−0.515156	−	0.857096i	$-0.672266\pi$
$149$	−12.5766	−1.03031	−0.515156	+	0.857096i	$0.672266\pi$
$150$	1.80782	0.147608
$151$	−19.9373	−1.62248	−0.811238	−	0.584716i	$-0.801206\pi$
$151$	−19.9373	−1.62248	−0.811238	+	0.584716i	$0.801206\pi$
$152$	−19.8387	−1.60913
$153$	−11.1594	−0.902182
$154$	26.3469	2.12310
$155$	0.372812	0.0299450
$156$	0.647111	0.0518103
$157$	12.5693	1.00314	0.501570	−	0.865117i	$-0.332756\pi$
$157$	12.5693	1.00314	0.501570	+	0.865117i	$0.332756\pi$
$158$	−4.55572	−0.362433
$159$	0.0962077	0.00762977
$160$	0.300473	0.0237545
$161$	31.8815	2.51262
$162$	−22.6539	−1.77986
$163$	−19.7424	−1.54634	−0.773172	−	0.634197i	$-0.781331\pi$
$163$	−19.7424	−1.54634	−0.773172	+	0.634197i	$0.781331\pi$
$164$	3.31315	0.258713
$165$	−0.0120599	−0.000938861 0
$166$	5.79276	0.449605
$167$	−5.08257	−0.393301	−0.196651	−	0.980474i	$-0.563006\pi$
$167$	−5.08257	−0.393301	−0.196651	+	0.980474i	$0.563006\pi$
$168$	4.81878	0.371777
$169$	1.00000	0.0769231
$170$	0.412093	0.0316061
$171$	−8.87465	−0.678662
$172$	11.0061	0.839211
$173$	17.5395	1.33351	0.666753	−	0.745278i	$-0.267683\pi$
$173$	17.5395	1.33351	0.666753	+	0.745278i	$0.267683\pi$
$174$	3.22316	0.244347
$175$	25.6676	1.94029
$176$	15.8210	1.19255
$177$	−1.05911	−0.0796078
$178$	23.6991	1.77632
$179$	−21.1125	−1.57802	−0.789011	−	0.614380i	$-0.789406\pi$
$179$	−21.1125	−1.57802	−0.789011	+	0.614380i	$0.789406\pi$
$180$	0.586771	0.0437353
$181$	−6.08251	−0.452109	−0.226055	−	0.974115i	$-0.572583\pi$
$181$	−6.08251	−0.452109	−0.226055	+	0.974115i	$0.572583\pi$
$182$	13.1874	0.977514
$183$	−0.644534	−0.0476453
$184$	41.3585	3.04899
$185$	0.157726	0.0115962
$186$	3.14647	0.230711
$187$	7.48120	0.547080
$188$	−27.4419	−2.00140
$189$	4.32563	0.314643
$190$	0.327723	0.0237755
$191$	−0.417139	−0.0301831	−0.0150916	−	0.999886i	$-0.504804\pi$
$191$	−0.417139	−0.0301831	−0.0150916	+	0.999886i	$0.504804\pi$
$192$	0.305175	0.0220241
$193$	−12.2242	−0.879920	−0.439960	−	0.898017i	$-0.645007\pi$
$193$	−12.2242	−0.879920	−0.439960	+	0.898017i	$0.645007\pi$
$194$	−1.63740	−0.117558
$195$	−0.00603632	−0.000432270 0
$196$	89.0024	6.35731
$197$	−0.324575	−0.0231250	−0.0115625	−	0.999933i	$-0.503681\pi$
$197$	−0.324575	−0.0231250	−0.0115625	+	0.999933i	$0.503681\pi$
$198$	15.2895	1.08658
$199$	3.18705	0.225924	0.112962	−	0.993599i	$-0.463966\pi$
$199$	3.18705	0.225924	0.112962	+	0.993599i	$0.463966\pi$
$200$	33.2975	2.35449
$201$	0.189451	0.0133628
$202$	8.34261	0.586984
$203$	45.7628	3.21192
$204$	2.42314	0.169654
$205$	−0.0309054	−0.00215853
$206$	−5.73476	−0.399560
$207$	18.5014	1.28593
$208$	7.91887	0.549075
$209$	5.94953	0.411538
$210$	−0.0796033	−0.00549315
$211$	4.97021	0.342163	0.171082	−	0.985257i	$-0.445274\pi$
$211$	4.97021	0.342163	0.171082	+	0.985257i	$0.445274\pi$
$212$	3.13810	0.215525
$213$	−1.08810	−0.0745552
$214$	−18.7753	−1.28345
$215$	−0.102666	−0.00700179
$216$	5.61145	0.381811
$217$	44.6739	3.03267
$218$	41.2355	2.79282
$219$	1.68874	0.114114
$220$	−0.393369	−0.0265209
$221$	3.74455	0.251886
$222$	1.33118	0.0893428
$223$	−28.5046	−1.90881	−0.954403	−	0.298521i	$-0.903507\pi$
$223$	−28.5046	−1.90881	−0.954403	+	0.298521i	$0.903507\pi$
$224$	36.0055	2.40572
$225$	14.8953	0.993022
$226$	−29.9437	−1.99183
$227$	9.88690	0.656217	0.328108	−	0.944640i	$-0.393589\pi$
$227$	9.88690	0.656217	0.328108	+	0.944640i	$0.393589\pi$
$228$	1.92704	0.127621
$229$	−14.4753	−0.956552	−0.478276	−	0.878210i	$-0.658738\pi$
$229$	−14.4753	−0.956552	−0.478276	+	0.878210i	$0.658738\pi$
$230$	−0.683217	−0.0450500
$231$	−1.44513	−0.0950826
$232$	59.3661	3.89758
$233$	8.35225	0.547174	0.273587	−	0.961847i	$-0.411790\pi$
$233$	8.35225	0.547174	0.273587	+	0.961847i	$0.411790\pi$
$234$	7.65285	0.500283
$235$	0.255980	0.0166983
$236$	−34.5461	−2.24876
$237$	0.249881	0.0162315
$238$	49.3809	3.20089
$239$	21.1259	1.36652	0.683259	−	0.730176i	$-0.260562\pi$
$239$	21.1259	1.36652	0.683259	+	0.730176i	$0.260562\pi$
$240$	−0.0478008	−0.00308553
$241$	−15.7498	−1.01453	−0.507266	−	0.861789i	$-0.669344\pi$
$241$	−15.7498	−1.01453	−0.507266	+	0.861789i	$0.669344\pi$
$242$	17.9972	1.15690
$243$	3.76951	0.241814
$244$	−21.0234	−1.34588
$245$	−0.830223	−0.0530410
$246$	−0.260836	−0.0166303
$247$	2.97791	0.189480
$248$	57.9536	3.68006
$249$	−0.317733	−0.0201355
$250$	−1.10031	−0.0695897
$251$	−21.9273	−1.38404	−0.692020	−	0.721878i	$-0.743279\pi$
$251$	−21.9273	−1.38404	−0.692020	+	0.721878i	$0.743279\pi$
$252$	70.3125	4.42927
$253$	−12.4032	−0.779785
$254$	−36.0116	−2.25957
$255$	−0.0226033	−0.00141548
$256$	−26.0546	−1.62841
$257$	5.36235	0.334494	0.167247	−	0.985915i	$-0.446512\pi$
$257$	5.36235	0.334494	0.167247	+	0.985915i	$0.446512\pi$
$258$	−0.866487	−0.0539452
$259$	18.9002	1.17440
$260$	−0.196892	−0.0122107
$261$	26.5569	1.64383
$262$	−21.4433	−1.32477
$263$	−7.48549	−0.461575	−0.230788	−	0.973004i	$-0.574130\pi$
$263$	−7.48549	−0.461575	−0.230788	+	0.973004i	$0.574130\pi$
$264$	−1.87471	−0.115380
$265$	−0.0292725	−0.00179820
$266$	39.2709	2.40785
$267$	−1.29990	−0.0795525
$268$	6.17949	0.377472
$269$	22.3505	1.36273	0.681367	−	0.731942i	$-0.261386\pi$
$269$	22.3505	1.36273	0.681367	+	0.731942i	$0.261386\pi$
$270$	−0.0926977	−0.00564140
$271$	1.97737	0.120117	0.0600583	−	0.998195i	$-0.480871\pi$
$271$	1.97737	0.120117	0.0600583	+	0.998195i	$0.480871\pi$
$272$	29.6526	1.79795
$273$	−0.723329	−0.0437779
$274$	−3.80564	−0.229907
$275$	−9.98577	−0.602165
$276$	−4.01738	−0.241818
$277$	−26.0452	−1.56490	−0.782451	−	0.622712i	$-0.786031\pi$
$277$	−26.0452	−1.56490	−0.782451	+	0.622712i	$0.786031\pi$
$278$	48.5240	2.91028
$279$	25.9250	1.55209
$280$	−1.46618	−0.0876210
$281$	−23.5557	−1.40522	−0.702608	−	0.711578i	$-0.747981\pi$
$281$	−23.5557	−1.40522	−0.702608	+	0.711578i	$0.747981\pi$
$282$	2.16043	0.128652
$283$	−2.94858	−0.175275	−0.0876374	−	0.996152i	$-0.527932\pi$
$283$	−2.94858	−0.175275	−0.0876374	+	0.996152i	$0.527932\pi$
$284$	−35.4915	−2.10603
$285$	−0.0179756	−0.00106478
$286$	−5.13044	−0.303370
$287$	−3.70338	−0.218603
$288$	20.8946	1.23123
$289$	−2.97831	−0.175195
$290$	−0.980691	−0.0575882
$291$	0.0898113	0.00526483
$292$	55.0831	3.22349
$293$	−29.0020	−1.69432	−0.847159	−	0.531340i	$-0.821689\pi$
$293$	−29.0020	−1.69432	−0.847159	+	0.531340i	$0.821689\pi$
$294$	−7.00695	−0.408653
$295$	0.322250	0.0187621
$296$	24.5184	1.42510
$297$	−1.68285	−0.0976488
$298$	32.2958	1.87084
$299$	−6.20818	−0.359028
$300$	−3.23437	−0.186736
$301$	−12.3025	−0.709103
$302$	51.1977	2.94610
$303$	−0.457593	−0.0262880
$304$	23.5817	1.35250
$305$	0.196108	0.0112291
$306$	28.6565	1.63818
$307$	−25.6455	−1.46367	−0.731833	−	0.681484i	$-0.761335\pi$
$307$	−25.6455	−1.46367	−0.731833	+	0.681484i	$0.761335\pi$
$308$	−47.1372	−2.68589
$309$	0.314552	0.0178943
$310$	−0.957358	−0.0543743
$311$	17.3718	0.985066	0.492533	−	0.870294i	$-0.336071\pi$
$311$	17.3718	0.985066	0.492533	+	0.870294i	$0.336071\pi$
$312$	−0.938344	−0.0531233
$313$	30.0703	1.69968	0.849838	−	0.527044i	$-0.176700\pi$
$313$	30.0703	1.69968	0.849838	+	0.527044i	$0.176700\pi$
$314$	−32.2772	−1.82151
$315$	−0.655882	−0.0369548
$316$	8.15062	0.458508
$317$	17.7873	0.999034	0.499517	−	0.866304i	$-0.333511\pi$
$317$	17.7873	0.999034	0.499517	+	0.866304i	$0.333511\pi$
$318$	−0.247055	−0.0138542
$319$	−17.8036	−0.996812
$320$	−0.0928537	−0.00519068
$321$	1.02983	0.0574793
$322$	−81.8696	−4.56242
$323$	11.1509	0.620455
$324$	40.5300	2.25167
$325$	−4.99816	−0.277248
$326$	50.6972	2.80786
$327$	−2.26177	−0.125076
$328$	−4.80423	−0.265269
$329$	30.6740	1.69111
$330$	0.0309690	0.00170479
$331$	−14.7780	−0.812271	−0.406135	−	0.913813i	$-0.633124\pi$
$331$	−14.7780	−0.812271	−0.406135	+	0.913813i	$0.633124\pi$
$332$	−10.3638	−0.568788
$333$	10.9681	0.601047
$334$	13.0517	0.714157
$335$	−0.0576429	−0.00314937
$336$	−5.72795	−0.312485
$337$	−14.1245	−0.769411	−0.384706	−	0.923039i	$-0.625697\pi$
$337$	−14.1245	−0.769411	−0.384706	+	0.923039i	$0.625697\pi$
$338$	−2.56793	−0.139677
$339$	1.64242	0.0892038
$340$	−0.737274	−0.0399843
$341$	−17.3800	−0.941181
$342$	22.7895	1.23232
$343$	−63.5374	−3.43070
$344$	−15.9595	−0.860477
$345$	0.0374745	0.00201756
$346$	−45.0404	−2.42139
$347$	23.7933	1.27729	0.638646	−	0.769501i	$-0.279495\pi$
$347$	23.7933	1.27729	0.638646	+	0.769501i	$0.279495\pi$
$348$	−5.76655	−0.309120
$349$	18.5230	0.991515	0.495758	−	0.868461i	$-0.334890\pi$
$349$	18.5230	0.991515	0.495758	+	0.868461i	$0.334890\pi$
$350$	−65.9127	−3.52318
$351$	−0.842314	−0.0449594
$352$	−14.0077	−0.746611
$353$	14.8575	0.790787	0.395394	−	0.918512i	$-0.370608\pi$
$353$	14.8575	0.790787	0.395394	+	0.918512i	$0.370608\pi$
$354$	2.71973	0.144552
$355$	0.331068	0.0175713
$356$	−42.4000	−2.24720
$357$	−2.70855	−0.143351
$358$	54.2155	2.86538
$359$	7.30934	0.385772	0.192886	−	0.981221i	$-0.438215\pi$
$359$	7.30934	0.385772	0.192886	+	0.981221i	$0.438215\pi$
$360$	−0.850848	−0.0448436
$361$	−10.1321	−0.533266
$362$	15.6195	0.820941
$363$	−0.987149	−0.0518119
$364$	−23.5935	−1.23664
$365$	−0.513821	−0.0268946
$366$	1.65512	0.0865146
$367$	4.78081	0.249556	0.124778	−	0.992185i	$-0.460178\pi$
$367$	4.78081	0.249556	0.124778	+	0.992185i	$0.460178\pi$
$368$	−49.1617	−2.56273
$369$	−2.14913	−0.111879
$370$	−0.405029	−0.0210564
$371$	−3.50771	−0.182111
$372$	−5.62935	−0.291868
$373$	−7.17387	−0.371449	−0.185724	−	0.982602i	$-0.559463\pi$
$373$	−7.17387	−0.371449	−0.185724	+	0.982602i	$0.559463\pi$
$374$	−19.2112	−0.993389
$375$	0.0603521	0.00311657
$376$	39.7921	2.05212
$377$	−8.91123	−0.458952
$378$	−11.1079	−0.571330
$379$	29.9864	1.54030	0.770149	−	0.637864i	$-0.220182\pi$
$379$	29.9864	1.54030	0.770149	+	0.637864i	$0.220182\pi$
$380$	−0.586328	−0.0300780
$381$	1.97524	0.101195
$382$	1.07119	0.0548066
$383$	−13.1806	−0.673500	−0.336750	−	0.941594i	$-0.609328\pi$
$383$	−13.1806	−0.673500	−0.336750	+	0.941594i	$0.609328\pi$
$384$	1.19141	0.0607991
$385$	0.439701	0.0224092
$386$	31.3910	1.59776
$387$	−7.13933	−0.362912
$388$	2.92946	0.148721
$389$	−11.3287	−0.574389	−0.287195	−	0.957872i	$-0.592723\pi$
$389$	−11.3287	−0.574389	−0.287195	+	0.957872i	$0.592723\pi$
$390$	0.0155009	0.000784917 0
$391$	−23.2469	−1.17564
$392$	−129.058	−6.51841
$393$	1.17617	0.0593298
$394$	0.833487	0.0419904
$395$	−0.0760298	−0.00382547
$396$	−27.3545	−1.37461
$397$	2.80225	0.140641	0.0703204	−	0.997524i	$-0.477598\pi$
$397$	2.80225	0.140641	0.0703204	+	0.997524i	$0.477598\pi$
$398$	−8.18413	−0.410234
$399$	−2.15401	−0.107835
$400$	−39.5798	−1.97899
$401$	29.8437	1.49033	0.745163	−	0.666883i	$-0.232372\pi$
$401$	29.8437	1.49033	0.745163	+	0.666883i	$0.232372\pi$
$402$	−0.486496	−0.0242642
$403$	−8.69920	−0.433338
$404$	−14.9257	−0.742584
$405$	−0.378068	−0.0187864
$406$	−117.516	−5.83221
$407$	−7.35296	−0.364473
$408$	−3.51368	−0.173953
$409$	14.4466	0.714337	0.357168	−	0.934040i	$-0.383742\pi$
$409$	14.4466	0.714337	0.357168	+	0.934040i	$0.383742\pi$
$410$	0.0793630	0.00391946
$411$	0.208740	0.0102964
$412$	10.2600	0.505476
$413$	38.6150	1.90012
$414$	−47.5103	−2.33500
$415$	0.0966746	0.00474557
$416$	−7.01123	−0.343754
$417$	−2.66155	−0.130336
$418$	−15.2780	−0.747271
$419$	26.5701	1.29803	0.649017	−	0.760774i	$-0.275181\pi$
$419$	26.5701	1.29803	0.649017	+	0.760774i	$0.275181\pi$
$420$	0.142418	0.00694929
$421$	−1.20863	−0.0589049	−0.0294524	−	0.999566i	$-0.509376\pi$
$421$	−1.20863	−0.0589049	−0.0294524	+	0.999566i	$0.509376\pi$
$422$	−12.7632	−0.621301
$423$	17.8006	0.865497
$424$	−4.55040	−0.220987
$425$	−18.7159	−0.907854
$426$	2.79416	0.135378
$427$	23.4996	1.13722
$428$	33.5908	1.62367
$429$	0.281405	0.0135864
$430$	0.263641	0.0127139
$431$	13.2674	0.639070	0.319535	−	0.947574i	$-0.396473\pi$
$431$	13.2674	0.639070	0.319535	+	0.947574i	$0.396473\pi$
$432$	−6.67017	−0.320919
$433$	1.70255	0.0818195	0.0409097	−	0.999163i	$-0.486974\pi$
$433$	1.70255	0.0818195	0.0409097	+	0.999163i	$0.486974\pi$
$434$	−114.720	−5.50672
$435$	0.0537910	0.00257908
$436$	−73.7743	−3.53315
$437$	−18.4874	−0.884372
$438$	−4.33656	−0.207209
$439$	−11.4169	−0.544900	−0.272450	−	0.962170i	$-0.587834\pi$
$439$	−11.4169	−0.544900	−0.272450	+	0.962170i	$0.587834\pi$
$440$	0.570405	0.0271930
$441$	−57.7329	−2.74919
$442$	−9.61577	−0.457375
$443$	17.9246	0.851625	0.425812	−	0.904811i	$-0.359988\pi$
$443$	17.9246	0.851625	0.425812	+	0.904811i	$0.359988\pi$
$444$	−2.38161	−0.113026
$445$	0.395512	0.0187491
$446$	73.1978	3.46602
$447$	−1.77143	−0.0837856
$448$	−11.1266	−0.525683
$449$	15.8500	0.748007	0.374003	−	0.927427i	$-0.377985\pi$
$449$	15.8500	0.748007	0.374003	+	0.927427i	$0.377985\pi$
$450$	−38.2502	−1.80313
$451$	1.44077	0.0678431
$452$	53.5723	2.51983
$453$	−2.80820	−0.131941
$454$	−25.3889	−1.19156
$455$	0.220083	0.0103176
$456$	−2.79430	−0.130855
$457$	32.9227	1.54006	0.770028	−	0.638010i	$-0.220242\pi$
$457$	32.9227	1.54006	0.770028	+	0.638010i	$0.220242\pi$
$458$	37.1715	1.73691
$459$	−3.15409	−0.147220
$460$	1.22234	0.0569920
$461$	−37.2003	−1.73259	−0.866294	−	0.499534i	$-0.833505\pi$
$461$	−37.2003	−1.73259	−0.866294	+	0.499534i	$0.833505\pi$
$462$	3.71100	0.172651
$463$	26.2034	1.21777	0.608887	−	0.793257i	$-0.291616\pi$
$463$	26.2034	1.21777	0.608887	+	0.793257i	$0.291616\pi$
$464$	−70.5668	−3.27598
$465$	0.0525111	0.00243515
$466$	−21.4480	−0.993560
$467$	17.9399	0.830162	0.415081	−	0.909785i	$-0.363753\pi$
$467$	17.9399	0.830162	0.415081	+	0.909785i	$0.363753\pi$
$468$	−13.6917	−0.632899
$469$	−6.90732	−0.318951
$470$	−0.657341	−0.0303209
$471$	1.77041	0.0815760
$472$	50.0936	2.30575
$473$	4.78618	0.220069
$474$	−0.641679	−0.0294733
$475$	−14.8841	−0.682928
$476$	−88.3472	−4.04939
$477$	−2.03558	−0.0932029
$478$	−54.2498	−2.48133
$479$	15.1653	0.692922	0.346461	−	0.938064i	$-0.387383\pi$
$479$	15.1653	0.692922	0.346461	+	0.938064i	$0.387383\pi$
$480$	0.0423220	0.00193173
$481$	−3.68037	−0.167810
$482$	40.4444	1.84219
$483$	4.49056	0.204327
$484$	−32.1988	−1.46358
$485$	−0.0273263	−0.00124082
$486$	−9.67985	−0.439087
$487$	−7.58539	−0.343727	−0.171863	−	0.985121i	$-0.554979\pi$
$487$	−7.58539	−0.343727	−0.171863	+	0.985121i	$0.554979\pi$
$488$	30.4850	1.37999
$489$	−2.78074	−0.125750
$490$	2.13196	0.0963121
$491$	1.30202	0.0587594	0.0293797	−	0.999568i	$-0.490647\pi$
$491$	1.30202	0.0587594	0.0293797	+	0.999568i	$0.490647\pi$
$492$	0.466661	0.0210387
$493$	−33.3686	−1.50285
$494$	−7.64707	−0.344058
$495$	0.255166	0.0114688
$496$	−68.8878	−3.09315
$497$	39.6718	1.77952
$498$	0.815917	0.0365621
$499$	38.2504	1.71232	0.856161	−	0.516709i	$-0.172843\pi$
$499$	38.2504	1.71232	0.856161	+	0.516709i	$0.172843\pi$
$500$	1.96856	0.0880367
$501$	−0.715887	−0.0319835
$502$	56.3079	2.51315
$503$	20.5204	0.914958	0.457479	−	0.889220i	$-0.348753\pi$
$503$	20.5204	0.914958	0.457479	+	0.889220i	$0.348753\pi$
$504$	−101.957	−4.54151
$505$	0.139229	0.00619560
$506$	31.8507	1.41594
$507$	0.140851	0.00625543
$508$	64.4282	2.85854
$509$	34.5465	1.53125	0.765624	−	0.643288i	$-0.222430\pi$
$509$	34.5465	1.53125	0.765624	+	0.643288i	$0.222430\pi$
$510$	0.0580438	0.00257022
$511$	−61.5709	−2.72374
$512$	49.9891	2.20923
$513$	−2.50833	−0.110746
$514$	−13.7701	−0.607375
$515$	−0.0957068	−0.00421735
$516$	1.55023	0.0682451
$517$	−11.9335	−0.524834
$518$	−48.5344	−2.13248
$519$	2.47047	0.108442
$520$	0.285504	0.0125202
$521$	31.5440	1.38197	0.690985	−	0.722869i	$-0.257177\pi$
$521$	31.5440	1.38197	0.690985	+	0.722869i	$0.257177\pi$
$522$	−68.1963	−2.98487
$523$	6.97069	0.304807	0.152403	−	0.988318i	$-0.451299\pi$
$523$	6.97069	0.304807	0.152403	+	0.988318i	$0.451299\pi$
$524$	38.3642	1.67595
$525$	3.61532	0.157785
$526$	19.2222	0.838130
$527$	−32.5746	−1.41897
$528$	2.22841	0.0969791
$529$	15.5415	0.675716
$530$	0.0751699	0.00326517
$531$	22.4089	0.972465
$532$	−70.2594	−3.04613
$533$	0.721146	0.0312363
$534$	3.33805	0.144452
$535$	−0.313339	−0.0135468
$536$	−8.96057	−0.387038
$537$	−2.97372	−0.128326
$538$	−57.3946	−2.47446
$539$	38.7039	1.66710
$540$	0.165845	0.00713684
$541$	17.5106	0.752840	0.376420	−	0.926449i	$-0.377155\pi$
$541$	17.5106	0.752840	0.376420	+	0.926449i	$0.377155\pi$
$542$	−5.07775	−0.218108
$543$	−0.856729	−0.0367658
$544$	−26.2539	−1.12563
$545$	0.688174	0.0294781
$546$	1.85746	0.0794920
$547$	−33.3301	−1.42509	−0.712546	−	0.701626i	$-0.752458\pi$
$547$	−33.3301	−1.42509	−0.712546	+	0.701626i	$0.752458\pi$
$548$	6.80866	0.290851
$549$	13.6372	0.582021
$550$	25.6428	1.09341
$551$	−26.5368	−1.13051
$552$	5.82541	0.247946
$553$	−9.11062	−0.387423
$554$	66.8822	2.84155
$555$	0.0222159	0.000943011 0
$556$	−86.8141	−3.68174
$557$	0.806026	0.0341524	0.0170762	−	0.999854i	$-0.494564\pi$
$557$	0.806026	0.0341524	0.0170762	+	0.999854i	$0.494564\pi$
$558$	−66.5737	−2.81829
$559$	2.39562	0.101324
$560$	1.74281	0.0736470
$561$	1.05374	0.0444888
$562$	60.4895	2.55159
$563$	12.7851	0.538829	0.269415	−	0.963024i	$-0.413170\pi$
$563$	12.7851	0.538829	0.269415	+	0.963024i	$0.413170\pi$
$564$	−3.86522	−0.162755
$565$	−0.499728	−0.0210237
$566$	7.57176	0.318265
$567$	−45.3038	−1.90258
$568$	51.4645	2.15940
$569$	19.5604	0.820015	0.410008	−	0.912082i	$-0.365526\pi$
$569$	19.5604	0.820015	0.410008	+	0.912082i	$0.365526\pi$
$570$	0.0461602	0.00193344
$571$	2.61429	0.109405	0.0547023	−	0.998503i	$-0.482579\pi$
$571$	2.61429	0.109405	0.0547023	+	0.998503i	$0.482579\pi$
$572$	9.17886	0.383788
$573$	−0.0587546	−0.00245451
$574$	9.51003	0.396941
$575$	31.0295	1.29402
$576$	−6.45695	−0.269040
$577$	−26.0405	−1.08408	−0.542041	−	0.840352i	$-0.682348\pi$
$577$	−26.0405	−1.08408	−0.542041	+	0.840352i	$0.682348\pi$
$578$	7.64810	0.318119
$579$	−1.72180	−0.0715556
$580$	1.75455	0.0728538
$581$	11.5845	0.480605
$582$	−0.230630	−0.00955990
$583$	1.36465	0.0565179
$584$	−79.8733	−3.30518
$585$	0.127718	0.00528047
$586$	74.4753	3.07655
$587$	−12.7992	−0.528279	−0.264139	−	0.964484i	$-0.585088\pi$
$587$	−12.7992	−0.528279	−0.264139	+	0.964484i	$0.585088\pi$
$588$	12.5361	0.516980
$589$	−25.9054	−1.06741
$590$	−0.827516	−0.0340683
$591$	−0.0457168	−0.00188054
$592$	−29.1443	−1.19782
$593$	35.5772	1.46098	0.730490	−	0.682924i	$-0.239292\pi$
$593$	35.5772	1.46098	0.730490	+	0.682924i	$0.239292\pi$
$594$	4.32144	0.177311
$595$	0.824112	0.0337853
$596$	−57.7803	−2.36677
$597$	0.448900	0.0183723
$598$	15.9422	0.651925
$599$	13.6381	0.557236	0.278618	−	0.960402i	$-0.410124\pi$
$599$	13.6381	0.557236	0.278618	+	0.960402i	$0.410124\pi$
$600$	4.69000	0.191468
$601$	36.4888	1.48841	0.744204	−	0.667952i	$-0.232829\pi$
$601$	36.4888	1.48841	0.744204	+	0.667952i	$0.232829\pi$
$602$	31.5919	1.28759
$603$	−4.00843	−0.163236
$604$	−91.5977	−3.72706
$605$	0.300353	0.0122111
$606$	1.17507	0.0477339
$607$	−16.4344	−0.667051	−0.333525	−	0.942741i	$-0.608238\pi$
$607$	−16.4344	−0.667051	−0.333525	+	0.942741i	$0.608238\pi$
$608$	−20.8788	−0.846748
$609$	6.44575	0.261195
$610$	−0.503593	−0.0203899
$611$	−5.97305	−0.241644
$612$	−51.2693	−2.07244
$613$	15.9775	0.645326	0.322663	−	0.946514i	$-0.395422\pi$
$613$	15.9775	0.645326	0.322663	+	0.946514i	$0.395422\pi$
$614$	65.8559	2.65773
$615$	−0.00435306	−0.000175533 0
$616$	68.3514	2.75396
$617$	14.7786	0.594964	0.297482	−	0.954727i	$-0.403853\pi$
$617$	14.7786	0.594964	0.297482	+	0.954727i	$0.403853\pi$
$618$	−0.807749	−0.0324924
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	1.71281	0.0687879
$621$	5.22923	0.209842
$622$	−44.6097	−1.78869
$623$	47.3940	1.89880
$624$	1.11538	0.0446511
$625$	24.9725	0.998898
$626$	−77.2186	−3.08628
$627$	0.838000	0.0334665
$628$	57.7470	2.30436
$629$	−13.7813	−0.549498
$630$	1.68426	0.0671026
$631$	−5.35950	−0.213358	−0.106679	−	0.994293i	$-0.534022\pi$
$631$	−5.35950	−0.213358	−0.106679	+	0.994293i	$0.534022\pi$
$632$	−11.8188	−0.470127
$633$	0.700061	0.0278249
$634$	−45.6766	−1.81405
$635$	−0.600993	−0.0238497
$636$	0.442005	0.0175267
$637$	19.3724	0.767563
$638$	45.7185	1.81001
$639$	23.0222	0.910743
$640$	−0.362504	−0.0143292
$641$	−14.4705	−0.571549	−0.285774	−	0.958297i	$-0.592251\pi$
$641$	−14.4705	−0.571549	−0.285774	+	0.958297i	$0.592251\pi$
$642$	−2.64453	−0.104371
$643$	−23.1600	−0.913340	−0.456670	−	0.889636i	$-0.650958\pi$
$643$	−23.1600	−0.913340	−0.456670	+	0.889636i	$0.650958\pi$
$644$	146.473	5.77183
$645$	−0.0144607	−0.000569390 0
$646$	−28.6349	−1.12662
$647$	−23.3562	−0.918227	−0.459113	−	0.888378i	$-0.651833\pi$
$647$	−23.3562	−0.918227	−0.459113	+	0.888378i	$0.651833\pi$
$648$	−58.7706	−2.30873
$649$	−15.0229	−0.589699
$650$	12.8350	0.503428
$651$	6.29239	0.246618
$652$	−90.7021	−3.55217
$653$	−5.90807	−0.231201	−0.115600	−	0.993296i	$-0.536879\pi$
$653$	−5.90807	−0.231201	−0.115600	+	0.993296i	$0.536879\pi$
$654$	5.80807	0.227114
$655$	−0.357865	−0.0139829
$656$	5.71066	0.222964
$657$	−35.7306	−1.39398
$658$	−78.7689	−3.07073
$659$	25.0062	0.974103	0.487051	−	0.873373i	$-0.338072\pi$
$659$	25.0062	0.974103	0.487051	+	0.873373i	$0.338072\pi$
$660$	−0.0554065	−0.00215670
$661$	−11.0449	−0.429597	−0.214799	−	0.976658i	$-0.568910\pi$
$661$	−11.0449	−0.429597	−0.214799	+	0.976658i	$0.568910\pi$
$662$	37.9489	1.47492
$663$	0.527426	0.0204835
$664$	15.0280	0.583201
$665$	0.655387	0.0254148
$666$	−28.1653	−1.09138
$667$	55.3225	2.14209
$668$	−23.3508	−0.903468
$669$	−4.01491	−0.155225
$670$	0.148023	0.00571863
$671$	−9.14231	−0.352935
$672$	5.07143	0.195635
$673$	−37.6692	−1.45204	−0.726021	−	0.687672i	$-0.758633\pi$
$673$	−37.6692	−1.45204	−0.726021	+	0.687672i	$0.758633\pi$
$674$	36.2708	1.39710
$675$	4.21002	0.162044
$676$	4.59428	0.176703
$677$	−22.6129	−0.869086	−0.434543	−	0.900651i	$-0.643090\pi$
$677$	−22.6129	−0.869086	−0.434543	+	0.900651i	$0.643090\pi$
$678$	−4.21762	−0.161977
$679$	−3.27450	−0.125664
$680$	1.06909	0.0409975
$681$	1.39258	0.0533639
$682$	44.6308	1.70900
$683$	−38.9611	−1.49080	−0.745402	−	0.666615i	$-0.767742\pi$
$683$	−38.9611	−1.49080	−0.745402	+	0.666615i	$0.767742\pi$
$684$	−40.7727	−1.55898
$685$	−0.0635119	−0.00242666
$686$	163.160	6.22947
$687$	−2.03886	−0.0777873
$688$	18.9706	0.723246
$689$	0.683044	0.0260219
$690$	−0.0962321	−0.00366349
$691$	39.7944	1.51385	0.756926	−	0.653501i	$-0.226700\pi$
$691$	39.7944	1.51385	0.756926	+	0.653501i	$0.226700\pi$
$692$	80.5816	3.06325
$693$	30.5764	1.16150
$694$	−61.0997	−2.31931
$695$	0.809811	0.0307179
$696$	8.36180	0.316953
$697$	2.70037	0.102284
$698$	−47.5659	−1.80040
$699$	1.17643	0.0444965
$700$	117.924	4.45712
$701$	36.6500	1.38425	0.692126	−	0.721776i	$-0.256674\pi$
$701$	36.6500	1.38425	0.692126	+	0.721776i	$0.256674\pi$
$702$	2.16301	0.0816374
$703$	−10.9598	−0.413357
$704$	4.32872	0.163145
$705$	0.0360552	0.00135792
$706$	−38.1532	−1.43591
$707$	16.6837	0.627456
$708$	−4.86587	−0.182870
$709$	22.9886	0.863354	0.431677	−	0.902028i	$-0.357922\pi$
$709$	22.9886	0.863354	0.431677	+	0.902028i	$0.357922\pi$
$710$	−0.850162	−0.0319060
$711$	−5.28704	−0.198279
$712$	61.4822	2.30414
$713$	54.0062	2.02255
$714$	6.95537	0.260298
$715$	−0.0856214	−0.00320206
$716$	−96.9967	−3.62494
$717$	2.97561	0.111126
$718$	−18.7699	−0.700486
$719$	13.9067	0.518633	0.259317	−	0.965792i	$-0.416503\pi$
$719$	13.9067	0.518633	0.259317	+	0.965792i	$0.416503\pi$
$720$	1.01138	0.0376919
$721$	−11.4685	−0.427109
$722$	26.0184	0.968306
$723$	−2.21838	−0.0825024
$724$	−27.9448	−1.03856
$725$	44.5398	1.65417
$726$	2.53493	0.0940802
$727$	33.4706	1.24135	0.620677	−	0.784066i	$-0.286858\pi$
$727$	33.4706	1.24135	0.620677	+	0.784066i	$0.286858\pi$
$728$	34.2118	1.26797
$729$	−25.9346	−0.960540
$730$	1.31946	0.0488353
$731$	8.97052	0.331787
$732$	−2.96117	−0.109448
$733$	−13.4359	−0.496265	−0.248132	−	0.968726i	$-0.579817\pi$
$733$	−13.4359	−0.496265	−0.248132	+	0.968726i	$0.579817\pi$
$734$	−12.2768	−0.453145
$735$	−0.116938	−0.00431333
$736$	43.5270	1.60443
$737$	2.68724	0.0989856
$738$	5.51882	0.203151
$739$	0.931923	0.0342813	0.0171407	−	0.999853i	$-0.494544\pi$
$739$	0.931923	0.0342813	0.0171407	+	0.999853i	$0.494544\pi$
$740$	0.724636	0.0266381
$741$	0.419443	0.0154086
$742$	9.00757	0.330678
$743$	−33.8213	−1.24078	−0.620392	−	0.784292i	$-0.713026\pi$
$743$	−33.8213	−1.24078	−0.620392	+	0.784292i	$0.713026\pi$
$744$	8.16284	0.299264
$745$	0.538981	0.0197467
$746$	18.4220	0.674478
$747$	6.72266	0.245969
$748$	34.3708	1.25672
$749$	−37.5473	−1.37195
$750$	−0.154980	−0.00565908
$751$	14.2305	0.519279	0.259640	−	0.965706i	$-0.416396\pi$
$751$	14.2305	0.519279	0.259640	+	0.965706i	$0.416396\pi$
$752$	−47.2997	−1.72484
$753$	−3.08849	−0.112551
$754$	22.8834	0.833366
$755$	0.854433	0.0310960
$756$	19.8731	0.722779
$757$	−10.4799	−0.380899	−0.190449	−	0.981697i	$-0.560995\pi$
$757$	−10.4799	−0.380899	−0.190449	+	0.981697i	$0.560995\pi$
$758$	−77.0031	−2.79688
$759$	−1.74701	−0.0634126
$760$	0.850205	0.0308402
$761$	−44.5000	−1.61312	−0.806561	−	0.591150i	$-0.798674\pi$
$761$	−44.5000	−1.61312	−0.806561	+	0.591150i	$0.798674\pi$
$762$	−5.07228	−0.183749
$763$	82.4636	2.98538
$764$	−1.91645	−0.0693349
$765$	0.478246	0.0172910
$766$	33.8470	1.22294
$767$	−7.51937	−0.271509
$768$	−3.66982	−0.132423
$769$	29.0530	1.04768	0.523840	−	0.851817i	$-0.324499\pi$
$769$	29.0530	1.04768	0.523840	+	0.851817i	$0.324499\pi$
$770$	−1.12912	−0.0406907
$771$	0.755294	0.0272012
$772$	−56.1616	−2.02130
$773$	2.00146	0.0719874	0.0359937	−	0.999352i	$-0.488540\pi$
$773$	2.00146	0.0719874	0.0359937	+	0.999352i	$0.488540\pi$
$774$	18.3333	0.658977
$775$	43.4800	1.56185
$776$	−4.24787	−0.152490
$777$	2.66212	0.0955029
$778$	29.0914	1.04298
$779$	2.14751	0.0769424
$780$	−0.0277326	−0.000992985 0
$781$	−15.4340	−0.552271
$782$	59.6964	2.13474
$783$	7.50605	0.268244
$784$	153.408	5.47884
$785$	−0.538670	−0.0192260
$786$	−3.02032	−0.107731
$787$	7.17399	0.255725	0.127863	−	0.991792i	$-0.459188\pi$
$787$	7.17399	0.255725	0.127863	+	0.991792i	$0.459188\pi$
$788$	−1.49119	−0.0531214
$789$	−1.05434	−0.0375356
$790$	0.195239	0.00694631
$791$	−59.8821	−2.12916
$792$	39.6654	1.40945
$793$	−4.57599	−0.162498
$794$	−7.19599	−0.255376
$795$	−0.00412307	−0.000146230 0
$796$	14.6422	0.518979
$797$	50.1759	1.77732	0.888662	−	0.458564i	$-0.151636\pi$
$797$	50.1759	1.77732	0.888662	+	0.458564i	$0.151636\pi$
$798$	5.53135	0.195808
$799$	−22.3664	−0.791266
$800$	35.0433	1.23897
$801$	27.5035	0.971788
$802$	−76.6368	−2.70614
$803$	23.9537	0.845306
$804$	0.870389	0.0306963
$805$	−1.36631	−0.0481562
$806$	22.3390	0.786857
$807$	3.14810	0.110818
$808$	21.6431	0.761401
$809$	−28.0194	−0.985111	−0.492556	−	0.870281i	$-0.663937\pi$
$809$	−28.0194	−0.985111	−0.492556	+	0.870281i	$0.663937\pi$
$810$	0.970854	0.0341123
$811$	−32.1245	−1.12804	−0.564022	−	0.825760i	$-0.690747\pi$
$811$	−32.1245	−1.12804	−0.564022	+	0.825760i	$0.690747\pi$
$812$	210.247	7.37823
$813$	0.278515	0.00976794
$814$	18.8819	0.661811
$815$	0.846079	0.0296369
$816$	4.17661	0.146211
$817$	7.13393	0.249585
$818$	−37.0978	−1.29709
$819$	15.3043	0.534777
$820$	−0.141988	−0.00495844
$821$	−46.8672	−1.63568	−0.817838	−	0.575449i	$-0.804827\pi$
$821$	−46.8672	−1.63568	−0.817838	+	0.575449i	$0.804827\pi$
$822$	−0.536029	−0.0186962
$823$	−44.9715	−1.56761	−0.783804	−	0.621009i	$-0.786723\pi$
$823$	−44.9715	−1.56761	−0.783804	+	0.621009i	$0.786723\pi$
$824$	−14.8776	−0.518286
$825$	−1.40651	−0.0489684
$826$	−99.1608	−3.45025
$827$	−42.8690	−1.49070	−0.745351	−	0.666672i	$-0.767718\pi$
$827$	−42.8690	−1.49070	−0.745351	+	0.666672i	$0.767718\pi$
$828$	85.0005	2.95397
$829$	28.0207	0.973200	0.486600	−	0.873625i	$-0.338237\pi$
$829$	28.0207	0.973200	0.486600	+	0.873625i	$0.338237\pi$
$830$	−0.248254	−0.00861702
$831$	−3.66850	−0.127259
$832$	2.16665	0.0751149
$833$	72.5411	2.51340
$834$	6.83467	0.236665
$835$	0.217818	0.00753791
$836$	27.3338	0.945360
$837$	7.32746	0.253274
$838$	−68.2302	−2.35697
$839$	25.5152	0.880881	0.440441	−	0.897782i	$-0.354822\pi$
$839$	25.5152	0.880881	0.440441	+	0.897782i	$0.354822\pi$
$840$	−0.206513	−0.00712539
$841$	50.4100	1.73827
$842$	3.10367	0.106960
$843$	−3.31785	−0.114273
$844$	22.8345	0.785997
$845$	−0.0428559	−0.00147429
$846$	−45.7108	−1.57157
$847$	35.9912	1.23667
$848$	5.40894	0.185744
$849$	−0.415311	−0.0142535
$850$	48.0612	1.64848
$851$	22.8484	0.783232
$852$	−4.99903	−0.171264
$853$	30.7328	1.05227	0.526135	−	0.850401i	$-0.323641\pi$
$853$	30.7328	1.05227	0.526135	+	0.850401i	$0.323641\pi$
$854$	−60.3453	−2.06497
$855$	0.380332	0.0130071
$856$	−48.7084	−1.66482
$857$	25.6194	0.875142	0.437571	−	0.899184i	$-0.355839\pi$
$857$	25.6194	0.875142	0.437571	+	0.899184i	$0.355839\pi$
$858$	−0.722630	−0.0246702
$859$	17.7623	0.606043	0.303021	−	0.952984i	$-0.402005\pi$
$859$	17.7623	0.606043	0.303021	+	0.952984i	$0.402005\pi$
$860$	−0.471679	−0.0160841
$861$	−0.521626	−0.0177770
$862$	−34.0699	−1.16043
$863$	49.1810	1.67414	0.837071	−	0.547094i	$-0.184266\pi$
$863$	49.1810	1.67414	0.837071	+	0.547094i	$0.184266\pi$
$864$	5.90566	0.200915
$865$	−0.751674	−0.0255577
$866$	−4.37204	−0.148568
$867$	−0.419499	−0.0142469
$868$	205.245	6.96646
$869$	3.54441	0.120236
$870$	−0.138132	−0.00468310
$871$	1.34504	0.0455749
$872$	106.976	3.62268
$873$	−1.90025	−0.0643136
$874$	47.4744	1.60585
$875$	−2.20042	−0.0743879
$876$	7.75853	0.262136
$877$	17.6724	0.596753	0.298377	−	0.954448i	$-0.403555\pi$
$877$	17.6724	0.596753	0.298377	+	0.954448i	$0.403555\pi$
$878$	29.3179	0.989432
$879$	−4.08498	−0.137783
$880$	−0.678024	−0.0228562
$881$	−19.2312	−0.647915	−0.323958	−	0.946072i	$-0.605014\pi$
$881$	−19.2312	−0.647915	−0.323958	+	0.946072i	$0.605014\pi$
$882$	148.254	4.99198
$883$	26.7449	0.900036	0.450018	−	0.893019i	$-0.351418\pi$
$883$	26.7449	0.900036	0.450018	+	0.893019i	$0.351418\pi$
$884$	17.2035	0.578618
$885$	0.0453893	0.00152574
$886$	−46.0292	−1.54638
$887$	−20.9349	−0.702924	−0.351462	−	0.936202i	$-0.614315\pi$
$887$	−20.9349	−0.702924	−0.351462	+	0.936202i	$0.614315\pi$
$888$	3.45345	0.115890
$889$	−72.0168	−2.41536
$890$	−1.01565	−0.0340446
$891$	17.6251	0.590461
$892$	−130.958	−4.38480
$893$	−17.7872	−0.595226
$894$	4.54891	0.152138
$895$	0.904795	0.0302440
$896$	−43.4387	−1.45118
$897$	−0.874430	−0.0291964
$898$	−40.7017	−1.35823
$899$	77.5205	2.58546
$900$	68.4334	2.28111
$901$	2.55770	0.0852092
$902$	−3.69980	−0.123190
$903$	−1.73282	−0.0576646
$904$	−77.6825	−2.58368
$905$	0.260672	0.00866502
$906$	7.21127	0.239578
$907$	−13.3085	−0.441903	−0.220952	−	0.975285i	$-0.570916\pi$
$907$	−13.3085	−0.441903	−0.220952	+	0.975285i	$0.570916\pi$
$908$	45.4232	1.50742
$909$	9.68184	0.321126
$910$	−0.565158	−0.0187348
$911$	−44.8275	−1.48520	−0.742600	−	0.669735i	$-0.766408\pi$
$911$	−44.8275	−1.48520	−0.742600	+	0.669735i	$0.766408\pi$
$912$	3.32151	0.109986
$913$	−4.50684	−0.149155
$914$	−84.5432	−2.79644
$915$	0.0276221	0.000913159 0
$916$	−66.5034	−2.19733
$917$	−42.8828	−1.41611
$918$	8.09949	0.267323
$919$	−29.1010	−0.959955	−0.479977	−	0.877281i	$-0.659355\pi$
$919$	−29.1010	−0.959955	−0.479977	+	0.877281i	$0.659355\pi$
$920$	−1.77246	−0.0584362
$921$	−3.61220	−0.119026
$922$	95.5278	3.14604
$923$	−7.72515	−0.254276
$924$	−6.63934	−0.218418
$925$	18.3951	0.604826
$926$	−67.2886	−2.21124
$927$	−6.65536	−0.218591
$928$	62.4787	2.05096
$929$	4.26440	0.139910	0.0699552	−	0.997550i	$-0.477714\pi$
$929$	4.26440	0.139910	0.0699552	+	0.997550i	$0.477714\pi$
$930$	−0.134845	−0.00442175
$931$	57.6893	1.89069
$932$	38.3726	1.25694
$933$	2.44685	0.0801061
$934$	−46.0686	−1.50741
$935$	−0.320614	−0.0104852
$936$	19.8537	0.648937
$937$	−32.6874	−1.06785	−0.533926	−	0.845531i	$-0.679284\pi$
$937$	−32.6874	−1.06785	−0.533926	+	0.845531i	$0.679284\pi$
$938$	17.7375	0.579151
$939$	4.23545	0.138219
$940$	1.17605	0.0383584
$941$	−37.6279	−1.22663	−0.613317	−	0.789837i	$-0.710165\pi$
$941$	−37.6279	−1.22663	−0.613317	+	0.789837i	$0.710165\pi$
$942$	−4.54628	−0.148126
$943$	−4.47700	−0.145791
$944$	−59.5449	−1.93802
$945$	−0.185379	−0.00603037
$946$	−12.2906	−0.399601
$947$	13.7651	0.447306	0.223653	−	0.974669i	$-0.428202\pi$
$947$	13.7651	0.447306	0.223653	+	0.974669i	$0.428202\pi$
$948$	1.14803	0.0372861
$949$	11.9895	0.389195
$950$	38.2213	1.24006
$951$	2.50536	0.0812420
$952$	128.108	4.15200
$953$	−37.1954	−1.20488	−0.602439	−	0.798165i	$-0.705804\pi$
$953$	−37.1954	−1.20488	−0.602439	+	0.798165i	$0.705804\pi$
$954$	5.22724	0.169238
$955$	0.0178769	0.000578483 0
$956$	97.0582	3.13909
$957$	−2.50767	−0.0810613
$958$	−38.9436	−1.25821
$959$	−7.61060	−0.245759
$960$	−0.0130786	−0.000422109 0
$961$	44.6761	1.44116
$962$	9.45094	0.304710
$963$	−21.7893	−0.702150
$964$	−72.3590	−2.33052
$965$	0.523881	0.0168643
$966$	−11.5315	−0.371018
$967$	20.8001	0.668886	0.334443	−	0.942416i	$-0.391452\pi$
$967$	20.8001	0.668886	0.334443	+	0.942416i	$0.391452\pi$
$968$	46.6899	1.50067
$969$	1.57063	0.0504558
$970$	0.0701722	0.00225309
$971$	−0.440086	−0.0141230	−0.00706152	−	0.999975i	$-0.502248\pi$
$971$	−0.440086	−0.0141230	−0.00706152	+	0.999975i	$0.502248\pi$
$972$	17.3182	0.555481
$973$	97.0393	3.11094
$974$	19.4788	0.624140
$975$	−0.703998	−0.0225460
$976$	−36.2366	−1.15991
$977$	40.4870	1.29529	0.647647	−	0.761940i	$-0.275753\pi$
$977$	40.4870	1.29529	0.647647	+	0.761940i	$0.275753\pi$
$978$	7.14076	0.228336
$979$	−18.4382	−0.589289
$980$	−3.81428	−0.121843
$981$	47.8550	1.52789
$982$	−3.34350	−0.106695
$983$	4.54212	0.144871	0.0724356	−	0.997373i	$-0.476923\pi$
$983$	4.54212	0.144871	0.0724356	+	0.997373i	$0.476923\pi$
$984$	−0.676683	−0.0215719
$985$	0.0139100	0.000443208 0
$986$	85.6883	2.72887
$987$	4.32048	0.137522
$988$	13.6814	0.435262
$989$	−14.8724	−0.472915
$990$	−0.655248	−0.0208251
$991$	22.0728	0.701166	0.350583	−	0.936532i	$-0.385984\pi$
$991$	22.0728	0.701166	0.350583	+	0.936532i	$0.385984\pi$
$992$	60.9921	1.93650
$993$	−2.08150	−0.0660543
$994$	−101.874	−3.23126
$995$	−0.136584	−0.00433001
$996$	−1.45976	−0.0462541
$997$	−19.0295	−0.602670	−0.301335	−	0.953518i	$-0.597432\pi$
$997$	−19.0295	−0.602670	−0.301335	+	0.953518i	$0.597432\pi$
$998$	−98.2244	−3.10924
$999$	3.10002	0.0980804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.8	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.8	✓	168	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)

\(f(q)\)	\(=\)	\(q-2.56793 q^{2} +0.140851 q^{3} +4.59428 q^{4} -0.0428559 q^{5} -0.361697 q^{6} -5.13541 q^{7} -6.66194 q^{8} -2.98016 q^{9} +O(q^{10})\)	\(q-2.56793 q^{2} +0.140851 q^{3} +4.59428 q^{4} -0.0428559 q^{5} -0.361697 q^{6} -5.13541 q^{7} -6.66194 q^{8} -2.98016 q^{9} +0.110051 q^{10} +1.99789 q^{11} +0.647111 q^{12} +1.00000 q^{13} +13.1874 q^{14} -0.00603632 q^{15} +7.91887 q^{16} +3.74455 q^{17} +7.65285 q^{18} +2.97791 q^{19} -0.196892 q^{20} -0.723329 q^{21} -5.13044 q^{22} -6.20818 q^{23} -0.938344 q^{24} -4.99816 q^{25} -2.56793 q^{26} -0.842314 q^{27} -23.5935 q^{28} -8.91123 q^{29} +0.0155009 q^{30} -8.69920 q^{31} -7.01123 q^{32} +0.281405 q^{33} -9.61577 q^{34} +0.220083 q^{35} -13.6917 q^{36} -3.68037 q^{37} -7.64707 q^{38} +0.140851 q^{39} +0.285504 q^{40} +0.721146 q^{41} +1.85746 q^{42} +2.39562 q^{43} +9.17886 q^{44} +0.127718 q^{45} +15.9422 q^{46} -5.97305 q^{47} +1.11538 q^{48} +19.3724 q^{49} +12.8350 q^{50} +0.527426 q^{51} +4.59428 q^{52} +0.683044 q^{53} +2.16301 q^{54} -0.0856214 q^{55} +34.2118 q^{56} +0.419443 q^{57} +22.8834 q^{58} -7.51937 q^{59} -0.0277326 q^{60} -4.57599 q^{61} +22.3390 q^{62} +15.3043 q^{63} +2.16665 q^{64} -0.0428559 q^{65} -0.722630 q^{66} +1.34504 q^{67} +17.2035 q^{68} -0.874430 q^{69} -0.565158 q^{70} -7.72515 q^{71} +19.8537 q^{72} +11.9895 q^{73} +9.45094 q^{74} -0.703998 q^{75} +13.6814 q^{76} -10.2600 q^{77} -0.361697 q^{78} +1.77408 q^{79} -0.339370 q^{80} +8.82184 q^{81} -1.85185 q^{82} -2.25580 q^{83} -3.32318 q^{84} -0.160476 q^{85} -6.15179 q^{86} -1.25516 q^{87} -13.3098 q^{88} -9.22886 q^{89} -0.327970 q^{90} -5.13541 q^{91} -28.5221 q^{92} -1.22529 q^{93} +15.3384 q^{94} -0.127621 q^{95} -0.987542 q^{96} +0.637632 q^{97} -49.7471 q^{98} -5.95403 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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