LMFDB - Embedded newform 8014.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8014,2,Mod(1,8014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8014 = 2 \cdot 4007 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8014.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:	$63.9921121799$
Analytic rank:	$0$
Dimension:	$91$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.21498 q^{3} +1.00000 q^{4} -2.38557 q^{5} +3.21498 q^{6} -0.918897 q^{7} -1.00000 q^{8} +7.33611 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.21498 q^{3} +1.00000 q^{4} -2.38557 q^{5} +3.21498 q^{6} -0.918897 q^{7} -1.00000 q^{8} +7.33611 q^{9} +2.38557 q^{10} +4.93270 q^{11} -3.21498 q^{12} -6.88935 q^{13} +0.918897 q^{14} +7.66955 q^{15} +1.00000 q^{16} -6.93832 q^{17} -7.33611 q^{18} -3.46562 q^{19} -2.38557 q^{20} +2.95424 q^{21} -4.93270 q^{22} +2.56909 q^{23} +3.21498 q^{24} +0.690924 q^{25} +6.88935 q^{26} -13.9405 q^{27} -0.918897 q^{28} -7.44452 q^{29} -7.66955 q^{30} +6.18390 q^{31} -1.00000 q^{32} -15.8586 q^{33} +6.93832 q^{34} +2.19209 q^{35} +7.33611 q^{36} -6.44744 q^{37} +3.46562 q^{38} +22.1492 q^{39} +2.38557 q^{40} +3.87129 q^{41} -2.95424 q^{42} +5.83970 q^{43} +4.93270 q^{44} -17.5008 q^{45} -2.56909 q^{46} -1.26512 q^{47} -3.21498 q^{48} -6.15563 q^{49} -0.690924 q^{50} +22.3066 q^{51} -6.88935 q^{52} -8.36796 q^{53} +13.9405 q^{54} -11.7673 q^{55} +0.918897 q^{56} +11.1419 q^{57} +7.44452 q^{58} -8.07140 q^{59} +7.66955 q^{60} +13.9140 q^{61} -6.18390 q^{62} -6.74113 q^{63} +1.00000 q^{64} +16.4350 q^{65} +15.8586 q^{66} +6.77225 q^{67} -6.93832 q^{68} -8.25957 q^{69} -2.19209 q^{70} +5.38938 q^{71} -7.33611 q^{72} -10.3384 q^{73} +6.44744 q^{74} -2.22131 q^{75} -3.46562 q^{76} -4.53265 q^{77} -22.1492 q^{78} -6.62558 q^{79} -2.38557 q^{80} +22.8102 q^{81} -3.87129 q^{82} -5.07082 q^{83} +2.95424 q^{84} +16.5518 q^{85} -5.83970 q^{86} +23.9340 q^{87} -4.93270 q^{88} -11.3299 q^{89} +17.5008 q^{90} +6.33061 q^{91} +2.56909 q^{92} -19.8811 q^{93} +1.26512 q^{94} +8.26746 q^{95} +3.21498 q^{96} -3.94306 q^{97} +6.15563 q^{98} +36.1869 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 91 q - 91 q^{2} - 2 q^{3} + 91 q^{4} + 22 q^{5} + 2 q^{6} - 14 q^{7} - 91 q^{8} + 123 q^{9}+O(q^{10}) $ 91 * q - 91 * q^2 - 2 * q^3 + 91 * q^4 + 22 * q^5 + 2 * q^6 - 14 * q^7 - 91 * q^8 + 123 * q^9	\( 91 q - 91 q^{2} - 2 q^{3} + 91 q^{4} + 22 q^{5} + 2 q^{6} - 14 q^{7} - 91 q^{8} + 123 q^{9} - 22 q^{10} + 59 q^{11} - 2 q^{12} - 15 q^{13} + 14 q^{14} + 23 q^{15} + 91 q^{16} + 25 q^{17} - 123 q^{18} + 2 q^{19} + 22 q^{20} + 20 q^{21} - 59 q^{22} + 36 q^{23} + 2 q^{24} + 121 q^{25} + 15 q^{26} - 8 q^{27} - 14 q^{28} + 71 q^{29} - 23 q^{30} + 13 q^{31} - 91 q^{32} + 24 q^{33} - 25 q^{34} + 27 q^{35} + 123 q^{36} + 5 q^{37} - 2 q^{38} + 48 q^{39} - 22 q^{40} + 77 q^{41} - 20 q^{42} - 36 q^{43} + 59 q^{44} + 62 q^{45} - 36 q^{46} + 41 q^{47} - 2 q^{48} + 137 q^{49} - 121 q^{50} + 13 q^{51} - 15 q^{52} + 33 q^{53} + 8 q^{54} + 12 q^{55} + 14 q^{56} + 52 q^{57} - 71 q^{58} + 76 q^{59} + 23 q^{60} + 18 q^{61} - 13 q^{62} - 18 q^{63} + 91 q^{64} + 84 q^{65} - 24 q^{66} - 59 q^{67} + 25 q^{68} + 64 q^{69} - 27 q^{70} + 124 q^{71} - 123 q^{72} + 43 q^{73} - 5 q^{74} + 9 q^{75} + 2 q^{76} + 50 q^{77} - 48 q^{78} - 20 q^{79} + 22 q^{80} + 227 q^{81} - 77 q^{82} + 29 q^{83} + 20 q^{84} + 3 q^{85} + 36 q^{86} - 25 q^{87} - 59 q^{88} + 148 q^{89} - 62 q^{90} + 27 q^{91} + 36 q^{92} + 65 q^{93} - 41 q^{94} + 54 q^{95} + 2 q^{96} + 38 q^{97} - 137 q^{98} + 157 q^{99}+O(q^{100}) \) 91 * q - 91 * q^2 - 2 * q^3 + 91 * q^4 + 22 * q^5 + 2 * q^6 - 14 * q^7 - 91 * q^8 + 123 * q^9 - 22 * q^10 + 59 * q^11 - 2 * q^12 - 15 * q^13 + 14 * q^14 + 23 * q^15 + 91 * q^16 + 25 * q^17 - 123 * q^18 + 2 * q^19 + 22 * q^20 + 20 * q^21 - 59 * q^22 + 36 * q^23 + 2 * q^24 + 121 * q^25 + 15 * q^26 - 8 * q^27 - 14 * q^28 + 71 * q^29 - 23 * q^30 + 13 * q^31 - 91 * q^32 + 24 * q^33 - 25 * q^34 + 27 * q^35 + 123 * q^36 + 5 * q^37 - 2 * q^38 + 48 * q^39 - 22 * q^40 + 77 * q^41 - 20 * q^42 - 36 * q^43 + 59 * q^44 + 62 * q^45 - 36 * q^46 + 41 * q^47 - 2 * q^48 + 137 * q^49 - 121 * q^50 + 13 * q^51 - 15 * q^52 + 33 * q^53 + 8 * q^54 + 12 * q^55 + 14 * q^56 + 52 * q^57 - 71 * q^58 + 76 * q^59 + 23 * q^60 + 18 * q^61 - 13 * q^62 - 18 * q^63 + 91 * q^64 + 84 * q^65 - 24 * q^66 - 59 * q^67 + 25 * q^68 + 64 * q^69 - 27 * q^70 + 124 * q^71 - 123 * q^72 + 43 * q^73 - 5 * q^74 + 9 * q^75 + 2 * q^76 + 50 * q^77 - 48 * q^78 - 20 * q^79 + 22 * q^80 + 227 * q^81 - 77 * q^82 + 29 * q^83 + 20 * q^84 + 3 * q^85 + 36 * q^86 - 25 * q^87 - 59 * q^88 + 148 * q^89 - 62 * q^90 + 27 * q^91 + 36 * q^92 + 65 * q^93 - 41 * q^94 + 54 * q^95 + 2 * q^96 + 38 * q^97 - 137 * q^98 + 157 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.21498	−1.85617	−0.928086	−	0.372367i	$-0.878546\pi$
$3$	−3.21498	−1.85617	−0.928086	+	0.372367i	$0.878546\pi$
$4$	1.00000	0.500000
$5$	−2.38557	−1.06686	−0.533429	−	0.845845i	$-0.679097\pi$
$5$	−2.38557	−1.06686	−0.533429	+	0.845845i	$0.679097\pi$
$6$	3.21498	1.31251
$7$	−0.918897	−0.347311	−0.173655	−	0.984807i	$-0.555558\pi$
$7$	−0.918897	−0.347311	−0.173655	+	0.984807i	$0.555558\pi$
$8$	−1.00000	−0.353553
$9$	7.33611	2.44537
$10$	2.38557	0.754382
$11$	4.93270	1.48727	0.743633	−	0.668588i	$-0.233101\pi$
$11$	4.93270	1.48727	0.743633	+	0.668588i	$0.233101\pi$
$12$	−3.21498	−0.928086
$13$	−6.88935	−1.91076	−0.955381	−	0.295375i	$-0.904555\pi$
$13$	−6.88935	−1.91076	−0.955381	+	0.295375i	$0.904555\pi$
$14$	0.918897	0.245586
$15$	7.66955	1.98027
$16$	1.00000	0.250000
$17$	−6.93832	−1.68279	−0.841395	−	0.540421i	$-0.818265\pi$
$17$	−6.93832	−1.68279	−0.841395	+	0.540421i	$0.818265\pi$
$18$	−7.33611	−1.72914
$19$	−3.46562	−0.795067	−0.397534	−	0.917588i	$-0.630134\pi$
$19$	−3.46562	−0.795067	−0.397534	+	0.917588i	$0.630134\pi$
$20$	−2.38557	−0.533429
$21$	2.95424	0.644668
$22$	−4.93270	−1.05166
$23$	2.56909	0.535692	0.267846	−	0.963462i	$-0.413688\pi$
$23$	2.56909	0.535692	0.267846	+	0.963462i	$0.413688\pi$
$24$	3.21498	0.656256
$25$	0.690924	0.138185
$26$	6.88935	1.35111
$27$	−13.9405	−2.68286
$28$	−0.918897	−0.173655
$29$	−7.44452	−1.38241	−0.691206	−	0.722658i	$-0.742920\pi$
$29$	−7.44452	−1.38241	−0.691206	+	0.722658i	$0.742920\pi$
$30$	−7.66955	−1.40026
$31$	6.18390	1.11066	0.555331	−	0.831629i	$-0.312592\pi$
$31$	6.18390	1.11066	0.555331	+	0.831629i	$0.312592\pi$
$32$	−1.00000	−0.176777
$33$	−15.8586	−2.76062
$34$	6.93832	1.18991
$35$	2.19209	0.370531
$36$	7.33611	1.22269
$37$	−6.44744	−1.05995	−0.529976	−	0.848013i	$-0.677799\pi$
$37$	−6.44744	−1.05995	−0.529976	+	0.848013i	$0.677799\pi$
$38$	3.46562	0.562197
$39$	22.1492	3.54670
$40$	2.38557	0.377191
$41$	3.87129	0.604593	0.302297	−	0.953214i	$-0.402247\pi$
$41$	3.87129	0.604593	0.302297	+	0.953214i	$0.402247\pi$
$42$	−2.95424	−0.455849
$43$	5.83970	0.890545	0.445273	−	0.895395i	$-0.353107\pi$
$43$	5.83970	0.890545	0.445273	+	0.895395i	$0.353107\pi$
$44$	4.93270	0.743633
$45$	−17.5008	−2.60886
$46$	−2.56909	−0.378791
$47$	−1.26512	−0.184536	−0.0922682	−	0.995734i	$-0.529412\pi$
$47$	−1.26512	−0.184536	−0.0922682	+	0.995734i	$0.529412\pi$
$48$	−3.21498	−0.464043
$49$	−6.15563	−0.879375
$50$	−0.690924	−0.0977115
$51$	22.3066	3.12354
$52$	−6.88935	−0.955381
$53$	−8.36796	−1.14943	−0.574714	−	0.818355i	$-0.694887\pi$
$53$	−8.36796	−1.14943	−0.574714	+	0.818355i	$0.694887\pi$
$54$	13.9405	1.89707
$55$	−11.7673	−1.58670
$56$	0.918897	0.122793
$57$	11.1419	1.47578
$58$	7.44452	0.977513
$59$	−8.07140	−1.05081	−0.525403	−	0.850853i	$-0.676086\pi$
$59$	−8.07140	−1.05081	−0.525403	+	0.850853i	$0.676086\pi$
$60$	7.66955	0.990135
$61$	13.9140	1.78150	0.890752	−	0.454489i	$-0.150178\pi$
$61$	13.9140	1.78150	0.890752	+	0.454489i	$0.150178\pi$
$62$	−6.18390	−0.785357
$63$	−6.74113	−0.849303
$64$	1.00000	0.125000
$65$	16.4350	2.03851
$66$	15.8586	1.95205
$67$	6.77225	0.827363	0.413681	−	0.910422i	$-0.364243\pi$
$67$	6.77225	0.827363	0.413681	+	0.910422i	$0.364243\pi$
$68$	−6.93832	−0.841395
$69$	−8.25957	−0.994336
$70$	−2.19209	−0.262005
$71$	5.38938	0.639602	0.319801	−	0.947485i	$-0.396384\pi$
$71$	5.38938	0.639602	0.319801	+	0.947485i	$0.396384\pi$
$72$	−7.33611	−0.864569
$73$	−10.3384	−1.21002	−0.605011	−	0.796217i	$-0.706831\pi$
$73$	−10.3384	−1.21002	−0.605011	+	0.796217i	$0.706831\pi$
$74$	6.44744	0.749499
$75$	−2.22131	−0.256495
$76$	−3.46562	−0.397534
$77$	−4.53265	−0.516543
$78$	−22.1492	−2.50790
$79$	−6.62558	−0.745436	−0.372718	−	0.927945i	$-0.621574\pi$
$79$	−6.62558	−0.745436	−0.372718	+	0.927945i	$0.621574\pi$
$80$	−2.38557	−0.266714
$81$	22.8102	2.53447
$82$	−3.87129	−0.427512
$83$	−5.07082	−0.556594	−0.278297	−	0.960495i	$-0.589770\pi$
$83$	−5.07082	−0.556594	−0.278297	+	0.960495i	$0.589770\pi$
$84$	2.95424	0.322334
$85$	16.5518	1.79530
$86$	−5.83970	−0.629711
$87$	23.9340	2.56599
$88$	−4.93270	−0.525828
$89$	−11.3299	−1.20097	−0.600486	−	0.799635i	$-0.705026\pi$
$89$	−11.3299	−1.20097	−0.600486	+	0.799635i	$0.705026\pi$
$90$	17.5008	1.84474
$91$	6.33061	0.663628
$92$	2.56909	0.267846
$93$	−19.8811	−2.06158
$94$	1.26512	0.130487
$95$	8.26746	0.848224
$96$	3.21498	0.328128
$97$	−3.94306	−0.400357	−0.200178	−	0.979759i	$-0.564152\pi$
$97$	−3.94306	−0.400357	−0.200178	+	0.979759i	$0.564152\pi$
$98$	6.15563	0.621812
$99$	36.1869	3.63692
$100$	0.690924	0.0690924
$101$	−7.03822	−0.700329	−0.350164	−	0.936688i	$-0.613874\pi$
$101$	−7.03822	−0.700329	−0.350164	+	0.936688i	$0.613874\pi$
$102$	−22.3066	−2.20868
$103$	−2.19288	−0.216071	−0.108035	−	0.994147i	$-0.534456\pi$
$103$	−2.19288	−0.216071	−0.108035	+	0.994147i	$0.534456\pi$
$104$	6.88935	0.675557
$105$	−7.04753	−0.687769
$106$	8.36796	0.812768
$107$	1.60720	0.155374	0.0776870	−	0.996978i	$-0.475247\pi$
$107$	1.60720	0.155374	0.0776870	+	0.996978i	$0.475247\pi$
$108$	−13.9405	−1.34143
$109$	−11.5184	−1.10326	−0.551632	−	0.834088i	$-0.685995\pi$
$109$	−11.5184	−1.10326	−0.551632	+	0.834088i	$0.685995\pi$
$110$	11.7673	1.12197
$111$	20.7284	1.96745
$112$	−0.918897	−0.0868276
$113$	−5.75546	−0.541428	−0.270714	−	0.962660i	$-0.587260\pi$
$113$	−5.75546	−0.541428	−0.270714	+	0.962660i	$0.587260\pi$
$114$	−11.1419	−1.04353
$115$	−6.12873	−0.571507
$116$	−7.44452	−0.691206
$117$	−50.5411	−4.67252
$118$	8.07140	0.743033
$119$	6.37560	0.584450
$120$	−7.66955	−0.700131
$121$	13.3316	1.21196
$122$	−13.9140	−1.25971
$123$	−12.4461	−1.12223
$124$	6.18390	0.555331
$125$	10.2796	0.919434
$126$	6.74113	0.600548
$127$	−19.4897	−1.72944	−0.864718	−	0.502258i	$-0.832503\pi$
$127$	−19.4897	−1.72944	−0.864718	+	0.502258i	$0.832503\pi$
$128$	−1.00000	−0.0883883
$129$	−18.7745	−1.65300
$130$	−16.4350	−1.44145
$131$	−14.2587	−1.24578	−0.622892	−	0.782308i	$-0.714043\pi$
$131$	−14.2587	−1.24578	−0.622892	+	0.782308i	$0.714043\pi$
$132$	−15.8586	−1.38031
$133$	3.18455	0.276135
$134$	−6.77225	−0.585034
$135$	33.2560	2.86222
$136$	6.93832	0.594956
$137$	22.0781	1.88626	0.943130	−	0.332425i	$-0.107867\pi$
$137$	22.0781	1.88626	0.943130	+	0.332425i	$0.107867\pi$
$138$	8.25957	0.703102
$139$	−11.3077	−0.959102	−0.479551	−	0.877514i	$-0.659201\pi$
$139$	−11.3077	−0.959102	−0.479551	+	0.877514i	$0.659201\pi$
$140$	2.19209	0.185265
$141$	4.06733	0.342531
$142$	−5.38938	−0.452267
$143$	−33.9831	−2.84181
$144$	7.33611	0.611343
$145$	17.7594	1.47484
$146$	10.3384	0.855615
$147$	19.7902	1.63227
$148$	−6.44744	−0.529976
$149$	−16.2946	−1.33491	−0.667453	−	0.744652i	$-0.732615\pi$
$149$	−16.2946	−1.33491	−0.667453	+	0.744652i	$0.732615\pi$
$150$	2.22131	0.181369
$151$	−14.7314	−1.19883	−0.599413	−	0.800440i	$-0.704599\pi$
$151$	−14.7314	−1.19883	−0.599413	+	0.800440i	$0.704599\pi$
$152$	3.46562	0.281099
$153$	−50.9003	−4.11504
$154$	4.53265	0.365251
$155$	−14.7521	−1.18492
$156$	22.1492	1.77335
$157$	18.8317	1.50293	0.751467	−	0.659770i	$-0.229346\pi$
$157$	18.8317	1.50293	0.751467	+	0.659770i	$0.229346\pi$
$158$	6.62558	0.527103
$159$	26.9028	2.13353
$160$	2.38557	0.188596
$161$	−2.36073	−0.186051
$162$	−22.8102	−1.79214
$163$	−11.0620	−0.866442	−0.433221	−	0.901288i	$-0.642623\pi$
$163$	−11.0620	−0.866442	−0.433221	+	0.901288i	$0.642623\pi$
$164$	3.87129	0.302297
$165$	37.8316	2.94519
$166$	5.07082	0.393572
$167$	−6.06060	−0.468983	−0.234491	−	0.972118i	$-0.575342\pi$
$167$	−6.06060	−0.468983	−0.234491	+	0.972118i	$0.575342\pi$
$168$	−2.95424	−0.227924
$169$	34.4632	2.65101
$170$	−16.5518	−1.26947
$171$	−25.4242	−1.94423
$172$	5.83970	0.445273
$173$	−16.7007	−1.26973	−0.634866	−	0.772622i	$-0.718945\pi$
$173$	−16.7007	−1.26973	−0.634866	+	0.772622i	$0.718945\pi$
$174$	−23.9340	−1.81443
$175$	−0.634889	−0.0479931
$176$	4.93270	0.371817
$177$	25.9494	1.95048
$178$	11.3299	0.849215
$179$	10.2224	0.764055	0.382027	−	0.924151i	$-0.375226\pi$
$179$	10.2224	0.764055	0.382027	+	0.924151i	$0.375226\pi$
$180$	−17.5008	−1.30443
$181$	−15.3513	−1.14105	−0.570525	−	0.821280i	$-0.693260\pi$
$181$	−15.3513	−1.14105	−0.570525	+	0.821280i	$0.693260\pi$
$182$	−6.33061	−0.469256
$183$	−44.7333	−3.30678
$184$	−2.56909	−0.189396
$185$	15.3808	1.13082
$186$	19.8811	1.45776
$187$	−34.2247	−2.50276
$188$	−1.26512	−0.0922682
$189$	12.8099	0.931784
$190$	−8.26746	−0.599785
$191$	−14.8012	−1.07098	−0.535488	−	0.844543i	$-0.679872\pi$
$191$	−14.8012	−1.07098	−0.535488	+	0.844543i	$0.679872\pi$
$192$	−3.21498	−0.232021
$193$	−2.52536	−0.181780	−0.0908898	−	0.995861i	$-0.528971\pi$
$193$	−2.52536	−0.181780	−0.0908898	+	0.995861i	$0.528971\pi$
$194$	3.94306	0.283095
$195$	−52.8383	−3.78383
$196$	−6.15563	−0.439688
$197$	−21.1507	−1.50692	−0.753461	−	0.657492i	$-0.771617\pi$
$197$	−21.1507	−1.50692	−0.753461	+	0.657492i	$0.771617\pi$
$198$	−36.1869	−2.57169
$199$	−27.0148	−1.91503	−0.957516	−	0.288381i	$-0.906883\pi$
$199$	−27.0148	−1.91503	−0.957516	+	0.288381i	$0.906883\pi$
$200$	−0.690924	−0.0488557
$201$	−21.7727	−1.53573
$202$	7.03822	0.495207
$203$	6.84075	0.480126
$204$	22.3066	1.56177
$205$	−9.23521	−0.645015
$206$	2.19288	0.152785
$207$	18.8471	1.30997
$208$	−6.88935	−0.477691
$209$	−17.0949	−1.18248
$210$	7.04753	0.486326
$211$	−15.9639	−1.09900	−0.549501	−	0.835493i	$-0.685182\pi$
$211$	−15.9639	−1.09900	−0.549501	+	0.835493i	$0.685182\pi$
$212$	−8.36796	−0.574714
$213$	−17.3268	−1.18721
$214$	−1.60720	−0.109866
$215$	−13.9310	−0.950085
$216$	13.9405	0.948533
$217$	−5.68237	−0.385745
$218$	11.5184	0.780125
$219$	33.2379	2.24601
$220$	−11.7673	−0.793351
$221$	47.8005	3.21541
$222$	−20.7284	−1.39120
$223$	−23.6108	−1.58110	−0.790548	−	0.612400i	$-0.790204\pi$
$223$	−23.6108	−1.58110	−0.790548	+	0.612400i	$0.790204\pi$
$224$	0.918897	0.0613964
$225$	5.06870	0.337913
$226$	5.75546	0.382847
$227$	−10.1893	−0.676286	−0.338143	−	0.941095i	$-0.609799\pi$
$227$	−10.1893	−0.676286	−0.338143	+	0.941095i	$0.609799\pi$
$228$	11.1419	0.737890
$229$	−20.2523	−1.33831	−0.669155	−	0.743123i	$-0.733344\pi$
$229$	−20.2523	−1.33831	−0.669155	+	0.743123i	$0.733344\pi$
$230$	6.12873	0.404116
$231$	14.5724	0.958793
$232$	7.44452	0.488756
$233$	−4.72594	−0.309607	−0.154803	−	0.987945i	$-0.549474\pi$
$233$	−4.72594	−0.309607	−0.154803	+	0.987945i	$0.549474\pi$
$234$	50.5411	3.30397
$235$	3.01802	0.196874
$236$	−8.07140	−0.525403
$237$	21.3011	1.38366
$238$	−6.37560	−0.413269
$239$	25.2003	1.63007	0.815035	−	0.579412i	$-0.196718\pi$
$239$	25.2003	1.63007	0.815035	+	0.579412i	$0.196718\pi$
$240$	7.66955	0.495067
$241$	18.0644	1.16363	0.581816	−	0.813321i	$-0.302343\pi$
$241$	18.0644	1.16363	0.581816	+	0.813321i	$0.302343\pi$
$242$	−13.3316	−0.856986
$243$	−31.5129	−2.02155
$244$	13.9140	0.890752
$245$	14.6847	0.938168
$246$	12.4461	0.793535
$247$	23.8759	1.51919
$248$	−6.18390	−0.392678
$249$	16.3026	1.03313
$250$	−10.2796	−0.650138
$251$	4.17623	0.263601	0.131801	−	0.991276i	$-0.457924\pi$
$251$	4.17623	0.263601	0.131801	+	0.991276i	$0.457924\pi$
$252$	−6.74113	−0.424652
$253$	12.6726	0.796717
$254$	19.4897	1.22290
$255$	−53.2138	−3.33238
$256$	1.00000	0.0625000
$257$	17.4740	1.09000	0.544999	−	0.838437i	$-0.316530\pi$
$257$	17.4740	1.09000	0.544999	+	0.838437i	$0.316530\pi$
$258$	18.7745	1.16885
$259$	5.92453	0.368133
$260$	16.4350	1.01926
$261$	−54.6138	−3.38051
$262$	14.2587	0.880903
$263$	25.5806	1.57737	0.788684	−	0.614798i	$-0.210763\pi$
$263$	25.5806	1.57737	0.788684	+	0.614798i	$0.210763\pi$
$264$	15.8586	0.976027
$265$	19.9623	1.22628
$266$	−3.18455	−0.195257
$267$	36.4256	2.22921
$268$	6.77225	0.413681
$269$	−26.8079	−1.63451	−0.817254	−	0.576277i	$-0.804505\pi$
$269$	−26.8079	−1.63451	−0.817254	+	0.576277i	$0.804505\pi$
$270$	−33.2560	−2.02390
$271$	−24.5085	−1.48879	−0.744393	−	0.667742i	$-0.767261\pi$
$271$	−24.5085	−1.48879	−0.744393	+	0.667742i	$0.767261\pi$
$272$	−6.93832	−0.420697
$273$	−20.3528	−1.23181
$274$	−22.0781	−1.33379
$275$	3.40813	0.205518
$276$	−8.25957	−0.497168
$277$	2.65358	0.159438	0.0797190	−	0.996817i	$-0.474598\pi$
$277$	2.65358	0.159438	0.0797190	+	0.996817i	$0.474598\pi$
$278$	11.3077	0.678188
$279$	45.3658	2.71598
$280$	−2.19209	−0.131002
$281$	−0.0913251	−0.00544800	−0.00272400	−	0.999996i	$-0.500867\pi$
$281$	−0.0913251	−0.00544800	−0.00272400	+	0.999996i	$0.500867\pi$
$282$	−4.06733	−0.242206
$283$	2.76051	0.164095	0.0820476	−	0.996628i	$-0.473854\pi$
$283$	2.76051	0.164095	0.0820476	+	0.996628i	$0.473854\pi$
$284$	5.38938	0.319801
$285$	−26.5797	−1.57445
$286$	33.9831	2.00947
$287$	−3.55731	−0.209982
$288$	−7.33611	−0.432285
$289$	31.1402	1.83178
$290$	−17.7594	−1.04287
$291$	12.6769	0.743130
$292$	−10.3384	−0.605011
$293$	−1.91845	−0.112077	−0.0560385	−	0.998429i	$-0.517847\pi$
$293$	−1.91845	−0.112077	−0.0560385	+	0.998429i	$0.517847\pi$
$294$	−19.7902	−1.15419
$295$	19.2549	1.12106
$296$	6.44744	0.374750
$297$	−68.7645	−3.99012
$298$	16.2946	0.943920
$299$	−17.6994	−1.02358
$300$	−2.22131	−0.128247
$301$	−5.36608	−0.309296
$302$	14.7314	0.847698
$303$	22.6277	1.29993
$304$	−3.46562	−0.198767
$305$	−33.1928	−1.90061
$306$	50.9003	2.90978
$307$	−9.52752	−0.543764	−0.271882	−	0.962331i	$-0.587646\pi$
$307$	−9.52752	−0.543764	−0.271882	+	0.962331i	$0.587646\pi$
$308$	−4.53265	−0.258272
$309$	7.05006	0.401064
$310$	14.7521	0.837864
$311$	−4.79818	−0.272080	−0.136040	−	0.990703i	$-0.543438\pi$
$311$	−4.79818	−0.272080	−0.136040	+	0.990703i	$0.543438\pi$
$312$	−22.1492	−1.25395
$313$	22.1085	1.24965	0.624823	−	0.780767i	$-0.285171\pi$
$313$	22.1085	1.24965	0.624823	+	0.780767i	$0.285171\pi$
$314$	−18.8317	−1.06274
$315$	16.0814	0.906085
$316$	−6.62558	−0.372718
$317$	16.0293	0.900298	0.450149	−	0.892953i	$-0.351371\pi$
$317$	16.0293	0.900298	0.450149	+	0.892953i	$0.351371\pi$
$318$	−26.9028	−1.50864
$319$	−36.7216	−2.05601
$320$	−2.38557	−0.133357
$321$	−5.16712	−0.288401
$322$	2.36073	0.131558
$323$	24.0456	1.33793
$324$	22.8102	1.26723
$325$	−4.76002	−0.264039
$326$	11.0620	0.612667
$327$	37.0315	2.04785
$328$	−3.87129	−0.213756
$329$	1.16251	0.0640914
$330$	−37.8316	−2.08256
$331$	−13.9719	−0.767967	−0.383983	−	0.923340i	$-0.625448\pi$
$331$	−13.9719	−0.767967	−0.383983	+	0.923340i	$0.625448\pi$
$332$	−5.07082	−0.278297
$333$	−47.2991	−2.59198
$334$	6.06060	0.331621
$335$	−16.1557	−0.882678
$336$	2.95424	0.161167
$337$	−27.5913	−1.50300	−0.751498	−	0.659735i	$-0.770668\pi$
$337$	−27.5913	−1.50300	−0.751498	+	0.659735i	$0.770668\pi$
$338$	−34.4632	−1.87455
$339$	18.5037	1.00498
$340$	16.5518	0.897648
$341$	30.5034	1.65185
$342$	25.4242	1.37478
$343$	12.0887	0.652727
$344$	−5.83970	−0.314855
$345$	19.7038	1.06081
$346$	16.7007	0.897836
$347$	−10.4357	−0.560219	−0.280110	−	0.959968i	$-0.590371\pi$
$347$	−10.4357	−0.560219	−0.280110	+	0.959968i	$0.590371\pi$
$348$	23.9340	1.28300
$349$	−29.1843	−1.56220	−0.781100	−	0.624405i	$-0.785341\pi$
$349$	−29.1843	−1.56220	−0.781100	+	0.624405i	$0.785341\pi$
$350$	0.634889	0.0339362
$351$	96.0412	5.12630
$352$	−4.93270	−0.262914
$353$	28.5078	1.51732	0.758658	−	0.651489i	$-0.225855\pi$
$353$	28.5078	1.51732	0.758658	+	0.651489i	$0.225855\pi$
$354$	−25.9494	−1.37920
$355$	−12.8567	−0.682364
$356$	−11.3299	−0.600486
$357$	−20.4974	−1.08484
$358$	−10.2224	−0.540268
$359$	−4.51364	−0.238221	−0.119111	−	0.992881i	$-0.538004\pi$
$359$	−4.51364	−0.238221	−0.119111	+	0.992881i	$0.538004\pi$
$360$	17.5008	0.922372
$361$	−6.98949	−0.367868
$362$	15.3513	0.806844
$363$	−42.8608	−2.24961
$364$	6.33061	0.331814
$365$	24.6630	1.29092
$366$	44.7333	2.33824
$367$	15.7353	0.821373	0.410687	−	0.911777i	$-0.365289\pi$
$367$	15.7353	0.821373	0.410687	+	0.911777i	$0.365289\pi$
$368$	2.56909	0.133923
$369$	28.4002	1.47845
$370$	−15.3808	−0.799609
$371$	7.68929	0.399208
$372$	−19.8811	−1.03079
$373$	−11.9823	−0.620422	−0.310211	−	0.950668i	$-0.600400\pi$
$373$	−11.9823	−0.620422	−0.310211	+	0.950668i	$0.600400\pi$
$374$	34.2247	1.76972
$375$	−33.0487	−1.70663
$376$	1.26512	0.0652435
$377$	51.2879	2.64146
$378$	−12.8099	−0.658871
$379$	4.61931	0.237278	0.118639	−	0.992937i	$-0.462147\pi$
$379$	4.61931	0.237278	0.118639	+	0.992937i	$0.462147\pi$
$380$	8.26746	0.424112
$381$	62.6592	3.21013
$382$	14.8012	0.757295
$383$	16.7558	0.856181	0.428090	−	0.903736i	$-0.359187\pi$
$383$	16.7558	0.856181	0.428090	+	0.903736i	$0.359187\pi$
$384$	3.21498	0.164064
$385$	10.8129	0.551078
$386$	2.52536	0.128538
$387$	42.8407	2.17771
$388$	−3.94306	−0.200178
$389$	−7.48847	−0.379681	−0.189840	−	0.981815i	$-0.560797\pi$
$389$	−7.48847	−0.379681	−0.189840	+	0.981815i	$0.560797\pi$
$390$	52.8383	2.67557
$391$	−17.8252	−0.901457
$392$	6.15563	0.310906
$393$	45.8413	2.31239
$394$	21.1507	1.06555
$395$	15.8058	0.795274
$396$	36.1869	1.81846
$397$	4.09325	0.205435	0.102717	−	0.994711i	$-0.467246\pi$
$397$	4.09325	0.205435	0.102717	+	0.994711i	$0.467246\pi$
$398$	27.0148	1.35413
$399$	−10.2383	−0.512554
$400$	0.690924	0.0345462
$401$	−15.3389	−0.765988	−0.382994	−	0.923751i	$-0.625107\pi$
$401$	−15.3389	−0.765988	−0.382994	+	0.923751i	$0.625107\pi$
$402$	21.7727	1.08592
$403$	−42.6031	−2.12221
$404$	−7.03822	−0.350164
$405$	−54.4153	−2.70392
$406$	−6.84075	−0.339501
$407$	−31.8033	−1.57643
$408$	−22.3066	−1.10434
$409$	−7.03593	−0.347904	−0.173952	−	0.984754i	$-0.555654\pi$
$409$	−7.03593	−0.347904	−0.173952	+	0.984754i	$0.555654\pi$
$410$	9.23521	0.456094
$411$	−70.9807	−3.50122
$412$	−2.19288	−0.108035
$413$	7.41679	0.364956
$414$	−18.8471	−0.926286
$415$	12.0968	0.593807
$416$	6.88935	0.337778
$417$	36.3539	1.78026
$418$	17.0949	0.836137
$419$	−17.1841	−0.839498	−0.419749	−	0.907640i	$-0.637882\pi$
$419$	−17.1841	−0.839498	−0.419749	+	0.907640i	$0.637882\pi$
$420$	−7.04753	−0.343884
$421$	−8.03305	−0.391507	−0.195753	−	0.980653i	$-0.562715\pi$
$421$	−8.03305	−0.391507	−0.195753	+	0.980653i	$0.562715\pi$
$422$	15.9639	0.777111
$423$	−9.28105	−0.451260
$424$	8.36796	0.406384
$425$	−4.79385	−0.232536
$426$	17.3268	0.839485
$427$	−12.7855	−0.618735
$428$	1.60720	0.0776870
$429$	109.255	5.27489
$430$	13.9310	0.671812
$431$	−35.1401	−1.69264	−0.846320	−	0.532676i	$-0.821187\pi$
$431$	−35.1401	−1.69264	−0.846320	+	0.532676i	$0.821187\pi$
$432$	−13.9405	−0.670714
$433$	−1.31680	−0.0632813	−0.0316407	−	0.999499i	$-0.510073\pi$
$433$	−1.31680	−0.0632813	−0.0316407	+	0.999499i	$0.510073\pi$
$434$	5.68237	0.272763
$435$	−57.0961	−2.73755
$436$	−11.5184	−0.551632
$437$	−8.90348	−0.425911
$438$	−33.2379	−1.58817
$439$	27.9854	1.33567	0.667834	−	0.744310i	$-0.267222\pi$
$439$	27.9854	1.33567	0.667834	+	0.744310i	$0.267222\pi$
$440$	11.7673	0.560984
$441$	−45.1584	−2.15040
$442$	−47.8005	−2.27364
$443$	8.69912	0.413308	0.206654	−	0.978414i	$-0.433743\pi$
$443$	8.69912	0.413308	0.206654	+	0.978414i	$0.433743\pi$
$444$	20.7284	0.983726
$445$	27.0283	1.28127
$446$	23.6108	1.11800
$447$	52.3868	2.47781
$448$	−0.918897	−0.0434138
$449$	−20.4221	−0.963778	−0.481889	−	0.876232i	$-0.660049\pi$
$449$	−20.4221	−0.963778	−0.481889	+	0.876232i	$0.660049\pi$
$450$	−5.06870	−0.238941
$451$	19.0959	0.899191
$452$	−5.75546	−0.270714
$453$	47.3612	2.22523
$454$	10.1893	0.478206
$455$	−15.1021	−0.707997
$456$	−11.1419	−0.521767
$457$	9.07725	0.424616	0.212308	−	0.977203i	$-0.431902\pi$
$457$	9.07725	0.424616	0.212308	+	0.977203i	$0.431902\pi$
$458$	20.2523	0.946327
$459$	96.7238	4.51468
$460$	−6.12873	−0.285753
$461$	30.4710	1.41918	0.709588	−	0.704616i	$-0.248881\pi$
$461$	30.4710	1.41918	0.709588	+	0.704616i	$0.248881\pi$
$462$	−14.5724	−0.677969
$463$	18.3635	0.853425	0.426712	−	0.904387i	$-0.359672\pi$
$463$	18.3635	0.853425	0.426712	+	0.904387i	$0.359672\pi$
$464$	−7.44452	−0.345603
$465$	47.4278	2.19941
$466$	4.72594	0.218925
$467$	5.34832	0.247491	0.123745	−	0.992314i	$-0.460509\pi$
$467$	5.34832	0.247491	0.123745	+	0.992314i	$0.460509\pi$
$468$	−50.5411	−2.33626
$469$	−6.22301	−0.287352
$470$	−3.01802	−0.139211
$471$	−60.5436	−2.78970
$472$	8.07140	0.371516
$473$	28.8055	1.32448
$474$	−21.3011	−0.978393
$475$	−2.39448	−0.109866
$476$	6.37560	0.292225
$477$	−61.3883	−2.81078
$478$	−25.2003	−1.15263
$479$	20.7977	0.950269	0.475135	−	0.879913i	$-0.342399\pi$
$479$	20.7977	0.950269	0.475135	+	0.879913i	$0.342399\pi$
$480$	−7.66955	−0.350066
$481$	44.4187	2.02532
$482$	−18.0644	−0.822811
$483$	7.58970	0.345343
$484$	13.3316	0.605981
$485$	9.40642	0.427124
$486$	31.5129	1.42945
$487$	33.6207	1.52350	0.761750	−	0.647872i	$-0.224341\pi$
$487$	33.6207	1.52350	0.761750	+	0.647872i	$0.224341\pi$
$488$	−13.9140	−0.629857
$489$	35.5641	1.60827
$490$	−14.6847	−0.663385
$491$	11.5202	0.519901	0.259951	−	0.965622i	$-0.416294\pi$
$491$	11.5202	0.519901	0.259951	+	0.965622i	$0.416294\pi$
$492$	−12.4461	−0.561114
$493$	51.6524	2.32631
$494$	−23.8759	−1.07423
$495$	−86.3262	−3.88007
$496$	6.18390	0.277665
$497$	−4.95229	−0.222141
$498$	−16.3026	−0.730536
$499$	32.7085	1.46423	0.732116	−	0.681180i	$-0.238533\pi$
$499$	32.7085	1.46423	0.732116	+	0.681180i	$0.238533\pi$
$500$	10.2796	0.459717
$501$	19.4847	0.870513
$502$	−4.17623	−0.186394
$503$	3.45400	0.154006	0.0770032	−	0.997031i	$-0.475465\pi$
$503$	3.45400	0.154006	0.0770032	+	0.997031i	$0.475465\pi$
$504$	6.74113	0.300274
$505$	16.7901	0.747151
$506$	−12.6726	−0.563364
$507$	−110.799	−4.92074
$508$	−19.4897	−0.864718
$509$	33.5324	1.48630	0.743148	−	0.669127i	$-0.233332\pi$
$509$	33.5324	1.48630	0.743148	+	0.669127i	$0.233332\pi$
$510$	53.2138	2.35635
$511$	9.49996	0.420254
$512$	−1.00000	−0.0441942
$513$	48.3125	2.13305
$514$	−17.4740	−0.770745
$515$	5.23126	0.230517
$516$	−18.7745	−0.826502
$517$	−6.24045	−0.274455
$518$	−5.92453	−0.260309
$519$	53.6925	2.35684
$520$	−16.4350	−0.720723
$521$	1.23694	0.0541915	0.0270958	−	0.999633i	$-0.491374\pi$
$521$	1.23694	0.0541915	0.0270958	+	0.999633i	$0.491374\pi$
$522$	54.6138	2.39038
$523$	−14.3098	−0.625723	−0.312862	−	0.949799i	$-0.601288\pi$
$523$	−14.3098	−0.625723	−0.312862	+	0.949799i	$0.601288\pi$
$524$	−14.2587	−0.622892
$525$	2.04116	0.0890833
$526$	−25.5806	−1.11537
$527$	−42.9059	−1.86901
$528$	−15.8586	−0.690155
$529$	−16.3998	−0.713034
$530$	−19.9623	−0.867107
$531$	−59.2127	−2.56961
$532$	3.18455	0.138068
$533$	−26.6707	−1.15523
$534$	−36.4256	−1.57629
$535$	−3.83408	−0.165762
$536$	−6.77225	−0.292517
$537$	−32.8647	−1.41822
$538$	26.8079	1.15577
$539$	−30.3639	−1.30787
$540$	33.2560	1.43111
$541$	−28.4772	−1.22433	−0.612165	−	0.790730i	$-0.709701\pi$
$541$	−28.4772	−1.22433	−0.612165	+	0.790730i	$0.709701\pi$
$542$	24.5085	1.05273
$543$	49.3540	2.11798
$544$	6.93832	0.297478
$545$	27.4779	1.17703
$546$	20.3528	0.871019
$547$	−26.3618	−1.12715	−0.563575	−	0.826065i	$-0.690575\pi$
$547$	−26.3618	−1.12715	−0.563575	+	0.826065i	$0.690575\pi$
$548$	22.0781	0.943130
$549$	102.075	4.35644
$550$	−3.40813	−0.145323
$551$	25.7998	1.09911
$552$	8.25957	0.351551
$553$	6.08823	0.258898
$554$	−2.65358	−0.112740
$555$	−49.4490	−2.09899
$556$	−11.3077	−0.479551
$557$	−39.0591	−1.65499	−0.827494	−	0.561474i	$-0.810234\pi$
$557$	−39.0591	−1.65499	−0.827494	+	0.561474i	$0.810234\pi$
$558$	−45.3658	−1.92049
$559$	−40.2317	−1.70162
$560$	2.19209	0.0926327
$561$	110.032	4.64554
$562$	0.0913251	0.00385232
$563$	29.7270	1.25284	0.626422	−	0.779484i	$-0.284519\pi$
$563$	29.7270	1.25284	0.626422	+	0.779484i	$0.284519\pi$
$564$	4.06733	0.171266
$565$	13.7300	0.577627
$566$	−2.76051	−0.116033
$567$	−20.9602	−0.880248
$568$	−5.38938	−0.226134
$569$	−14.6311	−0.613368	−0.306684	−	0.951811i	$-0.599220\pi$
$569$	−14.6311	−0.613368	−0.306684	+	0.951811i	$0.599220\pi$
$570$	26.5797	1.11330
$571$	13.1273	0.549361	0.274680	−	0.961536i	$-0.411428\pi$
$571$	13.1273	0.549361	0.274680	+	0.961536i	$0.411428\pi$
$572$	−33.9831	−1.42091
$573$	47.5856	1.98792
$574$	3.55731	0.148479
$575$	1.77505	0.0740245
$576$	7.33611	0.305671
$577$	−5.65432	−0.235393	−0.117696	−	0.993050i	$-0.537551\pi$
$577$	−5.65432	−0.235393	−0.117696	+	0.993050i	$0.537551\pi$
$578$	−31.1402	−1.29526
$579$	8.11900	0.337414
$580$	17.7594	0.737418
$581$	4.65956	0.193311
$582$	−12.6769	−0.525473
$583$	−41.2767	−1.70950
$584$	10.3384	0.427808
$585$	120.569	4.98492
$586$	1.91845	0.0792504
$587$	−11.6755	−0.481898	−0.240949	−	0.970538i	$-0.577459\pi$
$587$	−11.6755	−0.481898	−0.240949	+	0.970538i	$0.577459\pi$
$588$	19.7902	0.816136
$589$	−21.4310	−0.883051
$590$	−19.2549	−0.792710
$591$	67.9990	2.79711
$592$	−6.44744	−0.264988
$593$	−39.1850	−1.60913	−0.804567	−	0.593862i	$-0.797603\pi$
$593$	−39.1850	−1.60913	−0.804567	+	0.593862i	$0.797603\pi$
$594$	68.7645	2.82144
$595$	−15.2094	−0.623525
$596$	−16.2946	−0.667453
$597$	86.8522	3.55463
$598$	17.6994	0.723781
$599$	30.7546	1.25660	0.628299	−	0.777972i	$-0.283752\pi$
$599$	30.7546	1.25660	0.628299	+	0.777972i	$0.283752\pi$
$600$	2.22131	0.0906846
$601$	29.7169	1.21218	0.606088	−	0.795397i	$-0.292738\pi$
$601$	29.7169	1.21218	0.606088	+	0.795397i	$0.292738\pi$
$602$	5.36608	0.218705
$603$	49.6820	2.02321
$604$	−14.7314	−0.599413
$605$	−31.8034	−1.29299
$606$	−22.6277	−0.919189
$607$	−25.1175	−1.01949	−0.509744	−	0.860326i	$-0.670260\pi$
$607$	−25.1175	−1.01949	−0.509744	+	0.860326i	$0.670260\pi$
$608$	3.46562	0.140549
$609$	−21.9929	−0.891196
$610$	33.1928	1.34394
$611$	8.71584	0.352605
$612$	−50.9003	−2.05752
$613$	−1.16650	−0.0471146	−0.0235573	−	0.999722i	$-0.507499\pi$
$613$	−1.16650	−0.0471146	−0.0235573	+	0.999722i	$0.507499\pi$
$614$	9.52752	0.384499
$615$	29.6910	1.19726
$616$	4.53265	0.182626
$617$	8.36648	0.336822	0.168411	−	0.985717i	$-0.446136\pi$
$617$	8.36648	0.336822	0.168411	+	0.985717i	$0.446136\pi$
$618$	−7.05006	−0.283595
$619$	−25.9727	−1.04393	−0.521966	−	0.852966i	$-0.674801\pi$
$619$	−25.9727	−1.04393	−0.521966	+	0.852966i	$0.674801\pi$
$620$	−14.7521	−0.592459
$621$	−35.8144	−1.43718
$622$	4.79818	0.192389
$623$	10.4111	0.417110
$624$	22.1492	0.886676
$625$	−27.9772	−1.11909
$626$	−22.1085	−0.883633
$627$	54.9597	2.19488
$628$	18.8317	0.751467
$629$	44.7344	1.78368
$630$	−16.0814	−0.640699
$631$	−8.76685	−0.349003	−0.174501	−	0.984657i	$-0.555831\pi$
$631$	−8.76685	−0.349003	−0.174501	+	0.984657i	$0.555831\pi$
$632$	6.62558	0.263551
$633$	51.3237	2.03993
$634$	−16.0293	−0.636607
$635$	46.4941	1.84506
$636$	26.9028	1.06677
$637$	42.4083	1.68028
$638$	36.7216	1.45382
$639$	39.5371	1.56406
$640$	2.38557	0.0942978
$641$	−25.5436	−1.00891	−0.504456	−	0.863437i	$-0.668307\pi$
$641$	−25.5436	−1.00891	−0.504456	+	0.863437i	$0.668307\pi$
$642$	5.16712	0.203930
$643$	−0.596605	−0.0235278	−0.0117639	−	0.999931i	$-0.503745\pi$
$643$	−0.596605	−0.0235278	−0.0117639	+	0.999931i	$0.503745\pi$
$644$	−2.36073	−0.0930257
$645$	44.7879	1.76352
$646$	−24.0456	−0.946060
$647$	35.1583	1.38222	0.691108	−	0.722751i	$-0.257123\pi$
$647$	35.1583	1.38222	0.691108	+	0.722751i	$0.257123\pi$
$648$	−22.8102	−0.896070
$649$	−39.8138	−1.56283
$650$	4.76002	0.186703
$651$	18.2687	0.716008
$652$	−11.0620	−0.433221
$653$	−7.18231	−0.281065	−0.140533	−	0.990076i	$-0.544882\pi$
$653$	−7.18231	−0.281065	−0.140533	+	0.990076i	$0.544882\pi$
$654$	−37.0315	−1.44805
$655$	34.0150	1.32907
$656$	3.87129	0.151148
$657$	−75.8439	−2.95895
$658$	−1.16251	−0.0453195
$659$	0.940045	0.0366190	0.0183095	−	0.999832i	$-0.494172\pi$
$659$	0.940045	0.0366190	0.0183095	+	0.999832i	$0.494172\pi$
$660$	37.8316	1.47259
$661$	−27.1961	−1.05781	−0.528904	−	0.848682i	$-0.677397\pi$
$661$	−27.1961	−1.05781	−0.528904	+	0.848682i	$0.677397\pi$
$662$	13.9719	0.543035
$663$	−153.678	−5.96835
$664$	5.07082	0.196786
$665$	−7.59695	−0.294597
$666$	47.2991	1.83280
$667$	−19.1256	−0.740547
$668$	−6.06060	−0.234491
$669$	75.9083	2.93479
$670$	16.1557	0.624148
$671$	68.6336	2.64957
$672$	−2.95424	−0.113962
$673$	−0.600037	−0.0231297	−0.0115649	−	0.999933i	$-0.503681\pi$
$673$	−0.600037	−0.0231297	−0.0115649	+	0.999933i	$0.503681\pi$
$674$	27.5913	1.06278
$675$	−9.63185	−0.370730
$676$	34.4632	1.32551
$677$	24.6859	0.948756	0.474378	−	0.880321i	$-0.342673\pi$
$677$	24.6859	0.948756	0.474378	+	0.880321i	$0.342673\pi$
$678$	−18.5037	−0.710630
$679$	3.62326	0.139048
$680$	−16.5518	−0.634733
$681$	32.7583	1.25530
$682$	−30.5034	−1.16803
$683$	10.4164	0.398573	0.199286	−	0.979941i	$-0.436138\pi$
$683$	10.4164	0.398573	0.199286	+	0.979941i	$0.436138\pi$
$684$	−25.4242	−0.972117
$685$	−52.6688	−2.01237
$686$	−12.0887	−0.461548
$687$	65.1108	2.48413
$688$	5.83970	0.222636
$689$	57.6498	2.19628
$690$	−19.7038	−0.750109
$691$	21.8715	0.832030	0.416015	−	0.909358i	$-0.363426\pi$
$691$	21.8715	0.832030	0.416015	+	0.909358i	$0.363426\pi$
$692$	−16.7007	−0.634866
$693$	−33.2520	−1.26314
$694$	10.4357	0.396135
$695$	26.9751	1.02323
$696$	−23.9340	−0.907215
$697$	−26.8602	−1.01740
$698$	29.1843	1.10464
$699$	15.1938	0.574683
$700$	−0.634889	−0.0239965
$701$	7.48623	0.282751	0.141376	−	0.989956i	$-0.454847\pi$
$701$	7.48623	0.282751	0.141376	+	0.989956i	$0.454847\pi$
$702$	−96.0412	−3.62484
$703$	22.3444	0.842733
$704$	4.93270	0.185908
$705$	−9.70289	−0.365432
$706$	−28.5078	−1.07290
$707$	6.46740	0.243232
$708$	25.9494	0.975239
$709$	−49.3542	−1.85354	−0.926768	−	0.375633i	$-0.877425\pi$
$709$	−49.3542	−1.85354	−0.926768	+	0.375633i	$0.877425\pi$
$710$	12.8567	0.482505
$711$	−48.6060	−1.82287
$712$	11.3299	0.424608
$713$	15.8870	0.594973
$714$	20.4974	0.767098
$715$	81.0690	3.03181
$716$	10.2224	0.382027
$717$	−81.0184	−3.02569
$718$	4.51364	0.168448
$719$	−36.8033	−1.37253	−0.686265	−	0.727352i	$-0.740751\pi$
$719$	−36.8033	−1.37253	−0.686265	+	0.727352i	$0.740751\pi$
$720$	−17.5008	−0.652216
$721$	2.01503	0.0750436
$722$	6.98949	0.260122
$723$	−58.0768	−2.15990
$724$	−15.3513	−0.570525
$725$	−5.14360	−0.191028
$726$	42.8608	1.59071
$727$	9.40749	0.348904	0.174452	−	0.984666i	$-0.444185\pi$
$727$	9.40749	0.348904	0.174452	+	0.984666i	$0.444185\pi$
$728$	−6.33061	−0.234628
$729$	32.8826	1.21788
$730$	−24.6630	−0.912819
$731$	−40.5177	−1.49860
$732$	−44.7333	−1.65339
$733$	−27.5075	−1.01601	−0.508006	−	0.861353i	$-0.669617\pi$
$733$	−27.5075	−1.01601	−0.508006	+	0.861353i	$0.669617\pi$
$734$	−15.7353	−0.580799
$735$	−47.2109	−1.74140
$736$	−2.56909	−0.0946979
$737$	33.4055	1.23051
$738$	−28.4002	−1.04543
$739$	−17.9309	−0.659600	−0.329800	−	0.944051i	$-0.606981\pi$
$739$	−17.9309	−0.659600	−0.329800	+	0.944051i	$0.606981\pi$
$740$	15.3808	0.565409
$741$	−76.7605	−2.81987
$742$	−7.68929	−0.282283
$743$	−48.3634	−1.77428	−0.887140	−	0.461501i	$-0.847311\pi$
$743$	−48.3634	−1.77428	−0.887140	+	0.461501i	$0.847311\pi$
$744$	19.8811	0.728878
$745$	38.8718	1.42415
$746$	11.9823	0.438704
$747$	−37.2001	−1.36108
$748$	−34.2247	−1.25138
$749$	−1.47685	−0.0539630
$750$	33.0487	1.20677
$751$	25.3478	0.924953	0.462476	−	0.886632i	$-0.346961\pi$
$751$	25.3478	0.924953	0.462476	+	0.886632i	$0.346961\pi$
$752$	−1.26512	−0.0461341
$753$	−13.4265	−0.489289
$754$	−51.2879	−1.86780
$755$	35.1428	1.27898
$756$	12.8099	0.465892
$757$	2.61048	0.0948795	0.0474397	−	0.998874i	$-0.484894\pi$
$757$	2.61048	0.0948795	0.0474397	+	0.998874i	$0.484894\pi$
$758$	−4.61931	−0.167781
$759$	−40.7420	−1.47884
$760$	−8.26746	−0.299892
$761$	43.2160	1.56658	0.783289	−	0.621658i	$-0.213541\pi$
$761$	43.2160	1.56658	0.783289	+	0.621658i	$0.213541\pi$
$762$	−62.6592	−2.26990
$763$	10.5842	0.383175
$764$	−14.8012	−0.535488
$765$	121.426	4.39017
$766$	−16.7558	−0.605411
$767$	55.6067	2.00784
$768$	−3.21498	−0.116011
$769$	−41.4076	−1.49319	−0.746597	−	0.665276i	$-0.768314\pi$
$769$	−41.4076	−1.49319	−0.746597	+	0.665276i	$0.768314\pi$
$770$	−10.8129	−0.389671
$771$	−56.1786	−2.02322
$772$	−2.52536	−0.0908898
$773$	21.3348	0.767359	0.383679	−	0.923466i	$-0.374657\pi$
$773$	21.3348	0.767359	0.383679	+	0.923466i	$0.374657\pi$
$774$	−42.8407	−1.53988
$775$	4.27261	0.153477
$776$	3.94306	0.141547
$777$	−19.0473	−0.683317
$778$	7.48847	0.268475
$779$	−13.4164	−0.480692
$780$	−52.8383	−1.89191
$781$	26.5842	0.951259
$782$	17.8252	0.637426
$783$	103.780	3.70881
$784$	−6.15563	−0.219844
$785$	−44.9243	−1.60342
$786$	−45.8413	−1.63511
$787$	−1.63085	−0.0581336	−0.0290668	−	0.999577i	$-0.509254\pi$
$787$	−1.63085	−0.0581336	−0.0290668	+	0.999577i	$0.509254\pi$
$788$	−21.1507	−0.753461
$789$	−82.2413	−2.92787
$790$	−15.8058	−0.562344
$791$	5.28868	0.188044
$792$	−36.1869	−1.28584
$793$	−95.8584	−3.40403
$794$	−4.09325	−0.145264
$795$	−64.1785	−2.27618
$796$	−27.0148	−0.957516
$797$	47.8325	1.69431	0.847157	−	0.531343i	$-0.178313\pi$
$797$	47.8325	1.69431	0.847157	+	0.531343i	$0.178313\pi$
$798$	10.2383	0.362431
$799$	8.77779	0.310536
$800$	−0.690924	−0.0244279
$801$	−83.1177	−2.93682
$802$	15.3389	0.541635
$803$	−50.9965	−1.79963
$804$	−21.7727	−0.767863
$805$	5.63167	0.198490
$806$	42.6031	1.50063
$807$	86.1870	3.03393
$808$	7.03822	0.247604
$809$	−7.24229	−0.254625	−0.127313	−	0.991863i	$-0.540635\pi$
$809$	−7.24229	−0.254625	−0.127313	+	0.991863i	$0.540635\pi$
$810$	54.4153	1.91196
$811$	−20.0138	−0.702781	−0.351391	−	0.936229i	$-0.614291\pi$
$811$	−20.0138	−0.702781	−0.351391	+	0.936229i	$0.614291\pi$
$812$	6.84075	0.240063
$813$	78.7944	2.76344
$814$	31.8033	1.11471
$815$	26.3891	0.924370
$816$	22.3066	0.780886
$817$	−20.2382	−0.708044
$818$	7.03593	0.246005
$819$	46.4421	1.62282
$820$	−9.23521	−0.322507
$821$	10.9030	0.380517	0.190259	−	0.981734i	$-0.439067\pi$
$821$	10.9030	0.380517	0.190259	+	0.981734i	$0.439067\pi$
$822$	70.9807	2.47574
$823$	−44.5700	−1.55361	−0.776806	−	0.629740i	$-0.783161\pi$
$823$	−44.5700	−1.55361	−0.776806	+	0.629740i	$0.783161\pi$
$824$	2.19288	0.0763925
$825$	−10.9571	−0.381476
$826$	−7.41679	−0.258063
$827$	10.2716	0.357177	0.178589	−	0.983924i	$-0.442847\pi$
$827$	10.2716	0.357177	0.178589	+	0.983924i	$0.442847\pi$
$828$	18.8471	0.654983
$829$	−19.0214	−0.660639	−0.330319	−	0.943869i	$-0.607156\pi$
$829$	−19.0214	−0.660639	−0.330319	+	0.943869i	$0.607156\pi$
$830$	−12.0968	−0.419885
$831$	−8.53120	−0.295944
$832$	−6.88935	−0.238845
$833$	42.7097	1.47980
$834$	−36.3539	−1.25883
$835$	14.4579	0.500338
$836$	−17.0949	−0.591238
$837$	−86.2069	−2.97975
$838$	17.1841	0.593615
$839$	12.6294	0.436015	0.218007	−	0.975947i	$-0.430044\pi$
$839$	12.6294	0.436015	0.218007	+	0.975947i	$0.430044\pi$
$840$	7.04753	0.243163
$841$	26.4208	0.911063
$842$	8.03305	0.276837
$843$	0.293609	0.0101124
$844$	−15.9639	−0.549501
$845$	−82.2142	−2.82825
$846$	9.28105	0.319089
$847$	−12.2503	−0.420927
$848$	−8.36796	−0.287357
$849$	−8.87499	−0.304589
$850$	4.79385	0.164428
$851$	−16.5640	−0.567808
$852$	−17.3268	−0.593606
$853$	−21.2904	−0.728970	−0.364485	−	0.931209i	$-0.618755\pi$
$853$	−21.2904	−0.728970	−0.364485	+	0.931209i	$0.618755\pi$
$854$	12.7855	0.437512
$855$	60.6510	2.07422
$856$	−1.60720	−0.0549330
$857$	−24.2951	−0.829906	−0.414953	−	0.909843i	$-0.636202\pi$
$857$	−24.2951	−0.829906	−0.414953	+	0.909843i	$0.636202\pi$
$858$	−109.255	−3.72991
$859$	13.5812	0.463383	0.231692	−	0.972789i	$-0.425574\pi$
$859$	13.5812	0.463383	0.231692	+	0.972789i	$0.425574\pi$
$860$	−13.9310	−0.475043
$861$	11.4367	0.389762
$862$	35.1401	1.19688
$863$	1.83738	0.0625452	0.0312726	−	0.999511i	$-0.490044\pi$
$863$	1.83738	0.0625452	0.0312726	+	0.999511i	$0.490044\pi$
$864$	13.9405	0.474266
$865$	39.8407	1.35462
$866$	1.31680	0.0447467
$867$	−100.115	−3.40010
$868$	−5.68237	−0.192872
$869$	−32.6820	−1.10866
$870$	57.0961	1.93574
$871$	−46.6565	−1.58089
$872$	11.5184	0.390063
$873$	−28.9267	−0.979021
$874$	8.90348	0.301165
$875$	−9.44588	−0.319329
$876$	33.2379	1.12300
$877$	10.5853	0.357441	0.178720	−	0.983900i	$-0.442804\pi$
$877$	10.5853	0.357441	0.178720	+	0.983900i	$0.442804\pi$
$878$	−27.9854	−0.944460
$879$	6.16778	0.208034
$880$	−11.7673	−0.396675
$881$	13.3084	0.448370	0.224185	−	0.974547i	$-0.428028\pi$
$881$	13.3084	0.448370	0.224185	+	0.974547i	$0.428028\pi$
$882$	45.1584	1.52056
$883$	−3.65703	−0.123069	−0.0615345	−	0.998105i	$-0.519599\pi$
$883$	−3.65703	−0.123069	−0.0615345	+	0.998105i	$0.519599\pi$
$884$	47.8005	1.60771
$885$	−61.9040	−2.08088
$886$	−8.69912	−0.292253
$887$	−10.4505	−0.350893	−0.175447	−	0.984489i	$-0.556137\pi$
$887$	−10.4505	−0.350893	−0.175447	+	0.984489i	$0.556137\pi$
$888$	−20.7284	−0.695599
$889$	17.9091	0.600651
$890$	−27.0283	−0.905992
$891$	112.516	3.76943
$892$	−23.6108	−0.790548
$893$	4.38442	0.146719
$894$	−52.3868	−1.75208
$895$	−24.3861	−0.815138
$896$	0.918897	0.0306982
$897$	56.9031	1.89994
$898$	20.4221	0.681494
$899$	−46.0362	−1.53539
$900$	5.06870	0.168957
$901$	58.0595	1.93424
$902$	−19.0959	−0.635824
$903$	17.2519	0.574106
$904$	5.75546	0.191424
$905$	36.6214	1.21734
$906$	−47.3612	−1.57347
$907$	11.2637	0.374006	0.187003	−	0.982359i	$-0.440123\pi$
$907$	11.2637	0.374006	0.187003	+	0.982359i	$0.440123\pi$
$908$	−10.1893	−0.338143
$909$	−51.6332	−1.71256
$910$	15.1021	0.500629
$911$	36.0382	1.19400	0.596999	−	0.802242i	$-0.296360\pi$
$911$	36.0382	1.19400	0.596999	+	0.802242i	$0.296360\pi$
$912$	11.1419	0.368945
$913$	−25.0128	−0.827804
$914$	−9.07725	−0.300249
$915$	106.714	3.52786
$916$	−20.2523	−0.669155
$917$	13.1022	0.432674
$918$	−96.7238	−3.19236
$919$	40.5346	1.33711	0.668557	−	0.743661i	$-0.266912\pi$
$919$	40.5346	1.33711	0.668557	+	0.743661i	$0.266912\pi$
$920$	6.12873	0.202058
$921$	30.6308	1.00932
$922$	−30.4710	−1.00351
$923$	−37.1294	−1.22213
$924$	14.5724	0.479396
$925$	−4.45469	−0.146469
$926$	−18.3635	−0.603463
$927$	−16.0872	−0.528373
$928$	7.44452	0.244378
$929$	9.30128	0.305165	0.152583	−	0.988291i	$-0.451241\pi$
$929$	9.30128	0.305165	0.152583	+	0.988291i	$0.451241\pi$
$930$	−47.4278	−1.55522
$931$	21.3331	0.699163
$932$	−4.72594	−0.154803
$933$	15.4261	0.505026
$934$	−5.34832	−0.175002
$935$	81.6452	2.67008
$936$	50.5411	1.65199
$937$	54.8707	1.79255	0.896275	−	0.443500i	$-0.146263\pi$
$937$	54.8707	1.79255	0.896275	+	0.443500i	$0.146263\pi$
$938$	6.22301	0.203188
$939$	−71.0784	−2.31956
$940$	3.01802	0.0984370
$941$	34.1462	1.11314	0.556568	−	0.830802i	$-0.312118\pi$
$941$	34.1462	1.11314	0.556568	+	0.830802i	$0.312118\pi$
$942$	60.5436	1.97262
$943$	9.94568	0.323876
$944$	−8.07140	−0.262702
$945$	−30.5589	−0.994081
$946$	−28.8055	−0.936548
$947$	−29.0118	−0.942756	−0.471378	−	0.881931i	$-0.656243\pi$
$947$	−29.0118	−0.942756	−0.471378	+	0.881931i	$0.656243\pi$
$948$	21.3011	0.691828
$949$	71.2251	2.31207
$950$	2.39448	0.0776872
$951$	−51.5341	−1.67111
$952$	−6.37560	−0.206634
$953$	15.0317	0.486923	0.243462	−	0.969911i	$-0.421717\pi$
$953$	15.0317	0.486923	0.243462	+	0.969911i	$0.421717\pi$
$954$	61.3883	1.98752
$955$	35.3092	1.14258
$956$	25.2003	0.815035
$957$	118.059	3.81632
$958$	−20.7977	−0.671942
$959$	−20.2875	−0.655118
$960$	7.66955	0.247534
$961$	7.24066	0.233570
$962$	−44.4187	−1.43212
$963$	11.7906	0.379947
$964$	18.0644	0.581816
$965$	6.02442	0.193933
$966$	−7.58970	−0.244195
$967$	16.3291	0.525109	0.262554	−	0.964917i	$-0.415435\pi$
$967$	16.3291	0.525109	0.262554	+	0.964917i	$0.415435\pi$
$968$	−13.3316	−0.428493
$969$	−77.3060	−2.48343
$970$	−9.40642	−0.302022
$971$	36.6267	1.17541	0.587703	−	0.809077i	$-0.300032\pi$
$971$	36.6267	1.17541	0.587703	+	0.809077i	$0.300032\pi$
$972$	−31.5129	−1.01078
$973$	10.3906	0.333106
$974$	−33.6207	−1.07728
$975$	15.3034	0.490101
$976$	13.9140	0.445376
$977$	−57.4512	−1.83803	−0.919013	−	0.394227i	$-0.871012\pi$
$977$	−57.4512	−1.83803	−0.919013	+	0.394227i	$0.871012\pi$
$978$	−35.5641	−1.13722
$979$	−55.8873	−1.78616
$980$	14.6847	0.469084
$981$	−84.5004	−2.69789
$982$	−11.5202	−0.367626
$983$	3.98586	0.127129	0.0635646	−	0.997978i	$-0.479753\pi$
$983$	3.98586	0.127129	0.0635646	+	0.997978i	$0.479753\pi$
$984$	12.4461	0.396768
$985$	50.4563	1.60767
$986$	−51.6524	−1.64495
$987$	−3.73746	−0.118965
$988$	23.8759	0.759593
$989$	15.0027	0.477058
$990$	86.3262	2.74363
$991$	11.9118	0.378392	0.189196	−	0.981939i	$-0.439412\pi$
$991$	11.9118	0.378392	0.189196	+	0.981939i	$0.439412\pi$
$992$	−6.18390	−0.196339
$993$	44.9195	1.42548
$994$	4.95229	0.157077
$995$	64.4457	2.04307
$996$	16.3026	0.516567
$997$	−56.7337	−1.79677	−0.898387	−	0.439205i	$-0.855260\pi$
$997$	−56.7337	−1.79677	−0.898387	+	0.439205i	$0.855260\pi$
$998$	−32.7085	−1.03537
$999$	89.8807	2.84370

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8014.2.a.e.1.5	✓	91

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8014.2.a.e.1.5	✓	91	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:	\( N \)	\(=\)	\( 8014 = 2 \cdot 4007 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.21498 q^{3} +1.00000 q^{4} -2.38557 q^{5} +3.21498 q^{6} -0.918897 q^{7} -1.00000 q^{8} +7.33611 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.21498 q^{3} +1.00000 q^{4} -2.38557 q^{5} +3.21498 q^{6} -0.918897 q^{7} -1.00000 q^{8} +7.33611 q^{9} +2.38557 q^{10} +4.93270 q^{11} -3.21498 q^{12} -6.88935 q^{13} +0.918897 q^{14} +7.66955 q^{15} +1.00000 q^{16} -6.93832 q^{17} -7.33611 q^{18} -3.46562 q^{19} -2.38557 q^{20} +2.95424 q^{21} -4.93270 q^{22} +2.56909 q^{23} +3.21498 q^{24} +0.690924 q^{25} +6.88935 q^{26} -13.9405 q^{27} -0.918897 q^{28} -7.44452 q^{29} -7.66955 q^{30} +6.18390 q^{31} -1.00000 q^{32} -15.8586 q^{33} +6.93832 q^{34} +2.19209 q^{35} +7.33611 q^{36} -6.44744 q^{37} +3.46562 q^{38} +22.1492 q^{39} +2.38557 q^{40} +3.87129 q^{41} -2.95424 q^{42} +5.83970 q^{43} +4.93270 q^{44} -17.5008 q^{45} -2.56909 q^{46} -1.26512 q^{47} -3.21498 q^{48} -6.15563 q^{49} -0.690924 q^{50} +22.3066 q^{51} -6.88935 q^{52} -8.36796 q^{53} +13.9405 q^{54} -11.7673 q^{55} +0.918897 q^{56} +11.1419 q^{57} +7.44452 q^{58} -8.07140 q^{59} +7.66955 q^{60} +13.9140 q^{61} -6.18390 q^{62} -6.74113 q^{63} +1.00000 q^{64} +16.4350 q^{65} +15.8586 q^{66} +6.77225 q^{67} -6.93832 q^{68} -8.25957 q^{69} -2.19209 q^{70} +5.38938 q^{71} -7.33611 q^{72} -10.3384 q^{73} +6.44744 q^{74} -2.22131 q^{75} -3.46562 q^{76} -4.53265 q^{77} -22.1492 q^{78} -6.62558 q^{79} -2.38557 q^{80} +22.8102 q^{81} -3.87129 q^{82} -5.07082 q^{83} +2.95424 q^{84} +16.5518 q^{85} -5.83970 q^{86} +23.9340 q^{87} -4.93270 q^{88} -11.3299 q^{89} +17.5008 q^{90} +6.33061 q^{91} +2.56909 q^{92} -19.8811 q^{93} +1.26512 q^{94} +8.26746 q^{95} +3.21498 q^{96} -3.94306 q^{97} +6.15563 q^{98} +36.1869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 91 q - 91 q^{2} - 2 q^{3} + 91 q^{4} + 22 q^{5} + 2 q^{6} - 14 q^{7} - 91 q^{8} + 123 q^{9}+O(q^{10}) \) 91 * q - 91 * q^2 - 2 * q^3 + 91 * q^4 + 22 * q^5 + 2 * q^6 - 14 * q^7 - 91 * q^8 + 123 * q^9	\( 91 q - 91 q^{2} - 2 q^{3} + 91 q^{4} + 22 q^{5} + 2 q^{6} - 14 q^{7} - 91 q^{8} + 123 q^{9} - 22 q^{10} + 59 q^{11} - 2 q^{12} - 15 q^{13} + 14 q^{14} + 23 q^{15} + 91 q^{16} + 25 q^{17} - 123 q^{18} + 2 q^{19} + 22 q^{20} + 20 q^{21} - 59 q^{22} + 36 q^{23} + 2 q^{24} + 121 q^{25} + 15 q^{26} - 8 q^{27} - 14 q^{28} + 71 q^{29} - 23 q^{30} + 13 q^{31} - 91 q^{32} + 24 q^{33} - 25 q^{34} + 27 q^{35} + 123 q^{36} + 5 q^{37} - 2 q^{38} + 48 q^{39} - 22 q^{40} + 77 q^{41} - 20 q^{42} - 36 q^{43} + 59 q^{44} + 62 q^{45} - 36 q^{46} + 41 q^{47} - 2 q^{48} + 137 q^{49} - 121 q^{50} + 13 q^{51} - 15 q^{52} + 33 q^{53} + 8 q^{54} + 12 q^{55} + 14 q^{56} + 52 q^{57} - 71 q^{58} + 76 q^{59} + 23 q^{60} + 18 q^{61} - 13 q^{62} - 18 q^{63} + 91 q^{64} + 84 q^{65} - 24 q^{66} - 59 q^{67} + 25 q^{68} + 64 q^{69} - 27 q^{70} + 124 q^{71} - 123 q^{72} + 43 q^{73} - 5 q^{74} + 9 q^{75} + 2 q^{76} + 50 q^{77} - 48 q^{78} - 20 q^{79} + 22 q^{80} + 227 q^{81} - 77 q^{82} + 29 q^{83} + 20 q^{84} + 3 q^{85} + 36 q^{86} - 25 q^{87} - 59 q^{88} + 148 q^{89} - 62 q^{90} + 27 q^{91} + 36 q^{92} + 65 q^{93} - 41 q^{94} + 54 q^{95} + 2 q^{96} + 38 q^{97} - 137 q^{98} + 157 q^{99}+O(q^{100}) \) 91 * q - 91 * q^2 - 2 * q^3 + 91 * q^4 + 22 * q^5 + 2 * q^6 - 14 * q^7 - 91 * q^8 + 123 * q^9 - 22 * q^10 + 59 * q^11 - 2 * q^12 - 15 * q^13 + 14 * q^14 + 23 * q^15 + 91 * q^16 + 25 * q^17 - 123 * q^18 + 2 * q^19 + 22 * q^20 + 20 * q^21 - 59 * q^22 + 36 * q^23 + 2 * q^24 + 121 * q^25 + 15 * q^26 - 8 * q^27 - 14 * q^28 + 71 * q^29 - 23 * q^30 + 13 * q^31 - 91 * q^32 + 24 * q^33 - 25 * q^34 + 27 * q^35 + 123 * q^36 + 5 * q^37 - 2 * q^38 + 48 * q^39 - 22 * q^40 + 77 * q^41 - 20 * q^42 - 36 * q^43 + 59 * q^44 + 62 * q^45 - 36 * q^46 + 41 * q^47 - 2 * q^48 + 137 * q^49 - 121 * q^50 + 13 * q^51 - 15 * q^52 + 33 * q^53 + 8 * q^54 + 12 * q^55 + 14 * q^56 + 52 * q^57 - 71 * q^58 + 76 * q^59 + 23 * q^60 + 18 * q^61 - 13 * q^62 - 18 * q^63 + 91 * q^64 + 84 * q^65 - 24 * q^66 - 59 * q^67 + 25 * q^68 + 64 * q^69 - 27 * q^70 + 124 * q^71 - 123 * q^72 + 43 * q^73 - 5 * q^74 + 9 * q^75 + 2 * q^76 + 50 * q^77 - 48 * q^78 - 20 * q^79 + 22 * q^80 + 227 * q^81 - 77 * q^82 + 29 * q^83 + 20 * q^84 + 3 * q^85 + 36 * q^86 - 25 * q^87 - 59 * q^88 + 148 * q^89 - 62 * q^90 + 27 * q^91 + 36 * q^92 + 65 * q^93 - 41 * q^94 + 54 * q^95 + 2 * q^96 + 38 * q^97 - 137 * q^98 + 157 * q^99

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