LMFDB - Embedded newform 8038.2.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.14458 q^{3} +1.00000 q^{4} +3.80241 q^{5} +2.14458 q^{6} +4.02154 q^{7} -1.00000 q^{8} +1.59920 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.14458 q^{3} +1.00000 q^{4} +3.80241 q^{5} +2.14458 q^{6} +4.02154 q^{7} -1.00000 q^{8} +1.59920 q^{9} -3.80241 q^{10} -1.67663 q^{11} -2.14458 q^{12} +3.73728 q^{13} -4.02154 q^{14} -8.15455 q^{15} +1.00000 q^{16} -0.566494 q^{17} -1.59920 q^{18} +4.05893 q^{19} +3.80241 q^{20} -8.62449 q^{21} +1.67663 q^{22} +6.09025 q^{23} +2.14458 q^{24} +9.45829 q^{25} -3.73728 q^{26} +3.00411 q^{27} +4.02154 q^{28} -2.43026 q^{29} +8.15455 q^{30} +2.30202 q^{31} -1.00000 q^{32} +3.59567 q^{33} +0.566494 q^{34} +15.2915 q^{35} +1.59920 q^{36} -7.99124 q^{37} -4.05893 q^{38} -8.01488 q^{39} -3.80241 q^{40} -5.04828 q^{41} +8.62449 q^{42} -2.05490 q^{43} -1.67663 q^{44} +6.08082 q^{45} -6.09025 q^{46} -10.9054 q^{47} -2.14458 q^{48} +9.17277 q^{49} -9.45829 q^{50} +1.21489 q^{51} +3.73728 q^{52} +12.3247 q^{53} -3.00411 q^{54} -6.37525 q^{55} -4.02154 q^{56} -8.70468 q^{57} +2.43026 q^{58} +8.46140 q^{59} -8.15455 q^{60} +2.78539 q^{61} -2.30202 q^{62} +6.43126 q^{63} +1.00000 q^{64} +14.2107 q^{65} -3.59567 q^{66} -5.76739 q^{67} -0.566494 q^{68} -13.0610 q^{69} -15.2915 q^{70} +12.9623 q^{71} -1.59920 q^{72} -5.01418 q^{73} +7.99124 q^{74} -20.2840 q^{75} +4.05893 q^{76} -6.74265 q^{77} +8.01488 q^{78} -3.59161 q^{79} +3.80241 q^{80} -11.2402 q^{81} +5.04828 q^{82} -2.30177 q^{83} -8.62449 q^{84} -2.15404 q^{85} +2.05490 q^{86} +5.21187 q^{87} +1.67663 q^{88} +16.9221 q^{89} -6.08082 q^{90} +15.0296 q^{91} +6.09025 q^{92} -4.93686 q^{93} +10.9054 q^{94} +15.4337 q^{95} +2.14458 q^{96} +14.1082 q^{97} -9.17277 q^{98} -2.68128 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * 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
$3$	−2.14458	−1.23817	−0.619086	−	0.785324i	$-0.712497\pi$
$3$	−2.14458	−1.23817	−0.619086	+	0.785324i	$0.712497\pi$
$4$	1.00000	0.500000
$5$	3.80241	1.70049	0.850244	−	0.526389i	$-0.176454\pi$
$5$	3.80241	1.70049	0.850244	+	0.526389i	$0.176454\pi$
$6$	2.14458	0.875519
$7$	4.02154	1.52000	0.759999	−	0.649924i	$-0.225199\pi$
$7$	4.02154	1.52000	0.759999	+	0.649924i	$0.225199\pi$
$8$	−1.00000	−0.353553
$9$	1.59920	0.533068
$10$	−3.80241	−1.20243
$11$	−1.67663	−0.505524	−0.252762	−	0.967528i	$-0.581339\pi$
$11$	−1.67663	−0.505524	−0.252762	+	0.967528i	$0.581339\pi$
$12$	−2.14458	−0.619086
$13$	3.73728	1.03653	0.518267	−	0.855219i	$-0.326577\pi$
$13$	3.73728	1.03653	0.518267	+	0.855219i	$0.326577\pi$
$14$	−4.02154	−1.07480
$15$	−8.15455	−2.10549
$16$	1.00000	0.250000
$17$	−0.566494	−0.137395	−0.0686974	−	0.997638i	$-0.521884\pi$
$17$	−0.566494	−0.137395	−0.0686974	+	0.997638i	$0.521884\pi$
$18$	−1.59920	−0.376936
$19$	4.05893	0.931182	0.465591	−	0.885000i	$-0.345842\pi$
$19$	4.05893	0.931182	0.465591	+	0.885000i	$0.345842\pi$
$20$	3.80241	0.850244
$21$	−8.62449	−1.88202
$22$	1.67663	0.357460
$23$	6.09025	1.26991	0.634953	−	0.772551i	$-0.281020\pi$
$23$	6.09025	1.26991	0.634953	+	0.772551i	$0.281020\pi$
$24$	2.14458	0.437760
$25$	9.45829	1.89166
$26$	−3.73728	−0.732941
$27$	3.00411	0.578142
$28$	4.02154	0.759999
$29$	−2.43026	−0.451287	−0.225644	−	0.974210i	$-0.572448\pi$
$29$	−2.43026	−0.451287	−0.225644	+	0.974210i	$0.572448\pi$
$30$	8.15455	1.48881
$31$	2.30202	0.413455	0.206728	−	0.978399i	$-0.433719\pi$
$31$	2.30202	0.413455	0.206728	+	0.978399i	$0.433719\pi$
$32$	−1.00000	−0.176777
$33$	3.59567	0.625926
$34$	0.566494	0.0971529
$35$	15.2915	2.58474
$36$	1.59920	0.266534
$37$	−7.99124	−1.31375	−0.656876	−	0.753998i	$-0.728123\pi$
$37$	−7.99124	−1.31375	−0.656876	+	0.753998i	$0.728123\pi$
$38$	−4.05893	−0.658445
$39$	−8.01488	−1.28341
$40$	−3.80241	−0.601213
$41$	−5.04828	−0.788409	−0.394204	−	0.919023i	$-0.628980\pi$
$41$	−5.04828	−0.788409	−0.394204	+	0.919023i	$0.628980\pi$
$42$	8.62449	1.33079
$43$	−2.05490	−0.313369	−0.156684	−	0.987649i	$-0.550081\pi$
$43$	−2.05490	−0.313369	−0.156684	+	0.987649i	$0.550081\pi$
$44$	−1.67663	−0.252762
$45$	6.08082	0.906475
$46$	−6.09025	−0.897959
$47$	−10.9054	−1.59071	−0.795357	−	0.606141i	$-0.792717\pi$
$47$	−10.9054	−1.59071	−0.795357	+	0.606141i	$0.792717\pi$
$48$	−2.14458	−0.309543
$49$	9.17277	1.31040
$50$	−9.45829	−1.33760
$51$	1.21489	0.170118
$52$	3.73728	0.518267
$53$	12.3247	1.69293	0.846466	−	0.532442i	$-0.178726\pi$
$53$	12.3247	1.69293	0.846466	+	0.532442i	$0.178726\pi$
$54$	−3.00411	−0.408808
$55$	−6.37525	−0.859638
$56$	−4.02154	−0.537401
$57$	−8.70468	−1.15296
$58$	2.43026	0.319108
$59$	8.46140	1.10158	0.550790	−	0.834644i	$-0.314327\pi$
$59$	8.46140	1.10158	0.550790	+	0.834644i	$0.314327\pi$
$60$	−8.15455	−1.05275
$61$	2.78539	0.356633	0.178317	−	0.983973i	$-0.442935\pi$
$61$	2.78539	0.356633	0.178317	+	0.983973i	$0.442935\pi$
$62$	−2.30202	−0.292357
$63$	6.43126	0.810262
$64$	1.00000	0.125000
$65$	14.2107	1.76261
$66$	−3.59567	−0.442596
$67$	−5.76739	−0.704599	−0.352299	−	0.935887i	$-0.614600\pi$
$67$	−5.76739	−0.704599	−0.352299	+	0.935887i	$0.614600\pi$
$68$	−0.566494	−0.0686974
$69$	−13.0610	−1.57236
$70$	−15.2915	−1.82769
$71$	12.9623	1.53834	0.769169	−	0.639045i	$-0.220670\pi$
$71$	12.9623	1.53834	0.769169	+	0.639045i	$0.220670\pi$
$72$	−1.59920	−0.188468
$73$	−5.01418	−0.586866	−0.293433	−	0.955980i	$-0.594798\pi$
$73$	−5.01418	−0.586866	−0.293433	+	0.955980i	$0.594798\pi$
$74$	7.99124	0.928963
$75$	−20.2840	−2.34220
$76$	4.05893	0.465591
$77$	−6.74265	−0.768396
$78$	8.01488	0.907506
$79$	−3.59161	−0.404087	−0.202044	−	0.979377i	$-0.564758\pi$
$79$	−3.59161	−0.404087	−0.202044	+	0.979377i	$0.564758\pi$
$80$	3.80241	0.425122
$81$	−11.2402	−1.24891
$82$	5.04828	0.557489
$83$	−2.30177	−0.252652	−0.126326	−	0.991989i	$-0.540319\pi$
$83$	−2.30177	−0.252652	−0.126326	+	0.991989i	$0.540319\pi$
$84$	−8.62449	−0.941009
$85$	−2.15404	−0.233638
$86$	2.05490	0.221585
$87$	5.21187	0.558771
$88$	1.67663	0.178730
$89$	16.9221	1.79374	0.896870	−	0.442293i	$-0.145835\pi$
$89$	16.9221	1.79374	0.896870	+	0.442293i	$0.145835\pi$
$90$	−6.08082	−0.640975
$91$	15.0296	1.57553
$92$	6.09025	0.634953
$93$	−4.93686	−0.511929
$94$	10.9054	1.12481
$95$	15.4337	1.58346
$96$	2.14458	0.218880
$97$	14.1082	1.43247	0.716234	−	0.697860i	$-0.245864\pi$
$97$	14.1082	1.43247	0.716234	+	0.697860i	$0.245864\pi$
$98$	−9.17277	−0.926590
$99$	−2.68128	−0.269479
$100$	9.45829	0.945829
$101$	10.9113	1.08571	0.542856	−	0.839826i	$-0.317343\pi$
$101$	10.9113	1.08571	0.542856	+	0.839826i	$0.317343\pi$
$102$	−1.21489	−0.120292
$103$	14.2695	1.40602	0.703010	−	0.711180i	$-0.251839\pi$
$103$	14.2695	1.40602	0.703010	+	0.711180i	$0.251839\pi$
$104$	−3.73728	−0.366470
$105$	−32.7938	−3.20035
$106$	−12.3247	−1.19708
$107$	−7.12872	−0.689160	−0.344580	−	0.938757i	$-0.611979\pi$
$107$	−7.12872	−0.689160	−0.344580	+	0.938757i	$0.611979\pi$
$108$	3.00411	0.289071
$109$	−13.4218	−1.28558	−0.642790	−	0.766042i	$-0.722223\pi$
$109$	−13.4218	−1.28558	−0.642790	+	0.766042i	$0.722223\pi$
$110$	6.37525	0.607856
$111$	17.1378	1.62665
$112$	4.02154	0.380000
$113$	16.5582	1.55766	0.778832	−	0.627233i	$-0.215813\pi$
$113$	16.5582	1.55766	0.778832	+	0.627233i	$0.215813\pi$
$114$	8.70468	0.815268
$115$	23.1576	2.15946
$116$	−2.43026	−0.225644
$117$	5.97667	0.552543
$118$	−8.46140	−0.778935
$119$	−2.27818	−0.208840
$120$	8.15455	0.744405
$121$	−8.18890	−0.744445
$122$	−2.78539	−0.252178
$123$	10.8264	0.976185
$124$	2.30202	0.206728
$125$	16.9522	1.51625
$126$	−6.43126	−0.572942
$127$	−21.4887	−1.90682	−0.953408	−	0.301684i	$-0.902451\pi$
$127$	−21.4887	−1.90682	−0.953408	+	0.301684i	$0.902451\pi$
$128$	−1.00000	−0.0883883
$129$	4.40688	0.388004
$130$	−14.2107	−1.24636
$131$	−18.2869	−1.59773	−0.798865	−	0.601510i	$-0.794566\pi$
$131$	−18.2869	−1.59773	−0.798865	+	0.601510i	$0.794566\pi$
$132$	3.59567	0.312963
$133$	16.3231	1.41540
$134$	5.76739	0.498226
$135$	11.4229	0.983123
$136$	0.566494	0.0485764
$137$	16.5257	1.41189	0.705943	−	0.708268i	$-0.250523\pi$
$137$	16.5257	1.41189	0.705943	+	0.708268i	$0.250523\pi$
$138$	13.0610	1.11183
$139$	−3.51182	−0.297869	−0.148934	−	0.988847i	$-0.547584\pi$
$139$	−3.51182	−0.297869	−0.148934	+	0.988847i	$0.547584\pi$
$140$	15.2915	1.29237
$141$	23.3874	1.96958
$142$	−12.9623	−1.08777
$143$	−6.26605	−0.523993
$144$	1.59920	0.133267
$145$	−9.24082	−0.767408
$146$	5.01418	0.414977
$147$	−19.6717	−1.62249
$148$	−7.99124	−0.656876
$149$	19.4543	1.59376	0.796881	−	0.604137i	$-0.206482\pi$
$149$	19.4543	1.59376	0.796881	+	0.604137i	$0.206482\pi$
$150$	20.2840	1.65618
$151$	−10.8509	−0.883031	−0.441516	−	0.897254i	$-0.645559\pi$
$151$	−10.8509	−0.883031	−0.441516	+	0.897254i	$0.645559\pi$
$152$	−4.05893	−0.329223
$153$	−0.905939	−0.0732408
$154$	6.74265	0.543338
$155$	8.75322	0.703076
$156$	−8.01488	−0.641704
$157$	−14.0690	−1.12283	−0.561415	−	0.827535i	$-0.689743\pi$
$157$	−14.0690	−1.12283	−0.561415	+	0.827535i	$0.689743\pi$
$158$	3.59161	0.285733
$159$	−26.4313	−2.09614
$160$	−3.80241	−0.300607
$161$	24.4922	1.93025
$162$	11.2402	0.883110
$163$	2.34169	0.183415	0.0917077	−	0.995786i	$-0.470767\pi$
$163$	2.34169	0.183415	0.0917077	+	0.995786i	$0.470767\pi$
$164$	−5.04828	−0.394204
$165$	13.6722	1.06438
$166$	2.30177	0.178652
$167$	−1.77000	−0.136967	−0.0684834	−	0.997652i	$-0.521816\pi$
$167$	−1.77000	−0.136967	−0.0684834	+	0.997652i	$0.521816\pi$
$168$	8.62449	0.665394
$169$	0.967255	0.0744042
$170$	2.15404	0.165207
$171$	6.49106	0.496383
$172$	−2.05490	−0.156684
$173$	19.3228	1.46908	0.734541	−	0.678564i	$-0.237397\pi$
$173$	19.3228	1.46908	0.734541	+	0.678564i	$0.237397\pi$
$174$	−5.21187	−0.395111
$175$	38.0369	2.87532
$176$	−1.67663	−0.126381
$177$	−18.1461	−1.36394
$178$	−16.9221	−1.26837
$179$	20.0028	1.49508	0.747539	−	0.664218i	$-0.231236\pi$
$179$	20.0028	1.49508	0.747539	+	0.664218i	$0.231236\pi$
$180$	6.08082	0.453238
$181$	13.3438	0.991839	0.495919	−	0.868369i	$-0.334831\pi$
$181$	13.3438	0.991839	0.495919	+	0.868369i	$0.334831\pi$
$182$	−15.0296	−1.11407
$183$	−5.97349	−0.441573
$184$	−6.09025	−0.448979
$185$	−30.3860	−2.23402
$186$	4.93686	0.361988
$187$	0.949803	0.0694565
$188$	−10.9054	−0.795357
$189$	12.0812	0.878775
$190$	−15.4337	−1.11968
$191$	−20.1879	−1.46075	−0.730373	−	0.683048i	$-0.760654\pi$
$191$	−20.1879	−1.46075	−0.730373	+	0.683048i	$0.760654\pi$
$192$	−2.14458	−0.154771
$193$	−3.86640	−0.278310	−0.139155	−	0.990271i	$-0.544439\pi$
$193$	−3.86640	−0.278310	−0.139155	+	0.990271i	$0.544439\pi$
$194$	−14.1082	−1.01291
$195$	−30.4758	−2.18242
$196$	9.17277	0.655198
$197$	11.9027	0.848030	0.424015	−	0.905655i	$-0.360620\pi$
$197$	11.9027	0.848030	0.424015	+	0.905655i	$0.360620\pi$
$198$	2.68128	0.190550
$199$	−7.76370	−0.550354	−0.275177	−	0.961394i	$-0.588737\pi$
$199$	−7.76370	−0.550354	−0.275177	+	0.961394i	$0.588737\pi$
$200$	−9.45829	−0.668802
$201$	12.3686	0.872414
$202$	−10.9113	−0.767715
$203$	−9.77337	−0.685956
$204$	1.21489	0.0850592
$205$	−19.1956	−1.34068
$206$	−14.2695	−0.994207
$207$	9.73955	0.676946
$208$	3.73728	0.259134
$209$	−6.80534	−0.470735
$210$	32.7938	2.26299
$211$	21.9274	1.50955	0.754773	−	0.655986i	$-0.227747\pi$
$211$	21.9274	1.50955	0.754773	+	0.655986i	$0.227747\pi$
$212$	12.3247	0.846466
$213$	−27.7986	−1.90473
$214$	7.12872	0.487310
$215$	−7.81355	−0.532880
$216$	−3.00411	−0.204404
$217$	9.25767	0.628452
$218$	13.4218	0.909042
$219$	10.7533	0.726640
$220$	−6.37525	−0.429819
$221$	−2.11714	−0.142415
$222$	−17.1378	−1.15022
$223$	−20.5524	−1.37629	−0.688144	−	0.725574i	$-0.741574\pi$
$223$	−20.5524	−1.37629	−0.688144	+	0.725574i	$0.741574\pi$
$224$	−4.02154	−0.268700
$225$	15.1257	1.00838
$226$	−16.5582	−1.10143
$227$	−20.2884	−1.34659	−0.673293	−	0.739376i	$-0.735121\pi$
$227$	−20.2884	−1.34659	−0.673293	+	0.739376i	$0.735121\pi$
$228$	−8.70468	−0.576482
$229$	14.3829	0.950451	0.475226	−	0.879864i	$-0.342366\pi$
$229$	14.3829	0.950451	0.475226	+	0.879864i	$0.342366\pi$
$230$	−23.1576	−1.52697
$231$	14.4601	0.951406
$232$	2.43026	0.159554
$233$	−7.25106	−0.475033	−0.237516	−	0.971384i	$-0.576333\pi$
$233$	−7.25106	−0.475033	−0.237516	+	0.971384i	$0.576333\pi$
$234$	−5.97667	−0.390707
$235$	−41.4667	−2.70499
$236$	8.46140	0.550790
$237$	7.70247	0.500329
$238$	2.27818	0.147672
$239$	19.6198	1.26910	0.634549	−	0.772882i	$-0.281186\pi$
$239$	19.6198	1.26910	0.634549	+	0.772882i	$0.281186\pi$
$240$	−8.15455	−0.526374
$241$	−20.7252	−1.33503	−0.667514	−	0.744597i	$-0.732642\pi$
$241$	−20.7252	−1.33503	−0.667514	+	0.744597i	$0.732642\pi$
$242$	8.18890	0.526402
$243$	15.0930	0.968218
$244$	2.78539	0.178317
$245$	34.8786	2.22831
$246$	−10.8264	−0.690267
$247$	15.1694	0.965203
$248$	−2.30202	−0.146179
$249$	4.93632	0.312827
$250$	−16.9522	−1.07215
$251$	1.75114	0.110531	0.0552655	−	0.998472i	$-0.482399\pi$
$251$	1.75114	0.110531	0.0552655	+	0.998472i	$0.482399\pi$
$252$	6.43126	0.405131
$253$	−10.2111	−0.641968
$254$	21.4887	1.34832
$255$	4.61950	0.289284
$256$	1.00000	0.0625000
$257$	−16.3632	−1.02071	−0.510354	−	0.859964i	$-0.670486\pi$
$257$	−16.3632	−1.02071	−0.510354	+	0.859964i	$0.670486\pi$
$258$	−4.40688	−0.274360
$259$	−32.1371	−1.99690
$260$	14.2107	0.881307
$261$	−3.88647	−0.240567
$262$	18.2869	1.12977
$263$	−12.1968	−0.752084	−0.376042	−	0.926603i	$-0.622715\pi$
$263$	−12.1968	−0.752084	−0.376042	+	0.926603i	$0.622715\pi$
$264$	−3.59567	−0.221298
$265$	46.8637	2.87881
$266$	−16.3231	−1.00084
$267$	−36.2908	−2.22096
$268$	−5.76739	−0.352299
$269$	13.2941	0.810554	0.405277	−	0.914194i	$-0.367175\pi$
$269$	13.2941	0.810554	0.405277	+	0.914194i	$0.367175\pi$
$270$	−11.4229	−0.695173
$271$	−3.48280	−0.211565	−0.105783	−	0.994389i	$-0.533735\pi$
$271$	−3.48280	−0.211565	−0.105783	+	0.994389i	$0.533735\pi$
$272$	−0.566494	−0.0343487
$273$	−32.2321	−1.95078
$274$	−16.5257	−0.998354
$275$	−15.8581	−0.956279
$276$	−13.0610	−0.786180
$277$	−18.8098	−1.13017	−0.565086	−	0.825032i	$-0.691157\pi$
$277$	−18.8098	−1.13017	−0.565086	+	0.825032i	$0.691157\pi$
$278$	3.51182	0.210625
$279$	3.68140	0.220400
$280$	−15.2915	−0.913843
$281$	12.0031	0.716045	0.358022	−	0.933713i	$-0.383451\pi$
$281$	12.0031	0.716045	0.358022	+	0.933713i	$0.383451\pi$
$282$	−23.3874	−1.39270
$283$	−28.8762	−1.71651	−0.858257	−	0.513220i	$-0.828452\pi$
$283$	−28.8762	−1.71651	−0.858257	+	0.513220i	$0.828452\pi$
$284$	12.9623	0.769169
$285$	−33.0987	−1.96060
$286$	6.26605	0.370519
$287$	−20.3018	−1.19838
$288$	−1.59920	−0.0942340
$289$	−16.6791	−0.981123
$290$	9.24082	0.542640
$291$	−30.2560	−1.77364
$292$	−5.01418	−0.293433
$293$	−5.74415	−0.335577	−0.167788	−	0.985823i	$-0.553663\pi$
$293$	−5.74415	−0.335577	−0.167788	+	0.985823i	$0.553663\pi$
$294$	19.6717	1.14728
$295$	32.1737	1.87322
$296$	7.99124	0.464482
$297$	−5.03680	−0.292265
$298$	−19.4543	−1.12696
$299$	22.7610	1.31630
$300$	−20.2840	−1.17110
$301$	−8.26384	−0.476320
$302$	10.8509	0.624397
$303$	−23.4001	−1.34430
$304$	4.05893	0.232796
$305$	10.5912	0.606450
$306$	0.905939	0.0517891
$307$	−13.6003	−0.776212	−0.388106	−	0.921615i	$-0.626871\pi$
$307$	−13.6003	−0.776212	−0.388106	+	0.921615i	$0.626871\pi$
$308$	−6.74265	−0.384198
$309$	−30.6021	−1.74089
$310$	−8.75322	−0.497150
$311$	−5.56502	−0.315564	−0.157782	−	0.987474i	$-0.550434\pi$
$311$	−5.56502	−0.315564	−0.157782	+	0.987474i	$0.550434\pi$
$312$	8.01488	0.453753
$313$	2.95338	0.166935	0.0834673	−	0.996511i	$-0.473401\pi$
$313$	2.95338	0.166935	0.0834673	+	0.996511i	$0.473401\pi$
$314$	14.0690	0.793960
$315$	24.4543	1.37784
$316$	−3.59161	−0.202044
$317$	−18.9218	−1.06276	−0.531378	−	0.847135i	$-0.678325\pi$
$317$	−18.9218	−1.06276	−0.531378	+	0.847135i	$0.678325\pi$
$318$	26.4313	1.48220
$319$	4.07465	0.228137
$320$	3.80241	0.212561
$321$	15.2881	0.853298
$322$	−24.4922	−1.36490
$323$	−2.29936	−0.127940
$324$	−11.2402	−0.624453
$325$	35.3483	1.96077
$326$	−2.34169	−0.129694
$327$	28.7842	1.59177
$328$	5.04828	0.278745
$329$	−43.8564	−2.41788
$330$	−13.6722	−0.752630
$331$	−35.9527	−1.97614	−0.988070	−	0.154009i	$-0.950782\pi$
$331$	−35.9527	−1.97614	−0.988070	+	0.154009i	$0.950782\pi$
$332$	−2.30177	−0.126326
$333$	−12.7796	−0.700319
$334$	1.77000	0.0968502
$335$	−21.9300	−1.19816
$336$	−8.62449	−0.470505
$337$	−7.14143	−0.389018	−0.194509	−	0.980901i	$-0.562311\pi$
$337$	−7.14143	−0.389018	−0.194509	+	0.980901i	$0.562311\pi$
$338$	−0.967255	−0.0526117
$339$	−35.5103	−1.92865
$340$	−2.15404	−0.116819
$341$	−3.85965	−0.209012
$342$	−6.49106	−0.350996
$343$	8.73788	0.471801
$344$	2.05490	0.110793
$345$	−49.6632	−2.67378
$346$	−19.3228	−1.03880
$347$	0.969999	0.0520723	0.0260361	−	0.999661i	$-0.491712\pi$
$347$	0.969999	0.0520723	0.0260361	+	0.999661i	$0.491712\pi$
$348$	5.21187	0.279385
$349$	12.6351	0.676342	0.338171	−	0.941085i	$-0.390192\pi$
$349$	12.6351	0.676342	0.338171	+	0.941085i	$0.390192\pi$
$350$	−38.0369	−2.03316
$351$	11.2272	0.599264
$352$	1.67663	0.0893649
$353$	21.3924	1.13860	0.569302	−	0.822129i	$-0.307214\pi$
$353$	21.3924	1.13860	0.569302	+	0.822129i	$0.307214\pi$
$354$	18.1461	0.964454
$355$	49.2878	2.61593
$356$	16.9221	0.896870
$357$	4.88572	0.258580
$358$	−20.0028	−1.05718
$359$	−13.6078	−0.718195	−0.359097	−	0.933300i	$-0.616915\pi$
$359$	−13.6078	−0.718195	−0.359097	+	0.933300i	$0.616915\pi$
$360$	−6.08082	−0.320487
$361$	−2.52509	−0.132899
$362$	−13.3438	−0.701336
$363$	17.5617	0.921751
$364$	15.0296	0.787766
$365$	−19.0660	−0.997958
$366$	5.97349	0.312239
$367$	17.2977	0.902935	0.451467	−	0.892288i	$-0.350901\pi$
$367$	17.2977	0.902935	0.451467	+	0.892288i	$0.350901\pi$
$368$	6.09025	0.317476
$369$	−8.07323	−0.420275
$370$	30.3860	1.57969
$371$	49.5644	2.57326
$372$	−4.93686	−0.255964
$373$	35.4005	1.83297	0.916483	−	0.400073i	$-0.131015\pi$
$373$	35.4005	1.83297	0.916483	+	0.400073i	$0.131015\pi$
$374$	−0.949803	−0.0491131
$375$	−36.3554	−1.87738
$376$	10.9054	0.562403
$377$	−9.08255	−0.467775
$378$	−12.0812	−0.621388
$379$	1.49134	0.0766049	0.0383025	−	0.999266i	$-0.487805\pi$
$379$	1.49134	0.0766049	0.0383025	+	0.999266i	$0.487805\pi$
$380$	15.4337	0.791732
$381$	46.0842	2.36096
$382$	20.1879	1.03290
$383$	29.4597	1.50532	0.752660	−	0.658409i	$-0.228770\pi$
$383$	29.4597	1.50532	0.752660	+	0.658409i	$0.228770\pi$
$384$	2.14458	0.109440
$385$	−25.6383	−1.30665
$386$	3.86640	0.196795
$387$	−3.28620	−0.167047
$388$	14.1082	0.716234
$389$	−3.76958	−0.191125	−0.0955626	−	0.995423i	$-0.530465\pi$
$389$	−3.76958	−0.191125	−0.0955626	+	0.995423i	$0.530465\pi$
$390$	30.4758	1.54320
$391$	−3.45009	−0.174478
$392$	−9.17277	−0.463295
$393$	39.2175	1.97826
$394$	−11.9027	−0.599648
$395$	−13.6567	−0.687146
$396$	−2.68128	−0.134739
$397$	13.6134	0.683237	0.341619	−	0.939839i	$-0.389025\pi$
$397$	13.6134	0.683237	0.341619	+	0.939839i	$0.389025\pi$
$398$	7.76370	0.389159
$399$	−35.0062	−1.75250
$400$	9.45829	0.472915
$401$	14.6691	0.732540	0.366270	−	0.930509i	$-0.380635\pi$
$401$	14.6691	0.732540	0.366270	+	0.930509i	$0.380635\pi$
$402$	−12.3686	−0.616890
$403$	8.60330	0.428561
$404$	10.9113	0.542856
$405$	−42.7396	−2.12375
$406$	9.77337	0.485044
$407$	13.3984	0.664134
$408$	−1.21489	−0.0601459
$409$	−30.3036	−1.49841	−0.749207	−	0.662336i	$-0.769565\pi$
$409$	−30.3036	−1.49841	−0.749207	+	0.662336i	$0.769565\pi$
$410$	19.1956	0.948004
$411$	−35.4406	−1.74816
$412$	14.2695	0.703010
$413$	34.0278	1.67440
$414$	−9.73955	−0.478673
$415$	−8.75226	−0.429632
$416$	−3.73728	−0.183235
$417$	7.53137	0.368813
$418$	6.80534	0.332860
$419$	−21.6014	−1.05530	−0.527649	−	0.849462i	$-0.676926\pi$
$419$	−21.6014	−1.05530	−0.527649	+	0.849462i	$0.676926\pi$
$420$	−32.7938	−1.60017
$421$	10.5804	0.515655	0.257827	−	0.966191i	$-0.416993\pi$
$421$	10.5804	0.515655	0.257827	+	0.966191i	$0.416993\pi$
$422$	−21.9274	−1.06741
$423$	−17.4399	−0.847959
$424$	−12.3247	−0.598542
$425$	−5.35806	−0.259904
$426$	27.7986	1.34684
$427$	11.2016	0.542082
$428$	−7.12872	−0.344580
$429$	13.4380	0.648794
$430$	7.81355	0.376803
$431$	5.54248	0.266972	0.133486	−	0.991051i	$-0.457383\pi$
$431$	5.54248	0.266972	0.133486	+	0.991051i	$0.457383\pi$
$432$	3.00411	0.144535
$433$	11.9062	0.572176	0.286088	−	0.958203i	$-0.407645\pi$
$433$	11.9062	0.572176	0.286088	+	0.958203i	$0.407645\pi$
$434$	−9.25767	−0.444382
$435$	19.8176	0.950183
$436$	−13.4218	−0.642790
$437$	24.7199	1.18251
$438$	−10.7533	−0.513812
$439$	−25.4427	−1.21432	−0.607158	−	0.794581i	$-0.707690\pi$
$439$	−25.4427	−1.21432	−0.607158	+	0.794581i	$0.707690\pi$
$440$	6.37525	0.303928
$441$	14.6691	0.698530
$442$	2.11714	0.100702
$443$	−10.7967	−0.512966	−0.256483	−	0.966549i	$-0.582564\pi$
$443$	−10.7967	−0.512966	−0.256483	+	0.966549i	$0.582564\pi$
$444$	17.1378	0.813325
$445$	64.3448	3.05023
$446$	20.5524	0.973183
$447$	−41.7213	−1.97335
$448$	4.02154	0.190000
$449$	19.8857	0.938463	0.469231	−	0.883075i	$-0.344531\pi$
$449$	19.8857	0.938463	0.469231	+	0.883075i	$0.344531\pi$
$450$	−15.1257	−0.713034
$451$	8.46412	0.398560
$452$	16.5582	0.778832
$453$	23.2705	1.09334
$454$	20.2884	0.952180
$455$	57.1487	2.67917
$456$	8.70468	0.407634
$457$	21.8398	1.02162	0.510811	−	0.859693i	$-0.329345\pi$
$457$	21.8398	1.02162	0.510811	+	0.859693i	$0.329345\pi$
$458$	−14.3829	−0.672071
$459$	−1.70181	−0.0794337
$460$	23.1576	1.07973
$461$	−2.02737	−0.0944238	−0.0472119	−	0.998885i	$-0.515034\pi$
$461$	−2.02737	−0.0944238	−0.0472119	+	0.998885i	$0.515034\pi$
$462$	−14.4601	−0.672746
$463$	30.8671	1.43451	0.717257	−	0.696809i	$-0.245397\pi$
$463$	30.8671	1.43451	0.717257	+	0.696809i	$0.245397\pi$
$464$	−2.43026	−0.112822
$465$	−18.7719	−0.870528
$466$	7.25106	0.335899
$467$	14.0150	0.648537	0.324269	−	0.945965i	$-0.394882\pi$
$467$	14.0150	0.648537	0.324269	+	0.945965i	$0.394882\pi$
$468$	5.97667	0.276272
$469$	−23.1938	−1.07099
$470$	41.4667	1.91272
$471$	30.1721	1.39026
$472$	−8.46140	−0.389467
$473$	3.44531	0.158415
$474$	−7.70247	−0.353786
$475$	38.3905	1.76148
$476$	−2.27818	−0.104420
$477$	19.7098	0.902448
$478$	−19.6198	−0.897388
$479$	6.66110	0.304354	0.152177	−	0.988353i	$-0.451372\pi$
$479$	6.66110	0.304354	0.152177	+	0.988353i	$0.451372\pi$
$480$	8.15455	0.372202
$481$	−29.8655	−1.36175
$482$	20.7252	0.944007
$483$	−52.5253	−2.38999
$484$	−8.18890	−0.372223
$485$	53.6450	2.43589
$486$	−15.0930	−0.684634
$487$	35.9082	1.62716	0.813578	−	0.581456i	$-0.197517\pi$
$487$	35.9082	1.62716	0.813578	+	0.581456i	$0.197517\pi$
$488$	−2.78539	−0.126089
$489$	−5.02193	−0.227100
$490$	−34.8786	−1.57565
$491$	−11.0267	−0.497629	−0.248814	−	0.968551i	$-0.580041\pi$
$491$	−11.0267	−0.497629	−0.248814	+	0.968551i	$0.580041\pi$
$492$	10.8264	0.488093
$493$	1.37672	0.0620046
$494$	−15.1694	−0.682501
$495$	−10.1953	−0.458245
$496$	2.30202	0.103364
$497$	52.1283	2.33827
$498$	−4.93632	−0.221202
$499$	−25.5532	−1.14392	−0.571959	−	0.820282i	$-0.693816\pi$
$499$	−25.5532	−1.14392	−0.571959	+	0.820282i	$0.693816\pi$
$500$	16.9522	0.758127
$501$	3.79590	0.169588
$502$	−1.75114	−0.0781572
$503$	−37.6194	−1.67737	−0.838684	−	0.544619i	$-0.816674\pi$
$503$	−37.6194	−1.67737	−0.838684	+	0.544619i	$0.816674\pi$
$504$	−6.43126	−0.286471
$505$	41.4891	1.84624
$506$	10.2111	0.453940
$507$	−2.07435	−0.0921252
$508$	−21.4887	−0.953408
$509$	−17.1762	−0.761321	−0.380661	−	0.924715i	$-0.624303\pi$
$509$	−17.1762	−0.761321	−0.380661	+	0.924715i	$0.624303\pi$
$510$	−4.61950	−0.204555
$511$	−20.1647	−0.892035
$512$	−1.00000	−0.0441942
$513$	12.1935	0.538355
$514$	16.3632	0.721749
$515$	54.2586	2.39092
$516$	4.40688	0.194002
$517$	18.2844	0.804145
$518$	32.1371	1.41202
$519$	−41.4391	−1.81898
$520$	−14.2107	−0.623178
$521$	−0.162392	−0.00711452	−0.00355726	−	0.999994i	$-0.501132\pi$
$521$	−0.162392	−0.00711452	−0.00355726	+	0.999994i	$0.501132\pi$
$522$	3.88647	0.170106
$523$	7.68173	0.335898	0.167949	−	0.985796i	$-0.446286\pi$
$523$	7.68173	0.335898	0.167949	+	0.985796i	$0.446286\pi$
$524$	−18.2869	−0.798865
$525$	−81.5730	−3.56014
$526$	12.1968	0.531804
$527$	−1.30408	−0.0568067
$528$	3.59567	0.156481
$529$	14.0912	0.612659
$530$	−46.8637	−2.03563
$531$	13.5315	0.587217
$532$	16.3231	0.707698
$533$	−18.8668	−0.817213
$534$	36.2908	1.57045
$535$	−27.1063	−1.17191
$536$	5.76739	0.249113
$537$	−42.8974	−1.85116
$538$	−13.2941	−0.573148
$539$	−15.3794	−0.662437
$540$	11.4229	0.491562
$541$	4.96478	0.213453	0.106726	−	0.994288i	$-0.465963\pi$
$541$	4.96478	0.213453	0.106726	+	0.994288i	$0.465963\pi$
$542$	3.48280	0.149599
$543$	−28.6168	−1.22807
$544$	0.566494	0.0242882
$545$	−51.0353	−2.18611
$546$	32.2321	1.37941
$547$	−3.64414	−0.155812	−0.0779060	−	0.996961i	$-0.524823\pi$
$547$	−3.64414	−0.155812	−0.0779060	+	0.996961i	$0.524823\pi$
$548$	16.5257	0.705943
$549$	4.45441	0.190110
$550$	15.8581	0.676192
$551$	−9.86424	−0.420231
$552$	13.0610	0.555913
$553$	−14.4438	−0.614212
$554$	18.8098	0.799152
$555$	65.1650	2.76610
$556$	−3.51182	−0.148934
$557$	−36.1785	−1.53293	−0.766466	−	0.642285i	$-0.777987\pi$
$557$	−36.1785	−1.53293	−0.766466	+	0.642285i	$0.777987\pi$
$558$	−3.68140	−0.155846
$559$	−7.67972	−0.324817
$560$	15.2915	0.646185
$561$	−2.03692	−0.0859990
$562$	−12.0031	−0.506320
$563$	−23.5296	−0.991652	−0.495826	−	0.868422i	$-0.665135\pi$
$563$	−23.5296	−0.991652	−0.495826	+	0.868422i	$0.665135\pi$
$564$	23.3874	0.984789
$565$	62.9609	2.64879
$566$	28.8762	1.21376
$567$	−45.2027	−1.89834
$568$	−12.9623	−0.543885
$569$	−8.21593	−0.344430	−0.172215	−	0.985059i	$-0.555092\pi$
$569$	−8.21593	−0.344430	−0.172215	+	0.985059i	$0.555092\pi$
$570$	33.0987	1.38635
$571$	−27.2651	−1.14101	−0.570505	−	0.821294i	$-0.693252\pi$
$571$	−27.2651	−1.14101	−0.570505	+	0.821294i	$0.693252\pi$
$572$	−6.26605	−0.261997
$573$	43.2945	1.80865
$574$	20.3018	0.847383
$575$	57.6034	2.40223
$576$	1.59920	0.0666335
$577$	24.9121	1.03711	0.518553	−	0.855046i	$-0.326471\pi$
$577$	24.9121	1.03711	0.518553	+	0.855046i	$0.326471\pi$
$578$	16.6791	0.693758
$579$	8.29179	0.344595
$580$	−9.24082	−0.383704
$581$	−9.25666	−0.384031
$582$	30.2560	1.25415
$583$	−20.6641	−0.855819
$584$	5.01418	0.207488
$585$	22.7257	0.939593
$586$	5.74415	0.237289
$587$	−10.1499	−0.418930	−0.209465	−	0.977816i	$-0.567172\pi$
$587$	−10.1499	−0.418930	−0.209465	+	0.977816i	$0.567172\pi$
$588$	−19.6717	−0.811247
$589$	9.34375	0.385002
$590$	−32.1737	−1.32457
$591$	−25.5262	−1.05001
$592$	−7.99124	−0.328438
$593$	29.1757	1.19810	0.599051	−	0.800711i	$-0.295544\pi$
$593$	29.1757	1.19810	0.599051	+	0.800711i	$0.295544\pi$
$594$	5.03680	0.206662
$595$	−8.66255	−0.355130
$596$	19.4543	0.796881
$597$	16.6499	0.681433
$598$	−22.7610	−0.930765
$599$	32.9562	1.34656	0.673278	−	0.739390i	$-0.264886\pi$
$599$	32.9562	1.34656	0.673278	+	0.739390i	$0.264886\pi$
$600$	20.2840	0.828092
$601$	−21.6915	−0.884815	−0.442407	−	0.896814i	$-0.645875\pi$
$601$	−21.6915	−0.884815	−0.442407	+	0.896814i	$0.645875\pi$
$602$	8.26384	0.336809
$603$	−9.22323	−0.375599
$604$	−10.8509	−0.441516
$605$	−31.1375	−1.26592
$606$	23.4001	0.950562
$607$	8.33714	0.338394	0.169197	−	0.985582i	$-0.445883\pi$
$607$	8.33714	0.338394	0.169197	+	0.985582i	$0.445883\pi$
$608$	−4.05893	−0.164611
$609$	20.9597	0.849331
$610$	−10.5912	−0.428825
$611$	−40.7565	−1.64883
$612$	−0.905939	−0.0366204
$613$	16.6419	0.672158	0.336079	−	0.941834i	$-0.390899\pi$
$613$	16.6419	0.672158	0.336079	+	0.941834i	$0.390899\pi$
$614$	13.6003	0.548865
$615$	41.1664	1.65999
$616$	6.74265	0.271669
$617$	13.7281	0.552674	0.276337	−	0.961061i	$-0.410879\pi$
$617$	13.7281	0.552674	0.276337	+	0.961061i	$0.410879\pi$
$618$	30.6021	1.23100
$619$	−5.41918	−0.217815	−0.108908	−	0.994052i	$-0.534735\pi$
$619$	−5.41918	−0.217815	−0.108908	+	0.994052i	$0.534735\pi$
$620$	8.75322	0.351538
$621$	18.2958	0.734185
$622$	5.56502	0.223137
$623$	68.0529	2.72648
$624$	−8.01488	−0.320852
$625$	17.1678	0.686714
$626$	−2.95338	−0.118041
$627$	14.5946	0.582851
$628$	−14.0690	−0.561415
$629$	4.52699	0.180503
$630$	−24.4543	−0.974281
$631$	5.53359	0.220289	0.110144	−	0.993916i	$-0.464869\pi$
$631$	5.53359	0.220289	0.110144	+	0.993916i	$0.464869\pi$
$632$	3.59161	0.142866
$633$	−47.0250	−1.86908
$634$	18.9218	0.751481
$635$	−81.7089	−3.24252
$636$	−26.4313	−1.04807
$637$	34.2812	1.35827
$638$	−4.07465	−0.161317
$639$	20.7293	0.820039
$640$	−3.80241	−0.150303
$641$	32.7074	1.29186	0.645932	−	0.763395i	$-0.276469\pi$
$641$	32.7074	1.29186	0.645932	+	0.763395i	$0.276469\pi$
$642$	−15.2881	−0.603373
$643$	13.4273	0.529521	0.264761	−	0.964314i	$-0.414707\pi$
$643$	13.4273	0.529521	0.264761	+	0.964314i	$0.414707\pi$
$644$	24.4922	0.965127
$645$	16.7567	0.659796
$646$	2.29936	0.0904670
$647$	−19.6715	−0.773367	−0.386684	−	0.922212i	$-0.626379\pi$
$647$	−19.6715	−0.773367	−0.386684	+	0.922212i	$0.626379\pi$
$648$	11.2402	0.441555
$649$	−14.1867	−0.556875
$650$	−35.3483	−1.38647
$651$	−19.8538	−0.778131
$652$	2.34169	0.0917077
$653$	41.4355	1.62150	0.810748	−	0.585395i	$-0.199061\pi$
$653$	41.4355	1.62150	0.810748	+	0.585395i	$0.199061\pi$
$654$	−28.7842	−1.12555
$655$	−69.5341	−2.71692
$656$	−5.04828	−0.197102
$657$	−8.01870	−0.312839
$658$	43.8564	1.70970
$659$	−45.6366	−1.77775	−0.888876	−	0.458149i	$-0.848513\pi$
$659$	−45.6366	−1.77775	−0.888876	+	0.458149i	$0.848513\pi$
$660$	13.6722	0.532189
$661$	−44.5239	−1.73178	−0.865890	−	0.500234i	$-0.833247\pi$
$661$	−44.5239	−1.73178	−0.865890	+	0.500234i	$0.833247\pi$
$662$	35.9527	1.39734
$663$	4.54038	0.176334
$664$	2.30177	0.0893260
$665$	62.0672	2.40686
$666$	12.7796	0.495200
$667$	−14.8009	−0.573092
$668$	−1.77000	−0.0684834
$669$	44.0761	1.70408
$670$	21.9300	0.847228
$671$	−4.67009	−0.180287
$672$	8.62449	0.332697
$673$	−26.2398	−1.01147	−0.505734	−	0.862689i	$-0.668778\pi$
$673$	−26.2398	−1.01147	−0.505734	+	0.862689i	$0.668778\pi$
$674$	7.14143	0.275078
$675$	28.4138	1.09365
$676$	0.967255	0.0372021
$677$	29.3003	1.12610	0.563052	−	0.826422i	$-0.309627\pi$
$677$	29.3003	1.12610	0.563052	+	0.826422i	$0.309627\pi$
$678$	35.5103	1.36376
$679$	56.7366	2.17735
$680$	2.15404	0.0826036
$681$	43.5099	1.66730
$682$	3.85965	0.147794
$683$	−28.9710	−1.10854	−0.554272	−	0.832336i	$-0.687003\pi$
$683$	−28.9710	−1.10854	−0.554272	+	0.832336i	$0.687003\pi$
$684$	6.49106	0.248192
$685$	62.8375	2.40090
$686$	−8.73788	−0.333614
$687$	−30.8453	−1.17682
$688$	−2.05490	−0.0783422
$689$	46.0610	1.75478
$690$	49.6632	1.89065
$691$	34.6968	1.31993	0.659964	−	0.751298i	$-0.270572\pi$
$691$	34.6968	1.31993	0.659964	+	0.751298i	$0.270572\pi$
$692$	19.3228	0.734541
$693$	−10.7829	−0.409607
$694$	−0.969999	−0.0368206
$695$	−13.3534	−0.506522
$696$	−5.21187	−0.197555
$697$	2.85982	0.108323
$698$	−12.6351	−0.478246
$699$	15.5504	0.588172
$700$	38.0369	1.43766
$701$	37.2638	1.40744	0.703718	−	0.710480i	$-0.251522\pi$
$701$	37.2638	1.40744	0.703718	+	0.710480i	$0.251522\pi$
$702$	−11.2272	−0.423744
$703$	−32.4359	−1.22334
$704$	−1.67663	−0.0631905
$705$	88.9285	3.34924
$706$	−21.3924	−0.805114
$707$	43.8801	1.65028
$708$	−18.1461	−0.681972
$709$	41.8832	1.57296	0.786479	−	0.617617i	$-0.211902\pi$
$709$	41.8832	1.57296	0.786479	+	0.617617i	$0.211902\pi$
$710$	−49.2878	−1.84974
$711$	−5.74371	−0.215406
$712$	−16.9221	−0.634183
$713$	14.0199	0.525049
$714$	−4.88572	−0.182843
$715$	−23.8261	−0.891045
$716$	20.0028	0.747539
$717$	−42.0761	−1.57136
$718$	13.6078	0.507840
$719$	−9.36913	−0.349410	−0.174705	−	0.984621i	$-0.555897\pi$
$719$	−9.36913	−0.349410	−0.174705	+	0.984621i	$0.555897\pi$
$720$	6.08082	0.226619
$721$	57.3855	2.13715
$722$	2.52509	0.0939741
$723$	44.4468	1.65299
$724$	13.3438	0.495919
$725$	−22.9861	−0.853681
$726$	−17.5617	−0.651776
$727$	26.0321	0.965475	0.482738	−	0.875765i	$-0.339642\pi$
$727$	26.0321	0.965475	0.482738	+	0.875765i	$0.339642\pi$
$728$	−15.0296	−0.557034
$729$	1.35234	0.0500866
$730$	19.0660	0.705663
$731$	1.16409	0.0430552
$732$	−5.97349	−0.220786
$733$	1.44352	0.0533176	0.0266588	−	0.999645i	$-0.491513\pi$
$733$	1.44352	0.0533176	0.0266588	+	0.999645i	$0.491513\pi$
$734$	−17.2977	−0.638471
$735$	−74.7998	−2.75903
$736$	−6.09025	−0.224490
$737$	9.66980	0.356192
$738$	8.07323	0.297180
$739$	28.4360	1.04603	0.523017	−	0.852322i	$-0.324806\pi$
$739$	28.4360	1.04603	0.523017	+	0.852322i	$0.324806\pi$
$740$	−30.3860	−1.11701
$741$	−32.5318	−1.19509
$742$	−49.5644	−1.81957
$743$	−36.0367	−1.32206	−0.661030	−	0.750360i	$-0.729880\pi$
$743$	−36.0367	−1.32206	−0.661030	+	0.750360i	$0.729880\pi$
$744$	4.93686	0.180994
$745$	73.9733	2.71017
$746$	−35.4005	−1.29610
$747$	−3.68100	−0.134681
$748$	0.949803	0.0347282
$749$	−28.6684	−1.04752
$750$	36.3554	1.32751
$751$	19.1152	0.697523	0.348762	−	0.937212i	$-0.386602\pi$
$751$	19.1152	0.697523	0.348762	+	0.937212i	$0.386602\pi$
$752$	−10.9054	−0.397679
$753$	−3.75545	−0.136856
$754$	9.08255	0.330767
$755$	−41.2594	−1.50158
$756$	12.0812	0.439387
$757$	4.61651	0.167790	0.0838949	−	0.996475i	$-0.473264\pi$
$757$	4.61651	0.167790	0.0838949	+	0.996475i	$0.473264\pi$
$758$	−1.49134	−0.0541679
$759$	21.8985	0.794866
$760$	−15.4337	−0.559839
$761$	−19.5309	−0.707995	−0.353997	−	0.935246i	$-0.615178\pi$
$761$	−19.5309	−0.707995	−0.353997	+	0.935246i	$0.615178\pi$
$762$	−46.0842	−1.66945
$763$	−53.9765	−1.95408
$764$	−20.1879	−0.730373
$765$	−3.44475	−0.124545
$766$	−29.4597	−1.06442
$767$	31.6226	1.14183
$768$	−2.14458	−0.0773857
$769$	−28.7746	−1.03764	−0.518820	−	0.854884i	$-0.673628\pi$
$769$	−28.7746	−1.03764	−0.518820	+	0.854884i	$0.673628\pi$
$770$	25.6383	0.923940
$771$	35.0921	1.26381
$772$	−3.86640	−0.139155
$773$	−20.9197	−0.752428	−0.376214	−	0.926533i	$-0.622774\pi$
$773$	−20.9197	−0.752428	−0.376214	+	0.926533i	$0.622774\pi$
$774$	3.28620	0.118120
$775$	21.7732	0.782117
$776$	−14.1082	−0.506454
$777$	68.9204	2.47251
$778$	3.76958	0.135146
$779$	−20.4906	−0.734152
$780$	−30.4758	−1.09121
$781$	−21.7330	−0.777667
$782$	3.45009	0.123375
$783$	−7.30077	−0.260908
$784$	9.17277	0.327599
$785$	−53.4961	−1.90936
$786$	−39.2175	−1.39884
$787$	25.2755	0.900975	0.450487	−	0.892783i	$-0.351250\pi$
$787$	25.2755	0.900975	0.450487	+	0.892783i	$0.351250\pi$
$788$	11.9027	0.424015
$789$	26.1569	0.931209
$790$	13.6567	0.485885
$791$	66.5894	2.36765
$792$	2.68128	0.0952751
$793$	10.4098	0.369663
$794$	−13.6134	−0.483122
$795$	−100.503	−3.56446
$796$	−7.76370	−0.275177
$797$	−22.0981	−0.782754	−0.391377	−	0.920230i	$-0.628001\pi$
$797$	−22.0981	−0.782754	−0.391377	+	0.920230i	$0.628001\pi$
$798$	35.0062	1.23921
$799$	6.17783	0.218556
$800$	−9.45829	−0.334401
$801$	27.0619	0.956186
$802$	−14.6691	−0.517984
$803$	8.40695	0.296675
$804$	12.3686	0.436207
$805$	93.1292	3.28237
$806$	−8.60330	−0.303038
$807$	−28.5101	−1.00360
$808$	−10.9113	−0.383857
$809$	−34.3155	−1.20647	−0.603235	−	0.797564i	$-0.706122\pi$
$809$	−34.3155	−1.20647	−0.603235	+	0.797564i	$0.706122\pi$
$810$	42.7396	1.50172
$811$	−10.0304	−0.352215	−0.176108	−	0.984371i	$-0.556351\pi$
$811$	−10.0304	−0.352215	−0.176108	+	0.984371i	$0.556351\pi$
$812$	−9.77337	−0.342978
$813$	7.46913	0.261954
$814$	−13.3984	−0.469613
$815$	8.90406	0.311896
$816$	1.21489	0.0425296
$817$	−8.34068	−0.291803
$818$	30.3036	1.05954
$819$	24.0354	0.839865
$820$	−19.1956	−0.670340
$821$	−36.9564	−1.28979	−0.644893	−	0.764273i	$-0.723098\pi$
$821$	−36.9564	−1.28979	−0.644893	+	0.764273i	$0.723098\pi$
$822$	35.4406	1.23613
$823$	2.29207	0.0798967	0.0399484	−	0.999202i	$-0.487281\pi$
$823$	2.29207	0.0798967	0.0399484	+	0.999202i	$0.487281\pi$
$824$	−14.2695	−0.497103
$825$	34.0089	1.18404
$826$	−34.0278	−1.18398
$827$	49.6656	1.72704	0.863521	−	0.504312i	$-0.168254\pi$
$827$	49.6656	1.72704	0.863521	+	0.504312i	$0.168254\pi$
$828$	9.73955	0.338473
$829$	42.7658	1.48532	0.742659	−	0.669670i	$-0.233565\pi$
$829$	42.7658	1.48532	0.742659	+	0.669670i	$0.233565\pi$
$830$	8.75226	0.303796
$831$	40.3391	1.39935
$832$	3.73728	0.129567
$833$	−5.19632	−0.180042
$834$	−7.53137	−0.260790
$835$	−6.73026	−0.232910
$836$	−6.80534	−0.235368
$837$	6.91554	0.239036
$838$	21.6014	0.746209
$839$	27.2260	0.939947	0.469973	−	0.882681i	$-0.344264\pi$
$839$	27.2260	0.939947	0.469973	+	0.882681i	$0.344264\pi$
$840$	32.7938	1.13149
$841$	−23.0939	−0.796340
$842$	−10.5804	−0.364623
$843$	−25.7415	−0.886586
$844$	21.9274	0.754773
$845$	3.67790	0.126524
$846$	17.4399	0.599597
$847$	−32.9320	−1.13156
$848$	12.3247	0.423233
$849$	61.9273	2.12534
$850$	5.35806	0.183780
$851$	−48.6687	−1.66834
$852$	−27.7986	−0.952363
$853$	−10.4317	−0.357176	−0.178588	−	0.983924i	$-0.557153\pi$
$853$	−10.4317	−0.357176	−0.178588	+	0.983924i	$0.557153\pi$
$854$	−11.2016	−0.383310
$855$	24.6816	0.844094
$856$	7.12872	0.243655
$857$	−33.8038	−1.15472	−0.577359	−	0.816491i	$-0.695917\pi$
$857$	−33.8038	−1.15472	−0.577359	+	0.816491i	$0.695917\pi$
$858$	−13.4380	−0.458766
$859$	43.6866	1.49057	0.745283	−	0.666748i	$-0.232314\pi$
$859$	43.6866	1.49057	0.745283	+	0.666748i	$0.232314\pi$
$860$	−7.81355	−0.266440
$861$	43.5388	1.48380
$862$	−5.54248	−0.188778
$863$	3.89899	0.132723	0.0663616	−	0.997796i	$-0.478861\pi$
$863$	3.89899	0.132723	0.0663616	+	0.997796i	$0.478861\pi$
$864$	−3.00411	−0.102202
$865$	73.4730	2.49816
$866$	−11.9062	−0.404589
$867$	35.7696	1.21480
$868$	9.25767	0.314226
$869$	6.02181	0.204276
$870$	−19.8176	−0.671881
$871$	−21.5543	−0.730341
$872$	13.4218	0.454521
$873$	22.5618	0.763603
$874$	−24.7199	−0.836163
$875$	68.1741	2.30470
$876$	10.7533	0.363320
$877$	44.4430	1.50073	0.750367	−	0.661021i	$-0.229877\pi$
$877$	44.4430	1.50073	0.750367	+	0.661021i	$0.229877\pi$
$878$	25.4427	0.858651
$879$	12.3188	0.415502
$880$	−6.37525	−0.214909
$881$	−11.3124	−0.381125	−0.190563	−	0.981675i	$-0.561031\pi$
$881$	−11.3124	−0.381125	−0.190563	+	0.981675i	$0.561031\pi$
$882$	−14.6691	−0.493935
$883$	16.9105	0.569082	0.284541	−	0.958664i	$-0.408159\pi$
$883$	16.9105	0.569082	0.284541	+	0.958664i	$0.408159\pi$
$884$	−2.11714	−0.0712073
$885$	−68.9988	−2.31937
$886$	10.7967	0.362721
$887$	41.4472	1.39166	0.695830	−	0.718206i	$-0.255037\pi$
$887$	41.4472	1.39166	0.695830	+	0.718206i	$0.255037\pi$
$888$	−17.1378	−0.575108
$889$	−86.4177	−2.89836
$890$	−64.3448	−2.15684
$891$	18.8456	0.631353
$892$	−20.5524	−0.688144
$893$	−44.2642	−1.48125
$894$	41.7213	1.39537
$895$	76.0586	2.54236
$896$	−4.02154	−0.134350
$897$	−48.8126	−1.62981
$898$	−19.8857	−0.663593
$899$	−5.59450	−0.186587
$900$	15.1257	0.504191
$901$	−6.98189	−0.232600
$902$	−8.46412	−0.281824
$903$	17.7224	0.589766
$904$	−16.5582	−0.550717
$905$	50.7387	1.68661
$906$	−23.2705	−0.773111
$907$	19.4873	0.647064	0.323532	−	0.946217i	$-0.395130\pi$
$907$	19.4873	0.647064	0.323532	+	0.946217i	$0.395130\pi$
$908$	−20.2884	−0.673293
$909$	17.4494	0.578759
$910$	−57.1487	−1.89446
$911$	43.5801	1.44387	0.721937	−	0.691959i	$-0.243252\pi$
$911$	43.5801	1.44387	0.721937	+	0.691959i	$0.243252\pi$
$912$	−8.70468	−0.288241
$913$	3.85923	0.127722
$914$	−21.8398	−0.722395
$915$	−22.7136	−0.750889
$916$	14.3829	0.475226
$917$	−73.5413	−2.42855
$918$	1.70181	0.0561681
$919$	−27.4620	−0.905887	−0.452943	−	0.891539i	$-0.649626\pi$
$919$	−27.4620	−0.905887	−0.452943	+	0.891539i	$0.649626\pi$
$920$	−23.1576	−0.763484
$921$	29.1670	0.961084
$922$	2.02737	0.0667677
$923$	48.4436	1.59454
$924$	14.4601	0.475703
$925$	−75.5835	−2.48517
$926$	−30.8671	−1.01435
$927$	22.8199	0.749504
$928$	2.43026	0.0797771
$929$	39.8058	1.30599	0.652993	−	0.757364i	$-0.273513\pi$
$929$	39.8058	1.30599	0.652993	+	0.757364i	$0.273513\pi$
$930$	18.7719	0.615556
$931$	37.2316	1.22022
$932$	−7.25106	−0.237516
$933$	11.9346	0.390722
$934$	−14.0150	−0.458585
$935$	3.61154	0.118110
$936$	−5.97667	−0.195354
$937$	−37.5451	−1.22654	−0.613272	−	0.789872i	$-0.710147\pi$
$937$	−37.5451	−1.22654	−0.613272	+	0.789872i	$0.710147\pi$
$938$	23.1938	0.757304
$939$	−6.33374	−0.206694
$940$	−41.4667	−1.35250
$941$	−36.0265	−1.17443	−0.587215	−	0.809431i	$-0.699776\pi$
$941$	−36.0265	−1.17443	−0.587215	+	0.809431i	$0.699776\pi$
$942$	−30.1721	−0.983059
$943$	−30.7453	−1.00120
$944$	8.46140	0.275395
$945$	45.9375	1.49435
$946$	−3.44531	−0.112017
$947$	26.2670	0.853562	0.426781	−	0.904355i	$-0.359647\pi$
$947$	26.2670	0.853562	0.426781	+	0.904355i	$0.359647\pi$
$948$	7.70247	0.250165
$949$	−18.7394	−0.608307
$950$	−38.3905	−1.24555
$951$	40.5793	1.31587
$952$	2.27818	0.0738361
$953$	−18.8007	−0.609015	−0.304507	−	0.952510i	$-0.598492\pi$
$953$	−18.8007	−0.609015	−0.304507	+	0.952510i	$0.598492\pi$
$954$	−19.7098	−0.638127
$955$	−76.7627	−2.48398
$956$	19.6198	0.634549
$957$	−8.73840	−0.282472
$958$	−6.66110	−0.215210
$959$	66.4588	2.14607
$960$	−8.15455	−0.263187
$961$	−25.7007	−0.829055
$962$	29.8655	0.962903
$963$	−11.4003	−0.367369
$964$	−20.7252	−0.667514
$965$	−14.7016	−0.473262
$966$	52.5253	1.68997
$967$	27.0426	0.869630	0.434815	−	0.900520i	$-0.356814\pi$
$967$	27.0426	0.869630	0.434815	+	0.900520i	$0.356814\pi$
$968$	8.18890	0.263201
$969$	4.93115	0.158411
$970$	−53.6450	−1.72244
$971$	46.9072	1.50532	0.752662	−	0.658407i	$-0.228769\pi$
$971$	46.9072	1.50532	0.752662	+	0.658407i	$0.228769\pi$
$972$	15.0930	0.484109
$973$	−14.1229	−0.452760
$974$	−35.9082	−1.15057
$975$	−75.8071	−2.42777
$976$	2.78539	0.0891583
$977$	3.95609	0.126566	0.0632832	−	0.997996i	$-0.479843\pi$
$977$	3.95609	0.126566	0.0632832	+	0.997996i	$0.479843\pi$
$978$	5.02193	0.160584
$979$	−28.3722	−0.906780
$980$	34.8786	1.11416
$981$	−21.4643	−0.685301
$982$	11.0267	0.351877
$983$	−31.9289	−1.01837	−0.509187	−	0.860656i	$-0.670054\pi$
$983$	−31.9289	−1.01837	−0.509187	+	0.860656i	$0.670054\pi$
$984$	−10.8264	−0.345134
$985$	45.2588	1.44206
$986$	−1.37672	−0.0438438
$987$	94.0535	2.99375
$988$	15.1694	0.482601
$989$	−12.5148	−0.397948
$990$	10.1953	0.324028
$991$	−24.6810	−0.784017	−0.392009	−	0.919962i	$-0.628220\pi$
$991$	−24.6810	−0.784017	−0.392009	+	0.919962i	$0.628220\pi$
$992$	−2.30202	−0.0730893
$993$	77.1033	2.44680
$994$	−52.1283	−1.65341
$995$	−29.5208	−0.935871
$996$	4.93632	0.156413
$997$	40.2336	1.27421	0.637105	−	0.770777i	$-0.280132\pi$
$997$	40.2336	1.27421	0.637105	+	0.770777i	$0.280132\pi$
$998$	25.5532	0.808872
$999$	−24.0066	−0.759535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.12	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.12	✓	83	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.14458 q^{3} +1.00000 q^{4} +3.80241 q^{5} +2.14458 q^{6} +4.02154 q^{7} -1.00000 q^{8} +1.59920 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.14458 q^{3} +1.00000 q^{4} +3.80241 q^{5} +2.14458 q^{6} +4.02154 q^{7} -1.00000 q^{8} +1.59920 q^{9} -3.80241 q^{10} -1.67663 q^{11} -2.14458 q^{12} +3.73728 q^{13} -4.02154 q^{14} -8.15455 q^{15} +1.00000 q^{16} -0.566494 q^{17} -1.59920 q^{18} +4.05893 q^{19} +3.80241 q^{20} -8.62449 q^{21} +1.67663 q^{22} +6.09025 q^{23} +2.14458 q^{24} +9.45829 q^{25} -3.73728 q^{26} +3.00411 q^{27} +4.02154 q^{28} -2.43026 q^{29} +8.15455 q^{30} +2.30202 q^{31} -1.00000 q^{32} +3.59567 q^{33} +0.566494 q^{34} +15.2915 q^{35} +1.59920 q^{36} -7.99124 q^{37} -4.05893 q^{38} -8.01488 q^{39} -3.80241 q^{40} -5.04828 q^{41} +8.62449 q^{42} -2.05490 q^{43} -1.67663 q^{44} +6.08082 q^{45} -6.09025 q^{46} -10.9054 q^{47} -2.14458 q^{48} +9.17277 q^{49} -9.45829 q^{50} +1.21489 q^{51} +3.73728 q^{52} +12.3247 q^{53} -3.00411 q^{54} -6.37525 q^{55} -4.02154 q^{56} -8.70468 q^{57} +2.43026 q^{58} +8.46140 q^{59} -8.15455 q^{60} +2.78539 q^{61} -2.30202 q^{62} +6.43126 q^{63} +1.00000 q^{64} +14.2107 q^{65} -3.59567 q^{66} -5.76739 q^{67} -0.566494 q^{68} -13.0610 q^{69} -15.2915 q^{70} +12.9623 q^{71} -1.59920 q^{72} -5.01418 q^{73} +7.99124 q^{74} -20.2840 q^{75} +4.05893 q^{76} -6.74265 q^{77} +8.01488 q^{78} -3.59161 q^{79} +3.80241 q^{80} -11.2402 q^{81} +5.04828 q^{82} -2.30177 q^{83} -8.62449 q^{84} -2.15404 q^{85} +2.05490 q^{86} +5.21187 q^{87} +1.67663 q^{88} +16.9221 q^{89} -6.08082 q^{90} +15.0296 q^{91} +6.09025 q^{92} -4.93686 q^{93} +10.9054 q^{94} +15.4337 q^{95} +2.14458 q^{96} +14.1082 q^{97} -9.17277 q^{98} -2.68128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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