LMFDB - Embedded newform 8039.2.a.a.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, 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("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.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:	$64.1917381849$
Analytic rank:	$1$
Dimension:	$279$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61158 q^{2} -1.07466 q^{3} +4.82033 q^{4} -1.60450 q^{5} +2.80656 q^{6} -4.85582 q^{7} -7.36550 q^{8} -1.84510 q^{9} +O(q^{10})$	\(q-2.61158 q^{2} -1.07466 q^{3} +4.82033 q^{4} -1.60450 q^{5} +2.80656 q^{6} -4.85582 q^{7} -7.36550 q^{8} -1.84510 q^{9} +4.19028 q^{10} -4.48224 q^{11} -5.18022 q^{12} -1.93590 q^{13} +12.6814 q^{14} +1.72430 q^{15} +9.59492 q^{16} -0.556594 q^{17} +4.81863 q^{18} -1.76493 q^{19} -7.73424 q^{20} +5.21836 q^{21} +11.7057 q^{22} +5.18089 q^{23} +7.91542 q^{24} -2.42557 q^{25} +5.05574 q^{26} +5.20684 q^{27} -23.4067 q^{28} -6.94264 q^{29} -4.50313 q^{30} -1.91742 q^{31} -10.3268 q^{32} +4.81688 q^{33} +1.45359 q^{34} +7.79119 q^{35} -8.89401 q^{36} +1.60044 q^{37} +4.60925 q^{38} +2.08043 q^{39} +11.8180 q^{40} +5.46678 q^{41} -13.6282 q^{42} -11.7569 q^{43} -21.6059 q^{44} +2.96048 q^{45} -13.5303 q^{46} -9.24037 q^{47} -10.3113 q^{48} +16.5790 q^{49} +6.33455 q^{50} +0.598150 q^{51} -9.33165 q^{52} +4.34810 q^{53} -13.5981 q^{54} +7.19177 q^{55} +35.7656 q^{56} +1.89670 q^{57} +18.1312 q^{58} -10.9087 q^{59} +8.31168 q^{60} -14.2510 q^{61} +5.00749 q^{62} +8.95950 q^{63} +7.77950 q^{64} +3.10615 q^{65} -12.5797 q^{66} +8.20498 q^{67} -2.68297 q^{68} -5.56770 q^{69} -20.3473 q^{70} -0.888566 q^{71} +13.5901 q^{72} +1.43034 q^{73} -4.17968 q^{74} +2.60666 q^{75} -8.50754 q^{76} +21.7650 q^{77} -5.43320 q^{78} +2.79264 q^{79} -15.3951 q^{80} -0.0602743 q^{81} -14.2769 q^{82} +11.5917 q^{83} +25.1542 q^{84} +0.893058 q^{85} +30.7042 q^{86} +7.46098 q^{87} +33.0139 q^{88} -3.49553 q^{89} -7.73151 q^{90} +9.40037 q^{91} +24.9736 q^{92} +2.06058 q^{93} +24.1319 q^{94} +2.83184 q^{95} +11.0978 q^{96} -3.58864 q^{97} -43.2974 q^{98} +8.27020 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) $ 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.61158	−1.84666	−0.923332	−	0.384004i	$-0.874545\pi$
$2$	−2.61158	−1.84666	−0.923332	+	0.384004i	$0.874545\pi$
$3$	−1.07466	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$3$	−1.07466	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$4$	4.82033	2.41016
$5$	−1.60450	−0.717556	−0.358778	−	0.933423i	$-0.616807\pi$
$5$	−1.60450	−0.717556	−0.358778	+	0.933423i	$0.616807\pi$
$6$	2.80656	1.14577
$7$	−4.85582	−1.83533	−0.917665	−	0.397356i	$-0.869928\pi$
$7$	−4.85582	−1.83533	−0.917665	+	0.397356i	$0.869928\pi$
$8$	−7.36550	−2.60410
$9$	−1.84510	−0.615035
$10$	4.19028	1.32508
$11$	−4.48224	−1.35145	−0.675723	−	0.737156i	$-0.736168\pi$
$11$	−4.48224	−1.35145	−0.675723	+	0.737156i	$0.736168\pi$
$12$	−5.18022	−1.49540
$13$	−1.93590	−0.536921	−0.268460	−	0.963291i	$-0.586515\pi$
$13$	−1.93590	−0.536921	−0.268460	+	0.963291i	$0.586515\pi$
$14$	12.6814	3.38923
$15$	1.72430	0.445212
$16$	9.59492	2.39873
$17$	−0.556594	−0.134994	−0.0674970	−	0.997719i	$-0.521501\pi$
$17$	−0.556594	−0.134994	−0.0674970	+	0.997719i	$0.521501\pi$
$18$	4.81863	1.13576
$19$	−1.76493	−0.404902	−0.202451	−	0.979292i	$-0.564891\pi$
$19$	−1.76493	−0.404902	−0.202451	+	0.979292i	$0.564891\pi$
$20$	−7.73424	−1.72943
$21$	5.21836	1.13874
$22$	11.7057	2.49566
$23$	5.18089	1.08029	0.540145	−	0.841572i	$-0.318369\pi$
$23$	5.18089	1.08029	0.540145	+	0.841572i	$0.318369\pi$
$24$	7.91542	1.61573
$25$	−2.42557	−0.485113
$26$	5.05574	0.991512
$27$	5.20684	1.00206
$28$	−23.4067	−4.42345
$29$	−6.94264	−1.28922	−0.644608	−	0.764513i	$-0.722979\pi$
$29$	−6.94264	−1.28922	−0.644608	+	0.764513i	$0.722979\pi$
$30$	−4.50313	−0.822156
$31$	−1.91742	−0.344379	−0.172189	−	0.985064i	$-0.555084\pi$
$31$	−1.91742	−0.344379	−0.172189	+	0.985064i	$0.555084\pi$
$32$	−10.3268	−1.82555
$33$	4.81688	0.838512
$34$	1.45359	0.249288
$35$	7.79119	1.31695
$36$	−8.89401	−1.48234
$37$	1.60044	0.263111	0.131556	−	0.991309i	$-0.458003\pi$
$37$	1.60044	0.263111	0.131556	+	0.991309i	$0.458003\pi$
$38$	4.60925	0.747718
$39$	2.08043	0.333136
$40$	11.8180	1.86859
$41$	5.46678	0.853768	0.426884	−	0.904306i	$-0.359611\pi$
$41$	5.46678	0.853768	0.426884	+	0.904306i	$0.359611\pi$
$42$	−13.6282	−2.10287
$43$	−11.7569	−1.79292	−0.896459	−	0.443128i	$-0.853869\pi$
$43$	−11.7569	−1.79292	−0.896459	+	0.443128i	$0.853869\pi$
$44$	−21.6059	−3.25721
$45$	2.96048	0.441322
$46$	−13.5303	−1.99493
$47$	−9.24037	−1.34785	−0.673923	−	0.738801i	$-0.735392\pi$
$47$	−9.24037	−1.34785	−0.673923	+	0.738801i	$0.735392\pi$
$48$	−10.3113	−1.48830
$49$	16.5790	2.36843
$50$	6.33455	0.895841
$51$	0.598150	0.0837577
$52$	−9.33165	−1.29407
$53$	4.34810	0.597257	0.298629	−	0.954369i	$-0.403471\pi$
$53$	4.34810	0.597257	0.298629	+	0.954369i	$0.403471\pi$
$54$	−13.5981	−1.85046
$55$	7.19177	0.969738
$56$	35.7656	4.77938
$57$	1.89670	0.251224
$58$	18.1312	2.38075
$59$	−10.9087	−1.42020	−0.710099	−	0.704102i	$-0.751350\pi$
$59$	−10.9087	−1.42020	−0.710099	+	0.704102i	$0.751350\pi$
$60$	8.31168	1.07303
$61$	−14.2510	−1.82466	−0.912328	−	0.409460i	$-0.865717\pi$
$61$	−14.2510	−1.82466	−0.912328	+	0.409460i	$0.865717\pi$
$62$	5.00749	0.635952
$63$	8.95950	1.12879
$64$	7.77950	0.972438
$65$	3.10615	0.385271
$66$	−12.5797	−1.54845
$67$	8.20498	1.00240	0.501199	−	0.865332i	$-0.332893\pi$
$67$	8.20498	1.00240	0.501199	+	0.865332i	$0.332893\pi$
$68$	−2.68297	−0.325358
$69$	−5.56770	−0.670272
$70$	−20.3473	−2.43197
$71$	−0.888566	−0.105453	−0.0527267	−	0.998609i	$-0.516791\pi$
$71$	−0.888566	−0.105453	−0.0527267	+	0.998609i	$0.516791\pi$
$72$	13.5901	1.60161
$73$	1.43034	0.167408	0.0837042	−	0.996491i	$-0.473325\pi$
$73$	1.43034	0.167408	0.0837042	+	0.996491i	$0.473325\pi$
$74$	−4.17968	−0.485878
$75$	2.60666	0.300991
$76$	−8.50754	−0.975882
$77$	21.7650	2.48035
$78$	−5.43320	−0.615189
$79$	2.79264	0.314197	0.157098	−	0.987583i	$-0.449786\pi$
$79$	2.79264	0.314197	0.157098	+	0.987583i	$0.449786\pi$
$80$	−15.3951	−1.72122
$81$	−0.0602743	−0.00669714
$82$	−14.2769	−1.57662
$83$	11.5917	1.27236	0.636179	−	0.771542i	$-0.280514\pi$
$83$	11.5917	1.27236	0.636179	+	0.771542i	$0.280514\pi$
$84$	25.1542	2.74455
$85$	0.893058	0.0968657
$86$	30.7042	3.31091
$87$	7.46098	0.799901
$88$	33.0139	3.51930
$89$	−3.49553	−0.370525	−0.185263	−	0.982689i	$-0.559314\pi$
$89$	−3.49553	−0.370525	−0.185263	+	0.982689i	$0.559314\pi$
$90$	−7.73151	−0.814973
$91$	9.40037	0.985427
$92$	24.9736	2.60368
$93$	2.06058	0.213672
$94$	24.1319	2.48902
$95$	2.83184	0.290540
$96$	11.0978	1.13267
$97$	−3.58864	−0.364371	−0.182186	−	0.983264i	$-0.558317\pi$
$97$	−3.58864	−0.364371	−0.182186	+	0.983264i	$0.558317\pi$
$98$	−43.2974	−4.37370
$99$	8.27020	0.831186
$100$	−11.6920	−1.16920
$101$	10.8664	1.08125	0.540626	−	0.841263i	$-0.318187\pi$
$101$	10.8664	1.08125	0.540626	+	0.841263i	$0.318187\pi$
$102$	−1.56211	−0.154672
$103$	14.4809	1.42684	0.713422	−	0.700734i	$-0.247144\pi$
$103$	14.4809	1.42684	0.713422	+	0.700734i	$0.247144\pi$
$104$	14.2588	1.39820
$105$	−8.37289	−0.817110
$106$	−11.3554	−1.10293
$107$	12.3589	1.19478	0.597389	−	0.801952i	$-0.296205\pi$
$107$	12.3589	1.19478	0.597389	+	0.801952i	$0.296205\pi$
$108$	25.0987	2.41512
$109$	−2.69163	−0.257811	−0.128906	−	0.991657i	$-0.541146\pi$
$109$	−2.69163	−0.257811	−0.128906	+	0.991657i	$0.541146\pi$
$110$	−18.7818	−1.79078
$111$	−1.71993	−0.163249
$112$	−46.5912	−4.40246
$113$	−5.78533	−0.544238	−0.272119	−	0.962264i	$-0.587724\pi$
$113$	−5.78533	−0.544238	−0.272119	+	0.962264i	$0.587724\pi$
$114$	−4.95337	−0.463926
$115$	−8.31276	−0.775169
$116$	−33.4658	−3.10722
$117$	3.57193	0.330225
$118$	28.4890	2.62263
$119$	2.70272	0.247758
$120$	−12.7003	−1.15938
$121$	9.09045	0.826404
$122$	37.2176	3.36953
$123$	−5.87494	−0.529725
$124$	−9.24260	−0.830010
$125$	11.9144	1.06565
$126$	−23.3984	−2.08450
$127$	4.29730	0.381324	0.190662	−	0.981656i	$-0.438937\pi$
$127$	4.29730	0.381324	0.190662	+	0.981656i	$0.438937\pi$
$128$	0.336922	0.0297800
$129$	12.6347	1.11243
$130$	−8.11195	−0.711465
$131$	8.20706	0.717054	0.358527	−	0.933519i	$-0.383279\pi$
$131$	8.20706	0.717054	0.358527	+	0.933519i	$0.383279\pi$
$132$	23.2190	2.02095
$133$	8.57019	0.743129
$134$	−21.4279	−1.85109
$135$	−8.35440	−0.719032
$136$	4.09960	0.351538
$137$	4.95486	0.423322	0.211661	−	0.977343i	$-0.432113\pi$
$137$	4.95486	0.423322	0.211661	+	0.977343i	$0.432113\pi$
$138$	14.5405	1.23777
$139$	19.9388	1.69119	0.845595	−	0.533825i	$-0.179246\pi$
$139$	19.9388	1.69119	0.845595	+	0.533825i	$0.179246\pi$
$140$	37.5561	3.17407
$141$	9.93026	0.836279
$142$	2.32056	0.194737
$143$	8.67714	0.725619
$144$	−17.7036	−1.47530
$145$	11.1395	0.925085
$146$	−3.73544	−0.309147
$147$	−17.8168	−1.46951
$148$	7.71466	0.634141
$149$	−7.83128	−0.641564	−0.320782	−	0.947153i	$-0.603946\pi$
$149$	−7.83128	−0.641564	−0.320782	+	0.947153i	$0.603946\pi$
$150$	−6.80749	−0.555829
$151$	2.26481	0.184307	0.0921537	−	0.995745i	$-0.470625\pi$
$151$	2.26481	0.184307	0.0921537	+	0.995745i	$0.470625\pi$
$152$	12.9996	1.05441
$153$	1.02697	0.0830260
$154$	−56.8408	−4.58037
$155$	3.07651	0.247111
$156$	10.0284	0.802911
$157$	−5.31185	−0.423932	−0.211966	−	0.977277i	$-0.567987\pi$
$157$	−5.31185	−0.423932	−0.211966	+	0.977277i	$0.567987\pi$
$158$	−7.29319	−0.580215
$159$	−4.67273	−0.370571
$160$	16.5695	1.30993
$161$	−25.1575	−1.98269
$162$	0.157411	0.0123674
$163$	−5.31442	−0.416258	−0.208129	−	0.978101i	$-0.566737\pi$
$163$	−5.31442	−0.416258	−0.208129	+	0.978101i	$0.566737\pi$
$164$	26.3517	2.05772
$165$	−7.72871	−0.601679
$166$	−30.2727	−2.34962
$167$	8.51354	0.658798	0.329399	−	0.944191i	$-0.393154\pi$
$167$	8.51354	0.658798	0.329399	+	0.944191i	$0.393154\pi$
$168$	−38.4359	−2.96539
$169$	−9.25231	−0.711716
$170$	−2.33229	−0.178878
$171$	3.25648	0.249029
$172$	−56.6723	−4.32123
$173$	8.18816	0.622534	0.311267	−	0.950322i	$-0.399247\pi$
$173$	8.18816	0.622534	0.311267	+	0.950322i	$0.399247\pi$
$174$	−19.4849	−1.47715
$175$	11.7781	0.890343
$176$	−43.0067	−3.24175
$177$	11.7232	0.881169
$178$	9.12884	0.684236
$179$	14.5543	1.08784	0.543920	−	0.839137i	$-0.316939\pi$
$179$	14.5543	1.08784	0.543920	+	0.839137i	$0.316939\pi$
$180$	14.2705	1.06366
$181$	−7.75174	−0.576182	−0.288091	−	0.957603i	$-0.593021\pi$
$181$	−7.75174	−0.576182	−0.288091	+	0.957603i	$0.593021\pi$
$182$	−24.5498	−1.81975
$183$	15.3150	1.13212
$184$	−38.1599	−2.81318
$185$	−2.56792	−0.188797
$186$	−5.38135	−0.394580
$187$	2.49479	0.182437
$188$	−44.5416	−3.24853
$189$	−25.2835	−1.83911
$190$	−7.39555	−0.536530
$191$	9.07928	0.656954	0.328477	−	0.944512i	$-0.393465\pi$
$191$	9.07928	0.656954	0.328477	+	0.944512i	$0.393465\pi$
$192$	−8.36032	−0.603354
$193$	−7.17387	−0.516387	−0.258193	−	0.966093i	$-0.583127\pi$
$193$	−7.17387	−0.516387	−0.258193	+	0.966093i	$0.583127\pi$
$194$	9.37200	0.672871
$195$	−3.33806	−0.239043
$196$	79.9164	5.70831
$197$	2.95852	0.210786	0.105393	−	0.994431i	$-0.466390\pi$
$197$	2.95852	0.210786	0.105393	+	0.994431i	$0.466390\pi$
$198$	−21.5982	−1.53492
$199$	−21.7085	−1.53887	−0.769436	−	0.638723i	$-0.779463\pi$
$199$	−21.7085	−1.53887	−0.769436	+	0.638723i	$0.779463\pi$
$200$	17.8655	1.26328
$201$	−8.81756	−0.621943
$202$	−28.3786	−1.99671
$203$	33.7123	2.36614
$204$	2.88328	0.201870
$205$	−8.77148	−0.612626
$206$	−37.8179	−2.63490
$207$	−9.55929	−0.664417
$208$	−18.5748	−1.28793
$209$	7.91083	0.547204
$210$	21.8664	1.50893
$211$	−24.3791	−1.67833	−0.839163	−	0.543881i	$-0.816954\pi$
$211$	−24.3791	−1.67833	−0.839163	+	0.543881i	$0.816954\pi$
$212$	20.9593	1.43949
$213$	0.954907	0.0654292
$214$	−32.2761	−2.20635
$215$	18.8641	1.28652
$216$	−38.3510	−2.60946
$217$	9.31066	0.632049
$218$	7.02939	0.476090
$219$	−1.53713	−0.103870
$220$	34.6667	2.33723
$221$	1.07751	0.0724810
$222$	4.49174	0.301466
$223$	17.0919	1.14456	0.572279	−	0.820059i	$-0.306060\pi$
$223$	17.0919	1.14456	0.572279	+	0.820059i	$0.306060\pi$
$224$	50.1453	3.35048
$225$	4.47542	0.298362
$226$	15.1088	1.00502
$227$	16.3049	1.08220	0.541099	−	0.840959i	$-0.318009\pi$
$227$	16.3049	1.08220	0.541099	+	0.840959i	$0.318009\pi$
$228$	9.14272	0.605491
$229$	26.2786	1.73654	0.868269	−	0.496093i	$-0.165232\pi$
$229$	26.2786	1.73654	0.868269	+	0.496093i	$0.165232\pi$
$230$	21.7094	1.43148
$231$	−23.3899	−1.53894
$232$	51.1361	3.35725
$233$	−2.25041	−0.147429	−0.0737146	−	0.997279i	$-0.523485\pi$
$233$	−2.25041	−0.147429	−0.0737146	+	0.997279i	$0.523485\pi$
$234$	−9.32837	−0.609814
$235$	14.8262	0.967155
$236$	−52.5837	−3.42291
$237$	−3.00114	−0.194945
$238$	−7.05837	−0.457526
$239$	−18.0630	−1.16840	−0.584200	−	0.811610i	$-0.698592\pi$
$239$	−18.0630	−1.16840	−0.584200	+	0.811610i	$0.698592\pi$
$240$	16.5445	1.06794
$241$	−16.6208	−1.07064	−0.535321	−	0.844649i	$-0.679809\pi$
$241$	−16.6208	−1.07064	−0.535321	+	0.844649i	$0.679809\pi$
$242$	−23.7404	−1.52609
$243$	−15.5558	−0.997902
$244$	−68.6946	−4.39772
$245$	−26.6011	−1.69948
$246$	15.3428	0.978224
$247$	3.41672	0.217401
$248$	14.1228	0.896797
$249$	−12.4572	−0.789441
$250$	−31.1152	−1.96790
$251$	16.7675	1.05836	0.529179	−	0.848510i	$-0.322500\pi$
$251$	16.7675	1.05836	0.529179	+	0.848510i	$0.322500\pi$
$252$	43.1878	2.72057
$253$	−23.2220	−1.45995
$254$	−11.2227	−0.704176
$255$	−0.959734	−0.0601009
$256$	−16.4389	−1.02743
$257$	−24.2669	−1.51373	−0.756865	−	0.653572i	$-0.773270\pi$
$257$	−24.2669	−1.51373	−0.756865	+	0.653572i	$0.773270\pi$
$258$	−32.9965	−2.05427
$259$	−7.77147	−0.482896
$260$	14.9727	0.928566
$261$	12.8099	0.792913
$262$	−21.4334	−1.32416
$263$	16.7293	1.03157	0.515787	−	0.856717i	$-0.327499\pi$
$263$	16.7293	1.03157	0.515787	+	0.856717i	$0.327499\pi$
$264$	−35.4788	−2.18357
$265$	−6.97654	−0.428565
$266$	−22.3817	−1.37231
$267$	3.75651	0.229895
$268$	39.5507	2.41594
$269$	−3.81446	−0.232572	−0.116286	−	0.993216i	$-0.537099\pi$
$269$	−3.81446	−0.232572	−0.116286	+	0.993216i	$0.537099\pi$
$270$	21.8182	1.32781
$271$	−22.4132	−1.36150	−0.680752	−	0.732514i	$-0.738347\pi$
$271$	−22.4132	−1.36150	−0.680752	+	0.732514i	$0.738347\pi$
$272$	−5.34047	−0.323814
$273$	−10.1022	−0.611413
$274$	−12.9400	−0.781734
$275$	10.8720	0.655604
$276$	−26.8382	−1.61547
$277$	22.4830	1.35087	0.675436	−	0.737419i	$-0.263956\pi$
$277$	22.4830	1.35087	0.675436	+	0.737419i	$0.263956\pi$
$278$	−52.0718	−3.12306
$279$	3.53784	0.211805
$280$	−57.3860	−3.42947
$281$	24.8978	1.48528	0.742640	−	0.669691i	$-0.233573\pi$
$281$	24.8978	1.48528	0.742640	+	0.669691i	$0.233573\pi$
$282$	−25.9336	−1.54433
$283$	−13.6304	−0.810241	−0.405120	−	0.914263i	$-0.632770\pi$
$283$	−13.6304	−0.810241	−0.405120	+	0.914263i	$0.632770\pi$
$284$	−4.28318	−0.254160
$285$	−3.04326	−0.180267
$286$	−22.6610	−1.33997
$287$	−26.5457	−1.56695
$288$	19.0541	1.12277
$289$	−16.6902	−0.981777
$290$	−29.0916	−1.70832
$291$	3.85657	0.226076
$292$	6.89470	0.403482
$293$	−17.9641	−1.04947	−0.524737	−	0.851264i	$-0.675836\pi$
$293$	−17.9641	−1.04947	−0.524737	+	0.851264i	$0.675836\pi$
$294$	46.5300	2.71369
$295$	17.5031	1.01907
$296$	−11.7881	−0.685168
$297$	−23.3383	−1.35423
$298$	20.4520	1.18475
$299$	−10.0297	−0.580031
$300$	12.5650	0.725438
$301$	57.0897	3.29059
$302$	−5.91471	−0.340354
$303$	−11.6777	−0.670869
$304$	−16.9343	−0.971251
$305$	22.8658	1.30929
$306$	−2.68202	−0.153321
$307$	−5.44354	−0.310679	−0.155340	−	0.987861i	$-0.549647\pi$
$307$	−5.44354	−0.310679	−0.155340	+	0.987861i	$0.549647\pi$
$308$	104.914	5.97804
$309$	−15.5620	−0.885294
$310$	−8.03454	−0.456331
$311$	26.6450	1.51090	0.755451	−	0.655205i	$-0.227418\pi$
$311$	26.6450	1.51090	0.755451	+	0.655205i	$0.227418\pi$
$312$	−15.3234	−0.867518
$313$	17.5609	0.992600	0.496300	−	0.868151i	$-0.334692\pi$
$313$	17.5609	0.992600	0.496300	+	0.868151i	$0.334692\pi$
$314$	13.8723	0.782859
$315$	−14.3756	−0.809971
$316$	13.4614	0.757266
$317$	13.4913	0.757744	0.378872	−	0.925449i	$-0.376312\pi$
$317$	13.4913	0.757744	0.378872	+	0.925449i	$0.376312\pi$
$318$	12.2032	0.684321
$319$	31.1186	1.74231
$320$	−12.4822	−0.697779
$321$	−13.2816	−0.741307
$322$	65.7007	3.66136
$323$	0.982349	0.0546594
$324$	−0.290542	−0.0161412
$325$	4.69564	0.260467
$326$	13.8790	0.768688
$327$	2.89258	0.159960
$328$	−40.2656	−2.22330
$329$	44.8696	2.47374
$330$	20.1841	1.11110
$331$	−1.92501	−0.105808	−0.0529040	−	0.998600i	$-0.516848\pi$
$331$	−1.92501	−0.105808	−0.0529040	+	0.998600i	$0.516848\pi$
$332$	55.8759	3.06659
$333$	−2.95299	−0.161823
$334$	−22.2338	−1.21658
$335$	−13.1649	−0.719276
$336$	50.0698	2.73153
$337$	17.5549	0.956277	0.478138	−	0.878285i	$-0.341312\pi$
$337$	17.5549	0.956277	0.478138	+	0.878285i	$0.341312\pi$
$338$	24.1631	1.31430
$339$	6.21726	0.337675
$340$	4.30483	0.233462
$341$	8.59433	0.465409
$342$	−8.50454	−0.459873
$343$	−46.5141	−2.51153
$344$	86.5958	4.66893
$345$	8.93340	0.480958
$346$	−21.3840	−1.14961
$347$	−20.7742	−1.11522	−0.557610	−	0.830103i	$-0.688281\pi$
$347$	−20.7742	−1.11522	−0.557610	+	0.830103i	$0.688281\pi$
$348$	35.9644	1.92789
$349$	−26.2518	−1.40522	−0.702612	−	0.711573i	$-0.747983\pi$
$349$	−26.2518	−1.40522	−0.702612	+	0.711573i	$0.747983\pi$
$350$	−30.7595	−1.64416
$351$	−10.0799	−0.538026
$352$	46.2874	2.46712
$353$	9.20083	0.489711	0.244855	−	0.969560i	$-0.421259\pi$
$353$	9.20083	0.489711	0.244855	+	0.969560i	$0.421259\pi$
$354$	−30.6160	−1.62722
$355$	1.42571	0.0756687
$356$	−16.8496	−0.893027
$357$	−2.90451	−0.153723
$358$	−38.0097	−2.00887
$359$	1.75978	0.0928778	0.0464389	−	0.998921i	$-0.485213\pi$
$359$	1.75978	0.0928778	0.0464389	+	0.998921i	$0.485213\pi$
$360$	−21.8054	−1.14925
$361$	−15.8850	−0.836054
$362$	20.2443	1.06401
$363$	−9.76914	−0.512747
$364$	45.3129	2.37504
$365$	−2.29498	−0.120125
$366$	−39.9963	−2.09064
$367$	27.9025	1.45650	0.728249	−	0.685312i	$-0.240334\pi$
$367$	27.9025	1.45650	0.728249	+	0.685312i	$0.240334\pi$
$368$	49.7102	2.59132
$369$	−10.0868	−0.525097
$370$	6.70631	0.348645
$371$	−21.1136	−1.09616
$372$	9.93266	0.514984
$373$	13.1109	0.678859	0.339429	−	0.940632i	$-0.389766\pi$
$373$	13.1109	0.678859	0.339429	+	0.940632i	$0.389766\pi$
$374$	−6.51533	−0.336899
$375$	−12.8039	−0.661190
$376$	68.0600	3.50993
$377$	13.4402	0.692207
$378$	66.0298	3.39621
$379$	20.6418	1.06030	0.530149	−	0.847905i	$-0.322136\pi$
$379$	20.6418	1.06030	0.530149	+	0.847905i	$0.322136\pi$
$380$	13.6504	0.700250
$381$	−4.61814	−0.236594
$382$	−23.7112	−1.21317
$383$	−24.4068	−1.24713	−0.623565	−	0.781771i	$-0.714316\pi$
$383$	−24.4068	−1.24713	−0.623565	+	0.781771i	$0.714316\pi$
$384$	−0.362077	−0.0184772
$385$	−34.9220	−1.77979
$386$	18.7351	0.953592
$387$	21.6928	1.10271
$388$	−17.2984	−0.878194
$389$	35.6792	1.80900	0.904502	−	0.426468i	$-0.140242\pi$
$389$	35.6792	1.80900	0.904502	+	0.426468i	$0.140242\pi$
$390$	8.71760	0.441433
$391$	−2.88365	−0.145833
$392$	−122.113	−6.16763
$393$	−8.81980	−0.444900
$394$	−7.72640	−0.389251
$395$	−4.48080	−0.225454
$396$	39.8651	2.00330
$397$	−11.6822	−0.586313	−0.293157	−	0.956064i	$-0.594706\pi$
$397$	−11.6822	−0.586313	−0.293157	+	0.956064i	$0.594706\pi$
$398$	56.6933	2.84178
$399$	−9.21004	−0.461079
$400$	−23.2731	−1.16366
$401$	−37.4518	−1.87026	−0.935128	−	0.354311i	$-0.884715\pi$
$401$	−37.4518	−1.87026	−0.935128	+	0.354311i	$0.884715\pi$
$402$	23.0277	1.14852
$403$	3.71193	0.184904
$404$	52.3799	2.60600
$405$	0.0967103	0.00480558
$406$	−88.0421	−4.36946
$407$	−7.17357	−0.355580
$408$	−4.40567	−0.218113
$409$	−12.2508	−0.605763	−0.302882	−	0.953028i	$-0.597949\pi$
$409$	−12.2508	−0.605763	−0.302882	+	0.953028i	$0.597949\pi$
$410$	22.9074	1.13131
$411$	−5.32479	−0.262653
$412$	69.8027	3.43893
$413$	52.9709	2.60653
$414$	24.9648	1.22695
$415$	−18.5990	−0.912988
$416$	19.9917	0.980173
$417$	−21.4275	−1.04931
$418$	−20.6597	−1.01050
$419$	−9.83058	−0.480255	−0.240128	−	0.970741i	$-0.577189\pi$
$419$	−9.83058	−0.480255	−0.240128	+	0.970741i	$0.577189\pi$
$420$	−40.3601	−1.96937
$421$	−13.7215	−0.668745	−0.334373	−	0.942441i	$-0.608524\pi$
$421$	−13.7215	−0.668745	−0.334373	+	0.942441i	$0.608524\pi$
$422$	63.6678	3.09930
$423$	17.0495	0.828973
$424$	−32.0259	−1.55532
$425$	1.35006	0.0654873
$426$	−2.49381	−0.120826
$427$	69.2005	3.34884
$428$	59.5739	2.87961
$429$	−9.32498	−0.450214
$430$	−49.2649	−2.37577
$431$	6.41098	0.308806	0.154403	−	0.988008i	$-0.450655\pi$
$431$	6.41098	0.308806	0.154403	+	0.988008i	$0.450655\pi$
$432$	49.9592	2.40366
$433$	−0.343456	−0.0165054	−0.00825272	−	0.999966i	$-0.502627\pi$
$433$	−0.343456	−0.0165054	−0.00825272	+	0.999966i	$0.502627\pi$
$434$	−24.3155	−1.16718
$435$	−11.9712	−0.573974
$436$	−12.9745	−0.621367
$437$	−9.14391	−0.437412
$438$	4.01433	0.191812
$439$	−9.09387	−0.434027	−0.217013	−	0.976169i	$-0.569632\pi$
$439$	−9.09387	−0.434027	−0.217013	+	0.976169i	$0.569632\pi$
$440$	−52.9710	−2.52529
$441$	−30.5901	−1.45667
$442$	−2.81399	−0.133848
$443$	18.6093	0.884153	0.442077	−	0.896977i	$-0.354242\pi$
$443$	18.6093	0.884153	0.442077	+	0.896977i	$0.354242\pi$
$444$	−8.29064	−0.393457
$445$	5.60859	0.265873
$446$	−44.6368	−2.11361
$447$	8.41597	0.398062
$448$	−37.7759	−1.78474
$449$	20.3240	0.959148	0.479574	−	0.877502i	$-0.340791\pi$
$449$	20.3240	0.959148	0.479574	+	0.877502i	$0.340791\pi$
$450$	−11.6879	−0.550973
$451$	−24.5034	−1.15382
$452$	−27.8872	−1.31170
$453$	−2.43390	−0.114354
$454$	−42.5816	−1.99845
$455$	−15.0829	−0.707099
$456$	−13.9701	−0.654212
$457$	−12.2560	−0.573311	−0.286656	−	0.958034i	$-0.592544\pi$
$457$	−12.2560	−0.573311	−0.286656	+	0.958034i	$0.592544\pi$
$458$	−68.6286	−3.20680
$459$	−2.89810	−0.135272
$460$	−40.0703	−1.86829
$461$	5.04086	0.234776	0.117388	−	0.993086i	$-0.462548\pi$
$461$	5.04086	0.234776	0.117388	+	0.993086i	$0.462548\pi$
$462$	61.0846	2.84191
$463$	−2.33364	−0.108453	−0.0542266	−	0.998529i	$-0.517269\pi$
$463$	−2.33364	−0.108453	−0.0542266	+	0.998529i	$0.517269\pi$
$464$	−66.6141	−3.09248
$465$	−3.30620	−0.153321
$466$	5.87712	0.272252
$467$	23.3026	1.07832	0.539158	−	0.842205i	$-0.318742\pi$
$467$	23.3026	1.07832	0.539158	+	0.842205i	$0.318742\pi$
$468$	17.2179	0.795897
$469$	−39.8419	−1.83973
$470$	−38.7198	−1.78601
$471$	5.70843	0.263031
$472$	80.3484	3.69833
$473$	52.6974	2.42303
$474$	7.83771	0.359998
$475$	4.28095	0.196424
$476$	13.0280	0.597138
$477$	−8.02269	−0.367334
$478$	47.1730	2.15764
$479$	−23.0560	−1.05345	−0.526727	−	0.850035i	$-0.676581\pi$
$479$	−23.0560	−1.05345	−0.526727	+	0.850035i	$0.676581\pi$
$480$	−17.8065	−0.812754
$481$	−3.09829	−0.141270
$482$	43.4066	1.97711
$483$	27.0358	1.23017
$484$	43.8189	1.99177
$485$	5.75799	0.261457
$486$	40.6250	1.84279
$487$	20.5636	0.931825	0.465912	−	0.884831i	$-0.345726\pi$
$487$	20.5636	0.931825	0.465912	+	0.884831i	$0.345726\pi$
$488$	104.966	4.75159
$489$	5.71120	0.258269
$490$	69.4709	3.13837
$491$	30.8529	1.39237	0.696186	−	0.717862i	$-0.254879\pi$
$491$	30.8529	1.39237	0.696186	+	0.717862i	$0.254879\pi$
$492$	−28.3191	−1.27672
$493$	3.86423	0.174036
$494$	−8.92302	−0.401466
$495$	−13.2696	−0.596423
$496$	−18.3975	−0.826072
$497$	4.31472	0.193542
$498$	32.5328	1.45783
$499$	11.4801	0.513919	0.256960	−	0.966422i	$-0.417279\pi$
$499$	11.4801	0.513919	0.256960	+	0.966422i	$0.417279\pi$
$500$	57.4311	2.56840
$501$	−9.14917	−0.408755
$502$	−43.7897	−1.95443
$503$	−20.3665	−0.908099	−0.454049	−	0.890977i	$-0.650021\pi$
$503$	−20.3665	−0.908099	−0.454049	+	0.890977i	$0.650021\pi$
$504$	−65.9913	−2.93948
$505$	−17.4353	−0.775859
$506$	60.6460	2.69604
$507$	9.94309	0.441588
$508$	20.7144	0.919053
$509$	−7.10460	−0.314906	−0.157453	−	0.987526i	$-0.550328\pi$
$509$	−7.10460	−0.314906	−0.157453	+	0.987526i	$0.550328\pi$
$510$	2.50642	0.110986
$511$	−6.94547	−0.307250
$512$	42.2576	1.86754
$513$	−9.18971	−0.405736
$514$	63.3749	2.79535
$515$	−23.2347	−1.02384
$516$	60.9035	2.68113
$517$	41.4175	1.82154
$518$	20.2958	0.891746
$519$	−8.79949	−0.386255
$520$	−22.8784	−1.00328
$521$	−1.19800	−0.0524853	−0.0262427	−	0.999656i	$-0.508354\pi$
$521$	−1.19800	−0.0524853	−0.0262427	+	0.999656i	$0.508354\pi$
$522$	−33.4540	−1.46424
$523$	−40.0471	−1.75114	−0.875569	−	0.483094i	$-0.839513\pi$
$523$	−40.0471	−1.75114	−0.875569	+	0.483094i	$0.839513\pi$
$524$	39.5607	1.72822
$525$	−12.6575	−0.552418
$526$	−43.6899	−1.90497
$527$	1.06723	0.0464891
$528$	46.2176	2.01136
$529$	3.84165	0.167028
$530$	18.2198	0.791416
$531$	20.1278	0.873471
$532$	41.3111	1.79106
$533$	−10.5831	−0.458406
$534$	−9.81041	−0.424538
$535$	−19.8299	−0.857320
$536$	−60.4338	−2.61034
$537$	−15.6409	−0.674956
$538$	9.96176	0.429482
$539$	−74.3111	−3.20081
$540$	−40.2710	−1.73299
$541$	−18.2563	−0.784898	−0.392449	−	0.919774i	$-0.628372\pi$
$541$	−18.2563	−0.784898	−0.392449	+	0.919774i	$0.628372\pi$
$542$	58.5338	2.51424
$543$	8.33049	0.357495
$544$	5.74786	0.246437
$545$	4.31873	0.184994
$546$	26.3827	1.12907
$547$	2.15298	0.0920548	0.0460274	−	0.998940i	$-0.485344\pi$
$547$	2.15298	0.0920548	0.0460274	+	0.998940i	$0.485344\pi$
$548$	23.8841	1.02028
$549$	26.2946	1.12223
$550$	−28.3930	−1.21068
$551$	12.2533	0.522007
$552$	41.0089	1.74546
$553$	−13.5606	−0.576654
$554$	−58.7160	−2.49460
$555$	2.75964	0.117140
$556$	96.1118	4.07605
$557$	−2.45354	−0.103960	−0.0519800	−	0.998648i	$-0.516553\pi$
$557$	−2.45354	−0.103960	−0.0519800	+	0.998648i	$0.516553\pi$
$558$	−9.23934	−0.391133
$559$	22.7602	0.962655
$560$	74.7558	3.15901
$561$	−2.68105	−0.113194
$562$	−65.0226	−2.74281
$563$	−1.11672	−0.0470639	−0.0235320	−	0.999723i	$-0.507491\pi$
$563$	−1.11672	−0.0470639	−0.0235320	+	0.999723i	$0.507491\pi$
$564$	47.8671	2.01557
$565$	9.28258	0.390521
$566$	35.5967	1.49624
$567$	0.292681	0.0122915
$568$	6.54474	0.274611
$569$	−8.25308	−0.345987	−0.172994	−	0.984923i	$-0.555344\pi$
$569$	−8.25308	−0.345987	−0.172994	+	0.984923i	$0.555344\pi$
$570$	7.94771	0.332893
$571$	−36.5659	−1.53023	−0.765117	−	0.643891i	$-0.777319\pi$
$571$	−36.5659	−1.53023	−0.765117	+	0.643891i	$0.777319\pi$
$572$	41.8267	1.74886
$573$	−9.75715	−0.407611
$574$	69.3262	2.89362
$575$	−12.5666	−0.524063
$576$	−14.3540	−0.598083
$577$	−25.8697	−1.07697	−0.538485	−	0.842635i	$-0.681003\pi$
$577$	−25.8697	−1.07697	−0.538485	+	0.842635i	$0.681003\pi$
$578$	43.5877	1.81301
$579$	7.70948	0.320395
$580$	53.6961	2.22961
$581$	−56.2874	−2.33519
$582$	−10.0717	−0.417486
$583$	−19.4892	−0.807160
$584$	−10.5352	−0.435948
$585$	−5.73118	−0.236955
$586$	46.9146	1.93803
$587$	−10.1236	−0.417845	−0.208922	−	0.977932i	$-0.566996\pi$
$587$	−10.1236	−0.417845	−0.208922	+	0.977932i	$0.566996\pi$
$588$	−85.8830	−3.54175
$589$	3.38411	0.139440
$590$	−45.7107	−1.88188
$591$	−3.17941	−0.130783
$592$	15.3561	0.631132
$593$	−5.06594	−0.208033	−0.104017	−	0.994576i	$-0.533170\pi$
$593$	−5.06594	−0.208033	−0.104017	+	0.994576i	$0.533170\pi$
$594$	60.9498	2.50080
$595$	−4.33653	−0.177780
$596$	−37.7494	−1.54627
$597$	23.3292	0.954802
$598$	26.1932	1.07112
$599$	8.54623	0.349189	0.174595	−	0.984640i	$-0.444138\pi$
$599$	8.54623	0.349189	0.174595	+	0.984640i	$0.444138\pi$
$600$	−19.1994	−0.783811
$601$	20.0674	0.818566	0.409283	−	0.912408i	$-0.365779\pi$
$601$	20.0674	0.818566	0.409283	+	0.912408i	$0.365779\pi$
$602$	−149.094	−6.07662
$603$	−15.1390	−0.616509
$604$	10.9171	0.444211
$605$	−14.5857	−0.592991
$606$	30.4973	1.23887
$607$	23.9107	0.970506	0.485253	−	0.874374i	$-0.338728\pi$
$607$	23.9107	0.970506	0.485253	+	0.874374i	$0.338728\pi$
$608$	18.2261	0.739168
$609$	−36.2292	−1.46808
$610$	−59.7158	−2.41782
$611$	17.8884	0.723687
$612$	4.95036	0.200106
$613$	−21.2637	−0.858834	−0.429417	−	0.903106i	$-0.641281\pi$
$613$	−21.2637	−0.858834	−0.429417	+	0.903106i	$0.641281\pi$
$614$	14.2162	0.573720
$615$	9.42636	0.380107
$616$	−160.310	−6.45907
$617$	−23.5452	−0.947895	−0.473948	−	0.880553i	$-0.657171\pi$
$617$	−23.5452	−0.947895	−0.473948	+	0.880553i	$0.657171\pi$
$618$	40.6415	1.63484
$619$	15.7469	0.632922	0.316461	−	0.948605i	$-0.397505\pi$
$619$	15.7469	0.632922	0.316461	+	0.948605i	$0.397505\pi$
$620$	14.8298	0.595579
$621$	26.9761	1.08251
$622$	−69.5855	−2.79013
$623$	16.9737	0.680036
$624$	19.9616	0.799102
$625$	−6.98880	−0.279552
$626$	−45.8616	−1.83300
$627$	−8.50146	−0.339515
$628$	−25.6049	−1.02175
$629$	−0.890797	−0.0355184
$630$	37.5429	1.49574
$631$	−46.2708	−1.84201	−0.921005	−	0.389551i	$-0.872630\pi$
$631$	−46.2708	−1.84201	−0.921005	+	0.389551i	$0.872630\pi$
$632$	−20.5692	−0.818199
$633$	26.1992	1.04133
$634$	−35.2334	−1.39930
$635$	−6.89504	−0.273621
$636$	−22.5241	−0.893138
$637$	−32.0953	−1.27166
$638$	−81.2685	−3.21745
$639$	1.63950	0.0648575
$640$	−0.540593	−0.0213688
$641$	0.738453	0.0291671	0.0145836	−	0.999894i	$-0.495358\pi$
$641$	0.738453	0.0291671	0.0145836	+	0.999894i	$0.495358\pi$
$642$	34.6859	1.36894
$643$	11.4524	0.451639	0.225819	−	0.974169i	$-0.427494\pi$
$643$	11.4524	0.451639	0.225819	+	0.974169i	$0.427494\pi$
$644$	−121.267	−4.77861
$645$	−20.2725	−0.798228
$646$	−2.56548	−0.100937
$647$	32.7067	1.28583	0.642916	−	0.765937i	$-0.277725\pi$
$647$	32.7067	1.28583	0.642916	+	0.765937i	$0.277725\pi$
$648$	0.443951	0.0174400
$649$	48.8956	1.91932
$650$	−12.2630	−0.480996
$651$	−10.0058	−0.392158
$652$	−25.6173	−1.00325
$653$	32.8342	1.28490	0.642452	−	0.766326i	$-0.277917\pi$
$653$	32.8342	1.28490	0.642452	+	0.766326i	$0.277917\pi$
$654$	−7.55420	−0.295393
$655$	−13.1683	−0.514526
$656$	52.4533	2.04796
$657$	−2.63912	−0.102962
$658$	−117.180	−4.56817
$659$	−6.27674	−0.244507	−0.122254	−	0.992499i	$-0.539012\pi$
$659$	−6.27674	−0.244507	−0.122254	+	0.992499i	$0.539012\pi$
$660$	−37.2549	−1.45015
$661$	−22.6758	−0.881988	−0.440994	−	0.897510i	$-0.645374\pi$
$661$	−22.6758	−0.881988	−0.440994	+	0.897510i	$0.645374\pi$
$662$	5.02730	0.195392
$663$	−1.15796	−0.0449713
$664$	−85.3789	−3.31334
$665$	−13.7509	−0.533237
$666$	7.71195	0.298832
$667$	−35.9691	−1.39273
$668$	41.0381	1.58781
$669$	−18.3680	−0.710147
$670$	34.3812	1.32826
$671$	63.8765	2.46592
$672$	−53.8892	−2.07882
$673$	−10.9403	−0.421716	−0.210858	−	0.977517i	$-0.567626\pi$
$673$	−10.9403	−0.421716	−0.210858	+	0.977517i	$0.567626\pi$
$674$	−45.8460	−1.76592
$675$	−12.6295	−0.486111
$676$	−44.5992	−1.71535
$677$	−15.1150	−0.580918	−0.290459	−	0.956887i	$-0.593808\pi$
$677$	−15.1150	−0.580918	−0.290459	+	0.956887i	$0.593808\pi$
$678$	−16.2369	−0.623573
$679$	17.4258	0.668741
$680$	−6.57782	−0.252248
$681$	−17.5223	−0.671455
$682$	−22.4448	−0.859454
$683$	−0.319732	−0.0122342	−0.00611710	−	0.999981i	$-0.501947\pi$
$683$	−0.319732	−0.0122342	−0.00611710	+	0.999981i	$0.501947\pi$
$684$	15.6973	0.600201
$685$	−7.95010	−0.303758
$686$	121.475	4.63794
$687$	−28.2406	−1.07745
$688$	−112.807	−4.30072
$689$	−8.41746	−0.320680
$690$	−23.3303	−0.888168
$691$	6.36098	0.241983	0.120992	−	0.992654i	$-0.461393\pi$
$691$	6.36098	0.241983	0.120992	+	0.992654i	$0.461393\pi$
$692$	39.4696	1.50041
$693$	−40.1586	−1.52550
$694$	54.2535	2.05943
$695$	−31.9919	−1.21352
$696$	−54.9539	−2.08302
$697$	−3.04278	−0.115253
$698$	68.5585	2.59498
$699$	2.41843	0.0914733
$700$	56.7744	2.14587
$701$	−7.50817	−0.283580	−0.141790	−	0.989897i	$-0.545286\pi$
$701$	−7.50817	−0.283580	−0.141790	+	0.989897i	$0.545286\pi$
$702$	26.3244	0.993552
$703$	−2.82467	−0.106534
$704$	−34.8696	−1.31420
$705$	−15.9331	−0.600077
$706$	−24.0287	−0.904331
$707$	−52.7656	−1.98445
$708$	56.5096	2.12376
$709$	10.7051	0.402038	0.201019	−	0.979587i	$-0.435575\pi$
$709$	10.7051	0.402038	0.201019	+	0.979587i	$0.435575\pi$
$710$	−3.72335	−0.139735
$711$	−5.15271	−0.193242
$712$	25.7463	0.964885
$713$	−9.93395	−0.372029
$714$	7.58535	0.283875
$715$	−13.9225	−0.520672
$716$	70.1566	2.62187
$717$	19.4116	0.724940
$718$	−4.59581	−0.171514
$719$	−35.2559	−1.31482	−0.657411	−	0.753532i	$-0.728348\pi$
$719$	−35.2559	−1.31482	−0.657411	+	0.753532i	$0.728348\pi$
$720$	28.4055	1.05861
$721$	−70.3167	−2.61873
$722$	41.4850	1.54391
$723$	17.8618	0.664286
$724$	−37.3659	−1.38869
$725$	16.8398	0.625416
$726$	25.5129	0.946871
$727$	−21.7768	−0.807656	−0.403828	−	0.914835i	$-0.632321\pi$
$727$	−21.7768	−0.807656	−0.403828	+	0.914835i	$0.632321\pi$
$728$	−69.2385	−2.56615
$729$	16.8980	0.625851
$730$	5.99353	0.221830
$731$	6.54385	0.242033
$732$	73.8234	2.72859
$733$	−1.46872	−0.0542483	−0.0271242	−	0.999632i	$-0.508635\pi$
$733$	−1.46872	−0.0542483	−0.0271242	+	0.999632i	$0.508635\pi$
$734$	−72.8695	−2.68966
$735$	28.5872	1.05445
$736$	−53.5023	−1.97212
$737$	−36.7766	−1.35469
$738$	26.3424	0.969677
$739$	34.3577	1.26387	0.631934	−	0.775022i	$-0.282261\pi$
$739$	34.3577	1.26387	0.631934	+	0.775022i	$0.282261\pi$
$740$	−12.3782	−0.455032
$741$	−3.67181	−0.134887
$742$	55.1398	2.02424
$743$	−20.7120	−0.759849	−0.379925	−	0.925017i	$-0.624050\pi$
$743$	−20.7120	−0.759849	−0.379925	+	0.925017i	$0.624050\pi$
$744$	−15.1772	−0.556422
$745$	12.5653	0.460358
$746$	−34.2402	−1.25362
$747$	−21.3879	−0.782544
$748$	12.0257	0.439703
$749$	−60.0125	−2.19281
$750$	33.4383	1.22099
$751$	33.8764	1.23617	0.618083	−	0.786113i	$-0.287910\pi$
$751$	33.8764	1.23617	0.618083	+	0.786113i	$0.287910\pi$
$752$	−88.6606	−3.23312
$753$	−18.0194	−0.656664
$754$	−35.1002	−1.27827
$755$	−3.63389	−0.132251
$756$	−121.875	−4.43255
$757$	42.0596	1.52868	0.764341	−	0.644813i	$-0.223065\pi$
$757$	42.0596	1.52868	0.764341	+	0.644813i	$0.223065\pi$
$758$	−53.9076	−1.95801
$759$	24.9558	0.905837
$760$	−20.8579	−0.756596
$761$	−27.7762	−1.00689	−0.503444	−	0.864028i	$-0.667934\pi$
$761$	−27.7762	−1.00689	−0.503444	+	0.864028i	$0.667934\pi$
$762$	12.0606	0.436910
$763$	13.0701	0.473168
$764$	43.7651	1.58337
$765$	−1.64778	−0.0595758
$766$	63.7403	2.30303
$767$	21.1182	0.762533
$768$	17.6662	0.637476
$769$	5.80655	0.209390	0.104695	−	0.994504i	$-0.466613\pi$
$769$	5.80655	0.209390	0.104695	+	0.994504i	$0.466613\pi$
$770$	91.2014	3.28667
$771$	26.0787	0.939202
$772$	−34.5804	−1.24458
$773$	−12.3397	−0.443827	−0.221914	−	0.975066i	$-0.571230\pi$
$773$	−12.3397	−0.443827	−0.221914	+	0.975066i	$0.571230\pi$
$774$	−56.6524	−2.03633
$775$	4.65083	0.167063
$776$	26.4321	0.948858
$777$	8.35169	0.299615
$778$	−93.1788	−3.34062
$779$	−9.64848	−0.345693
$780$	−16.0905	−0.576134
$781$	3.98276	0.142515
$782$	7.53088	0.269304
$783$	−36.1493	−1.29187
$784$	159.074	5.68123
$785$	8.52288	0.304195
$786$	23.0336	0.821580
$787$	14.6120	0.520861	0.260431	−	0.965493i	$-0.416135\pi$
$787$	14.6120	0.520861	0.260431	+	0.965493i	$0.416135\pi$
$788$	14.2610	0.508029
$789$	−17.9783	−0.640046
$790$	11.7020	0.416337
$791$	28.0925	0.998856
$792$	−60.9142	−2.16449
$793$	27.5885	0.979696
$794$	30.5090	1.08272
$795$	7.49741	0.265906
$796$	−104.642	−3.70894
$797$	48.3761	1.71357	0.856785	−	0.515674i	$-0.172458\pi$
$797$	48.3761	1.71357	0.856785	+	0.515674i	$0.172458\pi$
$798$	24.0527	0.851457
$799$	5.14314	0.181951
$800$	25.0484	0.885596
$801$	6.44962	0.227886
$802$	97.8083	3.45373
$803$	−6.41112	−0.226243
$804$	−42.5036	−1.49898
$805$	40.3653	1.42269
$806$	−9.69398	−0.341456
$807$	4.09925	0.144301
$808$	−80.0369	−2.81569
$809$	13.4742	0.473726	0.236863	−	0.971543i	$-0.423881\pi$
$809$	13.4742	0.473726	0.236863	+	0.971543i	$0.423881\pi$
$810$	−0.252566	−0.00887428
$811$	−7.36671	−0.258680	−0.129340	−	0.991600i	$-0.541286\pi$
$811$	−7.36671	−0.258680	−0.129340	+	0.991600i	$0.541286\pi$
$812$	162.504	5.70278
$813$	24.0866	0.844753
$814$	18.7343	0.656637
$815$	8.52701	0.298688
$816$	5.73920	0.200912
$817$	20.7502	0.725957
$818$	31.9939	1.11864
$819$	−17.3447	−0.606072
$820$	−42.2814	−1.47653
$821$	45.4288	1.58548	0.792738	−	0.609563i	$-0.208655\pi$
$821$	45.4288	1.58548	0.792738	+	0.609563i	$0.208655\pi$
$822$	13.9061	0.485031
$823$	−0.598711	−0.0208697	−0.0104349	−	0.999946i	$-0.503322\pi$
$823$	−0.598711	−0.0208697	−0.0104349	+	0.999946i	$0.503322\pi$
$824$	−106.659	−3.71564
$825$	−11.6837	−0.406773
$826$	−138.338	−4.81338
$827$	−41.3768	−1.43881	−0.719406	−	0.694590i	$-0.755586\pi$
$827$	−41.3768	−1.43881	−0.719406	+	0.694590i	$0.755586\pi$
$828$	−46.0789	−1.60135
$829$	12.4548	0.432572	0.216286	−	0.976330i	$-0.430606\pi$
$829$	12.4548	0.432572	0.216286	+	0.976330i	$0.430606\pi$
$830$	48.5726	1.68598
$831$	−24.1616	−0.838156
$832$	−15.0603	−0.522122
$833$	−9.22779	−0.319724
$834$	55.9595	1.93772
$835$	−13.6600	−0.472724
$836$	38.1328	1.31885
$837$	−9.98371	−0.345087
$838$	25.6733	0.886870
$839$	−22.3347	−0.771080	−0.385540	−	0.922691i	$-0.625985\pi$
$839$	−22.3347	−0.771080	−0.385540	+	0.922691i	$0.625985\pi$
$840$	61.6705	2.12784
$841$	19.2003	0.662079
$842$	35.8348	1.23495
$843$	−26.7567	−0.921550
$844$	−117.515	−4.04504
$845$	14.8454	0.510696
$846$	−44.5259	−1.53083
$847$	−44.1416	−1.51672
$848$	41.7196	1.43266
$849$	14.6480	0.502718
$850$	−3.52577	−0.120933
$851$	8.29172	0.284237
$852$	4.60297	0.157695
$853$	−6.26770	−0.214602	−0.107301	−	0.994227i	$-0.534221\pi$
$853$	−6.26770	−0.214602	−0.107301	+	0.994227i	$0.534221\pi$
$854$	−180.722	−6.18419
$855$	−5.22503	−0.178692
$856$	−91.0294	−3.11132
$857$	−48.3104	−1.65025	−0.825125	−	0.564950i	$-0.808895\pi$
$857$	−48.3104	−1.65025	−0.825125	+	0.564950i	$0.808895\pi$
$858$	24.3529	0.831394
$859$	15.2106	0.518980	0.259490	−	0.965746i	$-0.416446\pi$
$859$	15.2106	0.518980	0.259490	+	0.965746i	$0.416446\pi$
$860$	90.9310	3.10072
$861$	28.5277	0.972220
$862$	−16.7428	−0.570261
$863$	−5.51633	−0.187778	−0.0938890	−	0.995583i	$-0.529930\pi$
$863$	−5.51633	−0.187778	−0.0938890	+	0.995583i	$0.529930\pi$
$864$	−53.7702	−1.82930
$865$	−13.1379	−0.446703
$866$	0.896961	0.0304800
$867$	17.9363	0.609149
$868$	44.8804	1.52334
$869$	−12.5173	−0.424620
$870$	31.2636	1.05994
$871$	−15.8840	−0.538208
$872$	19.8252	0.671365
$873$	6.62142	0.224101
$874$	23.8800	0.807753
$875$	−57.8540	−1.95582
$876$	−7.40947	−0.250343
$877$	−25.3505	−0.856027	−0.428014	−	0.903772i	$-0.640786\pi$
$877$	−25.3505	−0.856027	−0.428014	+	0.903772i	$0.640786\pi$
$878$	23.7493	0.801501
$879$	19.3053	0.651152
$880$	69.0044	2.32614
$881$	28.1503	0.948406	0.474203	−	0.880415i	$-0.342736\pi$
$881$	28.1503	0.948406	0.474203	+	0.880415i	$0.342736\pi$
$882$	79.8882	2.68998
$883$	39.5680	1.33157	0.665785	−	0.746144i	$-0.268097\pi$
$883$	39.5680	1.33157	0.665785	+	0.746144i	$0.268097\pi$
$884$	5.19394	0.174691
$885$	−18.8099	−0.632288
$886$	−48.5995	−1.63273
$887$	−14.5242	−0.487674	−0.243837	−	0.969816i	$-0.578406\pi$
$887$	−14.5242	−0.487674	−0.243837	+	0.969816i	$0.578406\pi$
$888$	12.6682	0.425116
$889$	−20.8669	−0.699855
$890$	−14.6473	−0.490977
$891$	0.270164	0.00905082
$892$	82.3885	2.75857
$893$	16.3086	0.545746
$894$	−21.9789	−0.735086
$895$	−23.3525	−0.780586
$896$	−1.63603	−0.0546561
$897$	10.7785	0.359883
$898$	−53.0776	−1.77122
$899$	13.3120	0.443979
$900$	21.5730	0.719101
$901$	−2.42013	−0.0806261
$902$	63.9925	2.13072
$903$	−61.3520	−2.04167
$904$	42.6119	1.41725
$905$	12.4377	0.413443
$906$	6.35631	0.211174
$907$	6.24417	0.207334	0.103667	−	0.994612i	$-0.466942\pi$
$907$	6.24417	0.207334	0.103667	+	0.994612i	$0.466942\pi$
$908$	78.5952	2.60827
$909$	−20.0497	−0.665008
$910$	39.3902	1.30577
$911$	9.12377	0.302284	0.151142	−	0.988512i	$-0.451705\pi$
$911$	9.12377	0.302284	0.151142	+	0.988512i	$0.451705\pi$
$912$	18.1987	0.602618
$913$	−51.9569	−1.71952
$914$	32.0075	1.05871
$915$	−24.5730	−0.812358
$916$	126.672	4.18534
$917$	−39.8520	−1.31603
$918$	7.56860	0.249801
$919$	−36.1303	−1.19183	−0.595914	−	0.803048i	$-0.703210\pi$
$919$	−36.1303	−1.19183	−0.595914	+	0.803048i	$0.703210\pi$
$920$	61.2277	2.01862
$921$	5.84996	0.192763
$922$	−13.1646	−0.433553
$923$	1.72017	0.0566201
$924$	−112.747	−3.70911
$925$	−3.88198	−0.127639
$926$	6.09447	0.200277
$927$	−26.7188	−0.877559
$928$	71.6956	2.35352
$929$	−29.8876	−0.980581	−0.490291	−	0.871559i	$-0.663109\pi$
$929$	−29.8876	−0.980581	−0.490291	+	0.871559i	$0.663109\pi$
$930$	8.63440	0.283133
$931$	−29.2608	−0.958984
$932$	−10.8477	−0.355329
$933$	−28.6344	−0.937447
$934$	−60.8565	−1.99129
$935$	−4.00290	−0.130909
$936$	−26.3091	−0.859939
$937$	46.6140	1.52281	0.761407	−	0.648274i	$-0.224509\pi$
$937$	46.6140	1.52281	0.761407	+	0.648274i	$0.224509\pi$
$938$	104.050	3.39736
$939$	−18.8720	−0.615864
$940$	71.4672	2.33100
$941$	43.4324	1.41586	0.707928	−	0.706284i	$-0.249630\pi$
$941$	43.4324	1.41586	0.707928	+	0.706284i	$0.249630\pi$
$942$	−14.9080	−0.485729
$943$	28.3228	0.922318
$944$	−104.668	−3.40667
$945$	40.5675	1.31966
$946$	−137.623	−4.47452
$947$	33.9560	1.10342	0.551710	−	0.834036i	$-0.313975\pi$
$947$	33.9560	1.10342	0.551710	+	0.834036i	$0.313975\pi$
$948$	−14.4665	−0.469850
$949$	−2.76899	−0.0898851
$950$	−11.1800	−0.362728
$951$	−14.4985	−0.470147
$952$	−19.9069	−0.645187
$953$	−55.0983	−1.78481	−0.892404	−	0.451237i	$-0.850983\pi$
$953$	−55.0983	−1.78481	−0.892404	+	0.451237i	$0.850983\pi$
$954$	20.9519	0.678342
$955$	−14.5677	−0.471401
$956$	−87.0697	−2.81604
$957$	−33.4419	−1.08102
$958$	60.2124	1.94537
$959$	−24.0599	−0.776936
$960$	13.4142	0.432941
$961$	−27.3235	−0.881403
$962$	8.09142	0.260878
$963$	−22.8034	−0.734830
$964$	−80.1179	−2.58042
$965$	11.5105	0.370536
$966$	−70.6060	−2.27171
$967$	31.3151	1.00703	0.503513	−	0.863988i	$-0.332041\pi$
$967$	31.3151	1.00703	0.503513	+	0.863988i	$0.332041\pi$
$968$	−66.9557	−2.15204
$969$	−1.05569	−0.0339137
$970$	−15.0374	−0.482822
$971$	−14.0500	−0.450887	−0.225443	−	0.974256i	$-0.572383\pi$
$971$	−14.0500	−0.450887	−0.225443	+	0.974256i	$0.572383\pi$
$972$	−74.9839	−2.40511
$973$	−96.8195	−3.10389
$974$	−53.7034	−1.72077
$975$	−5.04622	−0.161608
$976$	−136.737	−4.37686
$977$	26.6604	0.852942	0.426471	−	0.904501i	$-0.359757\pi$
$977$	26.6604	0.852942	0.426471	+	0.904501i	$0.359757\pi$
$978$	−14.9152	−0.476937
$979$	15.6678	0.500745
$980$	−128.226	−4.09604
$981$	4.96633	0.158563
$982$	−80.5747	−2.57124
$983$	15.3811	0.490580	0.245290	−	0.969450i	$-0.421117\pi$
$983$	15.3811	0.490580	0.245290	+	0.969450i	$0.421117\pi$
$984$	43.2719	1.37946
$985$	−4.74696	−0.151251
$986$	−10.0917	−0.321387
$987$	−48.2196	−1.53485
$988$	16.4697	0.523971
$989$	−60.9115	−1.93687
$990$	34.6545	1.10139
$991$	39.5955	1.25779	0.628896	−	0.777489i	$-0.283507\pi$
$991$	39.5955	1.25779	0.628896	+	0.777489i	$0.283507\pi$
$992$	19.8009	0.628679
$993$	2.06873	0.0656491
$994$	−11.2682	−0.357406
$995$	34.8313	1.10423
$996$	−60.0477	−1.90268
$997$	50.3979	1.59612	0.798059	−	0.602579i	$-0.205860\pi$
$997$	50.3979	1.59612	0.798059	+	0.602579i	$0.205860\pi$
$998$	−29.9811	−0.949036
$999$	8.33326	0.263653

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.a.1.12	✓	279

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.a.1.12	✓	279	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:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61158 q^{2} -1.07466 q^{3} +4.82033 q^{4} -1.60450 q^{5} +2.80656 q^{6} -4.85582 q^{7} -7.36550 q^{8} -1.84510 q^{9} +O(q^{10})\)	\(q-2.61158 q^{2} -1.07466 q^{3} +4.82033 q^{4} -1.60450 q^{5} +2.80656 q^{6} -4.85582 q^{7} -7.36550 q^{8} -1.84510 q^{9} +4.19028 q^{10} -4.48224 q^{11} -5.18022 q^{12} -1.93590 q^{13} +12.6814 q^{14} +1.72430 q^{15} +9.59492 q^{16} -0.556594 q^{17} +4.81863 q^{18} -1.76493 q^{19} -7.73424 q^{20} +5.21836 q^{21} +11.7057 q^{22} +5.18089 q^{23} +7.91542 q^{24} -2.42557 q^{25} +5.05574 q^{26} +5.20684 q^{27} -23.4067 q^{28} -6.94264 q^{29} -4.50313 q^{30} -1.91742 q^{31} -10.3268 q^{32} +4.81688 q^{33} +1.45359 q^{34} +7.79119 q^{35} -8.89401 q^{36} +1.60044 q^{37} +4.60925 q^{38} +2.08043 q^{39} +11.8180 q^{40} +5.46678 q^{41} -13.6282 q^{42} -11.7569 q^{43} -21.6059 q^{44} +2.96048 q^{45} -13.5303 q^{46} -9.24037 q^{47} -10.3113 q^{48} +16.5790 q^{49} +6.33455 q^{50} +0.598150 q^{51} -9.33165 q^{52} +4.34810 q^{53} -13.5981 q^{54} +7.19177 q^{55} +35.7656 q^{56} +1.89670 q^{57} +18.1312 q^{58} -10.9087 q^{59} +8.31168 q^{60} -14.2510 q^{61} +5.00749 q^{62} +8.95950 q^{63} +7.77950 q^{64} +3.10615 q^{65} -12.5797 q^{66} +8.20498 q^{67} -2.68297 q^{68} -5.56770 q^{69} -20.3473 q^{70} -0.888566 q^{71} +13.5901 q^{72} +1.43034 q^{73} -4.17968 q^{74} +2.60666 q^{75} -8.50754 q^{76} +21.7650 q^{77} -5.43320 q^{78} +2.79264 q^{79} -15.3951 q^{80} -0.0602743 q^{81} -14.2769 q^{82} +11.5917 q^{83} +25.1542 q^{84} +0.893058 q^{85} +30.7042 q^{86} +7.46098 q^{87} +33.0139 q^{88} -3.49553 q^{89} -7.73151 q^{90} +9.40037 q^{91} +24.9736 q^{92} +2.06058 q^{93} +24.1319 q^{94} +2.83184 q^{95} +11.0978 q^{96} -3.58864 q^{97} -43.2974 q^{98} +8.27020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * q^99

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