LMFDB - Embedded newform 4019.2.a.a.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4019,2,Mod(1,4019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4019, 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("4019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4019.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:	$32.0918765724$
Analytic rank:	$1$
Dimension:	$149$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42490 q^{2} -1.02548 q^{3} +3.88013 q^{4} -4.00476 q^{5} +2.48668 q^{6} -1.16451 q^{7} -4.55913 q^{8} -1.94839 q^{9} +O(q^{10})$	\(q-2.42490 q^{2} -1.02548 q^{3} +3.88013 q^{4} -4.00476 q^{5} +2.48668 q^{6} -1.16451 q^{7} -4.55913 q^{8} -1.94839 q^{9} +9.71114 q^{10} -0.587401 q^{11} -3.97899 q^{12} -7.00038 q^{13} +2.82381 q^{14} +4.10680 q^{15} +3.29515 q^{16} -4.27143 q^{17} +4.72466 q^{18} -6.39874 q^{19} -15.5390 q^{20} +1.19417 q^{21} +1.42439 q^{22} +8.15099 q^{23} +4.67529 q^{24} +11.0381 q^{25} +16.9752 q^{26} +5.07447 q^{27} -4.51843 q^{28} +5.84896 q^{29} -9.95856 q^{30} -4.93446 q^{31} +1.12784 q^{32} +0.602367 q^{33} +10.3578 q^{34} +4.66357 q^{35} -7.56002 q^{36} +6.49133 q^{37} +15.5163 q^{38} +7.17874 q^{39} +18.2582 q^{40} -2.68506 q^{41} -2.89575 q^{42} +4.89271 q^{43} -2.27919 q^{44} +7.80285 q^{45} -19.7653 q^{46} +2.55136 q^{47} -3.37911 q^{48} -5.64393 q^{49} -26.7663 q^{50} +4.38025 q^{51} -27.1624 q^{52} +2.52380 q^{53} -12.3051 q^{54} +2.35240 q^{55} +5.30913 q^{56} +6.56177 q^{57} -14.1831 q^{58} -2.71565 q^{59} +15.9349 q^{60} +1.48362 q^{61} +11.9656 q^{62} +2.26892 q^{63} -9.32520 q^{64} +28.0349 q^{65} -1.46068 q^{66} -6.72613 q^{67} -16.5737 q^{68} -8.35866 q^{69} -11.3087 q^{70} +2.17985 q^{71} +8.88297 q^{72} -13.7904 q^{73} -15.7408 q^{74} -11.3193 q^{75} -24.8280 q^{76} +0.684032 q^{77} -17.4077 q^{78} +13.6107 q^{79} -13.1963 q^{80} +0.641422 q^{81} +6.51100 q^{82} +1.35764 q^{83} +4.63355 q^{84} +17.1060 q^{85} -11.8643 q^{86} -5.99799 q^{87} +2.67804 q^{88} +9.53176 q^{89} -18.9211 q^{90} +8.15198 q^{91} +31.6269 q^{92} +5.06018 q^{93} -6.18679 q^{94} +25.6254 q^{95} -1.15657 q^{96} +12.1439 q^{97} +13.6859 q^{98} +1.14449 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 149 q - 8 q^{2} - 12 q^{3} + 124 q^{4} - 36 q^{5} - 45 q^{6} - 32 q^{7} - 21 q^{8} + 115 q^{9}+O(q^{10}) $ 149 * q - 8 * q^2 - 12 * q^3 + 124 * q^4 - 36 * q^5 - 45 * q^6 - 32 * q^7 - 21 * q^8 + 115 * q^9	\( 149 q - 8 q^{2} - 12 q^{3} + 124 q^{4} - 36 q^{5} - 45 q^{6} - 32 q^{7} - 21 q^{8} + 115 q^{9} - 58 q^{10} - 33 q^{11} - 33 q^{12} - 107 q^{13} - 28 q^{14} - 24 q^{15} + 74 q^{16} - 39 q^{17} - 33 q^{18} - 93 q^{19} - 63 q^{20} - 113 q^{21} - 38 q^{22} - 11 q^{23} - 130 q^{24} + 85 q^{25} - 33 q^{26} - 30 q^{27} - 94 q^{28} - 85 q^{29} - 16 q^{30} - 129 q^{31} - 35 q^{32} - 64 q^{33} - 78 q^{34} - 27 q^{35} + 79 q^{36} - 135 q^{37} - 11 q^{38} - 73 q^{39} - 146 q^{40} - 101 q^{41} + 4 q^{42} - 55 q^{43} - 82 q^{44} - 168 q^{45} - 113 q^{46} - 40 q^{47} - 65 q^{48} + 27 q^{49} - 5 q^{50} - 49 q^{51} - 177 q^{52} - 32 q^{53} - 155 q^{54} - 128 q^{55} - 44 q^{56} - 47 q^{57} - 46 q^{58} - 53 q^{59} - 11 q^{60} - 347 q^{61} - 11 q^{62} - 73 q^{63} + q^{64} - 31 q^{65} - 37 q^{66} - 40 q^{67} - 80 q^{68} - 175 q^{69} - 61 q^{70} - 31 q^{71} - 68 q^{72} - 193 q^{73} - 33 q^{74} - 56 q^{75} - 248 q^{76} - 84 q^{77} + 40 q^{78} - 111 q^{79} - 54 q^{80} + 49 q^{81} - 74 q^{82} - 24 q^{83} - 159 q^{84} - 258 q^{85} - q^{86} - 66 q^{87} - 97 q^{88} - 76 q^{89} - 75 q^{90} - 134 q^{91} + 31 q^{92} - 97 q^{93} - 111 q^{94} - 14 q^{95} - 216 q^{96} - 140 q^{97} - 13 q^{98} - 116 q^{99}+O(q^{100}) \) 149 * q - 8 * q^2 - 12 * q^3 + 124 * q^4 - 36 * q^5 - 45 * q^6 - 32 * q^7 - 21 * q^8 + 115 * q^9 - 58 * q^10 - 33 * q^11 - 33 * q^12 - 107 * q^13 - 28 * q^14 - 24 * q^15 + 74 * q^16 - 39 * q^17 - 33 * q^18 - 93 * q^19 - 63 * q^20 - 113 * q^21 - 38 * q^22 - 11 * q^23 - 130 * q^24 + 85 * q^25 - 33 * q^26 - 30 * q^27 - 94 * q^28 - 85 * q^29 - 16 * q^30 - 129 * q^31 - 35 * q^32 - 64 * q^33 - 78 * q^34 - 27 * q^35 + 79 * q^36 - 135 * q^37 - 11 * q^38 - 73 * q^39 - 146 * q^40 - 101 * q^41 + 4 * q^42 - 55 * q^43 - 82 * q^44 - 168 * q^45 - 113 * q^46 - 40 * q^47 - 65 * q^48 + 27 * q^49 - 5 * q^50 - 49 * q^51 - 177 * q^52 - 32 * q^53 - 155 * q^54 - 128 * q^55 - 44 * q^56 - 47 * q^57 - 46 * q^58 - 53 * q^59 - 11 * q^60 - 347 * q^61 - 11 * q^62 - 73 * q^63 + q^64 - 31 * q^65 - 37 * q^66 - 40 * q^67 - 80 * q^68 - 175 * q^69 - 61 * q^70 - 31 * q^71 - 68 * q^72 - 193 * q^73 - 33 * q^74 - 56 * q^75 - 248 * q^76 - 84 * q^77 + 40 * q^78 - 111 * q^79 - 54 * q^80 + 49 * q^81 - 74 * q^82 - 24 * q^83 - 159 * q^84 - 258 * q^85 - q^86 - 66 * q^87 - 97 * q^88 - 76 * q^89 - 75 * q^90 - 134 * q^91 + 31 * q^92 - 97 * q^93 - 111 * q^94 - 14 * q^95 - 216 * q^96 - 140 * q^97 - 13 * q^98 - 116 * 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 $

$2$	−2.42490	−1.71466	−0.857331	−	0.514766i	$-0.827879\pi$
$2$	−2.42490	−1.71466	−0.857331	+	0.514766i	$0.827879\pi$
$3$	−1.02548	−0.592060	−0.296030	−	0.955179i	$-0.595663\pi$
$3$	−1.02548	−0.592060	−0.296030	+	0.955179i	$0.595663\pi$
$4$	3.88013	1.94007
$5$	−4.00476	−1.79098	−0.895492	−	0.445078i	$-0.853176\pi$
$5$	−4.00476	−1.79098	−0.895492	+	0.445078i	$0.853176\pi$
$6$	2.48668	1.01518
$7$	−1.16451	−0.440142	−0.220071	−	0.975484i	$-0.570629\pi$
$7$	−1.16451	−0.440142	−0.220071	+	0.975484i	$0.570629\pi$
$8$	−4.55913	−1.61189
$9$	−1.94839	−0.649465
$10$	9.71114	3.07093
$11$	−0.587401	−0.177108	−0.0885541	−	0.996071i	$-0.528225\pi$
$11$	−0.587401	−0.177108	−0.0885541	+	0.996071i	$0.528225\pi$
$12$	−3.97899	−1.14864
$13$	−7.00038	−1.94156	−0.970779	−	0.239977i	$-0.922860\pi$
$13$	−7.00038	−1.94156	−0.970779	+	0.239977i	$0.922860\pi$
$14$	2.82381	0.754694
$15$	4.10680	1.06037
$16$	3.29515	0.823789
$17$	−4.27143	−1.03597	−0.517986	−	0.855389i	$-0.673318\pi$
$17$	−4.27143	−1.03597	−0.517986	+	0.855389i	$0.673318\pi$
$18$	4.72466	1.11361
$19$	−6.39874	−1.46797	−0.733986	−	0.679165i	$-0.762342\pi$
$19$	−6.39874	−1.46797	−0.733986	+	0.679165i	$0.762342\pi$
$20$	−15.5390	−3.47463
$21$	1.19417	0.260590
$22$	1.42439	0.303681
$23$	8.15099	1.69960	0.849799	−	0.527106i	$-0.176723\pi$
$23$	8.15099	1.69960	0.849799	+	0.527106i	$0.176723\pi$
$24$	4.67529	0.954339
$25$	11.0381	2.20762
$26$	16.9752	3.32911
$27$	5.07447	0.976582
$28$	−4.51843	−0.853904
$29$	5.84896	1.08613	0.543063	−	0.839692i	$-0.317265\pi$
$29$	5.84896	1.08613	0.543063	+	0.839692i	$0.317265\pi$
$30$	−9.95856	−1.81818
$31$	−4.93446	−0.886255	−0.443128	−	0.896458i	$-0.646131\pi$
$31$	−4.93446	−0.886255	−0.443128	+	0.896458i	$0.646131\pi$
$32$	1.12784	0.199375
$33$	0.602367	0.104859
$34$	10.3578	1.77634
$35$	4.66357	0.788286
$36$	−7.56002	−1.26000
$37$	6.49133	1.06717	0.533584	−	0.845747i	$-0.320845\pi$
$37$	6.49133	1.06717	0.533584	+	0.845747i	$0.320845\pi$
$38$	15.5163	2.51708
$39$	7.17874	1.14952
$40$	18.2582	2.88688
$41$	−2.68506	−0.419336	−0.209668	−	0.977773i	$-0.567238\pi$
$41$	−2.68506	−0.419336	−0.209668	+	0.977773i	$0.567238\pi$
$42$	−2.89575	−0.446824
$43$	4.89271	0.746131	0.373066	−	0.927805i	$-0.378307\pi$
$43$	4.89271	0.746131	0.373066	+	0.927805i	$0.378307\pi$
$44$	−2.27919	−0.343602
$45$	7.80285	1.16318
$46$	−19.7653	−2.91424
$47$	2.55136	0.372154	0.186077	−	0.982535i	$-0.440423\pi$
$47$	2.55136	0.372154	0.186077	+	0.982535i	$0.440423\pi$
$48$	−3.37911	−0.487733
$49$	−5.64393	−0.806275
$50$	−26.7663	−3.78533
$51$	4.38025	0.613358
$52$	−27.1624	−3.76675
$53$	2.52380	0.346671	0.173335	−	0.984863i	$-0.444546\pi$
$53$	2.52380	0.346671	0.173335	+	0.984863i	$0.444546\pi$
$54$	−12.3051	−1.67451
$55$	2.35240	0.317198
$56$	5.30913	0.709462
$57$	6.56177	0.869128
$58$	−14.1831	−1.86234
$59$	−2.71565	−0.353547	−0.176774	−	0.984252i	$-0.556566\pi$
$59$	−2.71565	−0.353547	−0.176774	+	0.984252i	$0.556566\pi$
$60$	15.9349	2.05719
$61$	1.48362	0.189958	0.0949791	−	0.995479i	$-0.469722\pi$
$61$	1.48362	0.189958	0.0949791	+	0.995479i	$0.469722\pi$
$62$	11.9656	1.51963
$63$	2.26892	0.285856
$64$	−9.32520	−1.16565
$65$	28.0349	3.47730
$66$	−1.46068	−0.179797
$67$	−6.72613	−0.821727	−0.410864	−	0.911697i	$-0.634773\pi$
$67$	−6.72613	−0.821727	−0.410864	+	0.911697i	$0.634773\pi$
$68$	−16.5737	−2.00986
$69$	−8.35866	−1.00626
$70$	−11.3087	−1.35164
$71$	2.17985	0.258700	0.129350	−	0.991599i	$-0.458711\pi$
$71$	2.17985	0.258700	0.129350	+	0.991599i	$0.458711\pi$
$72$	8.88297	1.04687
$73$	−13.7904	−1.61404	−0.807020	−	0.590524i	$-0.798921\pi$
$73$	−13.7904	−1.61404	−0.807020	+	0.590524i	$0.798921\pi$
$74$	−15.7408	−1.82983
$75$	−11.3193	−1.30705
$76$	−24.8280	−2.84796
$77$	0.684032	0.0779527
$78$	−17.4077	−1.97104
$79$	13.6107	1.53133	0.765664	−	0.643241i	$-0.222411\pi$
$79$	13.6107	1.53133	0.765664	+	0.643241i	$0.222411\pi$
$80$	−13.1963	−1.47539
$81$	0.641422	0.0712691
$82$	6.51100	0.719020
$83$	1.35764	0.149020	0.0745099	−	0.997220i	$-0.476261\pi$
$83$	1.35764	0.149020	0.0745099	+	0.997220i	$0.476261\pi$
$84$	4.63355	0.505562
$85$	17.1060	1.85541
$86$	−11.8643	−1.27936
$87$	−5.99799	−0.643052
$88$	2.67804	0.285480
$89$	9.53176	1.01036	0.505182	−	0.863013i	$-0.331425\pi$
$89$	9.53176	1.01036	0.505182	+	0.863013i	$0.331425\pi$
$90$	−18.9211	−1.99446
$91$	8.15198	0.854560
$92$	31.6269	3.29733
$93$	5.06018	0.524717
$94$	−6.18679	−0.638119
$95$	25.6254	2.62911
$96$	−1.15657	−0.118042
$97$	12.1439	1.23303	0.616515	−	0.787343i	$-0.288544\pi$
$97$	12.1439	1.23303	0.616515	+	0.787343i	$0.288544\pi$
$98$	13.6859	1.38249
$99$	1.14449	0.115026
$100$	42.8293	4.28293
$101$	12.6728	1.26099	0.630494	−	0.776194i	$-0.282852\pi$
$101$	12.6728	1.26099	0.630494	+	0.776194i	$0.282852\pi$
$102$	−10.6217	−1.05170
$103$	7.39377	0.728530	0.364265	−	0.931295i	$-0.381320\pi$
$103$	7.39377	0.728530	0.364265	+	0.931295i	$0.381320\pi$
$104$	31.9156	3.12959
$105$	−4.78239	−0.466713
$106$	−6.11996	−0.594424
$107$	−19.5311	−1.88815	−0.944074	−	0.329735i	$-0.893041\pi$
$107$	−19.5311	−1.88815	−0.944074	+	0.329735i	$0.893041\pi$
$108$	19.6896	1.89463
$109$	−4.23266	−0.405416	−0.202708	−	0.979239i	$-0.564974\pi$
$109$	−4.23266	−0.405416	−0.202708	+	0.979239i	$0.564974\pi$
$110$	−5.70434	−0.543887
$111$	−6.65672	−0.631828
$112$	−3.83722	−0.362584
$113$	−5.19796	−0.488983	−0.244492	−	0.969651i	$-0.578621\pi$
$113$	−5.19796	−0.488983	−0.244492	+	0.969651i	$0.578621\pi$
$114$	−15.9116	−1.49026
$115$	−32.6428	−3.04395
$116$	22.6947	2.10715
$117$	13.6395	1.26097
$118$	6.58517	0.606214
$119$	4.97410	0.455975
$120$	−18.7234	−1.70920
$121$	−10.6550	−0.968633
$122$	−3.59763	−0.325714
$123$	2.75347	0.248272
$124$	−19.1464	−1.71939
$125$	−24.1812	−2.16283
$126$	−5.50189	−0.490147
$127$	−2.71425	−0.240851	−0.120426	−	0.992722i	$-0.538426\pi$
$127$	−2.71425	−0.240851	−0.120426	+	0.992722i	$0.538426\pi$
$128$	20.3570	1.79932
$129$	−5.01737	−0.441755
$130$	−67.9817	−5.96239
$131$	10.1609	0.887761	0.443880	−	0.896086i	$-0.353601\pi$
$131$	10.1609	0.887761	0.443880	+	0.896086i	$0.353601\pi$
$132$	2.33726	0.203433
$133$	7.45137	0.646115
$134$	16.3102	1.40898
$135$	−20.3220	−1.74904
$136$	19.4740	1.66988
$137$	12.9098	1.10296	0.551480	−	0.834188i	$-0.314063\pi$
$137$	12.9098	1.10296	0.551480	+	0.834188i	$0.314063\pi$
$138$	20.2689	1.72540
$139$	−4.53415	−0.384581	−0.192291	−	0.981338i	$-0.561592\pi$
$139$	−4.53415	−0.384581	−0.192291	+	0.981338i	$0.561592\pi$
$140$	18.0952	1.52933
$141$	−2.61637	−0.220338
$142$	−5.28591	−0.443584
$143$	4.11204	0.343866
$144$	−6.42026	−0.535022
$145$	−23.4237	−1.94523
$146$	33.4402	2.76753
$147$	5.78773	0.477364
$148$	25.1872	2.07038
$149$	−16.8382	−1.37944	−0.689718	−	0.724078i	$-0.742265\pi$
$149$	−16.8382	−1.37944	−0.689718	+	0.724078i	$0.742265\pi$
$150$	27.4483	2.24114
$151$	−20.9919	−1.70829	−0.854147	−	0.520031i	$-0.825920\pi$
$151$	−20.9919	−1.70829	−0.854147	+	0.520031i	$0.825920\pi$
$152$	29.1727	2.36622
$153$	8.32242	0.672828
$154$	−1.65871	−0.133663
$155$	19.7613	1.58727
$156$	27.8545	2.23014
$157$	8.76717	0.699696	0.349848	−	0.936806i	$-0.386233\pi$
$157$	8.76717	0.699696	0.349848	+	0.936806i	$0.386233\pi$
$158$	−33.0046	−2.62571
$159$	−2.58811	−0.205250
$160$	−4.51672	−0.357078
$161$	−9.49187	−0.748064
$162$	−1.55538	−0.122202
$163$	12.5382	0.982066	0.491033	−	0.871141i	$-0.336619\pi$
$163$	12.5382	0.982066	0.491033	+	0.871141i	$0.336619\pi$
$164$	−10.4184	−0.813540
$165$	−2.41234	−0.187800
$166$	−3.29213	−0.255519
$167$	22.2888	1.72476	0.862381	−	0.506261i	$-0.168973\pi$
$167$	22.2888	1.72476	0.862381	+	0.506261i	$0.168973\pi$
$168$	−5.44439	−0.420044
$169$	36.0054	2.76964
$170$	−41.4804	−3.18140
$171$	12.4673	0.953396
$172$	18.9844	1.44754
$173$	10.6342	0.808500	0.404250	−	0.914648i	$-0.367533\pi$
$173$	10.6342	0.808500	0.404250	+	0.914648i	$0.367533\pi$
$174$	14.5445	1.10262
$175$	−12.8539	−0.971666
$176$	−1.93558	−0.145900
$177$	2.78484	0.209321
$178$	−23.1136	−1.73243
$179$	−16.2895	−1.21753	−0.608767	−	0.793349i	$-0.708336\pi$
$179$	−16.2895	−1.21753	−0.608767	+	0.793349i	$0.708336\pi$
$180$	30.2761	2.25665
$181$	−9.06314	−0.673658	−0.336829	−	0.941566i	$-0.609354\pi$
$181$	−9.06314	−0.673658	−0.336829	+	0.941566i	$0.609354\pi$
$182$	−19.7677	−1.46528
$183$	−1.52142	−0.112467
$184$	−37.1614	−2.73957
$185$	−25.9962	−1.91128
$186$	−12.2704	−0.899712
$187$	2.50904	0.183479
$188$	9.89962	0.722004
$189$	−5.90925	−0.429835
$190$	−62.1391	−4.50804
$191$	−2.19134	−0.158560	−0.0792799	−	0.996852i	$-0.525262\pi$
$191$	−2.19134	−0.158560	−0.0792799	+	0.996852i	$0.525262\pi$
$192$	9.56279	0.690135
$193$	15.9172	1.14575	0.572873	−	0.819644i	$-0.305829\pi$
$193$	15.9172	1.14575	0.572873	+	0.819644i	$0.305829\pi$
$194$	−29.4478	−2.11423
$195$	−28.7492	−2.05877
$196$	−21.8992	−1.56423
$197$	18.4831	1.31687	0.658434	−	0.752639i	$-0.271219\pi$
$197$	18.4831	1.31687	0.658434	+	0.752639i	$0.271219\pi$
$198$	−2.77527	−0.197230
$199$	−4.37321	−0.310009	−0.155004	−	0.987914i	$-0.549539\pi$
$199$	−4.37321	−0.310009	−0.155004	+	0.987914i	$0.549539\pi$
$200$	−50.3241	−3.55845
$201$	6.89750	0.486512
$202$	−30.7302	−2.16217
$203$	−6.81115	−0.478049
$204$	16.9960	1.18996
$205$	10.7530	0.751025
$206$	−17.9291	−1.24918
$207$	−15.8813	−1.10383
$208$	−23.0674	−1.59943
$209$	3.75863	0.259990
$210$	11.5968	0.800255
$211$	13.2357	0.911180	0.455590	−	0.890190i	$-0.349428\pi$
$211$	13.2357	0.911180	0.455590	+	0.890190i	$0.349428\pi$
$212$	9.79269	0.672564
$213$	−2.23539	−0.153166
$214$	47.3610	3.23753
$215$	−19.5941	−1.33631
$216$	−23.1352	−1.57415
$217$	5.74621	0.390078
$218$	10.2638	0.695151
$219$	14.1417	0.955609
$220$	9.12763	0.615385
$221$	29.9016	2.01140
$222$	16.1419	1.08337
$223$	−8.84357	−0.592209	−0.296105	−	0.955155i	$-0.595688\pi$
$223$	−8.84357	−0.592209	−0.296105	+	0.955155i	$0.595688\pi$
$224$	−1.31337	−0.0877534
$225$	−21.5066	−1.43377
$226$	12.6045	0.838441
$227$	−22.6153	−1.50103	−0.750516	−	0.660852i	$-0.770195\pi$
$227$	−22.6153	−1.50103	−0.750516	+	0.660852i	$0.770195\pi$
$228$	25.4605	1.68616
$229$	−15.4861	−1.02335	−0.511676	−	0.859179i	$-0.670975\pi$
$229$	−15.4861	−1.02335	−0.511676	+	0.859179i	$0.670975\pi$
$230$	79.1554	5.21935
$231$	−0.701460	−0.0461527
$232$	−26.6662	−1.75072
$233$	20.2211	1.32473	0.662364	−	0.749182i	$-0.269553\pi$
$233$	20.2211	1.32473	0.662364	+	0.749182i	$0.269553\pi$
$234$	−33.0744	−2.16214
$235$	−10.2176	−0.666522
$236$	−10.5371	−0.685905
$237$	−13.9575	−0.906638
$238$	−12.0617	−0.781843
$239$	−10.5402	−0.681791	−0.340895	−	0.940101i	$-0.610730\pi$
$239$	−10.5402	−0.681791	−0.340895	+	0.940101i	$0.610730\pi$
$240$	13.5325	0.873521
$241$	−11.5336	−0.742947	−0.371474	−	0.928443i	$-0.621147\pi$
$241$	−11.5336	−0.742947	−0.371474	+	0.928443i	$0.621147\pi$
$242$	25.8372	1.66088
$243$	−15.8812	−1.01878
$244$	5.75664	0.368531
$245$	22.6026	1.44403
$246$	−6.67689	−0.425703
$247$	44.7936	2.85015
$248$	22.4968	1.42855
$249$	−1.39223	−0.0882288
$250$	58.6369	3.70853
$251$	12.0298	0.759315	0.379657	−	0.925127i	$-0.376042\pi$
$251$	12.0298	0.759315	0.379657	+	0.925127i	$0.376042\pi$
$252$	8.80369	0.554580
$253$	−4.78790	−0.301013
$254$	6.58179	0.412978
$255$	−17.5419	−1.09851
$256$	−30.7132	−1.91958
$257$	11.4260	0.712732	0.356366	−	0.934346i	$-0.384016\pi$
$257$	11.4260	0.712732	0.356366	+	0.934346i	$0.384016\pi$
$258$	12.1666	0.757460
$259$	−7.55919	−0.469705
$260$	108.779	6.74618
$261$	−11.3961	−0.705400
$262$	−24.6391	−1.52221
$263$	2.16940	0.133771	0.0668855	−	0.997761i	$-0.478694\pi$
$263$	2.16940	0.133771	0.0668855	+	0.997761i	$0.478694\pi$
$264$	−2.74627	−0.169021
$265$	−10.1072	−0.620882
$266$	−18.0688	−1.10787
$267$	−9.77462	−0.598197
$268$	−26.0983	−1.59420
$269$	16.3744	0.998367	0.499183	−	0.866496i	$-0.333633\pi$
$269$	16.3744	0.998367	0.499183	+	0.866496i	$0.333633\pi$
$270$	49.2789	2.99902
$271$	23.8366	1.44797	0.723985	−	0.689816i	$-0.242308\pi$
$271$	23.8366	1.44797	0.723985	+	0.689816i	$0.242308\pi$
$272$	−14.0750	−0.853423
$273$	−8.35968	−0.505951
$274$	−31.3050	−1.89120
$275$	−6.48380	−0.390988
$276$	−32.4327	−1.95222
$277$	−22.4395	−1.34826	−0.674129	−	0.738614i	$-0.735481\pi$
$277$	−22.4395	−1.34826	−0.674129	+	0.738614i	$0.735481\pi$
$278$	10.9948	0.659427
$279$	9.61428	0.575592
$280$	−21.2618	−1.27063
$281$	−18.3431	−1.09426	−0.547128	−	0.837049i	$-0.684279\pi$
$281$	−18.3431	−1.09426	−0.547128	+	0.837049i	$0.684279\pi$
$282$	6.34442	0.377805
$283$	−13.3514	−0.793658	−0.396829	−	0.917893i	$-0.629889\pi$
$283$	−13.3514	−0.793658	−0.396829	+	0.917893i	$0.629889\pi$
$284$	8.45810	0.501896
$285$	−26.2783	−1.55659
$286$	−9.97127	−0.589613
$287$	3.12677	0.184567
$288$	−2.19747	−0.129487
$289$	1.24508	0.0732398
$290$	56.8001	3.33542
$291$	−12.4533	−0.730027
$292$	−53.5084	−3.13134
$293$	12.6268	0.737666	0.368833	−	0.929496i	$-0.379757\pi$
$293$	12.6268	0.737666	0.368833	+	0.929496i	$0.379757\pi$
$294$	−14.0346	−0.818517
$295$	10.8755	0.633197
$296$	−29.5948	−1.72016
$297$	−2.98075	−0.172961
$298$	40.8308	2.36527
$299$	−57.0601	−3.29987
$300$	−43.9205	−2.53575
$301$	−5.69759	−0.328403
$302$	50.9032	2.92915
$303$	−12.9957	−0.746581
$304$	−21.0848	−1.20930
$305$	−5.94155	−0.340212
$306$	−20.1810	−1.15367
$307$	−15.5753	−0.888931	−0.444465	−	0.895796i	$-0.646606\pi$
$307$	−15.5753	−0.888931	−0.444465	+	0.895796i	$0.646606\pi$
$308$	2.65413	0.151233
$309$	−7.58216	−0.431334
$310$	−47.9192	−2.72163
$311$	−2.09053	−0.118543	−0.0592716	−	0.998242i	$-0.518878\pi$
$311$	−2.09053	−0.118543	−0.0592716	+	0.998242i	$0.518878\pi$
$312$	−32.7288	−1.85290
$313$	−0.248548	−0.0140488	−0.00702439	−	0.999975i	$-0.502236\pi$
$313$	−0.248548	−0.0140488	−0.00702439	+	0.999975i	$0.502236\pi$
$314$	−21.2595	−1.19974
$315$	−9.08646	−0.511964
$316$	52.8114	2.97088
$317$	21.4679	1.20576	0.602879	−	0.797833i	$-0.294020\pi$
$317$	21.4679	1.20576	0.602879	+	0.797833i	$0.294020\pi$
$318$	6.27589	0.351935
$319$	−3.43569	−0.192362
$320$	37.3452	2.08766
$321$	20.0288	1.11790
$322$	23.0168	1.28268
$323$	27.3317	1.52078
$324$	2.48880	0.138267
$325$	−77.2710	−4.28623
$326$	−30.4038	−1.68391
$327$	4.34051	0.240030
$328$	12.2415	0.675926
$329$	−2.97107	−0.163801
$330$	5.84967	0.322014
$331$	−5.84423	−0.321228	−0.160614	−	0.987017i	$-0.551347\pi$
$331$	−5.84423	−0.321228	−0.160614	+	0.987017i	$0.551347\pi$
$332$	5.26781	0.289108
$333$	−12.6477	−0.693088
$334$	−54.0481	−2.95738
$335$	26.9365	1.47170
$336$	3.93499	0.214671
$337$	4.56268	0.248545	0.124273	−	0.992248i	$-0.460340\pi$
$337$	4.56268	0.248545	0.124273	+	0.992248i	$0.460340\pi$
$338$	−87.3094	−4.74900
$339$	5.33040	0.289508
$340$	66.3737	3.59962
$341$	2.89851	0.156963
$342$	−30.2319	−1.63475
$343$	14.7239	0.795017
$344$	−22.3065	−1.20268
$345$	33.4745	1.80220
$346$	−25.7868	−1.38630
$347$	−14.4727	−0.776936	−0.388468	−	0.921462i	$-0.626996\pi$
$347$	−14.4727	−0.776936	−0.388468	+	0.921462i	$0.626996\pi$
$348$	−23.2730	−1.24756
$349$	0.896493	0.0479882	0.0239941	−	0.999712i	$-0.492362\pi$
$349$	0.896493	0.0479882	0.0239941	+	0.999712i	$0.492362\pi$
$350$	31.1695	1.66608
$351$	−35.5232	−1.89609
$352$	−0.662493	−0.0353110
$353$	24.7662	1.31817	0.659087	−	0.752067i	$-0.270943\pi$
$353$	24.7662	1.31817	0.659087	+	0.752067i	$0.270943\pi$
$354$	−6.75295	−0.358915
$355$	−8.72977	−0.463328
$356$	36.9845	1.96017
$357$	−5.10083	−0.269965
$358$	39.5004	2.08766
$359$	−28.2940	−1.49330	−0.746649	−	0.665218i	$-0.768338\pi$
$359$	−28.2940	−1.49330	−0.746649	+	0.665218i	$0.768338\pi$
$360$	−35.5742	−1.87492
$361$	21.9439	1.15494
$362$	21.9772	1.15510
$363$	10.9264	0.573489
$364$	31.6308	1.65790
$365$	55.2271	2.89072
$366$	3.68929	0.192842
$367$	29.1829	1.52334	0.761668	−	0.647968i	$-0.224381\pi$
$367$	29.1829	1.52334	0.761668	+	0.647968i	$0.224381\pi$
$368$	26.8588	1.40011
$369$	5.23156	0.272344
$370$	63.0382	3.27720
$371$	−2.93898	−0.152584
$372$	19.6342	1.01798
$373$	30.4937	1.57890	0.789451	−	0.613814i	$-0.210365\pi$
$373$	30.4937	1.57890	0.789451	+	0.613814i	$0.210365\pi$
$374$	−6.08417	−0.314605
$375$	24.7973	1.28053
$376$	−11.6320	−0.599874
$377$	−40.9450	−2.10877
$378$	14.3293	0.737021
$379$	14.0825	0.723370	0.361685	−	0.932300i	$-0.382202\pi$
$379$	14.0825	0.723370	0.361685	+	0.932300i	$0.382202\pi$
$380$	99.4300	5.10065
$381$	2.78341	0.142598
$382$	5.31377	0.271876
$383$	−23.2726	−1.18917	−0.594586	−	0.804032i	$-0.702684\pi$
$383$	−23.2726	−1.18917	−0.594586	+	0.804032i	$0.702684\pi$
$384$	−20.8757	−1.06531
$385$	−2.73938	−0.139612
$386$	−38.5976	−1.96457
$387$	−9.53293	−0.484586
$388$	47.1200	2.39216
$389$	−11.1958	−0.567650	−0.283825	−	0.958876i	$-0.591603\pi$
$389$	−11.1958	−0.567650	−0.283825	+	0.958876i	$0.591603\pi$
$390$	69.7138	3.53009
$391$	−34.8163	−1.76074
$392$	25.7314	1.29963
$393$	−10.4198	−0.525608
$394$	−44.8197	−2.25798
$395$	−54.5077	−2.74258
$396$	4.44077	0.223157
$397$	−27.5576	−1.38308	−0.691539	−	0.722339i	$-0.743067\pi$
$397$	−27.5576	−1.38308	−0.691539	+	0.722339i	$0.743067\pi$
$398$	10.6046	0.531561
$399$	−7.64122	−0.382539
$400$	36.3723	1.81861
$401$	16.8766	0.842778	0.421389	−	0.906880i	$-0.361543\pi$
$401$	16.8766	0.842778	0.421389	+	0.906880i	$0.361543\pi$
$402$	−16.7257	−0.834204
$403$	34.5431	1.72072
$404$	49.1721	2.44640
$405$	−2.56874	−0.127642
$406$	16.5163	0.819692
$407$	−3.81302	−0.189004
$408$	−19.9701	−0.988669
$409$	−26.4055	−1.30567	−0.652835	−	0.757500i	$-0.726421\pi$
$409$	−26.4055	−1.30567	−0.652835	+	0.757500i	$0.726421\pi$
$410$	−26.0750	−1.28775
$411$	−13.2387	−0.653019
$412$	28.6888	1.41340
$413$	3.16239	0.155611
$414$	38.5106	1.89269
$415$	−5.43701	−0.266892
$416$	−7.89530	−0.387099
$417$	4.64967	0.227695
$418$	−9.11429	−0.445795
$419$	19.4998	0.952628	0.476314	−	0.879275i	$-0.341973\pi$
$419$	19.4998	0.952628	0.476314	+	0.879275i	$0.341973\pi$
$420$	−18.5563	−0.905454
$421$	−27.0270	−1.31722	−0.658608	−	0.752486i	$-0.728854\pi$
$421$	−27.0270	−1.31722	−0.658608	+	0.752486i	$0.728854\pi$
$422$	−32.0951	−1.56237
$423$	−4.97106	−0.241701
$424$	−11.5063	−0.558797
$425$	−47.1485	−2.28704
$426$	5.42059	0.262628
$427$	−1.72768	−0.0836085
$428$	−75.7834	−3.66313
$429$	−4.21680	−0.203589
$430$	47.5138	2.29132
$431$	5.77798	0.278315	0.139158	−	0.990270i	$-0.455561\pi$
$431$	5.77798	0.278315	0.139158	+	0.990270i	$0.455561\pi$
$432$	16.7212	0.804498
$433$	25.8541	1.24247	0.621235	−	0.783624i	$-0.286631\pi$
$433$	25.8541	1.24247	0.621235	+	0.783624i	$0.286631\pi$
$434$	−13.9340	−0.668852
$435$	24.0205	1.15169
$436$	−16.4233	−0.786533
$437$	−52.1561	−2.49496
$438$	−34.2922	−1.63855
$439$	−17.4329	−0.832026	−0.416013	−	0.909359i	$-0.636573\pi$
$439$	−17.4329	−0.832026	−0.416013	+	0.909359i	$0.636573\pi$
$440$	−10.7249	−0.511290
$441$	10.9966	0.523647
$442$	−72.5084	−3.44887
$443$	−19.1704	−0.910811	−0.455406	−	0.890284i	$-0.650506\pi$
$443$	−19.1704	−0.910811	−0.455406	+	0.890284i	$0.650506\pi$
$444$	−25.8289	−1.22579
$445$	−38.1724	−1.80955
$446$	21.4448	1.01544
$447$	17.2672	0.816709
$448$	10.8592	0.513051
$449$	30.9932	1.46266	0.731330	−	0.682023i	$-0.238900\pi$
$449$	30.9932	1.46266	0.731330	+	0.682023i	$0.238900\pi$
$450$	52.1513	2.45844
$451$	1.57721	0.0742679
$452$	−20.1688	−0.948660
$453$	21.5267	1.01141
$454$	54.8399	2.57376
$455$	−32.6467	−1.53050
$456$	−29.9159	−1.40094
$457$	−30.5107	−1.42723	−0.713615	−	0.700538i	$-0.752943\pi$
$457$	−30.5107	−1.42723	−0.713615	+	0.700538i	$0.752943\pi$
$458$	37.5522	1.75470
$459$	−21.6752	−1.01171
$460$	−126.658	−5.90547
$461$	0.548874	0.0255636	0.0127818	−	0.999918i	$-0.495931\pi$
$461$	0.548874	0.0255636	0.0127818	+	0.999918i	$0.495931\pi$
$462$	1.70097	0.0791362
$463$	−1.23838	−0.0575525	−0.0287763	−	0.999586i	$-0.509161\pi$
$463$	−1.23838	−0.0575525	−0.0287763	+	0.999586i	$0.509161\pi$
$464$	19.2732	0.894738
$465$	−20.2648	−0.939759
$466$	−49.0341	−2.27146
$467$	−1.13259	−0.0524101	−0.0262051	−	0.999657i	$-0.508342\pi$
$467$	−1.13259	−0.0524101	−0.0262051	+	0.999657i	$0.508342\pi$
$468$	52.9231	2.44637
$469$	7.83261	0.361676
$470$	24.7766	1.14286
$471$	−8.99054	−0.414262
$472$	12.3810	0.569881
$473$	−2.87398	−0.132146
$474$	33.8456	1.55458
$475$	−70.6300	−3.24073
$476$	19.3001	0.884621
$477$	−4.91736	−0.225151
$478$	25.5590	1.16904
$479$	−25.4461	−1.16266	−0.581330	−	0.813668i	$-0.697468\pi$
$479$	−25.4461	−1.16266	−0.581330	+	0.813668i	$0.697468\pi$
$480$	4.63180	0.211412
$481$	−45.4418	−2.07197
$482$	27.9679	1.27390
$483$	9.73371	0.442899
$484$	−41.3426	−1.87921
$485$	−48.6335	−2.20834
$486$	38.5102	1.74686
$487$	−2.08707	−0.0945741	−0.0472870	−	0.998881i	$-0.515058\pi$
$487$	−2.08707	−0.0945741	−0.0472870	+	0.998881i	$0.515058\pi$
$488$	−6.76401	−0.306193
$489$	−12.8576	−0.581442
$490$	−54.8090	−2.47602
$491$	−2.80254	−0.126477	−0.0632385	−	0.997998i	$-0.520143\pi$
$491$	−2.80254	−0.126477	−0.0632385	+	0.997998i	$0.520143\pi$
$492$	10.6838	0.481665
$493$	−24.9834	−1.12520
$494$	−108.620	−4.88705
$495$	−4.58341	−0.206009
$496$	−16.2598	−0.730087
$497$	−2.53845	−0.113865
$498$	3.37601	0.151282
$499$	32.1078	1.43734	0.718671	−	0.695350i	$-0.244751\pi$
$499$	32.1078	1.43734	0.718671	+	0.695350i	$0.244751\pi$
$500$	−93.8262	−4.19604
$501$	−22.8567	−1.02116
$502$	−29.1711	−1.30197
$503$	−1.93589	−0.0863172	−0.0431586	−	0.999068i	$-0.513742\pi$
$503$	−1.93589	−0.0863172	−0.0431586	+	0.999068i	$0.513742\pi$
$504$	−10.3443	−0.460770
$505$	−50.7515	−2.25841
$506$	11.6102	0.516135
$507$	−36.9227	−1.63980
$508$	−10.5317	−0.467267
$509$	−38.2274	−1.69440	−0.847201	−	0.531273i	$-0.821714\pi$
$509$	−38.2274	−1.69440	−0.847201	+	0.531273i	$0.821714\pi$
$510$	42.5373	1.88358
$511$	16.0589	0.710406
$512$	33.7625	1.49210
$513$	−32.4702	−1.43360
$514$	−27.7068	−1.22209
$515$	−29.6103	−1.30479
$516$	−19.4680	−0.857033
$517$	−1.49867	−0.0659116
$518$	18.3303	0.805385
$519$	−10.9051	−0.478681
$520$	−127.815	−5.60504
$521$	7.66301	0.335723	0.167861	−	0.985811i	$-0.446314\pi$
$521$	7.66301	0.335723	0.167861	+	0.985811i	$0.446314\pi$
$522$	27.6344	1.20952
$523$	−9.63594	−0.421350	−0.210675	−	0.977556i	$-0.567566\pi$
$523$	−9.63594	−0.421350	−0.210675	+	0.977556i	$0.567566\pi$
$524$	39.4256	1.72231
$525$	13.1814	0.575285
$526$	−5.26058	−0.229372
$527$	21.0772	0.918137
$528$	1.98489	0.0863814
$529$	43.4386	1.88864
$530$	24.5090	1.06460
$531$	5.29115	0.229617
$532$	28.9123	1.25351
$533$	18.7965	0.814166
$534$	23.7025	1.02571
$535$	78.2176	3.38164
$536$	30.6653	1.32454
$537$	16.7045	0.720854
$538$	−39.7063	−1.71186
$539$	3.31525	0.142798
$540$	−78.8522	−3.39326
$541$	−35.8910	−1.54308	−0.771538	−	0.636183i	$-0.780512\pi$
$541$	−35.8910	−1.54308	−0.771538	+	0.636183i	$0.780512\pi$
$542$	−57.8013	−2.48278
$543$	9.29406	0.398846
$544$	−4.81747	−0.206548
$545$	16.9508	0.726093
$546$	20.2714	0.867535
$547$	19.5391	0.835432	0.417716	−	0.908578i	$-0.362831\pi$
$547$	19.5391	0.835432	0.417716	+	0.908578i	$0.362831\pi$
$548$	50.0918	2.13982
$549$	−2.89068	−0.123371
$550$	15.7226	0.670412
$551$	−37.4260	−1.59440
$552$	38.1082	1.62199
$553$	−15.8498	−0.674001
$554$	54.4134	2.31181
$555$	26.6586	1.13159
$556$	−17.5931	−0.746113
$557$	23.8103	1.00887	0.504437	−	0.863449i	$-0.331700\pi$
$557$	23.8103	1.00887	0.504437	+	0.863449i	$0.331700\pi$
$558$	−23.3136	−0.986945
$559$	−34.2508	−1.44866
$560$	15.3672	0.649381
$561$	−2.57297	−0.108631
$562$	44.4800	1.87628
$563$	−12.5385	−0.528436	−0.264218	−	0.964463i	$-0.585114\pi$
$563$	−12.5385	−0.528436	−0.264218	+	0.964463i	$0.585114\pi$
$564$	−10.1518	−0.427470
$565$	20.8166	0.875761
$566$	32.3758	1.36086
$567$	−0.746939	−0.0313685
$568$	−9.93821	−0.416998
$569$	26.1592	1.09665	0.548326	−	0.836264i	$-0.315265\pi$
$569$	26.1592	1.09665	0.548326	+	0.836264i	$0.315265\pi$
$570$	63.7223	2.66903
$571$	10.7434	0.449597	0.224799	−	0.974405i	$-0.427828\pi$
$571$	10.7434	0.449597	0.224799	+	0.974405i	$0.427828\pi$
$572$	15.9552	0.667122
$573$	2.24717	0.0938769
$574$	−7.58210	−0.316471
$575$	89.9715	3.75207
$576$	18.1692	0.757049
$577$	38.0756	1.58511	0.792554	−	0.609802i	$-0.208751\pi$
$577$	38.0756	1.58511	0.792554	+	0.609802i	$0.208751\pi$
$578$	−3.01918	−0.125581
$579$	−16.3227	−0.678350
$580$	−90.8870	−3.77388
$581$	−1.58097	−0.0655899
$582$	30.1981	1.25175
$583$	−1.48249	−0.0613983
$584$	62.8720	2.60166
$585$	−54.6230	−2.25838
$586$	−30.6187	−1.26485
$587$	−7.76360	−0.320438	−0.160219	−	0.987081i	$-0.551220\pi$
$587$	−7.76360	−0.320438	−0.160219	+	0.987081i	$0.551220\pi$
$588$	22.4571	0.926117
$589$	31.5743	1.30100
$590$	−26.3720	−1.08572
$591$	−18.9540	−0.779665
$592$	21.3899	0.879121
$593$	12.3024	0.505201	0.252600	−	0.967571i	$-0.418714\pi$
$593$	12.3024	0.505201	0.252600	+	0.967571i	$0.418714\pi$
$594$	7.22802	0.296569
$595$	−19.9201	−0.816643
$596$	−65.3343	−2.67620
$597$	4.48464	0.183544
$598$	138.365	5.65816
$599$	−15.2696	−0.623898	−0.311949	−	0.950099i	$-0.600982\pi$
$599$	−15.2696	−0.623898	−0.311949	+	0.950099i	$0.600982\pi$
$600$	51.6063	2.10682
$601$	32.5262	1.32677	0.663386	−	0.748277i	$-0.269119\pi$
$601$	32.5262	1.32677	0.663386	+	0.748277i	$0.269119\pi$
$602$	13.8161	0.563101
$603$	13.1051	0.533683
$604$	−81.4512	−3.31420
$605$	42.6706	1.73481
$606$	31.5132	1.28013
$607$	−4.53370	−0.184017	−0.0920086	−	0.995758i	$-0.529329\pi$
$607$	−4.53370	−0.184017	−0.0920086	+	0.995758i	$0.529329\pi$
$608$	−7.21674	−0.292677
$609$	6.98469	0.283034
$610$	14.4076	0.583348
$611$	−17.8605	−0.722559
$612$	32.2921	1.30533
$613$	−39.9234	−1.61249	−0.806246	−	0.591581i	$-0.798504\pi$
$613$	−39.9234	−1.61249	−0.806246	+	0.591581i	$0.798504\pi$
$614$	37.7686	1.52422
$615$	−11.0270	−0.444652
$616$	−3.11859	−0.125652
$617$	23.9906	0.965826	0.482913	−	0.875668i	$-0.339579\pi$
$617$	23.9906	0.965826	0.482913	+	0.875668i	$0.339579\pi$
$618$	18.3860	0.739592
$619$	18.6217	0.748468	0.374234	−	0.927334i	$-0.377906\pi$
$619$	18.6217	0.748468	0.374234	+	0.927334i	$0.377906\pi$
$620$	76.6766	3.07941
$621$	41.3620	1.65980
$622$	5.06932	0.203261
$623$	−11.0998	−0.444704
$624$	23.6551	0.946961
$625$	41.6494	1.66597
$626$	0.602705	0.0240889
$627$	−3.85439	−0.153930
$628$	34.0178	1.35746
$629$	−27.7272	−1.10556
$630$	22.0337	0.877845
$631$	−48.4475	−1.92866	−0.964332	−	0.264696i	$-0.914728\pi$
$631$	−48.4475	−1.92866	−0.964332	+	0.264696i	$0.914728\pi$
$632$	−62.0531	−2.46834
$633$	−13.5729	−0.539474
$634$	−52.0575	−2.06747
$635$	10.8699	0.431360
$636$	−10.0422	−0.398199
$637$	39.5097	1.56543
$638$	8.33120	0.329835
$639$	−4.24720	−0.168017
$640$	−81.5249	−3.22255
$641$	−11.9499	−0.471993	−0.235997	−	0.971754i	$-0.575835\pi$
$641$	−11.9499	−0.471993	−0.235997	+	0.971754i	$0.575835\pi$
$642$	−48.5677	−1.91681
$643$	12.6344	0.498251	0.249126	−	0.968471i	$-0.419857\pi$
$643$	12.6344	0.498251	0.249126	+	0.968471i	$0.419857\pi$
$644$	−36.8297	−1.45129
$645$	20.0934	0.791175
$646$	−66.2767	−2.60762
$647$	28.2603	1.11103	0.555513	−	0.831508i	$-0.312522\pi$
$647$	28.2603	1.11103	0.555513	+	0.831508i	$0.312522\pi$
$648$	−2.92432	−0.114878
$649$	1.59518	0.0626161
$650$	187.374	7.34943
$651$	−5.89261	−0.230950
$652$	48.6498	1.90527
$653$	−36.1419	−1.41434	−0.707171	−	0.707043i	$-0.750029\pi$
$653$	−36.1419	−1.41434	−0.707171	+	0.707043i	$0.750029\pi$
$654$	−10.5253	−0.411571
$655$	−40.6919	−1.58997
$656$	−8.84770	−0.345445
$657$	26.8691	1.04826
$658$	7.20455	0.280863
$659$	−9.51200	−0.370535	−0.185267	−	0.982688i	$-0.559315\pi$
$659$	−9.51200	−0.370535	−0.185267	+	0.982688i	$0.559315\pi$
$660$	−9.36019	−0.364345
$661$	7.15427	0.278269	0.139134	−	0.990274i	$-0.455568\pi$
$661$	7.15427	0.278269	0.139134	+	0.990274i	$0.455568\pi$
$662$	14.1717	0.550797
$663$	−30.6635	−1.19087
$664$	−6.18963	−0.240204
$665$	−29.8409	−1.15718
$666$	30.6693	1.18841
$667$	47.6748	1.84598
$668$	86.4836	3.34615
$669$	9.06889	0.350624
$670$	−65.3184	−2.52347
$671$	−0.871481	−0.0336431
$672$	1.34684	0.0519553
$673$	−44.0011	−1.69612	−0.848059	−	0.529902i	$-0.822229\pi$
$673$	−44.0011	−1.69612	−0.848059	+	0.529902i	$0.822229\pi$
$674$	−11.0640	−0.426171
$675$	56.0126	2.15593
$676$	139.706	5.37329
$677$	−17.4546	−0.670834	−0.335417	−	0.942070i	$-0.608877\pi$
$677$	−17.4546	−0.670834	−0.335417	+	0.942070i	$0.608877\pi$
$678$	−12.9257	−0.496408
$679$	−14.1417	−0.542707
$680$	−77.9886	−2.99073
$681$	23.1915	0.888701
$682$	−7.02859	−0.269139
$683$	−12.7234	−0.486848	−0.243424	−	0.969920i	$-0.578271\pi$
$683$	−12.7234	−0.486848	−0.243424	+	0.969920i	$0.578271\pi$
$684$	48.3746	1.84965
$685$	−51.7008	−1.97538
$686$	−35.7040	−1.36319
$687$	15.8807	0.605885
$688$	16.1222	0.614654
$689$	−17.6676	−0.673082
$690$	−81.1721	−3.09017
$691$	31.1379	1.18454	0.592270	−	0.805739i	$-0.298232\pi$
$691$	31.1379	1.18454	0.592270	+	0.805739i	$0.298232\pi$
$692$	41.2619	1.56854
$693$	−1.33276	−0.0506275
$694$	35.0949	1.33218
$695$	18.1582	0.688779
$696$	27.3456	1.03653
$697$	11.4690	0.434421
$698$	−2.17390	−0.0822835
$699$	−20.7363	−0.784319
$700$	−49.8750	−1.88510
$701$	3.66122	0.138282	0.0691412	−	0.997607i	$-0.477974\pi$
$701$	3.66122	0.138282	0.0691412	+	0.997607i	$0.477974\pi$
$702$	86.1403	3.25115
$703$	−41.5363	−1.56657
$704$	5.47764	0.206446
$705$	10.4779	0.394621
$706$	−60.0556	−2.26022
$707$	−14.7575	−0.555014
$708$	10.8055	0.406097
$709$	43.6027	1.63753	0.818766	−	0.574127i	$-0.194658\pi$
$709$	43.6027	1.63753	0.818766	+	0.574127i	$0.194658\pi$
$710$	21.1688	0.794451
$711$	−26.5191	−0.994543
$712$	−43.4565	−1.62860
$713$	−40.2207	−1.50628
$714$	12.3690	0.462898
$715$	−16.4677	−0.615858
$716$	−63.2054	−2.36210
$717$	10.8088	0.403661
$718$	68.6100	2.56050
$719$	−20.4798	−0.763769	−0.381885	−	0.924210i	$-0.624725\pi$
$719$	−20.4798	−0.763769	−0.381885	+	0.924210i	$0.624725\pi$
$720$	25.7116	0.958215
$721$	−8.61009	−0.320656
$722$	−53.2117	−1.98033
$723$	11.8275	0.439870
$724$	−35.1662	−1.30694
$725$	64.5615	2.39775
$726$	−26.4955	−0.983340
$727$	−3.84205	−0.142494	−0.0712469	−	0.997459i	$-0.522698\pi$
$727$	−3.84205	−0.142494	−0.0712469	+	0.997459i	$0.522698\pi$
$728$	−37.1659	−1.37746
$729$	14.3615	0.531909
$730$	−133.920	−4.95661
$731$	−20.8988	−0.772972
$732$	−5.90331	−0.218193
$733$	39.7637	1.46871	0.734353	−	0.678768i	$-0.237486\pi$
$733$	39.7637	1.46871	0.734353	+	0.678768i	$0.237486\pi$
$734$	−70.7656	−2.61201
$735$	−23.1785	−0.854950
$736$	9.19299	0.338858
$737$	3.95094	0.145535
$738$	−12.6860	−0.466978
$739$	−35.1046	−1.29134	−0.645671	−	0.763616i	$-0.723422\pi$
$739$	−35.1046	−1.29134	−0.645671	+	0.763616i	$0.723422\pi$
$740$	−100.869	−3.70801
$741$	−45.9349	−1.68746
$742$	7.12673	0.261631
$743$	51.9646	1.90640	0.953198	−	0.302348i	$-0.0977706\pi$
$743$	51.9646	1.90640	0.953198	+	0.302348i	$0.0977706\pi$
$744$	−23.0700	−0.845788
$745$	67.4328	2.47055
$746$	−73.9440	−2.70728
$747$	−2.64521	−0.0967832
$748$	9.73541	0.355962
$749$	22.7441	0.831052
$750$	−60.1309	−2.19567
$751$	49.5928	1.80967	0.904833	−	0.425766i	$-0.139995\pi$
$751$	49.5928	1.80967	0.904833	+	0.425766i	$0.139995\pi$
$752$	8.40713	0.306577
$753$	−12.3363	−0.449560
$754$	99.2875	3.61584
$755$	84.0674	3.05953
$756$	−22.9287	−0.833907
$757$	−2.15837	−0.0784473	−0.0392237	−	0.999230i	$-0.512488\pi$
$757$	−2.15837	−0.0784473	−0.0392237	+	0.999230i	$0.512488\pi$
$758$	−34.1487	−1.24034
$759$	4.90989	0.178218
$760$	−116.830	−4.23785
$761$	7.47528	0.270979	0.135489	−	0.990779i	$-0.456739\pi$
$761$	7.47528	0.270979	0.135489	+	0.990779i	$0.456739\pi$
$762$	−6.74948	−0.244508
$763$	4.92896	0.178440
$764$	−8.50268	−0.307616
$765$	−33.3293	−1.20502
$766$	56.4336	2.03903
$767$	19.0106	0.686432
$768$	31.4957	1.13650
$769$	41.2914	1.48901	0.744503	−	0.667620i	$-0.232687\pi$
$769$	41.2914	1.48901	0.744503	+	0.667620i	$0.232687\pi$
$770$	6.64273	0.239387
$771$	−11.7171	−0.421980
$772$	61.7608	2.22282
$773$	−14.6846	−0.528168	−0.264084	−	0.964500i	$-0.585070\pi$
$773$	−14.6846	−0.528168	−0.264084	+	0.964500i	$0.585070\pi$
$774$	23.1164	0.830901
$775$	−54.4671	−1.95652
$776$	−55.3657	−1.98751
$777$	7.75178	0.278094
$778$	27.1487	0.973327
$779$	17.1810	0.615574
$780$	−111.550	−3.99415
$781$	−1.28045	−0.0458180
$782$	84.4261	3.01907
$783$	29.6804	1.06069
$784$	−18.5976	−0.664201
$785$	−35.1104	−1.25314
$786$	25.2669	0.901240
$787$	−3.46171	−0.123396	−0.0616982	−	0.998095i	$-0.519652\pi$
$787$	−3.46171	−0.123396	−0.0616982	+	0.998095i	$0.519652\pi$
$788$	71.7169	2.55481
$789$	−2.22467	−0.0792005
$790$	132.176	4.70260
$791$	6.05306	0.215222
$792$	−5.21787	−0.185409
$793$	−10.3859	−0.368815
$794$	66.8244	2.37151
$795$	10.3647	0.367600
$796$	−16.9686	−0.601438
$797$	21.2186	0.751601	0.375800	−	0.926701i	$-0.377368\pi$
$797$	21.2186	0.751601	0.375800	+	0.926701i	$0.377368\pi$
$798$	18.5292	0.655925
$799$	−10.8980	−0.385542
$800$	12.4492	0.440146
$801$	−18.5716	−0.656196
$802$	−40.9241	−1.44508
$803$	8.10048	0.285860
$804$	26.7632	0.943865
$805$	38.0127	1.33977
$806$	−83.7636	−2.95045
$807$	−16.7916	−0.591093
$808$	−57.7768	−2.03258
$809$	12.1084	0.425710	0.212855	−	0.977084i	$-0.431724\pi$
$809$	12.1084	0.425710	0.212855	+	0.977084i	$0.431724\pi$
$810$	6.22894	0.218863
$811$	44.7767	1.57232	0.786161	−	0.618022i	$-0.212066\pi$
$811$	44.7767	1.57232	0.786161	+	0.618022i	$0.212066\pi$
$812$	−26.4282	−0.927446
$813$	−24.4439	−0.857285
$814$	9.24618	0.324078
$815$	−50.2124	−1.75886
$816$	14.4336	0.505278
$817$	−31.3072	−1.09530
$818$	64.0308	2.23878
$819$	−15.8833	−0.555007
$820$	41.7232	1.45704
$821$	−32.2448	−1.12535	−0.562676	−	0.826678i	$-0.690228\pi$
$821$	−32.2448	−1.12535	−0.562676	+	0.826678i	$0.690228\pi$
$822$	32.1026	1.11971
$823$	55.3099	1.92798	0.963991	−	0.265935i	$-0.0856806\pi$
$823$	55.3099	1.92798	0.963991	+	0.265935i	$0.0856806\pi$
$824$	−33.7092	−1.17431
$825$	6.64900	0.231488
$826$	−7.66847	−0.266820
$827$	−4.83417	−0.168101	−0.0840503	−	0.996462i	$-0.526786\pi$
$827$	−4.83417	−0.168101	−0.0840503	+	0.996462i	$0.526786\pi$
$828$	−61.6217	−2.14150
$829$	−53.3201	−1.85188	−0.925941	−	0.377667i	$-0.876726\pi$
$829$	−53.3201	−1.85188	−0.925941	+	0.377667i	$0.876726\pi$
$830$	13.1842	0.457630
$831$	23.0112	0.798249
$832$	65.2800	2.26318
$833$	24.1076	0.835279
$834$	−11.2750	−0.390421
$835$	−89.2614	−3.08902
$836$	14.5840	0.504397
$837$	−25.0398	−0.865502
$838$	−47.2850	−1.63343
$839$	−17.7526	−0.612886	−0.306443	−	0.951889i	$-0.599139\pi$
$839$	−17.7526	−0.612886	−0.306443	+	0.951889i	$0.599139\pi$
$840$	21.8035	0.752292
$841$	5.21039	0.179668
$842$	65.5377	2.25858
$843$	18.8104	0.647865
$844$	51.3561	1.76775
$845$	−144.193	−4.96039
$846$	12.0543	0.414436
$847$	12.4078	0.426336
$848$	8.31632	0.285584
$849$	13.6916	0.469893
$850$	114.330	3.92150
$851$	52.9108	1.81376
$852$	−8.67360	−0.297153
$853$	−2.11763	−0.0725062	−0.0362531	−	0.999343i	$-0.511542\pi$
$853$	−2.11763	−0.0725062	−0.0362531	+	0.999343i	$0.511542\pi$
$854$	4.18946	0.143360
$855$	−49.9284	−1.70752
$856$	89.0450	3.04349
$857$	16.7983	0.573821	0.286910	−	0.957957i	$-0.407372\pi$
$857$	16.7983	0.573821	0.286910	+	0.957957i	$0.407372\pi$
$858$	10.2253	0.349087
$859$	21.6915	0.740105	0.370053	−	0.929011i	$-0.379340\pi$
$859$	21.6915	0.740105	0.370053	+	0.929011i	$0.379340\pi$
$860$	−76.0278	−2.59253
$861$	−3.20643	−0.109275
$862$	−14.0110	−0.477217
$863$	−7.76819	−0.264432	−0.132216	−	0.991221i	$-0.542209\pi$
$863$	−7.76819	−0.264432	−0.132216	+	0.991221i	$0.542209\pi$
$864$	5.72318	0.194707
$865$	−42.5873	−1.44801
$866$	−62.6936	−2.13042
$867$	−1.27680	−0.0433624
$868$	22.2960	0.756777
$869$	−7.99497	−0.271211
$870$	−58.2473	−1.97477
$871$	47.0855	1.59543
$872$	19.2972	0.653487
$873$	−23.6612	−0.800809
$874$	126.473	4.27802
$875$	28.1591	0.951952
$876$	54.8717	1.85394
$877$	−1.47533	−0.0498183	−0.0249092	−	0.999690i	$-0.507930\pi$
$877$	−1.47533	−0.0498183	−0.0249092	+	0.999690i	$0.507930\pi$
$878$	42.2729	1.42664
$879$	−12.9485	−0.436743
$880$	7.75153	0.261304
$881$	−38.1066	−1.28384	−0.641922	−	0.766770i	$-0.721863\pi$
$881$	−38.1066	−1.28384	−0.641922	+	0.766770i	$0.721863\pi$
$882$	−26.6656	−0.897878
$883$	47.6125	1.60229	0.801145	−	0.598471i	$-0.204225\pi$
$883$	47.6125	1.60229	0.801145	+	0.598471i	$0.204225\pi$
$884$	116.022	3.90225
$885$	−11.1526	−0.374891
$886$	46.4862	1.56173
$887$	46.0854	1.54740	0.773698	−	0.633555i	$-0.218405\pi$
$887$	46.0854	1.54740	0.773698	+	0.633555i	$0.218405\pi$
$888$	30.3488	1.01844
$889$	3.16076	0.106009
$890$	92.5643	3.10276
$891$	−0.376772	−0.0126223
$892$	−34.3142	−1.14893
$893$	−16.3255	−0.546312
$894$	−41.8711	−1.40038
$895$	65.2356	2.18058
$896$	−23.7058	−0.791956
$897$	58.5139	1.95372
$898$	−75.1554	−2.50797
$899$	−28.8615	−0.962585
$900$	−83.4484	−2.78161
$901$	−10.7802	−0.359142
$902$	−3.82457	−0.127344
$903$	5.84275	0.194435
$904$	23.6982	0.788190
$905$	36.2957	1.20651
$906$	−52.2001	−1.73423
$907$	−32.3971	−1.07573	−0.537865	−	0.843031i	$-0.680769\pi$
$907$	−32.3971	−1.07573	−0.537865	+	0.843031i	$0.680769\pi$
$908$	−87.7504	−2.91210
$909$	−24.6916	−0.818968
$910$	79.1650	2.62430
$911$	−9.47731	−0.313997	−0.156999	−	0.987599i	$-0.550182\pi$
$911$	−9.47731	−0.313997	−0.156999	+	0.987599i	$0.550182\pi$
$912$	21.6220	0.715978
$913$	−0.797477	−0.0263926
$914$	73.9853	2.44722
$915$	6.09293	0.201426
$916$	−60.0881	−1.98537
$917$	−11.8324	−0.390741
$918$	52.5602	1.73475
$919$	8.07580	0.266396	0.133198	−	0.991089i	$-0.457475\pi$
$919$	8.07580	0.266396	0.133198	+	0.991089i	$0.457475\pi$
$920$	148.822	4.90653
$921$	15.9722	0.526300
$922$	−1.33096	−0.0438329
$923$	−15.2598	−0.502282
$924$	−2.72176	−0.0895392
$925$	71.6520	2.35590
$926$	3.00295	0.0986831
$927$	−14.4060	−0.473155
$928$	6.59668	0.216547
$929$	−17.3936	−0.570665	−0.285333	−	0.958429i	$-0.592104\pi$
$929$	−17.3936	−0.570665	−0.285333	+	0.958429i	$0.592104\pi$
$930$	49.1401	1.61137
$931$	36.1140	1.18359
$932$	78.4605	2.57006
$933$	2.14379	0.0701847
$934$	2.74642	0.0898657
$935$	−10.0481	−0.328608
$936$	−62.1842	−2.03256
$937$	−49.8551	−1.62870	−0.814348	−	0.580377i	$-0.802905\pi$
$937$	−49.8551	−1.62870	−0.814348	+	0.580377i	$0.802905\pi$
$938$	−18.9933	−0.620153
$939$	0.254881	0.00831773
$940$	−39.6456	−1.29310
$941$	−43.2826	−1.41097	−0.705486	−	0.708724i	$-0.749271\pi$
$941$	−43.2826	−1.41097	−0.705486	+	0.708724i	$0.749271\pi$
$942$	21.8011	0.710320
$943$	−21.8859	−0.712704
$944$	−8.94848	−0.291248
$945$	23.6651	0.769827
$946$	6.96912	0.226586
$947$	−29.3534	−0.953857	−0.476929	−	0.878942i	$-0.658250\pi$
$947$	−29.3534	−0.953857	−0.476929	+	0.878942i	$0.658250\pi$
$948$	−54.1570	−1.75894
$949$	96.5378	3.13375
$950$	171.271	5.55675
$951$	−22.0149	−0.713881
$952$	−22.6775	−0.734983
$953$	8.94528	0.289766	0.144883	−	0.989449i	$-0.453719\pi$
$953$	8.94528	0.289766	0.144883	+	0.989449i	$0.453719\pi$
$954$	11.9241	0.386057
$955$	8.77579	0.283978
$956$	−40.8975	−1.32272
$957$	3.52323	0.113890
$958$	61.7041	1.99357
$959$	−15.0336	−0.485459
$960$	−38.2967	−1.23602
$961$	−6.65109	−0.214551
$962$	110.192	3.55272
$963$	38.0544	1.22628
$964$	−44.7521	−1.44137
$965$	−63.7446	−2.05201
$966$	−23.6032	−0.759422
$967$	−37.6248	−1.20993	−0.604966	−	0.796252i	$-0.706813\pi$
$967$	−37.6248	−1.20993	−0.604966	+	0.796252i	$0.706813\pi$
$968$	48.5773	1.56133
$969$	−28.0281	−0.900393
$970$	117.931	3.78655
$971$	37.1580	1.19246	0.596229	−	0.802815i	$-0.296665\pi$
$971$	37.1580	1.19246	0.596229	+	0.802815i	$0.296665\pi$
$972$	−61.6211	−1.97650
$973$	5.28004	0.169270
$974$	5.06093	0.162163
$975$	79.2398	2.53770
$976$	4.88876	0.156485
$977$	20.5797	0.658403	0.329202	−	0.944260i	$-0.393220\pi$
$977$	20.5797	0.658403	0.329202	+	0.944260i	$0.393220\pi$
$978$	31.1784	0.996977
$979$	−5.59897	−0.178944
$980$	87.7010	2.80151
$981$	8.24690	0.263303
$982$	6.79588	0.216865
$983$	9.93380	0.316839	0.158420	−	0.987372i	$-0.449360\pi$
$983$	9.93380	0.316839	0.158420	+	0.987372i	$0.449360\pi$
$984$	−12.5534	−0.400189
$985$	−74.0205	−2.35849
$986$	60.5822	1.92933
$987$	3.04677	0.0969798
$988$	173.805	5.52948
$989$	39.8804	1.26812
$990$	11.1143	0.353236
$991$	51.8749	1.64786	0.823930	−	0.566691i	$-0.191777\pi$
$991$	51.8749	1.64786	0.823930	+	0.566691i	$0.191777\pi$
$992$	−5.56527	−0.176698
$993$	5.99313	0.190186
$994$	6.15547	0.195240
$995$	17.5137	0.555221
$996$	−5.40202	−0.171170
$997$	−41.5540	−1.31603	−0.658015	−	0.753005i	$-0.728604\pi$
$997$	−41.5540	−1.31603	−0.658015	+	0.753005i	$0.728604\pi$
$998$	−77.8581	−2.46456
$999$	32.9401	1.04218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4019.2.a.a.1.13	✓	149

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4019.2.a.a.1.13	✓	149	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:	\( N \)	\(=\)	\( 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.42490 q^{2} -1.02548 q^{3} +3.88013 q^{4} -4.00476 q^{5} +2.48668 q^{6} -1.16451 q^{7} -4.55913 q^{8} -1.94839 q^{9} +O(q^{10})\)	\(q-2.42490 q^{2} -1.02548 q^{3} +3.88013 q^{4} -4.00476 q^{5} +2.48668 q^{6} -1.16451 q^{7} -4.55913 q^{8} -1.94839 q^{9} +9.71114 q^{10} -0.587401 q^{11} -3.97899 q^{12} -7.00038 q^{13} +2.82381 q^{14} +4.10680 q^{15} +3.29515 q^{16} -4.27143 q^{17} +4.72466 q^{18} -6.39874 q^{19} -15.5390 q^{20} +1.19417 q^{21} +1.42439 q^{22} +8.15099 q^{23} +4.67529 q^{24} +11.0381 q^{25} +16.9752 q^{26} +5.07447 q^{27} -4.51843 q^{28} +5.84896 q^{29} -9.95856 q^{30} -4.93446 q^{31} +1.12784 q^{32} +0.602367 q^{33} +10.3578 q^{34} +4.66357 q^{35} -7.56002 q^{36} +6.49133 q^{37} +15.5163 q^{38} +7.17874 q^{39} +18.2582 q^{40} -2.68506 q^{41} -2.89575 q^{42} +4.89271 q^{43} -2.27919 q^{44} +7.80285 q^{45} -19.7653 q^{46} +2.55136 q^{47} -3.37911 q^{48} -5.64393 q^{49} -26.7663 q^{50} +4.38025 q^{51} -27.1624 q^{52} +2.52380 q^{53} -12.3051 q^{54} +2.35240 q^{55} +5.30913 q^{56} +6.56177 q^{57} -14.1831 q^{58} -2.71565 q^{59} +15.9349 q^{60} +1.48362 q^{61} +11.9656 q^{62} +2.26892 q^{63} -9.32520 q^{64} +28.0349 q^{65} -1.46068 q^{66} -6.72613 q^{67} -16.5737 q^{68} -8.35866 q^{69} -11.3087 q^{70} +2.17985 q^{71} +8.88297 q^{72} -13.7904 q^{73} -15.7408 q^{74} -11.3193 q^{75} -24.8280 q^{76} +0.684032 q^{77} -17.4077 q^{78} +13.6107 q^{79} -13.1963 q^{80} +0.641422 q^{81} +6.51100 q^{82} +1.35764 q^{83} +4.63355 q^{84} +17.1060 q^{85} -11.8643 q^{86} -5.99799 q^{87} +2.67804 q^{88} +9.53176 q^{89} -18.9211 q^{90} +8.15198 q^{91} +31.6269 q^{92} +5.06018 q^{93} -6.18679 q^{94} +25.6254 q^{95} -1.15657 q^{96} +12.1439 q^{97} +13.6859 q^{98} +1.14449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 149 q - 8 q^{2} - 12 q^{3} + 124 q^{4} - 36 q^{5} - 45 q^{6} - 32 q^{7} - 21 q^{8} + 115 q^{9}+O(q^{10}) \) 149 * q - 8 * q^2 - 12 * q^3 + 124 * q^4 - 36 * q^5 - 45 * q^6 - 32 * q^7 - 21 * q^8 + 115 * q^9	\( 149 q - 8 q^{2} - 12 q^{3} + 124 q^{4} - 36 q^{5} - 45 q^{6} - 32 q^{7} - 21 q^{8} + 115 q^{9} - 58 q^{10} - 33 q^{11} - 33 q^{12} - 107 q^{13} - 28 q^{14} - 24 q^{15} + 74 q^{16} - 39 q^{17} - 33 q^{18} - 93 q^{19} - 63 q^{20} - 113 q^{21} - 38 q^{22} - 11 q^{23} - 130 q^{24} + 85 q^{25} - 33 q^{26} - 30 q^{27} - 94 q^{28} - 85 q^{29} - 16 q^{30} - 129 q^{31} - 35 q^{32} - 64 q^{33} - 78 q^{34} - 27 q^{35} + 79 q^{36} - 135 q^{37} - 11 q^{38} - 73 q^{39} - 146 q^{40} - 101 q^{41} + 4 q^{42} - 55 q^{43} - 82 q^{44} - 168 q^{45} - 113 q^{46} - 40 q^{47} - 65 q^{48} + 27 q^{49} - 5 q^{50} - 49 q^{51} - 177 q^{52} - 32 q^{53} - 155 q^{54} - 128 q^{55} - 44 q^{56} - 47 q^{57} - 46 q^{58} - 53 q^{59} - 11 q^{60} - 347 q^{61} - 11 q^{62} - 73 q^{63} + q^{64} - 31 q^{65} - 37 q^{66} - 40 q^{67} - 80 q^{68} - 175 q^{69} - 61 q^{70} - 31 q^{71} - 68 q^{72} - 193 q^{73} - 33 q^{74} - 56 q^{75} - 248 q^{76} - 84 q^{77} + 40 q^{78} - 111 q^{79} - 54 q^{80} + 49 q^{81} - 74 q^{82} - 24 q^{83} - 159 q^{84} - 258 q^{85} - q^{86} - 66 q^{87} - 97 q^{88} - 76 q^{89} - 75 q^{90} - 134 q^{91} + 31 q^{92} - 97 q^{93} - 111 q^{94} - 14 q^{95} - 216 q^{96} - 140 q^{97} - 13 q^{98} - 116 q^{99}+O(q^{100}) \) 149 * q - 8 * q^2 - 12 * q^3 + 124 * q^4 - 36 * q^5 - 45 * q^6 - 32 * q^7 - 21 * q^8 + 115 * q^9 - 58 * q^10 - 33 * q^11 - 33 * q^12 - 107 * q^13 - 28 * q^14 - 24 * q^15 + 74 * q^16 - 39 * q^17 - 33 * q^18 - 93 * q^19 - 63 * q^20 - 113 * q^21 - 38 * q^22 - 11 * q^23 - 130 * q^24 + 85 * q^25 - 33 * q^26 - 30 * q^27 - 94 * q^28 - 85 * q^29 - 16 * q^30 - 129 * q^31 - 35 * q^32 - 64 * q^33 - 78 * q^34 - 27 * q^35 + 79 * q^36 - 135 * q^37 - 11 * q^38 - 73 * q^39 - 146 * q^40 - 101 * q^41 + 4 * q^42 - 55 * q^43 - 82 * q^44 - 168 * q^45 - 113 * q^46 - 40 * q^47 - 65 * q^48 + 27 * q^49 - 5 * q^50 - 49 * q^51 - 177 * q^52 - 32 * q^53 - 155 * q^54 - 128 * q^55 - 44 * q^56 - 47 * q^57 - 46 * q^58 - 53 * q^59 - 11 * q^60 - 347 * q^61 - 11 * q^62 - 73 * q^63 + q^64 - 31 * q^65 - 37 * q^66 - 40 * q^67 - 80 * q^68 - 175 * q^69 - 61 * q^70 - 31 * q^71 - 68 * q^72 - 193 * q^73 - 33 * q^74 - 56 * q^75 - 248 * q^76 - 84 * q^77 + 40 * q^78 - 111 * q^79 - 54 * q^80 + 49 * q^81 - 74 * q^82 - 24 * q^83 - 159 * q^84 - 258 * q^85 - q^86 - 66 * q^87 - 97 * q^88 - 76 * q^89 - 75 * q^90 - 134 * q^91 + 31 * q^92 - 97 * q^93 - 111 * q^94 - 14 * q^95 - 216 * q^96 - 140 * q^97 - 13 * q^98 - 116 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more