LMFDB - Embedded newform 8041.2.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$78$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36526 q^{2} -1.70691 q^{3} +3.59444 q^{4} -0.509718 q^{5} +4.03728 q^{6} +0.312769 q^{7} -3.77127 q^{8} -0.0864584 q^{9} +O(q^{10})$	\(q-2.36526 q^{2} -1.70691 q^{3} +3.59444 q^{4} -0.509718 q^{5} +4.03728 q^{6} +0.312769 q^{7} -3.77127 q^{8} -0.0864584 q^{9} +1.20561 q^{10} +1.00000 q^{11} -6.13539 q^{12} +3.76096 q^{13} -0.739780 q^{14} +0.870043 q^{15} +1.73114 q^{16} -1.00000 q^{17} +0.204496 q^{18} -3.37007 q^{19} -1.83215 q^{20} -0.533869 q^{21} -2.36526 q^{22} +0.611649 q^{23} +6.43722 q^{24} -4.74019 q^{25} -8.89564 q^{26} +5.26831 q^{27} +1.12423 q^{28} -7.27340 q^{29} -2.05788 q^{30} -7.89935 q^{31} +3.44795 q^{32} -1.70691 q^{33} +2.36526 q^{34} -0.159424 q^{35} -0.310770 q^{36} -6.00809 q^{37} +7.97109 q^{38} -6.41962 q^{39} +1.92229 q^{40} +6.21231 q^{41} +1.26274 q^{42} -1.00000 q^{43} +3.59444 q^{44} +0.0440694 q^{45} -1.44671 q^{46} -7.02890 q^{47} -2.95490 q^{48} -6.90218 q^{49} +11.2118 q^{50} +1.70691 q^{51} +13.5186 q^{52} -4.25497 q^{53} -12.4609 q^{54} -0.509718 q^{55} -1.17954 q^{56} +5.75241 q^{57} +17.2035 q^{58} +9.41026 q^{59} +3.12732 q^{60} +14.0984 q^{61} +18.6840 q^{62} -0.0270415 q^{63} -11.6176 q^{64} -1.91703 q^{65} +4.03728 q^{66} +9.83006 q^{67} -3.59444 q^{68} -1.04403 q^{69} +0.377079 q^{70} -3.92911 q^{71} +0.326058 q^{72} +1.30459 q^{73} +14.2107 q^{74} +8.09107 q^{75} -12.1135 q^{76} +0.312769 q^{77} +15.1841 q^{78} -13.1289 q^{79} -0.882393 q^{80} -8.73315 q^{81} -14.6937 q^{82} +4.52049 q^{83} -1.91896 q^{84} +0.509718 q^{85} +2.36526 q^{86} +12.4150 q^{87} -3.77127 q^{88} -7.93708 q^{89} -0.104235 q^{90} +1.17631 q^{91} +2.19854 q^{92} +13.4835 q^{93} +16.6252 q^{94} +1.71779 q^{95} -5.88534 q^{96} +1.17482 q^{97} +16.3254 q^{98} -0.0864584 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) $ 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * 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 $

$2$	−2.36526	−1.67249	−0.836245	−	0.548356i	$-0.815254\pi$
$2$	−2.36526	−1.67249	−0.836245	+	0.548356i	$0.815254\pi$
$3$	−1.70691	−0.985485	−0.492742	−	0.870175i	$-0.664006\pi$
$3$	−1.70691	−0.985485	−0.492742	+	0.870175i	$0.664006\pi$
$4$	3.59444	1.79722
$5$	−0.509718	−0.227953	−0.113976	−	0.993483i	$-0.536359\pi$
$5$	−0.509718	−0.227953	−0.113976	+	0.993483i	$0.536359\pi$
$6$	4.03728	1.64821
$7$	0.312769	0.118216	0.0591079	−	0.998252i	$-0.481174\pi$
$7$	0.312769	0.118216	0.0591079	+	0.998252i	$0.481174\pi$
$8$	−3.77127	−1.33335
$9$	−0.0864584	−0.0288195
$10$	1.20561	0.381249
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−6.13539	−1.77114
$13$	3.76096	1.04310	0.521552	−	0.853220i	$-0.325353\pi$
$13$	3.76096	1.04310	0.521552	+	0.853220i	$0.325353\pi$
$14$	−0.739780	−0.197715
$15$	0.870043	0.224644
$16$	1.73114	0.432785
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.204496	0.0482002
$19$	−3.37007	−0.773148	−0.386574	−	0.922258i	$-0.626342\pi$
$19$	−3.37007	−0.773148	−0.386574	+	0.922258i	$0.626342\pi$
$20$	−1.83215	−0.409682
$21$	−0.533869	−0.116500
$22$	−2.36526	−0.504275
$23$	0.611649	0.127538	0.0637688	−	0.997965i	$-0.479688\pi$
$23$	0.611649	0.127538	0.0637688	+	0.997965i	$0.479688\pi$
$24$	6.43722	1.31399
$25$	−4.74019	−0.948037
$26$	−8.89564	−1.74458
$27$	5.26831	1.01389
$28$	1.12423	0.212460
$29$	−7.27340	−1.35064	−0.675318	−	0.737527i	$-0.735994\pi$
$29$	−7.27340	−1.35064	−0.675318	+	0.737527i	$0.735994\pi$
$30$	−2.05788	−0.375715
$31$	−7.89935	−1.41876	−0.709382	−	0.704824i	$-0.751026\pi$
$31$	−7.89935	−1.41876	−0.709382	+	0.704824i	$0.751026\pi$
$32$	3.44795	0.609517
$33$	−1.70691	−0.297135
$34$	2.36526	0.405638
$35$	−0.159424	−0.0269476
$36$	−0.310770	−0.0517950
$37$	−6.00809	−0.987724	−0.493862	−	0.869540i	$-0.664415\pi$
$37$	−6.00809	−0.987724	−0.493862	+	0.869540i	$0.664415\pi$
$38$	7.97109	1.29308
$39$	−6.41962	−1.02796
$40$	1.92229	0.303940
$41$	6.21231	0.970200	0.485100	−	0.874459i	$-0.338783\pi$
$41$	6.21231	0.970200	0.485100	+	0.874459i	$0.338783\pi$
$42$	1.26274	0.194845
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	3.59444	0.541883
$45$	0.0440694	0.00656948
$46$	−1.44671	−0.213305
$47$	−7.02890	−1.02527	−0.512635	−	0.858607i	$-0.671331\pi$
$47$	−7.02890	−1.02527	−0.512635	+	0.858607i	$0.671331\pi$
$48$	−2.95490	−0.426503
$49$	−6.90218	−0.986025
$50$	11.2118	1.58558
$51$	1.70691	0.239015
$52$	13.5186	1.87469
$53$	−4.25497	−0.584465	−0.292233	−	0.956347i	$-0.594398\pi$
$53$	−4.25497	−0.584465	−0.292233	+	0.956347i	$0.594398\pi$
$54$	−12.4609	−1.69571
$55$	−0.509718	−0.0687304
$56$	−1.17954	−0.157622
$57$	5.75241	0.761926
$58$	17.2035	2.25892
$59$	9.41026	1.22511	0.612556	−	0.790427i	$-0.290142\pi$
$59$	9.41026	1.22511	0.612556	+	0.790427i	$0.290142\pi$
$60$	3.12732	0.403735
$61$	14.0984	1.80511	0.902555	−	0.430574i	$-0.141689\pi$
$61$	14.0984	1.80511	0.902555	+	0.430574i	$0.141689\pi$
$62$	18.6840	2.37287
$63$	−0.0270415	−0.00340691
$64$	−11.6176	−1.45220
$65$	−1.91703	−0.237778
$66$	4.03728	0.496955
$67$	9.83006	1.20093	0.600467	−	0.799650i	$-0.294982\pi$
$67$	9.83006	1.20093	0.600467	+	0.799650i	$0.294982\pi$
$68$	−3.59444	−0.435890
$69$	−1.04403	−0.125686
$70$	0.377079	0.0450696
$71$	−3.92911	−0.466299	−0.233150	−	0.972441i	$-0.574903\pi$
$71$	−3.92911	−0.466299	−0.233150	+	0.972441i	$0.574903\pi$
$72$	0.326058	0.0384263
$73$	1.30459	0.152690	0.0763451	−	0.997081i	$-0.475675\pi$
$73$	1.30459	0.152690	0.0763451	+	0.997081i	$0.475675\pi$
$74$	14.2107	1.65196
$75$	8.09107	0.934277
$76$	−12.1135	−1.38952
$77$	0.312769	0.0356434
$78$	15.1841	1.71926
$79$	−13.1289	−1.47711	−0.738557	−	0.674191i	$-0.764492\pi$
$79$	−13.1289	−1.47711	−0.738557	+	0.674191i	$0.764492\pi$
$80$	−0.882393	−0.0986546
$81$	−8.73315	−0.970350
$82$	−14.6937	−1.62265
$83$	4.52049	0.496188	0.248094	−	0.968736i	$-0.420196\pi$
$83$	4.52049	0.496188	0.248094	+	0.968736i	$0.420196\pi$
$84$	−1.91896	−0.209376
$85$	0.509718	0.0552867
$86$	2.36526	0.255052
$87$	12.4150	1.33103
$88$	−3.77127	−0.402019
$89$	−7.93708	−0.841329	−0.420665	−	0.907216i	$-0.638203\pi$
$89$	−7.93708	−0.841329	−0.420665	+	0.907216i	$0.638203\pi$
$90$	−0.104235	−0.0109874
$91$	1.17631	0.123311
$92$	2.19854	0.229213
$93$	13.4835	1.39817
$94$	16.6252	1.71475
$95$	1.71779	0.176241
$96$	−5.88534	−0.600670
$97$	1.17482	0.119285	0.0596426	−	0.998220i	$-0.481004\pi$
$97$	1.17482	0.119285	0.0596426	+	0.998220i	$0.481004\pi$
$98$	16.3254	1.64912
$99$	−0.0864584	−0.00868939
$100$	−17.0383	−1.70383
$101$	−11.4548	−1.13979	−0.569896	−	0.821717i	$-0.693016\pi$
$101$	−11.4548	−1.13979	−0.569896	+	0.821717i	$0.693016\pi$
$102$	−4.03728	−0.399750
$103$	−3.30647	−0.325796	−0.162898	−	0.986643i	$-0.552084\pi$
$103$	−3.30647	−0.325796	−0.162898	+	0.986643i	$0.552084\pi$
$104$	−14.1836	−1.39082
$105$	0.272123	0.0265565
$106$	10.0641	0.977512
$107$	−13.3175	−1.28745	−0.643727	−	0.765255i	$-0.722613\pi$
$107$	−13.3175	−1.28745	−0.643727	+	0.765255i	$0.722613\pi$
$108$	18.9366	1.82218
$109$	7.19879	0.689519	0.344760	−	0.938691i	$-0.387960\pi$
$109$	7.19879	0.689519	0.344760	+	0.938691i	$0.387960\pi$
$110$	1.20561	0.114951
$111$	10.2553	0.973387
$112$	0.541448	0.0511620
$113$	3.29147	0.309635	0.154818	−	0.987943i	$-0.450521\pi$
$113$	3.29147	0.309635	0.154818	+	0.987943i	$0.450521\pi$
$114$	−13.6059	−1.27431
$115$	−0.311768	−0.0290726
$116$	−26.1438	−2.42739
$117$	−0.325167	−0.0300617
$118$	−22.2577	−2.04899
$119$	−0.312769	−0.0286715
$120$	−3.28117	−0.299528
$121$	1.00000	0.0909091
$122$	−33.3463	−3.01903
$123$	−10.6039	−0.956118
$124$	−28.3938	−2.54983
$125$	4.96475	0.444061
$126$	0.0639602	0.00569803
$127$	−18.4845	−1.64023	−0.820116	−	0.572198i	$-0.806091\pi$
$127$	−18.4845	−1.64023	−0.820116	+	0.572198i	$0.806091\pi$
$128$	20.5826	1.81927
$129$	1.70691	0.150285
$130$	4.53427	0.397682
$131$	18.7532	1.63848	0.819239	−	0.573452i	$-0.194396\pi$
$131$	18.7532	1.63848	0.819239	+	0.573452i	$0.194396\pi$
$132$	−6.13539	−0.534017
$133$	−1.05406	−0.0913983
$134$	−23.2506	−2.00855
$135$	−2.68535	−0.231118
$136$	3.77127	0.323384
$137$	7.06003	0.603179	0.301590	−	0.953438i	$-0.402483\pi$
$137$	7.06003	0.603179	0.301590	+	0.953438i	$0.402483\pi$
$138$	2.46940	0.210209
$139$	10.3144	0.874853	0.437427	−	0.899254i	$-0.355890\pi$
$139$	10.3144	0.874853	0.437427	+	0.899254i	$0.355890\pi$
$140$	−0.573042	−0.0484309
$141$	11.9977	1.01039
$142$	9.29335	0.779881
$143$	3.76096	0.314507
$144$	−0.149671	−0.0124726
$145$	3.70738	0.307881
$146$	−3.08568	−0.255373
$147$	11.7814	0.971713
$148$	−21.5958	−1.77516
$149$	9.73609	0.797611	0.398806	−	0.917035i	$-0.369425\pi$
$149$	9.73609	0.797611	0.398806	+	0.917035i	$0.369425\pi$
$150$	−19.1375	−1.56257
$151$	16.8568	1.37179	0.685893	−	0.727702i	$-0.259412\pi$
$151$	16.8568	1.37179	0.685893	+	0.727702i	$0.259412\pi$
$152$	12.7095	1.03087
$153$	0.0864584	0.00698974
$154$	−0.739780	−0.0596132
$155$	4.02644	0.323411
$156$	−23.0750	−1.84748
$157$	12.5896	1.00476	0.502379	−	0.864647i	$-0.332458\pi$
$157$	12.5896	1.00476	0.502379	+	0.864647i	$0.332458\pi$
$158$	31.0532	2.47046
$159$	7.26285	0.575982
$160$	−1.75748	−0.138941
$161$	0.191305	0.0150769
$162$	20.6561	1.62290
$163$	−24.8782	−1.94861	−0.974304	−	0.225236i	$-0.927685\pi$
$163$	−24.8782	−1.94861	−0.974304	+	0.225236i	$0.927685\pi$
$164$	22.3298	1.74367
$165$	0.870043	0.0677328
$166$	−10.6921	−0.829870
$167$	2.02938	0.157038	0.0785192	−	0.996913i	$-0.474981\pi$
$167$	2.02938	0.157038	0.0785192	+	0.996913i	$0.474981\pi$
$168$	2.01337	0.155335
$169$	1.14484	0.0880643
$170$	−1.20561	−0.0924664
$171$	0.291371	0.0222817
$172$	−3.59444	−0.274074
$173$	17.4523	1.32687	0.663436	−	0.748233i	$-0.269097\pi$
$173$	17.4523	1.32687	0.663436	+	0.748233i	$0.269097\pi$
$174$	−29.3647	−2.22614
$175$	−1.48259	−0.112073
$176$	1.73114	0.130490
$177$	−16.0625	−1.20733
$178$	18.7732	1.40711
$179$	16.3632	1.22304	0.611520	−	0.791229i	$-0.290558\pi$
$179$	16.3632	1.22304	0.611520	+	0.791229i	$0.290558\pi$
$180$	0.158405	0.0118068
$181$	11.9105	0.885303	0.442652	−	0.896694i	$-0.354038\pi$
$181$	11.9105	0.885303	0.442652	+	0.896694i	$0.354038\pi$
$182$	−2.78229	−0.206237
$183$	−24.0646	−1.77891
$184$	−2.30669	−0.170052
$185$	3.06243	0.225155
$186$	−31.8919	−2.33843
$187$	−1.00000	−0.0731272
$188$	−25.2650	−1.84264
$189$	1.64777	0.119857
$190$	−4.06301	−0.294762
$191$	14.2196	1.02890	0.514449	−	0.857521i	$-0.327997\pi$
$191$	14.2196	1.02890	0.514449	+	0.857521i	$0.327997\pi$
$192$	19.8301	1.43112
$193$	−7.28892	−0.524668	−0.262334	−	0.964977i	$-0.584492\pi$
$193$	−7.28892	−0.524668	−0.262334	+	0.964977i	$0.584492\pi$
$194$	−2.77876	−0.199503
$195$	3.27220	0.234327
$196$	−24.8095	−1.77211
$197$	−21.6754	−1.54431	−0.772156	−	0.635434i	$-0.780821\pi$
$197$	−21.6754	−1.54431	−0.772156	+	0.635434i	$0.780821\pi$
$198$	0.204496	0.0145329
$199$	−22.8038	−1.61652	−0.808259	−	0.588827i	$-0.799590\pi$
$199$	−22.8038	−1.61652	−0.808259	+	0.588827i	$0.799590\pi$
$200$	17.8765	1.26406
$201$	−16.7790	−1.18350
$202$	27.0935	1.90629
$203$	−2.27490	−0.159666
$204$	6.13539	0.429563
$205$	−3.16653	−0.221160
$206$	7.82066	0.544891
$207$	−0.0528821	−0.00367556
$208$	6.51075	0.451439
$209$	−3.37007	−0.233113
$210$	−0.643641	−0.0444154
$211$	−10.6408	−0.732540	−0.366270	−	0.930509i	$-0.619365\pi$
$211$	−10.6408	−0.732540	−0.366270	+	0.930509i	$0.619365\pi$
$212$	−15.2943	−1.05041
$213$	6.70663	0.459531
$214$	31.4994	2.15325
$215$	0.509718	0.0347625
$216$	−19.8682	−1.35186
$217$	−2.47067	−0.167720
$218$	−17.0270	−1.15321
$219$	−2.22681	−0.150474
$220$	−1.83215	−0.123524
$221$	−3.76096	−0.252990
$222$	−24.2564	−1.62798
$223$	−15.2857	−1.02361	−0.511805	−	0.859102i	$-0.671023\pi$
$223$	−15.2857	−1.02361	−0.511805	+	0.859102i	$0.671023\pi$
$224$	1.07841	0.0720545
$225$	0.409829	0.0273219
$226$	−7.78517	−0.517862
$227$	−0.255235	−0.0169406	−0.00847028	−	0.999964i	$-0.502696\pi$
$227$	−0.255235	−0.0169406	−0.00847028	+	0.999964i	$0.502696\pi$
$228$	20.6767	1.36935
$229$	−12.5246	−0.827648	−0.413824	−	0.910357i	$-0.635807\pi$
$229$	−12.5246	−0.827648	−0.413824	+	0.910357i	$0.635807\pi$
$230$	0.737413	0.0486235
$231$	−0.533869	−0.0351260
$232$	27.4299	1.80086
$233$	−9.08675	−0.595293	−0.297646	−	0.954676i	$-0.596202\pi$
$233$	−9.08675	−0.595293	−0.297646	+	0.954676i	$0.596202\pi$
$234$	0.769103	0.0502778
$235$	3.58276	0.233713
$236$	33.8246	2.20180
$237$	22.4098	1.45567
$238$	0.739780	0.0479528
$239$	−13.3197	−0.861578	−0.430789	−	0.902453i	$-0.641765\pi$
$239$	−13.3197	−0.861578	−0.430789	+	0.902453i	$0.641765\pi$
$240$	1.50617	0.0972226
$241$	0.314487	0.0202579	0.0101290	−	0.999949i	$-0.496776\pi$
$241$	0.314487	0.0202579	0.0101290	+	0.999949i	$0.496776\pi$
$242$	−2.36526	−0.152045
$243$	−0.898219	−0.0576208
$244$	50.6758	3.24418
$245$	3.51816	0.224767
$246$	25.0809	1.59910
$247$	−12.6747	−0.806473
$248$	29.7906	1.89170
$249$	−7.71607	−0.488986
$250$	−11.7429	−0.742687
$251$	23.1569	1.46165	0.730827	−	0.682563i	$-0.239135\pi$
$251$	23.1569	1.46165	0.730827	+	0.682563i	$0.239135\pi$
$252$	−0.0971993	−0.00612298
$253$	0.611649	0.0384540
$254$	43.7205	2.74327
$255$	−0.870043	−0.0544842
$256$	−25.4481	−1.59051
$257$	−18.8193	−1.17392	−0.586959	−	0.809616i	$-0.699675\pi$
$257$	−18.8193	−1.17392	−0.586959	+	0.809616i	$0.699675\pi$
$258$	−4.03728	−0.251350
$259$	−1.87915	−0.116765
$260$	−6.89066	−0.427341
$261$	0.628846	0.0389246
$262$	−44.3563	−2.74034
$263$	28.5766	1.76211	0.881054	−	0.473016i	$-0.156835\pi$
$263$	28.5766	1.76211	0.881054	+	0.473016i	$0.156835\pi$
$264$	6.43722	0.396183
$265$	2.16884	0.133231
$266$	2.49311	0.152863
$267$	13.5479	0.829117
$268$	35.3336	2.15834
$269$	27.8727	1.69943	0.849716	−	0.527241i	$-0.176774\pi$
$269$	27.8727	1.69943	0.849716	+	0.527241i	$0.176774\pi$
$270$	6.35155	0.386543
$271$	−15.3922	−0.935012	−0.467506	−	0.883990i	$-0.654847\pi$
$271$	−15.3922	−0.935012	−0.467506	+	0.883990i	$0.654847\pi$
$272$	−1.73114	−0.104966
$273$	−2.00786	−0.121521
$274$	−16.6988	−1.00881
$275$	−4.74019	−0.285844
$276$	−3.75270	−0.225886
$277$	−11.8593	−0.712554	−0.356277	−	0.934380i	$-0.615954\pi$
$277$	−11.8593	−0.712554	−0.356277	+	0.934380i	$0.615954\pi$
$278$	−24.3961	−1.46318
$279$	0.682965	0.0408880
$280$	0.601232	0.0359305
$281$	−26.2640	−1.56678	−0.783390	−	0.621531i	$-0.786511\pi$
$281$	−26.2640	−1.56678	−0.783390	+	0.621531i	$0.786511\pi$
$282$	−28.3777	−1.68986
$283$	−33.1837	−1.97257	−0.986284	−	0.165054i	$-0.947220\pi$
$283$	−33.1837	−1.97257	−0.986284	+	0.165054i	$0.947220\pi$
$284$	−14.1230	−0.838044
$285$	−2.93211	−0.173683
$286$	−8.89564	−0.526010
$287$	1.94302	0.114693
$288$	−0.298104	−0.0175660
$289$	1.00000	0.0588235
$290$	−8.76891	−0.514928
$291$	−2.00532	−0.117554
$292$	4.68926	0.274418
$293$	25.0005	1.46055	0.730273	−	0.683155i	$-0.239393\pi$
$293$	25.0005	1.46055	0.730273	+	0.683155i	$0.239393\pi$
$294$	−27.8660	−1.62518
$295$	−4.79658	−0.279268
$296$	22.6581	1.31698
$297$	5.26831	0.305698
$298$	−23.0284	−1.33400
$299$	2.30039	0.133035
$300$	29.0829	1.67910
$301$	−0.312769	−0.0180277
$302$	−39.8707	−2.29430
$303$	19.5522	1.12325
$304$	−5.83407	−0.334607
$305$	−7.18619	−0.411480
$306$	−0.204496	−0.0116903
$307$	0.0193220	0.00110276	0.000551381	−	1.00000i	$-0.499824\pi$
$307$	0.0193220	0.00110276	0.000551381		1.00000i	$0.499824\pi$
$308$	1.12423	0.0640591
$309$	5.64385	0.321068
$310$	−9.52357	−0.540902
$311$	22.7414	1.28955	0.644773	−	0.764374i	$-0.276952\pi$
$311$	22.7414	1.28955	0.644773	+	0.764374i	$0.276952\pi$
$312$	24.2101	1.37063
$313$	9.55587	0.540130	0.270065	−	0.962842i	$-0.412955\pi$
$313$	9.55587	0.540130	0.270065	+	0.962842i	$0.412955\pi$
$314$	−29.7776	−1.68045
$315$	0.0137836	0.000776616 0
$316$	−47.1910	−2.65470
$317$	−23.0615	−1.29526	−0.647632	−	0.761953i	$-0.724240\pi$
$317$	−23.0615	−1.29526	−0.647632	+	0.761953i	$0.724240\pi$
$318$	−17.1785	−0.963324
$319$	−7.27340	−0.407232
$320$	5.92169	0.331032
$321$	22.7318	1.26877
$322$	−0.452486	−0.0252160
$323$	3.37007	0.187516
$324$	−31.3908	−1.74393
$325$	−17.8277	−0.988901
$326$	58.8433	3.25903
$327$	−12.2877	−0.679511
$328$	−23.4283	−1.29361
$329$	−2.19843	−0.121203
$330$	−2.05788	−0.113282
$331$	−0.123516	−0.00678905	−0.00339453	−	0.999994i	$-0.501081\pi$
$331$	−0.123516	−0.00678905	−0.00339453	+	0.999994i	$0.501081\pi$
$332$	16.2487	0.891761
$333$	0.519450	0.0284657
$334$	−4.80001	−0.262645
$335$	−5.01056	−0.273756
$336$	−0.924202	−0.0504194
$337$	6.63459	0.361409	0.180705	−	0.983537i	$-0.442162\pi$
$337$	6.63459	0.361409	0.180705	+	0.983537i	$0.442162\pi$
$338$	−2.70783	−0.147287
$339$	−5.61824	−0.305141
$340$	1.83215	0.0993625
$341$	−7.89935	−0.427774
$342$	−0.689168	−0.0372659
$343$	−4.34818	−0.234779
$344$	3.77127	0.203333
$345$	0.532161	0.0286506
$346$	−41.2791	−2.21918
$347$	−10.9413	−0.587357	−0.293679	−	0.955904i	$-0.594880\pi$
$347$	−10.9413	−0.587357	−0.293679	+	0.955904i	$0.594880\pi$
$348$	44.6251	2.39216
$349$	19.7751	1.05854	0.529268	−	0.848455i	$-0.322467\pi$
$349$	19.7751	1.05854	0.529268	+	0.848455i	$0.322467\pi$
$350$	3.50670	0.187441
$351$	19.8139	1.05759
$352$	3.44795	0.183776
$353$	−29.5473	−1.57264	−0.786322	−	0.617816i	$-0.788018\pi$
$353$	−29.5473	−1.57264	−0.786322	+	0.617816i	$0.788018\pi$
$354$	37.9919	2.01924
$355$	2.00274	0.106294
$356$	−28.5294	−1.51205
$357$	0.533869	0.0282554
$358$	−38.7031	−2.04552
$359$	6.66030	0.351517	0.175758	−	0.984433i	$-0.443762\pi$
$359$	6.66030	0.351517	0.175758	+	0.984433i	$0.443762\pi$
$360$	−0.166198	−0.00875938
$361$	−7.64260	−0.402242
$362$	−28.1715	−1.48066
$363$	−1.70691	−0.0895895
$364$	4.22820	0.221618
$365$	−0.664971	−0.0348062
$366$	56.9191	2.97521
$367$	32.3388	1.68807	0.844036	−	0.536286i	$-0.180173\pi$
$367$	32.3388	1.68807	0.844036	+	0.536286i	$0.180173\pi$
$368$	1.05885	0.0551963
$369$	−0.537107	−0.0279606
$370$	−7.24344	−0.376569
$371$	−1.33083	−0.0690930
$372$	48.4656	2.51282
$373$	−7.23152	−0.374434	−0.187217	−	0.982319i	$-0.559947\pi$
$373$	−7.23152	−0.374434	−0.187217	+	0.982319i	$0.559947\pi$
$374$	2.36526	0.122305
$375$	−8.47438	−0.437615
$376$	26.5079	1.36704
$377$	−27.3550	−1.40885
$378$	−3.89739	−0.200460
$379$	−13.0354	−0.669582	−0.334791	−	0.942292i	$-0.608666\pi$
$379$	−13.0354	−0.669582	−0.334791	+	0.942292i	$0.608666\pi$
$380$	6.17449	0.316745
$381$	31.5513	1.61642
$382$	−33.6331	−1.72082
$383$	28.1019	1.43594	0.717970	−	0.696074i	$-0.245072\pi$
$383$	28.1019	1.43594	0.717970	+	0.696074i	$0.245072\pi$
$384$	−35.1327	−1.79286
$385$	−0.159424	−0.00812501
$386$	17.2402	0.877501
$387$	0.0864584	0.00439493
$388$	4.22284	0.214382
$389$	−3.32203	−0.168434	−0.0842169	−	0.996447i	$-0.526839\pi$
$389$	−3.32203	−0.168434	−0.0842169	+	0.996447i	$0.526839\pi$
$390$	−7.73959	−0.391910
$391$	−0.611649	−0.0309324
$392$	26.0300	1.31471
$393$	−32.0101	−1.61470
$394$	51.2680	2.58284
$395$	6.69202	0.336712
$396$	−0.310770	−0.0156168
$397$	−30.6687	−1.53922	−0.769609	−	0.638515i	$-0.779549\pi$
$397$	−30.6687	−1.53922	−0.769609	+	0.638515i	$0.779549\pi$
$398$	53.9369	2.70361
$399$	1.79918	0.0900716
$400$	−8.20593	−0.410296
$401$	12.5440	0.626418	0.313209	−	0.949684i	$-0.398596\pi$
$401$	12.5440	0.626418	0.313209	+	0.949684i	$0.398596\pi$
$402$	39.6867	1.97939
$403$	−29.7091	−1.47992
$404$	−41.1735	−2.04846
$405$	4.45144	0.221194
$406$	5.38072	0.267040
$407$	−6.00809	−0.297810
$408$	−6.43722	−0.318690
$409$	18.1340	0.896670	0.448335	−	0.893866i	$-0.352017\pi$
$409$	18.1340	0.896670	0.448335	+	0.893866i	$0.352017\pi$
$410$	7.48966	0.369888
$411$	−12.0508	−0.594424
$412$	−11.8849	−0.585529
$413$	2.94324	0.144827
$414$	0.125080	0.00614734
$415$	−2.30418	−0.113108
$416$	12.9676	0.635789
$417$	−17.6057	−0.862155
$418$	7.97109	0.389879
$419$	2.68353	0.131099	0.0655496	−	0.997849i	$-0.479120\pi$
$419$	2.68353	0.131099	0.0655496	+	0.997849i	$0.479120\pi$
$420$	0.978130	0.0477279
$421$	−21.5187	−1.04876	−0.524379	−	0.851485i	$-0.675702\pi$
$421$	−21.5187	−1.04876	−0.524379	+	0.851485i	$0.675702\pi$
$422$	25.1681	1.22517
$423$	0.607707	0.0295477
$424$	16.0467	0.779294
$425$	4.74019	0.229933
$426$	−15.8629	−0.768561
$427$	4.40954	0.213392
$428$	−47.8691	−2.31384
$429$	−6.41962	−0.309942
$430$	−1.20561	−0.0581399
$431$	30.9665	1.49160	0.745802	−	0.666168i	$-0.232067\pi$
$431$	30.9665	1.49160	0.745802	+	0.666168i	$0.232067\pi$
$432$	9.12017	0.438795
$433$	−31.4210	−1.51000	−0.754999	−	0.655726i	$-0.772363\pi$
$433$	−31.4210	−1.51000	−0.754999	+	0.655726i	$0.772363\pi$
$434$	5.84378	0.280510
$435$	−6.32817	−0.303412
$436$	25.8757	1.23922
$437$	−2.06130	−0.0986054
$438$	5.26698	0.251666
$439$	−12.7913	−0.610497	−0.305249	−	0.952273i	$-0.598740\pi$
$439$	−12.7913	−0.610497	−0.305249	+	0.952273i	$0.598740\pi$
$440$	1.92229	0.0916413
$441$	0.596751	0.0284167
$442$	8.89564	0.423123
$443$	8.83739	0.419877	0.209938	−	0.977715i	$-0.432674\pi$
$443$	8.83739	0.419877	0.209938	+	0.977715i	$0.432674\pi$
$444$	36.8620	1.74939
$445$	4.04567	0.191783
$446$	36.1547	1.71198
$447$	−16.6186	−0.786034
$448$	−3.63362	−0.171672
$449$	−4.73996	−0.223693	−0.111846	−	0.993726i	$-0.535676\pi$
$449$	−4.73996	−0.223693	−0.111846	+	0.993726i	$0.535676\pi$
$450$	−0.969351	−0.0456956
$451$	6.21231	0.292526
$452$	11.8310	0.556483
$453$	−28.7730	−1.35187
$454$	0.603697	0.0283329
$455$	−0.599589	−0.0281091
$456$	−21.6939	−1.01591
$457$	20.2014	0.944980	0.472490	−	0.881336i	$-0.343355\pi$
$457$	20.2014	0.944980	0.472490	+	0.881336i	$0.343355\pi$
$458$	29.6239	1.38423
$459$	−5.26831	−0.245903
$460$	−1.12063	−0.0522498
$461$	−14.2053	−0.661605	−0.330802	−	0.943700i	$-0.607319\pi$
$461$	−14.2053	−0.661605	−0.330802	+	0.943700i	$0.607319\pi$
$462$	1.26274	0.0587479
$463$	−16.3574	−0.760191	−0.380095	−	0.924947i	$-0.624109\pi$
$463$	−16.3574	−0.760191	−0.380095	+	0.924947i	$0.624109\pi$
$464$	−12.5913	−0.584535
$465$	−6.87277	−0.318717
$466$	21.4925	0.995621
$467$	12.0886	0.559395	0.279698	−	0.960088i	$-0.409766\pi$
$467$	12.0886	0.559395	0.279698	+	0.960088i	$0.409766\pi$
$468$	−1.16879	−0.0540275
$469$	3.07454	0.141969
$470$	−8.47414	−0.390883
$471$	−21.4893	−0.990175
$472$	−35.4886	−1.63350
$473$	−1.00000	−0.0459800
$474$	−53.0050	−2.43460
$475$	15.9748	0.732973
$476$	−1.12423	−0.0515291
$477$	0.367878	0.0168440
$478$	31.5045	1.44098
$479$	−28.5546	−1.30469	−0.652347	−	0.757921i	$-0.726215\pi$
$479$	−28.5546	−1.30469	−0.652347	+	0.757921i	$0.726215\pi$
$480$	2.99987	0.136924
$481$	−22.5962	−1.03030
$482$	−0.743843	−0.0338811
$483$	−0.326540	−0.0148581
$484$	3.59444	0.163384
$485$	−0.598829	−0.0271914
$486$	2.12452	0.0963702
$487$	−0.719499	−0.0326036	−0.0163018	−	0.999867i	$-0.505189\pi$
$487$	−0.719499	−0.0326036	−0.0163018	+	0.999867i	$0.505189\pi$
$488$	−53.1688	−2.40684
$489$	42.4648	1.92032
$490$	−8.32136	−0.375921
$491$	16.7778	0.757170	0.378585	−	0.925566i	$-0.376411\pi$
$491$	16.7778	0.757170	0.378585	+	0.925566i	$0.376411\pi$
$492$	−38.1150	−1.71836
$493$	7.27340	0.327577
$494$	29.9790	1.34882
$495$	0.0440694	0.00198077
$496$	−13.6749	−0.614020
$497$	−1.22891	−0.0551239
$498$	18.2505	0.817824
$499$	−25.1626	−1.12643	−0.563216	−	0.826309i	$-0.690436\pi$
$499$	−25.1626	−1.12643	−0.563216	+	0.826309i	$0.690436\pi$
$500$	17.8455	0.798076
$501$	−3.46397	−0.154759
$502$	−54.7721	−2.44460
$503$	1.50296	0.0670138	0.0335069	−	0.999438i	$-0.489332\pi$
$503$	1.50296	0.0670138	0.0335069	+	0.999438i	$0.489332\pi$
$504$	0.101981	0.00454259
$505$	5.83870	0.259819
$506$	−1.44671	−0.0643140
$507$	−1.95413	−0.0867860
$508$	−66.4414	−2.94786
$509$	10.1558	0.450146	0.225073	−	0.974342i	$-0.427738\pi$
$509$	10.1558	0.450146	0.225073	+	0.974342i	$0.427738\pi$
$510$	2.05788	0.0911243
$511$	0.408034	0.0180504
$512$	19.0261	0.840842
$513$	−17.7546	−0.783884
$514$	44.5126	1.96337
$515$	1.68537	0.0742662
$516$	6.13539	0.270096
$517$	−7.02890	−0.309131
$518$	4.44467	0.195288
$519$	−29.7895	−1.30761
$520$	7.22964	0.317041
$521$	15.9561	0.699051	0.349526	−	0.936927i	$-0.386343\pi$
$521$	15.9561	0.699051	0.349526	+	0.936927i	$0.386343\pi$
$522$	−1.48738	−0.0651010
$523$	−8.59964	−0.376036	−0.188018	−	0.982166i	$-0.560206\pi$
$523$	−8.59964	−0.376036	−0.188018	+	0.982166i	$0.560206\pi$
$524$	67.4075	2.94471
$525$	2.53064	0.110446
$526$	−67.5910	−2.94711
$527$	7.89935	0.344101
$528$	−2.95490	−0.128595
$529$	−22.6259	−0.983734
$530$	−5.12986	−0.222827
$531$	−0.813596	−0.0353070
$532$	−3.78875	−0.164263
$533$	23.3643	1.01202
$534$	−32.0442	−1.38669
$535$	6.78819	0.293479
$536$	−37.0718	−1.60126
$537$	−27.9305	−1.20529
$538$	−65.9262	−2.84228
$539$	−6.90218	−0.297298
$540$	−9.65234	−0.415371
$541$	29.5944	1.27236	0.636181	−	0.771540i	$-0.280513\pi$
$541$	29.5944	1.27236	0.636181	+	0.771540i	$0.280513\pi$
$542$	36.4066	1.56380
$543$	−20.3302	−0.872453
$544$	−3.44795	−0.147830
$545$	−3.66936	−0.157178
$546$	4.74911	0.203243
$547$	22.9565	0.981551	0.490776	−	0.871286i	$-0.336714\pi$
$547$	22.9565	0.981551	0.490776	+	0.871286i	$0.336714\pi$
$548$	25.3769	1.08405
$549$	−1.21892	−0.0520223
$550$	11.2118	0.478071
$551$	24.5119	1.04424
$552$	3.93732	0.167583
$553$	−4.10631	−0.174618
$554$	28.0502	1.19174
$555$	−5.22730	−0.221886
$556$	37.0744	1.57231
$557$	−10.2043	−0.432372	−0.216186	−	0.976352i	$-0.569362\pi$
$557$	−10.2043	−0.432372	−0.216186	+	0.976352i	$0.569362\pi$
$558$	−1.61539	−0.0683848
$559$	−3.76096	−0.159072
$560$	−0.275986	−0.0116625
$561$	1.70691	0.0720658
$562$	62.1212	2.62042
$563$	−22.8913	−0.964752	−0.482376	−	0.875964i	$-0.660226\pi$
$563$	−22.8913	−0.964752	−0.482376	+	0.875964i	$0.660226\pi$
$564$	43.1251	1.81589
$565$	−1.67772	−0.0705822
$566$	78.4881	3.29910
$567$	−2.73146	−0.114711
$568$	14.8177	0.621738
$569$	−17.4438	−0.731282	−0.365641	−	0.930756i	$-0.619150\pi$
$569$	−17.4438	−0.731282	−0.365641	+	0.930756i	$0.619150\pi$
$570$	6.93519	0.290483
$571$	35.6348	1.49127	0.745635	−	0.666355i	$-0.232146\pi$
$571$	35.6348	1.49127	0.745635	+	0.666355i	$0.232146\pi$
$572$	13.5186	0.565240
$573$	−24.2717	−1.01396
$574$	−4.59575	−0.191823
$575$	−2.89933	−0.120910
$576$	1.00444	0.0418515
$577$	25.8987	1.07818	0.539089	−	0.842249i	$-0.318769\pi$
$577$	25.8987	1.07818	0.539089	+	0.842249i	$0.318769\pi$
$578$	−2.36526	−0.0983818
$579$	12.4415	0.517052
$580$	13.3260	0.553331
$581$	1.41387	0.0586573
$582$	4.74309	0.196608
$583$	−4.25497	−0.176223
$584$	−4.91994	−0.203589
$585$	0.165743	0.00685264
$586$	−59.1327	−2.44275
$587$	−17.2978	−0.713957	−0.356978	−	0.934113i	$-0.616193\pi$
$587$	−17.2978	−0.713957	−0.356978	+	0.934113i	$0.616193\pi$
$588$	42.3476	1.74638
$589$	26.6214	1.09691
$590$	11.3451	0.467072
$591$	36.9980	1.52190
$592$	−10.4008	−0.427472
$593$	3.05149	0.125310	0.0626549	−	0.998035i	$-0.480043\pi$
$593$	3.05149	0.125310	0.0626549	+	0.998035i	$0.480043\pi$
$594$	−12.4609	−0.511277
$595$	0.159424	0.00653576
$596$	34.9958	1.43348
$597$	38.9240	1.59305
$598$	−5.44101	−0.222499
$599$	41.6329	1.70107	0.850537	−	0.525915i	$-0.176277\pi$
$599$	41.6329	1.70107	0.850537	+	0.525915i	$0.176277\pi$
$600$	−30.5136	−1.24571
$601$	43.9680	1.79349	0.896746	−	0.442545i	$-0.145924\pi$
$601$	43.9680	1.79349	0.896746	+	0.442545i	$0.145924\pi$
$602$	0.739780	0.0301512
$603$	−0.849891	−0.0346102
$604$	60.5908	2.46540
$605$	−0.509718	−0.0207230
$606$	−46.2461	−1.87862
$607$	16.5287	0.670881	0.335441	−	0.942061i	$-0.391115\pi$
$607$	16.5287	0.670881	0.335441	+	0.942061i	$0.391115\pi$
$608$	−11.6198	−0.471247
$609$	3.88304	0.157349
$610$	16.9972	0.688196
$611$	−26.4354	−1.06946
$612$	0.310770	0.0125621
$613$	−25.6612	−1.03644	−0.518222	−	0.855246i	$-0.673406\pi$
$613$	−25.6612	−1.03644	−0.518222	+	0.855246i	$0.673406\pi$
$614$	−0.0457014	−0.00184436
$615$	5.40498	0.217950
$616$	−1.17954	−0.0475250
$617$	7.89400	0.317800	0.158900	−	0.987295i	$-0.449205\pi$
$617$	7.89400	0.317800	0.158900	+	0.987295i	$0.449205\pi$
$618$	−13.3492	−0.536982
$619$	−12.7922	−0.514163	−0.257081	−	0.966390i	$-0.582761\pi$
$619$	−12.7922	−0.514163	−0.257081	+	0.966390i	$0.582761\pi$
$620$	14.4728	0.581242
$621$	3.22235	0.129309
$622$	−53.7893	−2.15675
$623$	−2.48248	−0.0994583
$624$	−11.1133	−0.444887
$625$	21.1703	0.846813
$626$	−22.6021	−0.903361
$627$	5.75241	0.229729
$628$	45.2526	1.80577
$629$	6.00809	0.239558
$630$	−0.0326017	−0.00129888
$631$	49.4310	1.96782	0.983908	−	0.178677i	$-0.0571818\pi$
$631$	49.4310	1.96782	0.983908	+	0.178677i	$0.0571818\pi$
$632$	49.5125	1.96950
$633$	18.1628	0.721907
$634$	54.5464	2.16631
$635$	9.42187	0.373896
$636$	26.1059	1.03517
$637$	−25.9588	−1.02853
$638$	17.2035	0.681091
$639$	0.339704	0.0134385
$640$	−10.4913	−0.414707
$641$	−38.3754	−1.51574	−0.757868	−	0.652408i	$-0.773759\pi$
$641$	−38.3754	−1.51574	−0.757868	+	0.652408i	$0.773759\pi$
$642$	−53.7666	−2.12200
$643$	30.5589	1.20513	0.602563	−	0.798071i	$-0.294146\pi$
$643$	30.5589	1.20513	0.602563	+	0.798071i	$0.294146\pi$
$644$	0.687635	0.0270966
$645$	−0.870043	−0.0342579
$646$	−7.97109	−0.313618
$647$	7.59646	0.298648	0.149324	−	0.988788i	$-0.452290\pi$
$647$	7.59646	0.298648	0.149324	+	0.988788i	$0.452290\pi$
$648$	32.9351	1.29381
$649$	9.41026	0.369385
$650$	42.1670	1.65393
$651$	4.21722	0.165286
$652$	−89.4232	−3.50208
$653$	−2.12704	−0.0832375	−0.0416188	−	0.999134i	$-0.513251\pi$
$653$	−2.12704	−0.0832375	−0.0416188	+	0.999134i	$0.513251\pi$
$654$	29.0636	1.13648
$655$	−9.55887	−0.373496
$656$	10.7544	0.419888
$657$	−0.112792	−0.00440045
$658$	5.19984	0.202711
$659$	−25.9194	−1.00968	−0.504839	−	0.863214i	$-0.668448\pi$
$659$	−25.9194	−1.00968	−0.504839	+	0.863214i	$0.668448\pi$
$660$	3.12732	0.121731
$661$	−34.0472	−1.32428	−0.662142	−	0.749379i	$-0.730352\pi$
$661$	−34.0472	−1.32428	−0.662142	+	0.749379i	$0.730352\pi$
$662$	0.292147	0.0113546
$663$	6.41962	0.249318
$664$	−17.0480	−0.661591
$665$	0.537272	0.0208345
$666$	−1.22863	−0.0476086
$667$	−4.44876	−0.172257
$668$	7.29450	0.282233
$669$	26.0914	1.00875
$670$	11.8513	0.457854
$671$	14.0984	0.544261
$672$	−1.84076	−0.0710087
$673$	−40.8975	−1.57648	−0.788241	−	0.615367i	$-0.789008\pi$
$673$	−40.8975	−1.57648	−0.788241	+	0.615367i	$0.789008\pi$
$674$	−15.6925	−0.604453
$675$	−24.9728	−0.961202
$676$	4.11505	0.158271
$677$	35.8741	1.37875	0.689376	−	0.724403i	$-0.257885\pi$
$677$	35.8741	1.37875	0.689376	+	0.724403i	$0.257885\pi$
$678$	13.2886	0.510345
$679$	0.367449	0.0141014
$680$	−1.92229	−0.0737163
$681$	0.435664	0.0166947
$682$	18.6840	0.715447
$683$	28.5667	1.09307	0.546537	−	0.837435i	$-0.315946\pi$
$683$	28.5667	1.09307	0.546537	+	0.837435i	$0.315946\pi$
$684$	1.04732	0.0400452
$685$	−3.59862	−0.137496
$686$	10.2846	0.392666
$687$	21.3783	0.815635
$688$	−1.73114	−0.0659991
$689$	−16.0028	−0.609658
$690$	−1.25870	−0.0479178
$691$	37.5901	1.42999	0.714997	−	0.699128i	$-0.246428\pi$
$691$	37.5901	1.42999	0.714997	+	0.699128i	$0.246428\pi$
$692$	62.7313	2.38468
$693$	−0.0270415	−0.00102722
$694$	25.8789	0.982349
$695$	−5.25742	−0.199425
$696$	−46.8204	−1.77472
$697$	−6.21231	−0.235308
$698$	−46.7732	−1.77039
$699$	15.5103	0.586652
$700$	−5.32907	−0.201420
$701$	26.6006	1.00469	0.502345	−	0.864667i	$-0.332471\pi$
$701$	26.6006	1.00469	0.502345	+	0.864667i	$0.332471\pi$
$702$	−46.8650	−1.76880
$703$	20.2477	0.763657
$704$	−11.6176	−0.437854
$705$	−6.11544	−0.230321
$706$	69.8870	2.63023
$707$	−3.58270	−0.134741
$708$	−57.7356	−2.16984
$709$	22.4671	0.843768	0.421884	−	0.906650i	$-0.361369\pi$
$709$	22.4671	0.843768	0.421884	+	0.906650i	$0.361369\pi$
$710$	−4.73699	−0.177776
$711$	1.13510	0.0425696
$712$	29.9329	1.12178
$713$	−4.83162	−0.180946
$714$	−1.26274	−0.0472568
$715$	−1.91703	−0.0716929
$716$	58.8165	2.19808
$717$	22.7355	0.849072
$718$	−15.7533	−0.587909
$719$	14.5171	0.541397	0.270699	−	0.962664i	$-0.412745\pi$
$719$	14.5171	0.541397	0.270699	+	0.962664i	$0.412745\pi$
$720$	0.0762903	0.00284317
$721$	−1.03416	−0.0385143
$722$	18.0767	0.672746
$723$	−0.536801	−0.0199639
$724$	42.8118	1.59109
$725$	34.4773	1.28045
$726$	4.03728	0.149838
$727$	21.5607	0.799642	0.399821	−	0.916593i	$-0.369072\pi$
$727$	21.5607	0.799642	0.399821	+	0.916593i	$0.369072\pi$
$728$	−4.43620	−0.164416
$729$	27.7326	1.02713
$730$	1.57283	0.0582129
$731$	1.00000	0.0369863
$732$	−86.4990	−3.19709
$733$	49.1129	1.81403	0.907013	−	0.421103i	$-0.138357\pi$
$733$	49.1129	1.81403	0.907013	+	0.421103i	$0.138357\pi$
$734$	−76.4896	−2.82328
$735$	−6.00519	−0.221505
$736$	2.10893	0.0777364
$737$	9.83006	0.362095
$738$	1.27040	0.0467639
$739$	34.3454	1.26342	0.631709	−	0.775206i	$-0.282354\pi$
$739$	34.3454	1.26342	0.631709	+	0.775206i	$0.282354\pi$
$740$	11.0077	0.404653
$741$	21.6346	0.794767
$742$	3.14774	0.115557
$743$	−24.7269	−0.907141	−0.453571	−	0.891220i	$-0.649850\pi$
$743$	−24.7269	−0.907141	−0.453571	+	0.891220i	$0.649850\pi$
$744$	−50.8498	−1.86425
$745$	−4.96266	−0.181818
$746$	17.1044	0.626237
$747$	−0.390834	−0.0142999
$748$	−3.59444	−0.131426
$749$	−4.16532	−0.152197
$750$	20.0441	0.731907
$751$	8.68683	0.316987	0.158493	−	0.987360i	$-0.449336\pi$
$751$	8.68683	0.316987	0.158493	+	0.987360i	$0.449336\pi$
$752$	−12.1680	−0.443722
$753$	−39.5268	−1.44044
$754$	64.7015	2.35629
$755$	−8.59221	−0.312703
$756$	5.92280	0.215410
$757$	−36.4264	−1.32394	−0.661970	−	0.749530i	$-0.730279\pi$
$757$	−36.4264	−1.32394	−0.661970	+	0.749530i	$0.730279\pi$
$758$	30.8320	1.11987
$759$	−1.04403	−0.0378959
$760$	−6.47824	−0.234991
$761$	−27.3623	−0.991884	−0.495942	−	0.868356i	$-0.665177\pi$
$761$	−27.3623	−0.991884	−0.495942	+	0.868356i	$0.665177\pi$
$762$	−74.6270	−2.70345
$763$	2.25156	0.0815120
$764$	51.1117	1.84916
$765$	−0.0440694	−0.00159333
$766$	−66.4682	−2.40159
$767$	35.3916	1.27792
$768$	43.4377	1.56742
$769$	1.11155	0.0400836	0.0200418	−	0.999799i	$-0.493620\pi$
$769$	1.11155	0.0400836	0.0200418	+	0.999799i	$0.493620\pi$
$770$	0.377079	0.0135890
$771$	32.1229	1.15688
$772$	−26.1996	−0.942944
$773$	24.6661	0.887179	0.443589	−	0.896230i	$-0.353705\pi$
$773$	24.6661	0.887179	0.443589	+	0.896230i	$0.353705\pi$
$774$	−0.204496	−0.00735047
$775$	37.4444	1.34504
$776$	−4.43058	−0.159048
$777$	3.20754	0.115070
$778$	7.85747	0.281704
$779$	−20.9360	−0.750109
$780$	11.7617	0.421138
$781$	−3.92911	−0.140595
$782$	1.44671	0.0517341
$783$	−38.3185	−1.36939
$784$	−11.9486	−0.426737
$785$	−6.41714	−0.229038
$786$	75.7122	2.70056
$787$	27.8373	0.992293	0.496146	−	0.868239i	$-0.334748\pi$
$787$	27.8373	0.992293	0.496146	+	0.868239i	$0.334748\pi$
$788$	−77.9111	−2.77547
$789$	−48.7777	−1.73653
$790$	−15.8284	−0.563148
$791$	1.02947	0.0366038
$792$	0.326058	0.0115860
$793$	53.0234	1.88292
$794$	72.5394	2.57433
$795$	−3.70201	−0.131297
$796$	−81.9670	−2.90524
$797$	−11.2323	−0.397868	−0.198934	−	0.980013i	$-0.563748\pi$
$797$	−11.2323	−0.397868	−0.198934	+	0.980013i	$0.563748\pi$
$798$	−4.25552	−0.150644
$799$	7.02890	0.248665
$800$	−16.3439	−0.577845
$801$	0.686227	0.0242466
$802$	−29.6698	−1.04768
$803$	1.30459	0.0460378
$804$	−60.3113	−2.12702
$805$	−0.0975116	−0.00343683
$806$	70.2698	2.47515
$807$	−47.5763	−1.67476
$808$	43.1990	1.51974
$809$	35.6915	1.25485	0.627424	−	0.778678i	$-0.284109\pi$
$809$	35.6915	1.25485	0.627424	+	0.778678i	$0.284109\pi$
$810$	−10.5288	−0.369945
$811$	−29.3774	−1.03158	−0.515789	−	0.856715i	$-0.672501\pi$
$811$	−29.3774	−1.03158	−0.515789	+	0.856715i	$0.672501\pi$
$812$	−8.17699	−0.286956
$813$	26.2732	0.921440
$814$	14.2107	0.498084
$815$	12.6809	0.444191
$816$	2.95490	0.103442
$817$	3.37007	0.117904
$818$	−42.8916	−1.49967
$819$	−0.101702	−0.00355376
$820$	−11.3819	−0.397474
$821$	−2.55006	−0.0889978	−0.0444989	−	0.999009i	$-0.514169\pi$
$821$	−2.55006	−0.0889978	−0.0444989	+	0.999009i	$0.514169\pi$
$822$	28.5033	0.994168
$823$	37.1014	1.29327	0.646637	−	0.762798i	$-0.276175\pi$
$823$	37.1014	1.29327	0.646637	+	0.762798i	$0.276175\pi$
$824$	12.4696	0.434399
$825$	8.09107	0.281695
$826$	−6.96152	−0.242222
$827$	−25.2824	−0.879157	−0.439578	−	0.898204i	$-0.644872\pi$
$827$	−25.2824	−0.879157	−0.439578	+	0.898204i	$0.644872\pi$
$828$	−0.190082	−0.00660580
$829$	4.68508	0.162720	0.0813598	−	0.996685i	$-0.474074\pi$
$829$	4.68508	0.162720	0.0813598	+	0.996685i	$0.474074\pi$
$830$	5.44997	0.189171
$831$	20.2427	0.702211
$832$	−43.6932	−1.51479
$833$	6.90218	0.239146
$834$	41.6420	1.44195
$835$	−1.03441	−0.0357973
$836$	−12.1135	−0.418956
$837$	−41.6162	−1.43847
$838$	−6.34725	−0.219262
$839$	41.7286	1.44063	0.720315	−	0.693647i	$-0.243997\pi$
$839$	41.7286	1.44063	0.720315	+	0.693647i	$0.243997\pi$
$840$	−1.02625	−0.0354090
$841$	23.9023	0.824217
$842$	50.8973	1.75404
$843$	44.8303	1.54404
$844$	−38.2476	−1.31654
$845$	−0.583543	−0.0200745
$846$	−1.43738	−0.0494183
$847$	0.312769	0.0107469
$848$	−7.36595	−0.252948
$849$	56.6417	1.94394
$850$	−11.2118	−0.384560
$851$	−3.67484	−0.125972
$852$	24.1066	0.825879
$853$	−2.82726	−0.0968035	−0.0484017	−	0.998828i	$-0.515413\pi$
$853$	−2.82726	−0.0968035	−0.0484017	+	0.998828i	$0.515413\pi$
$854$	−10.4297	−0.356897
$855$	−0.148517	−0.00507918
$856$	50.2240	1.71662
$857$	34.6563	1.18384	0.591919	−	0.805998i	$-0.298371\pi$
$857$	34.6563	1.18384	0.591919	+	0.805998i	$0.298371\pi$
$858$	15.1841	0.518375
$859$	−20.5264	−0.700350	−0.350175	−	0.936684i	$-0.613878\pi$
$859$	−20.5264	−0.700350	−0.350175	+	0.936684i	$0.613878\pi$
$860$	1.83215	0.0624759
$861$	−3.31656	−0.113028
$862$	−73.2437	−2.49469
$863$	34.2767	1.16679	0.583396	−	0.812188i	$-0.301724\pi$
$863$	34.2767	1.16679	0.583396	+	0.812188i	$0.301724\pi$
$864$	18.1649	0.617981
$865$	−8.89574	−0.302464
$866$	74.3188	2.52546
$867$	−1.70691	−0.0579697
$868$	−8.88070	−0.301431
$869$	−13.1289	−0.445367
$870$	14.9677	0.507454
$871$	36.9705	1.25270
$872$	−27.1486	−0.919368
$873$	−0.101573	−0.00343774
$874$	4.87551	0.164917
$875$	1.55282	0.0524950
$876$	−8.00414	−0.270435
$877$	−27.4380	−0.926516	−0.463258	−	0.886223i	$-0.653320\pi$
$877$	−27.4380	−0.926516	−0.463258	+	0.886223i	$0.653320\pi$
$878$	30.2548	1.02105
$879$	−42.6737	−1.43935
$880$	−0.882393	−0.0297455
$881$	47.7276	1.60798	0.803991	−	0.594641i	$-0.202706\pi$
$881$	47.7276	1.60798	0.803991	+	0.594641i	$0.202706\pi$
$882$	−1.41147	−0.0475266
$883$	30.4132	1.02349	0.511743	−	0.859139i	$-0.329000\pi$
$883$	30.4132	1.02349	0.511743	+	0.859139i	$0.329000\pi$
$884$	−13.5186	−0.454679
$885$	8.18733	0.275214
$886$	−20.9027	−0.702240
$887$	−4.22268	−0.141784	−0.0708918	−	0.997484i	$-0.522585\pi$
$887$	−4.22268	−0.141784	−0.0708918	+	0.997484i	$0.522585\pi$
$888$	−38.6754	−1.29786
$889$	−5.78138	−0.193901
$890$	−9.56906	−0.320756
$891$	−8.73315	−0.292572
$892$	−54.9438	−1.83965
$893$	23.6879	0.792686
$894$	39.3073	1.31463
$895$	−8.34060	−0.278796
$896$	6.43762	0.215066
$897$	−3.92655	−0.131104
$898$	11.2112	0.374124
$899$	57.4551	1.91623
$900$	1.47311	0.0491036
$901$	4.25497	0.141754
$902$	−14.6937	−0.489247
$903$	0.533869	0.0177661
$904$	−12.4130	−0.412851
$905$	−6.07102	−0.201807
$906$	68.0556	2.26100
$907$	15.3236	0.508813	0.254407	−	0.967097i	$-0.418120\pi$
$907$	15.3236	0.508813	0.254407	+	0.967097i	$0.418120\pi$
$908$	−0.917429	−0.0304460
$909$	0.990360	0.0328482
$910$	1.41818	0.0470123
$911$	12.3571	0.409409	0.204705	−	0.978824i	$-0.434377\pi$
$911$	12.3571	0.409409	0.204705	+	0.978824i	$0.434377\pi$
$912$	9.95823	0.329750
$913$	4.52049	0.149606
$914$	−47.7815	−1.58047
$915$	12.2662	0.405507
$916$	−45.0189	−1.48747
$917$	5.86544	0.193694
$918$	12.4609	0.411271
$919$	−32.9133	−1.08571	−0.542854	−	0.839827i	$-0.682656\pi$
$919$	−32.9133	−1.08571	−0.542854	+	0.839827i	$0.682656\pi$
$920$	1.17576	0.0387638
$921$	−0.0329809	−0.00108676
$922$	33.5991	1.10653
$923$	−14.7772	−0.486398
$924$	−1.91896	−0.0631293
$925$	28.4795	0.936400
$926$	38.6894	1.27141
$927$	0.285872	0.00938928
$928$	−25.0783	−0.823236
$929$	9.54867	0.313282	0.156641	−	0.987656i	$-0.449933\pi$
$929$	9.54867	0.313282	0.156641	+	0.987656i	$0.449933\pi$
$930$	16.2559	0.533051
$931$	23.2608	0.762343
$932$	−32.6618	−1.06987
$933$	−38.8175	−1.27083
$934$	−28.5927	−0.935583
$935$	0.509718	0.0166696
$936$	1.22629	0.0400826
$937$	−7.39221	−0.241493	−0.120746	−	0.992683i	$-0.538529\pi$
$937$	−7.39221	−0.241493	−0.120746	+	0.992683i	$0.538529\pi$
$938$	−7.27209	−0.237442
$939$	−16.3110	−0.532290
$940$	12.8780	0.420035
$941$	−10.3752	−0.338223	−0.169112	−	0.985597i	$-0.554090\pi$
$941$	−10.3752	−0.338223	−0.169112	+	0.985597i	$0.554090\pi$
$942$	50.8277	1.65606
$943$	3.79975	0.123737
$944$	16.2905	0.530210
$945$	−0.839896	−0.0273218
$946$	2.36526	0.0769012
$947$	−33.4153	−1.08585	−0.542925	−	0.839781i	$-0.682683\pi$
$947$	−33.4153	−1.08585	−0.542925	+	0.839781i	$0.682683\pi$
$948$	80.5508	2.61617
$949$	4.90650	0.159272
$950$	−37.7845	−1.22589
$951$	39.3639	1.27646
$952$	1.17954	0.0382291
$953$	−26.0232	−0.842974	−0.421487	−	0.906834i	$-0.638492\pi$
$953$	−26.0232	−0.842974	−0.421487	+	0.906834i	$0.638492\pi$
$954$	−0.870126	−0.0281714
$955$	−7.24801	−0.234540
$956$	−47.8768	−1.54845
$957$	12.4150	0.401321
$958$	67.5390	2.18209
$959$	2.20816	0.0713053
$960$	−10.1078	−0.326227
$961$	31.3997	1.01289
$962$	53.4459	1.72316
$963$	1.15141	0.0371037
$964$	1.13041	0.0364080
$965$	3.71529	0.119600
$966$	0.772352	0.0248500
$967$	−21.6294	−0.695555	−0.347778	−	0.937577i	$-0.613064\pi$
$967$	−21.6294	−0.695555	−0.347778	+	0.937577i	$0.613064\pi$
$968$	−3.77127	−0.121213
$969$	−5.75241	−0.184794
$970$	1.41638	0.0454774
$971$	−26.7593	−0.858746	−0.429373	−	0.903127i	$-0.641265\pi$
$971$	−26.7593	−0.858746	−0.429373	+	0.903127i	$0.641265\pi$
$972$	−3.22860	−0.103557
$973$	3.22602	0.103421
$974$	1.70180	0.0545292
$975$	30.4302	0.974547
$976$	24.4062	0.781225
$977$	41.8826	1.33994	0.669971	−	0.742387i	$-0.266306\pi$
$977$	41.8826	1.33994	0.669971	+	0.742387i	$0.266306\pi$
$978$	−100.440	−3.21172
$979$	−7.93708	−0.253670
$980$	12.6458	0.403957
$981$	−0.622396	−0.0198716
$982$	−39.6838	−1.26636
$983$	30.0906	0.959743	0.479871	−	0.877339i	$-0.340683\pi$
$983$	30.0906	0.959743	0.479871	+	0.877339i	$0.340683\pi$
$984$	39.9900	1.27484
$985$	11.0484	0.352030
$986$	−17.2035	−0.547870
$987$	3.75251	0.119444
$988$	−45.5586	−1.44941
$989$	−0.611649	−0.0194493
$990$	−0.104235	−0.00331282
$991$	28.9251	0.918837	0.459419	−	0.888220i	$-0.348058\pi$
$991$	28.9251	0.918837	0.459419	+	0.888220i	$0.348058\pi$
$992$	−27.2366	−0.864762
$993$	0.210831	0.00669051
$994$	2.90668	0.0921942
$995$	11.6235	0.368490
$996$	−27.7350	−0.878817
$997$	39.2762	1.24389	0.621945	−	0.783061i	$-0.286343\pi$
$997$	39.2762	1.24389	0.621945	+	0.783061i	$0.286343\pi$
$998$	59.5160	1.88395
$999$	−31.6525	−1.00144

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.i.1.7	✓	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.i.1.7	✓	78	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36526 q^{2} -1.70691 q^{3} +3.59444 q^{4} -0.509718 q^{5} +4.03728 q^{6} +0.312769 q^{7} -3.77127 q^{8} -0.0864584 q^{9} +O(q^{10})\)	\(q-2.36526 q^{2} -1.70691 q^{3} +3.59444 q^{4} -0.509718 q^{5} +4.03728 q^{6} +0.312769 q^{7} -3.77127 q^{8} -0.0864584 q^{9} +1.20561 q^{10} +1.00000 q^{11} -6.13539 q^{12} +3.76096 q^{13} -0.739780 q^{14} +0.870043 q^{15} +1.73114 q^{16} -1.00000 q^{17} +0.204496 q^{18} -3.37007 q^{19} -1.83215 q^{20} -0.533869 q^{21} -2.36526 q^{22} +0.611649 q^{23} +6.43722 q^{24} -4.74019 q^{25} -8.89564 q^{26} +5.26831 q^{27} +1.12423 q^{28} -7.27340 q^{29} -2.05788 q^{30} -7.89935 q^{31} +3.44795 q^{32} -1.70691 q^{33} +2.36526 q^{34} -0.159424 q^{35} -0.310770 q^{36} -6.00809 q^{37} +7.97109 q^{38} -6.41962 q^{39} +1.92229 q^{40} +6.21231 q^{41} +1.26274 q^{42} -1.00000 q^{43} +3.59444 q^{44} +0.0440694 q^{45} -1.44671 q^{46} -7.02890 q^{47} -2.95490 q^{48} -6.90218 q^{49} +11.2118 q^{50} +1.70691 q^{51} +13.5186 q^{52} -4.25497 q^{53} -12.4609 q^{54} -0.509718 q^{55} -1.17954 q^{56} +5.75241 q^{57} +17.2035 q^{58} +9.41026 q^{59} +3.12732 q^{60} +14.0984 q^{61} +18.6840 q^{62} -0.0270415 q^{63} -11.6176 q^{64} -1.91703 q^{65} +4.03728 q^{66} +9.83006 q^{67} -3.59444 q^{68} -1.04403 q^{69} +0.377079 q^{70} -3.92911 q^{71} +0.326058 q^{72} +1.30459 q^{73} +14.2107 q^{74} +8.09107 q^{75} -12.1135 q^{76} +0.312769 q^{77} +15.1841 q^{78} -13.1289 q^{79} -0.882393 q^{80} -8.73315 q^{81} -14.6937 q^{82} +4.52049 q^{83} -1.91896 q^{84} +0.509718 q^{85} +2.36526 q^{86} +12.4150 q^{87} -3.77127 q^{88} -7.93708 q^{89} -0.104235 q^{90} +1.17631 q^{91} +2.19854 q^{92} +13.4835 q^{93} +16.6252 q^{94} +1.71779 q^{95} -5.88534 q^{96} +1.17482 q^{97} +16.3254 q^{98} -0.0864584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more