LMFDB - Embedded newform 1339.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73240 q^{2} +1.73806 q^{3} +1.00120 q^{4} -1.26585 q^{5} -3.01101 q^{6} -1.31498 q^{7} +1.73032 q^{8} +0.0208578 q^{9} +O(q^{10})$	\(q-1.73240 q^{2} +1.73806 q^{3} +1.00120 q^{4} -1.26585 q^{5} -3.01101 q^{6} -1.31498 q^{7} +1.73032 q^{8} +0.0208578 q^{9} +2.19296 q^{10} -0.168710 q^{11} +1.74015 q^{12} +1.00000 q^{13} +2.27807 q^{14} -2.20013 q^{15} -5.00000 q^{16} +0.783074 q^{17} -0.0361340 q^{18} +4.19515 q^{19} -1.26737 q^{20} -2.28552 q^{21} +0.292272 q^{22} +3.36329 q^{23} +3.00740 q^{24} -3.39762 q^{25} -1.73240 q^{26} -5.17793 q^{27} -1.31656 q^{28} -3.38214 q^{29} +3.81150 q^{30} +4.82680 q^{31} +5.20134 q^{32} -0.293228 q^{33} -1.35660 q^{34} +1.66457 q^{35} +0.0208828 q^{36} +2.27401 q^{37} -7.26766 q^{38} +1.73806 q^{39} -2.19033 q^{40} +10.1995 q^{41} +3.95943 q^{42} +3.41089 q^{43} -0.168912 q^{44} -0.0264029 q^{45} -5.82656 q^{46} +1.90900 q^{47} -8.69031 q^{48} -5.27082 q^{49} +5.88602 q^{50} +1.36103 q^{51} +1.00120 q^{52} +7.98623 q^{53} +8.97023 q^{54} +0.213562 q^{55} -2.27534 q^{56} +7.29143 q^{57} +5.85920 q^{58} +5.17023 q^{59} -2.20277 q^{60} -1.18129 q^{61} -8.36193 q^{62} -0.0274276 q^{63} +0.989202 q^{64} -1.26585 q^{65} +0.507988 q^{66} -2.88500 q^{67} +0.784014 q^{68} +5.84561 q^{69} -2.88370 q^{70} +8.20953 q^{71} +0.0360907 q^{72} +8.14565 q^{73} -3.93948 q^{74} -5.90527 q^{75} +4.20018 q^{76} +0.221850 q^{77} -3.01101 q^{78} +10.1327 q^{79} +6.32926 q^{80} -9.06214 q^{81} -17.6696 q^{82} +15.0733 q^{83} -2.28826 q^{84} -0.991256 q^{85} -5.90902 q^{86} -5.87836 q^{87} -0.291922 q^{88} -9.54872 q^{89} +0.0457403 q^{90} -1.31498 q^{91} +3.36733 q^{92} +8.38927 q^{93} -3.30715 q^{94} -5.31044 q^{95} +9.04026 q^{96} -0.409630 q^{97} +9.13116 q^{98} -0.00351892 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) $ 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9 + 5 * q^10 + 9 * q^11 + 15 * q^12 + 28 * q^13 + 20 * q^14 + 4 * q^15 + 32 * q^16 - 7 * q^17 + 10 * q^18 - 3 * q^19 + 53 * q^20 + 25 * q^21 - 14 * q^22 - 6 * q^23 + 10 * q^24 + 45 * q^25 + 6 * q^26 + 8 * q^27 - 12 * q^28 + 25 * q^29 - 43 * q^30 + q^31 + 28 * q^32 + 17 * q^33 + 10 * q^34 + 11 * q^35 + 27 * q^36 + 18 * q^37 + 2 * q^38 + 5 * q^39 + 37 * q^40 + 56 * q^41 - 29 * q^42 - 30 * q^43 + 30 * q^44 + 38 * q^45 - 27 * q^46 + 40 * q^47 + 19 * q^48 + 36 * q^49 + 20 * q^50 - 18 * q^51 + 32 * q^52 + 51 * q^53 - 36 * q^54 - 5 * q^55 - 13 * q^56 + 31 * q^57 - 29 * q^58 + 29 * q^59 + 48 * q^60 + 16 * q^61 - 38 * q^62 - 23 * q^63 - 5 * q^64 + 27 * q^65 + 25 * q^66 - 2 * q^67 - 43 * q^68 + 38 * q^69 + 10 * q^70 + 34 * q^71 - 28 * q^72 + 14 * q^73 + 21 * q^74 + 19 * q^75 + 5 * q^76 + 34 * q^77 + 9 * q^78 + 3 * q^79 + 58 * q^80 + 4 * q^81 - 9 * q^82 + 38 * q^83 - 78 * q^84 - 27 * q^85 + 49 * q^86 + 8 * q^87 - 25 * q^88 + 125 * q^89 - 74 * q^90 + 6 * q^91 - 35 * q^92 + 36 * q^93 - q^94 - 12 * q^95 + 26 * q^96 - 2 * q^97 + 18 * q^98 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.73240	−1.22499	−0.612495	−	0.790475i	$-0.709834\pi$
$2$	−1.73240	−1.22499	−0.612495	+	0.790475i	$0.709834\pi$
$3$	1.73806	1.00347	0.501735	−	0.865021i	$-0.332695\pi$
$3$	1.73806	1.00347	0.501735	+	0.865021i	$0.332695\pi$
$4$	1.00120	0.500600
$5$	−1.26585	−0.566106	−0.283053	−	0.959104i	$-0.591347\pi$
$5$	−1.26585	−0.566106	−0.283053	+	0.959104i	$0.591347\pi$
$6$	−3.01101	−1.22924
$7$	−1.31498	−0.497016	−0.248508	−	0.968630i	$-0.579940\pi$
$7$	−1.31498	−0.497016	−0.248508	+	0.968630i	$0.579940\pi$
$8$	1.73032	0.611760
$9$	0.0208578	0.00695260
$10$	2.19296	0.693474
$11$	−0.168710	−0.0508679	−0.0254340	−	0.999677i	$-0.508097\pi$
$11$	−0.168710	−0.0508679	−0.0254340	+	0.999677i	$0.508097\pi$
$12$	1.74015	0.502337
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.27807	0.608840
$15$	−2.20013	−0.568071
$16$	−5.00000	−1.25000
$17$	0.783074	0.189923	0.0949617	−	0.995481i	$-0.469727\pi$
$17$	0.783074	0.189923	0.0949617	+	0.995481i	$0.469727\pi$
$18$	−0.0361340	−0.00851687
$19$	4.19515	0.962433	0.481216	−	0.876602i	$-0.340195\pi$
$19$	4.19515	0.962433	0.481216	+	0.876602i	$0.340195\pi$
$20$	−1.26737	−0.283393
$21$	−2.28552	−0.498741
$22$	0.292272	0.0623127
$23$	3.36329	0.701295	0.350647	−	0.936508i	$-0.385962\pi$
$23$	3.36329	0.701295	0.350647	+	0.936508i	$0.385962\pi$
$24$	3.00740	0.613883
$25$	−3.39762	−0.679524
$26$	−1.73240	−0.339751
$27$	−5.17793	−0.996494
$28$	−1.31656	−0.248806
$29$	−3.38214	−0.628047	−0.314023	−	0.949415i	$-0.601677\pi$
$29$	−3.38214	−0.628047	−0.314023	+	0.949415i	$0.601677\pi$
$30$	3.81150	0.695881
$31$	4.82680	0.866919	0.433459	−	0.901173i	$-0.357293\pi$
$31$	4.82680	0.866919	0.433459	+	0.901173i	$0.357293\pi$
$32$	5.20134	0.919477
$33$	−0.293228	−0.0510445
$34$	−1.35660	−0.232654
$35$	1.66457	0.281364
$36$	0.0208828	0.00348047
$37$	2.27401	0.373845	0.186922	−	0.982375i	$-0.440149\pi$
$37$	2.27401	0.373845	0.186922	+	0.982375i	$0.440149\pi$
$38$	−7.26766	−1.17897
$39$	1.73806	0.278313
$40$	−2.19033	−0.346321
$41$	10.1995	1.59289	0.796447	−	0.604708i	$-0.206710\pi$
$41$	10.1995	1.59289	0.796447	+	0.604708i	$0.206710\pi$
$42$	3.95943	0.610953
$43$	3.41089	0.520156	0.260078	−	0.965588i	$-0.416252\pi$
$43$	3.41089	0.520156	0.260078	+	0.965588i	$0.416252\pi$
$44$	−0.168912	−0.0254645
$45$	−0.0264029	−0.00393591
$46$	−5.82656	−0.859079
$47$	1.90900	0.278457	0.139228	−	0.990260i	$-0.455538\pi$
$47$	1.90900	0.278457	0.139228	+	0.990260i	$0.455538\pi$
$48$	−8.69031	−1.25434
$49$	−5.27082	−0.752975
$50$	5.88602	0.832410
$51$	1.36103	0.190582
$52$	1.00120	0.138841
$53$	7.98623	1.09699	0.548497	−	0.836153i	$-0.315200\pi$
$53$	7.98623	1.09699	0.548497	+	0.836153i	$0.315200\pi$
$54$	8.97023	1.22069
$55$	0.213562	0.0287967
$56$	−2.27534	−0.304055
$57$	7.29143	0.965773
$58$	5.85920	0.769351
$59$	5.17023	0.673106	0.336553	−	0.941664i	$-0.390739\pi$
$59$	5.17023	0.673106	0.336553	+	0.941664i	$0.390739\pi$
$60$	−2.20277	−0.284376
$61$	−1.18129	−0.151248	−0.0756242	−	0.997136i	$-0.524095\pi$
$61$	−1.18129	−0.151248	−0.0756242	+	0.997136i	$0.524095\pi$
$62$	−8.36193	−1.06197
$63$	−0.0274276	−0.00345556
$64$	0.989202	0.123650
$65$	−1.26585	−0.157010
$66$	0.507988	0.0625289
$67$	−2.88500	−0.352459	−0.176230	−	0.984349i	$-0.556390\pi$
$67$	−2.88500	−0.352459	−0.176230	+	0.984349i	$0.556390\pi$
$68$	0.784014	0.0950756
$69$	5.84561	0.703729
$70$	−2.88370	−0.344668
$71$	8.20953	0.974292	0.487146	−	0.873320i	$-0.338038\pi$
$71$	8.20953	0.974292	0.487146	+	0.873320i	$0.338038\pi$
$72$	0.0360907	0.00425333
$73$	8.14565	0.953376	0.476688	−	0.879072i	$-0.341837\pi$
$73$	8.14565	0.953376	0.476688	+	0.879072i	$0.341837\pi$
$74$	−3.93948	−0.457956
$75$	−5.90527	−0.681882
$76$	4.20018	0.481794
$77$	0.221850	0.0252822
$78$	−3.01101	−0.340930
$79$	10.1327	1.14001	0.570007	−	0.821640i	$-0.306940\pi$
$79$	10.1327	1.14001	0.570007	+	0.821640i	$0.306940\pi$
$80$	6.32926	0.707633
$81$	−9.06214	−1.00690
$82$	−17.6696	−1.95128
$83$	15.0733	1.65451	0.827256	−	0.561826i	$-0.189901\pi$
$83$	15.0733	1.65451	0.827256	+	0.561826i	$0.189901\pi$
$84$	−2.28826	−0.249670
$85$	−0.991256	−0.107517
$86$	−5.90902	−0.637186
$87$	−5.87836	−0.630226
$88$	−0.291922	−0.0311190
$89$	−9.54872	−1.01216	−0.506081	−	0.862486i	$-0.668906\pi$
$89$	−9.54872	−1.01216	−0.506081	+	0.862486i	$0.668906\pi$
$90$	0.0457403	0.00482145
$91$	−1.31498	−0.137848
$92$	3.36733	0.351068
$93$	8.38927	0.869927
$94$	−3.30715	−0.341106
$95$	−5.31044	−0.544839
$96$	9.04026	0.922667
$97$	−0.409630	−0.0415917	−0.0207958	−	0.999784i	$-0.506620\pi$
$97$	−0.409630	−0.0415917	−0.0207958	+	0.999784i	$0.506620\pi$
$98$	9.13116	0.922386
$99$	−0.00351892	−0.000353665 0
$100$	−3.40169	−0.340169
$101$	−1.82392	−0.181486	−0.0907432	−	0.995874i	$-0.528924\pi$
$101$	−1.82392	−0.181486	−0.0907432	+	0.995874i	$0.528924\pi$
$102$	−2.35785	−0.233462
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	1.73032	0.169672
$105$	2.89313	0.282340
$106$	−13.8353	−1.34381
$107$	−0.863704	−0.0834975	−0.0417487	−	0.999128i	$-0.513293\pi$
$107$	−0.863704	−0.0834975	−0.0417487	+	0.999128i	$0.513293\pi$
$108$	−5.18414	−0.498844
$109$	−1.64450	−0.157514	−0.0787571	−	0.996894i	$-0.525095\pi$
$109$	−1.64450	−0.157514	−0.0787571	+	0.996894i	$0.525095\pi$
$110$	−0.369974	−0.0352756
$111$	3.95236	0.375142
$112$	6.57491	0.621270
$113$	18.1540	1.70778	0.853891	−	0.520452i	$-0.174237\pi$
$113$	18.1540	1.70778	0.853891	+	0.520452i	$0.174237\pi$
$114$	−12.6316	−1.18306
$115$	−4.25743	−0.397007
$116$	−3.38619	−0.314400
$117$	0.0208578	0.00192831
$118$	−8.95689	−0.824549
$119$	−1.02973	−0.0943950
$120$	−3.80692	−0.347523
$121$	−10.9715	−0.997412
$122$	2.04646	0.185278
$123$	17.7274	1.59842
$124$	4.83259	0.433979
$125$	10.6301	0.950789
$126$	0.0475156	0.00423302
$127$	−8.48184	−0.752642	−0.376321	−	0.926489i	$-0.622811\pi$
$127$	−8.48184	−0.752642	−0.376321	+	0.926489i	$0.622811\pi$
$128$	−12.1164	−1.07095
$129$	5.92834	0.521962
$130$	2.19296	0.192335
$131$	−4.43930	−0.387863	−0.193932	−	0.981015i	$-0.562124\pi$
$131$	−4.43930	−0.387863	−0.193932	+	0.981015i	$0.562124\pi$
$132$	−0.293580	−0.0255528
$133$	−5.51654	−0.478345
$134$	4.99797	0.431759
$135$	6.55450	0.564121
$136$	1.35497	0.116188
$137$	−10.2680	−0.877257	−0.438629	−	0.898668i	$-0.644536\pi$
$137$	−10.2680	−0.877257	−0.438629	+	0.898668i	$0.644536\pi$
$138$	−10.1269	−0.862060
$139$	8.00297	0.678803	0.339402	−	0.940642i	$-0.389775\pi$
$139$	8.00297	0.678803	0.339402	+	0.940642i	$0.389775\pi$
$140$	1.66657	0.140851
$141$	3.31796	0.279423
$142$	−14.2222	−1.19350
$143$	−0.168710	−0.0141082
$144$	−0.104289	−0.00869075
$145$	4.28128	0.355541
$146$	−14.1115	−1.16788
$147$	−9.16101	−0.755588
$148$	2.27674	0.187146
$149$	14.4083	1.18037	0.590187	−	0.807266i	$-0.299054\pi$
$149$	14.4083	1.18037	0.590187	+	0.807266i	$0.299054\pi$
$150$	10.2303	0.835298
$151$	−3.83506	−0.312093	−0.156046	−	0.987750i	$-0.549875\pi$
$151$	−3.83506	−0.312093	−0.156046	+	0.987750i	$0.549875\pi$
$152$	7.25894	0.588778
$153$	0.0163332	0.00132046
$154$	−0.384333	−0.0309704
$155$	−6.11001	−0.490768
$156$	1.74015	0.139323
$157$	12.0233	0.959568	0.479784	−	0.877387i	$-0.340715\pi$
$157$	12.0233	0.959568	0.479784	+	0.877387i	$0.340715\pi$
$158$	−17.5538	−1.39651
$159$	13.8806	1.10080
$160$	−6.58413	−0.520521
$161$	−4.42267	−0.348555
$162$	15.6992	1.23345
$163$	−10.1436	−0.794510	−0.397255	−	0.917708i	$-0.630037\pi$
$163$	−10.1436	−0.794510	−0.397255	+	0.917708i	$0.630037\pi$
$164$	10.2117	0.797402
$165$	0.371183	0.0288966
$166$	−26.1130	−2.02676
$167$	19.9736	1.54560	0.772801	−	0.634649i	$-0.218855\pi$
$167$	19.9736	1.54560	0.772801	+	0.634649i	$0.218855\pi$
$168$	−3.95468	−0.305110
$169$	1.00000	0.0769231
$170$	1.71725	0.131707
$171$	0.0875016	0.00669141
$172$	3.41499	0.260390
$173$	−6.00262	−0.456371	−0.228185	−	0.973618i	$-0.573279\pi$
$173$	−6.00262	−0.456371	−0.228185	+	0.973618i	$0.573279\pi$
$174$	10.1837	0.772021
$175$	4.46781	0.337734
$176$	0.843549	0.0635849
$177$	8.98618	0.675442
$178$	16.5422	1.23989
$179$	23.2636	1.73880	0.869401	−	0.494107i	$-0.164505\pi$
$179$	23.2636	1.73880	0.869401	+	0.494107i	$0.164505\pi$
$180$	−0.0264346	−0.00197032
$181$	11.5612	0.859337	0.429669	−	0.902987i	$-0.358630\pi$
$181$	11.5612	0.859337	0.429669	+	0.902987i	$0.358630\pi$
$182$	2.27807	0.168862
$183$	−2.05315	−0.151773
$184$	5.81957	0.429024
$185$	−2.87856	−0.211636
$186$	−14.5336	−1.06565
$187$	−0.132112	−0.00966101
$188$	1.91129	0.139395
$189$	6.80889	0.495274
$190$	9.19978	0.667422
$191$	20.5149	1.48441	0.742204	−	0.670174i	$-0.233780\pi$
$191$	20.5149	1.48441	0.742204	+	0.670174i	$0.233780\pi$
$192$	1.71929	0.124079
$193$	21.0068	1.51210	0.756051	−	0.654513i	$-0.227126\pi$
$193$	21.0068	1.51210	0.756051	+	0.654513i	$0.227126\pi$
$194$	0.709643	0.0509494
$195$	−2.20013	−0.157554
$196$	−5.27715	−0.376939
$197$	22.5059	1.60348	0.801739	−	0.597675i	$-0.203909\pi$
$197$	22.5059	1.60348	0.801739	+	0.597675i	$0.203909\pi$
$198$	0.00609616	0.000433235 0
$199$	−7.18181	−0.509105	−0.254552	−	0.967059i	$-0.581928\pi$
$199$	−7.18181	−0.509105	−0.254552	+	0.967059i	$0.581928\pi$
$200$	−5.87896	−0.415706
$201$	−5.01431	−0.353682
$202$	3.15975	0.222319
$203$	4.44745	0.312150
$204$	1.36266	0.0954056
$205$	−12.9111	−0.901747
$206$	1.73240	0.120702
$207$	0.0701509	0.00487583
$208$	−5.00000	−0.346688
$209$	−0.707763	−0.0489570
$210$	−5.01205	−0.345864
$211$	−2.18244	−0.150245	−0.0751227	−	0.997174i	$-0.523935\pi$
$211$	−2.18244	−0.150245	−0.0751227	+	0.997174i	$0.523935\pi$
$212$	7.99581	0.549155
$213$	14.2687	0.977673
$214$	1.49628	0.102284
$215$	−4.31769	−0.294464
$216$	−8.95947	−0.609615
$217$	−6.34715	−0.430873
$218$	2.84892	0.192953
$219$	14.1576	0.956685
$220$	0.213818	0.0144156
$221$	0.783074	0.0526753
$222$	−6.84707	−0.459545
$223$	−10.8932	−0.729461	−0.364731	−	0.931113i	$-0.618839\pi$
$223$	−10.8932	−0.729461	−0.364731	+	0.931113i	$0.618839\pi$
$224$	−6.83967	−0.456995
$225$	−0.0708669	−0.00472446
$226$	−31.4499	−2.09202
$227$	−4.58492	−0.304312	−0.152156	−	0.988357i	$-0.548622\pi$
$227$	−4.58492	−0.304312	−0.152156	+	0.988357i	$0.548622\pi$
$228$	7.30017	0.483466
$229$	−22.1320	−1.46252	−0.731261	−	0.682098i	$-0.761068\pi$
$229$	−22.1320	−1.46252	−0.731261	+	0.682098i	$0.761068\pi$
$230$	7.37556	0.486330
$231$	0.385590	0.0253699
$232$	−5.85217	−0.384214
$233$	11.7278	0.768313	0.384156	−	0.923268i	$-0.374492\pi$
$233$	11.7278	0.768313	0.384156	+	0.923268i	$0.374492\pi$
$234$	−0.0361340	−0.00236215
$235$	−2.41651	−0.157636
$236$	5.17643	0.336957
$237$	17.6112	1.14397
$238$	1.78390	0.115633
$239$	13.6549	0.883264	0.441632	−	0.897196i	$-0.354400\pi$
$239$	13.6549	0.883264	0.441632	+	0.897196i	$0.354400\pi$
$240$	11.0006	0.710088
$241$	−15.2393	−0.981648	−0.490824	−	0.871259i	$-0.663304\pi$
$241$	−15.2393	−0.981648	−0.490824	+	0.871259i	$0.663304\pi$
$242$	19.0071	1.22182
$243$	−0.216757	−0.0139050
$244$	−1.18271	−0.0757149
$245$	6.67208	0.426264
$246$	−30.7108	−1.95805
$247$	4.19515	0.266931
$248$	8.35190	0.530346
$249$	26.1984	1.66025
$250$	−18.4156	−1.16471
$251$	−22.1601	−1.39873	−0.699365	−	0.714765i	$-0.746534\pi$
$251$	−22.1601	−1.39873	−0.699365	+	0.714765i	$0.746534\pi$
$252$	−0.0274605	−0.00172985
$253$	−0.567421	−0.0356734
$254$	14.6939	0.921979
$255$	−1.72286	−0.107890
$256$	19.0120	1.18825
$257$	−18.0406	−1.12534	−0.562671	−	0.826681i	$-0.690226\pi$
$257$	−18.0406	−1.12534	−0.562671	+	0.826681i	$0.690226\pi$
$258$	−10.2702	−0.639397
$259$	−2.99028	−0.185807
$260$	−1.26737	−0.0785990
$261$	−0.0705440	−0.00436656
$262$	7.69062	0.475128
$263$	−26.6232	−1.64165	−0.820827	−	0.571177i	$-0.806487\pi$
$263$	−26.6232	−1.64165	−0.820827	+	0.571177i	$0.806487\pi$
$264$	−0.507378	−0.0312270
$265$	−10.1094	−0.621015
$266$	9.55684	0.585968
$267$	−16.5963	−1.01567
$268$	−2.88846	−0.176441
$269$	−9.41772	−0.574208	−0.287104	−	0.957899i	$-0.592693\pi$
$269$	−9.41772	−0.574208	−0.287104	+	0.957899i	$0.592693\pi$
$270$	−11.3550	−0.691043
$271$	−20.1768	−1.22565	−0.612827	−	0.790217i	$-0.709968\pi$
$271$	−20.1768	−1.22565	−0.612827	+	0.790217i	$0.709968\pi$
$272$	−3.91537	−0.237404
$273$	−2.28552	−0.138326
$274$	17.7883	1.07463
$275$	0.573212	0.0345660
$276$	5.85262	0.352286
$277$	−16.2614	−0.977055	−0.488528	−	0.872548i	$-0.662466\pi$
$277$	−16.2614	−0.977055	−0.488528	+	0.872548i	$0.662466\pi$
$278$	−13.8643	−0.831527
$279$	0.100676	0.00602734
$280$	2.88024	0.172127
$281$	−5.28075	−0.315023	−0.157512	−	0.987517i	$-0.550347\pi$
$281$	−5.28075	−0.315023	−0.157512	+	0.987517i	$0.550347\pi$
$282$	−5.74803	−0.342290
$283$	−7.67756	−0.456383	−0.228192	−	0.973616i	$-0.573281\pi$
$283$	−7.67756	−0.456383	−0.228192	+	0.973616i	$0.573281\pi$
$284$	8.21938	0.487731
$285$	−9.22986	−0.546730
$286$	0.292272	0.0172824
$287$	−13.4122	−0.791694
$288$	0.108489	0.00639276
$289$	−16.3868	−0.963929
$290$	−7.41688	−0.435534
$291$	−0.711963	−0.0417360
$292$	8.15542	0.477260
$293$	−7.12342	−0.416155	−0.208077	−	0.978112i	$-0.566721\pi$
$293$	−7.12342	−0.416155	−0.208077	+	0.978112i	$0.566721\pi$
$294$	15.8705	0.925587
$295$	−6.54474	−0.381050
$296$	3.93476	0.228703
$297$	0.873568	0.0506896
$298$	−24.9609	−1.44595
$299$	3.36329	0.194504
$300$	−5.91235	−0.341350
$301$	−4.48526	−0.258526
$302$	6.64385	0.382311
$303$	−3.17008	−0.182116
$304$	−20.9757	−1.20304
$305$	1.49534	0.0856227
$306$	−0.0282956	−0.00161755
$307$	−21.2981	−1.21555	−0.607773	−	0.794111i	$-0.707937\pi$
$307$	−21.2981	−1.21555	−0.607773	+	0.794111i	$0.707937\pi$
$308$	0.222117	0.0126563
$309$	−1.73806	−0.0988749
$310$	10.5850	0.601186
$311$	34.5079	1.95676	0.978382	−	0.206806i	$-0.0663070\pi$
$311$	34.5079	1.95676	0.978382	+	0.206806i	$0.0663070\pi$
$312$	3.00740	0.170261
$313$	10.5867	0.598398	0.299199	−	0.954191i	$-0.403281\pi$
$313$	10.5867	0.598398	0.299199	+	0.954191i	$0.403281\pi$
$314$	−20.8292	−1.17546
$315$	0.0347193	0.00195621
$316$	10.1448	0.570691
$317$	−12.1324	−0.681425	−0.340713	−	0.940167i	$-0.610668\pi$
$317$	−12.1324	−0.681425	−0.340713	+	0.940167i	$0.610668\pi$
$318$	−24.0467	−1.34847
$319$	0.570600	0.0319475
$320$	−1.25218	−0.0699992
$321$	−1.50117	−0.0837872
$322$	7.66182	0.426976
$323$	3.28511	0.182789
$324$	−9.07301	−0.504056
$325$	−3.39762	−0.188466
$326$	17.5728	0.973266
$327$	−2.85824	−0.158061
$328$	17.6484	0.974469
$329$	−2.51030	−0.138397
$330$	−0.643037	−0.0353980
$331$	−0.970579	−0.0533479	−0.0266739	−	0.999644i	$-0.508492\pi$
$331$	−0.970579	−0.0533479	−0.0266739	+	0.999644i	$0.508492\pi$
$332$	15.0914	0.828248
$333$	0.0474308	0.00259919
$334$	−34.6022	−1.89335
$335$	3.65199	0.199529
$336$	11.4276	0.623426
$337$	−23.2589	−1.26699	−0.633495	−	0.773747i	$-0.718380\pi$
$337$	−23.2589	−1.26699	−0.633495	+	0.773747i	$0.718380\pi$
$338$	−1.73240	−0.0942300
$339$	31.5527	1.71371
$340$	−0.992445	−0.0538229
$341$	−0.814329	−0.0440984
$342$	−0.151588	−0.00819691
$343$	16.1359	0.871257
$344$	5.90193	0.318211
$345$	−7.39968	−0.398385
$346$	10.3989	0.559049
$347$	−17.9543	−0.963840	−0.481920	−	0.876215i	$-0.660060\pi$
$347$	−17.9543	−0.963840	−0.481920	+	0.876215i	$0.660060\pi$
$348$	−5.88541	−0.315491
$349$	−0.184792	−0.00989168	−0.00494584	−	0.999988i	$-0.501574\pi$
$349$	−0.184792	−0.00989168	−0.00494584	+	0.999988i	$0.501574\pi$
$350$	−7.74001	−0.413721
$351$	−5.17793	−0.276378
$352$	−0.877518	−0.0467719
$353$	−26.2698	−1.39820	−0.699100	−	0.715024i	$-0.746416\pi$
$353$	−26.2698	−1.39820	−0.699100	+	0.715024i	$0.746416\pi$
$354$	−15.5676	−0.827410
$355$	−10.3921	−0.551553
$356$	−9.56017	−0.506688
$357$	−1.78973	−0.0947226
$358$	−40.3018	−2.13002
$359$	1.35844	0.0716960	0.0358480	−	0.999357i	$-0.488587\pi$
$359$	1.35844	0.0716960	0.0358480	+	0.999357i	$0.488587\pi$
$360$	−0.0456854	−0.00240783
$361$	−1.40073	−0.0737229
$362$	−20.0286	−1.05268
$363$	−19.0692	−1.00087
$364$	−1.31656	−0.0690064
$365$	−10.3112	−0.539712
$366$	3.55687	0.185921
$367$	−4.87574	−0.254512	−0.127256	−	0.991870i	$-0.540617\pi$
$367$	−4.87574	−0.254512	−0.127256	+	0.991870i	$0.540617\pi$
$368$	−16.8165	−0.876618
$369$	0.212739	0.0110748
$370$	4.98680	0.259252
$371$	−10.5018	−0.545224
$372$	8.39934	0.435485
$373$	4.15389	0.215080	0.107540	−	0.994201i	$-0.465703\pi$
$373$	4.15389	0.215080	0.107540	+	0.994201i	$0.465703\pi$
$374$	0.228871	0.0118346
$375$	18.4758	0.954088
$376$	3.30318	0.170349
$377$	−3.38214	−0.174189
$378$	−11.7957	−0.606705
$379$	14.5147	0.745571	0.372785	−	0.927918i	$-0.378403\pi$
$379$	14.5147	0.745571	0.372785	+	0.927918i	$0.378403\pi$
$380$	−5.31681	−0.272746
$381$	−14.7420	−0.755254
$382$	−35.5400	−1.81839
$383$	4.90273	0.250518	0.125259	−	0.992124i	$-0.460024\pi$
$383$	4.90273	0.250518	0.125259	+	0.992124i	$0.460024\pi$
$384$	−21.0590	−1.07466
$385$	−0.280830	−0.0143124
$386$	−36.3921	−1.85231
$387$	0.0711438	0.00361644
$388$	−0.410122	−0.0208208
$389$	3.11855	0.158117	0.0790584	−	0.996870i	$-0.474809\pi$
$389$	3.11855	0.158117	0.0790584	+	0.996870i	$0.474809\pi$
$390$	3.81150	0.193003
$391$	2.63371	0.133192
$392$	−9.12021	−0.460640
$393$	−7.71577	−0.389209
$394$	−38.9891	−1.96424
$395$	−12.8265	−0.645370
$396$	−0.00352314	−0.000177044 0
$397$	−14.3870	−0.722065	−0.361033	−	0.932553i	$-0.617576\pi$
$397$	−14.3870	−0.722065	−0.361033	+	0.932553i	$0.617576\pi$
$398$	12.4417	0.623648
$399$	−9.58809	−0.480005
$400$	16.9881	0.849404
$401$	−0.0784816	−0.00391918	−0.00195959	−	0.999998i	$-0.500624\pi$
$401$	−0.0784816	−0.00391918	−0.00195959	+	0.999998i	$0.500624\pi$
$402$	8.68678	0.433257
$403$	4.82680	0.240440
$404$	−1.82610	−0.0908520
$405$	11.4713	0.570015
$406$	−7.70474	−0.382380
$407$	−0.383648	−0.0190167
$408$	2.35502	0.116591
$409$	22.0065	1.08815	0.544076	−	0.839036i	$-0.316880\pi$
$409$	22.0065	1.08815	0.544076	+	0.839036i	$0.316880\pi$
$410$	22.3671	1.10463
$411$	−17.8465	−0.880301
$412$	−1.00120	−0.0493256
$413$	−6.79876	−0.334545
$414$	−0.121529	−0.00597284
$415$	−19.0806	−0.936629
$416$	5.20134	0.255017
$417$	13.9097	0.681159
$418$	1.22613	0.0599718
$419$	−6.52001	−0.318523	−0.159262	−	0.987236i	$-0.550911\pi$
$419$	−6.52001	−0.318523	−0.159262	+	0.987236i	$0.550911\pi$
$420$	2.89660	0.141340
$421$	10.3189	0.502915	0.251457	−	0.967868i	$-0.419090\pi$
$421$	10.3189	0.502915	0.251457	+	0.967868i	$0.419090\pi$
$422$	3.78085	0.184049
$423$	0.0398176	0.00193600
$424$	13.8187	0.671097
$425$	−2.66059	−0.129057
$426$	−24.7190	−1.19764
$427$	1.55337	0.0751729
$428$	−0.864740	−0.0417988
$429$	−0.293228	−0.0141572
$430$	7.47995	0.360715
$431$	2.15229	0.103672	0.0518361	−	0.998656i	$-0.483493\pi$
$431$	2.15229	0.103672	0.0518361	+	0.998656i	$0.483493\pi$
$432$	25.8897	1.24562
$433$	−19.3652	−0.930630	−0.465315	−	0.885145i	$-0.654059\pi$
$433$	−19.3652	−0.930630	−0.465315	+	0.885145i	$0.654059\pi$
$434$	10.9958	0.527815
$435$	7.44114	0.356775
$436$	−1.64647	−0.0788515
$437$	14.1095	0.674949
$438$	−24.5267	−1.17193
$439$	27.4286	1.30909	0.654547	−	0.756022i	$-0.272860\pi$
$439$	27.4286	1.30909	0.654547	+	0.756022i	$0.272860\pi$
$440$	0.369530	0.0176166
$441$	−0.109938	−0.00523513
$442$	−1.35660	−0.0645267
$443$	2.70198	0.128375	0.0641874	−	0.997938i	$-0.479554\pi$
$443$	2.70198	0.128375	0.0641874	+	0.997938i	$0.479554\pi$
$444$	3.95711	0.187796
$445$	12.0873	0.572991
$446$	18.8713	0.893582
$447$	25.0425	1.18447
$448$	−1.30078	−0.0614562
$449$	−12.6156	−0.595369	−0.297685	−	0.954664i	$-0.596214\pi$
$449$	−12.6156	−0.595369	−0.297685	+	0.954664i	$0.596214\pi$
$450$	0.122770	0.00578741
$451$	−1.72076	−0.0810272
$452$	18.1757	0.854915
$453$	−6.66557	−0.313176
$454$	7.94290	0.372779
$455$	1.66457	0.0780363
$456$	12.6165	0.590821
$457$	−9.73221	−0.455254	−0.227627	−	0.973748i	$-0.573097\pi$
$457$	−9.73221	−0.455254	−0.227627	+	0.973748i	$0.573097\pi$
$458$	38.3414	1.79157
$459$	−4.05471	−0.189257
$460$	−4.26254	−0.198742
$461$	6.22220	0.289797	0.144898	−	0.989447i	$-0.453714\pi$
$461$	6.22220	0.289797	0.144898	+	0.989447i	$0.453714\pi$
$462$	−0.667994	−0.0310779
$463$	14.6090	0.678936	0.339468	−	0.940618i	$-0.389753\pi$
$463$	14.6090	0.678936	0.339468	+	0.940618i	$0.389753\pi$
$464$	16.9107	0.785058
$465$	−10.6196	−0.492471
$466$	−20.3172	−0.941175
$467$	22.1280	1.02396	0.511982	−	0.858996i	$-0.328911\pi$
$467$	22.1280	1.02396	0.511982	+	0.858996i	$0.328911\pi$
$468$	0.0208828	0.000965309 0
$469$	3.79373	0.175178
$470$	4.18636	0.193102
$471$	20.8973	0.962898
$472$	8.94615	0.411780
$473$	−0.575451	−0.0264593
$474$	−30.5096	−1.40135
$475$	−14.2535	−0.653996
$476$	−1.03096	−0.0472541
$477$	0.166575	0.00762696
$478$	−23.6558	−1.08199
$479$	6.12871	0.280028	0.140014	−	0.990150i	$-0.455285\pi$
$479$	6.12871	0.280028	0.140014	+	0.990150i	$0.455285\pi$
$480$	−11.4436	−0.522328
$481$	2.27401	0.103686
$482$	26.4005	1.20251
$483$	−7.68687	−0.349765
$484$	−10.9847	−0.499304
$485$	0.518531	0.0235453
$486$	0.375509	0.0170334
$487$	31.6954	1.43626	0.718128	−	0.695911i	$-0.244999\pi$
$487$	31.6954	1.43626	0.718128	+	0.695911i	$0.244999\pi$
$488$	−2.04400	−0.0925278
$489$	−17.6302	−0.797267
$490$	−11.5587	−0.522169
$491$	−2.61978	−0.118229	−0.0591144	−	0.998251i	$-0.518828\pi$
$491$	−2.61978	−0.118229	−0.0591144	+	0.998251i	$0.518828\pi$
$492$	17.7486	0.800170
$493$	−2.64846	−0.119281
$494$	−7.26766	−0.326988
$495$	0.00445443	0.000200212 0
$496$	−24.1340	−1.08365
$497$	−10.7954	−0.484239
$498$	−45.3859	−2.03379
$499$	3.32884	0.149019	0.0745096	−	0.997220i	$-0.476261\pi$
$499$	3.32884	0.149019	0.0745096	+	0.997220i	$0.476261\pi$
$500$	10.6429	0.475965
$501$	34.7153	1.55096
$502$	38.3900	1.71343
$503$	−17.9253	−0.799250	−0.399625	−	0.916679i	$-0.630860\pi$
$503$	−17.9253	−0.799250	−0.399625	+	0.916679i	$0.630860\pi$
$504$	−0.0474586	−0.00211397
$505$	2.30881	0.102741
$506$	0.982998	0.0436996
$507$	1.73806	0.0771900
$508$	−8.49202	−0.376772
$509$	−34.3916	−1.52438	−0.762191	−	0.647352i	$-0.775876\pi$
$509$	−34.3916	−1.52438	−0.762191	+	0.647352i	$0.775876\pi$
$510$	2.98468	0.132164
$511$	−10.7114	−0.473844
$512$	−8.70353	−0.384645
$513$	−21.7222	−0.959058
$514$	31.2535	1.37853
$515$	1.26585	0.0557801
$516$	5.93545	0.261294
$517$	−0.322068	−0.0141645
$518$	5.18035	0.227611
$519$	−10.4329	−0.457954
$520$	−2.19033	−0.0960522
$521$	45.5470	1.99545	0.997725	−	0.0674126i	$-0.0214744\pi$
$521$	45.5470	1.99545	0.997725	+	0.0674126i	$0.0214744\pi$
$522$	0.122210	0.00534899
$523$	−23.1676	−1.01305	−0.506524	−	0.862226i	$-0.669070\pi$
$523$	−23.1676	−1.01305	−0.506524	+	0.862226i	$0.669070\pi$
$524$	−4.44462	−0.194164
$525$	7.76532	0.338906
$526$	46.1219	2.01101
$527$	3.77974	0.164648
$528$	1.46614	0.0638056
$529$	−11.6883	−0.508185
$530$	17.5135	0.760737
$531$	0.107840	0.00467984
$532$	−5.52316	−0.239459
$533$	10.1995	0.441789
$534$	28.7513	1.24419
$535$	1.09332	0.0472684
$536$	−4.99197	−0.215620
$537$	40.4336	1.74484
$538$	16.3152	0.703399
$539$	0.889240	0.0383023
$540$	6.56236	0.282399
$541$	23.2698	1.00044	0.500222	−	0.865897i	$-0.333252\pi$
$541$	23.2698	1.00044	0.500222	+	0.865897i	$0.333252\pi$
$542$	34.9543	1.50141
$543$	20.0941	0.862319
$544$	4.07304	0.174630
$545$	2.08169	0.0891697
$546$	3.95943	0.169448
$547$	2.62104	0.112068	0.0560339	−	0.998429i	$-0.482155\pi$
$547$	2.62104	0.112068	0.0560339	+	0.998429i	$0.482155\pi$
$548$	−10.2803	−0.439155
$549$	−0.0246391	−0.00105157
$550$	−0.993030	−0.0423430
$551$	−14.1886	−0.604453
$552$	10.1148	0.430513
$553$	−13.3243	−0.566606
$554$	28.1713	1.19688
$555$	−5.00311	−0.212370
$556$	8.01257	0.339809
$557$	−14.5334	−0.615802	−0.307901	−	0.951418i	$-0.599627\pi$
$557$	−14.5334	−0.615802	−0.307901	+	0.951418i	$0.599627\pi$
$558$	−0.174412	−0.00738343
$559$	3.41089	0.144265
$560$	−8.32286	−0.351705
$561$	−0.229619	−0.00969454
$562$	9.14836	0.385900
$563$	−34.5066	−1.45428	−0.727140	−	0.686490i	$-0.759151\pi$
$563$	−34.5066	−1.45428	−0.727140	+	0.686490i	$0.759151\pi$
$564$	3.32194	0.139879
$565$	−22.9802	−0.966786
$566$	13.3006	0.559065
$567$	11.9165	0.500448
$568$	14.2051	0.596033
$569$	36.4507	1.52809	0.764046	−	0.645162i	$-0.223210\pi$
$569$	36.4507	1.52809	0.764046	+	0.645162i	$0.223210\pi$
$570$	15.9898	0.669739
$571$	−29.8975	−1.25117	−0.625586	−	0.780155i	$-0.715140\pi$
$571$	−29.8975	−1.25117	−0.625586	+	0.780155i	$0.715140\pi$
$572$	−0.168912	−0.00706258
$573$	35.6562	1.48956
$574$	23.2352	0.969818
$575$	−11.4272	−0.476547
$576$	0.0206326	0.000859692 0
$577$	23.9075	0.995281	0.497640	−	0.867383i	$-0.334200\pi$
$577$	23.9075	0.995281	0.497640	+	0.867383i	$0.334200\pi$
$578$	28.3884	1.18080
$579$	36.5111	1.51735
$580$	4.28642	0.177984
$581$	−19.8211	−0.822319
$582$	1.23340	0.0511262
$583$	−1.34736	−0.0558018
$584$	14.0946	0.583238
$585$	−0.0264029	−0.00109163
$586$	12.3406	0.509785
$587$	−17.5436	−0.724100	−0.362050	−	0.932159i	$-0.617923\pi$
$587$	−17.5436	−0.724100	−0.362050	+	0.932159i	$0.617923\pi$
$588$	−9.17200	−0.378247
$589$	20.2491	0.834351
$590$	11.3381	0.466782
$591$	39.1166	1.60904
$592$	−11.3700	−0.467306
$593$	−16.9251	−0.695031	−0.347515	−	0.937674i	$-0.612975\pi$
$593$	−16.9251	−0.695031	−0.347515	+	0.937674i	$0.612975\pi$
$594$	−1.51337	−0.0620942
$595$	1.30348	0.0534376
$596$	14.4256	0.590895
$597$	−12.4824	−0.510871
$598$	−5.82656	−0.238266
$599$	−11.5958	−0.473793	−0.236896	−	0.971535i	$-0.576130\pi$
$599$	−11.5958	−0.473793	−0.236896	+	0.971535i	$0.576130\pi$
$600$	−10.2180	−0.417148
$601$	−30.6259	−1.24926	−0.624628	−	0.780922i	$-0.714749\pi$
$601$	−30.6259	−1.24926	−0.624628	+	0.780922i	$0.714749\pi$
$602$	7.77026	0.316692
$603$	−0.0601748	−0.00245051
$604$	−3.83966	−0.156234
$605$	13.8883	0.564641
$606$	5.49183	0.223090
$607$	−22.7115	−0.921830	−0.460915	−	0.887444i	$-0.652479\pi$
$607$	−22.7115	−0.921830	−0.460915	+	0.887444i	$0.652479\pi$
$608$	21.8204	0.884935
$609$	7.72994	0.313233
$610$	−2.59052	−0.104887
$611$	1.90900	0.0772300
$612$	0.0163528	0.000661023 0
$613$	−10.5053	−0.424306	−0.212153	−	0.977236i	$-0.568048\pi$
$613$	−10.5053	−0.424306	−0.212153	+	0.977236i	$0.568048\pi$
$614$	36.8967	1.48903
$615$	−22.4402	−0.904877
$616$	0.383872	0.0154666
$617$	7.64751	0.307877	0.153939	−	0.988080i	$-0.450804\pi$
$617$	7.64751	0.307877	0.153939	+	0.988080i	$0.450804\pi$
$618$	3.01101	0.121121
$619$	−2.87523	−0.115565	−0.0577826	−	0.998329i	$-0.518403\pi$
$619$	−2.87523	−0.115565	−0.0577826	+	0.998329i	$0.518403\pi$
$620$	−6.11734	−0.245678
$621$	−17.4149	−0.698836
$622$	−59.7814	−2.39702
$623$	12.5564	0.503061
$624$	−8.69031	−0.347891
$625$	3.53191	0.141276
$626$	−18.3404	−0.733032
$627$	−1.23014	−0.0491269
$628$	12.0378	0.480359
$629$	1.78072	0.0710018
$630$	−0.0601477	−0.00239634
$631$	22.1558	0.882009	0.441005	−	0.897505i	$-0.354622\pi$
$631$	22.1558	0.882009	0.441005	+	0.897505i	$0.354622\pi$
$632$	17.5328	0.697416
$633$	−3.79321	−0.150767
$634$	21.0182	0.834739
$635$	10.7368	0.426075
$636$	13.8972	0.551061
$637$	−5.27082	−0.208838
$638$	−0.988505	−0.0391353
$639$	0.171233	0.00677387
$640$	15.3375	0.606270
$641$	37.0065	1.46167	0.730834	−	0.682555i	$-0.239131\pi$
$641$	37.0065	1.46167	0.730834	+	0.682555i	$0.239131\pi$
$642$	2.60062	0.102638
$643$	−13.0554	−0.514856	−0.257428	−	0.966297i	$-0.582875\pi$
$643$	−13.0554	−0.514856	−0.257428	+	0.966297i	$0.582875\pi$
$644$	−4.42797	−0.174487
$645$	−7.50441	−0.295486
$646$	−5.69112	−0.223914
$647$	3.12573	0.122885	0.0614426	−	0.998111i	$-0.480430\pi$
$647$	3.12573	0.122885	0.0614426	+	0.998111i	$0.480430\pi$
$648$	−15.6804	−0.615984
$649$	−0.872269	−0.0342395
$650$	5.88602	0.230869
$651$	−11.0317	−0.432368
$652$	−10.1558	−0.397731
$653$	2.29178	0.0896841	0.0448421	−	0.998994i	$-0.485722\pi$
$653$	2.29178	0.0896841	0.0448421	+	0.998994i	$0.485722\pi$
$654$	4.95160	0.193623
$655$	5.61949	0.219572
$656$	−50.9975	−1.99112
$657$	0.169900	0.00662845
$658$	4.34884	0.169535
$659$	−6.51424	−0.253759	−0.126879	−	0.991918i	$-0.540496\pi$
$659$	−6.51424	−0.253759	−0.126879	+	0.991918i	$0.540496\pi$
$660$	0.371629	0.0144656
$661$	32.3348	1.25768	0.628840	−	0.777535i	$-0.283530\pi$
$661$	32.3348	1.25768	0.628840	+	0.777535i	$0.283530\pi$
$662$	1.68143	0.0653506
$663$	1.36103	0.0528581
$664$	26.0816	1.01216
$665$	6.98313	0.270794
$666$	−0.0821690	−0.00318398
$667$	−11.3751	−0.440446
$668$	19.9975	0.773728
$669$	−18.9330	−0.731992
$670$	−6.32669	−0.244421
$671$	0.199295	0.00769370
$672$	−11.8878	−0.458581
$673$	−32.0204	−1.23430	−0.617149	−	0.786846i	$-0.711712\pi$
$673$	−32.0204	−1.23430	−0.617149	+	0.786846i	$0.711712\pi$
$674$	40.2936	1.55205
$675$	17.5926	0.677141
$676$	1.00120	0.0385077
$677$	−7.46247	−0.286806	−0.143403	−	0.989664i	$-0.545804\pi$
$677$	−7.46247	−0.286806	−0.143403	+	0.989664i	$0.545804\pi$
$678$	−54.6618	−2.09928
$679$	0.538656	0.0206717
$680$	−1.71519	−0.0657745
$681$	−7.96887	−0.305368
$682$	1.41074	0.0540200
$683$	−15.6256	−0.597896	−0.298948	−	0.954269i	$-0.596636\pi$
$683$	−15.6256	−0.597896	−0.298948	+	0.954269i	$0.596636\pi$
$684$	0.0876066	0.00334972
$685$	12.9978	0.496621
$686$	−27.9538	−1.06728
$687$	−38.4667	−1.46760
$688$	−17.0545	−0.650195
$689$	7.98623	0.304251
$690$	12.8192	0.488018
$691$	2.71455	0.103266	0.0516331	−	0.998666i	$-0.483557\pi$
$691$	2.71455	0.103266	0.0516331	+	0.998666i	$0.483557\pi$
$692$	−6.00982	−0.228459
$693$	0.00462731	0.000175777 0
$694$	31.1041	1.18069
$695$	−10.1306	−0.384275
$696$	−10.1714	−0.385547
$697$	7.98697	0.302528
$698$	0.320133	0.0121172
$699$	20.3836	0.770979
$700$	4.47317	0.169070
$701$	−0.430274	−0.0162512	−0.00812561	−	0.999967i	$-0.502586\pi$
$701$	−0.430274	−0.0162512	−0.00812561	+	0.999967i	$0.502586\pi$
$702$	8.97023	0.338560
$703$	9.53980	0.359800
$704$	−0.166888	−0.00628984
$705$	−4.20005	−0.158183
$706$	45.5097	1.71278
$707$	2.39842	0.0902017
$708$	8.99696	0.338126
$709$	26.8117	1.00693	0.503467	−	0.864015i	$-0.332058\pi$
$709$	26.8117	1.00693	0.503467	+	0.864015i	$0.332058\pi$
$710$	18.0032	0.675647
$711$	0.211345	0.00792607
$712$	−16.5223	−0.619200
$713$	16.2339	0.607966
$714$	3.10052	0.116034
$715$	0.213562	0.00798676
$716$	23.2915	0.870444
$717$	23.7331	0.886329
$718$	−2.35337	−0.0878268
$719$	−36.9270	−1.37715	−0.688573	−	0.725167i	$-0.741763\pi$
$719$	−36.9270	−1.37715	−0.688573	+	0.725167i	$0.741763\pi$
$720$	0.132014	0.00491989
$721$	1.31498	0.0489725
$722$	2.42663	0.0903097
$723$	−26.4868	−0.985055
$724$	11.5751	0.430184
$725$	11.4912	0.426773
$726$	33.0354	1.22606
$727$	52.7811	1.95754	0.978771	−	0.204957i	$-0.0657056\pi$
$727$	52.7811	1.95754	0.978771	+	0.204957i	$0.0657056\pi$
$728$	−2.27534	−0.0843296
$729$	26.8097	0.992951
$730$	17.8631	0.661142
$731$	2.67098	0.0987899
$732$	−2.05561	−0.0759777
$733$	16.5504	0.611303	0.305651	−	0.952143i	$-0.401126\pi$
$733$	16.5504	0.611303	0.305651	+	0.952143i	$0.401126\pi$
$734$	8.44672	0.311774
$735$	11.5965	0.427743
$736$	17.4936	0.644824
$737$	0.486728	0.0179289
$738$	−0.368549	−0.0135665
$739$	4.91793	0.180909	0.0904544	−	0.995901i	$-0.471168\pi$
$739$	4.91793	0.180909	0.0904544	+	0.995901i	$0.471168\pi$
$740$	−2.88201	−0.105945
$741$	7.29143	0.267857
$742$	18.1932	0.667894
$743$	−5.98644	−0.219621	−0.109811	−	0.993953i	$-0.535024\pi$
$743$	−5.98644	−0.219621	−0.109811	+	0.993953i	$0.535024\pi$
$744$	14.5161	0.532187
$745$	−18.2388	−0.668218
$746$	−7.19618	−0.263471
$747$	0.314396	0.0115032
$748$	−0.132271	−0.00483630
$749$	1.13576	0.0414996
$750$	−32.0075	−1.16875
$751$	−25.4630	−0.929159	−0.464579	−	0.885531i	$-0.653795\pi$
$751$	−25.4630	−0.929159	−0.464579	+	0.885531i	$0.653795\pi$
$752$	−9.54501	−0.348071
$753$	−38.5155	−1.40358
$754$	5.85920	0.213380
$755$	4.85462	0.176678
$756$	6.81705	0.247934
$757$	37.3518	1.35758	0.678788	−	0.734334i	$-0.262506\pi$
$757$	37.3518	1.35758	0.678788	+	0.734334i	$0.262506\pi$
$758$	−25.1453	−0.913317
$759$	−0.986212	−0.0357972
$760$	−9.18875	−0.333311
$761$	−39.6384	−1.43689	−0.718446	−	0.695583i	$-0.755146\pi$
$761$	−39.6384	−1.43689	−0.718446	+	0.695583i	$0.755146\pi$
$762$	25.5389	0.925178
$763$	2.16248	0.0782871
$764$	20.5395	0.743095
$765$	−0.0206754	−0.000747522 0
$766$	−8.49347	−0.306882
$767$	5.17023	0.186686
$768$	33.0440	1.19237
$769$	26.4608	0.954202	0.477101	−	0.878848i	$-0.341688\pi$
$769$	26.4608	0.954202	0.477101	+	0.878848i	$0.341688\pi$
$770$	0.486509	0.0175326
$771$	−31.3557	−1.12925
$772$	21.0320	0.756958
$773$	49.0402	1.76385	0.881927	−	0.471387i	$-0.156246\pi$
$773$	49.0402	1.76385	0.881927	+	0.471387i	$0.156246\pi$
$774$	−0.123249	−0.00443010
$775$	−16.3996	−0.589092
$776$	−0.708791	−0.0254441
$777$	−5.19729	−0.186452
$778$	−5.40257	−0.193691
$779$	42.7884	1.53305
$780$	−2.20277	−0.0788717
$781$	−1.38503	−0.0495602
$782$	−4.56263	−0.163159
$783$	17.5125	0.625845
$784$	26.3541	0.941218
$785$	−15.2198	−0.543217
$786$	13.3668	0.476777
$787$	−3.14890	−0.112246	−0.0561232	−	0.998424i	$-0.517874\pi$
$787$	−3.14890	−0.112246	−0.0561232	+	0.998424i	$0.517874\pi$
$788$	22.5329	0.802700
$789$	−46.2727	−1.64735
$790$	22.2205	0.790571
$791$	−23.8721	−0.848796
$792$	−0.00608885	−0.000216358 0
$793$	−1.18129	−0.0419488
$794$	24.9241	0.884522
$795$	−17.5707	−0.623170
$796$	−7.19042	−0.254858
$797$	−4.53573	−0.160664	−0.0803320	−	0.996768i	$-0.525598\pi$
$797$	−4.53573	−0.160664	−0.0803320	+	0.996768i	$0.525598\pi$
$798$	16.6104	0.588001
$799$	1.49489	0.0528854
$800$	−17.6722	−0.624806
$801$	−0.199165	−0.00703716
$802$	0.135961	0.00480096
$803$	−1.37425	−0.0484963
$804$	−5.02033	−0.177053
$805$	5.59844	0.197319
$806$	−8.36193	−0.294537
$807$	−16.3686	−0.576201
$808$	−3.15596	−0.111026
$809$	26.2751	0.923783	0.461891	−	0.886937i	$-0.347171\pi$
$809$	26.2751	0.923783	0.461891	+	0.886937i	$0.347171\pi$
$810$	−19.8729	−0.698262
$811$	−2.34618	−0.0823856	−0.0411928	−	0.999151i	$-0.513116\pi$
$811$	−2.34618	−0.0823856	−0.0411928	+	0.999151i	$0.513116\pi$
$812$	4.45278	0.156262
$813$	−35.0686	−1.22991
$814$	0.664630	0.0232953
$815$	12.8403	0.449777
$816$	−6.80515	−0.238228
$817$	14.3092	0.500616
$818$	−38.1240	−1.33298
$819$	−0.0274276	−0.000958399 0
$820$	−12.9265	−0.451415
$821$	47.3567	1.65276	0.826380	−	0.563113i	$-0.190396\pi$
$821$	47.3567	1.65276	0.826380	+	0.563113i	$0.190396\pi$
$822$	30.9172	1.07836
$823$	20.5829	0.717475	0.358738	−	0.933438i	$-0.383207\pi$
$823$	20.5829	0.717475	0.358738	+	0.933438i	$0.383207\pi$
$824$	−1.73032	−0.0602785
$825$	0.996277	0.0346859
$826$	11.7781	0.409814
$827$	−18.9440	−0.658748	−0.329374	−	0.944200i	$-0.606838\pi$
$827$	−18.9440	−0.658748	−0.329374	+	0.944200i	$0.606838\pi$
$828$	0.0702351	0.00244084
$829$	−0.641650	−0.0222854	−0.0111427	−	0.999938i	$-0.503547\pi$
$829$	−0.641650	−0.0222854	−0.0111427	+	0.999938i	$0.503547\pi$
$830$	33.0552	1.14736
$831$	−28.2634	−0.980446
$832$	0.989202	0.0342944
$833$	−4.12745	−0.143008
$834$	−24.0971	−0.834413
$835$	−25.2836	−0.874974
$836$	−0.708612	−0.0245079
$837$	−24.9928	−0.863879
$838$	11.2953	0.390188
$839$	26.1279	0.902036	0.451018	−	0.892515i	$-0.351061\pi$
$839$	26.1279	0.902036	0.451018	+	0.892515i	$0.351061\pi$
$840$	5.00604	0.172725
$841$	−17.5612	−0.605557
$842$	−17.8765	−0.616065
$843$	−9.17827	−0.316116
$844$	−2.18506	−0.0752128
$845$	−1.26585	−0.0435466
$846$	−0.0689799	−0.00237158
$847$	14.4274	0.495730
$848$	−39.9312	−1.37124
$849$	−13.3441	−0.457967
$850$	4.60919	0.158094
$851$	7.64815	0.262175
$852$	14.2858	0.489423
$853$	12.0234	0.411673	0.205836	−	0.978586i	$-0.434009\pi$
$853$	12.0234	0.411673	0.205836	+	0.978586i	$0.434009\pi$
$854$	−2.69106	−0.0920861
$855$	−0.110764	−0.00378805
$856$	−1.49448	−0.0510804
$857$	52.9324	1.80814	0.904068	−	0.427389i	$-0.140566\pi$
$857$	52.9324	1.80814	0.904068	+	0.427389i	$0.140566\pi$
$858$	0.507988	0.0173424
$859$	48.1789	1.64384	0.821921	−	0.569601i	$-0.192902\pi$
$859$	48.1789	1.64384	0.821921	+	0.569601i	$0.192902\pi$
$860$	−4.32287	−0.147409
$861$	−23.3112	−0.794442
$862$	−3.72863	−0.126997
$863$	−19.0463	−0.648344	−0.324172	−	0.945998i	$-0.605086\pi$
$863$	−19.0463	−0.648344	−0.324172	+	0.945998i	$0.605086\pi$
$864$	−26.9322	−0.916252
$865$	7.59843	0.258354
$866$	33.5482	1.14001
$867$	−28.4813	−0.967274
$868$	−6.35477	−0.215695
$869$	−1.70948	−0.0579902
$870$	−12.8910	−0.437046
$871$	−2.88500	−0.0977546
$872$	−2.84550	−0.0963609
$873$	−0.00854399	−0.000289170 0
$874$	−24.4433	−0.826806
$875$	−13.9784	−0.472558
$876$	14.1746	0.478916
$877$	−30.8727	−1.04250	−0.521248	−	0.853405i	$-0.674533\pi$
$877$	−30.8727	−1.04250	−0.521248	+	0.853405i	$0.674533\pi$
$878$	−47.5171	−1.60363
$879$	−12.3809	−0.417599
$880$	−1.06781	−0.0359958
$881$	41.1534	1.38649	0.693247	−	0.720700i	$-0.256179\pi$
$881$	41.1534	1.38649	0.693247	+	0.720700i	$0.256179\pi$
$882$	0.190456	0.00641299
$883$	−43.3357	−1.45836	−0.729180	−	0.684321i	$-0.760099\pi$
$883$	−43.3357	−1.45836	−0.729180	+	0.684321i	$0.760099\pi$
$884$	0.784014	0.0263692
$885$	−11.3752	−0.382372
$886$	−4.68090	−0.157258
$887$	−35.5000	−1.19197	−0.595987	−	0.802994i	$-0.703239\pi$
$887$	−35.5000	−1.19197	−0.595987	+	0.802994i	$0.703239\pi$
$888$	6.83885	0.229497
$889$	11.1535	0.374075
$890$	−20.9399	−0.701908
$891$	1.52887	0.0512191
$892$	−10.9062	−0.365168
$893$	8.00855	0.267996
$894$	−43.3836	−1.45096
$895$	−29.4483	−0.984347
$896$	15.9328	0.532278
$897$	5.84561	0.195179
$898$	21.8553	0.729321
$899$	−16.3249	−0.544466
$900$	−0.0709519	−0.00236506
$901$	6.25381	0.208345
$902$	2.98103	0.0992575
$903$	−7.79566	−0.259423
$904$	31.4122	1.04475
$905$	−14.6348	−0.486476
$906$	11.5474	0.383637
$907$	23.9714	0.795957	0.397979	−	0.917395i	$-0.369712\pi$
$907$	23.9714	0.795957	0.397979	+	0.917395i	$0.369712\pi$
$908$	−4.59042	−0.152338
$909$	−0.0380429	−0.00126180
$910$	−2.88370	−0.0955937
$911$	1.44734	0.0479526	0.0239763	−	0.999713i	$-0.492367\pi$
$911$	1.44734	0.0479526	0.0239763	+	0.999713i	$0.492367\pi$
$912$	−36.4571	−1.20722
$913$	−2.54302	−0.0841616
$914$	16.8601	0.557681
$915$	2.59899	0.0859198
$916$	−22.1585	−0.732138
$917$	5.83759	0.192774
$918$	7.02436	0.231838
$919$	−12.6270	−0.416525	−0.208263	−	0.978073i	$-0.566781\pi$
$919$	−12.6270	−0.416525	−0.208263	+	0.978073i	$0.566781\pi$
$920$	−7.36671	−0.242873
$921$	−37.0174	−1.21976
$922$	−10.7793	−0.354998
$923$	8.20953	0.270220
$924$	0.386052	0.0127002
$925$	−7.72621	−0.254036
$926$	−25.3085	−0.831690
$927$	−0.0208578	−0.000685060 0
$928$	−17.5917	−0.577474
$929$	5.13827	0.168581	0.0842907	−	0.996441i	$-0.473138\pi$
$929$	5.13827	0.168581	0.0842907	+	0.996441i	$0.473138\pi$
$930$	18.3973	0.603272
$931$	−22.1119	−0.724688
$932$	11.7419	0.384617
$933$	59.9769	1.96355
$934$	−38.3346	−1.25435
$935$	0.167235	0.00546916
$936$	0.0360907	0.00117966
$937$	−3.95578	−0.129230	−0.0646148	−	0.997910i	$-0.520582\pi$
$937$	−3.95578	−0.129230	−0.0646148	+	0.997910i	$0.520582\pi$
$938$	−6.57224	−0.214591
$939$	18.4004	0.600475
$940$	−2.41941	−0.0789125
$941$	21.0723	0.686938	0.343469	−	0.939164i	$-0.388398\pi$
$941$	21.0723	0.686938	0.343469	+	0.939164i	$0.388398\pi$
$942$	−36.2025	−1.17954
$943$	34.3039	1.11709
$944$	−25.8511	−0.841383
$945$	−8.61904	−0.280377
$946$	0.996910	0.0324124
$947$	−32.7145	−1.06308	−0.531540	−	0.847033i	$-0.678386\pi$
$947$	−32.7145	−1.06308	−0.531540	+	0.847033i	$0.678386\pi$
$948$	17.6323	0.572672
$949$	8.14565	0.264419
$950$	24.6927	0.801138
$951$	−21.0869	−0.683790
$952$	−1.78176	−0.0577471
$953$	17.6722	0.572458	0.286229	−	0.958161i	$-0.407598\pi$
$953$	17.6722	0.572458	0.286229	+	0.958161i	$0.407598\pi$
$954$	−0.288575	−0.00934295
$955$	−25.9689	−0.840333
$956$	13.6713	0.442162
$957$	0.991738	0.0320583
$958$	−10.6174	−0.343031
$959$	13.5023	0.436011
$960$	−2.17637	−0.0702421
$961$	−7.70201	−0.248452
$962$	−3.93948	−0.127014
$963$	−0.0180150	−0.000580525 0
$964$	−15.2576	−0.491413
$965$	−26.5915	−0.856010
$966$	13.3167	0.428458
$967$	−25.9980	−0.836039	−0.418019	−	0.908438i	$-0.637276\pi$
$967$	−25.9980	−0.836039	−0.418019	+	0.908438i	$0.637276\pi$
$968$	−18.9843	−0.610177
$969$	5.70973	0.183423
$970$	−0.898302	−0.0288428
$971$	−50.0845	−1.60729	−0.803644	−	0.595111i	$-0.797108\pi$
$971$	−50.0845	−1.60729	−0.803644	+	0.595111i	$0.797108\pi$
$972$	−0.217017	−0.00696082
$973$	−10.5238	−0.337376
$974$	−54.9090	−1.75940
$975$	−5.90527	−0.189120
$976$	5.90644	0.189060
$977$	24.2997	0.777417	0.388708	−	0.921361i	$-0.372921\pi$
$977$	24.2997	0.777417	0.388708	+	0.921361i	$0.372921\pi$
$978$	30.5426	0.976644
$979$	1.61096	0.0514866
$980$	6.68009	0.213388
$981$	−0.0343006	−0.00109513
$982$	4.53849	0.144829
$983$	6.12232	0.195272	0.0976358	−	0.995222i	$-0.468872\pi$
$983$	6.12232	0.195272	0.0976358	+	0.995222i	$0.468872\pi$
$984$	30.6740	0.977851
$985$	−28.4891	−0.907738
$986$	4.58819	0.146118
$987$	−4.36306	−0.138878
$988$	4.20018	0.133626
$989$	11.4718	0.364783
$990$	−0.00771684	−0.000245257 0
$991$	−19.1387	−0.607962	−0.303981	−	0.952678i	$-0.598316\pi$
$991$	−19.1387	−0.607962	−0.303981	+	0.952678i	$0.598316\pi$
$992$	25.1058	0.797111
$993$	−1.68693	−0.0535330
$994$	18.7019	0.593188
$995$	9.09110	0.288207
$996$	26.2298	0.831122
$997$	53.5674	1.69650	0.848248	−	0.529599i	$-0.177658\pi$
$997$	53.5674	1.69650	0.848248	+	0.529599i	$0.177658\pi$
$998$	−5.76687	−0.182547
$999$	−11.7747	−0.372534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.f.1.6	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.f.1.6	✓	28	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73240 q^{2} +1.73806 q^{3} +1.00120 q^{4} -1.26585 q^{5} -3.01101 q^{6} -1.31498 q^{7} +1.73032 q^{8} +0.0208578 q^{9} +O(q^{10})\)	\(q-1.73240 q^{2} +1.73806 q^{3} +1.00120 q^{4} -1.26585 q^{5} -3.01101 q^{6} -1.31498 q^{7} +1.73032 q^{8} +0.0208578 q^{9} +2.19296 q^{10} -0.168710 q^{11} +1.74015 q^{12} +1.00000 q^{13} +2.27807 q^{14} -2.20013 q^{15} -5.00000 q^{16} +0.783074 q^{17} -0.0361340 q^{18} +4.19515 q^{19} -1.26737 q^{20} -2.28552 q^{21} +0.292272 q^{22} +3.36329 q^{23} +3.00740 q^{24} -3.39762 q^{25} -1.73240 q^{26} -5.17793 q^{27} -1.31656 q^{28} -3.38214 q^{29} +3.81150 q^{30} +4.82680 q^{31} +5.20134 q^{32} -0.293228 q^{33} -1.35660 q^{34} +1.66457 q^{35} +0.0208828 q^{36} +2.27401 q^{37} -7.26766 q^{38} +1.73806 q^{39} -2.19033 q^{40} +10.1995 q^{41} +3.95943 q^{42} +3.41089 q^{43} -0.168912 q^{44} -0.0264029 q^{45} -5.82656 q^{46} +1.90900 q^{47} -8.69031 q^{48} -5.27082 q^{49} +5.88602 q^{50} +1.36103 q^{51} +1.00120 q^{52} +7.98623 q^{53} +8.97023 q^{54} +0.213562 q^{55} -2.27534 q^{56} +7.29143 q^{57} +5.85920 q^{58} +5.17023 q^{59} -2.20277 q^{60} -1.18129 q^{61} -8.36193 q^{62} -0.0274276 q^{63} +0.989202 q^{64} -1.26585 q^{65} +0.507988 q^{66} -2.88500 q^{67} +0.784014 q^{68} +5.84561 q^{69} -2.88370 q^{70} +8.20953 q^{71} +0.0360907 q^{72} +8.14565 q^{73} -3.93948 q^{74} -5.90527 q^{75} +4.20018 q^{76} +0.221850 q^{77} -3.01101 q^{78} +10.1327 q^{79} +6.32926 q^{80} -9.06214 q^{81} -17.6696 q^{82} +15.0733 q^{83} -2.28826 q^{84} -0.991256 q^{85} -5.90902 q^{86} -5.87836 q^{87} -0.291922 q^{88} -9.54872 q^{89} +0.0457403 q^{90} -1.31498 q^{91} +3.36733 q^{92} +8.38927 q^{93} -3.30715 q^{94} -5.31044 q^{95} +9.04026 q^{96} -0.409630 q^{97} +9.13116 q^{98} -0.00351892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9 + 5 * q^10 + 9 * q^11 + 15 * q^12 + 28 * q^13 + 20 * q^14 + 4 * q^15 + 32 * q^16 - 7 * q^17 + 10 * q^18 - 3 * q^19 + 53 * q^20 + 25 * q^21 - 14 * q^22 - 6 * q^23 + 10 * q^24 + 45 * q^25 + 6 * q^26 + 8 * q^27 - 12 * q^28 + 25 * q^29 - 43 * q^30 + q^31 + 28 * q^32 + 17 * q^33 + 10 * q^34 + 11 * q^35 + 27 * q^36 + 18 * q^37 + 2 * q^38 + 5 * q^39 + 37 * q^40 + 56 * q^41 - 29 * q^42 - 30 * q^43 + 30 * q^44 + 38 * q^45 - 27 * q^46 + 40 * q^47 + 19 * q^48 + 36 * q^49 + 20 * q^50 - 18 * q^51 + 32 * q^52 + 51 * q^53 - 36 * q^54 - 5 * q^55 - 13 * q^56 + 31 * q^57 - 29 * q^58 + 29 * q^59 + 48 * q^60 + 16 * q^61 - 38 * q^62 - 23 * q^63 - 5 * q^64 + 27 * q^65 + 25 * q^66 - 2 * q^67 - 43 * q^68 + 38 * q^69 + 10 * q^70 + 34 * q^71 - 28 * q^72 + 14 * q^73 + 21 * q^74 + 19 * q^75 + 5 * q^76 + 34 * q^77 + 9 * q^78 + 3 * q^79 + 58 * q^80 + 4 * q^81 - 9 * q^82 + 38 * q^83 - 78 * q^84 - 27 * q^85 + 49 * q^86 + 8 * q^87 - 25 * q^88 + 125 * q^89 - 74 * q^90 + 6 * q^91 - 35 * q^92 + 36 * q^93 - q^94 - 12 * q^95 + 26 * q^96 - 2 * q^97 + 18 * q^98 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more