LMFDB - Embedded newform 6009.2.a.c.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6009,2,Mod(1,6009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6009, 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("6009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6009 = 3 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6009.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:	$47.9821065746$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	6009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84272 q^{2} +1.00000 q^{3} +1.39563 q^{4} -0.293700 q^{5} -1.84272 q^{6} +0.646833 q^{7} +1.11368 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.84272 q^{2} +1.00000 q^{3} +1.39563 q^{4} -0.293700 q^{5} -1.84272 q^{6} +0.646833 q^{7} +1.11368 q^{8} +1.00000 q^{9} +0.541209 q^{10} +1.85159 q^{11} +1.39563 q^{12} +4.24609 q^{13} -1.19194 q^{14} -0.293700 q^{15} -4.84347 q^{16} -5.26800 q^{17} -1.84272 q^{18} -2.30132 q^{19} -0.409898 q^{20} +0.646833 q^{21} -3.41197 q^{22} +3.20067 q^{23} +1.11368 q^{24} -4.91374 q^{25} -7.82437 q^{26} +1.00000 q^{27} +0.902743 q^{28} -6.02422 q^{29} +0.541209 q^{30} -5.15844 q^{31} +6.69783 q^{32} +1.85159 q^{33} +9.70747 q^{34} -0.189975 q^{35} +1.39563 q^{36} -0.743401 q^{37} +4.24070 q^{38} +4.24609 q^{39} -0.327088 q^{40} +10.0919 q^{41} -1.19194 q^{42} +3.42695 q^{43} +2.58414 q^{44} -0.293700 q^{45} -5.89795 q^{46} -0.690882 q^{47} -4.84347 q^{48} -6.58161 q^{49} +9.05467 q^{50} -5.26800 q^{51} +5.92599 q^{52} +11.1792 q^{53} -1.84272 q^{54} -0.543813 q^{55} +0.720366 q^{56} -2.30132 q^{57} +11.1010 q^{58} +2.14994 q^{59} -0.409898 q^{60} +7.79285 q^{61} +9.50558 q^{62} +0.646833 q^{63} -2.65530 q^{64} -1.24708 q^{65} -3.41197 q^{66} +15.2231 q^{67} -7.35220 q^{68} +3.20067 q^{69} +0.350072 q^{70} +8.73331 q^{71} +1.11368 q^{72} -0.335823 q^{73} +1.36988 q^{74} -4.91374 q^{75} -3.21180 q^{76} +1.19767 q^{77} -7.82437 q^{78} -0.968713 q^{79} +1.42253 q^{80} +1.00000 q^{81} -18.5965 q^{82} -2.39580 q^{83} +0.902743 q^{84} +1.54721 q^{85} -6.31492 q^{86} -6.02422 q^{87} +2.06208 q^{88} +10.5407 q^{89} +0.541209 q^{90} +2.74651 q^{91} +4.46696 q^{92} -5.15844 q^{93} +1.27311 q^{94} +0.675899 q^{95} +6.69783 q^{96} +7.83508 q^{97} +12.1281 q^{98} +1.85159 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) $ 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9 + 13 * q^10 + 40 * q^11 + 107 * q^12 + 6 * q^13 + 37 * q^14 + 34 * q^15 + 133 * q^16 + 77 * q^17 + 17 * q^18 + 34 * q^19 + 55 * q^20 + 22 * q^21 + 8 * q^22 + 83 * q^23 + 51 * q^24 + 110 * q^25 + 22 * q^26 + 92 * q^27 + 32 * q^28 + 97 * q^29 + 13 * q^30 + 44 * q^31 + 104 * q^32 + 40 * q^33 + 20 * q^34 + 80 * q^35 + 107 * q^36 + 12 * q^37 + 54 * q^38 + 6 * q^39 + 23 * q^40 + 67 * q^41 + 37 * q^42 + 30 * q^43 + 87 * q^44 + 34 * q^45 + 33 * q^46 + 69 * q^47 + 133 * q^48 + 112 * q^49 + 58 * q^50 + 77 * q^51 - 3 * q^52 + 113 * q^53 + 17 * q^54 + 42 * q^55 + 92 * q^56 + 34 * q^57 - 30 * q^58 + 72 * q^59 + 55 * q^60 + 19 * q^61 + 60 * q^62 + 22 * q^63 + 147 * q^64 + 74 * q^65 + 8 * q^66 + 26 * q^67 + 171 * q^68 + 83 * q^69 - 35 * q^70 + 134 * q^71 + 51 * q^72 - 17 * q^73 + 95 * q^74 + 110 * q^75 + 27 * q^76 + 108 * q^77 + 22 * q^78 + 159 * q^79 + 79 * q^80 + 92 * q^81 - 64 * q^82 + 73 * q^83 + 32 * q^84 - 4 * q^85 + 22 * q^86 + 97 * q^87 - 16 * q^88 + 50 * q^89 + 13 * q^90 + 17 * q^91 + 154 * q^92 + 44 * q^93 + 8 * q^94 + 155 * q^95 + 104 * q^96 - 20 * q^97 + 63 * q^98 + 40 * 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 $

$2$	−1.84272	−1.30300	−0.651502	−	0.758647i	$-0.725861\pi$
$2$	−1.84272	−1.30300	−0.651502	+	0.758647i	$0.725861\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.39563	0.697817
$5$	−0.293700	−0.131347	−0.0656734	−	0.997841i	$-0.520920\pi$
$5$	−0.293700	−0.131347	−0.0656734	+	0.997841i	$0.520920\pi$
$6$	−1.84272	−0.752289
$7$	0.646833	0.244480	0.122240	−	0.992501i	$-0.460992\pi$
$7$	0.646833	0.244480	0.122240	+	0.992501i	$0.460992\pi$
$8$	1.11368	0.393745
$9$	1.00000	0.333333
$10$	0.541209	0.171145
$11$	1.85159	0.558276	0.279138	−	0.960251i	$-0.409951\pi$
$11$	1.85159	0.558276	0.279138	+	0.960251i	$0.409951\pi$
$12$	1.39563	0.402885
$13$	4.24609	1.17765	0.588827	−	0.808259i	$-0.299590\pi$
$13$	4.24609	1.17765	0.588827	+	0.808259i	$0.299590\pi$
$14$	−1.19194	−0.318558
$15$	−0.293700	−0.0758331
$16$	−4.84347	−1.21087
$17$	−5.26800	−1.27768	−0.638839	−	0.769341i	$-0.720585\pi$
$17$	−5.26800	−1.27768	−0.638839	+	0.769341i	$0.720585\pi$
$18$	−1.84272	−0.434334
$19$	−2.30132	−0.527960	−0.263980	−	0.964528i	$-0.585035\pi$
$19$	−2.30132	−0.527960	−0.263980	+	0.964528i	$0.585035\pi$
$20$	−0.409898	−0.0916560
$21$	0.646833	0.141151
$22$	−3.41197	−0.727435
$23$	3.20067	0.667386	0.333693	−	0.942682i	$-0.391705\pi$
$23$	3.20067	0.667386	0.333693	+	0.942682i	$0.391705\pi$
$24$	1.11368	0.227329
$25$	−4.91374	−0.982748
$26$	−7.82437	−1.53449
$27$	1.00000	0.192450
$28$	0.902743	0.170602
$29$	−6.02422	−1.11867	−0.559335	−	0.828942i	$-0.688943\pi$
$29$	−6.02422	−1.11867	−0.559335	+	0.828942i	$0.688943\pi$
$30$	0.541209	0.0988108
$31$	−5.15844	−0.926482	−0.463241	−	0.886232i	$-0.653314\pi$
$31$	−5.15844	−0.926482	−0.463241	+	0.886232i	$0.653314\pi$
$32$	6.69783	1.18402
$33$	1.85159	0.322321
$34$	9.70747	1.66482
$35$	−0.189975	−0.0321117
$36$	1.39563	0.232606
$37$	−0.743401	−0.122214	−0.0611072	−	0.998131i	$-0.519463\pi$
$37$	−0.743401	−0.122214	−0.0611072	+	0.998131i	$0.519463\pi$
$38$	4.24070	0.687933
$39$	4.24609	0.679918
$40$	−0.327088	−0.0517172
$41$	10.0919	1.57608	0.788042	−	0.615621i	$-0.211095\pi$
$41$	10.0919	1.57608	0.788042	+	0.615621i	$0.211095\pi$
$42$	−1.19194	−0.183920
$43$	3.42695	0.522605	0.261302	−	0.965257i	$-0.415848\pi$
$43$	3.42695	0.522605	0.261302	+	0.965257i	$0.415848\pi$
$44$	2.58414	0.389574
$45$	−0.293700	−0.0437823
$46$	−5.89795	−0.869605
$47$	−0.690882	−0.100776	−0.0503878	−	0.998730i	$-0.516046\pi$
$47$	−0.690882	−0.100776	−0.0503878	+	0.998730i	$0.516046\pi$
$48$	−4.84347	−0.699095
$49$	−6.58161	−0.940230
$50$	9.05467	1.28052
$51$	−5.26800	−0.737667
$52$	5.92599	0.821786
$53$	11.1792	1.53558	0.767789	−	0.640702i	$-0.221357\pi$
$53$	11.1792	1.53558	0.767789	+	0.640702i	$0.221357\pi$
$54$	−1.84272	−0.250763
$55$	−0.543813	−0.0733277
$56$	0.720366	0.0962629
$57$	−2.30132	−0.304818
$58$	11.1010	1.45763
$59$	2.14994	0.279898	0.139949	−	0.990159i	$-0.455306\pi$
$59$	2.14994	0.279898	0.139949	+	0.990159i	$0.455306\pi$
$60$	−0.409898	−0.0529176
$61$	7.79285	0.997772	0.498886	−	0.866668i	$-0.333743\pi$
$61$	7.79285	0.997772	0.498886	+	0.866668i	$0.333743\pi$
$62$	9.50558	1.20721
$63$	0.646833	0.0814933
$64$	−2.65530	−0.331913
$65$	−1.24708	−0.154681
$66$	−3.41197	−0.419985
$67$	15.2231	1.85980	0.929899	−	0.367816i	$-0.119894\pi$
$67$	15.2231	1.85980	0.929899	+	0.367816i	$0.119894\pi$
$68$	−7.35220	−0.891585
$69$	3.20067	0.385315
$70$	0.350072	0.0418416
$71$	8.73331	1.03645	0.518226	−	0.855244i	$-0.326592\pi$
$71$	8.73331	1.03645	0.518226	+	0.855244i	$0.326592\pi$
$72$	1.11368	0.131248
$73$	−0.335823	−0.0393051	−0.0196525	−	0.999807i	$-0.506256\pi$
$73$	−0.335823	−0.0393051	−0.0196525	+	0.999807i	$0.506256\pi$
$74$	1.36988	0.159246
$75$	−4.91374	−0.567390
$76$	−3.21180	−0.368419
$77$	1.19767	0.136487
$78$	−7.82437	−0.885936
$79$	−0.968713	−0.108989	−0.0544944	−	0.998514i	$-0.517355\pi$
$79$	−0.968713	−0.108989	−0.0544944	+	0.998514i	$0.517355\pi$
$80$	1.42253	0.159044
$81$	1.00000	0.111111
$82$	−18.5965	−2.05364
$83$	−2.39580	−0.262973	−0.131486	−	0.991318i	$-0.541975\pi$
$83$	−2.39580	−0.262973	−0.131486	+	0.991318i	$0.541975\pi$
$84$	0.902743	0.0984973
$85$	1.54721	0.167819
$86$	−6.31492	−0.680956
$87$	−6.02422	−0.645864
$88$	2.06208	0.219819
$89$	10.5407	1.11732	0.558658	−	0.829398i	$-0.311317\pi$
$89$	10.5407	1.11732	0.558658	+	0.829398i	$0.311317\pi$
$90$	0.541209	0.0570484
$91$	2.74651	0.287913
$92$	4.46696	0.465713
$93$	−5.15844	−0.534905
$94$	1.27311	0.131311
$95$	0.675899	0.0693458
$96$	6.69783	0.683594
$97$	7.83508	0.795532	0.397766	−	0.917487i	$-0.369786\pi$
$97$	7.83508	0.795532	0.397766	+	0.917487i	$0.369786\pi$
$98$	12.1281	1.22512
$99$	1.85159	0.186092
$100$	−6.85778	−0.685778
$101$	−0.528666	−0.0526042	−0.0263021	−	0.999654i	$-0.508373\pi$
$101$	−0.528666	−0.0526042	−0.0263021	+	0.999654i	$0.508373\pi$
$102$	9.70747	0.961183
$103$	−4.56400	−0.449704	−0.224852	−	0.974393i	$-0.572190\pi$
$103$	−4.56400	−0.449704	−0.224852	+	0.974393i	$0.572190\pi$
$104$	4.72879	0.463696
$105$	−0.189975	−0.0185397
$106$	−20.6002	−2.00086
$107$	8.94060	0.864321	0.432160	−	0.901797i	$-0.357751\pi$
$107$	8.94060	0.864321	0.432160	+	0.901797i	$0.357751\pi$
$108$	1.39563	0.134295
$109$	−4.26845	−0.408843	−0.204422	−	0.978883i	$-0.565531\pi$
$109$	−4.26845	−0.408843	−0.204422	+	0.978883i	$0.565531\pi$
$110$	1.00210	0.0955463
$111$	−0.743401	−0.0705605
$112$	−3.13292	−0.296033
$113$	2.52479	0.237512	0.118756	−	0.992923i	$-0.462109\pi$
$113$	2.52479	0.237512	0.118756	+	0.992923i	$0.462109\pi$
$114$	4.24070	0.397178
$115$	−0.940038	−0.0876590
$116$	−8.40761	−0.780627
$117$	4.24609	0.392551
$118$	−3.96174	−0.364708
$119$	−3.40752	−0.312367
$120$	−0.327088	−0.0298589
$121$	−7.57161	−0.688328
$122$	−14.3601	−1.30010
$123$	10.0919	0.909953
$124$	−7.19929	−0.646515
$125$	2.91167	0.260428
$126$	−1.19194	−0.106186
$127$	−7.31966	−0.649514	−0.324757	−	0.945797i	$-0.605283\pi$
$127$	−7.31966	−0.649514	−0.324757	+	0.945797i	$0.605283\pi$
$128$	−8.50266	−0.751536
$129$	3.42695	0.301726
$130$	2.29802	0.201550
$131$	−15.8398	−1.38393	−0.691966	−	0.721931i	$-0.743255\pi$
$131$	−15.8398	−1.38393	−0.691966	+	0.721931i	$0.743255\pi$
$132$	2.58414	0.224921
$133$	−1.48857	−0.129076
$134$	−28.0520	−2.42332
$135$	−0.293700	−0.0252777
$136$	−5.86687	−0.503080
$137$	−9.99139	−0.853622	−0.426811	−	0.904341i	$-0.640363\pi$
$137$	−9.99139	−0.853622	−0.426811	+	0.904341i	$0.640363\pi$
$138$	−5.89795	−0.502067
$139$	5.95444	0.505049	0.252524	−	0.967591i	$-0.418739\pi$
$139$	5.95444	0.505049	0.252524	+	0.967591i	$0.418739\pi$
$140$	−0.265136	−0.0224081
$141$	−0.690882	−0.0581828
$142$	−16.0931	−1.35050
$143$	7.86202	0.657455
$144$	−4.84347	−0.403623
$145$	1.76932	0.146934
$146$	0.618828	0.0512146
$147$	−6.58161	−0.542842
$148$	−1.03752	−0.0852833
$149$	13.2929	1.08900	0.544498	−	0.838762i	$-0.316720\pi$
$149$	13.2929	1.08900	0.544498	+	0.838762i	$0.316720\pi$
$150$	9.05467	0.739311
$151$	−19.0693	−1.55184	−0.775918	−	0.630833i	$-0.782713\pi$
$151$	−19.0693	−1.55184	−0.775918	+	0.630833i	$0.782713\pi$
$152$	−2.56294	−0.207882
$153$	−5.26800	−0.425893
$154$	−2.20698	−0.177843
$155$	1.51503	0.121691
$156$	5.92599	0.474459
$157$	−19.8832	−1.58685	−0.793426	−	0.608667i	$-0.791705\pi$
$157$	−19.8832	−1.58685	−0.793426	+	0.608667i	$0.791705\pi$
$158$	1.78507	0.142013
$159$	11.1792	0.886567
$160$	−1.96715	−0.155517
$161$	2.07030	0.163162
$162$	−1.84272	−0.144778
$163$	12.9989	1.01815	0.509074	−	0.860723i	$-0.329988\pi$
$163$	12.9989	1.01815	0.509074	+	0.860723i	$0.329988\pi$
$164$	14.0846	1.09982
$165$	−0.543813	−0.0423358
$166$	4.41479	0.342654
$167$	24.2783	1.87871	0.939357	−	0.342941i	$-0.111423\pi$
$167$	24.2783	1.87871	0.939357	+	0.342941i	$0.111423\pi$
$168$	0.720366	0.0555774
$169$	5.02927	0.386867
$170$	−2.85109	−0.218668
$171$	−2.30132	−0.175987
$172$	4.78277	0.364682
$173$	11.6932	0.889020	0.444510	−	0.895774i	$-0.353378\pi$
$173$	11.6932	0.889020	0.444510	+	0.895774i	$0.353378\pi$
$174$	11.1010	0.841563
$175$	−3.17837	−0.240262
$176$	−8.96813	−0.675999
$177$	2.14994	0.161599
$178$	−19.4237	−1.45587
$179$	−6.24157	−0.466517	−0.233258	−	0.972415i	$-0.574939\pi$
$179$	−6.24157	−0.466517	−0.233258	+	0.972415i	$0.574939\pi$
$180$	−0.409898	−0.0305520
$181$	16.7275	1.24334	0.621671	−	0.783278i	$-0.286454\pi$
$181$	16.7275	1.24334	0.621671	+	0.783278i	$0.286454\pi$
$182$	−5.06107	−0.375151
$183$	7.79285	0.576064
$184$	3.56452	0.262780
$185$	0.218337	0.0160525
$186$	9.50558	0.696983
$187$	−9.75418	−0.713296
$188$	−0.964219	−0.0703229
$189$	0.646833	0.0470502
$190$	−1.24550	−0.0903578
$191$	−14.1044	−1.02056	−0.510279	−	0.860009i	$-0.670458\pi$
$191$	−14.1044	−1.02056	−0.510279	+	0.860009i	$0.670458\pi$
$192$	−2.65530	−0.191630
$193$	8.51823	0.613156	0.306578	−	0.951846i	$-0.400816\pi$
$193$	8.51823	0.613156	0.306578	+	0.951846i	$0.400816\pi$
$194$	−14.4379	−1.03658
$195$	−1.24708	−0.0893051
$196$	−9.18551	−0.656108
$197$	24.9385	1.77679	0.888397	−	0.459077i	$-0.151820\pi$
$197$	24.9385	1.77679	0.888397	+	0.459077i	$0.151820\pi$
$198$	−3.41197	−0.242478
$199$	−9.17393	−0.650322	−0.325161	−	0.945659i	$-0.605419\pi$
$199$	−9.17393	−0.650322	−0.325161	+	0.945659i	$0.605419\pi$
$200$	−5.47234	−0.386953
$201$	15.2231	1.07375
$202$	0.974186	0.0685435
$203$	−3.89667	−0.273492
$204$	−7.35220	−0.514757
$205$	−2.96399	−0.207014
$206$	8.41019	0.585966
$207$	3.20067	0.222462
$208$	−20.5658	−1.42598
$209$	−4.26111	−0.294747
$210$	0.350072	0.0241573
$211$	2.63956	0.181715	0.0908575	−	0.995864i	$-0.471039\pi$
$211$	2.63956	0.181715	0.0908575	+	0.995864i	$0.471039\pi$
$212$	15.6020	1.07155
$213$	8.73331	0.598396
$214$	−16.4751	−1.12621
$215$	−1.00650	−0.0686425
$216$	1.11368	0.0757763
$217$	−3.33665	−0.226506
$218$	7.86558	0.532724
$219$	−0.335823	−0.0226928
$220$	−0.758964	−0.0511693
$221$	−22.3684	−1.50466
$222$	1.36988	0.0919405
$223$	5.18933	0.347503	0.173752	−	0.984789i	$-0.444411\pi$
$223$	5.18933	0.347503	0.173752	+	0.984789i	$0.444411\pi$
$224$	4.33238	0.289469
$225$	−4.91374	−0.327583
$226$	−4.65250	−0.309479
$227$	−11.7750	−0.781537	−0.390768	−	0.920489i	$-0.627791\pi$
$227$	−11.7750	−0.781537	−0.390768	+	0.920489i	$0.627791\pi$
$228$	−3.21180	−0.212707
$229$	7.56464	0.499886	0.249943	−	0.968261i	$-0.419588\pi$
$229$	7.56464	0.499886	0.249943	+	0.968261i	$0.419588\pi$
$230$	1.73223	0.114220
$231$	1.19767	0.0788010
$232$	−6.70906	−0.440471
$233$	17.3214	1.13476	0.567380	−	0.823456i	$-0.307957\pi$
$233$	17.3214	1.13476	0.567380	+	0.823456i	$0.307957\pi$
$234$	−7.82437	−0.511495
$235$	0.202912	0.0132365
$236$	3.00052	0.195317
$237$	−0.968713	−0.0629247
$238$	6.27912	0.407015
$239$	13.6066	0.880136	0.440068	−	0.897964i	$-0.354954\pi$
$239$	13.6066	0.880136	0.440068	+	0.897964i	$0.354954\pi$
$240$	1.42253	0.0918239
$241$	1.90399	0.122647	0.0613235	−	0.998118i	$-0.480468\pi$
$241$	1.90399	0.122647	0.0613235	+	0.998118i	$0.480468\pi$
$242$	13.9524	0.896894
$243$	1.00000	0.0641500
$244$	10.8760	0.696262
$245$	1.93302	0.123496
$246$	−18.5965	−1.18567
$247$	−9.77162	−0.621753
$248$	−5.74485	−0.364798
$249$	−2.39580	−0.151827
$250$	−5.36540	−0.339338
$251$	28.0823	1.77254	0.886270	−	0.463168i	$-0.153288\pi$
$251$	28.0823	1.77254	0.886270	+	0.463168i	$0.153288\pi$
$252$	0.902743	0.0568674
$253$	5.92633	0.372585
$254$	13.4881	0.846319
$255$	1.54721	0.0968903
$256$	20.9787	1.31117
$257$	−4.44220	−0.277097	−0.138548	−	0.990356i	$-0.544244\pi$
$257$	−4.44220	−0.277097	−0.138548	+	0.990356i	$0.544244\pi$
$258$	−6.31492	−0.393150
$259$	−0.480857	−0.0298790
$260$	−1.74046	−0.107939
$261$	−6.02422	−0.372890
$262$	29.1884	1.80327
$263$	3.85261	0.237562	0.118781	−	0.992920i	$-0.462101\pi$
$263$	3.85261	0.237562	0.118781	+	0.992920i	$0.462101\pi$
$264$	2.06208	0.126912
$265$	−3.28333	−0.201693
$266$	2.74303	0.168186
$267$	10.5407	0.645082
$268$	21.2459	1.29780
$269$	−14.5053	−0.884403	−0.442202	−	0.896916i	$-0.645802\pi$
$269$	−14.5053	−0.884403	−0.442202	+	0.896916i	$0.645802\pi$
$270$	0.541209	0.0329369
$271$	23.2270	1.41094	0.705470	−	0.708740i	$-0.250736\pi$
$271$	23.2270	1.41094	0.705470	+	0.708740i	$0.250736\pi$
$272$	25.5154	1.54710
$273$	2.74651	0.166226
$274$	18.4114	1.11227
$275$	−9.09824	−0.548644
$276$	4.46696	0.268880
$277$	12.7164	0.764053	0.382027	−	0.924151i	$-0.375226\pi$
$277$	12.7164	0.764053	0.382027	+	0.924151i	$0.375226\pi$
$278$	−10.9724	−0.658080
$279$	−5.15844	−0.308827
$280$	−0.211572	−0.0126438
$281$	−12.4246	−0.741192	−0.370596	−	0.928794i	$-0.620847\pi$
$281$	−12.4246	−0.741192	−0.370596	+	0.928794i	$0.620847\pi$
$282$	1.27311	0.0758123
$283$	19.2639	1.14512	0.572561	−	0.819862i	$-0.305950\pi$
$283$	19.2639	1.14512	0.572561	+	0.819862i	$0.305950\pi$
$284$	12.1885	0.723254
$285$	0.675899	0.0400368
$286$	−14.4875	−0.856666
$287$	6.52776	0.385321
$288$	6.69783	0.394673
$289$	10.7518	0.632460
$290$	−3.26036	−0.191455
$291$	7.83508	0.459301
$292$	−0.468685	−0.0274277
$293$	13.9557	0.815302	0.407651	−	0.913138i	$-0.366348\pi$
$293$	13.9557	0.815302	0.407651	+	0.913138i	$0.366348\pi$
$294$	12.1281	0.707324
$295$	−0.631437	−0.0367637
$296$	−0.827911	−0.0481214
$297$	1.85159	0.107440
$298$	−24.4951	−1.41897
$299$	13.5903	0.785949
$300$	−6.85778	−0.395934
$301$	2.21666	0.127766
$302$	35.1394	2.02205
$303$	−0.528666	−0.0303711
$304$	11.1464	0.639290
$305$	−2.28876	−0.131054
$306$	9.70747	0.554939
$307$	−18.8730	−1.07714	−0.538568	−	0.842582i	$-0.681035\pi$
$307$	−18.8730	−1.07714	−0.538568	+	0.842582i	$0.681035\pi$
$308$	1.67151	0.0952431
$309$	−4.56400	−0.259637
$310$	−2.79179	−0.158563
$311$	−19.1206	−1.08423	−0.542115	−	0.840304i	$-0.682376\pi$
$311$	−19.1206	−1.08423	−0.542115	+	0.840304i	$0.682376\pi$
$312$	4.72879	0.267715
$313$	15.8421	0.895449	0.447724	−	0.894172i	$-0.352235\pi$
$313$	15.8421	0.895449	0.447724	+	0.894172i	$0.352235\pi$
$314$	36.6393	2.06767
$315$	−0.189975	−0.0107039
$316$	−1.35197	−0.0760542
$317$	12.6076	0.708115	0.354058	−	0.935224i	$-0.384802\pi$
$317$	12.6076	0.708115	0.354058	+	0.935224i	$0.384802\pi$
$318$	−20.6002	−1.15520
$319$	−11.1544	−0.624526
$320$	0.779864	0.0435957
$321$	8.94060	0.499016
$322$	−3.81499	−0.212601
$323$	12.1234	0.674562
$324$	1.39563	0.0775352
$325$	−20.8642	−1.15734
$326$	−23.9533	−1.32665
$327$	−4.26845	−0.236046
$328$	11.2391	0.620576
$329$	−0.446886	−0.0246376
$330$	1.00210	0.0551637
$331$	−3.51329	−0.193108	−0.0965539	−	0.995328i	$-0.530782\pi$
$331$	−3.51329	−0.193108	−0.0965539	+	0.995328i	$0.530782\pi$
$332$	−3.34365	−0.183507
$333$	−0.743401	−0.0407381
$334$	−44.7383	−2.44797
$335$	−4.47103	−0.244278
$336$	−3.13292	−0.170915
$337$	−2.15555	−0.117420	−0.0587102	−	0.998275i	$-0.518699\pi$
$337$	−2.15555	−0.117420	−0.0587102	+	0.998275i	$0.518699\pi$
$338$	−9.26757	−0.504089
$339$	2.52479	0.137128
$340$	2.15934	0.117107
$341$	−9.55132	−0.517233
$342$	4.24070	0.229311
$343$	−8.78504	−0.474347
$344$	3.81653	0.205773
$345$	−0.940038	−0.0506099
$346$	−21.5474	−1.15840
$347$	29.6742	1.59299	0.796496	−	0.604644i	$-0.206684\pi$
$347$	29.6742	1.59299	0.796496	+	0.604644i	$0.206684\pi$
$348$	−8.40761	−0.450695
$349$	−24.8517	−1.33028	−0.665140	−	0.746719i	$-0.731628\pi$
$349$	−24.8517	−1.33028	−0.665140	+	0.746719i	$0.731628\pi$
$350$	5.85686	0.313062
$351$	4.24609	0.226639
$352$	12.4016	0.661010
$353$	24.7916	1.31952	0.659762	−	0.751475i	$-0.270657\pi$
$353$	24.7916	1.31952	0.659762	+	0.751475i	$0.270657\pi$
$354$	−3.96174	−0.210564
$355$	−2.56498	−0.136135
$356$	14.7110	0.779682
$357$	−3.40752	−0.180345
$358$	11.5015	0.607873
$359$	4.50904	0.237978	0.118989	−	0.992896i	$-0.462035\pi$
$359$	4.50904	0.237978	0.118989	+	0.992896i	$0.462035\pi$
$360$	−0.327088	−0.0172391
$361$	−13.7039	−0.721259
$362$	−30.8241	−1.62008
$363$	−7.57161	−0.397406
$364$	3.83313	0.200910
$365$	0.0986312	0.00516259
$366$	−14.3601	−0.750613
$367$	−12.0791	−0.630521	−0.315261	−	0.949005i	$-0.602092\pi$
$367$	−12.0791	−0.630521	−0.315261	+	0.949005i	$0.602092\pi$
$368$	−15.5024	−0.808116
$369$	10.0919	0.525362
$370$	−0.402335	−0.0209164
$371$	7.23107	0.375418
$372$	−7.19929	−0.373266
$373$	3.90152	0.202013	0.101007	−	0.994886i	$-0.467794\pi$
$373$	3.90152	0.202013	0.101007	+	0.994886i	$0.467794\pi$
$374$	17.9743	0.929427
$375$	2.91167	0.150358
$376$	−0.769422	−0.0396799
$377$	−25.5794	−1.31741
$378$	−1.19194	−0.0613066
$379$	13.0054	0.668044	0.334022	−	0.942565i	$-0.391594\pi$
$379$	13.0054	0.668044	0.334022	+	0.942565i	$0.391594\pi$
$380$	0.943308	0.0483907
$381$	−7.31966	−0.374997
$382$	25.9905	1.32979
$383$	−4.00315	−0.204552	−0.102276	−	0.994756i	$-0.532612\pi$
$383$	−4.00315	−0.204552	−0.102276	+	0.994756i	$0.532612\pi$
$384$	−8.50266	−0.433900
$385$	−0.351756	−0.0179272
$386$	−15.6968	−0.798944
$387$	3.42695	0.174202
$388$	10.9349	0.555136
$389$	33.5018	1.69861	0.849305	−	0.527903i	$-0.177021\pi$
$389$	33.5018	1.69861	0.849305	+	0.527903i	$0.177021\pi$
$390$	2.29802	0.116365
$391$	−16.8611	−0.852704
$392$	−7.32981	−0.370211
$393$	−15.8398	−0.799013
$394$	−45.9548	−2.31517
$395$	0.284511	0.0143153
$396$	2.58414	0.129858
$397$	5.18980	0.260469	0.130234	−	0.991483i	$-0.458427\pi$
$397$	5.18980	0.260469	0.130234	+	0.991483i	$0.458427\pi$
$398$	16.9050	0.847372
$399$	−1.48857	−0.0745218
$400$	23.7996	1.18998
$401$	24.8808	1.24249	0.621244	−	0.783617i	$-0.286628\pi$
$401$	24.8808	1.24249	0.621244	+	0.783617i	$0.286628\pi$
$402$	−28.0520	−1.39911
$403$	−21.9032	−1.09108
$404$	−0.737824	−0.0367081
$405$	−0.293700	−0.0145941
$406$	7.18048	0.356361
$407$	−1.37647	−0.0682293
$408$	−5.86687	−0.290453
$409$	−33.8508	−1.67381	−0.836906	−	0.547346i	$-0.815638\pi$
$409$	−33.8508	−1.67381	−0.836906	+	0.547346i	$0.815638\pi$
$410$	5.46181	0.269740
$411$	−9.99139	−0.492839
$412$	−6.36967	−0.313811
$413$	1.39065	0.0684294
$414$	−5.89795	−0.289868
$415$	0.703646	0.0345406
$416$	28.4396	1.39436
$417$	5.95444	0.291590
$418$	7.85205	0.384056
$419$	7.37396	0.360242	0.180121	−	0.983645i	$-0.442351\pi$
$419$	7.37396	0.360242	0.180121	+	0.983645i	$0.442351\pi$
$420$	−0.265136	−0.0129373
$421$	−22.8701	−1.11462	−0.557311	−	0.830304i	$-0.688167\pi$
$421$	−22.8701	−1.11462	−0.557311	+	0.830304i	$0.688167\pi$
$422$	−4.86399	−0.236775
$423$	−0.690882	−0.0335918
$424$	12.4500	0.604627
$425$	25.8856	1.25564
$426$	−16.0931	−0.779712
$427$	5.04067	0.243935
$428$	12.4778	0.603138
$429$	7.86202	0.379582
$430$	1.85470	0.0894413
$431$	−6.35806	−0.306257	−0.153129	−	0.988206i	$-0.548935\pi$
$431$	−6.35806	−0.306257	−0.153129	+	0.988206i	$0.548935\pi$
$432$	−4.84347	−0.233032
$433$	−16.5154	−0.793677	−0.396839	−	0.917888i	$-0.629893\pi$
$433$	−16.5154	−0.793677	−0.396839	+	0.917888i	$0.629893\pi$
$434$	6.14852	0.295139
$435$	1.76932	0.0848322
$436$	−5.95719	−0.285298
$437$	−7.36577	−0.352353
$438$	0.618828	0.0295688
$439$	11.0073	0.525348	0.262674	−	0.964885i	$-0.415396\pi$
$439$	11.0073	0.525348	0.262674	+	0.964885i	$0.415396\pi$
$440$	−0.605634	−0.0288725
$441$	−6.58161	−0.313410
$442$	41.2188	1.96058
$443$	−1.26932	−0.0603074	−0.0301537	−	0.999545i	$-0.509600\pi$
$443$	−1.26932	−0.0603074	−0.0301537	+	0.999545i	$0.509600\pi$
$444$	−1.03752	−0.0492383
$445$	−3.09582	−0.146756
$446$	−9.56251	−0.452798
$447$	13.2929	0.628732
$448$	−1.71754	−0.0811461
$449$	30.3277	1.43125	0.715627	−	0.698483i	$-0.246141\pi$
$449$	30.3277	1.43125	0.715627	+	0.698483i	$0.246141\pi$
$450$	9.05467	0.426841
$451$	18.6860	0.879890
$452$	3.52369	0.165740
$453$	−19.0693	−0.895953
$454$	21.6982	1.01834
$455$	−0.806652	−0.0378164
$456$	−2.56294	−0.120021
$457$	8.84302	0.413659	0.206830	−	0.978377i	$-0.433685\pi$
$457$	8.84302	0.413659	0.206830	+	0.978377i	$0.433685\pi$
$458$	−13.9395	−0.651352
$459$	−5.26800	−0.245889
$460$	−1.31195	−0.0611699
$461$	32.3015	1.50443	0.752215	−	0.658917i	$-0.228985\pi$
$461$	32.3015	1.50443	0.752215	+	0.658917i	$0.228985\pi$
$462$	−2.20698	−0.102678
$463$	−30.8173	−1.43220	−0.716100	−	0.697998i	$-0.754075\pi$
$463$	−30.8173	−1.43220	−0.716100	+	0.697998i	$0.754075\pi$
$464$	29.1782	1.35456
$465$	1.51503	0.0702580
$466$	−31.9185	−1.47860
$467$	−10.7823	−0.498946	−0.249473	−	0.968382i	$-0.580257\pi$
$467$	−10.7823	−0.498946	−0.249473	+	0.968382i	$0.580257\pi$
$468$	5.92599	0.273929
$469$	9.84681	0.454683
$470$	−0.373912	−0.0172473
$471$	−19.8832	−0.916170
$472$	2.39434	0.110208
$473$	6.34531	0.291758
$474$	1.78507	0.0819910
$475$	11.3081	0.518851
$476$	−4.75565	−0.217975
$477$	11.1792	0.511860
$478$	−25.0732	−1.14682
$479$	29.7721	1.36032	0.680161	−	0.733063i	$-0.261910\pi$
$479$	29.7721	1.36032	0.680161	+	0.733063i	$0.261910\pi$
$480$	−1.96715	−0.0897879
$481$	−3.15655	−0.143926
$482$	−3.50853	−0.159809
$483$	2.07030	0.0942019
$484$	−10.5672	−0.480327
$485$	−2.30117	−0.104491
$486$	−1.84272	−0.0835877
$487$	−7.15529	−0.324237	−0.162119	−	0.986771i	$-0.551833\pi$
$487$	−7.15529	−0.324237	−0.162119	+	0.986771i	$0.551833\pi$
$488$	8.67874	0.392868
$489$	12.9989	0.587829
$490$	−3.56202	−0.160916
$491$	−19.1505	−0.864248	−0.432124	−	0.901814i	$-0.642236\pi$
$491$	−19.1505	−0.864248	−0.432124	+	0.901814i	$0.642236\pi$
$492$	14.0846	0.634981
$493$	31.7356	1.42930
$494$	18.0064	0.810146
$495$	−0.543813	−0.0244426
$496$	24.9847	1.12185
$497$	5.64899	0.253392
$498$	4.41479	0.197831
$499$	24.7933	1.10990	0.554950	−	0.831884i	$-0.312737\pi$
$499$	24.7933	1.10990	0.554950	+	0.831884i	$0.312737\pi$
$500$	4.06362	0.181731
$501$	24.2783	1.08468
$502$	−51.7480	−2.30963
$503$	−43.6406	−1.94584	−0.972919	−	0.231148i	$-0.925752\pi$
$503$	−43.6406	−1.94584	−0.972919	+	0.231148i	$0.925752\pi$
$504$	0.720366	0.0320876
$505$	0.155269	0.00690940
$506$	−10.9206	−0.485480
$507$	5.02927	0.223358
$508$	−10.2156	−0.453242
$509$	25.5285	1.13153	0.565765	−	0.824566i	$-0.308581\pi$
$509$	25.5285	1.13153	0.565765	+	0.824566i	$0.308581\pi$
$510$	−2.85109	−0.126248
$511$	−0.217221	−0.00960930
$512$	−21.6526	−0.956918
$513$	−2.30132	−0.101606
$514$	8.18576	0.361058
$515$	1.34045	0.0590672
$516$	4.78277	0.210550
$517$	−1.27923	−0.0562605
$518$	0.886086	0.0389324
$519$	11.6932	0.513276
$520$	−1.38885	−0.0609049
$521$	2.77972	0.121782	0.0608909	−	0.998144i	$-0.480606\pi$
$521$	2.77972	0.121782	0.0608909	+	0.998144i	$0.480606\pi$
$522$	11.1010	0.485877
$523$	9.69648	0.423997	0.211999	−	0.977270i	$-0.432003\pi$
$523$	9.69648	0.423997	0.211999	+	0.977270i	$0.432003\pi$
$524$	−22.1066	−0.965731
$525$	−3.17837	−0.138715
$526$	−7.09929	−0.309544
$527$	27.1746	1.18375
$528$	−8.96813	−0.390288
$529$	−12.7557	−0.554596
$530$	6.05027	0.262807
$531$	2.14994	0.0932992
$532$	−2.07750	−0.0900711
$533$	42.8510	1.85608
$534$	−19.4237	−0.840544
$535$	−2.62586	−0.113526
$536$	16.9537	0.732287
$537$	−6.24157	−0.269344
$538$	26.7293	1.15238
$539$	−12.1864	−0.524907
$540$	−0.409898	−0.0176392
$541$	−20.4108	−0.877528	−0.438764	−	0.898602i	$-0.644584\pi$
$541$	−20.4108	−0.877528	−0.438764	+	0.898602i	$0.644584\pi$
$542$	−42.8010	−1.83846
$543$	16.7275	0.717844
$544$	−35.2842	−1.51280
$545$	1.25365	0.0537003
$546$	−5.06107	−0.216594
$547$	−5.42184	−0.231821	−0.115911	−	0.993260i	$-0.536979\pi$
$547$	−5.42184	−0.231821	−0.115911	+	0.993260i	$0.536979\pi$
$548$	−13.9443	−0.595672
$549$	7.79285	0.332591
$550$	16.7655	0.714885
$551$	13.8637	0.590612
$552$	3.56452	0.151716
$553$	−0.626596	−0.0266456
$554$	−23.4328	−0.995564
$555$	0.218337	0.00926790
$556$	8.31022	0.352432
$557$	−36.9812	−1.56694	−0.783471	−	0.621428i	$-0.786553\pi$
$557$	−36.9812	−1.56694	−0.783471	+	0.621428i	$0.786553\pi$
$558$	9.50558	0.402403
$559$	14.5511	0.615447
$560$	0.920140	0.0388830
$561$	−9.75418	−0.411822
$562$	22.8952	0.965776
$563$	−37.0545	−1.56166	−0.780830	−	0.624743i	$-0.785204\pi$
$563$	−37.0545	−1.56166	−0.780830	+	0.624743i	$0.785204\pi$
$564$	−0.964219	−0.0406009
$565$	−0.741532	−0.0311965
$566$	−35.4981	−1.49210
$567$	0.646833	0.0271644
$568$	9.72611	0.408099
$569$	−17.6690	−0.740725	−0.370362	−	0.928887i	$-0.620766\pi$
$569$	−17.6690	−0.740725	−0.370362	+	0.928887i	$0.620766\pi$
$570$	−1.24550	−0.0521681
$571$	−4.83324	−0.202265	−0.101132	−	0.994873i	$-0.532247\pi$
$571$	−4.83324	−0.202265	−0.101132	+	0.994873i	$0.532247\pi$
$572$	10.9725	0.458783
$573$	−14.1044	−0.589219
$574$	−12.0289	−0.502075
$575$	−15.7273	−0.655872
$576$	−2.65530	−0.110638
$577$	19.9202	0.829288	0.414644	−	0.909984i	$-0.363906\pi$
$577$	19.9202	0.829288	0.414644	+	0.909984i	$0.363906\pi$
$578$	−19.8126	−0.824097
$579$	8.51823	0.354006
$580$	2.46932	0.102533
$581$	−1.54968	−0.0642916
$582$	−14.4379	−0.598470
$583$	20.6993	0.857276
$584$	−0.373999	−0.0154762
$585$	−1.24708	−0.0515603
$586$	−25.7166	−1.06234
$587$	−35.0931	−1.44845	−0.724223	−	0.689566i	$-0.757801\pi$
$587$	−35.0931	−1.44845	−0.724223	+	0.689566i	$0.757801\pi$
$588$	−9.18551	−0.378804
$589$	11.8712	0.489145
$590$	1.16356	0.0479032
$591$	24.9385	1.02583
$592$	3.60064	0.147986
$593$	−34.8788	−1.43230	−0.716150	−	0.697946i	$-0.754098\pi$
$593$	−34.8788	−1.43230	−0.716150	+	0.697946i	$0.754098\pi$
$594$	−3.41197	−0.139995
$595$	1.00079	0.0410284
$596$	18.5520	0.759920
$597$	−9.17393	−0.375464
$598$	−25.0432	−1.02409
$599$	24.0173	0.981320	0.490660	−	0.871351i	$-0.336756\pi$
$599$	24.0173	0.981320	0.490660	+	0.871351i	$0.336756\pi$
$600$	−5.47234	−0.223407
$601$	−46.0316	−1.87767	−0.938833	−	0.344372i	$-0.888092\pi$
$601$	−46.0316	−1.87767	−0.938833	+	0.344372i	$0.888092\pi$
$602$	−4.08470	−0.166480
$603$	15.2231	0.619932
$604$	−26.6137	−1.08290
$605$	2.22378	0.0904097
$606$	0.974186	0.0395736
$607$	2.68396	0.108938	0.0544692	−	0.998515i	$-0.482653\pi$
$607$	2.68396	0.108938	0.0544692	+	0.998515i	$0.482653\pi$
$608$	−15.4139	−0.625115
$609$	−3.89667	−0.157901
$610$	4.21756	0.170764
$611$	−2.93355	−0.118679
$612$	−7.35220	−0.297195
$613$	21.0635	0.850746	0.425373	−	0.905018i	$-0.360143\pi$
$613$	21.0635	0.850746	0.425373	+	0.905018i	$0.360143\pi$
$614$	34.7777	1.40351
$615$	−2.96399	−0.119519
$616$	1.33382	0.0537413
$617$	7.13857	0.287388	0.143694	−	0.989622i	$-0.454102\pi$
$617$	7.13857	0.287388	0.143694	+	0.989622i	$0.454102\pi$
$618$	8.41019	0.338307
$619$	32.3353	1.29967	0.649834	−	0.760077i	$-0.274839\pi$
$619$	32.3353	1.29967	0.649834	+	0.760077i	$0.274839\pi$
$620$	2.11443	0.0849177
$621$	3.20067	0.128438
$622$	35.2340	1.41276
$623$	6.81810	0.273161
$624$	−20.5658	−0.823292
$625$	23.7135	0.948542
$626$	−29.1926	−1.16677
$627$	−4.26111	−0.170172
$628$	−27.7497	−1.10733
$629$	3.91624	0.156151
$630$	0.350072	0.0139472
$631$	−10.1537	−0.404212	−0.202106	−	0.979364i	$-0.564779\pi$
$631$	−10.1537	−0.404212	−0.202106	+	0.979364i	$0.564779\pi$
$632$	−1.07884	−0.0429138
$633$	2.63956	0.104913
$634$	−23.2324	−0.922676
$635$	2.14979	0.0853116
$636$	15.6020	0.618661
$637$	−27.9461	−1.10726
$638$	20.5545	0.813760
$639$	8.73331	0.345484
$640$	2.49723	0.0987119
$641$	−10.3943	−0.410549	−0.205275	−	0.978704i	$-0.565809\pi$
$641$	−10.3943	−0.410549	−0.205275	+	0.978704i	$0.565809\pi$
$642$	−16.4751	−0.650219
$643$	50.3179	1.98434	0.992172	−	0.124880i	$-0.0398545\pi$
$643$	50.3179	1.98434	0.992172	+	0.124880i	$0.0398545\pi$
$644$	2.88938	0.113858
$645$	−1.00650	−0.0396307
$646$	−22.3400	−0.878956
$647$	20.3227	0.798969	0.399484	−	0.916740i	$-0.369189\pi$
$647$	20.3227	0.798969	0.399484	+	0.916740i	$0.369189\pi$
$648$	1.11368	0.0437495
$649$	3.98080	0.156260
$650$	38.4469	1.50801
$651$	−3.33665	−0.130774
$652$	18.1416	0.710481
$653$	19.5617	0.765508	0.382754	−	0.923850i	$-0.374976\pi$
$653$	19.5617	0.765508	0.382754	+	0.923850i	$0.374976\pi$
$654$	7.86558	0.307568
$655$	4.65216	0.181775
$656$	−48.8797	−1.90843
$657$	−0.335823	−0.0131017
$658$	0.823487	0.0321029
$659$	13.5448	0.527630	0.263815	−	0.964573i	$-0.415019\pi$
$659$	13.5448	0.527630	0.263815	+	0.964573i	$0.415019\pi$
$660$	−0.758964	−0.0295426
$661$	4.42718	0.172197	0.0860987	−	0.996287i	$-0.472560\pi$
$661$	4.42718	0.172197	0.0860987	+	0.996287i	$0.472560\pi$
$662$	6.47402	0.251620
$663$	−22.3684	−0.868717
$664$	−2.66815	−0.103544
$665$	0.437194	0.0169537
$666$	1.36988	0.0530819
$667$	−19.2815	−0.746584
$668$	33.8837	1.31100
$669$	5.18933	0.200631
$670$	8.23888	0.318296
$671$	14.4292	0.557032
$672$	4.33238	0.167125
$673$	−22.3570	−0.861799	−0.430900	−	0.902400i	$-0.641804\pi$
$673$	−22.3570	−0.861799	−0.430900	+	0.902400i	$0.641804\pi$
$674$	3.97209	0.152999
$675$	−4.91374	−0.189130
$676$	7.01902	0.269962
$677$	−18.0570	−0.693985	−0.346993	−	0.937868i	$-0.612797\pi$
$677$	−18.0570	−0.693985	−0.346993	+	0.937868i	$0.612797\pi$
$678$	−4.65250	−0.178678
$679$	5.06799	0.194492
$680$	1.72310	0.0660779
$681$	−11.7750	−0.451221
$682$	17.6004	0.673956
$683$	29.9223	1.14494	0.572472	−	0.819924i	$-0.305984\pi$
$683$	29.9223	1.14494	0.572472	+	0.819924i	$0.305984\pi$
$684$	−3.21180	−0.122806
$685$	2.93447	0.112120
$686$	16.1884	0.618076
$687$	7.56464	0.288609
$688$	−16.5983	−0.632806
$689$	47.4678	1.80838
$690$	1.73223	0.0659449
$691$	10.7078	0.407345	0.203673	−	0.979039i	$-0.434712\pi$
$691$	10.7078	0.407345	0.203673	+	0.979039i	$0.434712\pi$
$692$	16.3195	0.620373
$693$	1.19767	0.0454958
$694$	−54.6813	−2.07567
$695$	−1.74882	−0.0663366
$696$	−6.70906	−0.254306
$697$	−53.1640	−2.01373
$698$	45.7948	1.73336
$699$	17.3214	0.655154
$700$	−4.43584	−0.167659
$701$	17.5395	0.662459	0.331230	−	0.943550i	$-0.392537\pi$
$701$	17.5395	0.662459	0.331230	+	0.943550i	$0.392537\pi$
$702$	−7.82437	−0.295312
$703$	1.71081	0.0645242
$704$	−4.91654	−0.185299
$705$	0.202912	0.00764212
$706$	−45.6841	−1.71934
$707$	−0.341959	−0.0128607
$708$	3.00052	0.112767
$709$	−1.46445	−0.0549984	−0.0274992	−	0.999622i	$-0.508754\pi$
$709$	−1.46445	−0.0549984	−0.0274992	+	0.999622i	$0.508754\pi$
$710$	4.72654	0.177384
$711$	−0.968713	−0.0363296
$712$	11.7390	0.439938
$713$	−16.5104	−0.618321
$714$	6.27912	0.234990
$715$	−2.30908	−0.0863547
$716$	−8.71095	−0.325543
$717$	13.6066	0.508147
$718$	−8.30892	−0.310086
$719$	21.9454	0.818424	0.409212	−	0.912439i	$-0.365804\pi$
$719$	21.9454	0.818424	0.409212	+	0.912439i	$0.365804\pi$
$720$	1.42253	0.0530146
$721$	−2.95215	−0.109944
$722$	25.2525	0.939802
$723$	1.90399	0.0708103
$724$	23.3454	0.867626
$725$	29.6015	1.09937
$726$	13.9524	0.517822
$727$	−2.47191	−0.0916781	−0.0458391	−	0.998949i	$-0.514596\pi$
$727$	−2.47191	−0.0916781	−0.0458391	+	0.998949i	$0.514596\pi$
$728$	3.05874	0.113364
$729$	1.00000	0.0370370
$730$	−0.181750	−0.00672688
$731$	−18.0532	−0.667720
$732$	10.8760	0.401987
$733$	15.1247	0.558645	0.279322	−	0.960197i	$-0.409890\pi$
$733$	15.1247	0.558645	0.279322	+	0.960197i	$0.409890\pi$
$734$	22.2584	0.821571
$735$	1.93302	0.0713005
$736$	21.4375	0.790198
$737$	28.1870	1.03828
$738$	−18.5965	−0.684548
$739$	2.39520	0.0881088	0.0440544	−	0.999029i	$-0.485973\pi$
$739$	2.39520	0.0881088	0.0440544	+	0.999029i	$0.485973\pi$
$740$	0.304719	0.0112017
$741$	−9.77162	−0.358969
$742$	−13.3249	−0.489171
$743$	−11.1720	−0.409861	−0.204931	−	0.978777i	$-0.565697\pi$
$743$	−11.1720	−0.409861	−0.204931	+	0.978777i	$0.565697\pi$
$744$	−5.74485	−0.210616
$745$	−3.90413	−0.143036
$746$	−7.18943	−0.263224
$747$	−2.39580	−0.0876576
$748$	−13.6133	−0.497750
$749$	5.78308	0.211309
$750$	−5.36540	−0.195917
$751$	−16.0936	−0.587262	−0.293631	−	0.955919i	$-0.594864\pi$
$751$	−16.0936	−0.587262	−0.293631	+	0.955919i	$0.594864\pi$
$752$	3.34627	0.122026
$753$	28.0823	1.02338
$754$	47.1358	1.71658
$755$	5.60066	0.203829
$756$	0.902743	0.0328324
$757$	−13.8286	−0.502610	−0.251305	−	0.967908i	$-0.580860\pi$
$757$	−13.8286	−0.502610	−0.251305	+	0.967908i	$0.580860\pi$
$758$	−23.9654	−0.870463
$759$	5.92633	0.215112
$760$	0.752736	0.0273046
$761$	28.0377	1.01636	0.508182	−	0.861250i	$-0.330318\pi$
$761$	28.0377	1.01636	0.508182	+	0.861250i	$0.330318\pi$
$762$	13.4881	0.488623
$763$	−2.76098	−0.0999540
$764$	−19.6845	−0.712162
$765$	1.54721	0.0559396
$766$	7.37671	0.266531
$767$	9.12882	0.329622
$768$	20.9787	0.757003
$769$	28.9359	1.04346	0.521728	−	0.853112i	$-0.325288\pi$
$769$	28.9359	1.04346	0.521728	+	0.853112i	$0.325288\pi$
$770$	0.648190	0.0233592
$771$	−4.44220	−0.159982
$772$	11.8883	0.427870
$773$	−27.0680	−0.973567	−0.486784	−	0.873523i	$-0.661830\pi$
$773$	−27.0680	−0.973567	−0.486784	+	0.873523i	$0.661830\pi$
$774$	−6.31492	−0.226985
$775$	25.3472	0.910499
$776$	8.72577	0.313237
$777$	−0.480857	−0.0172506
$778$	−61.7346	−2.21329
$779$	−23.2246	−0.832109
$780$	−1.74046	−0.0623186
$781$	16.1705	0.578626
$782$	31.0704	1.11108
$783$	−6.02422	−0.215288
$784$	31.8778	1.13849
$785$	5.83970	0.208428
$786$	29.1884	1.04112
$787$	23.3178	0.831190	0.415595	−	0.909550i	$-0.363573\pi$
$787$	23.3178	0.831190	0.415595	+	0.909550i	$0.363573\pi$
$788$	34.8050	1.23988
$789$	3.85261	0.137156
$790$	−0.524276	−0.0186529
$791$	1.63312	0.0580671
$792$	2.06208	0.0732729
$793$	33.0891	1.17503
$794$	−9.56338	−0.339391
$795$	−3.28333	−0.116448
$796$	−12.8034	−0.453806
$797$	−47.1247	−1.66924	−0.834621	−	0.550824i	$-0.814314\pi$
$797$	−47.1247	−1.66924	−0.834621	+	0.550824i	$0.814314\pi$
$798$	2.74303	0.0971022
$799$	3.63957	0.128759
$800$	−32.9114	−1.16359
$801$	10.5407	0.372438
$802$	−45.8485	−1.61897
$803$	−0.621806	−0.0219431
$804$	21.2459	0.749284
$805$	−0.608048	−0.0214309
$806$	40.3615	1.42167
$807$	−14.5053	−0.510610
$808$	−0.588765	−0.0207127
$809$	27.4950	0.966673	0.483337	−	0.875435i	$-0.339425\pi$
$809$	27.4950	0.966673	0.483337	+	0.875435i	$0.339425\pi$
$810$	0.541209	0.0190161
$811$	52.0755	1.82862	0.914310	−	0.405016i	$-0.132734\pi$
$811$	52.0755	1.82862	0.914310	+	0.405016i	$0.132734\pi$
$812$	−5.43832	−0.190848
$813$	23.2270	0.814607
$814$	2.53646	0.0889030
$815$	−3.81777	−0.133731
$816$	25.5154	0.893218
$817$	−7.88651	−0.275914
$818$	62.3776	2.18098
$819$	2.74651	0.0959709
$820$	−4.13664	−0.144458
$821$	−19.4300	−0.678110	−0.339055	−	0.940767i	$-0.610107\pi$
$821$	−19.4300	−0.678110	−0.339055	+	0.940767i	$0.610107\pi$
$822$	18.4114	0.642170
$823$	38.0131	1.32505	0.662527	−	0.749038i	$-0.269484\pi$
$823$	38.0131	1.32505	0.662527	+	0.749038i	$0.269484\pi$
$824$	−5.08283	−0.177069
$825$	−9.09824	−0.316760
$826$	−2.56258	−0.0891637
$827$	19.5155	0.678620	0.339310	−	0.940675i	$-0.389807\pi$
$827$	19.5155	0.678620	0.339310	+	0.940675i	$0.389807\pi$
$828$	4.46696	0.155238
$829$	−39.4862	−1.37141	−0.685707	−	0.727878i	$-0.740507\pi$
$829$	−39.4862	−1.37141	−0.685707	+	0.727878i	$0.740507\pi$
$830$	−1.29663	−0.0450065
$831$	12.7164	0.441126
$832$	−11.2747	−0.390878
$833$	34.6719	1.20131
$834$	−10.9724	−0.379943
$835$	−7.13056	−0.246763
$836$	−5.94695	−0.205679
$837$	−5.15844	−0.178302
$838$	−13.5882	−0.469396
$839$	−39.6675	−1.36947	−0.684736	−	0.728791i	$-0.740083\pi$
$839$	−39.6675	−1.36947	−0.684736	+	0.728791i	$0.740083\pi$
$840$	−0.211572	−0.00729992
$841$	7.29124	0.251422
$842$	42.1434	1.45236
$843$	−12.4246	−0.427927
$844$	3.68386	0.126804
$845$	−1.47710	−0.0508138
$846$	1.27311	0.0437703
$847$	−4.89757	−0.168282
$848$	−54.1461	−1.85938
$849$	19.2639	0.661136
$850$	−47.7000	−1.63610
$851$	−2.37938	−0.0815641
$852$	12.1885	0.417571
$853$	50.6834	1.73537	0.867683	−	0.497118i	$-0.165608\pi$
$853$	50.6834	1.73537	0.867683	+	0.497118i	$0.165608\pi$
$854$	−9.28858	−0.317849
$855$	0.675899	0.0231153
$856$	9.95698	0.340322
$857$	44.0941	1.50623	0.753113	−	0.657892i	$-0.228552\pi$
$857$	44.0941	1.50623	0.753113	+	0.657892i	$0.228552\pi$
$858$	−14.4875	−0.494596
$859$	−6.86685	−0.234294	−0.117147	−	0.993115i	$-0.537375\pi$
$859$	−6.86685	−0.234294	−0.117147	+	0.993115i	$0.537375\pi$
$860$	−1.40470	−0.0478999
$861$	6.52776	0.222465
$862$	11.7162	0.399054
$863$	14.7872	0.503362	0.251681	−	0.967810i	$-0.419017\pi$
$863$	14.7872	0.503362	0.251681	+	0.967810i	$0.419017\pi$
$864$	6.69783	0.227865
$865$	−3.43431	−0.116770
$866$	30.4333	1.03416
$867$	10.7518	0.365151
$868$	−4.65674	−0.158060
$869$	−1.79366	−0.0608458
$870$	−3.26036	−0.110537
$871$	64.6386	2.19020
$872$	−4.75369	−0.160980
$873$	7.83508	0.265177
$874$	13.5731	0.459116
$875$	1.88336	0.0636694
$876$	−0.468685	−0.0158354
$877$	29.8849	1.00914	0.504570	−	0.863371i	$-0.331651\pi$
$877$	29.8849	1.00914	0.504570	+	0.863371i	$0.331651\pi$
$878$	−20.2834	−0.684530
$879$	13.9557	0.470715
$880$	2.63394	0.0887903
$881$	−16.1970	−0.545690	−0.272845	−	0.962058i	$-0.587965\pi$
$881$	−16.1970	−0.545690	−0.272845	+	0.962058i	$0.587965\pi$
$882$	12.1281	0.408374
$883$	35.4480	1.19292	0.596461	−	0.802642i	$-0.296573\pi$
$883$	35.4480	1.19292	0.596461	+	0.802642i	$0.296573\pi$
$884$	−31.2181	−1.04998
$885$	−0.631437	−0.0212255
$886$	2.33901	0.0785807
$887$	13.4981	0.453222	0.226611	−	0.973985i	$-0.427235\pi$
$887$	13.4981	0.453222	0.226611	+	0.973985i	$0.427235\pi$
$888$	−0.827911	−0.0277829
$889$	−4.73460	−0.158793
$890$	5.70474	0.191223
$891$	1.85159	0.0620306
$892$	7.24241	0.242494
$893$	1.58994	0.0532054
$894$	−24.4951	−0.819240
$895$	1.83315	0.0612755
$896$	−5.49981	−0.183736
$897$	13.5903	0.453768
$898$	−55.8856	−1.86493
$899$	31.0756	1.03643
$900$	−6.85778	−0.228593
$901$	−58.8919	−1.96197
$902$	−34.4332	−1.14650
$903$	2.21666	0.0737660
$904$	2.81181	0.0935194
$905$	−4.91286	−0.163309
$906$	35.1394	1.16743
$907$	4.24136	0.140832	0.0704160	−	0.997518i	$-0.477567\pi$
$907$	4.24136	0.140832	0.0704160	+	0.997518i	$0.477567\pi$
$908$	−16.4336	−0.545370
$909$	−0.528666	−0.0175347
$910$	1.48644	0.0492749
$911$	17.7767	0.588970	0.294485	−	0.955656i	$-0.404852\pi$
$911$	17.7767	0.588970	0.294485	+	0.955656i	$0.404852\pi$
$912$	11.1464	0.369094
$913$	−4.43603	−0.146811
$914$	−16.2953	−0.538999
$915$	−2.28876	−0.0756642
$916$	10.5575	0.348829
$917$	−10.2457	−0.338344
$918$	9.70747	0.320394
$919$	−40.0724	−1.32187	−0.660933	−	0.750445i	$-0.729839\pi$
$919$	−40.0724	−1.32187	−0.660933	+	0.750445i	$0.729839\pi$
$920$	−1.04690	−0.0345153
$921$	−18.8730	−0.621885
$922$	−59.5228	−1.96028
$923$	37.0824	1.22058
$924$	1.67151	0.0549887
$925$	3.65288	0.120106
$926$	56.7878	1.86616
$927$	−4.56400	−0.149901
$928$	−40.3492	−1.32453
$929$	47.4743	1.55758	0.778790	−	0.627285i	$-0.215834\pi$
$929$	47.4743	1.55758	0.778790	+	0.627285i	$0.215834\pi$
$930$	−2.79179	−0.0915465
$931$	15.1464	0.496403
$932$	24.1743	0.791855
$933$	−19.1206	−0.625981
$934$	19.8688	0.650128
$935$	2.86481	0.0936892
$936$	4.72879	0.154565
$937$	−41.9253	−1.36964	−0.684820	−	0.728713i	$-0.740119\pi$
$937$	−41.9253	−1.36964	−0.684820	+	0.728713i	$0.740119\pi$
$938$	−18.1450	−0.592454
$939$	15.8421	0.516988
$940$	0.283191	0.00923668
$941$	−23.1455	−0.754523	−0.377261	−	0.926107i	$-0.623134\pi$
$941$	−23.1455	−0.754523	−0.377261	+	0.926107i	$0.623134\pi$
$942$	36.6393	1.19377
$943$	32.3007	1.05186
$944$	−10.4132	−0.338919
$945$	−0.189975	−0.00617989
$946$	−11.6927	−0.380161
$947$	20.5520	0.667849	0.333924	−	0.942600i	$-0.391627\pi$
$947$	20.5520	0.667849	0.333924	+	0.942600i	$0.391627\pi$
$948$	−1.35197	−0.0439099
$949$	−1.42593	−0.0462877
$950$	−20.8377	−0.676065
$951$	12.6076	0.408830
$952$	−3.79489	−0.122993
$953$	8.38876	0.271739	0.135869	−	0.990727i	$-0.456617\pi$
$953$	8.38876	0.271739	0.135869	+	0.990727i	$0.456617\pi$
$954$	−20.6002	−0.666955
$955$	4.14246	0.134047
$956$	18.9898	0.614174
$957$	−11.1544	−0.360570
$958$	−54.8618	−1.77250
$959$	−6.46276	−0.208693
$960$	0.779864	0.0251700
$961$	−4.39054	−0.141630
$962$	5.81665	0.187536
$963$	8.94060	0.288107
$964$	2.65728	0.0855851
$965$	−2.50181	−0.0805360
$966$	−3.81499	−0.122745
$967$	−19.7927	−0.636491	−0.318245	−	0.948008i	$-0.603094\pi$
$967$	−19.7927	−0.636491	−0.318245	+	0.948008i	$0.603094\pi$
$968$	−8.43235	−0.271026
$969$	12.1234	0.389459
$970$	4.24041	0.136152
$971$	−49.1134	−1.57612	−0.788062	−	0.615596i	$-0.788916\pi$
$971$	−49.1134	−1.57612	−0.788062	+	0.615596i	$0.788916\pi$
$972$	1.39563	0.0447650
$973$	3.85153	0.123474
$974$	13.1852	0.422482
$975$	−20.8642	−0.668188
$976$	−37.7445	−1.20817
$977$	−10.1368	−0.324304	−0.162152	−	0.986766i	$-0.551843\pi$
$977$	−10.1368	−0.324304	−0.162152	+	0.986766i	$0.551843\pi$
$978$	−23.9533	−0.765942
$979$	19.5171	0.623770
$980$	2.69779	0.0861777
$981$	−4.26845	−0.136281
$982$	35.2890	1.12612
$983$	−31.3583	−1.00017	−0.500087	−	0.865975i	$-0.666699\pi$
$983$	−31.3583	−1.00017	−0.500087	+	0.865975i	$0.666699\pi$
$984$	11.2391	0.358290
$985$	−7.32444	−0.233376
$986$	−58.4800	−1.86238
$987$	−0.446886	−0.0142245
$988$	−13.6376	−0.433870
$989$	10.9685	0.348779
$990$	1.00210	0.0318488
$991$	−7.44332	−0.236445	−0.118222	−	0.992987i	$-0.537720\pi$
$991$	−7.44332	−0.236445	−0.118222	+	0.992987i	$0.537720\pi$
$992$	−34.5503	−1.09697
$993$	−3.51329	−0.111491
$994$	−10.4095	−0.330171
$995$	2.69439	0.0854178
$996$	−3.34365	−0.105948
$997$	6.43538	0.203810	0.101905	−	0.994794i	$-0.467506\pi$
$997$	6.43538	0.203810	0.101905	+	0.994794i	$0.467506\pi$
$998$	−45.6872	−1.44620
$999$	−0.743401	−0.0235202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6009.2.a.c.1.17	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6009.2.a.c.1.17	✓	92	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:	\( N \)	\(=\)	\( 6009 = 3 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84272 q^{2} +1.00000 q^{3} +1.39563 q^{4} -0.293700 q^{5} -1.84272 q^{6} +0.646833 q^{7} +1.11368 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.84272 q^{2} +1.00000 q^{3} +1.39563 q^{4} -0.293700 q^{5} -1.84272 q^{6} +0.646833 q^{7} +1.11368 q^{8} +1.00000 q^{9} +0.541209 q^{10} +1.85159 q^{11} +1.39563 q^{12} +4.24609 q^{13} -1.19194 q^{14} -0.293700 q^{15} -4.84347 q^{16} -5.26800 q^{17} -1.84272 q^{18} -2.30132 q^{19} -0.409898 q^{20} +0.646833 q^{21} -3.41197 q^{22} +3.20067 q^{23} +1.11368 q^{24} -4.91374 q^{25} -7.82437 q^{26} +1.00000 q^{27} +0.902743 q^{28} -6.02422 q^{29} +0.541209 q^{30} -5.15844 q^{31} +6.69783 q^{32} +1.85159 q^{33} +9.70747 q^{34} -0.189975 q^{35} +1.39563 q^{36} -0.743401 q^{37} +4.24070 q^{38} +4.24609 q^{39} -0.327088 q^{40} +10.0919 q^{41} -1.19194 q^{42} +3.42695 q^{43} +2.58414 q^{44} -0.293700 q^{45} -5.89795 q^{46} -0.690882 q^{47} -4.84347 q^{48} -6.58161 q^{49} +9.05467 q^{50} -5.26800 q^{51} +5.92599 q^{52} +11.1792 q^{53} -1.84272 q^{54} -0.543813 q^{55} +0.720366 q^{56} -2.30132 q^{57} +11.1010 q^{58} +2.14994 q^{59} -0.409898 q^{60} +7.79285 q^{61} +9.50558 q^{62} +0.646833 q^{63} -2.65530 q^{64} -1.24708 q^{65} -3.41197 q^{66} +15.2231 q^{67} -7.35220 q^{68} +3.20067 q^{69} +0.350072 q^{70} +8.73331 q^{71} +1.11368 q^{72} -0.335823 q^{73} +1.36988 q^{74} -4.91374 q^{75} -3.21180 q^{76} +1.19767 q^{77} -7.82437 q^{78} -0.968713 q^{79} +1.42253 q^{80} +1.00000 q^{81} -18.5965 q^{82} -2.39580 q^{83} +0.902743 q^{84} +1.54721 q^{85} -6.31492 q^{86} -6.02422 q^{87} +2.06208 q^{88} +10.5407 q^{89} +0.541209 q^{90} +2.74651 q^{91} +4.46696 q^{92} -5.15844 q^{93} +1.27311 q^{94} +0.675899 q^{95} +6.69783 q^{96} +7.83508 q^{97} +12.1281 q^{98} +1.85159 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9 + 13 * q^10 + 40 * q^11 + 107 * q^12 + 6 * q^13 + 37 * q^14 + 34 * q^15 + 133 * q^16 + 77 * q^17 + 17 * q^18 + 34 * q^19 + 55 * q^20 + 22 * q^21 + 8 * q^22 + 83 * q^23 + 51 * q^24 + 110 * q^25 + 22 * q^26 + 92 * q^27 + 32 * q^28 + 97 * q^29 + 13 * q^30 + 44 * q^31 + 104 * q^32 + 40 * q^33 + 20 * q^34 + 80 * q^35 + 107 * q^36 + 12 * q^37 + 54 * q^38 + 6 * q^39 + 23 * q^40 + 67 * q^41 + 37 * q^42 + 30 * q^43 + 87 * q^44 + 34 * q^45 + 33 * q^46 + 69 * q^47 + 133 * q^48 + 112 * q^49 + 58 * q^50 + 77 * q^51 - 3 * q^52 + 113 * q^53 + 17 * q^54 + 42 * q^55 + 92 * q^56 + 34 * q^57 - 30 * q^58 + 72 * q^59 + 55 * q^60 + 19 * q^61 + 60 * q^62 + 22 * q^63 + 147 * q^64 + 74 * q^65 + 8 * q^66 + 26 * q^67 + 171 * q^68 + 83 * q^69 - 35 * q^70 + 134 * q^71 + 51 * q^72 - 17 * q^73 + 95 * q^74 + 110 * q^75 + 27 * q^76 + 108 * q^77 + 22 * q^78 + 159 * q^79 + 79 * q^80 + 92 * q^81 - 64 * q^82 + 73 * q^83 + 32 * q^84 - 4 * q^85 + 22 * q^86 + 97 * q^87 - 16 * q^88 + 50 * q^89 + 13 * q^90 + 17 * q^91 + 154 * q^92 + 44 * q^93 + 8 * q^94 + 155 * q^95 + 104 * q^96 - 20 * q^97 + 63 * q^98 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more