LMFDB - Embedded newform 6039.2.a.p.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.98740 q^{2} +1.94976 q^{4} -1.69625 q^{5} +1.47319 q^{7} -0.0998540 q^{8} +O(q^{10})$	\(q+1.98740 q^{2} +1.94976 q^{4} -1.69625 q^{5} +1.47319 q^{7} -0.0998540 q^{8} -3.37113 q^{10} -1.00000 q^{11} +1.08303 q^{13} +2.92781 q^{14} -4.09796 q^{16} +6.50371 q^{17} +1.99017 q^{19} -3.30728 q^{20} -1.98740 q^{22} +4.59431 q^{23} -2.12272 q^{25} +2.15241 q^{26} +2.87235 q^{28} -5.22348 q^{29} +5.02011 q^{31} -7.94458 q^{32} +12.9255 q^{34} -2.49890 q^{35} -0.631601 q^{37} +3.95527 q^{38} +0.169378 q^{40} +4.72010 q^{41} -0.586969 q^{43} -1.94976 q^{44} +9.13073 q^{46} +5.79589 q^{47} -4.82972 q^{49} -4.21870 q^{50} +2.11164 q^{52} -11.0820 q^{53} +1.69625 q^{55} -0.147104 q^{56} -10.3811 q^{58} +13.9032 q^{59} +1.00000 q^{61} +9.97696 q^{62} -7.59313 q^{64} -1.83709 q^{65} +0.913716 q^{67} +12.6807 q^{68} -4.96631 q^{70} +10.6391 q^{71} -4.55816 q^{73} -1.25524 q^{74} +3.88035 q^{76} -1.47319 q^{77} -5.39890 q^{79} +6.95119 q^{80} +9.38072 q^{82} +17.4446 q^{83} -11.0319 q^{85} -1.16654 q^{86} +0.0998540 q^{88} +8.43337 q^{89} +1.59550 q^{91} +8.95778 q^{92} +11.5187 q^{94} -3.37584 q^{95} +8.31582 q^{97} -9.59859 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

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 $

$2$	1.98740	1.40530	0.702652	−	0.711534i	$-0.251999\pi$
$2$	1.98740	1.40530	0.702652	+	0.711534i	$0.251999\pi$
$3$	0	0
$3$	0	0
$4$	1.94976	0.974878
$5$	−1.69625	−0.758588	−0.379294	−	0.925276i	$-0.623833\pi$
$5$	−1.69625	−0.758588	−0.379294	+	0.925276i	$0.623833\pi$
$6$	0	0
$7$	1.47319	0.556812	0.278406	−	0.960463i	$-0.410194\pi$
$7$	1.47319	0.556812	0.278406	+	0.960463i	$0.410194\pi$
$8$	−0.0998540	−0.0353037
$9$	0	0
$10$	−3.37113	−1.06605
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.08303	0.300378	0.150189	−	0.988657i	$-0.452012\pi$
$13$	1.08303	0.300378	0.150189	+	0.988657i	$0.452012\pi$
$14$	2.92781	0.782490
$15$	0	0
$16$	−4.09796	−1.02449
$17$	6.50371	1.57738	0.788691	−	0.614790i	$-0.210759\pi$
$17$	6.50371	1.57738	0.788691	+	0.614790i	$0.210759\pi$
$18$	0	0
$19$	1.99017	0.456577	0.228288	−	0.973594i	$-0.426687\pi$
$19$	1.99017	0.456577	0.228288	+	0.973594i	$0.426687\pi$
$20$	−3.30728	−0.739531
$21$	0	0
$22$	−1.98740	−0.423715
$23$	4.59431	0.957980	0.478990	−	0.877820i	$-0.341003\pi$
$23$	4.59431	0.957980	0.478990	+	0.877820i	$0.341003\pi$
$24$	0	0
$25$	−2.12272	−0.424544
$26$	2.15241	0.422122
$27$	0	0
$28$	2.87235	0.542824
$29$	−5.22348	−0.969976	−0.484988	−	0.874521i	$-0.661176\pi$
$29$	−5.22348	−0.969976	−0.484988	+	0.874521i	$0.661176\pi$
$30$	0	0
$31$	5.02011	0.901638	0.450819	−	0.892615i	$-0.351132\pi$
$31$	5.02011	0.901638	0.450819	+	0.892615i	$0.351132\pi$
$32$	−7.94458	−1.40442
$33$	0	0
$34$	12.9255	2.21670
$35$	−2.49890	−0.422391
$36$	0	0
$37$	−0.631601	−0.103835	−0.0519173	−	0.998651i	$-0.516533\pi$
$37$	−0.631601	−0.103835	−0.0519173	+	0.998651i	$0.516533\pi$
$38$	3.95527	0.641629
$39$	0	0
$40$	0.169378	0.0267810
$41$	4.72010	0.737156	0.368578	−	0.929597i	$-0.379845\pi$
$41$	4.72010	0.737156	0.368578	+	0.929597i	$0.379845\pi$
$42$	0	0
$43$	−0.586969	−0.0895119	−0.0447559	−	0.998998i	$-0.514251\pi$
$43$	−0.586969	−0.0895119	−0.0447559	+	0.998998i	$0.514251\pi$
$44$	−1.94976	−0.293937
$45$	0	0
$46$	9.13073	1.34625
$47$	5.79589	0.845417	0.422709	−	0.906266i	$-0.361079\pi$
$47$	5.79589	0.845417	0.422709	+	0.906266i	$0.361079\pi$
$48$	0	0
$49$	−4.82972	−0.689960
$50$	−4.21870	−0.596614
$51$	0	0
$52$	2.11164	0.292832
$53$	−11.0820	−1.52223	−0.761115	−	0.648617i	$-0.775348\pi$
$53$	−11.0820	−1.52223	−0.761115	+	0.648617i	$0.775348\pi$
$54$	0	0
$55$	1.69625	0.228723
$56$	−0.147104	−0.0196575
$57$	0	0
$58$	−10.3811	−1.36311
$59$	13.9032	1.81004	0.905019	−	0.425371i	$-0.139856\pi$
$59$	13.9032	1.81004	0.905019	+	0.425371i	$0.139856\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	9.97696	1.26707
$63$	0	0
$64$	−7.59313	−0.949141
$65$	−1.83709	−0.227863
$66$	0	0
$67$	0.913716	0.111628	0.0558141	−	0.998441i	$-0.482225\pi$
$67$	0.913716	0.111628	0.0558141	+	0.998441i	$0.482225\pi$
$68$	12.6807	1.53776
$69$	0	0
$70$	−4.96631	−0.593588
$71$	10.6391	1.26263	0.631315	−	0.775526i	$-0.282515\pi$
$71$	10.6391	1.26263	0.631315	+	0.775526i	$0.282515\pi$
$72$	0	0
$73$	−4.55816	−0.533492	−0.266746	−	0.963767i	$-0.585949\pi$
$73$	−4.55816	−0.533492	−0.266746	+	0.963767i	$0.585949\pi$
$74$	−1.25524	−0.145919
$75$	0	0
$76$	3.88035	0.445107
$77$	−1.47319	−0.167885
$78$	0	0
$79$	−5.39890	−0.607424	−0.303712	−	0.952764i	$-0.598226\pi$
$79$	−5.39890	−0.607424	−0.303712	+	0.952764i	$0.598226\pi$
$80$	6.95119	0.777166
$81$	0	0
$82$	9.38072	1.03593
$83$	17.4446	1.91479	0.957395	−	0.288780i	$-0.0932496\pi$
$83$	17.4446	1.91479	0.957395	+	0.288780i	$0.0932496\pi$
$84$	0	0
$85$	−11.0319	−1.19658
$86$	−1.16654	−0.125791
$87$	0	0
$88$	0.0998540	0.0106445
$89$	8.43337	0.893936	0.446968	−	0.894550i	$-0.352504\pi$
$89$	8.43337	0.893936	0.446968	+	0.894550i	$0.352504\pi$
$90$	0	0
$91$	1.59550	0.167254
$92$	8.95778	0.933913
$93$	0	0
$94$	11.5187	1.18807
$95$	−3.37584	−0.346354
$96$	0	0
$97$	8.31582	0.844343	0.422172	−	0.906516i	$-0.361268\pi$
$97$	8.31582	0.844343	0.422172	+	0.906516i	$0.361268\pi$
$98$	−9.59859	−0.969604
$99$	0	0
$100$	−4.13879	−0.413879
$101$	10.8424	1.07886	0.539429	−	0.842031i	$-0.318640\pi$
$101$	10.8424	1.07886	0.539429	+	0.842031i	$0.318640\pi$
$102$	0	0
$103$	17.2652	1.70119	0.850596	−	0.525819i	$-0.176241\pi$
$103$	17.2652	1.70119	0.850596	+	0.525819i	$0.176241\pi$
$104$	−0.108145	−0.0106044
$105$	0	0
$106$	−22.0244	−2.13920
$107$	6.97016	0.673831	0.336916	−	0.941535i	$-0.390616\pi$
$107$	6.97016	0.673831	0.336916	+	0.941535i	$0.390616\pi$
$108$	0	0
$109$	−9.70623	−0.929688	−0.464844	−	0.885393i	$-0.653890\pi$
$109$	−9.70623	−0.929688	−0.464844	+	0.885393i	$0.653890\pi$
$110$	3.37113	0.321425
$111$	0	0
$112$	−6.03706	−0.570449
$113$	12.9233	1.21572	0.607860	−	0.794044i	$-0.292028\pi$
$113$	12.9233	1.21572	0.607860	+	0.794044i	$0.292028\pi$
$114$	0	0
$115$	−7.79312	−0.726712
$116$	−10.1845	−0.945609
$117$	0	0
$118$	27.6312	2.54365
$119$	9.58118	0.878305
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.98740	0.179931
$123$	0	0
$124$	9.78798	0.878987
$125$	12.0819	1.08064
$126$	0	0
$127$	11.3829	1.01007	0.505033	−	0.863100i	$-0.331480\pi$
$127$	11.3829	1.01007	0.505033	+	0.863100i	$0.331480\pi$
$128$	0.798580	0.0705852
$129$	0	0
$130$	−3.65103	−0.320216
$131$	0.364456	0.0318426	0.0159213	−	0.999873i	$-0.494932\pi$
$131$	0.364456	0.0318426	0.0159213	+	0.999873i	$0.494932\pi$
$132$	0	0
$133$	2.93189	0.254227
$134$	1.81592	0.156872
$135$	0	0
$136$	−0.649422	−0.0556874
$137$	−12.2799	−1.04914	−0.524570	−	0.851367i	$-0.675774\pi$
$137$	−12.2799	−1.04914	−0.524570	+	0.851367i	$0.675774\pi$
$138$	0	0
$139$	−1.88780	−0.160121	−0.0800605	−	0.996790i	$-0.525511\pi$
$139$	−1.88780	−0.160121	−0.0800605	+	0.996790i	$0.525511\pi$
$140$	−4.87224	−0.411780
$141$	0	0
$142$	21.1442	1.77438
$143$	−1.08303	−0.0905672
$144$	0	0
$145$	8.86036	0.735813
$146$	−9.05889	−0.749719
$147$	0	0
$148$	−1.23147	−0.101226
$149$	−5.45639	−0.447005	−0.223503	−	0.974703i	$-0.571749\pi$
$149$	−5.45639	−0.447005	−0.223503	+	0.974703i	$0.571749\pi$
$150$	0	0
$151$	−1.61465	−0.131398	−0.0656992	−	0.997839i	$-0.520928\pi$
$151$	−1.61465	−0.131398	−0.0656992	+	0.997839i	$0.520928\pi$
$152$	−0.198727	−0.0161189
$153$	0	0
$154$	−2.92781	−0.235930
$155$	−8.51538	−0.683971
$156$	0	0
$157$	−6.70492	−0.535111	−0.267556	−	0.963542i	$-0.586216\pi$
$157$	−6.70492	−0.535111	−0.267556	+	0.963542i	$0.586216\pi$
$158$	−10.7298	−0.853615
$159$	0	0
$160$	13.4760	1.06537
$161$	6.76827	0.533415
$162$	0	0
$163$	19.9210	1.56033	0.780166	−	0.625573i	$-0.215135\pi$
$163$	19.9210	1.56033	0.780166	+	0.625573i	$0.215135\pi$
$164$	9.20304	0.718637
$165$	0	0
$166$	34.6693	2.69086
$167$	−1.84036	−0.142411	−0.0712055	−	0.997462i	$-0.522685\pi$
$167$	−1.84036	−0.142411	−0.0712055	+	0.997462i	$0.522685\pi$
$168$	0	0
$169$	−11.8271	−0.909773
$170$	−21.9249	−1.68156
$171$	0	0
$172$	−1.14445	−0.0872632
$173$	13.1406	0.999065	0.499532	−	0.866295i	$-0.333505\pi$
$173$	13.1406	0.999065	0.499532	+	0.866295i	$0.333505\pi$
$174$	0	0
$175$	−3.12716	−0.236391
$176$	4.09796	0.308896
$177$	0	0
$178$	16.7605	1.25625
$179$	−14.9918	−1.12054	−0.560268	−	0.828311i	$-0.689302\pi$
$179$	−14.9918	−1.12054	−0.560268	+	0.828311i	$0.689302\pi$
$180$	0	0
$181$	4.46504	0.331884	0.165942	−	0.986136i	$-0.446934\pi$
$181$	4.46504	0.331884	0.165942	+	0.986136i	$0.446934\pi$
$182$	3.17090	0.235042
$183$	0	0
$184$	−0.458760	−0.0338202
$185$	1.07136	0.0787677
$186$	0	0
$187$	−6.50371	−0.475598
$188$	11.3006	0.824179
$189$	0	0
$190$	−6.70914	−0.486732
$191$	6.85168	0.495770	0.247885	−	0.968789i	$-0.420264\pi$
$191$	6.85168	0.495770	0.247885	+	0.968789i	$0.420264\pi$
$192$	0	0
$193$	−3.98331	−0.286725	−0.143362	−	0.989670i	$-0.545791\pi$
$193$	−3.98331	−0.286725	−0.143362	+	0.989670i	$0.545791\pi$
$194$	16.5268	1.18656
$195$	0	0
$196$	−9.41678	−0.672627
$197$	7.04258	0.501763	0.250882	−	0.968018i	$-0.419280\pi$
$197$	7.04258	0.501763	0.250882	+	0.968018i	$0.419280\pi$
$198$	0	0
$199$	2.78368	0.197330	0.0986651	−	0.995121i	$-0.468543\pi$
$199$	2.78368	0.197330	0.0986651	+	0.995121i	$0.468543\pi$
$200$	0.211962	0.0149880
$201$	0	0
$202$	21.5481	1.51612
$203$	−7.69516	−0.540095
$204$	0	0
$205$	−8.00649	−0.559197
$206$	34.3129	2.39069
$207$	0	0
$208$	−4.43820	−0.307734
$209$	−1.99017	−0.137663
$210$	0	0
$211$	−1.15155	−0.0792758	−0.0396379	−	0.999214i	$-0.512620\pi$
$211$	−1.15155	−0.0792758	−0.0396379	+	0.999214i	$0.512620\pi$
$212$	−21.6072	−1.48399
$213$	0	0
$214$	13.8525	0.946937
$215$	0.995648	0.0679026
$216$	0	0
$217$	7.39555	0.502043
$218$	−19.2902	−1.30649
$219$	0	0
$220$	3.30728	0.222977
$221$	7.04369	0.473810
$222$	0	0
$223$	−12.4758	−0.835440	−0.417720	−	0.908576i	$-0.637171\pi$
$223$	−12.4758	−0.835440	−0.417720	+	0.908576i	$0.637171\pi$
$224$	−11.7038	−0.781996
$225$	0	0
$226$	25.6837	1.70846
$227$	−11.9482	−0.793029	−0.396515	−	0.918028i	$-0.629780\pi$
$227$	−11.9482	−0.793029	−0.396515	+	0.918028i	$0.629780\pi$
$228$	0	0
$229$	1.23617	0.0816887	0.0408444	−	0.999166i	$-0.486995\pi$
$229$	1.23617	0.0816887	0.0408444	+	0.999166i	$0.486995\pi$
$230$	−15.4880	−1.02125
$231$	0	0
$232$	0.521586	0.0342438
$233$	7.40871	0.485361	0.242680	−	0.970106i	$-0.421973\pi$
$233$	7.40871	0.485361	0.242680	+	0.970106i	$0.421973\pi$
$234$	0	0
$235$	−9.83130	−0.641323
$236$	27.1078	1.76457
$237$	0	0
$238$	19.0416	1.23429
$239$	14.1248	0.913656	0.456828	−	0.889555i	$-0.348986\pi$
$239$	14.1248	0.913656	0.456828	+	0.889555i	$0.348986\pi$
$240$	0	0
$241$	11.1006	0.715053	0.357526	−	0.933903i	$-0.383620\pi$
$241$	11.1006	0.715053	0.357526	+	0.933903i	$0.383620\pi$
$242$	1.98740	0.127755
$243$	0	0
$244$	1.94976	0.124820
$245$	8.19244	0.523396
$246$	0	0
$247$	2.15541	0.137145
$248$	−0.501278	−0.0318312
$249$	0	0
$250$	24.0117	1.51863
$251$	−10.9503	−0.691174	−0.345587	−	0.938387i	$-0.612320\pi$
$251$	−10.9503	−0.691174	−0.345587	+	0.938387i	$0.612320\pi$
$252$	0	0
$253$	−4.59431	−0.288842
$254$	22.6223	1.41945
$255$	0	0
$256$	16.7734	1.04833
$257$	−31.7877	−1.98286	−0.991432	−	0.130623i	$-0.958302\pi$
$257$	−31.7877	−1.98286	−0.991432	+	0.130623i	$0.958302\pi$
$258$	0	0
$259$	−0.930467	−0.0578164
$260$	−3.58188	−0.222138
$261$	0	0
$262$	0.724319	0.0447486
$263$	27.8343	1.71633	0.858167	−	0.513371i	$-0.171603\pi$
$263$	27.8343	1.71633	0.858167	+	0.513371i	$0.171603\pi$
$264$	0	0
$265$	18.7979	1.15475
$266$	5.82684	0.357267
$267$	0	0
$268$	1.78152	0.108824
$269$	20.4023	1.24395	0.621974	−	0.783038i	$-0.286331\pi$
$269$	20.4023	1.24395	0.621974	+	0.783038i	$0.286331\pi$
$270$	0	0
$271$	−27.2291	−1.65405	−0.827026	−	0.562164i	$-0.809969\pi$
$271$	−27.2291	−1.65405	−0.827026	+	0.562164i	$0.809969\pi$
$272$	−26.6520	−1.61601
$273$	0	0
$274$	−24.4050	−1.47436
$275$	2.12272	0.128005
$276$	0	0
$277$	−26.0515	−1.56528	−0.782640	−	0.622474i	$-0.786128\pi$
$277$	−26.0515	−1.56528	−0.782640	+	0.622474i	$0.786128\pi$
$278$	−3.75181	−0.225019
$279$	0	0
$280$	0.249525	0.0149120
$281$	−19.7880	−1.18046	−0.590228	−	0.807237i	$-0.700962\pi$
$281$	−19.7880	−1.18046	−0.590228	+	0.807237i	$0.700962\pi$
$282$	0	0
$283$	−5.53124	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$283$	−5.53124	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$284$	20.7437	1.23091
$285$	0	0
$286$	−2.15241	−0.127274
$287$	6.95359	0.410457
$288$	0	0
$289$	25.2983	1.48813
$290$	17.6091	1.03404
$291$	0	0
$292$	−8.88731	−0.520090
$293$	−32.5305	−1.90045	−0.950226	−	0.311561i	$-0.899148\pi$
$293$	−32.5305	−1.90045	−0.950226	+	0.311561i	$0.899148\pi$
$294$	0	0
$295$	−23.5833	−1.37307
$296$	0.0630679	0.00366575
$297$	0	0
$298$	−10.8440	−0.628178
$299$	4.97576	0.287756
$300$	0	0
$301$	−0.864714	−0.0498413
$302$	−3.20896	−0.184655
$303$	0	0
$304$	−8.15565	−0.467759
$305$	−1.69625	−0.0971272
$306$	0	0
$307$	−8.98148	−0.512600	−0.256300	−	0.966597i	$-0.582503\pi$
$307$	−8.98148	−0.512600	−0.256300	+	0.966597i	$0.582503\pi$
$308$	−2.87235	−0.163668
$309$	0	0
$310$	−16.9235	−0.961188
$311$	−21.0590	−1.19414	−0.597072	−	0.802188i	$-0.703669\pi$
$311$	−21.0590	−1.19414	−0.597072	+	0.802188i	$0.703669\pi$
$312$	0	0
$313$	18.0791	1.02189	0.510946	−	0.859613i	$-0.329295\pi$
$313$	18.0791	1.02189	0.510946	+	0.859613i	$0.329295\pi$
$314$	−13.3254	−0.751994
$315$	0	0
$316$	−10.5265	−0.592164
$317$	−29.6991	−1.66807	−0.834034	−	0.551713i	$-0.813974\pi$
$317$	−29.6991	−1.66807	−0.834034	+	0.551713i	$0.813974\pi$
$318$	0	0
$319$	5.22348	0.292459
$320$	12.8799	0.720007
$321$	0	0
$322$	13.4513	0.749610
$323$	12.9435	0.720196
$324$	0	0
$325$	−2.29896	−0.127524
$326$	39.5909	2.19274
$327$	0	0
$328$	−0.471321	−0.0260243
$329$	8.53842	0.470739
$330$	0	0
$331$	−7.68863	−0.422605	−0.211303	−	0.977421i	$-0.567771\pi$
$331$	−7.68863	−0.422605	−0.211303	+	0.977421i	$0.567771\pi$
$332$	34.0127	1.86669
$333$	0	0
$334$	−3.65752	−0.200131
$335$	−1.54990	−0.0846798
$336$	0	0
$337$	21.8639	1.19100	0.595500	−	0.803355i	$-0.296954\pi$
$337$	21.8639	1.19100	0.595500	+	0.803355i	$0.296954\pi$
$338$	−23.5051	−1.27851
$339$	0	0
$340$	−21.5096	−1.16652
$341$	−5.02011	−0.271854
$342$	0	0
$343$	−17.4274	−0.940990
$344$	0.0586112	0.00316010
$345$	0	0
$346$	26.1157	1.40399
$347$	13.3499	0.716659	0.358329	−	0.933595i	$-0.383346\pi$
$347$	13.3499	0.716659	0.358329	+	0.933595i	$0.383346\pi$
$348$	0	0
$349$	13.9200	0.745119	0.372559	−	0.928008i	$-0.378480\pi$
$349$	13.9200	0.745119	0.372559	+	0.928008i	$0.378480\pi$
$350$	−6.21492	−0.332202
$351$	0	0
$352$	7.94458	0.423448
$353$	16.7802	0.893118	0.446559	−	0.894754i	$-0.352649\pi$
$353$	16.7802	0.893118	0.446559	+	0.894754i	$0.352649\pi$
$354$	0	0
$355$	−18.0466	−0.957816
$356$	16.4430	0.871479
$357$	0	0
$358$	−29.7946	−1.57469
$359$	−5.31077	−0.280291	−0.140146	−	0.990131i	$-0.544757\pi$
$359$	−5.31077	−0.280291	−0.140146	+	0.990131i	$0.544757\pi$
$360$	0	0
$361$	−15.0392	−0.791538
$362$	8.87381	0.466397
$363$	0	0
$364$	3.11084	0.163052
$365$	7.73180	0.404701
$366$	0	0
$367$	−27.0741	−1.41325	−0.706627	−	0.707586i	$-0.749784\pi$
$367$	−27.0741	−1.41325	−0.706627	+	0.707586i	$0.749784\pi$
$368$	−18.8273	−0.981441
$369$	0	0
$370$	2.12921	0.110693
$371$	−16.3259	−0.847596
$372$	0	0
$373$	−20.5095	−1.06194	−0.530971	−	0.847390i	$-0.678173\pi$
$373$	−20.5095	−1.06194	−0.530971	+	0.847390i	$0.678173\pi$
$374$	−12.9255	−0.668360
$375$	0	0
$376$	−0.578743	−0.0298464
$377$	−5.65717	−0.291359
$378$	0	0
$379$	−9.93911	−0.510538	−0.255269	−	0.966870i	$-0.582164\pi$
$379$	−9.93911	−0.510538	−0.255269	+	0.966870i	$0.582164\pi$
$380$	−6.58206	−0.337653
$381$	0	0
$382$	13.6170	0.696707
$383$	−23.3964	−1.19550	−0.597751	−	0.801682i	$-0.703939\pi$
$383$	−23.3964	−1.19550	−0.597751	+	0.801682i	$0.703939\pi$
$384$	0	0
$385$	2.49890	0.127356
$386$	−7.91642	−0.402935
$387$	0	0
$388$	16.2138	0.823132
$389$	9.75823	0.494762	0.247381	−	0.968918i	$-0.420430\pi$
$389$	9.75823	0.494762	0.247381	+	0.968918i	$0.420430\pi$
$390$	0	0
$391$	29.8801	1.51110
$392$	0.482267	0.0243582
$393$	0	0
$394$	13.9964	0.705129
$395$	9.15791	0.460784
$396$	0	0
$397$	−30.0300	−1.50716	−0.753582	−	0.657354i	$-0.771676\pi$
$397$	−30.0300	−1.50716	−0.753582	+	0.657354i	$0.771676\pi$
$398$	5.53229	0.277309
$399$	0	0
$400$	8.69883	0.434942
$401$	3.91307	0.195409	0.0977046	−	0.995215i	$-0.468850\pi$
$401$	3.91307	0.195409	0.0977046	+	0.995215i	$0.468850\pi$
$402$	0	0
$403$	5.43691	0.270832
$404$	21.1400	1.05175
$405$	0	0
$406$	−15.2934	−0.758997
$407$	0.631601	0.0313073
$408$	0	0
$409$	13.9082	0.687717	0.343858	−	0.939022i	$-0.388266\pi$
$409$	13.9082	0.687717	0.343858	+	0.939022i	$0.388266\pi$
$410$	−15.9121	−0.785842
$411$	0	0
$412$	33.6630	1.65846
$413$	20.4820	1.00785
$414$	0	0
$415$	−29.5904	−1.45254
$416$	−8.60419	−0.421855
$417$	0	0
$418$	−3.95527	−0.193458
$419$	−24.4247	−1.19322	−0.596612	−	0.802530i	$-0.703487\pi$
$419$	−24.4247	−1.19322	−0.596612	+	0.802530i	$0.703487\pi$
$420$	0	0
$421$	11.4233	0.556735	0.278368	−	0.960475i	$-0.410207\pi$
$421$	11.4233	0.556735	0.278368	+	0.960475i	$0.410207\pi$
$422$	−2.28858	−0.111407
$423$	0	0
$424$	1.10658	0.0537404
$425$	−13.8056	−0.669668
$426$	0	0
$427$	1.47319	0.0712925
$428$	13.5901	0.656903
$429$	0	0
$430$	1.97875	0.0954238
$431$	−36.0825	−1.73803	−0.869016	−	0.494785i	$-0.835247\pi$
$431$	−36.0825	−1.73803	−0.869016	+	0.494785i	$0.835247\pi$
$432$	0	0
$433$	9.63856	0.463200	0.231600	−	0.972811i	$-0.425604\pi$
$433$	9.63856	0.463200	0.231600	+	0.972811i	$0.425604\pi$
$434$	14.6979	0.705523
$435$	0	0
$436$	−18.9248	−0.906333
$437$	9.14346	0.437391
$438$	0	0
$439$	−4.32711	−0.206522	−0.103261	−	0.994654i	$-0.532928\pi$
$439$	−4.32711	−0.206522	−0.103261	+	0.994654i	$0.532928\pi$
$440$	−0.169378	−0.00807477
$441$	0	0
$442$	13.9986	0.665847
$443$	17.6111	0.836727	0.418364	−	0.908280i	$-0.362604\pi$
$443$	17.6111	0.836727	0.418364	+	0.908280i	$0.362604\pi$
$444$	0	0
$445$	−14.3051	−0.678129
$446$	−24.7943	−1.17405
$447$	0	0
$448$	−11.1861	−0.528493
$449$	−13.5885	−0.641283	−0.320641	−	0.947201i	$-0.603898\pi$
$449$	−13.5885	−0.641283	−0.320641	+	0.947201i	$0.603898\pi$
$450$	0	0
$451$	−4.72010	−0.222261
$452$	25.1973	1.18518
$453$	0	0
$454$	−23.7458	−1.11445
$455$	−2.70637	−0.126877
$456$	0	0
$457$	17.9343	0.838932	0.419466	−	0.907771i	$-0.362217\pi$
$457$	17.9343	0.838932	0.419466	+	0.907771i	$0.362217\pi$
$458$	2.45677	0.114797
$459$	0	0
$460$	−15.1947	−0.708455
$461$	−1.41379	−0.0658469	−0.0329235	−	0.999458i	$-0.510482\pi$
$461$	−1.41379	−0.0658469	−0.0329235	+	0.999458i	$0.510482\pi$
$462$	0	0
$463$	17.1184	0.795561	0.397780	−	0.917481i	$-0.369781\pi$
$463$	17.1184	0.795561	0.397780	+	0.917481i	$0.369781\pi$
$464$	21.4056	0.993732
$465$	0	0
$466$	14.7241	0.682079
$467$	−35.8427	−1.65860	−0.829302	−	0.558801i	$-0.811262\pi$
$467$	−35.8427	−1.65860	−0.829302	+	0.558801i	$0.811262\pi$
$468$	0	0
$469$	1.34607	0.0621560
$470$	−19.5387	−0.901254
$471$	0	0
$472$	−1.38829	−0.0639011
$473$	0.586969	0.0269888
$474$	0	0
$475$	−4.22458	−0.193837
$476$	18.6810	0.856241
$477$	0	0
$478$	28.0716	1.28396
$479$	29.9152	1.36686	0.683430	−	0.730016i	$-0.260488\pi$
$479$	29.9152	1.36686	0.683430	+	0.730016i	$0.260488\pi$
$480$	0	0
$481$	−0.684041	−0.0311896
$482$	22.0613	1.00487
$483$	0	0
$484$	1.94976	0.0886253
$485$	−14.1057	−0.640509
$486$	0	0
$487$	19.4977	0.883525	0.441763	−	0.897132i	$-0.354353\pi$
$487$	19.4977	0.883525	0.441763	+	0.897132i	$0.354353\pi$
$488$	−0.0998540	−0.00452018
$489$	0	0
$490$	16.2816	0.735530
$491$	23.3492	1.05374	0.526868	−	0.849947i	$-0.323366\pi$
$491$	23.3492	1.05374	0.526868	+	0.849947i	$0.323366\pi$
$492$	0	0
$493$	−33.9720	−1.53002
$494$	4.28366	0.192731
$495$	0	0
$496$	−20.5722	−0.923719
$497$	15.6734	0.703048
$498$	0	0
$499$	−18.7542	−0.839551	−0.419776	−	0.907628i	$-0.637891\pi$
$499$	−18.7542	−0.839551	−0.419776	+	0.907628i	$0.637891\pi$
$500$	23.5569	1.05349
$501$	0	0
$502$	−21.7625	−0.971309
$503$	20.6970	0.922833	0.461417	−	0.887184i	$-0.347341\pi$
$503$	20.6970	0.922833	0.461417	+	0.887184i	$0.347341\pi$
$504$	0	0
$505$	−18.3914	−0.818408
$506$	−9.13073	−0.405910
$507$	0	0
$508$	22.1938	0.984691
$509$	22.8797	1.01413	0.507063	−	0.861909i	$-0.330731\pi$
$509$	22.8797	1.01413	0.507063	+	0.861909i	$0.330731\pi$
$510$	0	0
$511$	−6.71502	−0.297055
$512$	31.7382	1.40264
$513$	0	0
$514$	−63.1749	−2.78653
$515$	−29.2862	−1.29050
$516$	0	0
$517$	−5.79589	−0.254903
$518$	−1.84921	−0.0812496
$519$	0	0
$520$	0.183441	0.00804441
$521$	−13.6390	−0.597534	−0.298767	−	0.954326i	$-0.596575\pi$
$521$	−13.6390	−0.597534	−0.298767	+	0.954326i	$0.596575\pi$
$522$	0	0
$523$	23.8285	1.04195	0.520974	−	0.853572i	$-0.325569\pi$
$523$	23.8285	1.04195	0.520974	+	0.853572i	$0.325569\pi$
$524$	0.710600	0.0310427
$525$	0	0
$526$	55.3178	2.41197
$527$	32.6493	1.42223
$528$	0	0
$529$	−1.89233	−0.0822753
$530$	37.3589	1.62277
$531$	0	0
$532$	5.71648	0.247841
$533$	5.11199	0.221425
$534$	0	0
$535$	−11.8232	−0.511160
$536$	−0.0912382	−0.00394089
$537$	0	0
$538$	40.5474	1.74812
$539$	4.82972	0.208031
$540$	0	0
$541$	−32.6842	−1.40520	−0.702602	−	0.711583i	$-0.747979\pi$
$541$	−32.6842	−1.40520	−0.702602	+	0.711583i	$0.747979\pi$
$542$	−54.1151	−2.32444
$543$	0	0
$544$	−51.6693	−2.21530
$545$	16.4642	0.705250
$546$	0	0
$547$	−8.32193	−0.355820	−0.177910	−	0.984047i	$-0.556934\pi$
$547$	−8.32193	−0.355820	−0.177910	+	0.984047i	$0.556934\pi$
$548$	−23.9428	−1.02278
$549$	0	0
$550$	4.21870	0.179886
$551$	−10.3956	−0.442869
$552$	0	0
$553$	−7.95359	−0.338221
$554$	−51.7747	−2.19969
$555$	0	0
$556$	−3.68075	−0.156098
$557$	12.5811	0.533077	0.266539	−	0.963824i	$-0.414120\pi$
$557$	12.5811	0.533077	0.266539	+	0.963824i	$0.414120\pi$
$558$	0	0
$559$	−0.635703	−0.0268874
$560$	10.2404	0.432736
$561$	0	0
$562$	−39.3267	−1.65890
$563$	42.2063	1.77878	0.889391	−	0.457147i	$-0.151129\pi$
$563$	42.2063	1.77878	0.889391	+	0.457147i	$0.151129\pi$
$564$	0	0
$565$	−21.9212	−0.922231
$566$	−10.9928	−0.462061
$567$	0	0
$568$	−1.06236	−0.0445755
$569$	5.95645	0.249707	0.124854	−	0.992175i	$-0.460154\pi$
$569$	5.95645	0.249707	0.124854	+	0.992175i	$0.460154\pi$
$570$	0	0
$571$	−0.819100	−0.0342783	−0.0171391	−	0.999853i	$-0.505456\pi$
$571$	−0.819100	−0.0342783	−0.0171391	+	0.999853i	$0.505456\pi$
$572$	−2.11164	−0.0882920
$573$	0	0
$574$	13.8196	0.576817
$575$	−9.75244	−0.406705
$576$	0	0
$577$	−11.8580	−0.493655	−0.246827	−	0.969059i	$-0.579388\pi$
$577$	−11.8580	−0.493655	−0.246827	+	0.969059i	$0.579388\pi$
$578$	50.2777	2.09128
$579$	0	0
$580$	17.2755	0.717328
$581$	25.6991	1.06618
$582$	0	0
$583$	11.0820	0.458970
$584$	0.455151	0.0188343
$585$	0	0
$586$	−64.6511	−2.67071
$587$	−28.0996	−1.15979	−0.579897	−	0.814690i	$-0.696907\pi$
$587$	−28.0996	−1.15979	−0.579897	+	0.814690i	$0.696907\pi$
$588$	0	0
$589$	9.99087	0.411667
$590$	−46.8695	−1.92959
$591$	0	0
$592$	2.58828	0.106378
$593$	18.5874	0.763291	0.381646	−	0.924309i	$-0.375357\pi$
$593$	18.5874	0.763291	0.381646	+	0.924309i	$0.375357\pi$
$594$	0	0
$595$	−16.2521	−0.666272
$596$	−10.6386	−0.435776
$597$	0	0
$598$	9.88882	0.404384
$599$	32.2331	1.31701	0.658504	−	0.752577i	$-0.271190\pi$
$599$	32.2331	1.31701	0.658504	+	0.752577i	$0.271190\pi$
$600$	0	0
$601$	−20.3221	−0.828954	−0.414477	−	0.910060i	$-0.636035\pi$
$601$	−20.3221	−0.828954	−0.414477	+	0.910060i	$0.636035\pi$
$602$	−1.71853	−0.0700422
$603$	0	0
$604$	−3.14818	−0.128098
$605$	−1.69625	−0.0689625
$606$	0	0
$607$	3.03349	0.123126	0.0615628	−	0.998103i	$-0.480392\pi$
$607$	3.03349	0.123126	0.0615628	+	0.998103i	$0.480392\pi$
$608$	−15.8111	−0.641224
$609$	0	0
$610$	−3.37113	−0.136493
$611$	6.27710	0.253944
$612$	0	0
$613$	−47.3896	−1.91405	−0.957024	−	0.290010i	$-0.906341\pi$
$613$	−47.3896	−1.91405	−0.957024	+	0.290010i	$0.906341\pi$
$614$	−17.8498	−0.720359
$615$	0	0
$616$	0.147104	0.00592697
$617$	39.3278	1.58328	0.791638	−	0.610990i	$-0.209229\pi$
$617$	39.3278	1.58328	0.791638	+	0.610990i	$0.209229\pi$
$618$	0	0
$619$	5.30861	0.213371	0.106686	−	0.994293i	$-0.465976\pi$
$619$	5.30861	0.213371	0.106686	+	0.994293i	$0.465976\pi$
$620$	−16.6029	−0.666789
$621$	0	0
$622$	−41.8526	−1.67813
$623$	12.4239	0.497754
$624$	0	0
$625$	−9.88045	−0.395218
$626$	35.9304	1.43607
$627$	0	0
$628$	−13.0730	−0.521668
$629$	−4.10775	−0.163787
$630$	0	0
$631$	−30.3358	−1.20765	−0.603825	−	0.797117i	$-0.706357\pi$
$631$	−30.3358	−1.20765	−0.603825	+	0.797117i	$0.706357\pi$
$632$	0.539102	0.0214443
$633$	0	0
$634$	−59.0240	−2.34414
$635$	−19.3082	−0.766224
$636$	0	0
$637$	−5.23072	−0.207249
$638$	10.3811	0.410994
$639$	0	0
$640$	−1.35459	−0.0535451
$641$	11.6197	0.458949	0.229475	−	0.973315i	$-0.426299\pi$
$641$	11.6197	0.458949	0.229475	+	0.973315i	$0.426299\pi$
$642$	0	0
$643$	38.4190	1.51510	0.757549	−	0.652779i	$-0.226397\pi$
$643$	38.4190	1.51510	0.757549	+	0.652779i	$0.226397\pi$
$644$	13.1965	0.520014
$645$	0	0
$646$	25.7239	1.01209
$647$	−15.7690	−0.619942	−0.309971	−	0.950746i	$-0.600319\pi$
$647$	−15.7690	−0.619942	−0.309971	+	0.950746i	$0.600319\pi$
$648$	0	0
$649$	−13.9032	−0.545747
$650$	−4.56896	−0.179209
$651$	0	0
$652$	38.8410	1.52113
$653$	39.2465	1.53583	0.767917	−	0.640549i	$-0.221293\pi$
$653$	39.2465	1.53583	0.767917	+	0.640549i	$0.221293\pi$
$654$	0	0
$655$	−0.618210	−0.0241555
$656$	−19.3428	−0.755209
$657$	0	0
$658$	16.9693	0.661531
$659$	47.1026	1.83486	0.917428	−	0.397902i	$-0.130261\pi$
$659$	47.1026	1.83486	0.917428	+	0.397902i	$0.130261\pi$
$660$	0	0
$661$	30.4570	1.18464	0.592319	−	0.805703i	$-0.298212\pi$
$661$	30.4570	1.18464	0.592319	+	0.805703i	$0.298212\pi$
$662$	−15.2804	−0.593889
$663$	0	0
$664$	−1.74191	−0.0675992
$665$	−4.97324	−0.192854
$666$	0	0
$667$	−23.9983	−0.929218
$668$	−3.58825	−0.138833
$669$	0	0
$670$	−3.08026	−0.119001
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	45.2546	1.74444	0.872218	−	0.489117i	$-0.162681\pi$
$673$	45.2546	1.74444	0.872218	+	0.489117i	$0.162681\pi$
$674$	43.4522	1.67372
$675$	0	0
$676$	−23.0599	−0.886918
$677$	−18.6511	−0.716820	−0.358410	−	0.933564i	$-0.616681\pi$
$677$	−18.6511	−0.716820	−0.358410	+	0.933564i	$0.616681\pi$
$678$	0	0
$679$	12.2507	0.470141
$680$	1.10158	0.0422438
$681$	0	0
$682$	−9.97696	−0.382037
$683$	4.47064	0.171064	0.0855322	−	0.996335i	$-0.472741\pi$
$683$	4.47064	0.171064	0.0855322	+	0.996335i	$0.472741\pi$
$684$	0	0
$685$	20.8298	0.795865
$686$	−34.6352	−1.32238
$687$	0	0
$688$	2.40538	0.0917041
$689$	−12.0021	−0.457244
$690$	0	0
$691$	−5.26763	−0.200390	−0.100195	−	0.994968i	$-0.531947\pi$
$691$	−5.26763	−0.200390	−0.100195	+	0.994968i	$0.531947\pi$
$692$	25.6211	0.973967
$693$	0	0
$694$	26.5315	1.00712
$695$	3.20219	0.121466
$696$	0	0
$697$	30.6982	1.16278
$698$	27.6645	1.04712
$699$	0	0
$700$	−6.09721	−0.230453
$701$	−32.7360	−1.23642	−0.618210	−	0.786013i	$-0.712142\pi$
$701$	−32.7360	−1.23642	−0.618210	+	0.786013i	$0.712142\pi$
$702$	0	0
$703$	−1.25700	−0.0474085
$704$	7.59313	0.286177
$705$	0	0
$706$	33.3489	1.25510
$707$	15.9728	0.600721
$708$	0	0
$709$	−22.9084	−0.860342	−0.430171	−	0.902748i	$-0.641547\pi$
$709$	−22.9084	−0.860342	−0.430171	+	0.902748i	$0.641547\pi$
$710$	−35.8659	−1.34602
$711$	0	0
$712$	−0.842106	−0.0315593
$713$	23.0639	0.863750
$714$	0	0
$715$	1.83709	0.0687032
$716$	−29.2303	−1.09239
$717$	0	0
$718$	−10.5546	−0.393895
$719$	−30.3919	−1.13343	−0.566713	−	0.823915i	$-0.691785\pi$
$719$	−30.3919	−1.13343	−0.566713	+	0.823915i	$0.691785\pi$
$720$	0	0
$721$	25.4349	0.947245
$722$	−29.8889	−1.11235
$723$	0	0
$724$	8.70574	0.323546
$725$	11.0880	0.411798
$726$	0	0
$727$	17.6356	0.654067	0.327033	−	0.945013i	$-0.393951\pi$
$727$	17.6356	0.654067	0.327033	+	0.945013i	$0.393951\pi$
$728$	−0.159317	−0.00590468
$729$	0	0
$730$	15.3662	0.568728
$731$	−3.81747	−0.141194
$732$	0	0
$733$	33.9843	1.25524	0.627620	−	0.778520i	$-0.284029\pi$
$733$	33.9843	1.25524	0.627620	+	0.778520i	$0.284029\pi$
$734$	−53.8070	−1.98605
$735$	0	0
$736$	−36.4999	−1.34540
$737$	−0.913716	−0.0336572
$738$	0	0
$739$	−10.5128	−0.386720	−0.193360	−	0.981128i	$-0.561939\pi$
$739$	−10.5128	−0.386720	−0.193360	+	0.981128i	$0.561939\pi$
$740$	2.08888	0.0767889
$741$	0	0
$742$	−32.4460	−1.19113
$743$	−49.8571	−1.82908	−0.914540	−	0.404496i	$-0.867447\pi$
$743$	−49.8571	−1.82908	−0.914540	+	0.404496i	$0.867447\pi$
$744$	0	0
$745$	9.25543	0.339093
$746$	−40.7605	−1.49235
$747$	0	0
$748$	−12.6807	−0.463651
$749$	10.2684	0.375197
$750$	0	0
$751$	11.7670	0.429383	0.214692	−	0.976682i	$-0.431125\pi$
$751$	11.7670	0.429383	0.214692	+	0.976682i	$0.431125\pi$
$752$	−23.7513	−0.866122
$753$	0	0
$754$	−11.2431	−0.409448
$755$	2.73886	0.0996773
$756$	0	0
$757$	−52.5126	−1.90860	−0.954301	−	0.298848i	$-0.903398\pi$
$757$	−52.5126	−1.90860	−0.954301	+	0.298848i	$0.903398\pi$
$758$	−19.7530	−0.717461
$759$	0	0
$760$	0.337091	0.0122276
$761$	−21.8952	−0.793700	−0.396850	−	0.917883i	$-0.629897\pi$
$761$	−21.8952	−0.793700	−0.396850	+	0.917883i	$0.629897\pi$
$762$	0	0
$763$	−14.2991	−0.517662
$764$	13.3591	0.483315
$765$	0	0
$766$	−46.4980	−1.68004
$767$	15.0575	0.543695
$768$	0	0
$769$	−18.7637	−0.676635	−0.338317	−	0.941032i	$-0.609858\pi$
$769$	−18.7637	−0.676635	−0.338317	+	0.941032i	$0.609858\pi$
$770$	4.96631	0.178973
$771$	0	0
$772$	−7.76648	−0.279522
$773$	22.6135	0.813353	0.406676	−	0.913572i	$-0.366688\pi$
$773$	22.6135	0.813353	0.406676	+	0.913572i	$0.366688\pi$
$774$	0	0
$775$	−10.6563	−0.382785
$776$	−0.830368	−0.0298085
$777$	0	0
$778$	19.3935	0.695291
$779$	9.39381	0.336568
$780$	0	0
$781$	−10.6391	−0.380697
$782$	59.3836	2.12355
$783$	0	0
$784$	19.7920	0.706858
$785$	11.3733	0.405929
$786$	0	0
$787$	7.94959	0.283372	0.141686	−	0.989912i	$-0.454748\pi$
$787$	7.94959	0.283372	0.141686	+	0.989912i	$0.454748\pi$
$788$	13.7313	0.489158
$789$	0	0
$790$	18.2004	0.647542
$791$	19.0384	0.676928
$792$	0	0
$793$	1.08303	0.0384594
$794$	−59.6817	−2.11802
$795$	0	0
$796$	5.42751	0.192373
$797$	−47.4900	−1.68218	−0.841091	−	0.540893i	$-0.818086\pi$
$797$	−47.4900	−1.68218	−0.841091	+	0.540893i	$0.818086\pi$
$798$	0	0
$799$	37.6948	1.33355
$800$	16.8641	0.596237
$801$	0	0
$802$	7.77683	0.274609
$803$	4.55816	0.160854
$804$	0	0
$805$	−11.4807	−0.404642
$806$	10.8053	0.380601
$807$	0	0
$808$	−1.08266	−0.0380877
$809$	35.5115	1.24852	0.624259	−	0.781217i	$-0.285401\pi$
$809$	35.5115	1.24852	0.624259	+	0.781217i	$0.285401\pi$
$810$	0	0
$811$	−30.8176	−1.08215	−0.541077	−	0.840973i	$-0.681983\pi$
$811$	−30.8176	−1.08215	−0.541077	+	0.840973i	$0.681983\pi$
$812$	−15.0037	−0.526527
$813$	0	0
$814$	1.25524	0.0439963
$815$	−33.7910	−1.18365
$816$	0	0
$817$	−1.16817	−0.0408690
$818$	27.6412	0.966451
$819$	0	0
$820$	−15.6107	−0.545149
$821$	−14.9903	−0.523166	−0.261583	−	0.965181i	$-0.584245\pi$
$821$	−14.9903	−0.523166	−0.261583	+	0.965181i	$0.584245\pi$
$822$	0	0
$823$	23.9032	0.833212	0.416606	−	0.909087i	$-0.363219\pi$
$823$	23.9032	0.833212	0.416606	+	0.909087i	$0.363219\pi$
$824$	−1.72400	−0.0600584
$825$	0	0
$826$	40.7058	1.41634
$827$	−18.2190	−0.633536	−0.316768	−	0.948503i	$-0.602598\pi$
$827$	−18.2190	−0.633536	−0.316768	+	0.948503i	$0.602598\pi$
$828$	0	0
$829$	−45.8048	−1.59087	−0.795433	−	0.606041i	$-0.792757\pi$
$829$	−45.8048	−1.59087	−0.795433	+	0.606041i	$0.792757\pi$
$830$	−58.8080	−2.04126
$831$	0	0
$832$	−8.22356	−0.285101
$833$	−31.4111	−1.08833
$834$	0	0
$835$	3.12171	0.108031
$836$	−3.88035	−0.134205
$837$	0	0
$838$	−48.5416	−1.67684
$839$	3.50925	0.121153	0.0605764	−	0.998164i	$-0.480706\pi$
$839$	3.50925	0.121153	0.0605764	+	0.998164i	$0.480706\pi$
$840$	0	0
$841$	−1.71522	−0.0591456
$842$	22.7026	0.782382
$843$	0	0
$844$	−2.24524	−0.0772842
$845$	20.0617	0.690143
$846$	0	0
$847$	1.47319	0.0506193
$848$	45.4136	1.55951
$849$	0	0
$850$	−27.4372	−0.941087
$851$	−2.90177	−0.0994715
$852$	0	0
$853$	−24.5602	−0.840925	−0.420462	−	0.907310i	$-0.638132\pi$
$853$	−24.5602	−0.840925	−0.420462	+	0.907310i	$0.638132\pi$
$854$	2.92781	0.100188
$855$	0	0
$856$	−0.695999	−0.0237888
$857$	28.5171	0.974127	0.487063	−	0.873367i	$-0.338068\pi$
$857$	28.5171	0.974127	0.487063	+	0.873367i	$0.338068\pi$
$858$	0	0
$859$	−21.1415	−0.721340	−0.360670	−	0.932693i	$-0.617452\pi$
$859$	−21.1415	−0.721340	−0.360670	+	0.932693i	$0.617452\pi$
$860$	1.94127	0.0661968
$861$	0	0
$862$	−71.7103	−2.44246
$863$	11.5621	0.393579	0.196789	−	0.980446i	$-0.436948\pi$
$863$	11.5621	0.393579	0.196789	+	0.980446i	$0.436948\pi$
$864$	0	0
$865$	−22.2899	−0.757879
$866$	19.1557	0.650936
$867$	0	0
$868$	14.4195	0.489431
$869$	5.39890	0.183145
$870$	0	0
$871$	0.989579	0.0335306
$872$	0.969206	0.0328215
$873$	0	0
$874$	18.1717	0.614667
$875$	17.7990	0.601715
$876$	0	0
$877$	8.37864	0.282926	0.141463	−	0.989944i	$-0.454819\pi$
$877$	8.37864	0.282926	0.141463	+	0.989944i	$0.454819\pi$
$878$	−8.59969	−0.290225
$879$	0	0
$880$	−6.95119	−0.234324
$881$	−7.84326	−0.264246	−0.132123	−	0.991233i	$-0.542179\pi$
$881$	−7.84326	−0.264246	−0.132123	+	0.991233i	$0.542179\pi$
$882$	0	0
$883$	−33.5962	−1.13060	−0.565301	−	0.824885i	$-0.691240\pi$
$883$	−33.5962	−1.13060	−0.565301	+	0.824885i	$0.691240\pi$
$884$	13.7335	0.461907
$885$	0	0
$886$	35.0002	1.17586
$887$	20.8584	0.700356	0.350178	−	0.936683i	$-0.386121\pi$
$887$	20.8584	0.700356	0.350178	+	0.936683i	$0.386121\pi$
$888$	0	0
$889$	16.7691	0.562417
$890$	−28.4300	−0.952977
$891$	0	0
$892$	−24.3247	−0.814452
$893$	11.5348	0.385998
$894$	0	0
$895$	25.4298	0.850026
$896$	1.17646	0.0393027
$897$	0	0
$898$	−27.0059	−0.901197
$899$	−26.2224	−0.874567
$900$	0	0
$901$	−72.0742	−2.40114
$902$	−9.38072	−0.312344
$903$	0	0
$904$	−1.29044	−0.0429194
$905$	−7.57384	−0.251763
$906$	0	0
$907$	−10.1615	−0.337406	−0.168703	−	0.985667i	$-0.553958\pi$
$907$	−10.1615	−0.337406	−0.168703	+	0.985667i	$0.553958\pi$
$908$	−23.2961	−0.773107
$909$	0	0
$910$	−5.37865	−0.178300
$911$	−10.1200	−0.335291	−0.167646	−	0.985847i	$-0.553616\pi$
$911$	−10.1200	−0.335291	−0.167646	+	0.985847i	$0.553616\pi$
$912$	0	0
$913$	−17.4446	−0.577331
$914$	35.6427	1.17895
$915$	0	0
$916$	2.41024	0.0796366
$917$	0.536911	0.0177304
$918$	0	0
$919$	0.868507	0.0286494	0.0143247	−	0.999897i	$-0.495440\pi$
$919$	0.868507	0.0286494	0.0143247	+	0.999897i	$0.495440\pi$
$920$	0.778174	0.0256556
$921$	0	0
$922$	−2.80977	−0.0925349
$923$	11.5224	0.379266
$924$	0	0
$925$	1.34071	0.0440824
$926$	34.0211	1.11800
$927$	0	0
$928$	41.4984	1.36225
$929$	−45.3136	−1.48669	−0.743346	−	0.668907i	$-0.766762\pi$
$929$	−45.3136	−1.48669	−0.743346	+	0.668907i	$0.766762\pi$
$930$	0	0
$931$	−9.61198	−0.315020
$932$	14.4452	0.473168
$933$	0	0
$934$	−71.2338	−2.33084
$935$	11.0319	0.360783
$936$	0	0
$937$	−10.5251	−0.343841	−0.171920	−	0.985111i	$-0.554997\pi$
$937$	−10.5251	−0.343841	−0.171920	+	0.985111i	$0.554997\pi$
$938$	2.67519	0.0873480
$939$	0	0
$940$	−19.1686	−0.625212
$941$	45.3209	1.47742	0.738710	−	0.674024i	$-0.235436\pi$
$941$	45.3209	1.47742	0.738710	+	0.674024i	$0.235436\pi$
$942$	0	0
$943$	21.6856	0.706180
$944$	−56.9747	−1.85437
$945$	0	0
$946$	1.16654	0.0379275
$947$	1.40363	0.0456120	0.0228060	−	0.999740i	$-0.492740\pi$
$947$	1.40363	0.0456120	0.0228060	+	0.999740i	$0.492740\pi$
$948$	0	0
$949$	−4.93661	−0.160249
$950$	−8.39593	−0.272400
$951$	0	0
$952$	−0.956719	−0.0310074
$953$	9.19599	0.297887	0.148944	−	0.988846i	$-0.452413\pi$
$953$	9.19599	0.297887	0.148944	+	0.988846i	$0.452413\pi$
$954$	0	0
$955$	−11.6222	−0.376085
$956$	27.5399	0.890703
$957$	0	0
$958$	59.4534	1.92085
$959$	−18.0905	−0.584174
$960$	0	0
$961$	−5.79854	−0.187050
$962$	−1.35946	−0.0438309
$963$	0	0
$964$	21.6435	0.697089
$965$	6.75670	0.217506
$966$	0	0
$967$	37.9062	1.21898	0.609490	−	0.792794i	$-0.291374\pi$
$967$	37.9062	1.21898	0.609490	+	0.792794i	$0.291374\pi$
$968$	−0.0998540	−0.00320943
$969$	0	0
$970$	−28.0337	−0.900109
$971$	2.56099	0.0821861	0.0410930	−	0.999155i	$-0.486916\pi$
$971$	2.56099	0.0821861	0.0410930	+	0.999155i	$0.486916\pi$
$972$	0	0
$973$	−2.78108	−0.0891573
$974$	38.7497	1.24162
$975$	0	0
$976$	−4.09796	−0.131173
$977$	−12.0243	−0.384692	−0.192346	−	0.981327i	$-0.561610\pi$
$977$	−12.0243	−0.384692	−0.192346	+	0.981327i	$0.561610\pi$
$978$	0	0
$979$	−8.43337	−0.269532
$980$	15.9733	0.510247
$981$	0	0
$982$	46.4042	1.48082
$983$	9.71713	0.309928	0.154964	−	0.987920i	$-0.450474\pi$
$983$	9.71713	0.309928	0.154964	+	0.987920i	$0.450474\pi$
$984$	0	0
$985$	−11.9460	−0.380631
$986$	−67.5160	−2.15015
$987$	0	0
$988$	4.20252	0.133700
$989$	−2.69671	−0.0857505
$990$	0	0
$991$	0.152915	0.00485750	0.00242875	−	0.999997i	$-0.499227\pi$
$991$	0.152915	0.00485750	0.00242875	+	0.999997i	$0.499227\pi$
$992$	−39.8826	−1.26627
$993$	0	0
$994$	31.1493	0.987996
$995$	−4.72184	−0.149692
$996$	0	0
$997$	9.30232	0.294607	0.147304	−	0.989091i	$-0.452941\pi$
$997$	9.30232	0.294607	0.147304	+	0.989091i	$0.452941\pi$
$998$	−37.2720	−1.17982
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.20	yes	25
3.2	odd	2		6039.2.a.m.1.6	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.6	✓	25	3.2	odd	2
6039.2.a.p.1.20	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+1.98740 q^{2} +1.94976 q^{4} -1.69625 q^{5} +1.47319 q^{7} -0.0998540 q^{8} +O(q^{10})\)	\(q+1.98740 q^{2} +1.94976 q^{4} -1.69625 q^{5} +1.47319 q^{7} -0.0998540 q^{8} -3.37113 q^{10} -1.00000 q^{11} +1.08303 q^{13} +2.92781 q^{14} -4.09796 q^{16} +6.50371 q^{17} +1.99017 q^{19} -3.30728 q^{20} -1.98740 q^{22} +4.59431 q^{23} -2.12272 q^{25} +2.15241 q^{26} +2.87235 q^{28} -5.22348 q^{29} +5.02011 q^{31} -7.94458 q^{32} +12.9255 q^{34} -2.49890 q^{35} -0.631601 q^{37} +3.95527 q^{38} +0.169378 q^{40} +4.72010 q^{41} -0.586969 q^{43} -1.94976 q^{44} +9.13073 q^{46} +5.79589 q^{47} -4.82972 q^{49} -4.21870 q^{50} +2.11164 q^{52} -11.0820 q^{53} +1.69625 q^{55} -0.147104 q^{56} -10.3811 q^{58} +13.9032 q^{59} +1.00000 q^{61} +9.97696 q^{62} -7.59313 q^{64} -1.83709 q^{65} +0.913716 q^{67} +12.6807 q^{68} -4.96631 q^{70} +10.6391 q^{71} -4.55816 q^{73} -1.25524 q^{74} +3.88035 q^{76} -1.47319 q^{77} -5.39890 q^{79} +6.95119 q^{80} +9.38072 q^{82} +17.4446 q^{83} -11.0319 q^{85} -1.16654 q^{86} +0.0998540 q^{88} +8.43337 q^{89} +1.59550 q^{91} +8.95778 q^{92} +11.5187 q^{94} -3.37584 q^{95} +8.31582 q^{97} -9.59859 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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