LMFDB - Embedded newform 6039.2.a.m.1.6

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:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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	$-0.748001\pi$
$2$	−1.98740	−1.40530	−0.702652	+	0.711534i	$0.748001\pi$
$3$	0	0
$3$	0	0
$4$	1.94976	0.974878
$5$	1.69625	0.758588	0.379294	−	0.925276i	$-0.376167\pi$
$5$	1.69625	0.758588	0.379294	+	0.925276i	$0.376167\pi$
$6$	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	$-0.789241\pi$
$17$	−6.50371	−1.57738	−0.788691	+	0.614790i	$0.789241\pi$
$18$	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	$-0.658997\pi$
$23$	−4.59431	−0.957980	−0.478990	+	0.877820i	$0.658997\pi$
$24$	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	$-0.338824\pi$
$29$	5.22348	0.969976	0.484988	+	0.874521i	$0.338824\pi$
$30$	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	$-0.620155\pi$
$41$	−4.72010	−0.737156	−0.368578	+	0.929597i	$0.620155\pi$
$42$	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	$-0.638921\pi$
$47$	−5.79589	−0.845417	−0.422709	+	0.906266i	$0.638921\pi$
$48$	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	$-0.224652\pi$
$53$	11.0820	1.52223	0.761115	+	0.648617i	$0.224652\pi$
$54$	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	$-0.860144\pi$
$59$	−13.9032	−1.81004	−0.905019	+	0.425371i	$0.860144\pi$
$60$	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	$-0.717485\pi$
$71$	−10.6391	−1.26263	−0.631315	+	0.775526i	$0.717485\pi$
$72$	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	$-0.906750\pi$
$83$	−17.4446	−1.91479	−0.957395	+	0.288780i	$0.906750\pi$
$84$	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	$-0.647496\pi$
$89$	−8.43337	−0.893936	−0.446968	+	0.894550i	$0.647496\pi$
$90$	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	$-0.681360\pi$
$101$	−10.8424	−1.07886	−0.539429	+	0.842031i	$0.681360\pi$
$102$	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	$-0.609384\pi$
$107$	−6.97016	−0.673831	−0.336916	+	0.941535i	$0.609384\pi$
$108$	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	$-0.707972\pi$
$113$	−12.9233	−1.21572	−0.607860	+	0.794044i	$0.707972\pi$
$114$	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	$-0.505068\pi$
$131$	−0.364456	−0.0318426	−0.0159213	+	0.999873i	$0.505068\pi$
$132$	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	$-0.324226\pi$
$137$	12.2799	1.04914	0.524570	+	0.851367i	$0.324226\pi$
$138$	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	$-0.428251\pi$
$149$	5.45639	0.447005	0.223503	+	0.974703i	$0.428251\pi$
$150$	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	$-0.477315\pi$
$167$	1.84036	0.142411	0.0712055	+	0.997462i	$0.477315\pi$
$168$	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	$-0.666495\pi$
$173$	−13.1406	−0.999065	−0.499532	+	0.866295i	$0.666495\pi$
$174$	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	$-0.310698\pi$
$179$	14.9918	1.12054	0.560268	+	0.828311i	$0.310698\pi$
$180$	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	$-0.579736\pi$
$191$	−6.85168	−0.495770	−0.247885	+	0.968789i	$0.579736\pi$
$192$	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	$-0.580720\pi$
$197$	−7.04258	−0.501763	−0.250882	+	0.968018i	$0.580720\pi$
$198$	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	$-0.370220\pi$
$227$	11.9482	0.793029	0.396515	+	0.918028i	$0.370220\pi$
$228$	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	$-0.578027\pi$
$233$	−7.40871	−0.485361	−0.242680	+	0.970106i	$0.578027\pi$
$234$	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	$-0.651014\pi$
$239$	−14.1248	−0.913656	−0.456828	+	0.889555i	$0.651014\pi$
$240$	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	$-0.387680\pi$
$251$	10.9503	0.691174	0.345587	+	0.938387i	$0.387680\pi$
$252$	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	$-0.0416978\pi$
$257$	31.7877	1.98286	0.991432	+	0.130623i	$0.0416978\pi$
$258$	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	$-0.828397\pi$
$263$	−27.8343	−1.71633	−0.858167	+	0.513371i	$0.828397\pi$
$264$	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	$-0.713669\pi$
$269$	−20.4023	−1.24395	−0.621974	+	0.783038i	$0.713669\pi$
$270$	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	$-0.299038\pi$
$281$	19.7880	1.18046	0.590228	+	0.807237i	$0.299038\pi$
$282$	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	$-0.100852\pi$
$293$	32.5305	1.90045	0.950226	+	0.311561i	$0.100852\pi$
$294$	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	$-0.296331\pi$
$311$	21.0590	1.19414	0.597072	+	0.802188i	$0.296331\pi$
$312$	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	$-0.186026\pi$
$317$	29.6991	1.66807	0.834034	+	0.551713i	$0.186026\pi$
$318$	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	$-0.616654\pi$
$347$	−13.3499	−0.716659	−0.358329	+	0.933595i	$0.616654\pi$
$348$	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	$-0.647351\pi$
$353$	−16.7802	−0.893118	−0.446559	+	0.894754i	$0.647351\pi$
$354$	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	$-0.455243\pi$
$359$	5.31077	0.280291	0.140146	+	0.990131i	$0.455243\pi$
$360$	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	$-0.296061\pi$
$383$	23.3964	1.19550	0.597751	+	0.801682i	$0.296061\pi$
$384$	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	$-0.579570\pi$
$389$	−9.75823	−0.494762	−0.247381	+	0.968918i	$0.579570\pi$
$390$	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	$-0.531150\pi$
$401$	−3.91307	−0.195409	−0.0977046	+	0.995215i	$0.531150\pi$
$402$	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	$-0.296513\pi$
$419$	24.4247	1.19322	0.596612	+	0.802530i	$0.296513\pi$
$420$	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	$-0.164753\pi$
$431$	36.0825	1.73803	0.869016	+	0.494785i	$0.164753\pi$
$432$	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	$-0.637396\pi$
$443$	−17.6111	−0.836727	−0.418364	+	0.908280i	$0.637396\pi$
$444$	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	$-0.396102\pi$
$449$	13.5885	0.641283	0.320641	+	0.947201i	$0.396102\pi$
$450$	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	$-0.489518\pi$
$461$	1.41379	0.0658469	0.0329235	+	0.999458i	$0.489518\pi$
$462$	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	$-0.188738\pi$
$467$	35.8427	1.65860	0.829302	+	0.558801i	$0.188738\pi$
$468$	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	$-0.739512\pi$
$479$	−29.9152	−1.36686	−0.683430	+	0.730016i	$0.739512\pi$
$480$	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	$-0.676634\pi$
$491$	−23.3492	−1.05374	−0.526868	+	0.849947i	$0.676634\pi$
$492$	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	$-0.652659\pi$
$503$	−20.6970	−0.922833	−0.461417	+	0.887184i	$0.652659\pi$
$504$	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	$-0.669269\pi$
$509$	−22.8797	−1.01413	−0.507063	+	0.861909i	$0.669269\pi$
$510$	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	$-0.403425\pi$
$521$	13.6390	0.597534	0.298767	+	0.954326i	$0.403425\pi$
$522$	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	$-0.585880\pi$
$557$	−12.5811	−0.533077	−0.266539	+	0.963824i	$0.585880\pi$
$558$	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	$-0.848871\pi$
$563$	−42.2063	−1.77878	−0.889391	+	0.457147i	$0.848871\pi$
$564$	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	$-0.539846\pi$
$569$	−5.95645	−0.249707	−0.124854	+	0.992175i	$0.539846\pi$
$570$	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	$-0.303093\pi$
$587$	28.0996	1.15979	0.579897	+	0.814690i	$0.303093\pi$
$588$	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	$-0.624643\pi$
$593$	−18.5874	−0.763291	−0.381646	+	0.924309i	$0.624643\pi$
$594$	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	$-0.728810\pi$
$599$	−32.2331	−1.31701	−0.658504	+	0.752577i	$0.728810\pi$
$600$	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	$-0.790771\pi$
$617$	−39.3278	−1.58328	−0.791638	+	0.610990i	$0.790771\pi$
$618$	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	$-0.573701\pi$
$641$	−11.6197	−0.458949	−0.229475	+	0.973315i	$0.573701\pi$
$642$	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	$-0.399681\pi$
$647$	15.7690	0.619942	0.309971	+	0.950746i	$0.399681\pi$
$648$	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	$-0.778707\pi$
$653$	−39.2465	−1.53583	−0.767917	+	0.640549i	$0.778707\pi$
$654$	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	$-0.869739\pi$
$659$	−47.1026	−1.83486	−0.917428	+	0.397902i	$0.869739\pi$
$660$	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	$-0.383319\pi$
$677$	18.6511	0.716820	0.358410	+	0.933564i	$0.383319\pi$
$678$	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	$-0.527259\pi$
$683$	−4.47064	−0.171064	−0.0855322	+	0.996335i	$0.527259\pi$
$684$	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	$-0.287858\pi$
$701$	32.7360	1.23642	0.618210	+	0.786013i	$0.287858\pi$
$702$	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	$-0.308215\pi$
$719$	30.3919	1.13343	0.566713	+	0.823915i	$0.308215\pi$
$720$	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	$-0.132553\pi$
$743$	49.8571	1.82908	0.914540	+	0.404496i	$0.132553\pi$
$744$	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	$-0.370103\pi$
$761$	21.8952	0.793700	0.396850	+	0.917883i	$0.370103\pi$
$762$	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	$-0.633312\pi$
$773$	−22.6135	−0.813353	−0.406676	+	0.913572i	$0.633312\pi$
$774$	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	$-0.181914\pi$
$797$	47.4900	1.68218	0.841091	+	0.540893i	$0.181914\pi$
$798$	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	$-0.714599\pi$
$809$	−35.5115	−1.24852	−0.624259	+	0.781217i	$0.714599\pi$
$810$	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	$-0.415755\pi$
$821$	14.9903	0.523166	0.261583	+	0.965181i	$0.415755\pi$
$822$	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	$-0.397402\pi$
$827$	18.2190	0.633536	0.316768	+	0.948503i	$0.397402\pi$
$828$	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	$-0.519294\pi$
$839$	−3.50925	−0.121153	−0.0605764	+	0.998164i	$0.519294\pi$
$840$	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	$-0.661932\pi$
$857$	−28.5171	−0.974127	−0.487063	+	0.873367i	$0.661932\pi$
$858$	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	$-0.563052\pi$
$863$	−11.5621	−0.393579	−0.196789	+	0.980446i	$0.563052\pi$
$864$	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	$-0.457821\pi$
$881$	7.84326	0.264246	0.132123	+	0.991233i	$0.457821\pi$
$882$	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	$-0.613879\pi$
$887$	−20.8584	−0.700356	−0.350178	+	0.936683i	$0.613879\pi$
$888$	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	$-0.446384\pi$
$911$	10.1200	0.335291	0.167646	+	0.985847i	$0.446384\pi$
$912$	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	$-0.233238\pi$
$929$	45.3136	1.48669	0.743346	+	0.668907i	$0.233238\pi$
$930$	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	$-0.764564\pi$
$941$	−45.3209	−1.47742	−0.738710	+	0.674024i	$0.764564\pi$
$942$	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	$-0.507260\pi$
$947$	−1.40363	−0.0456120	−0.0228060	+	0.999740i	$0.507260\pi$
$948$	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	$-0.547587\pi$
$953$	−9.19599	−0.297887	−0.148944	+	0.988846i	$0.547587\pi$
$954$	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	$-0.513084\pi$
$971$	−2.56099	−0.0821861	−0.0410930	+	0.999155i	$0.513084\pi$
$972$	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	$-0.438390\pi$
$977$	12.0243	0.384692	0.192346	+	0.981327i	$0.438390\pi$
$978$	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	$-0.549526\pi$
$983$	−9.71713	−0.309928	−0.154964	+	0.987920i	$0.549526\pi$
$984$	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.m.1.6	✓	25
3.2	odd	2		6039.2.a.p.1.20	yes	25

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

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

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