LMFDB - Embedded newform 6039.2.a.p.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.21594 q^{2} +2.91037 q^{4} +4.37397 q^{5} +0.199638 q^{7} -2.01732 q^{8} +O(q^{10})$	\(q-2.21594 q^{2} +2.91037 q^{4} +4.37397 q^{5} +0.199638 q^{7} -2.01732 q^{8} -9.69243 q^{10} -1.00000 q^{11} +6.55576 q^{13} -0.442385 q^{14} -1.35048 q^{16} +0.167373 q^{17} -1.49539 q^{19} +12.7299 q^{20} +2.21594 q^{22} +8.33777 q^{23} +14.1316 q^{25} -14.5272 q^{26} +0.581021 q^{28} -1.38251 q^{29} -1.87059 q^{31} +7.02723 q^{32} -0.370888 q^{34} +0.873211 q^{35} +8.78153 q^{37} +3.31369 q^{38} -8.82371 q^{40} +3.86672 q^{41} -11.0953 q^{43} -2.91037 q^{44} -18.4760 q^{46} +1.72912 q^{47} -6.96014 q^{49} -31.3147 q^{50} +19.0797 q^{52} +8.30007 q^{53} -4.37397 q^{55} -0.402735 q^{56} +3.06355 q^{58} -1.27479 q^{59} +1.00000 q^{61} +4.14510 q^{62} -12.8709 q^{64} +28.6747 q^{65} +8.70975 q^{67} +0.487117 q^{68} -1.93498 q^{70} -13.6961 q^{71} -6.82811 q^{73} -19.4593 q^{74} -4.35215 q^{76} -0.199638 q^{77} -13.5956 q^{79} -5.90696 q^{80} -8.56840 q^{82} +13.8714 q^{83} +0.732083 q^{85} +24.5865 q^{86} +2.01732 q^{88} -1.60785 q^{89} +1.30878 q^{91} +24.2660 q^{92} -3.83161 q^{94} -6.54080 q^{95} +6.07276 q^{97} +15.4232 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.21594	−1.56690	−0.783452	−	0.621453i	$-0.786543\pi$
$2$	−2.21594	−1.56690	−0.783452	+	0.621453i	$0.786543\pi$
$3$	0	0
$3$	0	0
$4$	2.91037	1.45519
$5$	4.37397	1.95610	0.978049	−	0.208376i	$-0.0668177\pi$
$5$	4.37397	1.95610	0.978049	+	0.208376i	$0.0668177\pi$
$6$	0	0
$7$	0.199638	0.0754561	0.0377281	−	0.999288i	$-0.487988\pi$
$7$	0.199638	0.0754561	0.0377281	+	0.999288i	$0.487988\pi$
$8$	−2.01732	−0.713232
$9$	0	0
$10$	−9.69243	−3.06502
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.55576	1.81824	0.909121	−	0.416532i	$-0.136755\pi$
$13$	6.55576	1.81824	0.909121	+	0.416532i	$0.136755\pi$
$14$	−0.442385	−0.118232
$15$	0	0
$16$	−1.35048	−0.337620
$17$	0.167373	0.0405939	0.0202969	−	0.999794i	$-0.493539\pi$
$17$	0.167373	0.0405939	0.0202969	+	0.999794i	$0.493539\pi$
$18$	0	0
$19$	−1.49539	−0.343067	−0.171533	−	0.985178i	$-0.554872\pi$
$19$	−1.49539	−0.343067	−0.171533	+	0.985178i	$0.554872\pi$
$20$	12.7299	2.84649
$21$	0	0
$22$	2.21594	0.472439
$23$	8.33777	1.73854	0.869272	−	0.494334i	$-0.164588\pi$
$23$	8.33777	1.73854	0.869272	+	0.494334i	$0.164588\pi$
$24$	0	0
$25$	14.1316	2.82632
$26$	−14.5272	−2.84901
$27$	0	0
$28$	0.581021	0.109803
$29$	−1.38251	−0.256725	−0.128363	−	0.991727i	$-0.540972\pi$
$29$	−1.38251	−0.256725	−0.128363	+	0.991727i	$0.540972\pi$
$30$	0	0
$31$	−1.87059	−0.335967	−0.167984	−	0.985790i	$-0.553726\pi$
$31$	−1.87059	−0.335967	−0.167984	+	0.985790i	$0.553726\pi$
$32$	7.02723	1.24225
$33$	0	0
$34$	−0.370888	−0.0636067
$35$	0.873211	0.147600
$36$	0	0
$37$	8.78153	1.44367	0.721837	−	0.692063i	$-0.243298\pi$
$37$	8.78153	1.44367	0.721837	+	0.692063i	$0.243298\pi$
$38$	3.31369	0.537552
$39$	0	0
$40$	−8.82371	−1.39515
$41$	3.86672	0.603880	0.301940	−	0.953327i	$-0.402366\pi$
$41$	3.86672	0.603880	0.301940	+	0.953327i	$0.402366\pi$
$42$	0	0
$43$	−11.0953	−1.69202	−0.846010	−	0.533168i	$-0.821001\pi$
$43$	−11.0953	−1.69202	−0.846010	+	0.533168i	$0.821001\pi$
$44$	−2.91037	−0.438755
$45$	0	0
$46$	−18.4760	−2.72413
$47$	1.72912	0.252217	0.126109	−	0.992016i	$-0.459751\pi$
$47$	1.72912	0.252217	0.126109	+	0.992016i	$0.459751\pi$
$48$	0	0
$49$	−6.96014	−0.994306
$50$	−31.3147	−4.42857
$51$	0	0
$52$	19.0797	2.64588
$53$	8.30007	1.14010	0.570051	−	0.821609i	$-0.306923\pi$
$53$	8.30007	1.14010	0.570051	+	0.821609i	$0.306923\pi$
$54$	0	0
$55$	−4.37397	−0.589786
$56$	−0.402735	−0.0538177
$57$	0	0
$58$	3.06355	0.402263
$59$	−1.27479	−0.165964	−0.0829818	−	0.996551i	$-0.526444\pi$
$59$	−1.27479	−0.165964	−0.0829818	+	0.996551i	$0.526444\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	4.14510	0.526428
$63$	0	0
$64$	−12.8709	−1.60887
$65$	28.6747	3.55666
$66$	0	0
$67$	8.70975	1.06407	0.532033	−	0.846724i	$-0.321428\pi$
$67$	8.70975	1.06407	0.532033	+	0.846724i	$0.321428\pi$
$68$	0.487117	0.0590716
$69$	0	0
$70$	−1.93498	−0.231274
$71$	−13.6961	−1.62542	−0.812712	−	0.582665i	$-0.802010\pi$
$71$	−13.6961	−1.62542	−0.812712	+	0.582665i	$0.802010\pi$
$72$	0	0
$73$	−6.82811	−0.799170	−0.399585	−	0.916696i	$-0.630846\pi$
$73$	−6.82811	−0.799170	−0.399585	+	0.916696i	$0.630846\pi$
$74$	−19.4593	−2.26210
$75$	0	0
$76$	−4.35215	−0.499226
$77$	−0.199638	−0.0227509
$78$	0	0
$79$	−13.5956	−1.52962	−0.764810	−	0.644255i	$-0.777167\pi$
$79$	−13.5956	−1.52962	−0.764810	+	0.644255i	$0.777167\pi$
$80$	−5.90696	−0.660418
$81$	0	0
$82$	−8.56840	−0.946222
$83$	13.8714	1.52258	0.761291	−	0.648410i	$-0.224566\pi$
$83$	13.8714	1.52258	0.761291	+	0.648410i	$0.224566\pi$
$84$	0	0
$85$	0.732083	0.0794056
$86$	24.5865	2.65123
$87$	0	0
$88$	2.01732	0.215048
$89$	−1.60785	−0.170432	−0.0852158	−	0.996363i	$-0.527158\pi$
$89$	−1.60785	−0.170432	−0.0852158	+	0.996363i	$0.527158\pi$
$90$	0	0
$91$	1.30878	0.137198
$92$	24.2660	2.52990
$93$	0	0
$94$	−3.83161	−0.395200
$95$	−6.54080	−0.671072
$96$	0	0
$97$	6.07276	0.616595	0.308298	−	0.951290i	$-0.400241\pi$
$97$	6.07276	0.616595	0.308298	+	0.951290i	$0.400241\pi$
$98$	15.4232	1.55798
$99$	0	0
$100$	41.1282	4.11282
$101$	11.4789	1.14220	0.571098	−	0.820882i	$-0.306518\pi$
$101$	11.4789	1.14220	0.571098	+	0.820882i	$0.306518\pi$
$102$	0	0
$103$	0.411386	0.0405351	0.0202675	−	0.999795i	$-0.493548\pi$
$103$	0.411386	0.0405351	0.0202675	+	0.999795i	$0.493548\pi$
$104$	−13.2251	−1.29683
$105$	0	0
$106$	−18.3924	−1.78643
$107$	13.0633	1.26288	0.631441	−	0.775424i	$-0.282464\pi$
$107$	13.0633	1.26288	0.631441	+	0.775424i	$0.282464\pi$
$108$	0	0
$109$	−18.4974	−1.77173	−0.885864	−	0.463945i	$-0.846433\pi$
$109$	−18.4974	−1.77173	−0.885864	+	0.463945i	$0.846433\pi$
$110$	9.69243	0.924137
$111$	0	0
$112$	−0.269608	−0.0254755
$113$	−4.30032	−0.404540	−0.202270	−	0.979330i	$-0.564832\pi$
$113$	−4.30032	−0.404540	−0.202270	+	0.979330i	$0.564832\pi$
$114$	0	0
$115$	36.4691	3.40076
$116$	−4.02361	−0.373583
$117$	0	0
$118$	2.82485	0.260049
$119$	0.0334140	0.00306306
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.21594	−0.200621
$123$	0	0
$124$	−5.44410	−0.488894
$125$	39.9413	3.57246
$126$	0	0
$127$	0.914225	0.0811244	0.0405622	−	0.999177i	$-0.487085\pi$
$127$	0.914225	0.0811244	0.0405622	+	0.999177i	$0.487085\pi$
$128$	14.4667	1.27869
$129$	0	0
$130$	−63.5413	−5.57294
$131$	6.91112	0.603827	0.301914	−	0.953335i	$-0.402375\pi$
$131$	6.91112	0.603827	0.301914	+	0.953335i	$0.402375\pi$
$132$	0	0
$133$	−0.298537	−0.0258865
$134$	−19.3002	−1.66729
$135$	0	0
$136$	−0.337645	−0.0289529
$137$	14.2854	1.22048	0.610240	−	0.792216i	$-0.291073\pi$
$137$	14.2854	1.22048	0.610240	+	0.792216i	$0.291073\pi$
$138$	0	0
$139$	2.66484	0.226029	0.113014	−	0.993593i	$-0.463949\pi$
$139$	2.66484	0.226029	0.113014	+	0.993593i	$0.463949\pi$
$140$	2.54137	0.214785
$141$	0	0
$142$	30.3496	2.54688
$143$	−6.55576	−0.548221
$144$	0	0
$145$	−6.04704	−0.502179
$146$	15.1307	1.25222
$147$	0	0
$148$	25.5575	2.10081
$149$	12.6712	1.03807	0.519034	−	0.854754i	$-0.326292\pi$
$149$	12.6712	1.03807	0.519034	+	0.854754i	$0.326292\pi$
$150$	0	0
$151$	−13.3712	−1.08814	−0.544068	−	0.839041i	$-0.683117\pi$
$151$	−13.3712	−1.08814	−0.544068	+	0.839041i	$0.683117\pi$
$152$	3.01669	0.244686
$153$	0	0
$154$	0.442385	0.0356484
$155$	−8.18188	−0.657184
$156$	0	0
$157$	−22.5501	−1.79970	−0.899848	−	0.436203i	$-0.856323\pi$
$157$	−22.5501	−1.79970	−0.899848	+	0.436203i	$0.856323\pi$
$158$	30.1269	2.39677
$159$	0	0
$160$	30.7369	2.42996
$161$	1.66454	0.131184
$162$	0	0
$163$	−4.81974	−0.377511	−0.188755	−	0.982024i	$-0.560445\pi$
$163$	−4.81974	−0.377511	−0.188755	+	0.982024i	$0.560445\pi$
$164$	11.2536	0.878758
$165$	0	0
$166$	−30.7381	−2.38574
$167$	−21.9543	−1.69887	−0.849436	−	0.527691i	$-0.823058\pi$
$167$	−21.9543	−1.69887	−0.849436	+	0.527691i	$0.823058\pi$
$168$	0	0
$169$	29.9780	2.30600
$170$	−1.62225	−0.124421
$171$	0	0
$172$	−32.2915	−2.46220
$173$	7.61604	0.579037	0.289518	−	0.957172i	$-0.406505\pi$
$173$	7.61604	0.579037	0.289518	+	0.957172i	$0.406505\pi$
$174$	0	0
$175$	2.82121	0.213263
$176$	1.35048	0.101796
$177$	0	0
$178$	3.56289	0.267050
$179$	−4.73309	−0.353768	−0.176884	−	0.984232i	$-0.556602\pi$
$179$	−4.73309	−0.353768	−0.176884	+	0.984232i	$0.556602\pi$
$180$	0	0
$181$	12.1229	0.901087	0.450544	−	0.892754i	$-0.351230\pi$
$181$	12.1229	0.901087	0.450544	+	0.892754i	$0.351230\pi$
$182$	−2.90017	−0.214975
$183$	0	0
$184$	−16.8200	−1.23999
$185$	38.4101	2.82397
$186$	0	0
$187$	−0.167373	−0.0122395
$188$	5.03237	0.367023
$189$	0	0
$190$	14.4940	1.05150
$191$	2.92453	0.211612	0.105806	−	0.994387i	$-0.466258\pi$
$191$	2.92453	0.211612	0.105806	+	0.994387i	$0.466258\pi$
$192$	0	0
$193$	−12.5595	−0.904053	−0.452026	−	0.892005i	$-0.649299\pi$
$193$	−12.5595	−0.904053	−0.452026	+	0.892005i	$0.649299\pi$
$194$	−13.4568	−0.966145
$195$	0	0
$196$	−20.2566	−1.44690
$197$	−11.9924	−0.854423	−0.427211	−	0.904152i	$-0.640504\pi$
$197$	−11.9924	−0.854423	−0.427211	+	0.904152i	$0.640504\pi$
$198$	0	0
$199$	5.90551	0.418631	0.209315	−	0.977848i	$-0.432877\pi$
$199$	5.90551	0.418631	0.209315	+	0.977848i	$0.432877\pi$
$200$	−28.5080	−2.01582
$201$	0	0
$202$	−25.4366	−1.78971
$203$	−0.276001	−0.0193715
$204$	0	0
$205$	16.9129	1.18125
$206$	−0.911606	−0.0635146
$207$	0	0
$208$	−8.85343	−0.613875
$209$	1.49539	0.103438
$210$	0	0
$211$	7.27267	0.500671	0.250336	−	0.968159i	$-0.419459\pi$
$211$	7.27267	0.500671	0.250336	+	0.968159i	$0.419459\pi$
$212$	24.1563	1.65906
$213$	0	0
$214$	−28.9475	−1.97881
$215$	−48.5305	−3.30975
$216$	0	0
$217$	−0.373440	−0.0253508
$218$	40.9890	2.77613
$219$	0	0
$220$	−12.7299	−0.858248
$221$	1.09726	0.0738095
$222$	0	0
$223$	16.5209	1.10632	0.553159	−	0.833076i	$-0.313422\pi$
$223$	16.5209	1.10632	0.553159	+	0.833076i	$0.313422\pi$
$224$	1.40290	0.0937354
$225$	0	0
$226$	9.52923	0.633875
$227$	−23.0603	−1.53057	−0.765284	−	0.643693i	$-0.777401\pi$
$227$	−23.0603	−1.53057	−0.765284	+	0.643693i	$0.777401\pi$
$228$	0	0
$229$	7.37984	0.487674	0.243837	−	0.969816i	$-0.421594\pi$
$229$	7.37984	0.487674	0.243837	+	0.969816i	$0.421594\pi$
$230$	−80.8132	−5.32867
$231$	0	0
$232$	2.78896	0.183105
$233$	−15.6156	−1.02301	−0.511506	−	0.859280i	$-0.670912\pi$
$233$	−15.6156	−1.02301	−0.511506	+	0.859280i	$0.670912\pi$
$234$	0	0
$235$	7.56310	0.493362
$236$	−3.71011	−0.241508
$237$	0	0
$238$	−0.0740433	−0.00479952
$239$	−3.08699	−0.199681	−0.0998404	−	0.995003i	$-0.531833\pi$
$239$	−3.08699	−0.199681	−0.0998404	+	0.995003i	$0.531833\pi$
$240$	0	0
$241$	−5.95131	−0.383357	−0.191679	−	0.981458i	$-0.561393\pi$
$241$	−5.95131	−0.383357	−0.191679	+	0.981458i	$0.561393\pi$
$242$	−2.21594	−0.142446
$243$	0	0
$244$	2.91037	0.186317
$245$	−30.4434	−1.94496
$246$	0	0
$247$	−9.80344	−0.623778
$248$	3.77358	0.239622
$249$	0	0
$250$	−88.5073	−5.59769
$251$	−20.8339	−1.31502	−0.657512	−	0.753444i	$-0.728391\pi$
$251$	−20.8339	−1.31502	−0.657512	+	0.753444i	$0.728391\pi$
$252$	0	0
$253$	−8.33777	−0.524191
$254$	−2.02586	−0.127114
$255$	0	0
$256$	−6.31540	−0.394712
$257$	1.44393	0.0900696	0.0450348	−	0.998985i	$-0.485660\pi$
$257$	1.44393	0.0900696	0.0450348	+	0.998985i	$0.485660\pi$
$258$	0	0
$259$	1.75313	0.108934
$260$	83.4540	5.17560
$261$	0	0
$262$	−15.3146	−0.946139
$263$	8.16704	0.503601	0.251800	−	0.967779i	$-0.418977\pi$
$263$	8.16704	0.503601	0.251800	+	0.967779i	$0.418977\pi$
$264$	0	0
$265$	36.3043	2.23015
$266$	0.661540	0.0405616
$267$	0	0
$268$	25.3486	1.54841
$269$	14.8984	0.908371	0.454186	−	0.890907i	$-0.349930\pi$
$269$	14.8984	0.908371	0.454186	+	0.890907i	$0.349930\pi$
$270$	0	0
$271$	−3.89552	−0.236636	−0.118318	−	0.992976i	$-0.537750\pi$
$271$	−3.89552	−0.236636	−0.118318	+	0.992976i	$0.537750\pi$
$272$	−0.226034	−0.0137053
$273$	0	0
$274$	−31.6554	−1.91238
$275$	−14.1316	−0.852167
$276$	0	0
$277$	−13.3340	−0.801160	−0.400580	−	0.916262i	$-0.631191\pi$
$277$	−13.3340	−0.801160	−0.400580	+	0.916262i	$0.631191\pi$
$278$	−5.90511	−0.354165
$279$	0	0
$280$	−1.76155	−0.105273
$281$	28.2362	1.68443	0.842215	−	0.539142i	$-0.181252\pi$
$281$	28.2362	1.68443	0.842215	+	0.539142i	$0.181252\pi$
$282$	0	0
$283$	18.0491	1.07291	0.536455	−	0.843929i	$-0.319763\pi$
$283$	18.0491	1.07291	0.536455	+	0.843929i	$0.319763\pi$
$284$	−39.8606	−2.36529
$285$	0	0
$286$	14.5272	0.859009
$287$	0.771945	0.0455665
$288$	0	0
$289$	−16.9720	−0.998352
$290$	13.3999	0.786866
$291$	0	0
$292$	−19.8723	−1.16294
$293$	5.19996	0.303785	0.151892	−	0.988397i	$-0.451463\pi$
$293$	5.19996	0.303785	0.151892	+	0.988397i	$0.451463\pi$
$294$	0	0
$295$	−5.57589	−0.324641
$296$	−17.7152	−1.02967
$297$	0	0
$298$	−28.0787	−1.62655
$299$	54.6604	3.16109
$300$	0	0
$301$	−2.21505	−0.127673
$302$	29.6298	1.70500
$303$	0	0
$304$	2.01950	0.115826
$305$	4.37397	0.250453
$306$	0	0
$307$	−5.26021	−0.300216	−0.150108	−	0.988670i	$-0.547962\pi$
$307$	−5.26021	−0.300216	−0.150108	+	0.988670i	$0.547962\pi$
$308$	−0.581021	−0.0331068
$309$	0	0
$310$	18.1305	1.02974
$311$	−25.6078	−1.45208	−0.726041	−	0.687651i	$-0.758642\pi$
$311$	−25.6078	−1.45208	−0.726041	+	0.687651i	$0.758642\pi$
$312$	0	0
$313$	−11.3576	−0.641971	−0.320985	−	0.947084i	$-0.604014\pi$
$313$	−11.3576	−0.641971	−0.320985	+	0.947084i	$0.604014\pi$
$314$	49.9696	2.81995
$315$	0	0
$316$	−39.5682	−2.22588
$317$	−14.7577	−0.828875	−0.414437	−	0.910078i	$-0.636022\pi$
$317$	−14.7577	−0.828875	−0.414437	+	0.910078i	$0.636022\pi$
$318$	0	0
$319$	1.38251	0.0774055
$320$	−56.2970	−3.14710
$321$	0	0
$322$	−3.68851	−0.205552
$323$	−0.250288	−0.0139264
$324$	0	0
$325$	92.6434	5.13893
$326$	10.6802	0.591523
$327$	0	0
$328$	−7.80043	−0.430707
$329$	0.345198	0.0190314
$330$	0	0
$331$	−24.2608	−1.33349	−0.666746	−	0.745285i	$-0.732313\pi$
$331$	−24.2608	−1.33349	−0.666746	+	0.745285i	$0.732313\pi$
$332$	40.3709	2.21564
$333$	0	0
$334$	48.6493	2.66197
$335$	38.0962	2.08142
$336$	0	0
$337$	−24.8766	−1.35511	−0.677557	−	0.735470i	$-0.736961\pi$
$337$	−24.8766	−1.35511	−0.677557	+	0.735470i	$0.736961\pi$
$338$	−66.4294	−3.61328
$339$	0	0
$340$	2.13063	0.115550
$341$	1.87059	0.101298
$342$	0	0
$343$	−2.78698	−0.150483
$344$	22.3828	1.20680
$345$	0	0
$346$	−16.8767	−0.907295
$347$	−3.40627	−0.182858	−0.0914291	−	0.995812i	$-0.529143\pi$
$347$	−3.40627	−0.182858	−0.0914291	+	0.995812i	$0.529143\pi$
$348$	0	0
$349$	−13.2063	−0.706915	−0.353458	−	0.935451i	$-0.614994\pi$
$349$	−13.2063	−0.706915	−0.353458	+	0.935451i	$0.614994\pi$
$350$	−6.25161	−0.334163
$351$	0	0
$352$	−7.02723	−0.374553
$353$	−19.6701	−1.04693	−0.523466	−	0.852047i	$-0.675361\pi$
$353$	−19.6701	−1.04693	−0.523466	+	0.852047i	$0.675361\pi$
$354$	0	0
$355$	−59.9062	−3.17949
$356$	−4.67944	−0.248010
$357$	0	0
$358$	10.4882	0.554320
$359$	−32.9968	−1.74150	−0.870752	−	0.491723i	$-0.836367\pi$
$359$	−32.9968	−1.74150	−0.870752	+	0.491723i	$0.836367\pi$
$360$	0	0
$361$	−16.7638	−0.882305
$362$	−26.8635	−1.41192
$363$	0	0
$364$	3.80904	0.199648
$365$	−29.8659	−1.56326
$366$	0	0
$367$	−2.05423	−0.107230	−0.0536151	−	0.998562i	$-0.517074\pi$
$367$	−2.05423	−0.107230	−0.0536151	+	0.998562i	$0.517074\pi$
$368$	−11.2600	−0.586968
$369$	0	0
$370$	−85.1143	−4.42488
$371$	1.65701	0.0860278
$372$	0	0
$373$	−3.43639	−0.177929	−0.0889647	−	0.996035i	$-0.528356\pi$
$373$	−3.43639	−0.177929	−0.0889647	+	0.996035i	$0.528356\pi$
$374$	0.370888	0.0191781
$375$	0	0
$376$	−3.48819	−0.179890
$377$	−9.06339	−0.466788
$378$	0	0
$379$	20.1353	1.03428	0.517140	−	0.855901i	$-0.326997\pi$
$379$	20.1353	1.03428	0.517140	+	0.855901i	$0.326997\pi$
$380$	−19.0362	−0.976534
$381$	0	0
$382$	−6.48058	−0.331575
$383$	22.9082	1.17056	0.585278	−	0.810833i	$-0.300986\pi$
$383$	22.9082	1.17056	0.585278	+	0.810833i	$0.300986\pi$
$384$	0	0
$385$	−0.873211	−0.0445030
$386$	27.8311	1.41656
$387$	0	0
$388$	17.6740	0.897260
$389$	10.7991	0.547538	0.273769	−	0.961795i	$-0.411730\pi$
$389$	10.7991	0.547538	0.273769	+	0.961795i	$0.411730\pi$
$390$	0	0
$391$	1.39552	0.0705743
$392$	14.0409	0.709171
$393$	0	0
$394$	26.5744	1.33880
$395$	−59.4666	−2.99209
$396$	0	0
$397$	32.4566	1.62895	0.814475	−	0.580199i	$-0.197025\pi$
$397$	32.4566	1.62895	0.814475	+	0.580199i	$0.197025\pi$
$398$	−13.0862	−0.655954
$399$	0	0
$400$	−19.0844	−0.954222
$401$	−12.3634	−0.617397	−0.308699	−	0.951160i	$-0.599893\pi$
$401$	−12.3634	−0.617397	−0.308699	+	0.951160i	$0.599893\pi$
$402$	0	0
$403$	−12.2631	−0.610869
$404$	33.4080	1.66211
$405$	0	0
$406$	0.611601	0.0303532
$407$	−8.78153	−0.435284
$408$	0	0
$409$	−0.881095	−0.0435673	−0.0217837	−	0.999763i	$-0.506935\pi$
$409$	−0.881095	−0.0435673	−0.0217837	+	0.999763i	$0.506935\pi$
$410$	−37.4779	−1.85090
$411$	0	0
$412$	1.19729	0.0589861
$413$	−0.254497	−0.0125230
$414$	0	0
$415$	60.6730	2.97832
$416$	46.0689	2.25871
$417$	0	0
$418$	−3.31369	−0.162078
$419$	−5.86563	−0.286555	−0.143277	−	0.989683i	$-0.545764\pi$
$419$	−5.86563	−0.286555	−0.143277	+	0.989683i	$0.545764\pi$
$420$	0	0
$421$	−12.4450	−0.606534	−0.303267	−	0.952906i	$-0.598077\pi$
$421$	−12.4450	−0.606534	−0.303267	+	0.952906i	$0.598077\pi$
$422$	−16.1158	−0.784503
$423$	0	0
$424$	−16.7439	−0.813158
$425$	2.36525	0.114731
$426$	0	0
$427$	0.199638	0.00966117
$428$	38.0192	1.83773
$429$	0	0
$430$	107.541	5.18607
$431$	−13.6748	−0.658693	−0.329346	−	0.944209i	$-0.606828\pi$
$431$	−13.6748	−0.658693	−0.329346	+	0.944209i	$0.606828\pi$
$432$	0	0
$433$	−6.93462	−0.333256	−0.166628	−	0.986020i	$-0.553288\pi$
$433$	−6.93462	−0.333256	−0.166628	+	0.986020i	$0.553288\pi$
$434$	0.827520	0.0397222
$435$	0	0
$436$	−53.8342	−2.57819
$437$	−12.4682	−0.596436
$438$	0	0
$439$	18.5997	0.887713	0.443857	−	0.896098i	$-0.353610\pi$
$439$	18.5997	0.887713	0.443857	+	0.896098i	$0.353610\pi$
$440$	8.82371	0.420654
$441$	0	0
$442$	−2.43145	−0.115652
$443$	14.1141	0.670579	0.335289	−	0.942115i	$-0.391166\pi$
$443$	14.1141	0.670579	0.335289	+	0.942115i	$0.391166\pi$
$444$	0	0
$445$	−7.03268	−0.333381
$446$	−36.6091	−1.73349
$447$	0	0
$448$	−2.56953	−0.121399
$449$	0.649171	0.0306363	0.0153181	−	0.999883i	$-0.495124\pi$
$449$	0.649171	0.0306363	0.0153181	+	0.999883i	$0.495124\pi$
$450$	0	0
$451$	−3.86672	−0.182077
$452$	−12.5155	−0.588681
$453$	0	0
$454$	51.1002	2.39825
$455$	5.72457	0.268372
$456$	0	0
$457$	−28.8884	−1.35134	−0.675672	−	0.737202i	$-0.736147\pi$
$457$	−28.8884	−1.35134	−0.675672	+	0.737202i	$0.736147\pi$
$458$	−16.3533	−0.764137
$459$	0	0
$460$	106.139	4.94874
$461$	−17.9520	−0.836109	−0.418055	−	0.908422i	$-0.637288\pi$
$461$	−17.9520	−0.836109	−0.418055	+	0.908422i	$0.637288\pi$
$462$	0	0
$463$	23.0623	1.07180	0.535898	−	0.844283i	$-0.319973\pi$
$463$	23.0623	1.07180	0.535898	+	0.844283i	$0.319973\pi$
$464$	1.86705	0.0866756
$465$	0	0
$466$	34.6032	1.60296
$467$	−5.24953	−0.242919	−0.121460	−	0.992596i	$-0.538758\pi$
$467$	−5.24953	−0.242919	−0.121460	+	0.992596i	$0.538758\pi$
$468$	0	0
$469$	1.73880	0.0802903
$470$	−16.7593	−0.773051
$471$	0	0
$472$	2.57167	0.118371
$473$	11.0953	0.510163
$474$	0	0
$475$	−21.1323	−0.969615
$476$	0.0972472	0.00445732
$477$	0	0
$478$	6.84057	0.312881
$479$	34.0171	1.55428	0.777141	−	0.629327i	$-0.216669\pi$
$479$	34.0171	1.55428	0.777141	+	0.629327i	$0.216669\pi$
$480$	0	0
$481$	57.5696	2.62495
$482$	13.1877	0.600684
$483$	0	0
$484$	2.91037	0.132290
$485$	26.5620	1.20612
$486$	0	0
$487$	37.1252	1.68231	0.841153	−	0.540798i	$-0.181878\pi$
$487$	37.1252	1.68231	0.841153	+	0.540798i	$0.181878\pi$
$488$	−2.01732	−0.0913200
$489$	0	0
$490$	67.4607	3.04756
$491$	−21.8342	−0.985366	−0.492683	−	0.870209i	$-0.663984\pi$
$491$	−21.8342	−0.985366	−0.492683	+	0.870209i	$0.663984\pi$
$492$	0	0
$493$	−0.231394	−0.0104215
$494$	21.7238	0.977400
$495$	0	0
$496$	2.52619	0.113429
$497$	−2.73426	−0.122648
$498$	0	0
$499$	42.6043	1.90723	0.953616	−	0.301025i	$-0.0973287\pi$
$499$	42.6043	1.90723	0.953616	+	0.301025i	$0.0973287\pi$
$500$	116.244	5.19859
$501$	0	0
$502$	46.1666	2.06052
$503$	−22.8485	−1.01877	−0.509383	−	0.860540i	$-0.670126\pi$
$503$	−22.8485	−1.01877	−0.509383	+	0.860540i	$0.670126\pi$
$504$	0	0
$505$	50.2085	2.23425
$506$	18.4760	0.821356
$507$	0	0
$508$	2.66073	0.118051
$509$	20.1128	0.891484	0.445742	−	0.895162i	$-0.352940\pi$
$509$	20.1128	0.891484	0.445742	+	0.895162i	$0.352940\pi$
$510$	0	0
$511$	−1.36315	−0.0603023
$512$	−14.9389	−0.660210
$513$	0	0
$514$	−3.19965	−0.141130
$515$	1.79939	0.0792906
$516$	0	0
$517$	−1.72912	−0.0760464
$518$	−3.88482	−0.170689
$519$	0	0
$520$	−57.8462	−2.53672
$521$	−31.5055	−1.38028	−0.690140	−	0.723676i	$-0.742451\pi$
$521$	−31.5055	−1.38028	−0.690140	+	0.723676i	$0.742451\pi$
$522$	0	0
$523$	41.6877	1.82287	0.911437	−	0.411440i	$-0.134974\pi$
$523$	41.6877	1.82287	0.911437	+	0.411440i	$0.134974\pi$
$524$	20.1139	0.878681
$525$	0	0
$526$	−18.0976	−0.789094
$527$	−0.313085	−0.0136382
$528$	0	0
$529$	46.5183	2.02254
$530$	−80.4479	−3.49443
$531$	0	0
$532$	−0.868855	−0.0376696
$533$	25.3493	1.09800
$534$	0	0
$535$	57.1387	2.47032
$536$	−17.5704	−0.758925
$537$	0	0
$538$	−33.0139	−1.42333
$539$	6.96014	0.299795
$540$	0	0
$541$	26.1904	1.12601	0.563006	−	0.826453i	$-0.309645\pi$
$541$	26.1904	1.12601	0.563006	+	0.826453i	$0.309645\pi$
$542$	8.63222	0.370786
$543$	0	0
$544$	1.17617	0.0504278
$545$	−80.9069	−3.46567
$546$	0	0
$547$	3.88465	0.166096	0.0830478	−	0.996546i	$-0.473535\pi$
$547$	3.88465	0.166096	0.0830478	+	0.996546i	$0.473535\pi$
$548$	41.5757	1.77603
$549$	0	0
$550$	31.3147	1.33526
$551$	2.06739	0.0880738
$552$	0	0
$553$	−2.71419	−0.115419
$554$	29.5472	1.25534
$555$	0	0
$556$	7.75567	0.328914
$557$	33.1647	1.40524	0.702618	−	0.711567i	$-0.252014\pi$
$557$	33.1647	1.40524	0.702618	+	0.711567i	$0.252014\pi$
$558$	0	0
$559$	−72.7382	−3.07650
$560$	−1.17925	−0.0498326
$561$	0	0
$562$	−62.5696	−2.63934
$563$	34.0638	1.43562	0.717808	−	0.696241i	$-0.245145\pi$
$563$	34.0638	1.43562	0.717808	+	0.696241i	$0.245145\pi$
$564$	0	0
$565$	−18.8094	−0.791319
$566$	−39.9957	−1.68115
$567$	0	0
$568$	27.6294	1.15930
$569$	30.5969	1.28269	0.641344	−	0.767254i	$-0.278377\pi$
$569$	30.5969	1.28269	0.641344	+	0.767254i	$0.278377\pi$
$570$	0	0
$571$	−28.4799	−1.19185	−0.595924	−	0.803041i	$-0.703214\pi$
$571$	−28.4799	−1.19185	−0.595924	+	0.803041i	$0.703214\pi$
$572$	−19.0797	−0.797763
$573$	0	0
$574$	−1.71058	−0.0713982
$575$	117.826	4.91368
$576$	0	0
$577$	6.75123	0.281057	0.140529	−	0.990077i	$-0.455120\pi$
$577$	6.75123	0.281057	0.140529	+	0.990077i	$0.455120\pi$
$578$	37.6088	1.56432
$579$	0	0
$580$	−17.5991	−0.730764
$581$	2.76926	0.114888
$582$	0	0
$583$	−8.30007	−0.343754
$584$	13.7745	0.569994
$585$	0	0
$586$	−11.5228	−0.476002
$587$	−6.23936	−0.257526	−0.128763	−	0.991675i	$-0.541101\pi$
$587$	−6.23936	−0.257526	−0.128763	+	0.991675i	$0.541101\pi$
$588$	0	0
$589$	2.79726	0.115259
$590$	12.3558	0.508681
$591$	0	0
$592$	−11.8593	−0.487414
$593$	−36.0169	−1.47904	−0.739519	−	0.673136i	$-0.764947\pi$
$593$	−36.0169	−1.47904	−0.739519	+	0.673136i	$0.764947\pi$
$594$	0	0
$595$	0.146152	0.00599164
$596$	36.8780	1.51058
$597$	0	0
$598$	−121.124	−4.95313
$599$	17.5057	0.715262	0.357631	−	0.933863i	$-0.383585\pi$
$599$	17.5057	0.715262	0.357631	+	0.933863i	$0.383585\pi$
$600$	0	0
$601$	26.7958	1.09302	0.546511	−	0.837452i	$-0.315956\pi$
$601$	26.7958	1.09302	0.546511	+	0.837452i	$0.315956\pi$
$602$	4.90840	0.200052
$603$	0	0
$604$	−38.9153	−1.58344
$605$	4.37397	0.177827
$606$	0	0
$607$	−25.8975	−1.05115	−0.525574	−	0.850748i	$-0.676149\pi$
$607$	−25.8975	−1.05115	−0.525574	+	0.850748i	$0.676149\pi$
$608$	−10.5085	−0.426175
$609$	0	0
$610$	−9.69243	−0.392435
$611$	11.3357	0.458592
$612$	0	0
$613$	13.4550	0.543442	0.271721	−	0.962376i	$-0.412407\pi$
$613$	13.4550	0.543442	0.271721	+	0.962376i	$0.412407\pi$
$614$	11.6563	0.470410
$615$	0	0
$616$	0.402735	0.0162267
$617$	−31.5476	−1.27006	−0.635029	−	0.772488i	$-0.719012\pi$
$617$	−31.5476	−1.27006	−0.635029	+	0.772488i	$0.719012\pi$
$618$	0	0
$619$	−2.43152	−0.0977312	−0.0488656	−	0.998805i	$-0.515561\pi$
$619$	−2.43152	−0.0977312	−0.0488656	+	0.998805i	$0.515561\pi$
$620$	−23.8123	−0.956325
$621$	0	0
$622$	56.7452	2.27527
$623$	−0.320988	−0.0128601
$624$	0	0
$625$	104.044	4.16176
$626$	25.1678	1.00591
$627$	0	0
$628$	−65.6293	−2.61889
$629$	1.46979	0.0586043
$630$	0	0
$631$	−16.7329	−0.666125	−0.333062	−	0.942905i	$-0.608082\pi$
$631$	−16.7329	−0.666125	−0.333062	+	0.942905i	$0.608082\pi$
$632$	27.4267	1.09097
$633$	0	0
$634$	32.7021	1.29877
$635$	3.99879	0.158687
$636$	0	0
$637$	−45.6291	−1.80789
$638$	−3.06355	−0.121287
$639$	0	0
$640$	63.2768	2.50124
$641$	31.8084	1.25636	0.628179	−	0.778069i	$-0.283801\pi$
$641$	31.8084	1.25636	0.628179	+	0.778069i	$0.283801\pi$
$642$	0	0
$643$	20.4840	0.807812	0.403906	−	0.914801i	$-0.367652\pi$
$643$	20.4840	0.807812	0.403906	+	0.914801i	$0.367652\pi$
$644$	4.84442	0.190897
$645$	0	0
$646$	0.554622	0.0218213
$647$	30.4688	1.19785	0.598927	−	0.800804i	$-0.295594\pi$
$647$	30.4688	1.19785	0.598927	+	0.800804i	$0.295594\pi$
$648$	0	0
$649$	1.27479	0.0500399
$650$	−205.292	−8.05221
$651$	0	0
$652$	−14.0272	−0.549348
$653$	40.7130	1.59322	0.796612	−	0.604491i	$-0.206623\pi$
$653$	40.7130	1.59322	0.796612	+	0.604491i	$0.206623\pi$
$654$	0	0
$655$	30.2290	1.18115
$656$	−5.22193	−0.203882
$657$	0	0
$658$	−0.764936	−0.0298203
$659$	−34.9149	−1.36009	−0.680045	−	0.733170i	$-0.738040\pi$
$659$	−34.9149	−1.36009	−0.680045	+	0.733170i	$0.738040\pi$
$660$	0	0
$661$	24.4554	0.951204	0.475602	−	0.879661i	$-0.342230\pi$
$661$	24.4554	0.951204	0.475602	+	0.879661i	$0.342230\pi$
$662$	53.7603	2.08945
$663$	0	0
$664$	−27.9831	−1.08595
$665$	−1.30579	−0.0506365
$666$	0	0
$667$	−11.5270	−0.446328
$668$	−63.8951	−2.47218
$669$	0	0
$670$	−84.4186	−3.26138
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−29.9385	−1.15404	−0.577021	−	0.816729i	$-0.695785\pi$
$673$	−29.9385	−1.15404	−0.577021	+	0.816729i	$0.695785\pi$
$674$	55.1249	2.12333
$675$	0	0
$676$	87.2472	3.35566
$677$	−29.0393	−1.11607	−0.558036	−	0.829817i	$-0.688445\pi$
$677$	−29.0393	−1.11607	−0.558036	+	0.829817i	$0.688445\pi$
$678$	0	0
$679$	1.21235	0.0465259
$680$	−1.47685	−0.0566346
$681$	0	0
$682$	−4.14510	−0.158724
$683$	−19.4790	−0.745342	−0.372671	−	0.927964i	$-0.621558\pi$
$683$	−19.4790	−0.745342	−0.372671	+	0.927964i	$0.621558\pi$
$684$	0	0
$685$	62.4837	2.38738
$686$	6.17576	0.235792
$687$	0	0
$688$	14.9840	0.571260
$689$	54.4133	2.07298
$690$	0	0
$691$	−15.6764	−0.596357	−0.298179	−	0.954510i	$-0.596379\pi$
$691$	−15.6764	−0.596357	−0.298179	+	0.954510i	$0.596379\pi$
$692$	22.1655	0.842606
$693$	0	0
$694$	7.54807	0.286521
$695$	11.6559	0.442134
$696$	0	0
$697$	0.647184	0.0245138
$698$	29.2642	1.10767
$699$	0	0
$700$	8.21076	0.310337
$701$	−8.39143	−0.316940	−0.158470	−	0.987364i	$-0.550656\pi$
$701$	−8.39143	−0.316940	−0.158470	+	0.987364i	$0.550656\pi$
$702$	0	0
$703$	−13.1318	−0.495276
$704$	12.8709	0.485091
$705$	0	0
$706$	43.5876	1.64044
$707$	2.29163	0.0861857
$708$	0	0
$709$	−41.4966	−1.55844	−0.779218	−	0.626753i	$-0.784384\pi$
$709$	−41.4966	−1.55844	−0.779218	+	0.626753i	$0.784384\pi$
$710$	132.748	4.98195
$711$	0	0
$712$	3.24355	0.121557
$713$	−15.5965	−0.584094
$714$	0	0
$715$	−28.6747	−1.07237
$716$	−13.7751	−0.514798
$717$	0	0
$718$	73.1187	2.72877
$719$	7.00388	0.261201	0.130600	−	0.991435i	$-0.458310\pi$
$719$	7.00388	0.261201	0.130600	+	0.991435i	$0.458310\pi$
$720$	0	0
$721$	0.0821284	0.00305862
$722$	37.1475	1.38249
$723$	0	0
$724$	35.2821	1.31125
$725$	−19.5370	−0.725587
$726$	0	0
$727$	16.9284	0.627841	0.313921	−	0.949449i	$-0.398357\pi$
$727$	16.9284	0.627841	0.313921	+	0.949449i	$0.398357\pi$
$728$	−2.64024	−0.0978537
$729$	0	0
$730$	66.1810	2.44947
$731$	−1.85705	−0.0686856
$732$	0	0
$733$	40.5440	1.49753	0.748763	−	0.662838i	$-0.230648\pi$
$733$	40.5440	1.49753	0.748763	+	0.662838i	$0.230648\pi$
$734$	4.55205	0.168019
$735$	0	0
$736$	58.5914	2.15971
$737$	−8.70975	−0.320828
$738$	0	0
$739$	−32.2921	−1.18788	−0.593942	−	0.804508i	$-0.702429\pi$
$739$	−32.2921	−1.18788	−0.593942	+	0.804508i	$0.702429\pi$
$740$	111.788	4.10940
$741$	0	0
$742$	−3.67183	−0.134797
$743$	34.0107	1.24773	0.623866	−	0.781531i	$-0.285561\pi$
$743$	34.0107	1.24773	0.623866	+	0.781531i	$0.285561\pi$
$744$	0	0
$745$	55.4236	2.03056
$746$	7.61481	0.278798
$747$	0	0
$748$	−0.487117	−0.0178108
$749$	2.60794	0.0952922
$750$	0	0
$751$	12.0090	0.438214	0.219107	−	0.975701i	$-0.429686\pi$
$751$	12.0090	0.438214	0.219107	+	0.975701i	$0.429686\pi$
$752$	−2.33514	−0.0851537
$753$	0	0
$754$	20.0839	0.731412
$755$	−58.4854	−2.12850
$756$	0	0
$757$	32.2357	1.17163	0.585814	−	0.810446i	$-0.300775\pi$
$757$	32.2357	1.17163	0.585814	+	0.810446i	$0.300775\pi$
$758$	−44.6185	−1.62062
$759$	0	0
$760$	13.1949	0.478630
$761$	4.61746	0.167383	0.0836914	−	0.996492i	$-0.473329\pi$
$761$	4.61746	0.167383	0.0836914	+	0.996492i	$0.473329\pi$
$762$	0	0
$763$	−3.69278	−0.133688
$764$	8.51148	0.307935
$765$	0	0
$766$	−50.7631	−1.83415
$767$	−8.35723	−0.301762
$768$	0	0
$769$	−30.8074	−1.11094	−0.555472	−	0.831535i	$-0.687463\pi$
$769$	−30.8074	−1.11094	−0.555472	+	0.831535i	$0.687463\pi$
$770$	1.93498	0.0697318
$771$	0	0
$772$	−36.5528	−1.31556
$773$	20.2468	0.728227	0.364113	−	0.931355i	$-0.381372\pi$
$773$	20.2468	0.728227	0.364113	+	0.931355i	$0.381372\pi$
$774$	0	0
$775$	−26.4343	−0.949550
$776$	−12.2507	−0.439775
$777$	0	0
$778$	−23.9302	−0.857940
$779$	−5.78226	−0.207171
$780$	0	0
$781$	13.6961	0.490084
$782$	−3.09237	−0.110583
$783$	0	0
$784$	9.39954	0.335698
$785$	−98.6335	−3.52038
$786$	0	0
$787$	−32.6493	−1.16382	−0.581910	−	0.813253i	$-0.697694\pi$
$787$	−32.6493	−1.16382	−0.581910	+	0.813253i	$0.697694\pi$
$788$	−34.9023	−1.24334
$789$	0	0
$790$	131.774	4.68831
$791$	−0.858508	−0.0305250
$792$	0	0
$793$	6.55576	0.232802
$794$	−71.9217	−2.55241
$795$	0	0
$796$	17.1872	0.609185
$797$	27.1829	0.962867	0.481433	−	0.876483i	$-0.340116\pi$
$797$	27.1829	0.962867	0.481433	+	0.876483i	$0.340116\pi$
$798$	0	0
$799$	0.289407	0.0102385
$800$	99.3059	3.51099
$801$	0	0
$802$	27.3964	0.967402
$803$	6.82811	0.240959
$804$	0	0
$805$	7.28063	0.256608
$806$	27.1743	0.957173
$807$	0	0
$808$	−23.1567	−0.814651
$809$	20.2140	0.710688	0.355344	−	0.934736i	$-0.384364\pi$
$809$	20.2140	0.710688	0.355344	+	0.934736i	$0.384364\pi$
$810$	0	0
$811$	−36.9453	−1.29732	−0.648662	−	0.761077i	$-0.724671\pi$
$811$	−36.9453	−1.29732	−0.648662	+	0.761077i	$0.724671\pi$
$812$	−0.803266	−0.0281891
$813$	0	0
$814$	19.4593	0.682048
$815$	−21.0814	−0.738448
$816$	0	0
$817$	16.5918	0.580475
$818$	1.95245	0.0682658
$819$	0	0
$820$	49.2228	1.71894
$821$	−21.4004	−0.746878	−0.373439	−	0.927655i	$-0.621822\pi$
$821$	−21.4004	−0.746878	−0.373439	+	0.927655i	$0.621822\pi$
$822$	0	0
$823$	1.02414	0.0356993	0.0178497	−	0.999841i	$-0.494318\pi$
$823$	1.02414	0.0356993	0.0178497	+	0.999841i	$0.494318\pi$
$824$	−0.829900	−0.0289109
$825$	0	0
$826$	0.563949	0.0196223
$827$	−1.01657	−0.0353495	−0.0176747	−	0.999844i	$-0.505626\pi$
$827$	−1.01657	−0.0353495	−0.0176747	+	0.999844i	$0.505626\pi$
$828$	0	0
$829$	10.4037	0.361337	0.180669	−	0.983544i	$-0.442174\pi$
$829$	10.4037	0.361337	0.180669	+	0.983544i	$0.442174\pi$
$830$	−134.447	−4.66674
$831$	0	0
$832$	−84.3788	−2.92531
$833$	−1.16494	−0.0403628
$834$	0	0
$835$	−96.0273	−3.32316
$836$	4.35215	0.150522
$837$	0	0
$838$	12.9979	0.449003
$839$	−17.5516	−0.605947	−0.302973	−	0.952999i	$-0.597979\pi$
$839$	−17.5516	−0.605947	−0.302973	+	0.952999i	$0.597979\pi$
$840$	0	0
$841$	−27.0887	−0.934092
$842$	27.5774	0.950380
$843$	0	0
$844$	21.1662	0.728569
$845$	131.123	4.51077
$846$	0	0
$847$	0.199638	0.00685965
$848$	−11.2091	−0.384922
$849$	0	0
$850$	−5.24123	−0.179773
$851$	73.2183	2.50989
$852$	0	0
$853$	−25.8638	−0.885558	−0.442779	−	0.896631i	$-0.646007\pi$
$853$	−25.8638	−0.885558	−0.442779	+	0.896631i	$0.646007\pi$
$854$	−0.442385	−0.0151381
$855$	0	0
$856$	−26.3530	−0.900727
$857$	8.12435	0.277522	0.138761	−	0.990326i	$-0.455688\pi$
$857$	8.12435	0.277522	0.138761	+	0.990326i	$0.455688\pi$
$858$	0	0
$859$	−43.2781	−1.47663	−0.738315	−	0.674456i	$-0.764378\pi$
$859$	−43.2781	−1.47663	−0.738315	+	0.674456i	$0.764378\pi$
$860$	−141.242	−4.81631
$861$	0	0
$862$	30.3025	1.03211
$863$	18.4292	0.627338	0.313669	−	0.949532i	$-0.398442\pi$
$863$	18.4292	0.627338	0.313669	+	0.949532i	$0.398442\pi$
$864$	0	0
$865$	33.3123	1.13265
$866$	15.3667	0.522181
$867$	0	0
$868$	−1.08685	−0.0368901
$869$	13.5956	0.461198
$870$	0	0
$871$	57.0991	1.93473
$872$	37.3152	1.26365
$873$	0	0
$874$	27.6288	0.934558
$875$	7.97381	0.269564
$876$	0	0
$877$	10.3096	0.348131	0.174066	−	0.984734i	$-0.444310\pi$
$877$	10.3096	0.348131	0.174066	+	0.984734i	$0.444310\pi$
$878$	−41.2157	−1.39096
$879$	0	0
$880$	5.90696	0.199124
$881$	−14.8142	−0.499104	−0.249552	−	0.968361i	$-0.580283\pi$
$881$	−14.8142	−0.499104	−0.249552	+	0.968361i	$0.580283\pi$
$882$	0	0
$883$	23.4486	0.789107	0.394554	−	0.918873i	$-0.370899\pi$
$883$	23.4486	0.789107	0.394554	+	0.918873i	$0.370899\pi$
$884$	3.19343	0.107407
$885$	0	0
$886$	−31.2758	−1.05073
$887$	−9.12803	−0.306489	−0.153245	−	0.988188i	$-0.548972\pi$
$887$	−9.12803	−0.306489	−0.153245	+	0.988188i	$0.548972\pi$
$888$	0	0
$889$	0.182514	0.00612133
$890$	15.5840	0.522376
$891$	0	0
$892$	48.0818	1.60990
$893$	−2.58571	−0.0865274
$894$	0	0
$895$	−20.7024	−0.692005
$896$	2.88810	0.0964848
$897$	0	0
$898$	−1.43852	−0.0480041
$899$	2.58610	0.0862512
$900$	0	0
$901$	1.38921	0.0462812
$902$	8.56840	0.285297
$903$	0	0
$904$	8.67514	0.288531
$905$	53.0251	1.76261
$906$	0	0
$907$	12.7594	0.423670	0.211835	−	0.977305i	$-0.432056\pi$
$907$	12.7594	0.423670	0.211835	+	0.977305i	$0.432056\pi$
$908$	−67.1141	−2.22726
$909$	0	0
$910$	−12.6853	−0.420513
$911$	−46.5307	−1.54163	−0.770815	−	0.637059i	$-0.780151\pi$
$911$	−46.5307	−1.54163	−0.770815	+	0.637059i	$0.780151\pi$
$912$	0	0
$913$	−13.8714	−0.459076
$914$	64.0149	2.11743
$915$	0	0
$916$	21.4781	0.709656
$917$	1.37972	0.0455625
$918$	0	0
$919$	22.3991	0.738877	0.369439	−	0.929255i	$-0.379550\pi$
$919$	22.3991	0.738877	0.369439	+	0.929255i	$0.379550\pi$
$920$	−73.5701	−2.42553
$921$	0	0
$922$	39.7805	1.31010
$923$	−89.7882	−2.95541
$924$	0	0
$925$	124.097	4.08028
$926$	−51.1046	−1.67940
$927$	0	0
$928$	−9.71519	−0.318917
$929$	35.4953	1.16456	0.582282	−	0.812987i	$-0.302160\pi$
$929$	35.4953	1.16456	0.582282	+	0.812987i	$0.302160\pi$
$930$	0	0
$931$	10.4081	0.341113
$932$	−45.4472	−1.48867
$933$	0	0
$934$	11.6326	0.380631
$935$	−0.732083	−0.0239417
$936$	0	0
$937$	18.7095	0.611213	0.305606	−	0.952158i	$-0.401141\pi$
$937$	18.7095	0.611213	0.305606	+	0.952158i	$0.401141\pi$
$938$	−3.85307	−0.125807
$939$	0	0
$940$	22.0114	0.717933
$941$	−35.2603	−1.14945	−0.574726	−	0.818346i	$-0.694891\pi$
$941$	−35.2603	−1.14945	−0.574726	+	0.818346i	$0.694891\pi$
$942$	0	0
$943$	32.2398	1.04987
$944$	1.72158	0.0560327
$945$	0	0
$946$	−24.5865	−0.799376
$947$	46.0805	1.49741	0.748707	−	0.662901i	$-0.230675\pi$
$947$	46.0805	1.49741	0.748707	+	0.662901i	$0.230675\pi$
$948$	0	0
$949$	−44.7635	−1.45308
$950$	46.8278	1.51929
$951$	0	0
$952$	−0.0674069	−0.00218467
$953$	55.4806	1.79719	0.898596	−	0.438777i	$-0.144588\pi$
$953$	55.4806	1.79719	0.898596	+	0.438777i	$0.144588\pi$
$954$	0	0
$955$	12.7918	0.413934
$956$	−8.98429	−0.290573
$957$	0	0
$958$	−75.3798	−2.43541
$959$	2.85190	0.0920928
$960$	0	0
$961$	−27.5009	−0.887126
$962$	−127.571	−4.11304
$963$	0	0
$964$	−17.3205	−0.557856
$965$	−54.9349	−1.76842
$966$	0	0
$967$	−36.9590	−1.18852	−0.594261	−	0.804272i	$-0.702556\pi$
$967$	−36.9590	−1.18852	−0.594261	+	0.804272i	$0.702556\pi$
$968$	−2.01732	−0.0648393
$969$	0	0
$970$	−58.8598	−1.88987
$971$	−1.63235	−0.0523847	−0.0261924	−	0.999657i	$-0.508338\pi$
$971$	−1.63235	−0.0523847	−0.0261924	+	0.999657i	$0.508338\pi$
$972$	0	0
$973$	0.532003	0.0170552
$974$	−82.2672	−2.63601
$975$	0	0
$976$	−1.35048	−0.0432278
$977$	42.7546	1.36784	0.683921	−	0.729556i	$-0.260273\pi$
$977$	42.7546	1.36784	0.683921	+	0.729556i	$0.260273\pi$
$978$	0	0
$979$	1.60785	0.0513871
$980$	−88.6017	−2.83028
$981$	0	0
$982$	48.3833	1.54397
$983$	5.28988	0.168721	0.0843604	−	0.996435i	$-0.473115\pi$
$983$	5.28988	0.168721	0.0843604	+	0.996435i	$0.473115\pi$
$984$	0	0
$985$	−52.4544	−1.67133
$986$	0.512755	0.0163294
$987$	0	0
$988$	−28.5317	−0.907713
$989$	−92.5101	−2.94165
$990$	0	0
$991$	−57.8840	−1.83875	−0.919373	−	0.393387i	$-0.871303\pi$
$991$	−57.8840	−1.83875	−0.919373	+	0.393387i	$0.871303\pi$
$992$	−13.1450	−0.417355
$993$	0	0
$994$	6.05894	0.192178
$995$	25.8305	0.818882
$996$	0	0
$997$	3.61825	0.114591	0.0572956	−	0.998357i	$-0.481752\pi$
$997$	3.61825	0.114591	0.0572956	+	0.998357i	$0.481752\pi$
$998$	−94.4085	−2.98845
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.3	yes	25
3.2	odd	2		6039.2.a.m.1.23	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.23	✓	25	3.2	odd	2
6039.2.a.p.1.3	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)

\(f(q)\)	\(=\)	\(q-2.21594 q^{2} +2.91037 q^{4} +4.37397 q^{5} +0.199638 q^{7} -2.01732 q^{8} +O(q^{10})\)	\(q-2.21594 q^{2} +2.91037 q^{4} +4.37397 q^{5} +0.199638 q^{7} -2.01732 q^{8} -9.69243 q^{10} -1.00000 q^{11} +6.55576 q^{13} -0.442385 q^{14} -1.35048 q^{16} +0.167373 q^{17} -1.49539 q^{19} +12.7299 q^{20} +2.21594 q^{22} +8.33777 q^{23} +14.1316 q^{25} -14.5272 q^{26} +0.581021 q^{28} -1.38251 q^{29} -1.87059 q^{31} +7.02723 q^{32} -0.370888 q^{34} +0.873211 q^{35} +8.78153 q^{37} +3.31369 q^{38} -8.82371 q^{40} +3.86672 q^{41} -11.0953 q^{43} -2.91037 q^{44} -18.4760 q^{46} +1.72912 q^{47} -6.96014 q^{49} -31.3147 q^{50} +19.0797 q^{52} +8.30007 q^{53} -4.37397 q^{55} -0.402735 q^{56} +3.06355 q^{58} -1.27479 q^{59} +1.00000 q^{61} +4.14510 q^{62} -12.8709 q^{64} +28.6747 q^{65} +8.70975 q^{67} +0.487117 q^{68} -1.93498 q^{70} -13.6961 q^{71} -6.82811 q^{73} -19.4593 q^{74} -4.35215 q^{76} -0.199638 q^{77} -13.5956 q^{79} -5.90696 q^{80} -8.56840 q^{82} +13.8714 q^{83} +0.732083 q^{85} +24.5865 q^{86} +2.01732 q^{88} -1.60785 q^{89} +1.30878 q^{91} +24.2660 q^{92} -3.83161 q^{94} -6.54080 q^{95} +6.07276 q^{97} +15.4232 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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

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Database

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