LMFDB - Embedded newform 6039.2.a.p.1.8

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			1.8
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)

$f(q)$	$=$	$q-1.04320 q^{2} -0.911728 q^{4} -1.64856 q^{5} +5.01087 q^{7} +3.03752 q^{8} +O(q^{10})$	\(q-1.04320 q^{2} -0.911728 q^{4} -1.64856 q^{5} +5.01087 q^{7} +3.03752 q^{8} +1.71978 q^{10} -1.00000 q^{11} -3.01798 q^{13} -5.22736 q^{14} -1.34530 q^{16} +3.18804 q^{17} -6.03685 q^{19} +1.50303 q^{20} +1.04320 q^{22} +4.89559 q^{23} -2.28226 q^{25} +3.14836 q^{26} -4.56855 q^{28} +0.904141 q^{29} +5.23680 q^{31} -4.67163 q^{32} -3.32577 q^{34} -8.26071 q^{35} -5.38300 q^{37} +6.29766 q^{38} -5.00753 q^{40} +11.7027 q^{41} -11.2235 q^{43} +0.911728 q^{44} -5.10709 q^{46} +7.10722 q^{47} +18.1089 q^{49} +2.38086 q^{50} +2.75158 q^{52} +13.0696 q^{53} +1.64856 q^{55} +15.2206 q^{56} -0.943203 q^{58} -0.468335 q^{59} +1.00000 q^{61} -5.46305 q^{62} +7.56405 q^{64} +4.97531 q^{65} -3.15644 q^{67} -2.90662 q^{68} +8.61759 q^{70} -9.71441 q^{71} -3.32256 q^{73} +5.61556 q^{74} +5.50397 q^{76} -5.01087 q^{77} -5.59101 q^{79} +2.21780 q^{80} -12.2083 q^{82} +0.444183 q^{83} -5.25566 q^{85} +11.7084 q^{86} -3.03752 q^{88} -3.27073 q^{89} -15.1227 q^{91} -4.46345 q^{92} -7.41427 q^{94} +9.95209 q^{95} +7.38020 q^{97} -18.8912 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.04320	−0.737656	−0.368828	−	0.929498i	$-0.620241\pi$
$2$	−1.04320	−0.737656	−0.368828	+	0.929498i	$0.620241\pi$
$3$	0	0
$3$	0	0
$4$	−0.911728	−0.455864
$5$	−1.64856	−0.737257	−0.368628	−	0.929577i	$-0.620172\pi$
$5$	−1.64856	−0.737257	−0.368628	+	0.929577i	$0.620172\pi$
$6$	0	0
$7$	5.01087	1.89393	0.946966	−	0.321334i	$-0.104131\pi$
$7$	5.01087	1.89393	0.946966	+	0.321334i	$0.104131\pi$
$8$	3.03752	1.07393
$9$	0	0
$10$	1.71978	0.543842
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.01798	−0.837037	−0.418518	−	0.908208i	$-0.637450\pi$
$13$	−3.01798	−0.837037	−0.418518	+	0.908208i	$0.637450\pi$
$14$	−5.22736	−1.39707
$15$	0	0
$16$	−1.34530	−0.336324
$17$	3.18804	0.773213	0.386606	−	0.922245i	$-0.373647\pi$
$17$	3.18804	0.773213	0.386606	+	0.922245i	$0.373647\pi$
$18$	0	0
$19$	−6.03685	−1.38495	−0.692475	−	0.721442i	$-0.743480\pi$
$19$	−6.03685	−1.38495	−0.692475	+	0.721442i	$0.743480\pi$
$20$	1.50303	0.336089
$21$	0	0
$22$	1.04320	0.222412
$23$	4.89559	1.02080	0.510401	−	0.859937i	$-0.329497\pi$
$23$	4.89559	1.02080	0.510401	+	0.859937i	$0.329497\pi$
$24$	0	0
$25$	−2.28226	−0.456453
$26$	3.14836	0.617445
$27$	0	0
$28$	−4.56855	−0.863375
$29$	0.904141	0.167895	0.0839474	−	0.996470i	$-0.473247\pi$
$29$	0.904141	0.167895	0.0839474	+	0.996470i	$0.473247\pi$
$30$	0	0
$31$	5.23680	0.940558	0.470279	−	0.882518i	$-0.344153\pi$
$31$	5.23680	0.940558	0.470279	+	0.882518i	$0.344153\pi$
$32$	−4.67163	−0.825835
$33$	0	0
$34$	−3.32577	−0.570365
$35$	−8.26071	−1.39631
$36$	0	0
$37$	−5.38300	−0.884960	−0.442480	−	0.896778i	$-0.645901\pi$
$37$	−5.38300	−0.884960	−0.442480	+	0.896778i	$0.645901\pi$
$38$	6.29766	1.02162
$39$	0	0
$40$	−5.00753	−0.791759
$41$	11.7027	1.82766	0.913828	−	0.406101i	$-0.133112\pi$
$41$	11.7027	1.82766	0.913828	+	0.406101i	$0.133112\pi$
$42$	0	0
$43$	−11.2235	−1.71157	−0.855785	−	0.517331i	$-0.826926\pi$
$43$	−11.2235	−1.71157	−0.855785	+	0.517331i	$0.826926\pi$
$44$	0.911728	0.137448
$45$	0	0
$46$	−5.10709	−0.753000
$47$	7.10722	1.03669	0.518347	−	0.855170i	$-0.326548\pi$
$47$	7.10722	1.03669	0.518347	+	0.855170i	$0.326548\pi$
$48$	0	0
$49$	18.1089	2.58698
$50$	2.38086	0.336705
$51$	0	0
$52$	2.75158	0.381575
$53$	13.0696	1.79525	0.897625	−	0.440760i	$-0.145291\pi$
$53$	13.0696	1.79525	0.897625	+	0.440760i	$0.145291\pi$
$54$	0	0
$55$	1.64856	0.222291
$56$	15.2206	2.03394
$57$	0	0
$58$	−0.943203	−0.123849
$59$	−0.468335	−0.0609720	−0.0304860	−	0.999535i	$-0.509705\pi$
$59$	−0.468335	−0.0609720	−0.0304860	+	0.999535i	$0.509705\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−5.46305	−0.693808
$63$	0	0
$64$	7.56405	0.945506
$65$	4.97531	0.617111
$66$	0	0
$67$	−3.15644	−0.385621	−0.192810	−	0.981236i	$-0.561760\pi$
$67$	−3.15644	−0.385621	−0.192810	+	0.981236i	$0.561760\pi$
$68$	−2.90662	−0.352480
$69$	0	0
$70$	8.61759	1.03000
$71$	−9.71441	−1.15289	−0.576444	−	0.817136i	$-0.695560\pi$
$71$	−9.71441	−1.15289	−0.576444	+	0.817136i	$0.695560\pi$
$72$	0	0
$73$	−3.32256	−0.388877	−0.194438	−	0.980915i	$-0.562288\pi$
$73$	−3.32256	−0.388877	−0.194438	+	0.980915i	$0.562288\pi$
$74$	5.61556	0.652796
$75$	0	0
$76$	5.50397	0.631348
$77$	−5.01087	−0.571042
$78$	0	0
$79$	−5.59101	−0.629038	−0.314519	−	0.949251i	$-0.601843\pi$
$79$	−5.59101	−0.629038	−0.314519	+	0.949251i	$0.601843\pi$
$80$	2.21780	0.247957
$81$	0	0
$82$	−12.2083	−1.34818
$83$	0.444183	0.0487554	0.0243777	−	0.999703i	$-0.492240\pi$
$83$	0.444183	0.0487554	0.0243777	+	0.999703i	$0.492240\pi$
$84$	0	0
$85$	−5.25566	−0.570056
$86$	11.7084	1.26255
$87$	0	0
$88$	−3.03752	−0.323801
$89$	−3.27073	−0.346697	−0.173348	−	0.984861i	$-0.555459\pi$
$89$	−3.27073	−0.346697	−0.173348	+	0.984861i	$0.555459\pi$
$90$	0	0
$91$	−15.1227	−1.58529
$92$	−4.46345	−0.465347
$93$	0	0
$94$	−7.41427	−0.764723
$95$	9.95209	1.02106
$96$	0	0
$97$	7.38020	0.749346	0.374673	−	0.927157i	$-0.377755\pi$
$97$	7.38020	0.749346	0.374673	+	0.927157i	$0.377755\pi$
$98$	−18.8912	−1.90830
$99$	0	0
$100$	2.08080	0.208080
$101$	−1.95613	−0.194643	−0.0973213	−	0.995253i	$-0.531027\pi$
$101$	−1.95613	−0.194643	−0.0973213	+	0.995253i	$0.531027\pi$
$102$	0	0
$103$	4.90048	0.482858	0.241429	−	0.970418i	$-0.422384\pi$
$103$	4.90048	0.482858	0.241429	+	0.970418i	$0.422384\pi$
$104$	−9.16718	−0.898916
$105$	0	0
$106$	−13.6343	−1.32428
$107$	1.67145	0.161585	0.0807927	−	0.996731i	$-0.474255\pi$
$107$	1.67145	0.161585	0.0807927	+	0.996731i	$0.474255\pi$
$108$	0	0
$109$	17.3299	1.65991	0.829954	−	0.557832i	$-0.188367\pi$
$109$	17.3299	1.65991	0.829954	+	0.557832i	$0.188367\pi$
$110$	−1.71978	−0.163974
$111$	0	0
$112$	−6.74111	−0.636975
$113$	−8.43203	−0.793219	−0.396609	−	0.917987i	$-0.629813\pi$
$113$	−8.43203	−0.793219	−0.396609	+	0.917987i	$0.629813\pi$
$114$	0	0
$115$	−8.07066	−0.752593
$116$	−0.824331	−0.0765372
$117$	0	0
$118$	0.488568	0.0449763
$119$	15.9748	1.46441
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.04320	−0.0944471
$123$	0	0
$124$	−4.77454	−0.428766
$125$	12.0052	1.07378
$126$	0	0
$127$	14.7903	1.31242	0.656212	−	0.754577i	$-0.272158\pi$
$127$	14.7903	1.31242	0.656212	+	0.754577i	$0.272158\pi$
$128$	1.45242	0.128377
$129$	0	0
$130$	−5.19025	−0.455215
$131$	0.343726	0.0300315	0.0150157	−	0.999887i	$-0.495220\pi$
$131$	0.343726	0.0300315	0.0150157	+	0.999887i	$0.495220\pi$
$132$	0	0
$133$	−30.2499	−2.62300
$134$	3.29281	0.284455
$135$	0	0
$136$	9.68373	0.830373
$137$	−13.7289	−1.17294	−0.586469	−	0.809972i	$-0.699482\pi$
$137$	−13.7289	−1.17294	−0.586469	+	0.809972i	$0.699482\pi$
$138$	0	0
$139$	−16.3286	−1.38498	−0.692488	−	0.721429i	$-0.743486\pi$
$139$	−16.3286	−1.38498	−0.692488	+	0.721429i	$0.743486\pi$
$140$	7.53152	0.636529
$141$	0	0
$142$	10.1341	0.850435
$143$	3.01798	0.252376
$144$	0	0
$145$	−1.49053	−0.123782
$146$	3.46611	0.286857
$147$	0	0
$148$	4.90783	0.403421
$149$	0.633092	0.0518649	0.0259325	−	0.999664i	$-0.491745\pi$
$149$	0.633092	0.0518649	0.0259325	+	0.999664i	$0.491745\pi$
$150$	0	0
$151$	−10.3808	−0.844777	−0.422388	−	0.906415i	$-0.638808\pi$
$151$	−10.3808	−0.844777	−0.422388	+	0.906415i	$0.638808\pi$
$152$	−18.3371	−1.48733
$153$	0	0
$154$	5.22736	0.421232
$155$	−8.63316	−0.693432
$156$	0	0
$157$	20.2141	1.61326	0.806631	−	0.591056i	$-0.201289\pi$
$157$	20.2141	1.61326	0.806631	+	0.591056i	$0.201289\pi$
$158$	5.83256	0.464013
$159$	0	0
$160$	7.70144	0.608852
$161$	24.5312	1.93333
$162$	0	0
$163$	11.6042	0.908909	0.454455	−	0.890770i	$-0.349834\pi$
$163$	11.6042	0.908909	0.454455	+	0.890770i	$0.349834\pi$
$164$	−10.6697	−0.833163
$165$	0	0
$166$	−0.463373	−0.0359647
$167$	12.8356	0.993252	0.496626	−	0.867965i	$-0.334572\pi$
$167$	12.8356	0.993252	0.496626	+	0.867965i	$0.334572\pi$
$168$	0	0
$169$	−3.89181	−0.299370
$170$	5.48272	0.420505
$171$	0	0
$172$	10.2328	0.780243
$173$	−13.6455	−1.03745	−0.518726	−	0.854941i	$-0.673593\pi$
$173$	−13.6455	−1.03745	−0.518726	+	0.854941i	$0.673593\pi$
$174$	0	0
$175$	−11.4361	−0.864490
$176$	1.34530	0.101406
$177$	0	0
$178$	3.41204	0.255743
$179$	4.63889	0.346727	0.173364	−	0.984858i	$-0.444536\pi$
$179$	4.63889	0.346727	0.173364	+	0.984858i	$0.444536\pi$
$180$	0	0
$181$	16.0481	1.19284	0.596422	−	0.802671i	$-0.296589\pi$
$181$	16.0481	1.19284	0.596422	+	0.802671i	$0.296589\pi$
$182$	15.7761	1.16940
$183$	0	0
$184$	14.8705	1.09627
$185$	8.87418	0.652443
$186$	0	0
$187$	−3.18804	−0.233132
$188$	−6.47985	−0.472591
$189$	0	0
$190$	−10.3821	−0.753193
$191$	−18.4234	−1.33307	−0.666536	−	0.745473i	$-0.732224\pi$
$191$	−18.4234	−1.33307	−0.666536	+	0.745473i	$0.732224\pi$
$192$	0	0
$193$	−17.6427	−1.26995	−0.634976	−	0.772532i	$-0.718990\pi$
$193$	−17.6427	−1.26995	−0.634976	+	0.772532i	$0.718990\pi$
$194$	−7.69904	−0.552759
$195$	0	0
$196$	−16.5103	−1.17931
$197$	−17.6732	−1.25916	−0.629581	−	0.776935i	$-0.716773\pi$
$197$	−17.6732	−1.25916	−0.629581	+	0.776935i	$0.716773\pi$
$198$	0	0
$199$	−13.5925	−0.963544	−0.481772	−	0.876297i	$-0.660007\pi$
$199$	−13.5925	−0.963544	−0.481772	+	0.876297i	$0.660007\pi$
$200$	−6.93242	−0.490196
$201$	0	0
$202$	2.04065	0.143579
$203$	4.53054	0.317981
$204$	0	0
$205$	−19.2926	−1.34745
$206$	−5.11219	−0.356183
$207$	0	0
$208$	4.06008	0.281516
$209$	6.03685	0.417578
$210$	0	0
$211$	19.5411	1.34527	0.672633	−	0.739976i	$-0.265163\pi$
$211$	19.5411	1.34527	0.672633	+	0.739976i	$0.265163\pi$
$212$	−11.9159	−0.818390
$213$	0	0
$214$	−1.74366	−0.119194
$215$	18.5026	1.26187
$216$	0	0
$217$	26.2410	1.78135
$218$	−18.0786	−1.22444
$219$	0	0
$220$	−1.50303	−0.101335
$221$	−9.62143	−0.647207
$222$	0	0
$223$	−11.7071	−0.783966	−0.391983	−	0.919972i	$-0.628211\pi$
$223$	−11.7071	−0.783966	−0.391983	+	0.919972i	$0.628211\pi$
$224$	−23.4089	−1.56408
$225$	0	0
$226$	8.79632	0.585123
$227$	−8.66900	−0.575382	−0.287691	−	0.957723i	$-0.592888\pi$
$227$	−8.66900	−0.575382	−0.287691	+	0.957723i	$0.592888\pi$
$228$	0	0
$229$	1.24221	0.0820876	0.0410438	−	0.999157i	$-0.486932\pi$
$229$	1.24221	0.0820876	0.0410438	+	0.999157i	$0.486932\pi$
$230$	8.41933	0.555154
$231$	0	0
$232$	2.74635	0.180307
$233$	−8.15946	−0.534544	−0.267272	−	0.963621i	$-0.586122\pi$
$233$	−8.15946	−0.534544	−0.267272	+	0.963621i	$0.586122\pi$
$234$	0	0
$235$	−11.7166	−0.764310
$236$	0.426994	0.0277949
$237$	0	0
$238$	−16.6650	−1.08023
$239$	29.0899	1.88167	0.940833	−	0.338870i	$-0.110045\pi$
$239$	29.0899	1.88167	0.940833	+	0.338870i	$0.110045\pi$
$240$	0	0
$241$	−14.5444	−0.936887	−0.468443	−	0.883493i	$-0.655185\pi$
$241$	−14.5444	−0.936887	−0.468443	+	0.883493i	$0.655185\pi$
$242$	−1.04320	−0.0670596
$243$	0	0
$244$	−0.911728	−0.0583674
$245$	−29.8535	−1.90727
$246$	0	0
$247$	18.2191	1.15925
$248$	15.9069	1.01009
$249$	0	0
$250$	−12.5239	−0.792080
$251$	−7.59268	−0.479246	−0.239623	−	0.970866i	$-0.577024\pi$
$251$	−7.59268	−0.479246	−0.239623	+	0.970866i	$0.577024\pi$
$252$	0	0
$253$	−4.89559	−0.307783
$254$	−15.4292	−0.968117
$255$	0	0
$256$	−16.6433	−1.04020
$257$	9.70834	0.605590	0.302795	−	0.953056i	$-0.402080\pi$
$257$	9.70834	0.605590	0.302795	+	0.953056i	$0.402080\pi$
$258$	0	0
$259$	−26.9735	−1.67605
$260$	−4.53613	−0.281319
$261$	0	0
$262$	−0.358576	−0.0221529
$263$	27.9737	1.72493	0.862465	−	0.506116i	$-0.168919\pi$
$263$	27.9737	1.72493	0.862465	+	0.506116i	$0.168919\pi$
$264$	0	0
$265$	−21.5460	−1.32356
$266$	31.5568	1.93487
$267$	0	0
$268$	2.87782	0.175791
$269$	5.48461	0.334403	0.167201	−	0.985923i	$-0.446527\pi$
$269$	5.48461	0.334403	0.167201	+	0.985923i	$0.446527\pi$
$270$	0	0
$271$	−8.15441	−0.495345	−0.247673	−	0.968844i	$-0.579666\pi$
$271$	−8.15441	−0.495345	−0.247673	+	0.968844i	$0.579666\pi$
$272$	−4.28885	−0.260050
$273$	0	0
$274$	14.3220	0.865225
$275$	2.28226	0.137626
$276$	0	0
$277$	15.1489	0.910209	0.455105	−	0.890438i	$-0.349602\pi$
$277$	15.1489	0.910209	0.455105	+	0.890438i	$0.349602\pi$
$278$	17.0341	1.02164
$279$	0	0
$280$	−25.0921	−1.49954
$281$	28.0473	1.67316	0.836581	−	0.547843i	$-0.184551\pi$
$281$	28.0473	1.67316	0.836581	+	0.547843i	$0.184551\pi$
$282$	0	0
$283$	12.2114	0.725892	0.362946	−	0.931810i	$-0.381771\pi$
$283$	12.2114	0.725892	0.362946	+	0.931810i	$0.381771\pi$
$284$	8.85690	0.525560
$285$	0	0
$286$	−3.14836	−0.186167
$287$	58.6408	3.46146
$288$	0	0
$289$	−6.83642	−0.402142
$290$	1.55492	0.0913082
$291$	0	0
$292$	3.02927	0.177275
$293$	10.3409	0.604122	0.302061	−	0.953289i	$-0.402325\pi$
$293$	10.3409	0.604122	0.302061	+	0.953289i	$0.402325\pi$
$294$	0	0
$295$	0.772076	0.0449520
$296$	−16.3510	−0.950382
$297$	0	0
$298$	−0.660443	−0.0382584
$299$	−14.7748	−0.854448
$300$	0	0
$301$	−56.2396	−3.24160
$302$	10.8293	0.623154
$303$	0	0
$304$	8.12136	0.465792
$305$	−1.64856	−0.0943960
$306$	0	0
$307$	27.4302	1.56553	0.782763	−	0.622320i	$-0.213810\pi$
$307$	27.4302	1.56553	0.782763	+	0.622320i	$0.213810\pi$
$308$	4.56855	0.260317
$309$	0	0
$310$	9.00614	0.511514
$311$	26.8048	1.51996	0.759982	−	0.649944i	$-0.225208\pi$
$311$	26.8048	1.51996	0.759982	+	0.649944i	$0.225208\pi$
$312$	0	0
$313$	18.6081	1.05179	0.525897	−	0.850548i	$-0.323730\pi$
$313$	18.6081	1.05179	0.525897	+	0.850548i	$0.323730\pi$
$314$	−21.0874	−1.19003
$315$	0	0
$316$	5.09748	0.286756
$317$	27.1197	1.52319	0.761597	−	0.648051i	$-0.224416\pi$
$317$	27.1197	1.52319	0.761597	+	0.648051i	$0.224416\pi$
$318$	0	0
$319$	−0.904141	−0.0506222
$320$	−12.4698	−0.697081
$321$	0	0
$322$	−25.5910	−1.42613
$323$	−19.2457	−1.07086
$324$	0	0
$325$	6.88782	0.382067
$326$	−12.1055	−0.670462
$327$	0	0
$328$	35.5473	1.96277
$329$	35.6134	1.96343
$330$	0	0
$331$	−14.5870	−0.801774	−0.400887	−	0.916128i	$-0.631298\pi$
$331$	−14.5870	−0.801774	−0.400887	+	0.916128i	$0.631298\pi$
$332$	−0.404974	−0.0222258
$333$	0	0
$334$	−13.3902	−0.732678
$335$	5.20357	0.284302
$336$	0	0
$337$	−2.98497	−0.162602	−0.0813008	−	0.996690i	$-0.525907\pi$
$337$	−2.98497	−0.162602	−0.0813008	+	0.996690i	$0.525907\pi$
$338$	4.05994	0.220832
$339$	0	0
$340$	4.79173	0.259868
$341$	−5.23680	−0.283589
$342$	0	0
$343$	55.6650	3.00563
$344$	−34.0917	−1.83810
$345$	0	0
$346$	14.2351	0.765282
$347$	23.5999	1.26691	0.633455	−	0.773780i	$-0.281636\pi$
$347$	23.5999	1.26691	0.633455	+	0.773780i	$0.281636\pi$
$348$	0	0
$349$	23.9001	1.27934	0.639672	−	0.768648i	$-0.279070\pi$
$349$	23.9001	1.27934	0.639672	+	0.768648i	$0.279070\pi$
$350$	11.9302	0.637696
$351$	0	0
$352$	4.67163	0.248999
$353$	34.4485	1.83351	0.916755	−	0.399449i	$-0.130799\pi$
$353$	34.4485	1.83351	0.916755	+	0.399449i	$0.130799\pi$
$354$	0	0
$355$	16.0148	0.849975
$356$	2.98202	0.158047
$357$	0	0
$358$	−4.83931	−0.255765
$359$	8.98660	0.474294	0.237147	−	0.971474i	$-0.423788\pi$
$359$	8.98660	0.474294	0.237147	+	0.971474i	$0.423788\pi$
$360$	0	0
$361$	17.4436	0.918085
$362$	−16.7414	−0.879908
$363$	0	0
$364$	13.7878	0.722677
$365$	5.47743	0.286702
$366$	0	0
$367$	12.8108	0.668717	0.334359	−	0.942446i	$-0.391480\pi$
$367$	12.8108	0.668717	0.334359	+	0.942446i	$0.391480\pi$
$368$	−6.58602	−0.343320
$369$	0	0
$370$	−9.25757	−0.481278
$371$	65.4902	3.40008
$372$	0	0
$373$	13.0797	0.677241	0.338621	−	0.940923i	$-0.390040\pi$
$373$	13.0797	0.677241	0.338621	+	0.940923i	$0.390040\pi$
$374$	3.32577	0.171971
$375$	0	0
$376$	21.5883	1.11333
$377$	−2.72868	−0.140534
$378$	0	0
$379$	2.92093	0.150038	0.0750192	−	0.997182i	$-0.476098\pi$
$379$	2.92093	0.150038	0.0750192	+	0.997182i	$0.476098\pi$
$380$	−9.07360	−0.465466
$381$	0	0
$382$	19.2194	0.983348
$383$	−9.46576	−0.483678	−0.241839	−	0.970316i	$-0.577751\pi$
$383$	−9.46576	−0.483678	−0.241839	+	0.970316i	$0.577751\pi$
$384$	0	0
$385$	8.26071	0.421005
$386$	18.4049	0.936787
$387$	0	0
$388$	−6.72873	−0.341600
$389$	−16.2760	−0.825226	−0.412613	−	0.910907i	$-0.635384\pi$
$389$	−16.2760	−0.825226	−0.412613	+	0.910907i	$0.635384\pi$
$390$	0	0
$391$	15.6073	0.789296
$392$	55.0060	2.77822
$393$	0	0
$394$	18.4367	0.928828
$395$	9.21709	0.463762
$396$	0	0
$397$	21.0178	1.05486	0.527428	−	0.849600i	$-0.323157\pi$
$397$	21.0178	1.05486	0.527428	+	0.849600i	$0.323157\pi$
$398$	14.1797	0.710764
$399$	0	0
$400$	3.07032	0.153516
$401$	1.15040	0.0574484	0.0287242	−	0.999587i	$-0.490856\pi$
$401$	1.15040	0.0574484	0.0287242	+	0.999587i	$0.490856\pi$
$402$	0	0
$403$	−15.8046	−0.787281
$404$	1.78346	0.0887306
$405$	0	0
$406$	−4.72627	−0.234561
$407$	5.38300	0.266826
$408$	0	0
$409$	−17.9590	−0.888015	−0.444007	−	0.896023i	$-0.646444\pi$
$409$	−17.9590	−0.888015	−0.444007	+	0.896023i	$0.646444\pi$
$410$	20.1261	0.993956
$411$	0	0
$412$	−4.46790	−0.220118
$413$	−2.34677	−0.115477
$414$	0	0
$415$	−0.732260	−0.0359453
$416$	14.0989	0.691254
$417$	0	0
$418$	−6.29766	−0.308029
$419$	−30.0432	−1.46771	−0.733853	−	0.679308i	$-0.762280\pi$
$419$	−30.0432	−1.46771	−0.733853	+	0.679308i	$0.762280\pi$
$420$	0	0
$421$	−6.11518	−0.298036	−0.149018	−	0.988834i	$-0.547611\pi$
$421$	−6.11518	−0.298036	−0.149018	+	0.988834i	$0.547611\pi$
$422$	−20.3854	−0.992343
$423$	0	0
$424$	39.6993	1.92797
$425$	−7.27594	−0.352935
$426$	0	0
$427$	5.01087	0.242493
$428$	−1.52391	−0.0736610
$429$	0	0
$430$	−19.3020	−0.930824
$431$	32.2060	1.55131	0.775654	−	0.631158i	$-0.217420\pi$
$431$	32.2060	1.55131	0.775654	+	0.631158i	$0.217420\pi$
$432$	0	0
$433$	−28.5878	−1.37384	−0.686921	−	0.726733i	$-0.741038\pi$
$433$	−28.5878	−1.37384	−0.686921	+	0.726733i	$0.741038\pi$
$434$	−27.3746	−1.31402
$435$	0	0
$436$	−15.8002	−0.756692
$437$	−29.5540	−1.41376
$438$	0	0
$439$	−19.8151	−0.945723	−0.472861	−	0.881137i	$-0.656779\pi$
$439$	−19.8151	−0.945723	−0.472861	+	0.881137i	$0.656779\pi$
$440$	5.00753	0.238724
$441$	0	0
$442$	10.0371	0.477416
$443$	−27.8588	−1.32361	−0.661806	−	0.749675i	$-0.730210\pi$
$443$	−27.8588	−1.32361	−0.661806	+	0.749675i	$0.730210\pi$
$444$	0	0
$445$	5.39199	0.255605
$446$	12.2129	0.578297
$447$	0	0
$448$	37.9025	1.79072
$449$	−29.3342	−1.38437	−0.692183	−	0.721722i	$-0.743351\pi$
$449$	−29.3342	−1.38437	−0.692183	+	0.721722i	$0.743351\pi$
$450$	0	0
$451$	−11.7027	−0.551059
$452$	7.68772	0.361600
$453$	0	0
$454$	9.04352	0.424434
$455$	24.9306	1.16877
$456$	0	0
$457$	−20.0757	−0.939101	−0.469550	−	0.882906i	$-0.655584\pi$
$457$	−20.0757	−0.939101	−0.469550	+	0.882906i	$0.655584\pi$
$458$	−1.29588	−0.0605524
$459$	0	0
$460$	7.35824	0.343080
$461$	37.3900	1.74143	0.870713	−	0.491792i	$-0.163658\pi$
$461$	37.3900	1.74143	0.870713	+	0.491792i	$0.163658\pi$
$462$	0	0
$463$	3.21469	0.149399	0.0746996	−	0.997206i	$-0.476200\pi$
$463$	3.21469	0.149399	0.0746996	+	0.997206i	$0.476200\pi$
$464$	−1.21634	−0.0564671
$465$	0	0
$466$	8.51197	0.394309
$467$	32.0410	1.48268	0.741341	−	0.671129i	$-0.234190\pi$
$467$	32.0410	1.48268	0.741341	+	0.671129i	$0.234190\pi$
$468$	0	0
$469$	−15.8165	−0.730340
$470$	12.2228	0.563797
$471$	0	0
$472$	−1.42258	−0.0654794
$473$	11.2235	0.516058
$474$	0	0
$475$	13.7777	0.632164
$476$	−14.5647	−0.667573
$477$	0	0
$478$	−30.3466	−1.38802
$479$	−35.8435	−1.63773	−0.818867	−	0.573984i	$-0.805397\pi$
$479$	−35.8435	−1.63773	−0.818867	+	0.573984i	$0.805397\pi$
$480$	0	0
$481$	16.2458	0.740744
$482$	15.1728	0.691100
$483$	0	0
$484$	−0.911728	−0.0414422
$485$	−12.1667	−0.552460
$486$	0	0
$487$	27.6269	1.25190	0.625948	−	0.779865i	$-0.284712\pi$
$487$	27.6269	1.25190	0.625948	+	0.779865i	$0.284712\pi$
$488$	3.03752	0.137502
$489$	0	0
$490$	31.1432	1.40691
$491$	1.38118	0.0623318	0.0311659	−	0.999514i	$-0.490078\pi$
$491$	1.38118	0.0623318	0.0311659	+	0.999514i	$0.490078\pi$
$492$	0	0
$493$	2.88244	0.129818
$494$	−19.0062	−0.855130
$495$	0	0
$496$	−7.04505	−0.316332
$497$	−48.6777	−2.18349
$498$	0	0
$499$	32.7390	1.46560	0.732799	−	0.680445i	$-0.238213\pi$
$499$	32.7390	1.46560	0.732799	+	0.680445i	$0.238213\pi$
$500$	−10.9455	−0.489497
$501$	0	0
$502$	7.92071	0.353519
$503$	28.8264	1.28530	0.642652	−	0.766158i	$-0.277834\pi$
$503$	28.8264	1.28530	0.642652	+	0.766158i	$0.277834\pi$
$504$	0	0
$505$	3.22480	0.143502
$506$	5.10709	0.227038
$507$	0	0
$508$	−13.4847	−0.598287
$509$	22.0483	0.977275	0.488638	−	0.872487i	$-0.337494\pi$
$509$	22.0483	0.977275	0.488638	+	0.872487i	$0.337494\pi$
$510$	0	0
$511$	−16.6489	−0.736506
$512$	14.4575	0.638936
$513$	0	0
$514$	−10.1278	−0.446717
$515$	−8.07871	−0.355991
$516$	0	0
$517$	−7.10722	−0.312575
$518$	28.1389	1.23635
$519$	0	0
$520$	15.1126	0.662732
$521$	0.419315	0.0183705	0.00918527	−	0.999958i	$-0.497076\pi$
$521$	0.419315	0.0183705	0.00918527	+	0.999958i	$0.497076\pi$
$522$	0	0
$523$	32.9826	1.44223	0.721114	−	0.692816i	$-0.243630\pi$
$523$	32.9826	1.44223	0.721114	+	0.692816i	$0.243630\pi$
$524$	−0.313384	−0.0136903
$525$	0	0
$526$	−29.1822	−1.27240
$527$	16.6951	0.727251
$528$	0	0
$529$	0.966817	0.0420355
$530$	22.4768	0.976332
$531$	0	0
$532$	27.5797	1.19573
$533$	−35.3185	−1.52982
$534$	0	0
$535$	−2.75548	−0.119130
$536$	−9.58777	−0.414128
$537$	0	0
$538$	−5.72156	−0.246674
$539$	−18.1089	−0.780003
$540$	0	0
$541$	−21.7042	−0.933136	−0.466568	−	0.884485i	$-0.654510\pi$
$541$	−21.7042	−0.933136	−0.466568	+	0.884485i	$0.654510\pi$
$542$	8.50670	0.365394
$543$	0	0
$544$	−14.8933	−0.638546
$545$	−28.5694	−1.22378
$546$	0	0
$547$	−35.6177	−1.52290	−0.761451	−	0.648223i	$-0.775513\pi$
$547$	−35.6177	−1.52290	−0.761451	+	0.648223i	$0.775513\pi$
$548$	12.5170	0.534700
$549$	0	0
$550$	−2.38086	−0.101520
$551$	−5.45817	−0.232526
$552$	0	0
$553$	−28.0158	−1.19135
$554$	−15.8034	−0.671421
$555$	0	0
$556$	14.8873	0.631361
$557$	2.05471	0.0870607	0.0435303	−	0.999052i	$-0.486139\pi$
$557$	2.05471	0.0870607	0.0435303	+	0.999052i	$0.486139\pi$
$558$	0	0
$559$	33.8723	1.43265
$560$	11.1131	0.469614
$561$	0	0
$562$	−29.2590	−1.23422
$563$	−11.9315	−0.502854	−0.251427	−	0.967876i	$-0.580900\pi$
$563$	−11.9315	−0.502854	−0.251427	+	0.967876i	$0.580900\pi$
$564$	0	0
$565$	13.9007	0.584806
$566$	−12.7390	−0.535459
$567$	0	0
$568$	−29.5078	−1.23812
$569$	39.3292	1.64877	0.824384	−	0.566031i	$-0.191522\pi$
$569$	39.3292	1.64877	0.824384	+	0.566031i	$0.191522\pi$
$570$	0	0
$571$	41.1563	1.72234	0.861168	−	0.508320i	$-0.169733\pi$
$571$	41.1563	1.72234	0.861168	+	0.508320i	$0.169733\pi$
$572$	−2.75158	−0.115049
$573$	0	0
$574$	−61.1742	−2.55336
$575$	−11.1730	−0.465947
$576$	0	0
$577$	21.1444	0.880253	0.440127	−	0.897936i	$-0.354934\pi$
$577$	21.1444	0.880253	0.440127	+	0.897936i	$0.354934\pi$
$578$	7.13177	0.296643
$579$	0	0
$580$	1.35896	0.0564275
$581$	2.22574	0.0923394
$582$	0	0
$583$	−13.0696	−0.541288
$584$	−10.0924	−0.417625
$585$	0	0
$586$	−10.7877	−0.445634
$587$	16.6914	0.688927	0.344464	−	0.938800i	$-0.388061\pi$
$587$	16.6914	0.688927	0.344464	+	0.938800i	$0.388061\pi$
$588$	0	0
$589$	−31.6138	−1.30262
$590$	−0.805432	−0.0331591
$591$	0	0
$592$	7.24173	0.297633
$593$	11.4836	0.471573	0.235787	−	0.971805i	$-0.424233\pi$
$593$	11.4836	0.471573	0.235787	+	0.971805i	$0.424233\pi$
$594$	0	0
$595$	−26.3354	−1.07965
$596$	−0.577207	−0.0236433
$597$	0	0
$598$	15.4131	0.630289
$599$	36.1765	1.47813	0.739067	−	0.673632i	$-0.235267\pi$
$599$	36.1765	1.47813	0.739067	+	0.673632i	$0.235267\pi$
$600$	0	0
$601$	17.6510	0.719999	0.359999	−	0.932953i	$-0.382777\pi$
$601$	17.6510	0.719999	0.359999	+	0.932953i	$0.382777\pi$
$602$	58.6694	2.39118
$603$	0	0
$604$	9.46446	0.385103
$605$	−1.64856	−0.0670233
$606$	0	0
$607$	28.3146	1.14925	0.574627	−	0.818415i	$-0.305147\pi$
$607$	28.3146	1.14925	0.574627	+	0.818415i	$0.305147\pi$
$608$	28.2019	1.14374
$609$	0	0
$610$	1.71978	0.0696318
$611$	−21.4494	−0.867751
$612$	0	0
$613$	3.45999	0.139747	0.0698737	−	0.997556i	$-0.477740\pi$
$613$	3.45999	0.139747	0.0698737	+	0.997556i	$0.477740\pi$
$614$	−28.6153	−1.15482
$615$	0	0
$616$	−15.2206	−0.613257
$617$	13.0700	0.526180	0.263090	−	0.964771i	$-0.415258\pi$
$617$	13.0700	0.526180	0.263090	+	0.964771i	$0.415258\pi$
$618$	0	0
$619$	8.11491	0.326166	0.163083	−	0.986612i	$-0.447856\pi$
$619$	8.11491	0.326166	0.163083	+	0.986612i	$0.447856\pi$
$620$	7.87110	0.316111
$621$	0	0
$622$	−27.9629	−1.12121
$623$	−16.3892	−0.656620
$624$	0	0
$625$	−8.37996	−0.335199
$626$	−19.4121	−0.775862
$627$	0	0
$628$	−18.4298	−0.735428
$629$	−17.1612	−0.684262
$630$	0	0
$631$	1.68038	0.0668947	0.0334474	−	0.999440i	$-0.489351\pi$
$631$	1.68038	0.0668947	0.0334474	+	0.999440i	$0.489351\pi$
$632$	−16.9828	−0.675540
$633$	0	0
$634$	−28.2913	−1.12359
$635$	−24.3826	−0.967593
$636$	0	0
$637$	−54.6521	−2.16540
$638$	0.943203	0.0373417
$639$	0	0
$640$	−2.39440	−0.0946468
$641$	39.1788	1.54747	0.773735	−	0.633509i	$-0.218386\pi$
$641$	39.1788	1.54747	0.773735	+	0.633509i	$0.218386\pi$
$642$	0	0
$643$	6.09680	0.240434	0.120217	−	0.992748i	$-0.461641\pi$
$643$	6.09680	0.240434	0.120217	+	0.992748i	$0.461641\pi$
$644$	−22.3658	−0.881335
$645$	0	0
$646$	20.0772	0.789926
$647$	41.2844	1.62306	0.811529	−	0.584312i	$-0.198636\pi$
$647$	41.2844	1.62306	0.811529	+	0.584312i	$0.198636\pi$
$648$	0	0
$649$	0.468335	0.0183837
$650$	−7.18539	−0.281834
$651$	0	0
$652$	−10.5798	−0.414339
$653$	−16.4695	−0.644500	−0.322250	−	0.946655i	$-0.604439\pi$
$653$	−16.4695	−0.644500	−0.322250	+	0.946655i	$0.604439\pi$
$654$	0	0
$655$	−0.566651	−0.0221409
$656$	−15.7436	−0.614685
$657$	0	0
$658$	−37.1520	−1.44833
$659$	−46.5604	−1.81374	−0.906868	−	0.421415i	$-0.861534\pi$
$659$	−46.5604	−1.81374	−0.906868	+	0.421415i	$0.861534\pi$
$660$	0	0
$661$	−28.4805	−1.10776	−0.553882	−	0.832595i	$-0.686854\pi$
$661$	−28.4805	−1.10776	−0.553882	+	0.832595i	$0.686854\pi$
$662$	15.2172	0.591433
$663$	0	0
$664$	1.34922	0.0523597
$665$	49.8687	1.93382
$666$	0	0
$667$	4.42631	0.171387
$668$	−11.7026	−0.452788
$669$	0	0
$670$	−5.42838	−0.209717
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	31.6459	1.21986	0.609930	−	0.792456i	$-0.291198\pi$
$673$	31.6459	1.21986	0.609930	+	0.792456i	$0.291198\pi$
$674$	3.11393	0.119944
$675$	0	0
$676$	3.54827	0.136472
$677$	−40.6813	−1.56351	−0.781755	−	0.623586i	$-0.785675\pi$
$677$	−40.6813	−1.56351	−0.781755	+	0.623586i	$0.785675\pi$
$678$	0	0
$679$	36.9812	1.41921
$680$	−15.9642	−0.612198
$681$	0	0
$682$	5.46305	0.209191
$683$	−3.16227	−0.121001	−0.0605005	−	0.998168i	$-0.519270\pi$
$683$	−3.16227	−0.121001	−0.0605005	+	0.998168i	$0.519270\pi$
$684$	0	0
$685$	22.6328	0.864757
$686$	−58.0699	−2.21712
$687$	0	0
$688$	15.0990	0.575643
$689$	−39.4438	−1.50269
$690$	0	0
$691$	−5.23615	−0.199192	−0.0995962	−	0.995028i	$-0.531755\pi$
$691$	−5.23615	−0.199192	−0.0995962	+	0.995028i	$0.531755\pi$
$692$	12.4410	0.472937
$693$	0	0
$694$	−24.6195	−0.934543
$695$	26.9187	1.02108
$696$	0	0
$697$	37.3087	1.41317
$698$	−24.9327	−0.943716
$699$	0	0
$700$	10.4266	0.394090
$701$	−26.1863	−0.989043	−0.494521	−	0.869165i	$-0.664657\pi$
$701$	−26.1863	−0.989043	−0.494521	+	0.869165i	$0.664657\pi$
$702$	0	0
$703$	32.4964	1.22562
$704$	−7.56405	−0.285081
$705$	0	0
$706$	−35.9368	−1.35250
$707$	−9.80194	−0.368640
$708$	0	0
$709$	11.6344	0.436938	0.218469	−	0.975844i	$-0.429894\pi$
$709$	11.6344	0.436938	0.218469	+	0.975844i	$0.429894\pi$
$710$	−16.7066	−0.626989
$711$	0	0
$712$	−9.93492	−0.372327
$713$	25.6373	0.960123
$714$	0	0
$715$	−4.97531	−0.186066
$716$	−4.22941	−0.158060
$717$	0	0
$718$	−9.37484	−0.349866
$719$	10.3289	0.385205	0.192602	−	0.981277i	$-0.438307\pi$
$719$	10.3289	0.385205	0.192602	+	0.981277i	$0.438307\pi$
$720$	0	0
$721$	24.5557	0.914501
$722$	−18.1972	−0.677231
$723$	0	0
$724$	−14.6315	−0.543774
$725$	−2.06349	−0.0766360
$726$	0	0
$727$	1.38619	0.0514109	0.0257055	−	0.999670i	$-0.491817\pi$
$727$	1.38619	0.0514109	0.0257055	+	0.999670i	$0.491817\pi$
$728$	−45.9356	−1.70249
$729$	0	0
$730$	−5.71407	−0.211487
$731$	−35.7810	−1.32341
$732$	0	0
$733$	20.8739	0.770996	0.385498	−	0.922709i	$-0.374030\pi$
$733$	20.8739	0.770996	0.385498	+	0.922709i	$0.374030\pi$
$734$	−13.3642	−0.493283
$735$	0	0
$736$	−22.8704	−0.843014
$737$	3.15644	0.116269
$738$	0	0
$739$	−48.2442	−1.77469	−0.887347	−	0.461103i	$-0.847454\pi$
$739$	−48.2442	−1.77469	−0.887347	+	0.461103i	$0.847454\pi$
$740$	−8.09084	−0.297425
$741$	0	0
$742$	−68.3196	−2.50809
$743$	−10.1761	−0.373325	−0.186663	−	0.982424i	$-0.559767\pi$
$743$	−10.1761	−0.373325	−0.186663	+	0.982424i	$0.559767\pi$
$744$	0	0
$745$	−1.04369	−0.0382377
$746$	−13.6448	−0.499571
$747$	0	0
$748$	2.90662	0.106277
$749$	8.37544	0.306032
$750$	0	0
$751$	9.83366	0.358835	0.179418	−	0.983773i	$-0.442579\pi$
$751$	9.83366	0.358835	0.179418	+	0.983773i	$0.442579\pi$
$752$	−9.56131	−0.348665
$753$	0	0
$754$	2.84656	0.103666
$755$	17.1133	0.622817
$756$	0	0
$757$	21.7125	0.789153	0.394576	−	0.918863i	$-0.370891\pi$
$757$	21.7125	0.789153	0.394576	+	0.918863i	$0.370891\pi$
$758$	−3.04713	−0.110677
$759$	0	0
$760$	30.2297	1.09655
$761$	−28.6061	−1.03697	−0.518484	−	0.855087i	$-0.673504\pi$
$761$	−28.6061	−1.03697	−0.518484	+	0.855087i	$0.673504\pi$
$762$	0	0
$763$	86.8382	3.14375
$764$	16.7971	0.607699
$765$	0	0
$766$	9.87471	0.356788
$767$	1.41342	0.0510358
$768$	0	0
$769$	18.3697	0.662429	0.331214	−	0.943556i	$-0.392542\pi$
$769$	18.3697	0.662429	0.331214	+	0.943556i	$0.392542\pi$
$770$	−8.61759	−0.310556
$771$	0	0
$772$	16.0854	0.578925
$773$	28.0562	1.00911	0.504556	−	0.863379i	$-0.331656\pi$
$773$	28.0562	1.00911	0.504556	+	0.863379i	$0.331656\pi$
$774$	0	0
$775$	−11.9518	−0.429320
$776$	22.4175	0.804742
$777$	0	0
$778$	16.9792	0.608732
$779$	−70.6476	−2.53121
$780$	0	0
$781$	9.71441	0.347609
$782$	−16.2816	−0.582229
$783$	0	0
$784$	−24.3618	−0.870063
$785$	−33.3241	−1.18939
$786$	0	0
$787$	−17.8283	−0.635511	−0.317756	−	0.948173i	$-0.602929\pi$
$787$	−17.8283	−0.635511	−0.317756	+	0.948173i	$0.602929\pi$
$788$	16.1131	0.574007
$789$	0	0
$790$	−9.61530	−0.342097
$791$	−42.2518	−1.50230
$792$	0	0
$793$	−3.01798	−0.107172
$794$	−21.9259	−0.778120
$795$	0	0
$796$	12.3926	0.439245
$797$	−18.9394	−0.670867	−0.335434	−	0.942064i	$-0.608883\pi$
$797$	−18.9394	−0.670867	−0.335434	+	0.942064i	$0.608883\pi$
$798$	0	0
$799$	22.6581	0.801585
$800$	10.6619	0.376954
$801$	0	0
$802$	−1.20010	−0.0423772
$803$	3.32256	0.117251
$804$	0	0
$805$	−40.4410	−1.42536
$806$	16.4874	0.580743
$807$	0	0
$808$	−5.94180	−0.209032
$809$	−25.7275	−0.904529	−0.452265	−	0.891884i	$-0.649384\pi$
$809$	−25.7275	−0.904529	−0.452265	+	0.891884i	$0.649384\pi$
$810$	0	0
$811$	−46.4287	−1.63033	−0.815166	−	0.579227i	$-0.803355\pi$
$811$	−46.4287	−1.63033	−0.815166	+	0.579227i	$0.803355\pi$
$812$	−4.13062	−0.144956
$813$	0	0
$814$	−5.61556	−0.196825
$815$	−19.1301	−0.670099
$816$	0	0
$817$	67.7548	2.37044
$818$	18.7349	0.655049
$819$	0	0
$820$	17.5896	0.614255
$821$	24.6784	0.861281	0.430641	−	0.902524i	$-0.358288\pi$
$821$	24.6784	0.861281	0.430641	+	0.902524i	$0.358288\pi$
$822$	0	0
$823$	4.02139	0.140177	0.0700885	−	0.997541i	$-0.477672\pi$
$823$	4.02139	0.140177	0.0700885	+	0.997541i	$0.477672\pi$
$824$	14.8853	0.518554
$825$	0	0
$826$	2.44815	0.0851821
$827$	0.943492	0.0328084	0.0164042	−	0.999865i	$-0.494778\pi$
$827$	0.943492	0.0328084	0.0164042	+	0.999865i	$0.494778\pi$
$828$	0	0
$829$	9.46783	0.328831	0.164416	−	0.986391i	$-0.447426\pi$
$829$	9.46783	0.328831	0.164416	+	0.986391i	$0.447426\pi$
$830$	0.763896	0.0265152
$831$	0	0
$832$	−22.8281	−0.791423
$833$	57.7317	2.00028
$834$	0	0
$835$	−21.1603	−0.732282
$836$	−5.50397	−0.190359
$837$	0	0
$838$	31.3412	1.08266
$839$	−19.2334	−0.664011	−0.332005	−	0.943278i	$-0.607725\pi$
$839$	−19.2334	−0.664011	−0.332005	+	0.943278i	$0.607725\pi$
$840$	0	0
$841$	−28.1825	−0.971811
$842$	6.37938	0.219848
$843$	0	0
$844$	−17.8162	−0.613258
$845$	6.41586	0.220712
$846$	0	0
$847$	5.01087	0.172176
$848$	−17.5825	−0.603786
$849$	0	0
$850$	7.59028	0.260344
$851$	−26.3530	−0.903369
$852$	0	0
$853$	−42.0074	−1.43831	−0.719153	−	0.694852i	$-0.755470\pi$
$853$	−42.0074	−1.43831	−0.719153	+	0.694852i	$0.755470\pi$
$854$	−5.22736	−0.178876
$855$	0	0
$856$	5.07708	0.173531
$857$	−25.4294	−0.868652	−0.434326	−	0.900756i	$-0.643013\pi$
$857$	−25.4294	−0.868652	−0.434326	+	0.900756i	$0.643013\pi$
$858$	0	0
$859$	5.15518	0.175893	0.0879463	−	0.996125i	$-0.471970\pi$
$859$	5.15518	0.175893	0.0879463	+	0.996125i	$0.471970\pi$
$860$	−16.8693	−0.575240
$861$	0	0
$862$	−33.5974	−1.14433
$863$	−24.1318	−0.821457	−0.410729	−	0.911758i	$-0.634726\pi$
$863$	−24.1318	−0.821457	−0.410729	+	0.911758i	$0.634726\pi$
$864$	0	0
$865$	22.4955	0.764868
$866$	29.8229	1.01342
$867$	0	0
$868$	−23.9246	−0.812054
$869$	5.59101	0.189662
$870$	0	0
$871$	9.52607	0.322779
$872$	52.6401	1.78262
$873$	0	0
$874$	30.8308	1.04287
$875$	60.1566	2.03367
$876$	0	0
$877$	11.3805	0.384291	0.192146	−	0.981366i	$-0.438455\pi$
$877$	11.3805	0.384291	0.192146	+	0.981366i	$0.438455\pi$
$878$	20.6712	0.697618
$879$	0	0
$880$	−2.21780	−0.0747619
$881$	−34.9613	−1.17788	−0.588938	−	0.808178i	$-0.700454\pi$
$881$	−34.9613	−1.17788	−0.588938	+	0.808178i	$0.700454\pi$
$882$	0	0
$883$	−44.0098	−1.48105	−0.740523	−	0.672031i	$-0.765422\pi$
$883$	−44.0098	−1.48105	−0.740523	+	0.672031i	$0.765422\pi$
$884$	8.77212	0.295038
$885$	0	0
$886$	29.0624	0.976370
$887$	6.63679	0.222842	0.111421	−	0.993773i	$-0.464460\pi$
$887$	6.63679	0.222842	0.111421	+	0.993773i	$0.464460\pi$
$888$	0	0
$889$	74.1122	2.48564
$890$	−5.62493	−0.188548
$891$	0	0
$892$	10.6737	0.357382
$893$	−42.9052	−1.43577
$894$	0	0
$895$	−7.64748	−0.255627
$896$	7.27789	0.243137
$897$	0	0
$898$	30.6015	1.02119
$899$	4.73481	0.157915
$900$	0	0
$901$	41.6664	1.38811
$902$	12.2083	0.406492
$903$	0	0
$904$	−25.6125	−0.851859
$905$	−26.4561	−0.879432
$906$	0	0
$907$	−49.3540	−1.63877	−0.819386	−	0.573242i	$-0.805686\pi$
$907$	−49.3540	−1.63877	−0.819386	+	0.573242i	$0.805686\pi$
$908$	7.90377	0.262296
$909$	0	0
$910$	−26.0077	−0.862147
$911$	46.6081	1.54420	0.772098	−	0.635504i	$-0.219208\pi$
$911$	46.6081	1.54420	0.772098	+	0.635504i	$0.219208\pi$
$912$	0	0
$913$	−0.444183	−0.0147003
$914$	20.9430	0.692733
$915$	0	0
$916$	−1.13256	−0.0374208
$917$	1.72237	0.0568775
$918$	0	0
$919$	−0.500503	−0.0165101	−0.00825504	−	0.999966i	$-0.502628\pi$
$919$	−0.500503	−0.0165101	−0.00825504	+	0.999966i	$0.502628\pi$
$920$	−24.5148	−0.808229
$921$	0	0
$922$	−39.0053	−1.28457
$923$	29.3179	0.965010
$924$	0	0
$925$	12.2854	0.403942
$926$	−3.35357	−0.110205
$927$	0	0
$928$	−4.22381	−0.138653
$929$	30.9867	1.01664	0.508321	−	0.861168i	$-0.330266\pi$
$929$	30.9867	1.01664	0.508321	+	0.861168i	$0.330266\pi$
$930$	0	0
$931$	−109.320	−3.58283
$932$	7.43921	0.243679
$933$	0	0
$934$	−33.4253	−1.09371
$935$	5.25566	0.171878
$936$	0	0
$937$	18.8113	0.614538	0.307269	−	0.951623i	$-0.400585\pi$
$937$	18.8113	0.614538	0.307269	+	0.951623i	$0.400585\pi$
$938$	16.4999	0.538739
$939$	0	0
$940$	10.6824	0.348421
$941$	−13.0851	−0.426561	−0.213281	−	0.976991i	$-0.568415\pi$
$941$	−13.0851	−0.426561	−0.213281	+	0.976991i	$0.568415\pi$
$942$	0	0
$943$	57.2917	1.86567
$944$	0.630049	0.0205063
$945$	0	0
$946$	−11.7084	−0.380673
$947$	−10.6578	−0.346331	−0.173165	−	0.984893i	$-0.555400\pi$
$947$	−10.6578	−0.346331	−0.173165	+	0.984893i	$0.555400\pi$
$948$	0	0
$949$	10.0274	0.325504
$950$	−14.3729	−0.466319
$951$	0	0
$952$	48.5240	1.57267
$953$	−27.5269	−0.891684	−0.445842	−	0.895112i	$-0.647096\pi$
$953$	−27.5269	−0.891684	−0.445842	+	0.895112i	$0.647096\pi$
$954$	0	0
$955$	30.3720	0.982816
$956$	−26.5220	−0.857784
$957$	0	0
$958$	37.3921	1.20808
$959$	−68.7937	−2.22147
$960$	0	0
$961$	−3.57589	−0.115351
$962$	−16.9476	−0.546414
$963$	0	0
$964$	13.2605	0.427093
$965$	29.0850	0.936280
$966$	0	0
$967$	−7.68136	−0.247016	−0.123508	−	0.992344i	$-0.539414\pi$
$967$	−7.68136	−0.247016	−0.123508	+	0.992344i	$0.539414\pi$
$968$	3.03752	0.0976297
$969$	0	0
$970$	12.6923	0.407525
$971$	−42.1602	−1.35299	−0.676493	−	0.736449i	$-0.736501\pi$
$971$	−42.1602	−1.35299	−0.676493	+	0.736449i	$0.736501\pi$
$972$	0	0
$973$	−81.8207	−2.62305
$974$	−28.8205	−0.923469
$975$	0	0
$976$	−1.34530	−0.0430619
$977$	−2.82716	−0.0904488	−0.0452244	−	0.998977i	$-0.514400\pi$
$977$	−2.82716	−0.0904488	−0.0452244	+	0.998977i	$0.514400\pi$
$978$	0	0
$979$	3.27073	0.104533
$980$	27.2182	0.869454
$981$	0	0
$982$	−1.44085	−0.0459794
$983$	24.3097	0.775359	0.387680	−	0.921794i	$-0.373277\pi$
$983$	24.3097	0.775359	0.387680	+	0.921794i	$0.373277\pi$
$984$	0	0
$985$	29.1352	0.928326
$986$	−3.00696	−0.0957613
$987$	0	0
$988$	−16.6109	−0.528462
$989$	−54.9458	−1.74717
$990$	0	0
$991$	39.6676	1.26008	0.630042	−	0.776561i	$-0.283038\pi$
$991$	39.6676	1.26008	0.630042	+	0.776561i	$0.283038\pi$
$992$	−24.4644	−0.776746
$993$	0	0
$994$	50.7807	1.61067
$995$	22.4079	0.710379
$996$	0	0
$997$	27.7285	0.878170	0.439085	−	0.898445i	$-0.355303\pi$
$997$	27.7285	0.878170	0.439085	+	0.898445i	$0.355303\pi$
$998$	−34.1534	−1.08111
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.8	yes	25
3.2	odd	2		6039.2.a.m.1.18	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.18	✓	25	3.2	odd	2
6039.2.a.p.1.8	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)

\(f(q)\)	\(=\)	\(q-1.04320 q^{2} -0.911728 q^{4} -1.64856 q^{5} +5.01087 q^{7} +3.03752 q^{8} +O(q^{10})\)	\(q-1.04320 q^{2} -0.911728 q^{4} -1.64856 q^{5} +5.01087 q^{7} +3.03752 q^{8} +1.71978 q^{10} -1.00000 q^{11} -3.01798 q^{13} -5.22736 q^{14} -1.34530 q^{16} +3.18804 q^{17} -6.03685 q^{19} +1.50303 q^{20} +1.04320 q^{22} +4.89559 q^{23} -2.28226 q^{25} +3.14836 q^{26} -4.56855 q^{28} +0.904141 q^{29} +5.23680 q^{31} -4.67163 q^{32} -3.32577 q^{34} -8.26071 q^{35} -5.38300 q^{37} +6.29766 q^{38} -5.00753 q^{40} +11.7027 q^{41} -11.2235 q^{43} +0.911728 q^{44} -5.10709 q^{46} +7.10722 q^{47} +18.1089 q^{49} +2.38086 q^{50} +2.75158 q^{52} +13.0696 q^{53} +1.64856 q^{55} +15.2206 q^{56} -0.943203 q^{58} -0.468335 q^{59} +1.00000 q^{61} -5.46305 q^{62} +7.56405 q^{64} +4.97531 q^{65} -3.15644 q^{67} -2.90662 q^{68} +8.61759 q^{70} -9.71441 q^{71} -3.32256 q^{73} +5.61556 q^{74} +5.50397 q^{76} -5.01087 q^{77} -5.59101 q^{79} +2.21780 q^{80} -12.2083 q^{82} +0.444183 q^{83} -5.25566 q^{85} +11.7084 q^{86} -3.03752 q^{88} -3.27073 q^{89} -15.1227 q^{91} -4.46345 q^{92} -7.41427 q^{94} +9.95209 q^{95} +7.38020 q^{97} -18.8912 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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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Database

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