LMFDB - Embedded newform 6039.2.a.p.1.5

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			1.5
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)

$f(q)$	$=$	$q-1.87622 q^{2} +1.52020 q^{4} -0.769125 q^{5} -0.850176 q^{7} +0.900213 q^{8} +O(q^{10})$	\(q-1.87622 q^{2} +1.52020 q^{4} -0.769125 q^{5} -0.850176 q^{7} +0.900213 q^{8} +1.44305 q^{10} -1.00000 q^{11} -0.486521 q^{13} +1.59512 q^{14} -4.72939 q^{16} -2.67691 q^{17} -7.96625 q^{19} -1.16922 q^{20} +1.87622 q^{22} -7.17905 q^{23} -4.40845 q^{25} +0.912819 q^{26} -1.29244 q^{28} +4.07022 q^{29} -7.87104 q^{31} +7.07295 q^{32} +5.02246 q^{34} +0.653891 q^{35} +2.50533 q^{37} +14.9464 q^{38} -0.692376 q^{40} -1.48908 q^{41} -6.73850 q^{43} -1.52020 q^{44} +13.4695 q^{46} -12.0321 q^{47} -6.27720 q^{49} +8.27121 q^{50} -0.739608 q^{52} +11.5877 q^{53} +0.769125 q^{55} -0.765339 q^{56} -7.63663 q^{58} +4.12976 q^{59} +1.00000 q^{61} +14.7678 q^{62} -3.81162 q^{64} +0.374195 q^{65} +6.50801 q^{67} -4.06943 q^{68} -1.22684 q^{70} +7.19240 q^{71} -7.31828 q^{73} -4.70054 q^{74} -12.1103 q^{76} +0.850176 q^{77} +8.91585 q^{79} +3.63749 q^{80} +2.79383 q^{82} +9.65801 q^{83} +2.05888 q^{85} +12.6429 q^{86} -0.900213 q^{88} -7.33234 q^{89} +0.413628 q^{91} -10.9136 q^{92} +22.5748 q^{94} +6.12704 q^{95} +5.47164 q^{97} +11.7774 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.87622	−1.32669	−0.663344	−	0.748315i	$-0.730863\pi$
$2$	−1.87622	−1.32669	−0.663344	+	0.748315i	$0.730863\pi$
$3$	0	0
$3$	0	0
$4$	1.52020	0.760099
$5$	−0.769125	−0.343963	−0.171982	−	0.985100i	$-0.555017\pi$
$5$	−0.769125	−0.343963	−0.171982	+	0.985100i	$0.555017\pi$
$6$	0	0
$7$	−0.850176	−0.321336	−0.160668	−	0.987008i	$-0.551365\pi$
$7$	−0.850176	−0.321336	−0.160668	+	0.987008i	$0.551365\pi$
$8$	0.900213	0.318273
$9$	0	0
$10$	1.44305	0.456331
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.486521	−0.134937	−0.0674683	−	0.997721i	$-0.521492\pi$
$13$	−0.486521	−0.134937	−0.0674683	+	0.997721i	$0.521492\pi$
$14$	1.59512	0.426313
$15$	0	0
$16$	−4.72939	−1.18235
$17$	−2.67691	−0.649245	−0.324623	−	0.945844i	$-0.605237\pi$
$17$	−2.67691	−0.649245	−0.324623	+	0.945844i	$0.605237\pi$
$18$	0	0
$19$	−7.96625	−1.82758	−0.913791	−	0.406184i	$-0.866859\pi$
$19$	−7.96625	−1.82758	−0.913791	+	0.406184i	$0.866859\pi$
$20$	−1.16922	−0.261446
$21$	0	0
$22$	1.87622	0.400011
$23$	−7.17905	−1.49694	−0.748468	−	0.663171i	$-0.769210\pi$
$23$	−7.17905	−1.49694	−0.748468	+	0.663171i	$0.769210\pi$
$24$	0	0
$25$	−4.40845	−0.881689
$26$	0.912819	0.179019
$27$	0	0
$28$	−1.29244	−0.244248
$29$	4.07022	0.755821	0.377911	−	0.925842i	$-0.376643\pi$
$29$	4.07022	0.755821	0.377911	+	0.925842i	$0.376643\pi$
$30$	0	0
$31$	−7.87104	−1.41368	−0.706840	−	0.707373i	$-0.749880\pi$
$31$	−7.87104	−1.41368	−0.706840	+	0.707373i	$0.749880\pi$
$32$	7.07295	1.25033
$33$	0	0
$34$	5.02246	0.861345
$35$	0.653891	0.110528
$36$	0	0
$37$	2.50533	0.411873	0.205937	−	0.978565i	$-0.433976\pi$
$37$	2.50533	0.411873	0.205937	+	0.978565i	$0.433976\pi$
$38$	14.9464	2.42463
$39$	0	0
$40$	−0.692376	−0.109474
$41$	−1.48908	−0.232555	−0.116277	−	0.993217i	$-0.537096\pi$
$41$	−1.48908	−0.232555	−0.116277	+	0.993217i	$0.537096\pi$
$42$	0	0
$43$	−6.73850	−1.02761	−0.513806	−	0.857906i	$-0.671765\pi$
$43$	−6.73850	−1.02761	−0.513806	+	0.857906i	$0.671765\pi$
$44$	−1.52020	−0.229179
$45$	0	0
$46$	13.4695	1.98597
$47$	−12.0321	−1.75506	−0.877529	−	0.479524i	$-0.840809\pi$
$47$	−12.0321	−1.75506	−0.877529	+	0.479524i	$0.840809\pi$
$48$	0	0
$49$	−6.27720	−0.896743
$50$	8.27121	1.16973
$51$	0	0
$52$	−0.739608	−0.102565
$53$	11.5877	1.59170	0.795848	−	0.605497i	$-0.207026\pi$
$53$	11.5877	1.59170	0.795848	+	0.605497i	$0.207026\pi$
$54$	0	0
$55$	0.769125	0.103709
$56$	−0.765339	−0.102273
$57$	0	0
$58$	−7.63663	−1.00274
$59$	4.12976	0.537649	0.268824	−	0.963189i	$-0.413365\pi$
$59$	4.12976	0.537649	0.268824	+	0.963189i	$0.413365\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	14.7678	1.87551
$63$	0	0
$64$	−3.81162	−0.476453
$65$	0.374195	0.0464132
$66$	0	0
$67$	6.50801	0.795080	0.397540	−	0.917585i	$-0.369864\pi$
$67$	6.50801	0.795080	0.397540	+	0.917585i	$0.369864\pi$
$68$	−4.06943	−0.493491
$69$	0	0
$70$	−1.22684	−0.146636
$71$	7.19240	0.853580	0.426790	−	0.904351i	$-0.359644\pi$
$71$	7.19240	0.853580	0.426790	+	0.904351i	$0.359644\pi$
$72$	0	0
$73$	−7.31828	−0.856539	−0.428270	−	0.903651i	$-0.640877\pi$
$73$	−7.31828	−0.856539	−0.428270	+	0.903651i	$0.640877\pi$
$74$	−4.70054	−0.546427
$75$	0	0
$76$	−12.1103	−1.38914
$77$	0.850176	0.0968866
$78$	0	0
$79$	8.91585	1.00311	0.501556	−	0.865125i	$-0.332761\pi$
$79$	8.91585	1.00311	0.501556	+	0.865125i	$0.332761\pi$
$80$	3.63749	0.406684
$81$	0	0
$82$	2.79383	0.308527
$83$	9.65801	1.06010	0.530052	−	0.847965i	$-0.322172\pi$
$83$	9.65801	1.06010	0.530052	+	0.847965i	$0.322172\pi$
$84$	0	0
$85$	2.05888	0.223316
$86$	12.6429	1.36332
$87$	0	0
$88$	−0.900213	−0.0959630
$89$	−7.33234	−0.777226	−0.388613	−	0.921401i	$-0.627046\pi$
$89$	−7.33234	−0.777226	−0.388613	+	0.921401i	$0.627046\pi$
$90$	0	0
$91$	0.413628	0.0433600
$92$	−10.9136	−1.13782
$93$	0	0
$94$	22.5748	2.32841
$95$	6.12704	0.628621
$96$	0	0
$97$	5.47164	0.555560	0.277780	−	0.960645i	$-0.410401\pi$
$97$	5.47164	0.555560	0.277780	+	0.960645i	$0.410401\pi$
$98$	11.7774	1.18970
$99$	0	0
$100$	−6.70172	−0.670172
$101$	11.9824	1.19229	0.596146	−	0.802876i	$-0.296698\pi$
$101$	11.9824	1.19229	0.596146	+	0.802876i	$0.296698\pi$
$102$	0	0
$103$	−17.6317	−1.73730	−0.868652	−	0.495422i	$-0.835013\pi$
$103$	−17.6317	−1.73730	−0.868652	+	0.495422i	$0.835013\pi$
$104$	−0.437972	−0.0429467
$105$	0	0
$106$	−21.7411	−2.11168
$107$	−6.79965	−0.657347	−0.328674	−	0.944444i	$-0.606602\pi$
$107$	−6.79965	−0.657347	−0.328674	+	0.944444i	$0.606602\pi$
$108$	0	0
$109$	5.41437	0.518602	0.259301	−	0.965797i	$-0.416508\pi$
$109$	5.41437	0.518602	0.259301	+	0.965797i	$0.416508\pi$
$110$	−1.44305	−0.137589
$111$	0	0
$112$	4.02082	0.379932
$113$	11.0498	1.03948	0.519739	−	0.854325i	$-0.326029\pi$
$113$	11.0498	1.03948	0.519739	+	0.854325i	$0.326029\pi$
$114$	0	0
$115$	5.52159	0.514890
$116$	6.18755	0.574499
$117$	0	0
$118$	−7.74833	−0.713292
$119$	2.27584	0.208626
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.87622	−0.169865
$123$	0	0
$124$	−11.9655	−1.07454
$125$	7.23627	0.647232
$126$	0	0
$127$	−21.3031	−1.89035	−0.945173	−	0.326571i	$-0.894107\pi$
$127$	−21.3031	−1.89035	−0.945173	+	0.326571i	$0.894107\pi$
$128$	−6.99446	−0.618229
$129$	0	0
$130$	−0.702072	−0.0615758
$131$	−16.3115	−1.42515	−0.712573	−	0.701598i	$-0.752470\pi$
$131$	−16.3115	−1.42515	−0.712573	+	0.701598i	$0.752470\pi$
$132$	0	0
$133$	6.77271	0.587269
$134$	−12.2104	−1.05482
$135$	0	0
$136$	−2.40979	−0.206637
$137$	−6.16767	−0.526940	−0.263470	−	0.964668i	$-0.584867\pi$
$137$	−6.16767	−0.526940	−0.263470	+	0.964668i	$0.584867\pi$
$138$	0	0
$139$	−7.40889	−0.628414	−0.314207	−	0.949354i	$-0.601739\pi$
$139$	−7.40889	−0.628414	−0.314207	+	0.949354i	$0.601739\pi$
$140$	0.994045	0.0840121
$141$	0	0
$142$	−13.4945	−1.13243
$143$	0.486521	0.0406849
$144$	0	0
$145$	−3.13051	−0.259975
$146$	13.7307	1.13636
$147$	0	0
$148$	3.80860	0.313065
$149$	−14.8737	−1.21850	−0.609249	−	0.792979i	$-0.708529\pi$
$149$	−14.8737	−1.21850	−0.609249	+	0.792979i	$0.708529\pi$
$150$	0	0
$151$	−3.78514	−0.308030	−0.154015	−	0.988069i	$-0.549220\pi$
$151$	−3.78514	−0.308030	−0.154015	+	0.988069i	$0.549220\pi$
$152$	−7.17132	−0.581671
$153$	0	0
$154$	−1.59512	−0.128538
$155$	6.05381	0.486254
$156$	0	0
$157$	16.9525	1.35296	0.676479	−	0.736462i	$-0.263505\pi$
$157$	16.9525	1.35296	0.676479	+	0.736462i	$0.263505\pi$
$158$	−16.7281	−1.33082
$159$	0	0
$160$	−5.43998	−0.430068
$161$	6.10346	0.481020
$162$	0	0
$163$	6.32294	0.495251	0.247625	−	0.968856i	$-0.420350\pi$
$163$	6.32294	0.495251	0.247625	+	0.968856i	$0.420350\pi$
$164$	−2.26369	−0.176765
$165$	0	0
$166$	−18.1205	−1.40643
$167$	0.662031	0.0512295	0.0256147	−	0.999672i	$-0.491846\pi$
$167$	0.662031	0.0512295	0.0256147	+	0.999672i	$0.491846\pi$
$168$	0	0
$169$	−12.7633	−0.981792
$170$	−3.86290	−0.296271
$171$	0	0
$172$	−10.2439	−0.781087
$173$	20.1492	1.53192	0.765959	−	0.642890i	$-0.222265\pi$
$173$	20.1492	1.53192	0.765959	+	0.642890i	$0.222265\pi$
$174$	0	0
$175$	3.74796	0.283319
$176$	4.72939	0.356491
$177$	0	0
$178$	13.7571	1.03114
$179$	17.6388	1.31839	0.659194	−	0.751973i	$-0.270897\pi$
$179$	17.6388	1.31839	0.659194	+	0.751973i	$0.270897\pi$
$180$	0	0
$181$	−1.79338	−0.133301	−0.0666503	−	0.997776i	$-0.521231\pi$
$181$	−1.79338	−0.133301	−0.0666503	+	0.997776i	$0.521231\pi$
$182$	−0.776057	−0.0575252
$183$	0	0
$184$	−6.46267	−0.476435
$185$	−1.92691	−0.141669
$186$	0	0
$187$	2.67691	0.195755
$188$	−18.2911	−1.33402
$189$	0	0
$190$	−11.4957	−0.833983
$191$	2.49324	0.180405	0.0902023	−	0.995923i	$-0.471249\pi$
$191$	2.49324	0.180405	0.0902023	+	0.995923i	$0.471249\pi$
$192$	0	0
$193$	−5.72570	−0.412145	−0.206072	−	0.978537i	$-0.566068\pi$
$193$	−5.72570	−0.412145	−0.206072	+	0.978537i	$0.566068\pi$
$194$	−10.2660	−0.737055
$195$	0	0
$196$	−9.54259	−0.681614
$197$	18.9594	1.35080	0.675399	−	0.737452i	$-0.263971\pi$
$197$	18.9594	1.35080	0.675399	+	0.737452i	$0.263971\pi$
$198$	0	0
$199$	1.22929	0.0871423	0.0435711	−	0.999050i	$-0.486126\pi$
$199$	1.22929	0.0871423	0.0435711	+	0.999050i	$0.486126\pi$
$200$	−3.96854	−0.280618
$201$	0	0
$202$	−22.4816	−1.58180
$203$	−3.46041	−0.242873
$204$	0	0
$205$	1.14529	0.0799902
$206$	33.0810	2.30486
$207$	0	0
$208$	2.30095	0.159542
$209$	7.96625	0.551037
$210$	0	0
$211$	0.127788	0.00879728	0.00439864	−	0.999990i	$-0.498600\pi$
$211$	0.127788	0.00879728	0.00439864	+	0.999990i	$0.498600\pi$
$212$	17.6156	1.20985
$213$	0	0
$214$	12.7576	0.872095
$215$	5.18275	0.353461
$216$	0	0
$217$	6.69177	0.454267
$218$	−10.1585	−0.688023
$219$	0	0
$220$	1.16922	0.0788290
$221$	1.30237	0.0876069
$222$	0	0
$223$	0.499136	0.0334246	0.0167123	−	0.999860i	$-0.494680\pi$
$223$	0.499136	0.0334246	0.0167123	+	0.999860i	$0.494680\pi$
$224$	−6.01326	−0.401778
$225$	0	0
$226$	−20.7319	−1.37906
$227$	3.65059	0.242298	0.121149	−	0.992634i	$-0.461342\pi$
$227$	3.65059	0.242298	0.121149	+	0.992634i	$0.461342\pi$
$228$	0	0
$229$	7.86713	0.519875	0.259937	−	0.965625i	$-0.416298\pi$
$229$	7.86713	0.519875	0.259937	+	0.965625i	$0.416298\pi$
$230$	−10.3597	−0.683099
$231$	0	0
$232$	3.66407	0.240558
$233$	5.58116	0.365634	0.182817	−	0.983147i	$-0.441478\pi$
$233$	5.58116	0.365634	0.182817	+	0.983147i	$0.441478\pi$
$234$	0	0
$235$	9.25416	0.603675
$236$	6.27805	0.408666
$237$	0	0
$238$	−4.26998	−0.276782
$239$	−7.54659	−0.488148	−0.244074	−	0.969757i	$-0.578484\pi$
$239$	−7.54659	−0.488148	−0.244074	+	0.969757i	$0.578484\pi$
$240$	0	0
$241$	−2.50601	−0.161426	−0.0807131	−	0.996737i	$-0.525720\pi$
$241$	−2.50601	−0.161426	−0.0807131	+	0.996737i	$0.525720\pi$
$242$	−1.87622	−0.120608
$243$	0	0
$244$	1.52020	0.0973207
$245$	4.82795	0.308446
$246$	0	0
$247$	3.87574	0.246608
$248$	−7.08561	−0.449937
$249$	0	0
$250$	−13.5768	−0.858674
$251$	−21.8824	−1.38120	−0.690601	−	0.723236i	$-0.742654\pi$
$251$	−21.8824	−1.38120	−0.690601	+	0.723236i	$0.742654\pi$
$252$	0	0
$253$	7.17905	0.451343
$254$	39.9693	2.50790
$255$	0	0
$256$	20.7464	1.29665
$257$	−7.44037	−0.464118	−0.232059	−	0.972702i	$-0.574546\pi$
$257$	−7.44037	−0.464118	−0.232059	+	0.972702i	$0.574546\pi$
$258$	0	0
$259$	−2.12997	−0.132350
$260$	0.568851	0.0352786
$261$	0	0
$262$	30.6040	1.89072
$263$	−6.09108	−0.375592	−0.187796	−	0.982208i	$-0.560134\pi$
$263$	−6.09108	−0.375592	−0.187796	+	0.982208i	$0.560134\pi$
$264$	0	0
$265$	−8.91240	−0.547484
$266$	−12.7071	−0.779122
$267$	0	0
$268$	9.89346	0.604339
$269$	4.61827	0.281581	0.140791	−	0.990039i	$-0.455036\pi$
$269$	4.61827	0.281581	0.140791	+	0.990039i	$0.455036\pi$
$270$	0	0
$271$	13.4329	0.815992	0.407996	−	0.912984i	$-0.366228\pi$
$271$	13.4329	0.815992	0.407996	+	0.912984i	$0.366228\pi$
$272$	12.6601	0.767634
$273$	0	0
$274$	11.5719	0.699085
$275$	4.40845	0.265839
$276$	0	0
$277$	4.15123	0.249423	0.124712	−	0.992193i	$-0.460199\pi$
$277$	4.15123	0.249423	0.124712	+	0.992193i	$0.460199\pi$
$278$	13.9007	0.833709
$279$	0	0
$280$	0.588641	0.0351780
$281$	−4.01510	−0.239520	−0.119760	−	0.992803i	$-0.538213\pi$
$281$	−4.01510	−0.239520	−0.119760	+	0.992803i	$0.538213\pi$
$282$	0	0
$283$	10.1947	0.606014	0.303007	−	0.952988i	$-0.402009\pi$
$283$	10.1947	0.606014	0.303007	+	0.952988i	$0.402009\pi$
$284$	10.9339	0.648806
$285$	0	0
$286$	−0.912819	−0.0539761
$287$	1.26598	0.0747283
$288$	0	0
$289$	−9.83417	−0.578481
$290$	5.87352	0.344905
$291$	0	0
$292$	−11.1252	−0.651055
$293$	−8.29294	−0.484479	−0.242239	−	0.970216i	$-0.577882\pi$
$293$	−8.29294	−0.484479	−0.242239	+	0.970216i	$0.577882\pi$
$294$	0	0
$295$	−3.17630	−0.184931
$296$	2.25533	0.131088
$297$	0	0
$298$	27.9062	1.61657
$299$	3.49276	0.201991
$300$	0	0
$301$	5.72891	0.330209
$302$	7.10174	0.408659
$303$	0	0
$304$	37.6755	2.16084
$305$	−0.769125	−0.0440400
$306$	0	0
$307$	−30.2048	−1.72388	−0.861940	−	0.507011i	$-0.830750\pi$
$307$	−30.2048	−1.72388	−0.861940	+	0.507011i	$0.830750\pi$
$308$	1.29244	0.0736434
$309$	0	0
$310$	−11.3583	−0.645107
$311$	−3.18263	−0.180470	−0.0902352	−	0.995920i	$-0.528762\pi$
$311$	−3.18263	−0.180470	−0.0902352	+	0.995920i	$0.528762\pi$
$312$	0	0
$313$	28.9554	1.63665	0.818327	−	0.574753i	$-0.194902\pi$
$313$	28.9554	1.63665	0.818327	+	0.574753i	$0.194902\pi$
$314$	−31.8066	−1.79495
$315$	0	0
$316$	13.5539	0.762465
$317$	21.9392	1.23223	0.616114	−	0.787657i	$-0.288706\pi$
$317$	21.9392	1.23223	0.616114	+	0.787657i	$0.288706\pi$
$318$	0	0
$319$	−4.07022	−0.227889
$320$	2.93161	0.163882
$321$	0	0
$322$	−11.4514	−0.638163
$323$	21.3249	1.18655
$324$	0	0
$325$	2.14480	0.118972
$326$	−11.8632	−0.657043
$327$	0	0
$328$	−1.34049	−0.0740159
$329$	10.2294	0.563964
$330$	0	0
$331$	9.42763	0.518189	0.259095	−	0.965852i	$-0.416576\pi$
$331$	9.42763	0.518189	0.259095	+	0.965852i	$0.416576\pi$
$332$	14.6821	0.805785
$333$	0	0
$334$	−1.24212	−0.0679655
$335$	−5.00547	−0.273478
$336$	0	0
$337$	−24.9097	−1.35692	−0.678459	−	0.734639i	$-0.737352\pi$
$337$	−24.9097	−1.35692	−0.678459	+	0.734639i	$0.737352\pi$
$338$	23.9467	1.30253
$339$	0	0
$340$	3.12990	0.169743
$341$	7.87104	0.426241
$342$	0	0
$343$	11.2880	0.609492
$344$	−6.06608	−0.327061
$345$	0	0
$346$	−37.8044	−2.03238
$347$	−1.87324	−0.100561	−0.0502803	−	0.998735i	$-0.516011\pi$
$347$	−1.87324	−0.100561	−0.0502803	+	0.998735i	$0.516011\pi$
$348$	0	0
$349$	4.49899	0.240826	0.120413	−	0.992724i	$-0.461578\pi$
$349$	4.49899	0.240826	0.120413	+	0.992724i	$0.461578\pi$
$350$	−7.03199	−0.375876
$351$	0	0
$352$	−7.07295	−0.376990
$353$	2.48662	0.132350	0.0661748	−	0.997808i	$-0.478920\pi$
$353$	2.48662	0.132350	0.0661748	+	0.997808i	$0.478920\pi$
$354$	0	0
$355$	−5.53185	−0.293600
$356$	−11.1466	−0.590769
$357$	0	0
$358$	−33.0943	−1.74909
$359$	−14.6012	−0.770620	−0.385310	−	0.922787i	$-0.625905\pi$
$359$	−14.6012	−0.770620	−0.385310	+	0.922787i	$0.625905\pi$
$360$	0	0
$361$	44.4611	2.34006
$362$	3.36477	0.176848
$363$	0	0
$364$	0.628797	0.0329579
$365$	5.62867	0.294618
$366$	0	0
$367$	−25.9792	−1.35610	−0.678051	−	0.735015i	$-0.737175\pi$
$367$	−25.9792	−1.35610	−0.678051	+	0.735015i	$0.737175\pi$
$368$	33.9526	1.76990
$369$	0	0
$370$	3.61530	0.187951
$371$	−9.85160	−0.511469
$372$	0	0
$373$	−15.5082	−0.802983	−0.401491	−	0.915863i	$-0.631508\pi$
$373$	−15.5082	−0.802983	−0.401491	+	0.915863i	$0.631508\pi$
$374$	−5.02246	−0.259705
$375$	0	0
$376$	−10.8314	−0.558588
$377$	−1.98025	−0.101988
$378$	0	0
$379$	32.2899	1.65862	0.829311	−	0.558788i	$-0.188733\pi$
$379$	32.2899	1.65862	0.829311	+	0.558788i	$0.188733\pi$
$380$	9.31432	0.477814
$381$	0	0
$382$	−4.67787	−0.239340
$383$	−10.1071	−0.516447	−0.258224	−	0.966085i	$-0.583137\pi$
$383$	−10.1071	−0.516447	−0.258224	+	0.966085i	$0.583137\pi$
$384$	0	0
$385$	−0.653891	−0.0333254
$386$	10.7427	0.546787
$387$	0	0
$388$	8.31797	0.422281
$389$	2.49677	0.126591	0.0632957	−	0.997995i	$-0.479839\pi$
$389$	2.49677	0.126591	0.0632957	+	0.997995i	$0.479839\pi$
$390$	0	0
$391$	19.2177	0.971878
$392$	−5.65082	−0.285409
$393$	0	0
$394$	−35.5719	−1.79209
$395$	−6.85740	−0.345033
$396$	0	0
$397$	−2.05342	−0.103058	−0.0515291	−	0.998671i	$-0.516410\pi$
$397$	−2.05342	−0.103058	−0.0515291	+	0.998671i	$0.516410\pi$
$398$	−2.30642	−0.115611
$399$	0	0
$400$	20.8493	1.04246
$401$	−4.15456	−0.207469	−0.103734	−	0.994605i	$-0.533079\pi$
$401$	−4.15456	−0.207469	−0.103734	+	0.994605i	$0.533079\pi$
$402$	0	0
$403$	3.82942	0.190757
$404$	18.2156	0.906260
$405$	0	0
$406$	6.49248	0.322216
$407$	−2.50533	−0.124184
$408$	0	0
$409$	−11.3164	−0.559559	−0.279779	−	0.960064i	$-0.590261\pi$
$409$	−11.3164	−0.559559	−0.279779	+	0.960064i	$0.590261\pi$
$410$	−2.14881	−0.106122
$411$	0	0
$412$	−26.8037	−1.32052
$413$	−3.51102	−0.172766
$414$	0	0
$415$	−7.42821	−0.364637
$416$	−3.44114	−0.168716
$417$	0	0
$418$	−14.9464	−0.731054
$419$	−18.1932	−0.888794	−0.444397	−	0.895830i	$-0.646582\pi$
$419$	−18.1932	−0.888794	−0.444397	+	0.895830i	$0.646582\pi$
$420$	0	0
$421$	−5.49672	−0.267894	−0.133947	−	0.990989i	$-0.542765\pi$
$421$	−5.49672	−0.267894	−0.133947	+	0.990989i	$0.542765\pi$
$422$	−0.239758	−0.0116712
$423$	0	0
$424$	10.4314	0.506594
$425$	11.8010	0.572433
$426$	0	0
$427$	−0.850176	−0.0411429
$428$	−10.3368	−0.499649
$429$	0	0
$430$	−9.72397	−0.468932
$431$	−10.7579	−0.518189	−0.259095	−	0.965852i	$-0.583424\pi$
$431$	−10.7579	−0.518189	−0.259095	+	0.965852i	$0.583424\pi$
$432$	0	0
$433$	0.659986	0.0317169	0.0158584	−	0.999874i	$-0.494952\pi$
$433$	0.659986	0.0317169	0.0158584	+	0.999874i	$0.494952\pi$
$434$	−12.5552	−0.602670
$435$	0	0
$436$	8.23091	0.394189
$437$	57.1901	2.73577
$438$	0	0
$439$	−6.46783	−0.308692	−0.154346	−	0.988017i	$-0.549327\pi$
$439$	−6.46783	−0.308692	−0.154346	+	0.988017i	$0.549327\pi$
$440$	0.692376	0.0330077
$441$	0	0
$442$	−2.44353	−0.116227
$443$	−32.9602	−1.56598	−0.782992	−	0.622031i	$-0.786308\pi$
$443$	−32.9602	−1.56598	−0.782992	+	0.622031i	$0.786308\pi$
$444$	0	0
$445$	5.63948	0.267337
$446$	−0.936488	−0.0443440
$447$	0	0
$448$	3.24055	0.153102
$449$	32.5610	1.53665	0.768323	−	0.640062i	$-0.221091\pi$
$449$	32.5610	1.53665	0.768323	+	0.640062i	$0.221091\pi$
$450$	0	0
$451$	1.48908	0.0701179
$452$	16.7979	0.790107
$453$	0	0
$454$	−6.84931	−0.321454
$455$	−0.318132	−0.0149142
$456$	0	0
$457$	−11.8562	−0.554609	−0.277304	−	0.960782i	$-0.589441\pi$
$457$	−11.8562	−0.554609	−0.277304	+	0.960782i	$0.589441\pi$
$458$	−14.7605	−0.689711
$459$	0	0
$460$	8.39391	0.391368
$461$	−34.1302	−1.58960	−0.794800	−	0.606871i	$-0.792424\pi$
$461$	−34.1302	−1.58960	−0.794800	+	0.606871i	$0.792424\pi$
$462$	0	0
$463$	−0.481777	−0.0223901	−0.0111950	−	0.999937i	$-0.503564\pi$
$463$	−0.481777	−0.0223901	−0.0111950	+	0.999937i	$0.503564\pi$
$464$	−19.2497	−0.893644
$465$	0	0
$466$	−10.4715	−0.485082
$467$	−15.5227	−0.718304	−0.359152	−	0.933279i	$-0.616934\pi$
$467$	−15.5227	−0.718304	−0.359152	+	0.933279i	$0.616934\pi$
$468$	0	0
$469$	−5.53295	−0.255488
$470$	−17.3628	−0.800888
$471$	0	0
$472$	3.71766	0.171119
$473$	6.73850	0.309837
$474$	0	0
$475$	35.1188	1.61136
$476$	3.45973	0.158577
$477$	0	0
$478$	14.1591	0.647620
$479$	11.3271	0.517547	0.258774	−	0.965938i	$-0.416682\pi$
$479$	11.3271	0.517547	0.258774	+	0.965938i	$0.416682\pi$
$480$	0	0
$481$	−1.21889	−0.0555768
$482$	4.70182	0.214162
$483$	0	0
$484$	1.52020	0.0690999
$485$	−4.20837	−0.191092
$486$	0	0
$487$	24.9454	1.13038	0.565192	−	0.824959i	$-0.308802\pi$
$487$	24.9454	1.13038	0.565192	+	0.824959i	$0.308802\pi$
$488$	0.900213	0.0407507
$489$	0	0
$490$	−9.05829	−0.409212
$491$	33.4996	1.51181	0.755907	−	0.654679i	$-0.227196\pi$
$491$	33.4996	1.51181	0.755907	+	0.654679i	$0.227196\pi$
$492$	0	0
$493$	−10.8956	−0.490713
$494$	−7.27175	−0.327171
$495$	0	0
$496$	37.2252	1.67146
$497$	−6.11480	−0.274286
$498$	0	0
$499$	25.8005	1.15499	0.577495	−	0.816394i	$-0.304030\pi$
$499$	25.8005	1.15499	0.577495	+	0.816394i	$0.304030\pi$
$500$	11.0006	0.491960
$501$	0	0
$502$	41.0561	1.83242
$503$	34.6660	1.54568	0.772841	−	0.634600i	$-0.218835\pi$
$503$	34.6660	1.54568	0.772841	+	0.634600i	$0.218835\pi$
$504$	0	0
$505$	−9.21595	−0.410104
$506$	−13.4695	−0.598791
$507$	0	0
$508$	−32.3850	−1.43685
$509$	0.688397	0.0305127	0.0152563	−	0.999884i	$-0.495144\pi$
$509$	0.688397	0.0305127	0.0152563	+	0.999884i	$0.495144\pi$
$510$	0	0
$511$	6.22182	0.275237
$512$	−24.9359	−1.10202
$513$	0	0
$514$	13.9598	0.615739
$515$	13.5610	0.597569
$516$	0	0
$517$	12.0321	0.529170
$518$	3.99629	0.175587
$519$	0	0
$520$	0.336855	0.0147721
$521$	5.22647	0.228976	0.114488	−	0.993425i	$-0.463477\pi$
$521$	5.22647	0.228976	0.114488	+	0.993425i	$0.463477\pi$
$522$	0	0
$523$	16.2044	0.708570	0.354285	−	0.935138i	$-0.384724\pi$
$523$	16.2044	0.708570	0.354285	+	0.935138i	$0.384724\pi$
$524$	−24.7968	−1.08325
$525$	0	0
$526$	11.4282	0.498293
$527$	21.0700	0.917825
$528$	0	0
$529$	28.5388	1.24082
$530$	16.7216	0.726340
$531$	0	0
$532$	10.2959	0.446383
$533$	0.724467	0.0313801
$534$	0	0
$535$	5.22978	0.226103
$536$	5.85859	0.253053
$537$	0	0
$538$	−8.66490	−0.373570
$539$	6.27720	0.270378
$540$	0	0
$541$	36.8970	1.58632	0.793162	−	0.609010i	$-0.208433\pi$
$541$	36.8970	1.58632	0.793162	+	0.609010i	$0.208433\pi$
$542$	−25.2031	−1.08257
$543$	0	0
$544$	−18.9336	−0.811773
$545$	−4.16432	−0.178380
$546$	0	0
$547$	37.1624	1.58895	0.794474	−	0.607298i	$-0.207747\pi$
$547$	37.1624	1.58895	0.794474	+	0.607298i	$0.207747\pi$
$548$	−9.37609	−0.400527
$549$	0	0
$550$	−8.27121	−0.352686
$551$	−32.4244	−1.38133
$552$	0	0
$553$	−7.58005	−0.322336
$554$	−7.78861	−0.330906
$555$	0	0
$556$	−11.2630	−0.477657
$557$	−1.65849	−0.0702726	−0.0351363	−	0.999383i	$-0.511187\pi$
$557$	−1.65849	−0.0702726	−0.0351363	+	0.999383i	$0.511187\pi$
$558$	0	0
$559$	3.27842	0.138662
$560$	−3.09251	−0.130682
$561$	0	0
$562$	7.53320	0.317769
$563$	−22.9792	−0.968457	−0.484228	−	0.874942i	$-0.660900\pi$
$563$	−22.9792	−0.968457	−0.484228	+	0.874942i	$0.660900\pi$
$564$	0	0
$565$	−8.49868	−0.357542
$566$	−19.1276	−0.803992
$567$	0	0
$568$	6.47469	0.271672
$569$	−9.90895	−0.415405	−0.207702	−	0.978192i	$-0.566599\pi$
$569$	−9.90895	−0.415405	−0.207702	+	0.978192i	$0.566599\pi$
$570$	0	0
$571$	25.6468	1.07328	0.536642	−	0.843810i	$-0.319692\pi$
$571$	25.6468	1.07328	0.536642	+	0.843810i	$0.319692\pi$
$572$	0.739608	0.0309246
$573$	0	0
$574$	−2.37525	−0.0991411
$575$	31.6485	1.31983
$576$	0	0
$577$	−21.8912	−0.911344	−0.455672	−	0.890148i	$-0.650601\pi$
$577$	−21.8912	−0.911344	−0.455672	+	0.890148i	$0.650601\pi$
$578$	18.4511	0.767463
$579$	0	0
$580$	−4.75900	−0.197607
$581$	−8.21101	−0.340650
$582$	0	0
$583$	−11.5877	−0.479914
$584$	−6.58800	−0.272614
$585$	0	0
$586$	15.5594	0.642752
$587$	−23.1670	−0.956205	−0.478102	−	0.878304i	$-0.658675\pi$
$587$	−23.1670	−0.956205	−0.478102	+	0.878304i	$0.658675\pi$
$588$	0	0
$589$	62.7027	2.58362
$590$	5.95943	0.245346
$591$	0	0
$592$	−11.8487	−0.486978
$593$	−12.9029	−0.529860	−0.264930	−	0.964268i	$-0.585349\pi$
$593$	−12.9029	−0.529860	−0.264930	+	0.964268i	$0.585349\pi$
$594$	0	0
$595$	−1.75041	−0.0717597
$596$	−22.6109	−0.926179
$597$	0	0
$598$	−6.55318	−0.267979
$599$	17.2261	0.703840	0.351920	−	0.936030i	$-0.385529\pi$
$599$	17.2261	0.703840	0.351920	+	0.936030i	$0.385529\pi$
$600$	0	0
$601$	−37.7259	−1.53887	−0.769436	−	0.638724i	$-0.779463\pi$
$601$	−37.7259	−1.53887	−0.769436	+	0.638724i	$0.779463\pi$
$602$	−10.7487	−0.438084
$603$	0	0
$604$	−5.75416	−0.234133
$605$	−0.769125	−0.0312694
$606$	0	0
$607$	−18.1220	−0.735549	−0.367774	−	0.929915i	$-0.619880\pi$
$607$	−18.1220	−0.735549	−0.367774	+	0.929915i	$0.619880\pi$
$608$	−56.3449	−2.28509
$609$	0	0
$610$	1.44305	0.0584272
$611$	5.85385	0.236821
$612$	0	0
$613$	14.7891	0.597328	0.298664	−	0.954358i	$-0.403459\pi$
$613$	14.7891	0.597328	0.298664	+	0.954358i	$0.403459\pi$
$614$	56.6708	2.28705
$615$	0	0
$616$	0.765339	0.0308364
$617$	21.6484	0.871531	0.435766	−	0.900060i	$-0.356478\pi$
$617$	21.6484	0.871531	0.435766	+	0.900060i	$0.356478\pi$
$618$	0	0
$619$	40.5540	1.63000	0.815001	−	0.579460i	$-0.196736\pi$
$619$	40.5540	1.63000	0.815001	+	0.579460i	$0.196736\pi$
$620$	9.20300	0.369601
$621$	0	0
$622$	5.97131	0.239428
$623$	6.23378	0.249751
$624$	0	0
$625$	16.4766	0.659066
$626$	−54.3266	−2.17133
$627$	0	0
$628$	25.7712	1.02838
$629$	−6.70653	−0.267407
$630$	0	0
$631$	−0.399645	−0.0159096	−0.00795481	−	0.999968i	$-0.502532\pi$
$631$	−0.399645	−0.0159096	−0.00795481	+	0.999968i	$0.502532\pi$
$632$	8.02616	0.319264
$633$	0	0
$634$	−41.1627	−1.63478
$635$	16.3847	0.650209
$636$	0	0
$637$	3.05399	0.121003
$638$	7.63663	0.302337
$639$	0	0
$640$	5.37961	0.212648
$641$	−0.567229	−0.0224042	−0.0112021	−	0.999937i	$-0.503566\pi$
$641$	−0.567229	−0.0224042	−0.0112021	+	0.999937i	$0.503566\pi$
$642$	0	0
$643$	−0.146710	−0.00578566	−0.00289283	−	0.999996i	$-0.500921\pi$
$643$	−0.146710	−0.00578566	−0.00289283	+	0.999996i	$0.500921\pi$
$644$	9.27847	0.365623
$645$	0	0
$646$	−40.0102	−1.57418
$647$	36.4434	1.43274	0.716368	−	0.697722i	$-0.245803\pi$
$647$	36.4434	1.43274	0.716368	+	0.697722i	$0.245803\pi$
$648$	0	0
$649$	−4.12976	−0.162107
$650$	−4.02412	−0.157839
$651$	0	0
$652$	9.61212	0.376440
$653$	−9.17320	−0.358975	−0.179488	−	0.983760i	$-0.557444\pi$
$653$	−9.17320	−0.358975	−0.179488	+	0.983760i	$0.557444\pi$
$654$	0	0
$655$	12.5456	0.490198
$656$	7.04243	0.274961
$657$	0	0
$658$	−19.1926	−0.748204
$659$	−21.5384	−0.839019	−0.419509	−	0.907751i	$-0.637798\pi$
$659$	−21.5384	−0.839019	−0.419509	+	0.907751i	$0.637798\pi$
$660$	0	0
$661$	−13.1809	−0.512676	−0.256338	−	0.966587i	$-0.582516\pi$
$661$	−13.1809	−0.512676	−0.256338	+	0.966587i	$0.582516\pi$
$662$	−17.6883	−0.687475
$663$	0	0
$664$	8.69426	0.337403
$665$	−5.20906	−0.201999
$666$	0	0
$667$	−29.2203	−1.13142
$668$	1.00642	0.0389395
$669$	0	0
$670$	9.39136	0.362820
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−38.5935	−1.48767	−0.743836	−	0.668363i	$-0.766995\pi$
$673$	−38.5935	−1.48767	−0.743836	+	0.668363i	$0.766995\pi$
$674$	46.7360	1.80021
$675$	0	0
$676$	−19.4027	−0.746260
$677$	−21.2526	−0.816803	−0.408401	−	0.912802i	$-0.633914\pi$
$677$	−21.2526	−0.816803	−0.408401	+	0.912802i	$0.633914\pi$
$678$	0	0
$679$	−4.65185	−0.178522
$680$	1.85343	0.0710756
$681$	0	0
$682$	−14.7678	−0.565488
$683$	33.1546	1.26862	0.634312	−	0.773077i	$-0.281283\pi$
$683$	33.1546	1.26862	0.634312	+	0.773077i	$0.281283\pi$
$684$	0	0
$685$	4.74371	0.181248
$686$	−21.1787	−0.808606
$687$	0	0
$688$	31.8690	1.21500
$689$	−5.63766	−0.214778
$690$	0	0
$691$	37.4173	1.42342	0.711710	−	0.702473i	$-0.247921\pi$
$691$	37.4173	1.42342	0.711710	+	0.702473i	$0.247921\pi$
$692$	30.6308	1.16441
$693$	0	0
$694$	3.51460	0.133412
$695$	5.69836	0.216151
$696$	0	0
$697$	3.98612	0.150985
$698$	−8.44110	−0.319500
$699$	0	0
$700$	5.69764	0.215350
$701$	−1.40367	−0.0530160	−0.0265080	−	0.999649i	$-0.508439\pi$
$701$	−1.40367	−0.0530160	−0.0265080	+	0.999649i	$0.508439\pi$
$702$	0	0
$703$	−19.9581	−0.752733
$704$	3.81162	0.143656
$705$	0	0
$706$	−4.66545	−0.175587
$707$	−10.1871	−0.383127
$708$	0	0
$709$	−1.64388	−0.0617374	−0.0308687	−	0.999523i	$-0.509827\pi$
$709$	−1.64388	−0.0617374	−0.0308687	+	0.999523i	$0.509827\pi$
$710$	10.3790	0.389516
$711$	0	0
$712$	−6.60066	−0.247370
$713$	56.5066	2.11619
$714$	0	0
$715$	−0.374195	−0.0139941
$716$	26.8145	1.00211
$717$	0	0
$718$	27.3950	1.02237
$719$	53.5366	1.99658	0.998289	−	0.0584757i	$-0.0186240\pi$
$719$	53.5366	1.99658	0.998289	+	0.0584757i	$0.0186240\pi$
$720$	0	0
$721$	14.9901	0.558259
$722$	−83.4188	−3.10453
$723$	0	0
$724$	−2.72629	−0.101322
$725$	−17.9434	−0.666400
$726$	0	0
$727$	38.5425	1.42946	0.714730	−	0.699400i	$-0.246549\pi$
$727$	38.5425	1.42946	0.714730	+	0.699400i	$0.246549\pi$
$728$	0.372353	0.0138003
$729$	0	0
$730$	−10.5606	−0.390866
$731$	18.0383	0.667172
$732$	0	0
$733$	−28.9121	−1.06789	−0.533946	−	0.845519i	$-0.679291\pi$
$733$	−28.9121	−1.06789	−0.533946	+	0.845519i	$0.679291\pi$
$734$	48.7426	1.79912
$735$	0	0
$736$	−50.7771	−1.87167
$737$	−6.50801	−0.239725
$738$	0	0
$739$	40.6687	1.49602	0.748012	−	0.663685i	$-0.231009\pi$
$739$	40.6687	1.49602	0.748012	+	0.663685i	$0.231009\pi$
$740$	−2.92929	−0.107683
$741$	0	0
$742$	18.4838	0.678560
$743$	33.9551	1.24569	0.622846	−	0.782344i	$-0.285976\pi$
$743$	33.9551	1.24569	0.622846	+	0.782344i	$0.285976\pi$
$744$	0	0
$745$	11.4397	0.419118
$746$	29.0967	1.06531
$747$	0	0
$748$	4.06943	0.148793
$749$	5.78090	0.211230
$750$	0	0
$751$	−28.4606	−1.03854	−0.519271	−	0.854610i	$-0.673796\pi$
$751$	−28.4606	−1.03854	−0.519271	+	0.854610i	$0.673796\pi$
$752$	56.9044	2.07509
$753$	0	0
$754$	3.71538	0.135306
$755$	2.91124	0.105951
$756$	0	0
$757$	20.7133	0.752839	0.376420	−	0.926449i	$-0.377155\pi$
$757$	20.7133	0.752839	0.376420	+	0.926449i	$0.377155\pi$
$758$	−60.5830	−2.20047
$759$	0	0
$760$	5.51564	0.200073
$761$	14.9737	0.542795	0.271398	−	0.962467i	$-0.412514\pi$
$761$	14.9737	0.542795	0.271398	+	0.962467i	$0.412514\pi$
$762$	0	0
$763$	−4.60316	−0.166646
$764$	3.79022	0.137125
$765$	0	0
$766$	18.9631	0.685164
$767$	−2.00921	−0.0725485
$768$	0	0
$769$	14.3690	0.518161	0.259080	−	0.965856i	$-0.416581\pi$
$769$	14.3690	0.518161	0.259080	+	0.965856i	$0.416581\pi$
$770$	1.22684	0.0442124
$771$	0	0
$772$	−8.70420	−0.313271
$773$	−41.2642	−1.48417	−0.742085	−	0.670306i	$-0.766163\pi$
$773$	−41.2642	−1.48417	−0.742085	+	0.670306i	$0.766163\pi$
$774$	0	0
$775$	34.6991	1.24643
$776$	4.92564	0.176820
$777$	0	0
$778$	−4.68449	−0.167947
$779$	11.8624	0.425013
$780$	0	0
$781$	−7.19240	−0.257364
$782$	−36.0565	−1.28938
$783$	0	0
$784$	29.6874	1.06026
$785$	−13.0386	−0.465368
$786$	0	0
$787$	−46.6666	−1.66349	−0.831743	−	0.555161i	$-0.812657\pi$
$787$	−46.6666	−1.66349	−0.831743	+	0.555161i	$0.812657\pi$
$788$	28.8220	1.02674
$789$	0	0
$790$	12.8660	0.457751
$791$	−9.39428	−0.334022
$792$	0	0
$793$	−0.486521	−0.0172769
$794$	3.85267	0.136726
$795$	0	0
$796$	1.86877	0.0662368
$797$	−26.3191	−0.932272	−0.466136	−	0.884713i	$-0.654354\pi$
$797$	−26.3191	−0.932272	−0.466136	+	0.884713i	$0.654354\pi$
$798$	0	0
$799$	32.2087	1.13946
$800$	−31.1807	−1.10241
$801$	0	0
$802$	7.79486	0.275246
$803$	7.31828	0.258256
$804$	0	0
$805$	−4.69432	−0.165453
$806$	−7.18484	−0.253075
$807$	0	0
$808$	10.7867	0.379475
$809$	−23.7556	−0.835202	−0.417601	−	0.908630i	$-0.637129\pi$
$809$	−23.7556	−0.835202	−0.417601	+	0.908630i	$0.637129\pi$
$810$	0	0
$811$	−1.23871	−0.0434969	−0.0217484	−	0.999763i	$-0.506923\pi$
$811$	−1.23871	−0.0434969	−0.0217484	+	0.999763i	$0.506923\pi$
$812$	−5.26050	−0.184608
$813$	0	0
$814$	4.70054	0.164754
$815$	−4.86313	−0.170348
$816$	0	0
$817$	53.6806	1.87805
$818$	21.2320	0.742360
$819$	0	0
$820$	1.74106	0.0608005
$821$	42.0798	1.46859	0.734297	−	0.678828i	$-0.237512\pi$
$821$	42.0798	1.46859	0.734297	+	0.678828i	$0.237512\pi$
$822$	0	0
$823$	−39.8864	−1.39035	−0.695176	−	0.718839i	$-0.744674\pi$
$823$	−39.8864	−1.39035	−0.695176	+	0.718839i	$0.744674\pi$
$824$	−15.8723	−0.552938
$825$	0	0
$826$	6.58745	0.229207
$827$	−33.9411	−1.18025	−0.590124	−	0.807312i	$-0.700921\pi$
$827$	−33.9411	−1.18025	−0.590124	+	0.807312i	$0.700921\pi$
$828$	0	0
$829$	−15.6166	−0.542388	−0.271194	−	0.962525i	$-0.587419\pi$
$829$	−15.6166	−0.542388	−0.271194	+	0.962525i	$0.587419\pi$
$830$	13.9370	0.483759
$831$	0	0
$832$	1.85443	0.0642909
$833$	16.8035	0.582206
$834$	0	0
$835$	−0.509184	−0.0176211
$836$	12.1103	0.418843
$837$	0	0
$838$	34.1344	1.17915
$839$	−0.0521388	−0.00180003	−0.000900016	−	1.00000i	$-0.500286\pi$
$839$	−0.0521388	−0.00180003	−0.000900016		1.00000i	$0.500286\pi$
$840$	0	0
$841$	−12.4333	−0.428734
$842$	10.3130	0.355411
$843$	0	0
$844$	0.194263	0.00668681
$845$	9.81657	0.337700
$846$	0	0
$847$	−0.850176	−0.0292124
$848$	−54.8029	−1.88194
$849$	0	0
$850$	−22.1413	−0.759439
$851$	−17.9859	−0.616548
$852$	0	0
$853$	−28.5465	−0.977413	−0.488707	−	0.872448i	$-0.662531\pi$
$853$	−28.5465	−0.977413	−0.488707	+	0.872448i	$0.662531\pi$
$854$	1.59512	0.0545838
$855$	0	0
$856$	−6.12114	−0.209216
$857$	−1.76911	−0.0604317	−0.0302159	−	0.999543i	$-0.509619\pi$
$857$	−1.76911	−0.0604317	−0.0302159	+	0.999543i	$0.509619\pi$
$858$	0	0
$859$	15.7366	0.536927	0.268463	−	0.963290i	$-0.413484\pi$
$859$	15.7366	0.536927	0.268463	+	0.963290i	$0.413484\pi$
$860$	7.87881	0.268665
$861$	0	0
$862$	20.1842	0.687475
$863$	12.1406	0.413269	0.206635	−	0.978418i	$-0.433749\pi$
$863$	12.1406	0.413269	0.206635	+	0.978418i	$0.433749\pi$
$864$	0	0
$865$	−15.4973	−0.526923
$866$	−1.23828	−0.0420784
$867$	0	0
$868$	10.1728	0.345288
$869$	−8.91585	−0.302450
$870$	0	0
$871$	−3.16628	−0.107285
$872$	4.87408	0.165057
$873$	0	0
$874$	−107.301	−3.62952
$875$	−6.15210	−0.207979
$876$	0	0
$877$	−26.8909	−0.908040	−0.454020	−	0.890992i	$-0.650011\pi$
$877$	−26.8909	−0.908040	−0.454020	+	0.890992i	$0.650011\pi$
$878$	12.1351	0.409538
$879$	0	0
$880$	−3.63749	−0.122620
$881$	−8.75900	−0.295098	−0.147549	−	0.989055i	$-0.547138\pi$
$881$	−8.75900	−0.295098	−0.147549	+	0.989055i	$0.547138\pi$
$882$	0	0
$883$	23.1635	0.779515	0.389758	−	0.920918i	$-0.372559\pi$
$883$	23.1635	0.779515	0.389758	+	0.920918i	$0.372559\pi$
$884$	1.97986	0.0665899
$885$	0	0
$886$	61.8405	2.07757
$887$	38.9121	1.30654	0.653271	−	0.757124i	$-0.273396\pi$
$887$	38.9121	1.30654	0.653271	+	0.757124i	$0.273396\pi$
$888$	0	0
$889$	18.1114	0.607437
$890$	−10.5809	−0.354673
$891$	0	0
$892$	0.758786	0.0254060
$893$	95.8504	3.20751
$894$	0	0
$895$	−13.5665	−0.453477
$896$	5.94653	0.198659
$897$	0	0
$898$	−61.0915	−2.03865
$899$	−32.0369	−1.06849
$900$	0	0
$901$	−31.0192	−1.03340
$902$	−2.79383	−0.0930245
$903$	0	0
$904$	9.94717	0.330838
$905$	1.37933	0.0458505
$906$	0	0
$907$	−22.0198	−0.731156	−0.365578	−	0.930781i	$-0.619129\pi$
$907$	−22.0198	−0.731156	−0.365578	+	0.930781i	$0.619129\pi$
$908$	5.54962	0.184171
$909$	0	0
$910$	0.596885	0.0197865
$911$	32.6346	1.08123	0.540616	−	0.841269i	$-0.318191\pi$
$911$	32.6346	1.08123	0.540616	+	0.841269i	$0.318191\pi$
$912$	0	0
$913$	−9.65801	−0.319633
$914$	22.2448	0.735792
$915$	0	0
$916$	11.9596	0.395156
$917$	13.8677	0.457951
$918$	0	0
$919$	22.9321	0.756459	0.378229	−	0.925712i	$-0.376533\pi$
$919$	22.9321	0.756459	0.378229	+	0.925712i	$0.376533\pi$
$920$	4.97060	0.163876
$921$	0	0
$922$	64.0357	2.10890
$923$	−3.49925	−0.115179
$924$	0	0
$925$	−11.0446	−0.363144
$926$	0.903919	0.0297046
$927$	0	0
$928$	28.7885	0.945029
$929$	−30.2281	−0.991751	−0.495876	−	0.868394i	$-0.665153\pi$
$929$	−30.2281	−0.991751	−0.495876	+	0.868394i	$0.665153\pi$
$930$	0	0
$931$	50.0057	1.63887
$932$	8.48447	0.277918
$933$	0	0
$934$	29.1240	0.952965
$935$	−2.05888	−0.0673324
$936$	0	0
$937$	−27.6590	−0.903580	−0.451790	−	0.892124i	$-0.649214\pi$
$937$	−27.6590	−0.903580	−0.451790	+	0.892124i	$0.649214\pi$
$938$	10.3810	0.338953
$939$	0	0
$940$	14.0682	0.458853
$941$	0.607129	0.0197919	0.00989593	−	0.999951i	$-0.496850\pi$
$941$	0.607129	0.0197919	0.00989593	+	0.999951i	$0.496850\pi$
$942$	0	0
$943$	10.6902	0.348119
$944$	−19.5313	−0.635688
$945$	0	0
$946$	−12.6429	−0.411056
$947$	−3.14463	−0.102187	−0.0510934	−	0.998694i	$-0.516271\pi$
$947$	−3.14463	−0.102187	−0.0510934	+	0.998694i	$0.516271\pi$
$948$	0	0
$949$	3.56049	0.115578
$950$	−65.8905	−2.13777
$951$	0	0
$952$	2.04874	0.0664001
$953$	16.1840	0.524251	0.262126	−	0.965034i	$-0.415577\pi$
$953$	16.1840	0.524251	0.262126	+	0.965034i	$0.415577\pi$
$954$	0	0
$955$	−1.91761	−0.0620525
$956$	−11.4723	−0.371041
$957$	0	0
$958$	−21.2521	−0.686623
$959$	5.24361	0.169325
$960$	0	0
$961$	30.9533	0.998493
$962$	2.28691	0.0737330
$963$	0	0
$964$	−3.80963	−0.122700
$965$	4.40378	0.141763
$966$	0	0
$967$	−40.1314	−1.29054	−0.645269	−	0.763955i	$-0.723255\pi$
$967$	−40.1314	−1.29054	−0.645269	+	0.763955i	$0.723255\pi$
$968$	0.900213	0.0289339
$969$	0	0
$970$	7.89583	0.253520
$971$	41.3913	1.32831	0.664154	−	0.747596i	$-0.268792\pi$
$971$	41.3913	1.32831	0.664154	+	0.747596i	$0.268792\pi$
$972$	0	0
$973$	6.29886	0.201932
$974$	−46.8031	−1.49967
$975$	0	0
$976$	−4.72939	−0.151384
$977$	−45.3750	−1.45167	−0.725837	−	0.687867i	$-0.758547\pi$
$977$	−45.3750	−1.45167	−0.725837	+	0.687867i	$0.758547\pi$
$978$	0	0
$979$	7.33234	0.234343
$980$	7.33944	0.234450
$981$	0	0
$982$	−62.8525	−2.00570
$983$	0.801122	0.0255518	0.0127759	−	0.999918i	$-0.495933\pi$
$983$	0.801122	0.0255518	0.0127759	+	0.999918i	$0.495933\pi$
$984$	0	0
$985$	−14.5821	−0.464625
$986$	20.4425	0.651023
$987$	0	0
$988$	5.89190	0.187446
$989$	48.3760	1.53827
$990$	0	0
$991$	21.4923	0.682726	0.341363	−	0.939931i	$-0.389111\pi$
$991$	21.4923	0.682726	0.341363	+	0.939931i	$0.389111\pi$
$992$	−55.6715	−1.76757
$993$	0	0
$994$	11.4727	0.363892
$995$	−0.945479	−0.0299737
$996$	0	0
$997$	41.6368	1.31865	0.659325	−	0.751858i	$-0.270842\pi$
$997$	41.6368	1.31865	0.659325	+	0.751858i	$0.270842\pi$
$998$	−48.4075	−1.53231
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.5	yes	25
3.2	odd	2		6039.2.a.m.1.21	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.21	✓	25	3.2	odd	2
6039.2.a.p.1.5	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)

\(f(q)\)	\(=\)	\(q-1.87622 q^{2} +1.52020 q^{4} -0.769125 q^{5} -0.850176 q^{7} +0.900213 q^{8} +O(q^{10})\)	\(q-1.87622 q^{2} +1.52020 q^{4} -0.769125 q^{5} -0.850176 q^{7} +0.900213 q^{8} +1.44305 q^{10} -1.00000 q^{11} -0.486521 q^{13} +1.59512 q^{14} -4.72939 q^{16} -2.67691 q^{17} -7.96625 q^{19} -1.16922 q^{20} +1.87622 q^{22} -7.17905 q^{23} -4.40845 q^{25} +0.912819 q^{26} -1.29244 q^{28} +4.07022 q^{29} -7.87104 q^{31} +7.07295 q^{32} +5.02246 q^{34} +0.653891 q^{35} +2.50533 q^{37} +14.9464 q^{38} -0.692376 q^{40} -1.48908 q^{41} -6.73850 q^{43} -1.52020 q^{44} +13.4695 q^{46} -12.0321 q^{47} -6.27720 q^{49} +8.27121 q^{50} -0.739608 q^{52} +11.5877 q^{53} +0.769125 q^{55} -0.765339 q^{56} -7.63663 q^{58} +4.12976 q^{59} +1.00000 q^{61} +14.7678 q^{62} -3.81162 q^{64} +0.374195 q^{65} +6.50801 q^{67} -4.06943 q^{68} -1.22684 q^{70} +7.19240 q^{71} -7.31828 q^{73} -4.70054 q^{74} -12.1103 q^{76} +0.850176 q^{77} +8.91585 q^{79} +3.63749 q^{80} +2.79383 q^{82} +9.65801 q^{83} +2.05888 q^{85} +12.6429 q^{86} -0.900213 q^{88} -7.33234 q^{89} +0.413628 q^{91} -10.9136 q^{92} +22.5748 q^{94} +6.12704 q^{95} +5.47164 q^{97} +11.7774 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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