LMFDB - Embedded newform 6039.2.a.i.1.11

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.46794$ of defining polynomial
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)

$f(q)$	$=$	$q+1.46794 q^{2} +0.154851 q^{4} -3.84216 q^{5} +2.46223 q^{7} -2.70857 q^{8} +O(q^{10})$	\(q+1.46794 q^{2} +0.154851 q^{4} -3.84216 q^{5} +2.46223 q^{7} -2.70857 q^{8} -5.64006 q^{10} -1.00000 q^{11} -4.64696 q^{13} +3.61441 q^{14} -4.28572 q^{16} -5.70692 q^{17} +6.71196 q^{19} -0.594961 q^{20} -1.46794 q^{22} -2.93787 q^{23} +9.76218 q^{25} -6.82147 q^{26} +0.381279 q^{28} -5.67137 q^{29} +0.232595 q^{31} -0.874048 q^{32} -8.37742 q^{34} -9.46029 q^{35} +5.55034 q^{37} +9.85276 q^{38} +10.4068 q^{40} +4.35928 q^{41} -9.35832 q^{43} -0.154851 q^{44} -4.31262 q^{46} -10.7428 q^{47} -0.937404 q^{49} +14.3303 q^{50} -0.719586 q^{52} +11.3834 q^{53} +3.84216 q^{55} -6.66913 q^{56} -8.32523 q^{58} +13.2037 q^{59} -1.00000 q^{61} +0.341436 q^{62} +7.28839 q^{64} +17.8544 q^{65} +12.5899 q^{67} -0.883721 q^{68} -13.8871 q^{70} +0.299770 q^{71} -2.65468 q^{73} +8.14757 q^{74} +1.03935 q^{76} -2.46223 q^{77} -12.9557 q^{79} +16.4664 q^{80} +6.39917 q^{82} +7.42505 q^{83} +21.9269 q^{85} -13.7375 q^{86} +2.70857 q^{88} +3.40420 q^{89} -11.4419 q^{91} -0.454931 q^{92} -15.7698 q^{94} -25.7884 q^{95} -15.8330 q^{97} -1.37605 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	1.46794	1.03799	0.518996	−	0.854777i	$-0.326306\pi$
$2$	1.46794	1.03799	0.518996	+	0.854777i	$0.326306\pi$
$3$	0	0
$3$	0	0
$4$	0.154851	0.0774254
$5$	−3.84216	−1.71827	−0.859133	−	0.511753i	$-0.828996\pi$
$5$	−3.84216	−1.71827	−0.859133	+	0.511753i	$0.828996\pi$
$6$	0	0
$7$	2.46223	0.930637	0.465318	−	0.885143i	$-0.345940\pi$
$7$	2.46223	0.930637	0.465318	+	0.885143i	$0.345940\pi$
$8$	−2.70857	−0.957624
$9$	0	0
$10$	−5.64006	−1.78354
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.64696	−1.28884	−0.644418	−	0.764674i	$-0.722900\pi$
$13$	−4.64696	−1.28884	−0.644418	+	0.764674i	$0.722900\pi$
$14$	3.61441	0.965993
$15$	0	0
$16$	−4.28572	−1.07143
$17$	−5.70692	−1.38413	−0.692066	−	0.721834i	$-0.743299\pi$
$17$	−5.70692	−1.38413	−0.692066	+	0.721834i	$0.743299\pi$
$18$	0	0
$19$	6.71196	1.53983	0.769914	−	0.638147i	$-0.220299\pi$
$19$	6.71196	1.53983	0.769914	+	0.638147i	$0.220299\pi$
$20$	−0.594961	−0.133037
$21$	0	0
$22$	−1.46794	−0.312966
$23$	−2.93787	−0.612588	−0.306294	−	0.951937i	$-0.599089\pi$
$23$	−2.93787	−0.612588	−0.306294	+	0.951937i	$0.599089\pi$
$24$	0	0
$25$	9.76218	1.95244
$26$	−6.82147	−1.33780
$27$	0	0
$28$	0.381279	0.0720549
$29$	−5.67137	−1.05315	−0.526573	−	0.850130i	$-0.676523\pi$
$29$	−5.67137	−1.05315	−0.526573	+	0.850130i	$0.676523\pi$
$30$	0	0
$31$	0.232595	0.0417754	0.0208877	−	0.999782i	$-0.493351\pi$
$31$	0.232595	0.0417754	0.0208877	+	0.999782i	$0.493351\pi$
$32$	−0.874048	−0.154511
$33$	0	0
$34$	−8.37742	−1.43672
$35$	−9.46029	−1.59908
$36$	0	0
$37$	5.55034	0.912470	0.456235	−	0.889859i	$-0.349198\pi$
$37$	5.55034	0.912470	0.456235	+	0.889859i	$0.349198\pi$
$38$	9.85276	1.59833
$39$	0	0
$40$	10.4068	1.64545
$41$	4.35928	0.680806	0.340403	−	0.940280i	$-0.389437\pi$
$41$	4.35928	0.680806	0.340403	+	0.940280i	$0.389437\pi$
$42$	0	0
$43$	−9.35832	−1.42713	−0.713565	−	0.700589i	$-0.752921\pi$
$43$	−9.35832	−1.42713	−0.713565	+	0.700589i	$0.752921\pi$
$44$	−0.154851	−0.0233446
$45$	0	0
$46$	−4.31262	−0.635861
$47$	−10.7428	−1.56699	−0.783497	−	0.621395i	$-0.786566\pi$
$47$	−10.7428	−1.56699	−0.783497	+	0.621395i	$0.786566\pi$
$48$	0	0
$49$	−0.937404	−0.133915
$50$	14.3303	2.02661
$51$	0	0
$52$	−0.719586	−0.0997886
$53$	11.3834	1.56363	0.781816	−	0.623510i	$-0.214294\pi$
$53$	11.3834	1.56363	0.781816	+	0.623510i	$0.214294\pi$
$54$	0	0
$55$	3.84216	0.518076
$56$	−6.66913	−0.891200
$57$	0	0
$58$	−8.32523	−1.09316
$59$	13.2037	1.71897	0.859485	−	0.511161i	$-0.170784\pi$
$59$	13.2037	1.71897	0.859485	+	0.511161i	$0.170784\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0.341436	0.0433624
$63$	0	0
$64$	7.28839	0.911049
$65$	17.8544	2.21456
$66$	0	0
$67$	12.5899	1.53810	0.769049	−	0.639190i	$-0.220730\pi$
$67$	12.5899	1.53810	0.769049	+	0.639190i	$0.220730\pi$
$68$	−0.883721	−0.107167
$69$	0	0
$70$	−13.8871	−1.65983
$71$	0.299770	0.0355761	0.0177881	−	0.999842i	$-0.494338\pi$
$71$	0.299770	0.0355761	0.0177881	+	0.999842i	$0.494338\pi$
$72$	0	0
$73$	−2.65468	−0.310707	−0.155353	−	0.987859i	$-0.549652\pi$
$73$	−2.65468	−0.310707	−0.155353	+	0.987859i	$0.549652\pi$
$74$	8.14757	0.947135
$75$	0	0
$76$	1.03935	0.119222
$77$	−2.46223	−0.280598
$78$	0	0
$79$	−12.9557	−1.45763	−0.728815	−	0.684710i	$-0.759929\pi$
$79$	−12.9557	−1.45763	−0.728815	+	0.684710i	$0.759929\pi$
$80$	16.4664	1.84100
$81$	0	0
$82$	6.39917	0.706670
$83$	7.42505	0.815005	0.407502	−	0.913204i	$-0.366400\pi$
$83$	7.42505	0.815005	0.407502	+	0.913204i	$0.366400\pi$
$84$	0	0
$85$	21.9269	2.37831
$86$	−13.7375	−1.48135
$87$	0	0
$88$	2.70857	0.288735
$89$	3.40420	0.360845	0.180422	−	0.983589i	$-0.442254\pi$
$89$	3.40420	0.360845	0.180422	+	0.983589i	$0.442254\pi$
$90$	0	0
$91$	−11.4419	−1.19944
$92$	−0.454931	−0.0474299
$93$	0	0
$94$	−15.7698	−1.62653
$95$	−25.7884	−2.64583
$96$	0	0
$97$	−15.8330	−1.60760	−0.803798	−	0.594902i	$-0.797191\pi$
$97$	−15.8330	−1.60760	−0.803798	+	0.594902i	$0.797191\pi$
$98$	−1.37605	−0.139002
$99$	0	0
$100$	1.51168	0.151168
$101$	−7.72797	−0.768962	−0.384481	−	0.923133i	$-0.625620\pi$
$101$	−7.72797	−0.768962	−0.384481	+	0.923133i	$0.625620\pi$
$102$	0	0
$103$	16.1339	1.58972	0.794860	−	0.606793i	$-0.207544\pi$
$103$	16.1339	1.58972	0.794860	+	0.606793i	$0.207544\pi$
$104$	12.5866	1.23422
$105$	0	0
$106$	16.7102	1.62304
$107$	−1.14677	−0.110863	−0.0554314	−	0.998462i	$-0.517653\pi$
$107$	−1.14677	−0.110863	−0.0554314	+	0.998462i	$0.517653\pi$
$108$	0	0
$109$	11.7557	1.12600	0.562998	−	0.826458i	$-0.309648\pi$
$109$	11.7557	1.12600	0.562998	+	0.826458i	$0.309648\pi$
$110$	5.64006	0.537759
$111$	0	0
$112$	−10.5525	−0.997113
$113$	16.4246	1.54509	0.772546	−	0.634959i	$-0.218983\pi$
$113$	16.4246	1.54509	0.772546	+	0.634959i	$0.218983\pi$
$114$	0	0
$115$	11.2878	1.05259
$116$	−0.878216	−0.0815403
$117$	0	0
$118$	19.3822	1.78428
$119$	−14.0518	−1.28812
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.46794	−0.132901
$123$	0	0
$124$	0.0360176	0.00323447
$125$	−18.2970	−1.63654
$126$	0	0
$127$	11.9297	1.05859	0.529295	−	0.848438i	$-0.322457\pi$
$127$	11.9297	1.05859	0.529295	+	0.848438i	$0.322457\pi$
$128$	12.4470	1.10017
$129$	0	0
$130$	26.2091	2.29869
$131$	−20.8633	−1.82283	−0.911417	−	0.411485i	$-0.865010\pi$
$131$	−20.8633	−1.82283	−0.911417	+	0.411485i	$0.865010\pi$
$132$	0	0
$133$	16.5264	1.43302
$134$	18.4812	1.59653
$135$	0	0
$136$	15.4576	1.32548
$137$	19.1597	1.63692	0.818460	−	0.574564i	$-0.194828\pi$
$137$	19.1597	1.63692	0.818460	+	0.574564i	$0.194828\pi$
$138$	0	0
$139$	−2.58202	−0.219004	−0.109502	−	0.993987i	$-0.534926\pi$
$139$	−2.58202	−0.219004	−0.109502	+	0.993987i	$0.534926\pi$
$140$	−1.46493	−0.123809
$141$	0	0
$142$	0.440044	0.0369277
$143$	4.64696	0.388598
$144$	0	0
$145$	21.7903	1.80959
$146$	−3.89691	−0.322511
$147$	0	0
$148$	0.859474	0.0706483
$149$	−9.45719	−0.774763	−0.387382	−	0.921919i	$-0.626620\pi$
$149$	−9.45719	−0.774763	−0.387382	+	0.921919i	$0.626620\pi$
$150$	0	0
$151$	6.16589	0.501773	0.250887	−	0.968016i	$-0.419278\pi$
$151$	6.16589	0.501773	0.250887	+	0.968016i	$0.419278\pi$
$152$	−18.1798	−1.47458
$153$	0	0
$154$	−3.61441	−0.291258
$155$	−0.893668	−0.0717811
$156$	0	0
$157$	0.475244	0.0379286	0.0189643	−	0.999820i	$-0.493963\pi$
$157$	0.475244	0.0379286	0.0189643	+	0.999820i	$0.493963\pi$
$158$	−19.0182	−1.51301
$159$	0	0
$160$	3.35823	0.265491
$161$	−7.23372	−0.570097
$162$	0	0
$163$	17.6623	1.38341	0.691707	−	0.722178i	$-0.256859\pi$
$163$	17.6623	1.38341	0.691707	+	0.722178i	$0.256859\pi$
$164$	0.675038	0.0527116
$165$	0	0
$166$	10.8995	0.845968
$167$	15.9865	1.23707	0.618536	−	0.785756i	$-0.287726\pi$
$167$	15.9865	1.23707	0.618536	+	0.785756i	$0.287726\pi$
$168$	0	0
$169$	8.59425	0.661096
$170$	32.1874	2.46866
$171$	0	0
$172$	−1.44914	−0.110496
$173$	2.22261	0.168982	0.0844909	−	0.996424i	$-0.473074\pi$
$173$	2.22261	0.168982	0.0844909	+	0.996424i	$0.473074\pi$
$174$	0	0
$175$	24.0368	1.81701
$176$	4.28572	0.323049
$177$	0	0
$178$	4.99717	0.374553
$179$	24.5110	1.83204	0.916020	−	0.401132i	$-0.131383\pi$
$179$	24.5110	1.83204	0.916020	+	0.401132i	$0.131383\pi$
$180$	0	0
$181$	−14.1308	−1.05034	−0.525168	−	0.850999i	$-0.675997\pi$
$181$	−14.1308	−1.05034	−0.525168	+	0.850999i	$0.675997\pi$
$182$	−16.7960	−1.24501
$183$	0	0
$184$	7.95743	0.586629
$185$	−21.3253	−1.56787
$186$	0	0
$187$	5.70692	0.417331
$188$	−1.66353	−0.121325
$189$	0	0
$190$	−37.8558	−2.74635
$191$	21.1658	1.53151	0.765753	−	0.643135i	$-0.222367\pi$
$191$	21.1658	1.53151	0.765753	+	0.643135i	$0.222367\pi$
$192$	0	0
$193$	−10.1194	−0.728414	−0.364207	−	0.931318i	$-0.618660\pi$
$193$	−10.1194	−0.728414	−0.364207	+	0.931318i	$0.618660\pi$
$194$	−23.2419	−1.66867
$195$	0	0
$196$	−0.145158	−0.0103684
$197$	−4.80836	−0.342582	−0.171291	−	0.985221i	$-0.554794\pi$
$197$	−4.80836	−0.342582	−0.171291	+	0.985221i	$0.554794\pi$
$198$	0	0
$199$	13.0978	0.928480	0.464240	−	0.885710i	$-0.346328\pi$
$199$	13.0978	0.928480	0.464240	+	0.885710i	$0.346328\pi$
$200$	−26.4415	−1.86970
$201$	0	0
$202$	−11.3442	−0.798176
$203$	−13.9642	−0.980097
$204$	0	0
$205$	−16.7491	−1.16980
$206$	23.6836	1.65011
$207$	0	0
$208$	19.9156	1.38090
$209$	−6.71196	−0.464276
$210$	0	0
$211$	−6.36135	−0.437933	−0.218967	−	0.975732i	$-0.570269\pi$
$211$	−6.36135	−0.437933	−0.218967	+	0.975732i	$0.570269\pi$
$212$	1.76273	0.121065
$213$	0	0
$214$	−1.68340	−0.115075
$215$	35.9561	2.45219
$216$	0	0
$217$	0.572704	0.0388777
$218$	17.2567	1.16877
$219$	0	0
$220$	0.594961	0.0401123
$221$	26.5198	1.78392
$222$	0	0
$223$	22.5356	1.50910	0.754548	−	0.656244i	$-0.227856\pi$
$223$	22.5356	1.50910	0.754548	+	0.656244i	$0.227856\pi$
$224$	−2.15211	−0.143794
$225$	0	0
$226$	24.1103	1.60379
$227$	−27.1685	−1.80323	−0.901617	−	0.432535i	$-0.857619\pi$
$227$	−27.1685	−1.80323	−0.901617	+	0.432535i	$0.857619\pi$
$228$	0	0
$229$	−24.6096	−1.62625	−0.813124	−	0.582090i	$-0.802235\pi$
$229$	−24.6096	−1.62625	−0.813124	+	0.582090i	$0.802235\pi$
$230$	16.5698	1.09258
$231$	0	0
$232$	15.3613	1.00852
$233$	−24.0471	−1.57538	−0.787690	−	0.616071i	$-0.788723\pi$
$233$	−24.0471	−1.57538	−0.787690	+	0.616071i	$0.788723\pi$
$234$	0	0
$235$	41.2754	2.69251
$236$	2.04460	0.133092
$237$	0	0
$238$	−20.6272	−1.33706
$239$	−16.2704	−1.05245	−0.526224	−	0.850346i	$-0.676393\pi$
$239$	−16.2704	−1.05245	−0.526224	+	0.850346i	$0.676393\pi$
$240$	0	0
$241$	1.53799	0.0990706	0.0495353	−	0.998772i	$-0.484226\pi$
$241$	1.53799	0.0990706	0.0495353	+	0.998772i	$0.484226\pi$
$242$	1.46794	0.0943628
$243$	0	0
$244$	−0.154851	−0.00991330
$245$	3.60165	0.230101
$246$	0	0
$247$	−31.1902	−1.98458
$248$	−0.630001	−0.0400051
$249$	0	0
$250$	−26.8590	−1.69871
$251$	22.8738	1.44378	0.721891	−	0.692006i	$-0.243273\pi$
$251$	22.8738	1.44378	0.721891	+	0.692006i	$0.243273\pi$
$252$	0	0
$253$	2.93787	0.184702
$254$	17.5121	1.09881
$255$	0	0
$256$	3.69471	0.230920
$257$	10.1026	0.630186	0.315093	−	0.949061i	$-0.397964\pi$
$257$	10.1026	0.630186	0.315093	+	0.949061i	$0.397964\pi$
$258$	0	0
$259$	13.6662	0.849178
$260$	2.76476	0.171463
$261$	0	0
$262$	−30.6261	−1.89208
$263$	11.1552	0.687859	0.343929	−	0.938996i	$-0.388242\pi$
$263$	11.1552	0.687859	0.343929	+	0.938996i	$0.388242\pi$
$264$	0	0
$265$	−43.7368	−2.68673
$266$	24.2598	1.48746
$267$	0	0
$268$	1.94955	0.119088
$269$	−6.03952	−0.368236	−0.184118	−	0.982904i	$-0.558943\pi$
$269$	−6.03952	−0.368236	−0.184118	+	0.982904i	$0.558943\pi$
$270$	0	0
$271$	4.26042	0.258802	0.129401	−	0.991592i	$-0.458695\pi$
$271$	4.26042	0.258802	0.129401	+	0.991592i	$0.458695\pi$
$272$	24.4583	1.48300
$273$	0	0
$274$	28.1252	1.69911
$275$	−9.76218	−0.588681
$276$	0	0
$277$	8.39998	0.504706	0.252353	−	0.967635i	$-0.418796\pi$
$277$	8.39998	0.504706	0.252353	+	0.967635i	$0.418796\pi$
$278$	−3.79026	−0.227325
$279$	0	0
$280$	25.6239	1.53132
$281$	−12.3489	−0.736671	−0.368335	−	0.929693i	$-0.620072\pi$
$281$	−12.3489	−0.736671	−0.368335	+	0.929693i	$0.620072\pi$
$282$	0	0
$283$	7.31966	0.435109	0.217554	−	0.976048i	$-0.430192\pi$
$283$	7.31966	0.435109	0.217554	+	0.976048i	$0.430192\pi$
$284$	0.0464196	0.00275450
$285$	0	0
$286$	6.82147	0.403362
$287$	10.7336	0.633583
$288$	0	0
$289$	15.5689	0.915820
$290$	31.9869	1.87833
$291$	0	0
$292$	−0.411079	−0.0240566
$293$	−16.8649	−0.985255	−0.492628	−	0.870240i	$-0.663964\pi$
$293$	−16.8649	−0.985255	−0.492628	+	0.870240i	$0.663964\pi$
$294$	0	0
$295$	−50.7305	−2.95365
$296$	−15.0335	−0.873803
$297$	0	0
$298$	−13.8826	−0.804197
$299$	13.6522	0.789525
$300$	0	0
$301$	−23.0424	−1.32814
$302$	9.05117	0.520836
$303$	0	0
$304$	−28.7656	−1.64982
$305$	3.84216	0.220001
$306$	0	0
$307$	20.2128	1.15360	0.576802	−	0.816884i	$-0.304300\pi$
$307$	20.2128	1.15360	0.576802	+	0.816884i	$0.304300\pi$
$308$	−0.381279	−0.0217254
$309$	0	0
$310$	−1.31185	−0.0745082
$311$	27.5538	1.56243	0.781215	−	0.624262i	$-0.214600\pi$
$311$	27.5538	1.56243	0.781215	+	0.624262i	$0.214600\pi$
$312$	0	0
$313$	18.0023	1.01755	0.508776	−	0.860899i	$-0.330098\pi$
$313$	18.0023	1.01755	0.508776	+	0.860899i	$0.330098\pi$
$314$	0.697630	0.0393695
$315$	0	0
$316$	−2.00620	−0.112858
$317$	−29.5066	−1.65726	−0.828628	−	0.559799i	$-0.810878\pi$
$317$	−29.5066	−1.65726	−0.828628	+	0.559799i	$0.810878\pi$
$318$	0	0
$319$	5.67137	0.317536
$320$	−28.0032	−1.56542
$321$	0	0
$322$	−10.6187	−0.591756
$323$	−38.3046	−2.13133
$324$	0	0
$325$	−45.3645	−2.51637
$326$	25.9271	1.43597
$327$	0	0
$328$	−11.8074	−0.651956
$329$	−26.4512	−1.45830
$330$	0	0
$331$	17.8450	0.980852	0.490426	−	0.871483i	$-0.336841\pi$
$331$	17.8450	0.980852	0.490426	+	0.871483i	$0.336841\pi$
$332$	1.14977	0.0631021
$333$	0	0
$334$	23.4672	1.28407
$335$	−48.3723	−2.64286
$336$	0	0
$337$	−15.1235	−0.823832	−0.411916	−	0.911222i	$-0.635140\pi$
$337$	−15.1235	−0.823832	−0.411916	+	0.911222i	$0.635140\pi$
$338$	12.6159	0.686212
$339$	0	0
$340$	3.39540	0.184141
$341$	−0.232595	−0.0125957
$342$	0	0
$343$	−19.5437	−1.05526
$344$	25.3477	1.36665
$345$	0	0
$346$	3.26266	0.175402
$347$	−11.2684	−0.604920	−0.302460	−	0.953162i	$-0.597808\pi$
$347$	−11.2684	−0.604920	−0.302460	+	0.953162i	$0.597808\pi$
$348$	0	0
$349$	6.44520	0.345004	0.172502	−	0.985009i	$-0.444815\pi$
$349$	6.44520	0.345004	0.172502	+	0.985009i	$0.444815\pi$
$350$	35.2845	1.88604
$351$	0	0
$352$	0.874048	0.0465869
$353$	22.9363	1.22078	0.610389	−	0.792102i	$-0.291013\pi$
$353$	22.9363	1.22078	0.610389	+	0.792102i	$0.291013\pi$
$354$	0	0
$355$	−1.15176	−0.0611292
$356$	0.527143	0.0279385
$357$	0	0
$358$	35.9807	1.90164
$359$	−11.2684	−0.594724	−0.297362	−	0.954765i	$-0.596107\pi$
$359$	−11.2684	−0.594724	−0.297362	+	0.954765i	$0.596107\pi$
$360$	0	0
$361$	26.0504	1.37107
$362$	−20.7432	−1.09024
$363$	0	0
$364$	−1.77179	−0.0928669
$365$	10.1997	0.533877
$366$	0	0
$367$	10.1541	0.530039	0.265020	−	0.964243i	$-0.414622\pi$
$367$	10.1541	0.530039	0.265020	+	0.964243i	$0.414622\pi$
$368$	12.5909	0.656346
$369$	0	0
$370$	−31.3042	−1.62743
$371$	28.0286	1.45517
$372$	0	0
$373$	−24.8297	−1.28563	−0.642817	−	0.766020i	$-0.722234\pi$
$373$	−24.8297	−1.28563	−0.642817	+	0.766020i	$0.722234\pi$
$374$	8.37742	0.433186
$375$	0	0
$376$	29.0976	1.50059
$377$	26.3546	1.35733
$378$	0	0
$379$	12.2701	0.630271	0.315136	−	0.949047i	$-0.397950\pi$
$379$	12.2701	0.630271	0.315136	+	0.949047i	$0.397950\pi$
$380$	−3.99335	−0.204855
$381$	0	0
$382$	31.0702	1.58969
$383$	22.3129	1.14014	0.570068	−	0.821597i	$-0.306917\pi$
$383$	22.3129	1.14014	0.570068	+	0.821597i	$0.306917\pi$
$384$	0	0
$385$	9.46029	0.482141
$386$	−14.8547	−0.756087
$387$	0	0
$388$	−2.45175	−0.124469
$389$	−24.6635	−1.25049	−0.625245	−	0.780429i	$-0.715001\pi$
$389$	−24.6635	−1.25049	−0.625245	+	0.780429i	$0.715001\pi$
$390$	0	0
$391$	16.7662	0.847903
$392$	2.53902	0.128240
$393$	0	0
$394$	−7.05839	−0.355597
$395$	49.7779	2.50460
$396$	0	0
$397$	−3.99794	−0.200651	−0.100325	−	0.994955i	$-0.531988\pi$
$397$	−3.99794	−0.200651	−0.100325	+	0.994955i	$0.531988\pi$
$398$	19.2268	0.963754
$399$	0	0
$400$	−41.8380	−2.09190
$401$	−1.45172	−0.0724952	−0.0362476	−	0.999343i	$-0.511540\pi$
$401$	−1.45172	−0.0724952	−0.0362476	+	0.999343i	$0.511540\pi$
$402$	0	0
$403$	−1.08086	−0.0538416
$404$	−1.19668	−0.0595372
$405$	0	0
$406$	−20.4987	−1.01733
$407$	−5.55034	−0.275120
$408$	0	0
$409$	13.2091	0.653147	0.326574	−	0.945172i	$-0.394106\pi$
$409$	13.2091	0.653147	0.326574	+	0.945172i	$0.394106\pi$
$410$	−24.5866	−1.21425
$411$	0	0
$412$	2.49835	0.123085
$413$	32.5105	1.59974
$414$	0	0
$415$	−28.5282	−1.40039
$416$	4.06167	0.199140
$417$	0	0
$418$	−9.85276	−0.481914
$419$	12.6761	0.619267	0.309634	−	0.950856i	$-0.399794\pi$
$419$	12.6761	0.619267	0.309634	+	0.950856i	$0.399794\pi$
$420$	0	0
$421$	−4.36098	−0.212541	−0.106271	−	0.994337i	$-0.533891\pi$
$421$	−4.36098	−0.212541	−0.106271	+	0.994337i	$0.533891\pi$
$422$	−9.33809	−0.454571
$423$	0	0
$424$	−30.8328	−1.49737
$425$	−55.7120	−2.70243
$426$	0	0
$427$	−2.46223	−0.119156
$428$	−0.177579	−0.00858359
$429$	0	0
$430$	52.7815	2.54535
$431$	1.72411	0.0830477	0.0415238	−	0.999138i	$-0.486779\pi$
$431$	1.72411	0.0830477	0.0415238	+	0.999138i	$0.486779\pi$
$432$	0	0
$433$	−6.68214	−0.321123	−0.160561	−	0.987026i	$-0.551330\pi$
$433$	−6.68214	−0.321123	−0.160561	+	0.987026i	$0.551330\pi$
$434$	0.840696	0.0403547
$435$	0	0
$436$	1.82039	0.0871807
$437$	−19.7189	−0.943281
$438$	0	0
$439$	−4.62223	−0.220607	−0.110303	−	0.993898i	$-0.535182\pi$
$439$	−4.62223	−0.220607	−0.110303	+	0.993898i	$0.535182\pi$
$440$	−10.4068	−0.496123
$441$	0	0
$442$	38.9296	1.85169
$443$	−24.7151	−1.17425	−0.587125	−	0.809497i	$-0.699740\pi$
$443$	−24.7151	−1.17425	−0.587125	+	0.809497i	$0.699740\pi$
$444$	0	0
$445$	−13.0795	−0.620027
$446$	33.0810	1.56643
$447$	0	0
$448$	17.9457	0.847856
$449$	17.0015	0.802350	0.401175	−	0.916002i	$-0.368602\pi$
$449$	17.0015	0.802350	0.401175	+	0.916002i	$0.368602\pi$
$450$	0	0
$451$	−4.35928	−0.205271
$452$	2.54335	0.119629
$453$	0	0
$454$	−39.8817	−1.87174
$455$	43.9616	2.06095
$456$	0	0
$457$	−31.3936	−1.46853	−0.734265	−	0.678863i	$-0.762473\pi$
$457$	−31.3936	−1.46853	−0.734265	+	0.678863i	$0.762473\pi$
$458$	−36.1254	−1.68803
$459$	0	0
$460$	1.74792	0.0814971
$461$	−4.61978	−0.215165	−0.107582	−	0.994196i	$-0.534311\pi$
$461$	−4.61978	−0.215165	−0.107582	+	0.994196i	$0.534311\pi$
$462$	0	0
$463$	27.6857	1.28666	0.643331	−	0.765588i	$-0.277552\pi$
$463$	27.6857	1.28666	0.643331	+	0.765588i	$0.277552\pi$
$464$	24.3059	1.12837
$465$	0	0
$466$	−35.2998	−1.63523
$467$	−3.83606	−0.177512	−0.0887558	−	0.996053i	$-0.528289\pi$
$467$	−3.83606	−0.177512	−0.0887558	+	0.996053i	$0.528289\pi$
$468$	0	0
$469$	30.9992	1.43141
$470$	60.5899	2.79480
$471$	0	0
$472$	−35.7630	−1.64613
$473$	9.35832	0.430296
$474$	0	0
$475$	65.5233	3.00641
$476$	−2.17593	−0.0997335
$477$	0	0
$478$	−23.8841	−1.09243
$479$	−19.8463	−0.906801	−0.453400	−	0.891307i	$-0.649789\pi$
$479$	−19.8463	−0.906801	−0.453400	+	0.891307i	$0.649789\pi$
$480$	0	0
$481$	−25.7922	−1.17602
$482$	2.25768	0.102834
$483$	0	0
$484$	0.154851	0.00703867
$485$	60.8329	2.76228
$486$	0	0
$487$	−18.7149	−0.848054	−0.424027	−	0.905650i	$-0.639384\pi$
$487$	−18.7149	−0.848054	−0.424027	+	0.905650i	$0.639384\pi$
$488$	2.70857	0.122611
$489$	0	0
$490$	5.28701	0.238843
$491$	33.1187	1.49463	0.747313	−	0.664472i	$-0.231344\pi$
$491$	33.1187	1.49463	0.747313	+	0.664472i	$0.231344\pi$
$492$	0	0
$493$	32.3661	1.45769
$494$	−45.7854	−2.05998
$495$	0	0
$496$	−0.996839	−0.0447594
$497$	0.738103	0.0331085
$498$	0	0
$499$	3.36699	0.150727	0.0753635	−	0.997156i	$-0.475988\pi$
$499$	3.36699	0.150727	0.0753635	+	0.997156i	$0.475988\pi$
$500$	−2.83331	−0.126709
$501$	0	0
$502$	33.5774	1.49863
$503$	−15.7130	−0.700609	−0.350305	−	0.936636i	$-0.613922\pi$
$503$	−15.7130	−0.700609	−0.350305	+	0.936636i	$0.613922\pi$
$504$	0	0
$505$	29.6921	1.32128
$506$	4.31262	0.191719
$507$	0	0
$508$	1.84732	0.0819617
$509$	7.97241	0.353371	0.176685	−	0.984267i	$-0.443462\pi$
$509$	7.97241	0.353371	0.176685	+	0.984267i	$0.443462\pi$
$510$	0	0
$511$	−6.53644	−0.289155
$512$	−19.4704	−0.860480
$513$	0	0
$514$	14.8301	0.654127
$515$	−61.9889	−2.73156
$516$	0	0
$517$	10.7428	0.472467
$518$	20.0612	0.881439
$519$	0	0
$520$	−48.3598	−2.12072
$521$	19.8637	0.870244	0.435122	−	0.900372i	$-0.356705\pi$
$521$	19.8637	0.870244	0.435122	+	0.900372i	$0.356705\pi$
$522$	0	0
$523$	10.5930	0.463197	0.231599	−	0.972811i	$-0.425604\pi$
$523$	10.5930	0.463197	0.231599	+	0.972811i	$0.425604\pi$
$524$	−3.23070	−0.141134
$525$	0	0
$526$	16.3752	0.713991
$527$	−1.32740	−0.0578226
$528$	0	0
$529$	−14.3689	−0.624736
$530$	−64.2031	−2.78880
$531$	0	0
$532$	2.55913	0.110952
$533$	−20.2574	−0.877446
$534$	0	0
$535$	4.40608	0.190492
$536$	−34.1006	−1.47292
$537$	0	0
$538$	−8.86566	−0.382226
$539$	0.937404	0.0403768
$540$	0	0
$541$	0.967556	0.0415985	0.0207993	−	0.999784i	$-0.493379\pi$
$541$	0.967556	0.0415985	0.0207993	+	0.999784i	$0.493379\pi$
$542$	6.25404	0.268634
$543$	0	0
$544$	4.98812	0.213864
$545$	−45.1674	−1.93476
$546$	0	0
$547$	2.94593	0.125959	0.0629794	−	0.998015i	$-0.479940\pi$
$547$	2.94593	0.125959	0.0629794	+	0.998015i	$0.479940\pi$
$548$	2.96689	0.126739
$549$	0	0
$550$	−14.3303	−0.611046
$551$	−38.0660	−1.62167
$552$	0	0
$553$	−31.9000	−1.35653
$554$	12.3307	0.523880
$555$	0	0
$556$	−0.399828	−0.0169565
$557$	−0.0922439	−0.00390850	−0.00195425	−	0.999998i	$-0.500622\pi$
$557$	−0.0922439	−0.00390850	−0.00195425	+	0.999998i	$0.500622\pi$
$558$	0	0
$559$	43.4877	1.83934
$560$	40.5442	1.71330
$561$	0	0
$562$	−18.1274	−0.764658
$563$	−16.5637	−0.698079	−0.349039	−	0.937108i	$-0.613492\pi$
$563$	−16.5637	−0.698079	−0.349039	+	0.937108i	$0.613492\pi$
$564$	0	0
$565$	−63.1057	−2.65488
$566$	10.7448	0.451639
$567$	0	0
$568$	−0.811948	−0.0340686
$569$	44.9171	1.88302	0.941512	−	0.336980i	$-0.109406\pi$
$569$	44.9171	1.88302	0.941512	+	0.336980i	$0.109406\pi$
$570$	0	0
$571$	11.9694	0.500903	0.250452	−	0.968129i	$-0.419421\pi$
$571$	11.9694	0.500903	0.250452	+	0.968129i	$0.419421\pi$
$572$	0.719586	0.0300874
$573$	0	0
$574$	15.7563	0.657653
$575$	−28.6800	−1.19604
$576$	0	0
$577$	1.73895	0.0723935	0.0361967	−	0.999345i	$-0.488476\pi$
$577$	1.73895	0.0723935	0.0361967	+	0.999345i	$0.488476\pi$
$578$	22.8543	0.950613
$579$	0	0
$580$	3.37424	0.140108
$581$	18.2822	0.758474
$582$	0	0
$583$	−11.3834	−0.471453
$584$	7.19039	0.297540
$585$	0	0
$586$	−24.7566	−1.02269
$587$	−34.0791	−1.40660	−0.703298	−	0.710895i	$-0.748290\pi$
$587$	−34.0791	−1.40660	−0.703298	+	0.710895i	$0.748290\pi$
$588$	0	0
$589$	1.56117	0.0643269
$590$	−74.4694	−3.06586
$591$	0	0
$592$	−23.7872	−0.977648
$593$	19.1334	0.785714	0.392857	−	0.919600i	$-0.371487\pi$
$593$	19.1334	0.785714	0.392857	+	0.919600i	$0.371487\pi$
$594$	0	0
$595$	53.9891	2.21334
$596$	−1.46445	−0.0599863
$597$	0	0
$598$	20.0406	0.819520
$599$	−11.7116	−0.478521	−0.239261	−	0.970955i	$-0.576905\pi$
$599$	−11.7116	−0.478521	−0.239261	+	0.970955i	$0.576905\pi$
$600$	0	0
$601$	−20.3403	−0.829698	−0.414849	−	0.909890i	$-0.636166\pi$
$601$	−20.3403	−0.829698	−0.414849	+	0.909890i	$0.636166\pi$
$602$	−33.8248	−1.37860
$603$	0	0
$604$	0.954793	0.0388500
$605$	−3.84216	−0.156206
$606$	0	0
$607$	18.2830	0.742085	0.371042	−	0.928616i	$-0.379000\pi$
$607$	18.2830	0.742085	0.371042	+	0.928616i	$0.379000\pi$
$608$	−5.86657	−0.237921
$609$	0	0
$610$	5.64006	0.228359
$611$	49.9213	2.01960
$612$	0	0
$613$	−33.4060	−1.34925	−0.674627	−	0.738158i	$-0.735696\pi$
$613$	−33.4060	−1.34925	−0.674627	+	0.738158i	$0.735696\pi$
$614$	29.6712	1.19743
$615$	0	0
$616$	6.66913	0.268707
$617$	37.0969	1.49347	0.746733	−	0.665124i	$-0.231621\pi$
$617$	37.0969	1.49347	0.746733	+	0.665124i	$0.231621\pi$
$618$	0	0
$619$	−35.5774	−1.42998	−0.714988	−	0.699137i	$-0.753568\pi$
$619$	−35.5774	−1.42998	−0.714988	+	0.699137i	$0.753568\pi$
$620$	−0.138385	−0.00555768
$621$	0	0
$622$	40.4473	1.62179
$623$	8.38194	0.335815
$624$	0	0
$625$	21.4892	0.859568
$626$	26.4263	1.05621
$627$	0	0
$628$	0.0735919	0.00293663
$629$	−31.6753	−1.26298
$630$	0	0
$631$	3.96808	0.157967	0.0789834	−	0.996876i	$-0.474833\pi$
$631$	3.96808	0.157967	0.0789834	+	0.996876i	$0.474833\pi$
$632$	35.0914	1.39586
$633$	0	0
$634$	−43.3140	−1.72022
$635$	−45.8358	−1.81894
$636$	0	0
$637$	4.35608	0.172594
$638$	8.32523	0.329599
$639$	0	0
$640$	−47.8235	−1.89039
$641$	4.20491	0.166084	0.0830419	−	0.996546i	$-0.473536\pi$
$641$	4.20491	0.166084	0.0830419	+	0.996546i	$0.473536\pi$
$642$	0	0
$643$	29.2183	1.15226	0.576129	−	0.817359i	$-0.304563\pi$
$643$	29.2183	1.15226	0.576129	+	0.817359i	$0.304563\pi$
$644$	−1.12015	−0.0441400
$645$	0	0
$646$	−56.2289	−2.21230
$647$	−8.11234	−0.318929	−0.159464	−	0.987204i	$-0.550977\pi$
$647$	−8.11234	−0.318929	−0.159464	+	0.987204i	$0.550977\pi$
$648$	0	0
$649$	−13.2037	−0.518289
$650$	−66.5923	−2.61197
$651$	0	0
$652$	2.73501	0.107111
$653$	23.5399	0.921189	0.460595	−	0.887611i	$-0.347636\pi$
$653$	23.5399	0.921189	0.460595	+	0.887611i	$0.347636\pi$
$654$	0	0
$655$	80.1600	3.13211
$656$	−18.6827	−0.729436
$657$	0	0
$658$	−38.8288	−1.51371
$659$	46.8281	1.82416	0.912082	−	0.410007i	$-0.134474\pi$
$659$	46.8281	1.82416	0.912082	+	0.410007i	$0.134474\pi$
$660$	0	0
$661$	−45.2831	−1.76131	−0.880655	−	0.473758i	$-0.842897\pi$
$661$	−45.2831	−1.76131	−0.880655	+	0.473758i	$0.842897\pi$
$662$	26.1955	1.01812
$663$	0	0
$664$	−20.1113	−0.780468
$665$	−63.4971	−2.46231
$666$	0	0
$667$	16.6617	0.645145
$668$	2.47552	0.0957808
$669$	0	0
$670$	−71.0077	−2.74327
$671$	1.00000	0.0386046
$672$	0	0
$673$	42.7822	1.64913	0.824566	−	0.565765i	$-0.191419\pi$
$673$	42.7822	1.64913	0.824566	+	0.565765i	$0.191419\pi$
$674$	−22.2005	−0.855130
$675$	0	0
$676$	1.33083	0.0511856
$677$	−32.8918	−1.26414	−0.632068	−	0.774913i	$-0.717794\pi$
$677$	−32.8918	−1.26414	−0.632068	+	0.774913i	$0.717794\pi$
$678$	0	0
$679$	−38.9845	−1.49609
$680$	−59.3905	−2.27752
$681$	0	0
$682$	−0.341436	−0.0130743
$683$	29.9574	1.14629	0.573144	−	0.819455i	$-0.305724\pi$
$683$	29.9574	1.14629	0.573144	+	0.819455i	$0.305724\pi$
$684$	0	0
$685$	−73.6144	−2.81266
$686$	−28.6891	−1.09535
$687$	0	0
$688$	40.1072	1.52907
$689$	−52.8983	−2.01526
$690$	0	0
$691$	−46.0051	−1.75011	−0.875057	−	0.484019i	$-0.839177\pi$
$691$	−46.0051	−1.75011	−0.875057	+	0.484019i	$0.839177\pi$
$692$	0.344173	0.0130835
$693$	0	0
$694$	−16.5414	−0.627901
$695$	9.92054	0.376308
$696$	0	0
$697$	−24.8781	−0.942324
$698$	9.46117	0.358111
$699$	0	0
$700$	3.72211	0.140683
$701$	4.97041	0.187730	0.0938650	−	0.995585i	$-0.470078\pi$
$701$	4.97041	0.187730	0.0938650	+	0.995585i	$0.470078\pi$
$702$	0	0
$703$	37.2536	1.40505
$704$	−7.28839	−0.274692
$705$	0	0
$706$	33.6692	1.26716
$707$	−19.0281	−0.715625
$708$	0	0
$709$	−31.0750	−1.16704	−0.583522	−	0.812097i	$-0.698326\pi$
$709$	−31.0750	−1.16704	−0.583522	+	0.812097i	$0.698326\pi$
$710$	−1.69072	−0.0634516
$711$	0	0
$712$	−9.22052	−0.345554
$713$	−0.683335	−0.0255911
$714$	0	0
$715$	−17.8544	−0.667715
$716$	3.79555	0.141846
$717$	0	0
$718$	−16.5414	−0.617318
$719$	37.4717	1.39746	0.698729	−	0.715386i	$-0.253749\pi$
$719$	37.4717	1.39746	0.698729	+	0.715386i	$0.253749\pi$
$720$	0	0
$721$	39.7254	1.47945
$722$	38.2404	1.42316
$723$	0	0
$724$	−2.18817	−0.0813226
$725$	−55.3649	−2.05620
$726$	0	0
$727$	−7.56311	−0.280500	−0.140250	−	0.990116i	$-0.544791\pi$
$727$	−7.56311	−0.280500	−0.140250	+	0.990116i	$0.544791\pi$
$728$	30.9912	1.14861
$729$	0	0
$730$	14.9726	0.554159
$731$	53.4072	1.97534
$732$	0	0
$733$	−12.6942	−0.468869	−0.234435	−	0.972132i	$-0.575324\pi$
$733$	−12.6942	−0.468869	−0.234435	+	0.972132i	$0.575324\pi$
$734$	14.9056	0.550176
$735$	0	0
$736$	2.56784	0.0946518
$737$	−12.5899	−0.463754
$738$	0	0
$739$	13.3928	0.492662	0.246331	−	0.969186i	$-0.420775\pi$
$739$	13.3928	0.492662	0.246331	+	0.969186i	$0.420775\pi$
$740$	−3.30223	−0.121393
$741$	0	0
$742$	41.1444	1.51046
$743$	3.77240	0.138396	0.0691980	−	0.997603i	$-0.477956\pi$
$743$	3.77240	0.138396	0.0691980	+	0.997603i	$0.477956\pi$
$744$	0	0
$745$	36.3360	1.33125
$746$	−36.4486	−1.33448
$747$	0	0
$748$	0.883721	0.0323120
$749$	−2.82362	−0.103173
$750$	0	0
$751$	−9.51843	−0.347332	−0.173666	−	0.984805i	$-0.555561\pi$
$751$	−9.51843	−0.347332	−0.173666	+	0.984805i	$0.555561\pi$
$752$	46.0406	1.67893
$753$	0	0
$754$	38.6870	1.40890
$755$	−23.6903	−0.862179
$756$	0	0
$757$	−10.7916	−0.392229	−0.196114	−	0.980581i	$-0.562832\pi$
$757$	−10.7916	−0.392229	−0.196114	+	0.980581i	$0.562832\pi$
$758$	18.0117	0.654216
$759$	0	0
$760$	69.8497	2.53371
$761$	22.1374	0.802480	0.401240	−	0.915973i	$-0.368579\pi$
$761$	22.1374	0.802480	0.401240	+	0.915973i	$0.368579\pi$
$762$	0	0
$763$	28.9454	1.04789
$764$	3.27755	0.118577
$765$	0	0
$766$	32.7540	1.18345
$767$	−61.3569	−2.21547
$768$	0	0
$769$	19.4521	0.701460	0.350730	−	0.936477i	$-0.385934\pi$
$769$	19.4521	0.701460	0.350730	+	0.936477i	$0.385934\pi$
$770$	13.8871	0.500458
$771$	0	0
$772$	−1.56700	−0.0563977
$773$	−39.4207	−1.41786	−0.708932	−	0.705277i	$-0.750823\pi$
$773$	−39.4207	−1.41786	−0.708932	+	0.705277i	$0.750823\pi$
$774$	0	0
$775$	2.27064	0.0815637
$776$	42.8848	1.53947
$777$	0	0
$778$	−36.2046	−1.29800
$779$	29.2593	1.04832
$780$	0	0
$781$	−0.299770	−0.0107266
$782$	24.6118	0.880116
$783$	0	0
$784$	4.01745	0.143480
$785$	−1.82596	−0.0651714
$786$	0	0
$787$	−37.7502	−1.34565	−0.672824	−	0.739802i	$-0.734919\pi$
$787$	−37.7502	−1.34565	−0.672824	+	0.739802i	$0.734919\pi$
$788$	−0.744578	−0.0265245
$789$	0	0
$790$	73.0710	2.59975
$791$	40.4411	1.43792
$792$	0	0
$793$	4.64696	0.165018
$794$	−5.86874	−0.208274
$795$	0	0
$796$	2.02821	0.0718879
$797$	20.3463	0.720705	0.360352	−	0.932816i	$-0.382656\pi$
$797$	20.3463	0.720705	0.360352	+	0.932816i	$0.382656\pi$
$798$	0	0
$799$	61.3082	2.16893
$800$	−8.53261	−0.301673
$801$	0	0
$802$	−2.13103	−0.0752494
$803$	2.65468	0.0936816
$804$	0	0
$805$	27.7931	0.979578
$806$	−1.58664	−0.0558870
$807$	0	0
$808$	20.9318	0.736377
$809$	3.16766	0.111369	0.0556844	−	0.998448i	$-0.482266\pi$
$809$	3.16766	0.111369	0.0556844	+	0.998448i	$0.482266\pi$
$810$	0	0
$811$	−14.3129	−0.502593	−0.251296	−	0.967910i	$-0.580857\pi$
$811$	−14.3129	−0.502593	−0.251296	+	0.967910i	$0.580857\pi$
$812$	−2.16237	−0.0758844
$813$	0	0
$814$	−8.14757	−0.285572
$815$	−67.8612	−2.37707
$816$	0	0
$817$	−62.8126	−2.19754
$818$	19.3902	0.677961
$819$	0	0
$820$	−2.59360	−0.0905726
$821$	14.3547	0.500982	0.250491	−	0.968119i	$-0.419408\pi$
$821$	14.3547	0.500982	0.250491	+	0.968119i	$0.419408\pi$
$822$	0	0
$823$	21.5880	0.752509	0.376255	−	0.926516i	$-0.377212\pi$
$823$	21.5880	0.752509	0.376255	+	0.926516i	$0.377212\pi$
$824$	−43.6998	−1.52235
$825$	0	0
$826$	47.7235	1.66051
$827$	−27.7946	−0.966513	−0.483257	−	0.875479i	$-0.660546\pi$
$827$	−27.7946	−0.966513	−0.483257	+	0.875479i	$0.660546\pi$
$828$	0	0
$829$	18.2361	0.633368	0.316684	−	0.948531i	$-0.397431\pi$
$829$	18.2361	0.633368	0.316684	+	0.948531i	$0.397431\pi$
$830$	−41.8777	−1.45360
$831$	0	0
$832$	−33.8689	−1.17419
$833$	5.34969	0.185356
$834$	0	0
$835$	−61.4227	−2.12562
$836$	−1.03935	−0.0359467
$837$	0	0
$838$	18.6078	0.642794
$839$	−25.3815	−0.876268	−0.438134	−	0.898910i	$-0.644360\pi$
$839$	−25.3815	−0.876268	−0.438134	+	0.898910i	$0.644360\pi$
$840$	0	0
$841$	3.16442	0.109118
$842$	−6.40166	−0.220616
$843$	0	0
$844$	−0.985060	−0.0339072
$845$	−33.0205	−1.13594
$846$	0	0
$847$	2.46223	0.0846034
$848$	−48.7861	−1.67532
$849$	0	0
$850$	−81.7819	−2.80510
$851$	−16.3062	−0.558968
$852$	0	0
$853$	−50.9440	−1.74429	−0.872144	−	0.489249i	$-0.837271\pi$
$853$	−50.9440	−1.74429	−0.872144	+	0.489249i	$0.837271\pi$
$854$	−3.61441	−0.123683
$855$	0	0
$856$	3.10612	0.106165
$857$	−25.5195	−0.871729	−0.435865	−	0.900012i	$-0.643557\pi$
$857$	−25.5195	−0.871729	−0.435865	+	0.900012i	$0.643557\pi$
$858$	0	0
$859$	19.5049	0.665497	0.332748	−	0.943016i	$-0.392024\pi$
$859$	19.5049	0.665497	0.332748	+	0.943016i	$0.392024\pi$
$860$	5.56784	0.189862
$861$	0	0
$862$	2.53090	0.0862027
$863$	23.2335	0.790876	0.395438	−	0.918493i	$-0.370593\pi$
$863$	23.2335	0.790876	0.395438	+	0.918493i	$0.370593\pi$
$864$	0	0
$865$	−8.53962	−0.290356
$866$	−9.80898	−0.333323
$867$	0	0
$868$	0.0886837	0.00301012
$869$	12.9557	0.439492
$870$	0	0
$871$	−58.5047	−1.98235
$872$	−31.8413	−1.07828
$873$	0	0
$874$	−28.9461	−0.979117
$875$	−45.0516	−1.52302
$876$	0	0
$877$	7.94166	0.268171	0.134085	−	0.990970i	$-0.457190\pi$
$877$	7.94166	0.268171	0.134085	+	0.990970i	$0.457190\pi$
$878$	−6.78516	−0.228988
$879$	0	0
$880$	−16.4664	−0.555083
$881$	−8.19164	−0.275983	−0.137992	−	0.990433i	$-0.544065\pi$
$881$	−8.19164	−0.275983	−0.137992	+	0.990433i	$0.544065\pi$
$882$	0	0
$883$	16.4576	0.553844	0.276922	−	0.960892i	$-0.410686\pi$
$883$	16.4576	0.553844	0.276922	+	0.960892i	$0.410686\pi$
$884$	4.10662	0.138121
$885$	0	0
$886$	−36.2803	−1.21886
$887$	−13.3032	−0.446679	−0.223340	−	0.974741i	$-0.571696\pi$
$887$	−13.3032	−0.446679	−0.223340	+	0.974741i	$0.571696\pi$
$888$	0	0
$889$	29.3737	0.985162
$890$	−19.1999	−0.643582
$891$	0	0
$892$	3.48966	0.116842
$893$	−72.1050	−2.41290
$894$	0	0
$895$	−94.1752	−3.14793
$896$	30.6475	1.02386
$897$	0	0
$898$	24.9572	0.832832
$899$	−1.31913	−0.0439956
$900$	0	0
$901$	−64.9642	−2.16427
$902$	−6.39917	−0.213069
$903$	0	0
$904$	−44.4871	−1.47962
$905$	54.2928	1.80476
$906$	0	0
$907$	6.59934	0.219128	0.109564	−	0.993980i	$-0.465055\pi$
$907$	6.59934	0.219128	0.109564	+	0.993980i	$0.465055\pi$
$908$	−4.20706	−0.139616
$909$	0	0
$910$	64.5330	2.13925
$911$	45.1362	1.49543	0.747714	−	0.664021i	$-0.231152\pi$
$911$	45.1362	1.49543	0.747714	+	0.664021i	$0.231152\pi$
$912$	0	0
$913$	−7.42505	−0.245733
$914$	−46.0840	−1.52432
$915$	0	0
$916$	−3.81082	−0.125913
$917$	−51.3703	−1.69640
$918$	0	0
$919$	22.6447	0.746979	0.373490	−	0.927634i	$-0.378161\pi$
$919$	22.6447	0.746979	0.373490	+	0.927634i	$0.378161\pi$
$920$	−30.5737	−1.00798
$921$	0	0
$922$	−6.78156	−0.223339
$923$	−1.39302	−0.0458518
$924$	0	0
$925$	54.1834	1.78154
$926$	40.6409	1.33554
$927$	0	0
$928$	4.95705	0.162723
$929$	11.5176	0.377879	0.188940	−	0.981989i	$-0.439495\pi$
$929$	11.5176	0.377879	0.188940	+	0.981989i	$0.439495\pi$
$930$	0	0
$931$	−6.29181	−0.206206
$932$	−3.72372	−0.121974
$933$	0	0
$934$	−5.63111	−0.184255
$935$	−21.9269	−0.717086
$936$	0	0
$937$	−7.74043	−0.252869	−0.126434	−	0.991975i	$-0.540353\pi$
$937$	−7.74043	−0.252869	−0.126434	+	0.991975i	$0.540353\pi$
$938$	45.5050	1.48579
$939$	0	0
$940$	6.39153	0.208469
$941$	31.6675	1.03233	0.516166	−	0.856489i	$-0.327359\pi$
$941$	31.6675	1.03233	0.516166	+	0.856489i	$0.327359\pi$
$942$	0	0
$943$	−12.8070	−0.417053
$944$	−56.5872	−1.84176
$945$	0	0
$946$	13.7375	0.446643
$947$	−32.2734	−1.04874	−0.524372	−	0.851489i	$-0.675700\pi$
$947$	−32.2734	−1.04874	−0.524372	+	0.851489i	$0.675700\pi$
$948$	0	0
$949$	12.3362	0.400450
$950$	96.1843	3.12063
$951$	0	0
$952$	38.0602	1.23354
$953$	−32.3994	−1.04952	−0.524759	−	0.851251i	$-0.675845\pi$
$953$	−32.3994	−1.04952	−0.524759	+	0.851251i	$0.675845\pi$
$954$	0	0
$955$	−81.3225	−2.63153
$956$	−2.51949	−0.0814862
$957$	0	0
$958$	−29.1332	−0.941251
$959$	47.1755	1.52338
$960$	0	0
$961$	−30.9459	−0.998255
$962$	−37.8614	−1.22070
$963$	0	0
$964$	0.238159	0.00767058
$965$	38.8805	1.25161
$966$	0	0
$967$	−54.5762	−1.75505	−0.877526	−	0.479529i	$-0.840808\pi$
$967$	−54.5762	−1.75505	−0.877526	+	0.479529i	$0.840808\pi$
$968$	−2.70857	−0.0870567
$969$	0	0
$970$	89.2990	2.86722
$971$	−3.84654	−0.123441	−0.0617207	−	0.998093i	$-0.519659\pi$
$971$	−3.84654	−0.123441	−0.0617207	+	0.998093i	$0.519659\pi$
$972$	0	0
$973$	−6.35755	−0.203814
$974$	−27.4724	−0.880272
$975$	0	0
$976$	4.28572	0.137183
$977$	3.97466	0.127161	0.0635804	−	0.997977i	$-0.479748\pi$
$977$	3.97466	0.127161	0.0635804	+	0.997977i	$0.479748\pi$
$978$	0	0
$979$	−3.40420	−0.108799
$980$	0.557719	0.0178157
$981$	0	0
$982$	48.6163	1.55141
$983$	−3.16990	−0.101104	−0.0505521	−	0.998721i	$-0.516098\pi$
$983$	−3.16990	−0.101104	−0.0505521	+	0.998721i	$0.516098\pi$
$984$	0	0
$985$	18.4745	0.588646
$986$	47.5115	1.51307
$987$	0	0
$988$	−4.82983	−0.153657
$989$	27.4935	0.874243
$990$	0	0
$991$	25.3534	0.805378	0.402689	−	0.915337i	$-0.368076\pi$
$991$	25.3534	0.805378	0.402689	+	0.915337i	$0.368076\pi$
$992$	−0.203299	−0.00645476
$993$	0	0
$994$	1.08349	0.0343663
$995$	−50.3239	−1.59537
$996$	0	0
$997$	5.71711	0.181063	0.0905314	−	0.995894i	$-0.471143\pi$
$997$	5.71711	0.181063	0.0905314	+	0.995894i	$0.471143\pi$
$998$	4.94254	0.156453
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.11		13
3.2	odd	2		2013.2.a.e.1.3	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.3	✓	13	3.2	odd	2
6039.2.a.i.1.11		13	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)

\(f(q)\)	\(=\)	\(q+1.46794 q^{2} +0.154851 q^{4} -3.84216 q^{5} +2.46223 q^{7} -2.70857 q^{8} +O(q^{10})\)	\(q+1.46794 q^{2} +0.154851 q^{4} -3.84216 q^{5} +2.46223 q^{7} -2.70857 q^{8} -5.64006 q^{10} -1.00000 q^{11} -4.64696 q^{13} +3.61441 q^{14} -4.28572 q^{16} -5.70692 q^{17} +6.71196 q^{19} -0.594961 q^{20} -1.46794 q^{22} -2.93787 q^{23} +9.76218 q^{25} -6.82147 q^{26} +0.381279 q^{28} -5.67137 q^{29} +0.232595 q^{31} -0.874048 q^{32} -8.37742 q^{34} -9.46029 q^{35} +5.55034 q^{37} +9.85276 q^{38} +10.4068 q^{40} +4.35928 q^{41} -9.35832 q^{43} -0.154851 q^{44} -4.31262 q^{46} -10.7428 q^{47} -0.937404 q^{49} +14.3303 q^{50} -0.719586 q^{52} +11.3834 q^{53} +3.84216 q^{55} -6.66913 q^{56} -8.32523 q^{58} +13.2037 q^{59} -1.00000 q^{61} +0.341436 q^{62} +7.28839 q^{64} +17.8544 q^{65} +12.5899 q^{67} -0.883721 q^{68} -13.8871 q^{70} +0.299770 q^{71} -2.65468 q^{73} +8.14757 q^{74} +1.03935 q^{76} -2.46223 q^{77} -12.9557 q^{79} +16.4664 q^{80} +6.39917 q^{82} +7.42505 q^{83} +21.9269 q^{85} -13.7375 q^{86} +2.70857 q^{88} +3.40420 q^{89} -11.4419 q^{91} -0.454931 q^{92} -15.7698 q^{94} -25.7884 q^{95} -15.8330 q^{97} -1.37605 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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