LMFDB - Embedded newform 6039.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34443$ 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)

$f(q)$	$=$	$q-2.34443 q^{2} +3.49635 q^{4} -3.40234 q^{5} -2.95885 q^{7} -3.50809 q^{8} +O(q^{10})$	\(q-2.34443 q^{2} +3.49635 q^{4} -3.40234 q^{5} -2.95885 q^{7} -3.50809 q^{8} +7.97654 q^{10} -1.00000 q^{11} +5.60121 q^{13} +6.93681 q^{14} +1.23176 q^{16} +3.44142 q^{17} +6.93678 q^{19} -11.8958 q^{20} +2.34443 q^{22} -0.836947 q^{23} +6.57591 q^{25} -13.1316 q^{26} -10.3452 q^{28} -6.68910 q^{29} +0.117341 q^{31} +4.12839 q^{32} -8.06817 q^{34} +10.0670 q^{35} -0.152675 q^{37} -16.2628 q^{38} +11.9357 q^{40} +10.6206 q^{41} +5.87242 q^{43} -3.49635 q^{44} +1.96216 q^{46} -0.889406 q^{47} +1.75477 q^{49} -15.4168 q^{50} +19.5838 q^{52} -3.79401 q^{53} +3.40234 q^{55} +10.3799 q^{56} +15.6821 q^{58} +9.37456 q^{59} -1.00000 q^{61} -0.275097 q^{62} -12.1423 q^{64} -19.0572 q^{65} +2.66764 q^{67} +12.0324 q^{68} -23.6014 q^{70} -1.13373 q^{71} -15.2793 q^{73} +0.357937 q^{74} +24.2534 q^{76} +2.95885 q^{77} +5.23851 q^{79} -4.19087 q^{80} -24.8993 q^{82} +17.7327 q^{83} -11.7089 q^{85} -13.7675 q^{86} +3.50809 q^{88} -2.71742 q^{89} -16.5731 q^{91} -2.92626 q^{92} +2.08515 q^{94} -23.6013 q^{95} -16.6565 q^{97} -4.11394 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	$ 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) $ 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.34443	−1.65776	−0.828881	−	0.559425i	$-0.811022\pi$
$2$	−2.34443	−1.65776	−0.828881	+	0.559425i	$0.811022\pi$
$3$	0	0
$3$	0	0
$4$	3.49635	1.74817
$5$	−3.40234	−1.52157	−0.760786	−	0.649003i	$-0.775186\pi$
$5$	−3.40234	−1.52157	−0.760786	+	0.649003i	$0.775186\pi$
$6$	0	0
$7$	−2.95885	−1.11834	−0.559169	−	0.829053i	$-0.688880\pi$
$7$	−2.95885	−1.11834	−0.559169	+	0.829053i	$0.688880\pi$
$8$	−3.50809	−1.24030
$9$	0	0
$10$	7.97654	2.52240
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.60121	1.55349	0.776747	−	0.629812i	$-0.216868\pi$
$13$	5.60121	1.55349	0.776747	+	0.629812i	$0.216868\pi$
$14$	6.93681	1.85394
$15$	0	0
$16$	1.23176	0.307941
$17$	3.44142	0.834667	0.417334	−	0.908753i	$-0.362965\pi$
$17$	3.44142	0.834667	0.417334	+	0.908753i	$0.362965\pi$
$18$	0	0
$19$	6.93678	1.59141	0.795703	−	0.605687i	$-0.207102\pi$
$19$	6.93678	1.59141	0.795703	+	0.605687i	$0.207102\pi$
$20$	−11.8958	−2.65997
$21$	0	0
$22$	2.34443	0.499834
$23$	−0.836947	−0.174516	−0.0872578	−	0.996186i	$-0.527810\pi$
$23$	−0.836947	−0.174516	−0.0872578	+	0.996186i	$0.527810\pi$
$24$	0	0
$25$	6.57591	1.31518
$26$	−13.1316	−2.57532
$27$	0	0
$28$	−10.3452	−1.95505
$29$	−6.68910	−1.24213	−0.621067	−	0.783757i	$-0.713301\pi$
$29$	−6.68910	−1.24213	−0.621067	+	0.783757i	$0.713301\pi$
$30$	0	0
$31$	0.117341	0.0210750	0.0105375	−	0.999944i	$-0.496646\pi$
$31$	0.117341	0.0210750	0.0105375	+	0.999944i	$0.496646\pi$
$32$	4.12839	0.729804
$33$	0	0
$34$	−8.06817	−1.38368
$35$	10.0670	1.70163
$36$	0	0
$37$	−0.152675	−0.0250997	−0.0125498	−	0.999921i	$-0.503995\pi$
$37$	−0.152675	−0.0250997	−0.0125498	+	0.999921i	$0.503995\pi$
$38$	−16.2628	−2.63817
$39$	0	0
$40$	11.9357	1.88720
$41$	10.6206	1.65866	0.829330	−	0.558759i	$-0.188722\pi$
$41$	10.6206	1.65866	0.829330	+	0.558759i	$0.188722\pi$
$42$	0	0
$43$	5.87242	0.895536	0.447768	−	0.894150i	$-0.352219\pi$
$43$	5.87242	0.895536	0.447768	+	0.894150i	$0.352219\pi$
$44$	−3.49635	−0.527095
$45$	0	0
$46$	1.96216	0.289305
$47$	−0.889406	−0.129733	−0.0648666	−	0.997894i	$-0.520662\pi$
$47$	−0.889406	−0.129733	−0.0648666	+	0.997894i	$0.520662\pi$
$48$	0	0
$49$	1.75477	0.250682
$50$	−15.4168	−2.18026
$51$	0	0
$52$	19.5838	2.71578
$53$	−3.79401	−0.521148	−0.260574	−	0.965454i	$-0.583912\pi$
$53$	−3.79401	−0.521148	−0.260574	+	0.965454i	$0.583912\pi$
$54$	0	0
$55$	3.40234	0.458771
$56$	10.3799	1.38707
$57$	0	0
$58$	15.6821	2.05916
$59$	9.37456	1.22046	0.610232	−	0.792223i	$-0.291076\pi$
$59$	9.37456	1.22046	0.610232	+	0.792223i	$0.291076\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.275097	−0.0349373
$63$	0	0
$64$	−12.1423	−1.51778
$65$	−19.0572	−2.36375
$66$	0	0
$67$	2.66764	0.325904	0.162952	−	0.986634i	$-0.447898\pi$
$67$	2.66764	0.325904	0.162952	+	0.986634i	$0.447898\pi$
$68$	12.0324	1.45914
$69$	0	0
$70$	−23.6014	−2.82090
$71$	−1.13373	−0.134549	−0.0672745	−	0.997735i	$-0.521430\pi$
$71$	−1.13373	−0.134549	−0.0672745	+	0.997735i	$0.521430\pi$
$72$	0	0
$73$	−15.2793	−1.78830	−0.894151	−	0.447766i	$-0.852220\pi$
$73$	−15.2793	−1.78830	−0.894151	+	0.447766i	$0.852220\pi$
$74$	0.357937	0.0416093
$75$	0	0
$76$	24.2534	2.78206
$77$	2.95885	0.337192
$78$	0	0
$79$	5.23851	0.589378	0.294689	−	0.955593i	$-0.404784\pi$
$79$	5.23851	0.589378	0.294689	+	0.955593i	$0.404784\pi$
$80$	−4.19087	−0.468554
$81$	0	0
$82$	−24.8993	−2.74966
$83$	17.7327	1.94641	0.973207	−	0.229929i	$-0.0738493\pi$
$83$	17.7327	1.94641	0.973207	+	0.229929i	$0.0738493\pi$
$84$	0	0
$85$	−11.7089	−1.27001
$86$	−13.7675	−1.48458
$87$	0	0
$88$	3.50809	0.373963
$89$	−2.71742	−0.288046	−0.144023	−	0.989574i	$-0.546004\pi$
$89$	−2.71742	−0.288046	−0.144023	+	0.989574i	$0.546004\pi$
$90$	0	0
$91$	−16.5731	−1.73733
$92$	−2.92626	−0.305084
$93$	0	0
$94$	2.08515	0.215067
$95$	−23.6013	−2.42144
$96$	0	0
$97$	−16.6565	−1.69121	−0.845604	−	0.533811i	$-0.820759\pi$
$97$	−16.6565	−1.69121	−0.845604	+	0.533811i	$0.820759\pi$
$98$	−4.11394	−0.415571
$99$	0	0
$100$	22.9917	2.29917
$101$	−5.98398	−0.595429	−0.297714	−	0.954655i	$-0.596224\pi$
$101$	−5.98398	−0.595429	−0.297714	+	0.954655i	$0.596224\pi$
$102$	0	0
$103$	−6.59746	−0.650067	−0.325033	−	0.945703i	$-0.605376\pi$
$103$	−6.59746	−0.650067	−0.325033	+	0.945703i	$0.605376\pi$
$104$	−19.6495	−1.92679
$105$	0	0
$106$	8.89480	0.863939
$107$	−9.19690	−0.889098	−0.444549	−	0.895754i	$-0.646636\pi$
$107$	−9.19690	−0.889098	−0.444549	+	0.895754i	$0.646636\pi$
$108$	0	0
$109$	−8.38506	−0.803143	−0.401571	−	0.915828i	$-0.631536\pi$
$109$	−8.38506	−0.803143	−0.401571	+	0.915828i	$0.631536\pi$
$110$	−7.97654	−0.760534
$111$	0	0
$112$	−3.64460	−0.344382
$113$	17.3735	1.63437	0.817183	−	0.576379i	$-0.195535\pi$
$113$	17.3735	1.63437	0.817183	+	0.576379i	$0.195535\pi$
$114$	0	0
$115$	2.84758	0.265538
$116$	−23.3874	−2.17147
$117$	0	0
$118$	−21.9780	−2.02324
$119$	−10.1826	−0.933441
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.34443	0.212255
$123$	0	0
$124$	0.410264	0.0368428
$125$	−5.36178	−0.479572
$126$	0	0
$127$	−11.1692	−0.991102	−0.495551	−	0.868579i	$-0.665034\pi$
$127$	−11.1692	−0.991102	−0.495551	+	0.868579i	$0.665034\pi$
$128$	20.2099	1.78632
$129$	0	0
$130$	44.6783	3.91854
$131$	0.0869930	0.00760061	0.00380031	−	0.999993i	$-0.498790\pi$
$131$	0.0869930	0.00760061	0.00380031	+	0.999993i	$0.498790\pi$
$132$	0	0
$133$	−20.5249	−1.77973
$134$	−6.25409	−0.540271
$135$	0	0
$136$	−12.0728	−1.03523
$137$	−10.4873	−0.895989	−0.447994	−	0.894036i	$-0.647862\pi$
$137$	−10.4873	−0.895989	−0.447994	+	0.894036i	$0.647862\pi$
$138$	0	0
$139$	7.79620	0.661265	0.330633	−	0.943760i	$-0.392738\pi$
$139$	7.79620	0.661265	0.330633	+	0.943760i	$0.392738\pi$
$140$	35.1978	2.97475
$141$	0	0
$142$	2.65795	0.223050
$143$	−5.60121	−0.468396
$144$	0	0
$145$	22.7586	1.89000
$146$	35.8211	2.96458
$147$	0	0
$148$	−0.533806	−0.0438786
$149$	−10.3850	−0.850770	−0.425385	−	0.905012i	$-0.639861\pi$
$149$	−10.3850	−0.850770	−0.425385	+	0.905012i	$0.639861\pi$
$150$	0	0
$151$	−3.38814	−0.275723	−0.137862	−	0.990452i	$-0.544023\pi$
$151$	−3.38814	−0.275723	−0.137862	+	0.990452i	$0.544023\pi$
$152$	−24.3348	−1.97381
$153$	0	0
$154$	−6.93681	−0.558984
$155$	−0.399233	−0.0320671
$156$	0	0
$157$	6.15931	0.491567	0.245783	−	0.969325i	$-0.420955\pi$
$157$	6.15931	0.491567	0.245783	+	0.969325i	$0.420955\pi$
$158$	−12.2813	−0.977049
$159$	0	0
$160$	−14.0462	−1.11045
$161$	2.47640	0.195168
$162$	0	0
$163$	−0.500868	−0.0392310	−0.0196155	−	0.999808i	$-0.506244\pi$
$163$	−0.500868	−0.0392310	−0.0196155	+	0.999808i	$0.506244\pi$
$164$	37.1333	2.89963
$165$	0	0
$166$	−41.5730	−3.22669
$167$	−4.82285	−0.373203	−0.186602	−	0.982436i	$-0.559747\pi$
$167$	−4.82285	−0.373203	−0.186602	+	0.982436i	$0.559747\pi$
$168$	0	0
$169$	18.3735	1.41335
$170$	27.4507	2.10537
$171$	0	0
$172$	20.5320	1.56555
$173$	−5.75856	−0.437816	−0.218908	−	0.975746i	$-0.570249\pi$
$173$	−5.75856	−0.437816	−0.218908	+	0.975746i	$0.570249\pi$
$174$	0	0
$175$	−19.4571	−1.47082
$176$	−1.23176	−0.0928476
$177$	0	0
$178$	6.37079	0.477511
$179$	−12.1227	−0.906090	−0.453045	−	0.891488i	$-0.649662\pi$
$179$	−12.1227	−0.906090	−0.453045	+	0.891488i	$0.649662\pi$
$180$	0	0
$181$	26.0278	1.93463	0.967317	−	0.253571i	$-0.0816051\pi$
$181$	26.0278	1.93463	0.967317	+	0.253571i	$0.0816051\pi$
$182$	38.8545	2.88009
$183$	0	0
$184$	2.93608	0.216451
$185$	0.519453	0.0381910
$186$	0	0
$187$	−3.44142	−0.251662
$188$	−3.10967	−0.226796
$189$	0	0
$190$	55.3315	4.01417
$191$	7.55414	0.546599	0.273299	−	0.961929i	$-0.411885\pi$
$191$	7.55414	0.546599	0.273299	+	0.961929i	$0.411885\pi$
$192$	0	0
$193$	7.78558	0.560418	0.280209	−	0.959939i	$-0.409596\pi$
$193$	7.78558	0.560418	0.280209	+	0.959939i	$0.409596\pi$
$194$	39.0499	2.80362
$195$	0	0
$196$	6.13530	0.438236
$197$	−10.6392	−0.758011	−0.379006	−	0.925394i	$-0.623734\pi$
$197$	−10.6392	−0.758011	−0.379006	+	0.925394i	$0.623734\pi$
$198$	0	0
$199$	−14.2510	−1.01023	−0.505113	−	0.863053i	$-0.668549\pi$
$199$	−14.2510	−1.01023	−0.505113	+	0.863053i	$0.668549\pi$
$200$	−23.0689	−1.63122
$201$	0	0
$202$	14.0290	0.987079
$203$	19.7920	1.38913
$204$	0	0
$205$	−36.1349	−2.52377
$206$	15.4673	1.07766
$207$	0	0
$208$	6.89935	0.478384
$209$	−6.93678	−0.479827
$210$	0	0
$211$	12.0462	0.829293	0.414646	−	0.909983i	$-0.363905\pi$
$211$	12.0462	0.829293	0.414646	+	0.909983i	$0.363905\pi$
$212$	−13.2652	−0.911058
$213$	0	0
$214$	21.5615	1.47391
$215$	−19.9800	−1.36262
$216$	0	0
$217$	−0.347193	−0.0235690
$218$	19.6582	1.33142
$219$	0	0
$220$	11.8958	0.802012
$221$	19.2761	1.29665
$222$	0	0
$223$	13.9356	0.933199	0.466600	−	0.884469i	$-0.345479\pi$
$223$	13.9356	0.933199	0.466600	+	0.884469i	$0.345479\pi$
$224$	−12.2153	−0.816168
$225$	0	0
$226$	−40.7311	−2.70939
$227$	6.52993	0.433407	0.216703	−	0.976237i	$-0.430470\pi$
$227$	6.52993	0.433407	0.216703	+	0.976237i	$0.430470\pi$
$228$	0	0
$229$	−11.8163	−0.780841	−0.390420	−	0.920637i	$-0.627670\pi$
$229$	−11.8163	−0.780841	−0.390420	+	0.920637i	$0.627670\pi$
$230$	−6.67595	−0.440199
$231$	0	0
$232$	23.4659	1.54061
$233$	24.9037	1.63149	0.815747	−	0.578409i	$-0.196326\pi$
$233$	24.9037	1.63149	0.815747	+	0.578409i	$0.196326\pi$
$234$	0	0
$235$	3.02606	0.197398
$236$	32.7768	2.13358
$237$	0	0
$238$	23.8725	1.54742
$239$	−9.06407	−0.586306	−0.293153	−	0.956066i	$-0.594704\pi$
$239$	−9.06407	−0.586306	−0.293153	+	0.956066i	$0.594704\pi$
$240$	0	0
$241$	18.2687	1.17679	0.588396	−	0.808573i	$-0.299760\pi$
$241$	18.2687	1.17679	0.588396	+	0.808573i	$0.299760\pi$
$242$	−2.34443	−0.150706
$243$	0	0
$244$	−3.49635	−0.223831
$245$	−5.97033	−0.381431
$246$	0	0
$247$	38.8543	2.47224
$248$	−0.411641	−0.0261392
$249$	0	0
$250$	12.5703	0.795017
$251$	21.8990	1.38225	0.691126	−	0.722734i	$-0.257115\pi$
$251$	21.8990	1.38225	0.691126	+	0.722734i	$0.257115\pi$
$252$	0	0
$253$	0.836947	0.0526184
$254$	26.1853	1.64301
$255$	0	0
$256$	−23.0961	−1.44351
$257$	−3.24672	−0.202525	−0.101263	−	0.994860i	$-0.532288\pi$
$257$	−3.24672	−0.202525	−0.101263	+	0.994860i	$0.532288\pi$
$258$	0	0
$259$	0.451743	0.0280699
$260$	−66.6306	−4.13226
$261$	0	0
$262$	−0.203949	−0.0126000
$263$	−26.4462	−1.63074	−0.815372	−	0.578938i	$-0.803467\pi$
$263$	−26.4462	−1.63074	−0.815372	+	0.578938i	$0.803467\pi$
$264$	0	0
$265$	12.9085	0.792964
$266$	48.1191	2.95037
$267$	0	0
$268$	9.32700	0.569737
$269$	6.95561	0.424091	0.212046	−	0.977260i	$-0.431987\pi$
$269$	6.95561	0.424091	0.212046	+	0.977260i	$0.431987\pi$
$270$	0	0
$271$	11.3494	0.689428	0.344714	−	0.938708i	$-0.387976\pi$
$271$	11.3494	0.689428	0.344714	+	0.938708i	$0.387976\pi$
$272$	4.23901	0.257028
$273$	0	0
$274$	24.5867	1.48534
$275$	−6.57591	−0.396542
$276$	0	0
$277$	−18.8926	−1.13515	−0.567574	−	0.823322i	$-0.692118\pi$
$277$	−18.8926	−1.13515	−0.567574	+	0.823322i	$0.692118\pi$
$278$	−18.2776	−1.09622
$279$	0	0
$280$	−35.3159	−2.11053
$281$	15.2462	0.909514	0.454757	−	0.890615i	$-0.349726\pi$
$281$	15.2462	0.909514	0.454757	+	0.890615i	$0.349726\pi$
$282$	0	0
$283$	−10.3061	−0.612631	−0.306316	−	0.951930i	$-0.599096\pi$
$283$	−10.3061	−0.612631	−0.306316	+	0.951930i	$0.599096\pi$
$284$	−3.96391	−0.235215
$285$	0	0
$286$	13.1316	0.776490
$287$	−31.4247	−1.85494
$288$	0	0
$289$	−5.15662	−0.303330
$290$	−53.3559	−3.13317
$291$	0	0
$292$	−53.4216	−3.12626
$293$	17.6417	1.03064	0.515320	−	0.856998i	$-0.327673\pi$
$293$	17.6417	1.03064	0.515320	+	0.856998i	$0.327673\pi$
$294$	0	0
$295$	−31.8954	−1.85702
$296$	0.535598	0.0311310
$297$	0	0
$298$	24.3468	1.41037
$299$	−4.68791	−0.271109
$300$	0	0
$301$	−17.3756	−1.00151
$302$	7.94326	0.457083
$303$	0	0
$304$	8.54446	0.490058
$305$	3.40234	0.194817
$306$	0	0
$307$	−8.18683	−0.467247	−0.233624	−	0.972327i	$-0.575058\pi$
$307$	−8.18683	−0.467247	−0.233624	+	0.972327i	$0.575058\pi$
$308$	10.3452	0.589470
$309$	0	0
$310$	0.935973	0.0531597
$311$	9.35473	0.530458	0.265229	−	0.964185i	$-0.414552\pi$
$311$	9.35473	0.530458	0.265229	+	0.964185i	$0.414552\pi$
$312$	0	0
$313$	−8.48617	−0.479667	−0.239833	−	0.970814i	$-0.577093\pi$
$313$	−8.48617	−0.479667	−0.239833	+	0.970814i	$0.577093\pi$
$314$	−14.4401	−0.814900
$315$	0	0
$316$	18.3157	1.03034
$317$	16.8444	0.946074	0.473037	−	0.881043i	$-0.343158\pi$
$317$	16.8444	0.946074	0.473037	+	0.881043i	$0.343158\pi$
$318$	0	0
$319$	6.68910	0.374518
$320$	41.3121	2.30941
$321$	0	0
$322$	−5.80574	−0.323541
$323$	23.8724	1.32829
$324$	0	0
$325$	36.8330	2.04313
$326$	1.17425	0.0650357
$327$	0	0
$328$	−37.2580	−2.05723
$329$	2.63162	0.145086
$330$	0	0
$331$	−3.26480	−0.179450	−0.0897249	−	0.995967i	$-0.528599\pi$
$331$	−3.26480	−0.179450	−0.0897249	+	0.995967i	$0.528599\pi$
$332$	61.9997	3.40267
$333$	0	0
$334$	11.3068	0.618683
$335$	−9.07621	−0.495886
$336$	0	0
$337$	−10.7004	−0.582888	−0.291444	−	0.956588i	$-0.594136\pi$
$337$	−10.7004	−0.582888	−0.291444	+	0.956588i	$0.594136\pi$
$338$	−43.0754	−2.34299
$339$	0	0
$340$	−40.9384	−2.22019
$341$	−0.117341	−0.00635435
$342$	0	0
$343$	15.5198	0.837992
$344$	−20.6010	−1.11073
$345$	0	0
$346$	13.5005	0.725794
$347$	−26.7520	−1.43612	−0.718061	−	0.695981i	$-0.754970\pi$
$347$	−26.7520	−1.43612	−0.718061	+	0.695981i	$0.754970\pi$
$348$	0	0
$349$	36.1274	1.93386	0.966928	−	0.255048i	$-0.0820914\pi$
$349$	36.1274	1.93386	0.966928	+	0.255048i	$0.0820914\pi$
$350$	45.6158	2.43827
$351$	0	0
$352$	−4.12839	−0.220044
$353$	11.3578	0.604514	0.302257	−	0.953227i	$-0.402260\pi$
$353$	11.3578	0.604514	0.302257	+	0.953227i	$0.402260\pi$
$354$	0	0
$355$	3.85733	0.204726
$356$	−9.50104	−0.503554
$357$	0	0
$358$	28.4207	1.50208
$359$	−21.9983	−1.16102	−0.580512	−	0.814252i	$-0.697148\pi$
$359$	−21.9983	−1.16102	−0.580512	+	0.814252i	$0.697148\pi$
$360$	0	0
$361$	29.1189	1.53257
$362$	−61.0204	−3.20716
$363$	0	0
$364$	−57.9454	−3.03716
$365$	51.9852	2.72103
$366$	0	0
$367$	−22.0610	−1.15158	−0.575788	−	0.817599i	$-0.695305\pi$
$367$	−22.0610	−1.15158	−0.575788	+	0.817599i	$0.695305\pi$
$368$	−1.03092	−0.0537404
$369$	0	0
$370$	−1.21782	−0.0633115
$371$	11.2259	0.582820
$372$	0	0
$373$	−17.4877	−0.905477	−0.452738	−	0.891643i	$-0.649553\pi$
$373$	−17.4877	−0.905477	−0.452738	+	0.891643i	$0.649553\pi$
$374$	8.06817	0.417195
$375$	0	0
$376$	3.12011	0.160908
$377$	−37.4670	−1.92965
$378$	0	0
$379$	−21.5859	−1.10879	−0.554396	−	0.832253i	$-0.687051\pi$
$379$	−21.5859	−1.10879	−0.554396	+	0.832253i	$0.687051\pi$
$380$	−82.5183	−4.23310
$381$	0	0
$382$	−17.7102	−0.906130
$383$	18.2673	0.933414	0.466707	−	0.884412i	$-0.345440\pi$
$383$	18.2673	0.933414	0.466707	+	0.884412i	$0.345440\pi$
$384$	0	0
$385$	−10.0670	−0.513062
$386$	−18.2527	−0.929040
$387$	0	0
$388$	−58.2368	−2.95653
$389$	33.0652	1.67647	0.838236	−	0.545307i	$-0.183587\pi$
$389$	33.0652	1.67647	0.838236	+	0.545307i	$0.183587\pi$
$390$	0	0
$391$	−2.88029	−0.145662
$392$	−6.15590	−0.310920
$393$	0	0
$394$	24.9428	1.25660
$395$	−17.8232	−0.896781
$396$	0	0
$397$	30.1806	1.51472	0.757359	−	0.652998i	$-0.226489\pi$
$397$	30.1806	1.51472	0.757359	+	0.652998i	$0.226489\pi$
$398$	33.4104	1.67471
$399$	0	0
$400$	8.09996	0.404998
$401$	−19.5558	−0.976569	−0.488284	−	0.872685i	$-0.662377\pi$
$401$	−19.5558	−0.976569	−0.488284	+	0.872685i	$0.662377\pi$
$402$	0	0
$403$	0.657249	0.0327399
$404$	−20.9221	−1.04091
$405$	0	0
$406$	−46.4010	−2.30284
$407$	0.152675	0.00756784
$408$	0	0
$409$	23.6297	1.16841	0.584206	−	0.811605i	$-0.301406\pi$
$409$	23.6297	1.16841	0.584206	+	0.811605i	$0.301406\pi$
$410$	84.7157	4.18381
$411$	0	0
$412$	−23.0670	−1.13643
$413$	−27.7379	−1.36489
$414$	0	0
$415$	−60.3326	−2.96161
$416$	23.1240	1.13375
$417$	0	0
$418$	16.2628	0.795439
$419$	−2.18707	−0.106845	−0.0534227	−	0.998572i	$-0.517013\pi$
$419$	−2.18707	−0.106845	−0.0534227	+	0.998572i	$0.517013\pi$
$420$	0	0
$421$	−18.1274	−0.883476	−0.441738	−	0.897144i	$-0.645638\pi$
$421$	−18.1274	−0.883476	−0.441738	+	0.897144i	$0.645638\pi$
$422$	−28.2414	−1.37477
$423$	0	0
$424$	13.3097	0.646378
$425$	22.6305	1.09774
$426$	0	0
$427$	2.95885	0.143189
$428$	−32.1556	−1.55430
$429$	0	0
$430$	46.8416	2.25890
$431$	−6.17100	−0.297247	−0.148623	−	0.988894i	$-0.547484\pi$
$431$	−6.17100	−0.297247	−0.148623	+	0.988894i	$0.547484\pi$
$432$	0	0
$433$	39.8873	1.91686	0.958431	−	0.285326i	$-0.0921018\pi$
$433$	39.8873	1.91686	0.958431	+	0.285326i	$0.0921018\pi$
$434$	0.813970	0.0390718
$435$	0	0
$436$	−29.3171	−1.40403
$437$	−5.80572	−0.277725
$438$	0	0
$439$	1.12159	0.0535304	0.0267652	−	0.999642i	$-0.491479\pi$
$439$	1.12159	0.0535304	0.0267652	+	0.999642i	$0.491479\pi$
$440$	−11.9357	−0.569012
$441$	0	0
$442$	−45.1915	−2.14954
$443$	−19.2213	−0.913231	−0.456616	−	0.889664i	$-0.650939\pi$
$443$	−19.2213	−0.913231	−0.456616	+	0.889664i	$0.650939\pi$
$444$	0	0
$445$	9.24557	0.438282
$446$	−32.6711	−1.54702
$447$	0	0
$448$	35.9271	1.69739
$449$	29.0769	1.37222	0.686111	−	0.727497i	$-0.259316\pi$
$449$	29.0769	1.37222	0.686111	+	0.727497i	$0.259316\pi$
$450$	0	0
$451$	−10.6206	−0.500105
$452$	60.7440	2.85716
$453$	0	0
$454$	−15.3090	−0.718485
$455$	56.3873	2.64348
$456$	0	0
$457$	−19.8618	−0.929096	−0.464548	−	0.885548i	$-0.653783\pi$
$457$	−19.8618	−0.929096	−0.464548	+	0.885548i	$0.653783\pi$
$458$	27.7024	1.29445
$459$	0	0
$460$	9.95613	0.464207
$461$	7.77290	0.362020	0.181010	−	0.983481i	$-0.442063\pi$
$461$	7.77290	0.362020	0.181010	+	0.983481i	$0.442063\pi$
$462$	0	0
$463$	−6.57813	−0.305712	−0.152856	−	0.988248i	$-0.548847\pi$
$463$	−6.57813	−0.305712	−0.152856	+	0.988248i	$0.548847\pi$
$464$	−8.23938	−0.382504
$465$	0	0
$466$	−58.3849	−2.70463
$467$	26.7754	1.23902	0.619509	−	0.784990i	$-0.287332\pi$
$467$	26.7754	1.23902	0.619509	+	0.784990i	$0.287332\pi$
$468$	0	0
$469$	−7.89313	−0.364471
$470$	−7.09439	−0.327240
$471$	0	0
$472$	−32.8868	−1.51374
$473$	−5.87242	−0.270014
$474$	0	0
$475$	45.6156	2.09299
$476$	−35.6021	−1.63182
$477$	0	0
$478$	21.2501	0.971955
$479$	18.7776	0.857970	0.428985	−	0.903312i	$-0.358871\pi$
$479$	18.7776	0.857970	0.428985	+	0.903312i	$0.358871\pi$
$480$	0	0
$481$	−0.855166	−0.0389922
$482$	−42.8297	−1.95084
$483$	0	0
$484$	3.49635	0.158925
$485$	56.6709	2.57329
$486$	0	0
$487$	33.6902	1.52665	0.763324	−	0.646016i	$-0.223566\pi$
$487$	33.6902	1.52665	0.763324	+	0.646016i	$0.223566\pi$
$488$	3.50809	0.158804
$489$	0	0
$490$	13.9970	0.632321
$491$	5.96513	0.269203	0.134601	−	0.990900i	$-0.457025\pi$
$491$	5.96513	0.269203	0.134601	+	0.990900i	$0.457025\pi$
$492$	0	0
$493$	−23.0200	−1.03677
$494$	−91.0912	−4.09839
$495$	0	0
$496$	0.144536	0.00648985
$497$	3.35453	0.150471
$498$	0	0
$499$	−29.5064	−1.32089	−0.660445	−	0.750874i	$-0.729632\pi$
$499$	−29.5064	−1.32089	−0.660445	+	0.750874i	$0.729632\pi$
$500$	−18.7467	−0.838377
$501$	0	0
$502$	−51.3407	−2.29145
$503$	−22.2297	−0.991173	−0.495586	−	0.868559i	$-0.665047\pi$
$503$	−22.2297	−0.991173	−0.495586	+	0.868559i	$0.665047\pi$
$504$	0	0
$505$	20.3595	0.905988
$506$	−1.96216	−0.0872288
$507$	0	0
$508$	−39.0513	−1.73262
$509$	22.8655	1.01350	0.506749	−	0.862094i	$-0.330847\pi$
$509$	22.8655	1.01350	0.506749	+	0.862094i	$0.330847\pi$
$510$	0	0
$511$	45.2090	1.99993
$512$	13.7275	0.606674
$513$	0	0
$514$	7.61171	0.335738
$515$	22.4468	0.989124
$516$	0	0
$517$	0.889406	0.0391160
$518$	−1.05908	−0.0465333
$519$	0	0
$520$	66.8543	2.93176
$521$	−10.0702	−0.441182	−0.220591	−	0.975366i	$-0.570799\pi$
$521$	−10.0702	−0.441182	−0.220591	+	0.975366i	$0.570799\pi$
$522$	0	0
$523$	28.0628	1.22710	0.613551	−	0.789655i	$-0.289740\pi$
$523$	28.0628	1.22710	0.613551	+	0.789655i	$0.289740\pi$
$524$	0.304158	0.0132872
$525$	0	0
$526$	62.0013	2.70338
$527$	0.403819	0.0175906
$528$	0	0
$529$	−22.2995	−0.969544
$530$	−30.2631	−1.31455
$531$	0	0
$532$	−71.7621	−3.11128
$533$	59.4882	2.57672
$534$	0	0
$535$	31.2910	1.35283
$536$	−9.35831	−0.404217
$537$	0	0
$538$	−16.3069	−0.703042
$539$	−1.75477	−0.0755834
$540$	0	0
$541$	28.2703	1.21544	0.607718	−	0.794153i	$-0.292085\pi$
$541$	28.2703	1.21544	0.607718	+	0.794153i	$0.292085\pi$
$542$	−26.6079	−1.14291
$543$	0	0
$544$	14.2075	0.609143
$545$	28.5288	1.22204
$546$	0	0
$547$	−7.09345	−0.303294	−0.151647	−	0.988435i	$-0.548458\pi$
$547$	−7.09345	−0.303294	−0.151647	+	0.988435i	$0.548458\pi$
$548$	−36.6672	−1.56635
$549$	0	0
$550$	15.4168	0.657373
$551$	−46.4008	−1.97674
$552$	0	0
$553$	−15.4999	−0.659124
$554$	44.2924	1.88180
$555$	0	0
$556$	27.2582	1.15601
$557$	−26.9056	−1.14002	−0.570012	−	0.821636i	$-0.693062\pi$
$557$	−26.9056	−1.14002	−0.570012	+	0.821636i	$0.693062\pi$
$558$	0	0
$559$	32.8926	1.39121
$560$	12.4001	0.524002
$561$	0	0
$562$	−35.7437	−1.50776
$563$	16.1624	0.681164	0.340582	−	0.940215i	$-0.389376\pi$
$563$	16.1624	0.681164	0.340582	+	0.940215i	$0.389376\pi$
$564$	0	0
$565$	−59.1107	−2.48681
$566$	24.1618	1.01560
$567$	0	0
$568$	3.97722	0.166880
$569$	−6.60974	−0.277095	−0.138547	−	0.990356i	$-0.544243\pi$
$569$	−6.60974	−0.277095	−0.138547	+	0.990356i	$0.544243\pi$
$570$	0	0
$571$	−15.9453	−0.667291	−0.333646	−	0.942699i	$-0.608279\pi$
$571$	−15.9453	−0.667291	−0.333646	+	0.942699i	$0.608279\pi$
$572$	−19.5838	−0.818839
$573$	0	0
$574$	73.6731	3.07506
$575$	−5.50369	−0.229520
$576$	0	0
$577$	−3.98449	−0.165876	−0.0829382	−	0.996555i	$-0.526430\pi$
$577$	−3.98449	−0.165876	−0.0829382	+	0.996555i	$0.526430\pi$
$578$	12.0893	0.502850
$579$	0	0
$580$	79.5720	3.30405
$581$	−52.4683	−2.17675
$582$	0	0
$583$	3.79401	0.157132
$584$	53.6010	2.21802
$585$	0	0
$586$	−41.3597	−1.70856
$587$	11.1945	0.462046	0.231023	−	0.972948i	$-0.425793\pi$
$587$	11.1945	0.462046	0.231023	+	0.972948i	$0.425793\pi$
$588$	0	0
$589$	0.813966	0.0335389
$590$	74.7766	3.07850
$591$	0	0
$592$	−0.188060	−0.00772921
$593$	1.90798	0.0783514	0.0391757	−	0.999232i	$-0.487527\pi$
$593$	1.90798	0.0783514	0.0391757	+	0.999232i	$0.487527\pi$
$594$	0	0
$595$	34.6448	1.42030
$596$	−36.3095	−1.48730
$597$	0	0
$598$	10.9905	0.449434
$599$	0.764630	0.0312419	0.0156210	−	0.999878i	$-0.495027\pi$
$599$	0.764630	0.0312419	0.0156210	+	0.999878i	$0.495027\pi$
$600$	0	0
$601$	31.5123	1.28541	0.642706	−	0.766113i	$-0.277812\pi$
$601$	31.5123	1.28541	0.642706	+	0.766113i	$0.277812\pi$
$602$	40.7358	1.66027
$603$	0	0
$604$	−11.8461	−0.482012
$605$	−3.40234	−0.138325
$606$	0	0
$607$	−21.1448	−0.858242	−0.429121	−	0.903247i	$-0.641177\pi$
$607$	−21.1448	−0.858242	−0.429121	+	0.903247i	$0.641177\pi$
$608$	28.6377	1.16141
$609$	0	0
$610$	−7.97654	−0.322961
$611$	−4.98175	−0.201540
$612$	0	0
$613$	3.15314	0.127354	0.0636771	−	0.997971i	$-0.479717\pi$
$613$	3.15314	0.127354	0.0636771	+	0.997971i	$0.479717\pi$
$614$	19.1934	0.774584
$615$	0	0
$616$	−10.3799	−0.418218
$617$	7.47829	0.301065	0.150532	−	0.988605i	$-0.451901\pi$
$617$	7.47829	0.301065	0.150532	+	0.988605i	$0.451901\pi$
$618$	0	0
$619$	12.4737	0.501359	0.250679	−	0.968070i	$-0.419346\pi$
$619$	12.4737	0.501359	0.250679	+	0.968070i	$0.419346\pi$
$620$	−1.39586	−0.0560590
$621$	0	0
$622$	−21.9315	−0.879373
$623$	8.04042	0.322133
$624$	0	0
$625$	−14.6370	−0.585478
$626$	19.8952	0.795173
$627$	0	0
$628$	21.5351	0.859344
$629$	−0.525420	−0.0209499
$630$	0	0
$631$	31.4546	1.25219	0.626095	−	0.779747i	$-0.284652\pi$
$631$	31.4546	1.25219	0.626095	+	0.779747i	$0.284652\pi$
$632$	−18.3771	−0.731003
$633$	0	0
$634$	−39.4904	−1.56837
$635$	38.0013	1.50803
$636$	0	0
$637$	9.82885	0.389433
$638$	−15.6821	−0.620861
$639$	0	0
$640$	−68.7608	−2.71801
$641$	−20.2670	−0.800500	−0.400250	−	0.916406i	$-0.631077\pi$
$641$	−20.2670	−0.800500	−0.400250	+	0.916406i	$0.631077\pi$
$642$	0	0
$643$	−30.4684	−1.20156	−0.600779	−	0.799415i	$-0.705143\pi$
$643$	−30.4684	−1.20156	−0.600779	+	0.799415i	$0.705143\pi$
$644$	8.65835	0.341187
$645$	0	0
$646$	−55.9671	−2.20200
$647$	−14.1881	−0.557792	−0.278896	−	0.960321i	$-0.589968\pi$
$647$	−14.1881	−0.557792	−0.278896	+	0.960321i	$0.589968\pi$
$648$	0	0
$649$	−9.37456	−0.367984
$650$	−86.3524	−3.38702
$651$	0	0
$652$	−1.75121	−0.0685826
$653$	−2.19721	−0.0859833	−0.0429917	−	0.999075i	$-0.513689\pi$
$653$	−2.19721	−0.0859833	−0.0429917	+	0.999075i	$0.513689\pi$
$654$	0	0
$655$	−0.295980	−0.0115649
$656$	13.0821	0.510769
$657$	0	0
$658$	−6.16964	−0.240518
$659$	17.7527	0.691547	0.345774	−	0.938318i	$-0.387616\pi$
$659$	17.7527	0.691547	0.345774	+	0.938318i	$0.387616\pi$
$660$	0	0
$661$	30.9265	1.20290	0.601451	−	0.798910i	$-0.294590\pi$
$661$	30.9265	1.20290	0.601451	+	0.798910i	$0.294590\pi$
$662$	7.65410	0.297485
$663$	0	0
$664$	−62.2078	−2.41413
$665$	69.8325	2.70799
$666$	0	0
$667$	5.59842	0.216772
$668$	−16.8624	−0.652425
$669$	0	0
$670$	21.2785	0.822062
$671$	1.00000	0.0386046
$672$	0	0
$673$	−40.4599	−1.55961	−0.779806	−	0.626021i	$-0.784682\pi$
$673$	−40.4599	−1.55961	−0.779806	+	0.626021i	$0.784682\pi$
$674$	25.0863	0.966289
$675$	0	0
$676$	64.2402	2.47078
$677$	−39.8009	−1.52967	−0.764837	−	0.644224i	$-0.777180\pi$
$677$	−39.8009	−1.52967	−0.764837	+	0.644224i	$0.777180\pi$
$678$	0	0
$679$	49.2839	1.89134
$680$	41.0758	1.57518
$681$	0	0
$682$	0.275097	0.0105340
$683$	6.44784	0.246720	0.123360	−	0.992362i	$-0.460633\pi$
$683$	6.44784	0.246720	0.123360	+	0.992362i	$0.460633\pi$
$684$	0	0
$685$	35.6813	1.36331
$686$	−36.3851	−1.38919
$687$	0	0
$688$	7.23342	0.275772
$689$	−21.2511	−0.809601
$690$	0	0
$691$	15.7322	0.598481	0.299241	−	0.954178i	$-0.403267\pi$
$691$	15.7322	0.598481	0.299241	+	0.954178i	$0.403267\pi$
$692$	−20.1340	−0.765378
$693$	0	0
$694$	62.7181	2.38075
$695$	−26.5253	−1.00616
$696$	0	0
$697$	36.5500	1.38443
$698$	−84.6982	−3.20587
$699$	0	0
$700$	−68.0289	−2.57125
$701$	2.82672	0.106764	0.0533818	−	0.998574i	$-0.483000\pi$
$701$	2.82672	0.106764	0.0533818	+	0.998574i	$0.483000\pi$
$702$	0	0
$703$	−1.05907	−0.0399438
$704$	12.1423	0.457628
$705$	0	0
$706$	−26.6275	−1.00214
$707$	17.7057	0.665891
$708$	0	0
$709$	46.2111	1.73549	0.867747	−	0.497007i	$-0.165568\pi$
$709$	46.2111	1.73549	0.867747	+	0.497007i	$0.165568\pi$
$710$	−9.04324	−0.339387
$711$	0	0
$712$	9.53293	0.357262
$713$	−0.0982080	−0.00367792
$714$	0	0
$715$	19.0572	0.712699
$716$	−42.3851	−1.58400
$717$	0	0
$718$	51.5734	1.92470
$719$	26.3866	0.984054	0.492027	−	0.870580i	$-0.336256\pi$
$719$	26.3866	0.984054	0.492027	+	0.870580i	$0.336256\pi$
$720$	0	0
$721$	19.5209	0.726995
$722$	−68.2672	−2.54064
$723$	0	0
$724$	91.0024	3.38208
$725$	−43.9869	−1.63363
$726$	0	0
$727$	17.8295	0.661261	0.330631	−	0.943760i	$-0.392739\pi$
$727$	17.8295	0.661261	0.330631	+	0.943760i	$0.392739\pi$
$728$	58.1399	2.15481
$729$	0	0
$730$	−121.876	−4.51082
$731$	20.2095	0.747474
$732$	0	0
$733$	−4.62017	−0.170650	−0.0853250	−	0.996353i	$-0.527193\pi$
$733$	−4.62017	−0.170650	−0.0853250	+	0.996353i	$0.527193\pi$
$734$	51.7205	1.90904
$735$	0	0
$736$	−3.45525	−0.127362
$737$	−2.66764	−0.0982637
$738$	0	0
$739$	−0.782472	−0.0287837	−0.0143919	−	0.999896i	$-0.504581\pi$
$739$	−0.782472	−0.0287837	−0.0143919	+	0.999896i	$0.504581\pi$
$740$	1.81619	0.0667645
$741$	0	0
$742$	−26.3183	−0.966177
$743$	−18.1688	−0.666547	−0.333274	−	0.942830i	$-0.608153\pi$
$743$	−18.1688	−0.666547	−0.333274	+	0.942830i	$0.608153\pi$
$744$	0	0
$745$	35.3332	1.29451
$746$	40.9986	1.50106
$747$	0	0
$748$	−12.0324	−0.439949
$749$	27.2122	0.994313
$750$	0	0
$751$	−21.7126	−0.792305	−0.396152	−	0.918185i	$-0.629655\pi$
$751$	−21.7126	−0.792305	−0.396152	+	0.918185i	$0.629655\pi$
$752$	−1.09554	−0.0399501
$753$	0	0
$754$	87.8388	3.19890
$755$	11.5276	0.419533
$756$	0	0
$757$	21.2185	0.771201	0.385600	−	0.922666i	$-0.373994\pi$
$757$	21.2185	0.771201	0.385600	+	0.922666i	$0.373994\pi$
$758$	50.6066	1.83811
$759$	0	0
$760$	82.7953	3.00330
$761$	49.6485	1.79976	0.899878	−	0.436142i	$-0.143656\pi$
$761$	49.6485	1.79976	0.899878	+	0.436142i	$0.143656\pi$
$762$	0	0
$763$	24.8101	0.898186
$764$	26.4119	0.955550
$765$	0	0
$766$	−42.8264	−1.54738
$767$	52.5089	1.89599
$768$	0	0
$769$	38.0830	1.37331	0.686654	−	0.726984i	$-0.259079\pi$
$769$	38.0830	1.37331	0.686654	+	0.726984i	$0.259079\pi$
$770$	23.6014	0.850534
$771$	0	0
$772$	27.2211	0.979709
$773$	−14.0384	−0.504925	−0.252462	−	0.967607i	$-0.581240\pi$
$773$	−14.0384	−0.504925	−0.252462	+	0.967607i	$0.581240\pi$
$774$	0	0
$775$	0.771622	0.0277175
$776$	58.4323	2.09760
$777$	0	0
$778$	−77.5190	−2.77919
$779$	73.6728	2.63960
$780$	0	0
$781$	1.13373	0.0405680
$782$	6.75263	0.241474
$783$	0	0
$784$	2.16146	0.0771951
$785$	−20.9561	−0.747954
$786$	0	0
$787$	35.9870	1.28280	0.641398	−	0.767208i	$-0.278355\pi$
$787$	35.9870	1.28280	0.641398	+	0.767208i	$0.278355\pi$
$788$	−37.1983	−1.32514
$789$	0	0
$790$	41.7852	1.48665
$791$	−51.4057	−1.82777
$792$	0	0
$793$	−5.60121	−0.198905
$794$	−70.7562	−2.51104
$795$	0	0
$796$	−49.8264	−1.76605
$797$	0.425077	0.0150570	0.00752850	−	0.999972i	$-0.497604\pi$
$797$	0.425077	0.0150570	0.00752850	+	0.999972i	$0.497604\pi$
$798$	0	0
$799$	−3.06082	−0.108284
$800$	27.1479	0.959825
$801$	0	0
$802$	45.8471	1.61892
$803$	15.2793	0.539193
$804$	0	0
$805$	−8.42555	−0.296962
$806$	−1.54087	−0.0542750
$807$	0	0
$808$	20.9923	0.738508
$809$	2.50013	0.0879000	0.0439500	−	0.999034i	$-0.486006\pi$
$809$	2.50013	0.0879000	0.0439500	+	0.999034i	$0.486006\pi$
$810$	0	0
$811$	−4.72280	−0.165840	−0.0829199	−	0.996556i	$-0.526425\pi$
$811$	−4.72280	−0.165840	−0.0829199	+	0.996556i	$0.526425\pi$
$812$	69.1998	2.42844
$813$	0	0
$814$	−0.357937	−0.0125457
$815$	1.70412	0.0596928
$816$	0	0
$817$	40.7357	1.42516
$818$	−55.3981	−1.93695
$819$	0	0
$820$	−126.340	−4.41199
$821$	47.3691	1.65319	0.826596	−	0.562796i	$-0.190274\pi$
$821$	47.3691	1.65319	0.826596	+	0.562796i	$0.190274\pi$
$822$	0	0
$823$	39.5678	1.37925	0.689623	−	0.724169i	$-0.257776\pi$
$823$	39.5678	1.37925	0.689623	+	0.724169i	$0.257776\pi$
$824$	23.1445	0.806275
$825$	0	0
$826$	65.0295	2.26267
$827$	−47.3942	−1.64806	−0.824029	−	0.566547i	$-0.808279\pi$
$827$	−47.3942	−1.64806	−0.824029	+	0.566547i	$0.808279\pi$
$828$	0	0
$829$	−9.95290	−0.345679	−0.172839	−	0.984950i	$-0.555294\pi$
$829$	−9.95290	−0.345679	−0.172839	+	0.984950i	$0.555294\pi$
$830$	141.446	4.90965
$831$	0	0
$832$	−68.0113	−2.35787
$833$	6.03891	0.209236
$834$	0	0
$835$	16.4090	0.567856
$836$	−24.2534	−0.838821
$837$	0	0
$838$	5.12743	0.177124
$839$	28.5042	0.984074	0.492037	−	0.870574i	$-0.336252\pi$
$839$	28.5042	0.984074	0.492037	+	0.870574i	$0.336252\pi$
$840$	0	0
$841$	15.7441	0.542899
$842$	42.4984	1.46459
$843$	0	0
$844$	42.1176	1.44975
$845$	−62.5129	−2.15051
$846$	0	0
$847$	−2.95885	−0.101667
$848$	−4.67332	−0.160483
$849$	0	0
$850$	−53.0556	−1.81979
$851$	0.127781	0.00438028
$852$	0	0
$853$	15.0590	0.515610	0.257805	−	0.966197i	$-0.417001\pi$
$853$	15.0590	0.515610	0.257805	+	0.966197i	$0.417001\pi$
$854$	−6.93681	−0.237373
$855$	0	0
$856$	32.2635	1.10274
$857$	16.3111	0.557176	0.278588	−	0.960411i	$-0.410134\pi$
$857$	16.3111	0.557176	0.278588	+	0.960411i	$0.410134\pi$
$858$	0	0
$859$	18.4124	0.628223	0.314111	−	0.949386i	$-0.398293\pi$
$859$	18.4124	0.628223	0.314111	+	0.949386i	$0.398293\pi$
$860$	−69.8569	−2.38210
$861$	0	0
$862$	14.4675	0.492764
$863$	−12.4595	−0.424127	−0.212064	−	0.977256i	$-0.568018\pi$
$863$	−12.4595	−0.424127	−0.212064	+	0.977256i	$0.568018\pi$
$864$	0	0
$865$	19.5926	0.666168
$866$	−93.5130	−3.17770
$867$	0	0
$868$	−1.21391	−0.0412027
$869$	−5.23851	−0.177704
$870$	0	0
$871$	14.9420	0.506290
$872$	29.4155	0.996135
$873$	0	0
$874$	13.6111	0.460402
$875$	15.8647	0.536325
$876$	0	0
$877$	53.9500	1.82176	0.910880	−	0.412671i	$-0.135404\pi$
$877$	53.9500	1.82176	0.910880	+	0.412671i	$0.135404\pi$
$878$	−2.62948	−0.0887407
$879$	0	0
$880$	4.19087	0.141274
$881$	43.0352	1.44989	0.724946	−	0.688805i	$-0.241865\pi$
$881$	43.0352	1.44989	0.724946	+	0.688805i	$0.241865\pi$
$882$	0	0
$883$	9.74848	0.328063	0.164031	−	0.986455i	$-0.447550\pi$
$883$	9.74848	0.328063	0.164031	+	0.986455i	$0.447550\pi$
$884$	67.3960	2.26677
$885$	0	0
$886$	45.0630	1.51392
$887$	−38.3592	−1.28797	−0.643987	−	0.765036i	$-0.722721\pi$
$887$	−38.3592	−1.28797	−0.643987	+	0.765036i	$0.722721\pi$
$888$	0	0
$889$	33.0478	1.10839
$890$	−21.6756	−0.726568
$891$	0	0
$892$	48.7238	1.63140
$893$	−6.16961	−0.206458
$894$	0	0
$895$	41.2454	1.37868
$896$	−59.7979	−1.99771
$897$	0	0
$898$	−68.1687	−2.27482
$899$	−0.784904	−0.0261780
$900$	0	0
$901$	−13.0568	−0.434985
$902$	24.8993	0.829055
$903$	0	0
$904$	−60.9479	−2.02710
$905$	−88.5555	−2.94368
$906$	0	0
$907$	−49.6502	−1.64861	−0.824304	−	0.566148i	$-0.808433\pi$
$907$	−49.6502	−1.64861	−0.824304	+	0.566148i	$0.808433\pi$
$908$	22.8309	0.757671
$909$	0	0
$910$	−132.196	−4.38226
$911$	−43.8140	−1.45162	−0.725812	−	0.687893i	$-0.758536\pi$
$911$	−43.8140	−1.45162	−0.725812	+	0.687893i	$0.758536\pi$
$912$	0	0
$913$	−17.7327	−0.586866
$914$	46.5646	1.54022
$915$	0	0
$916$	−41.3138	−1.36505
$917$	−0.257399	−0.00850006
$918$	0	0
$919$	5.04128	0.166296	0.0831482	−	0.996537i	$-0.473503\pi$
$919$	5.04128	0.166296	0.0831482	+	0.996537i	$0.473503\pi$
$920$	−9.98955	−0.329346
$921$	0	0
$922$	−18.2230	−0.600143
$923$	−6.35025	−0.209021
$924$	0	0
$925$	−1.00398	−0.0330106
$926$	15.4220	0.506797
$927$	0	0
$928$	−27.6152	−0.906515
$929$	17.0037	0.557873	0.278936	−	0.960310i	$-0.410018\pi$
$929$	17.0037	0.557873	0.278936	+	0.960310i	$0.410018\pi$
$930$	0	0
$931$	12.1725	0.398937
$932$	87.0719	2.85214
$933$	0	0
$934$	−62.7730	−2.05400
$935$	11.7089	0.382921
$936$	0	0
$937$	19.4635	0.635846	0.317923	−	0.948117i	$-0.397015\pi$
$937$	19.4635	0.635846	0.317923	+	0.948117i	$0.397015\pi$
$938$	18.5049	0.604206
$939$	0	0
$940$	10.5802	0.345087
$941$	−19.4710	−0.634736	−0.317368	−	0.948302i	$-0.602799\pi$
$941$	−19.4710	−0.634736	−0.317368	+	0.948302i	$0.602799\pi$
$942$	0	0
$943$	−8.88888	−0.289462
$944$	11.5472	0.375830
$945$	0	0
$946$	13.7675	0.447619
$947$	−30.1329	−0.979187	−0.489593	−	0.871951i	$-0.662855\pi$
$947$	−30.1329	−0.979187	−0.489593	+	0.871951i	$0.662855\pi$
$948$	0	0
$949$	−85.5823	−2.77812
$950$	−106.943	−3.46968
$951$	0	0
$952$	35.7216	1.15774
$953$	35.5897	1.15286	0.576432	−	0.817145i	$-0.304445\pi$
$953$	35.5897	1.15286	0.576432	+	0.817145i	$0.304445\pi$
$954$	0	0
$955$	−25.7018	−0.831689
$956$	−31.6911	−1.02496
$957$	0	0
$958$	−44.0227	−1.42231
$959$	31.0303	1.00202
$960$	0	0
$961$	−30.9862	−0.999556
$962$	2.00488	0.0646398
$963$	0	0
$964$	63.8738	2.05724
$965$	−26.4892	−0.852717
$966$	0	0
$967$	−0.0995826	−0.00320236	−0.00160118	−	0.999999i	$-0.500510\pi$
$967$	−0.0995826	−0.00320236	−0.00160118	+	0.999999i	$0.500510\pi$
$968$	−3.50809	−0.112754
$969$	0	0
$970$	−132.861	−4.26591
$971$	15.9640	0.512310	0.256155	−	0.966636i	$-0.417544\pi$
$971$	15.9640	0.512310	0.256155	+	0.966636i	$0.417544\pi$
$972$	0	0
$973$	−23.0678	−0.739519
$974$	−78.9842	−2.53082
$975$	0	0
$976$	−1.23176	−0.0394277
$977$	−23.2681	−0.744412	−0.372206	−	0.928150i	$-0.621399\pi$
$977$	−23.2681	−0.744412	−0.372206	+	0.928150i	$0.621399\pi$
$978$	0	0
$979$	2.71742	0.0868490
$980$	−20.8744	−0.666807
$981$	0	0
$982$	−13.9848	−0.446274
$983$	38.0596	1.21391	0.606956	−	0.794735i	$-0.292390\pi$
$983$	38.0596	1.21391	0.606956	+	0.794735i	$0.292390\pi$
$984$	0	0
$985$	36.1981	1.15337
$986$	53.9688	1.71872
$987$	0	0
$988$	135.848	4.32191
$989$	−4.91491	−0.156285
$990$	0	0
$991$	−13.6967	−0.435090	−0.217545	−	0.976050i	$-0.569805\pi$
$991$	−13.6967	−0.435090	−0.217545	+	0.976050i	$0.569805\pi$
$992$	0.484428	0.0153806
$993$	0	0
$994$	−7.86446	−0.249446
$995$	48.4867	1.53713
$996$	0	0
$997$	−18.8111	−0.595753	−0.297877	−	0.954604i	$-0.596278\pi$
$997$	−18.8111	−0.595753	−0.297877	+	0.954604i	$0.596278\pi$
$998$	69.1758	2.18972
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.k.1.3		19
3.2	odd	2		671.2.a.c.1.17	✓	19
33.32	even	2		7381.2.a.i.1.3		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.17	✓	19	3.2	odd	2
6039.2.a.k.1.3		19	1.1	even	1	trivial
7381.2.a.i.1.3		19	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34443 q^{2} +3.49635 q^{4} -3.40234 q^{5} -2.95885 q^{7} -3.50809 q^{8} +O(q^{10})\)	\(q-2.34443 q^{2} +3.49635 q^{4} -3.40234 q^{5} -2.95885 q^{7} -3.50809 q^{8} +7.97654 q^{10} -1.00000 q^{11} +5.60121 q^{13} +6.93681 q^{14} +1.23176 q^{16} +3.44142 q^{17} +6.93678 q^{19} -11.8958 q^{20} +2.34443 q^{22} -0.836947 q^{23} +6.57591 q^{25} -13.1316 q^{26} -10.3452 q^{28} -6.68910 q^{29} +0.117341 q^{31} +4.12839 q^{32} -8.06817 q^{34} +10.0670 q^{35} -0.152675 q^{37} -16.2628 q^{38} +11.9357 q^{40} +10.6206 q^{41} +5.87242 q^{43} -3.49635 q^{44} +1.96216 q^{46} -0.889406 q^{47} +1.75477 q^{49} -15.4168 q^{50} +19.5838 q^{52} -3.79401 q^{53} +3.40234 q^{55} +10.3799 q^{56} +15.6821 q^{58} +9.37456 q^{59} -1.00000 q^{61} -0.275097 q^{62} -12.1423 q^{64} -19.0572 q^{65} +2.66764 q^{67} +12.0324 q^{68} -23.6014 q^{70} -1.13373 q^{71} -15.2793 q^{73} +0.357937 q^{74} +24.2534 q^{76} +2.95885 q^{77} +5.23851 q^{79} -4.19087 q^{80} -24.8993 q^{82} +17.7327 q^{83} -11.7089 q^{85} -13.7675 q^{86} +3.50809 q^{88} -2.71742 q^{89} -16.5731 q^{91} -2.92626 q^{92} +2.08515 q^{94} -23.6013 q^{95} -16.6565 q^{97} -4.11394 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8	\( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 19 * q - 5 * q^2 + 23 * q^4 + 9 * q^7 - 9 * q^8 + 7 * q^10 - 19 * q^11 + 8 * q^13 + 11 * q^14 + 31 * q^16 - 9 * q^17 + 17 * q^19 + 6 * q^20 + 5 * q^22 + 10 * q^23 + 45 * q^25 - 5 * q^26 + 36 * q^28 - 27 * q^29 + 7 * q^31 - 8 * q^32 - 5 * q^34 - 17 * q^35 + 20 * q^37 + 37 * q^38 + 10 * q^40 - 19 * q^41 + 20 * q^43 - 23 * q^44 + 41 * q^46 + 19 * q^47 + 42 * q^49 - 36 * q^50 - 28 * q^52 - 3 * q^53 + 44 * q^56 + 23 * q^58 + 28 * q^59 - 19 * q^61 + 11 * q^62 + 47 * q^64 - 25 * q^65 + 3 * q^67 - 38 * q^68 + 3 * q^70 + 19 * q^71 + 20 * q^73 + 22 * q^74 - 25 * q^76 - 9 * q^77 + 69 * q^79 + 36 * q^80 - 61 * q^82 - q^83 + 24 * q^85 + 27 * q^86 + 9 * q^88 + 24 * q^91 + 67 * q^92 + 64 * q^94 + 3 * q^95 + 21 * q^97 + 87 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more