LMFDB - Embedded newform 6039.2.a.c.1.1

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:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2x^{10} - 14x^{9} + 27x^{8} + 66x^{7} - 125x^{6} - 115x^{5} + 227x^{4} + 40x^{3} - 129x^{2} + 26x - 1 $ x^11 - 2x^10 - 14x^9 + 27x^8 + 66x^7 - 125x^6 - 115x^5 + 227x^4 + 40x^3 - 129x^2 + 26x - 1
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			1.1
Root			$-2.39095$ 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)

$f(q)$	$=$	$q-2.39095 q^{2} +3.71663 q^{4} +2.34211 q^{5} -0.680854 q^{7} -4.10437 q^{8} +O(q^{10})$	\(q-2.39095 q^{2} +3.71663 q^{4} +2.34211 q^{5} -0.680854 q^{7} -4.10437 q^{8} -5.59986 q^{10} -1.00000 q^{11} -4.08597 q^{13} +1.62789 q^{14} +2.38007 q^{16} +4.92668 q^{17} -2.37841 q^{19} +8.70475 q^{20} +2.39095 q^{22} -8.55411 q^{23} +0.485471 q^{25} +9.76934 q^{26} -2.53048 q^{28} +4.22552 q^{29} +2.09530 q^{31} +2.51811 q^{32} -11.7794 q^{34} -1.59463 q^{35} -9.47790 q^{37} +5.68666 q^{38} -9.61288 q^{40} +9.28832 q^{41} +4.84525 q^{43} -3.71663 q^{44} +20.4524 q^{46} +6.86282 q^{47} -6.53644 q^{49} -1.16074 q^{50} -15.1860 q^{52} +11.6635 q^{53} -2.34211 q^{55} +2.79447 q^{56} -10.1030 q^{58} -2.43364 q^{59} +1.00000 q^{61} -5.00974 q^{62} -10.7808 q^{64} -9.56978 q^{65} +13.7392 q^{67} +18.3106 q^{68} +3.81268 q^{70} -11.1807 q^{71} -0.142993 q^{73} +22.6612 q^{74} -8.83968 q^{76} +0.680854 q^{77} +6.05595 q^{79} +5.57439 q^{80} -22.2079 q^{82} +8.41175 q^{83} +11.5388 q^{85} -11.5847 q^{86} +4.10437 q^{88} -13.1648 q^{89} +2.78195 q^{91} -31.7924 q^{92} -16.4086 q^{94} -5.57050 q^{95} -2.72647 q^{97} +15.6283 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8}+O(q^{10}) $ 11 * q + 2 * q^2 + 10 * q^4 + q^5 - 11 * q^7 + 3 * q^8	$ 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8} - 8 q^{10} - 11 q^{11} - 13 q^{13} - 5 q^{14} + 4 q^{16} + 13 q^{17} - 12 q^{19} + 7 q^{20} - 2 q^{22} + 3 q^{23} + 12 q^{25} - 12 q^{26} - 13 q^{28} - 2 q^{29} + q^{31} + 23 q^{32} - 14 q^{34} + 4 q^{35} - 14 q^{37} + 8 q^{38} - 34 q^{40} - 3 q^{41} - 21 q^{43} - 10 q^{44} - 12 q^{46} + 16 q^{47} - 18 q^{49} + 13 q^{50} - 33 q^{52} - q^{55} - 16 q^{56} - 17 q^{58} - 3 q^{59} + 11 q^{61} + 21 q^{62} - 7 q^{64} + q^{65} - 24 q^{67} - 2 q^{68} + 4 q^{70} - 7 q^{71} - 42 q^{73} + 16 q^{74} - 13 q^{76} + 11 q^{77} - 11 q^{79} - 42 q^{80} - 38 q^{82} + 34 q^{83} - 14 q^{85} - 42 q^{86} - 3 q^{88} - 29 q^{89} + 9 q^{91} - 42 q^{92} - 33 q^{94} + 31 q^{95} - 45 q^{97} + 33 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 10 * q^4 + q^5 - 11 * q^7 + 3 * q^8 - 8 * q^10 - 11 * q^11 - 13 * q^13 - 5 * q^14 + 4 * q^16 + 13 * q^17 - 12 * q^19 + 7 * q^20 - 2 * q^22 + 3 * q^23 + 12 * q^25 - 12 * q^26 - 13 * q^28 - 2 * q^29 + q^31 + 23 * q^32 - 14 * q^34 + 4 * q^35 - 14 * q^37 + 8 * q^38 - 34 * q^40 - 3 * q^41 - 21 * q^43 - 10 * q^44 - 12 * q^46 + 16 * q^47 - 18 * q^49 + 13 * q^50 - 33 * q^52 - q^55 - 16 * q^56 - 17 * q^58 - 3 * q^59 + 11 * q^61 + 21 * q^62 - 7 * q^64 + q^65 - 24 * q^67 - 2 * q^68 + 4 * q^70 - 7 * q^71 - 42 * q^73 + 16 * q^74 - 13 * q^76 + 11 * q^77 - 11 * q^79 - 42 * q^80 - 38 * q^82 + 34 * q^83 - 14 * q^85 - 42 * q^86 - 3 * q^88 - 29 * q^89 + 9 * q^91 - 42 * q^92 - 33 * q^94 + 31 * q^95 - 45 * q^97 + 33 * 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 $

$2$	−2.39095	−1.69066	−0.845328	−	0.534248i	$-0.820595\pi$
$2$	−2.39095	−1.69066	−0.845328	+	0.534248i	$0.820595\pi$
$3$	0	0
$3$	0	0
$4$	3.71663	1.85831
$5$	2.34211	1.04742	0.523711	−	0.851896i	$-0.324547\pi$
$5$	2.34211	1.04742	0.523711	+	0.851896i	$0.324547\pi$
$6$	0	0
$7$	−0.680854	−0.257338	−0.128669	−	0.991688i	$-0.541071\pi$
$7$	−0.680854	−0.257338	−0.128669	+	0.991688i	$0.541071\pi$
$8$	−4.10437	−1.45111
$9$	0	0
$10$	−5.59986	−1.77083
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.08597	−1.13324	−0.566622	−	0.823978i	$-0.691750\pi$
$13$	−4.08597	−1.13324	−0.566622	+	0.823978i	$0.691750\pi$
$14$	1.62789	0.435071
$15$	0	0
$16$	2.38007	0.595018
$17$	4.92668	1.19489	0.597447	−	0.801908i	$-0.296182\pi$
$17$	4.92668	1.19489	0.597447	+	0.801908i	$0.296182\pi$
$18$	0	0
$19$	−2.37841	−0.545646	−0.272823	−	0.962064i	$-0.587957\pi$
$19$	−2.37841	−0.545646	−0.272823	+	0.962064i	$0.587957\pi$
$20$	8.70475	1.94644
$21$	0	0
$22$	2.39095	0.509752
$23$	−8.55411	−1.78365	−0.891827	−	0.452376i	$-0.850576\pi$
$23$	−8.55411	−1.78365	−0.891827	+	0.452376i	$0.850576\pi$
$24$	0	0
$25$	0.485471	0.0970942
$26$	9.76934	1.91593
$27$	0	0
$28$	−2.53048	−0.478216
$29$	4.22552	0.784659	0.392330	−	0.919825i	$-0.371669\pi$
$29$	4.22552	0.784659	0.392330	+	0.919825i	$0.371669\pi$
$30$	0	0
$31$	2.09530	0.376326	0.188163	−	0.982138i	$-0.439747\pi$
$31$	2.09530	0.376326	0.188163	+	0.982138i	$0.439747\pi$
$32$	2.51811	0.445143
$33$	0	0
$34$	−11.7794	−2.02016
$35$	−1.59463	−0.269542
$36$	0	0
$37$	−9.47790	−1.55816	−0.779078	−	0.626927i	$-0.784313\pi$
$37$	−9.47790	−1.55816	−0.779078	+	0.626927i	$0.784313\pi$
$38$	5.68666	0.922498
$39$	0	0
$40$	−9.61288	−1.51993
$41$	9.28832	1.45059	0.725296	−	0.688437i	$-0.241703\pi$
$41$	9.28832	1.45059	0.725296	+	0.688437i	$0.241703\pi$
$42$	0	0
$43$	4.84525	0.738894	0.369447	−	0.929252i	$-0.379547\pi$
$43$	4.84525	0.738894	0.369447	+	0.929252i	$0.379547\pi$
$44$	−3.71663	−0.560303
$45$	0	0
$46$	20.4524	3.01554
$47$	6.86282	1.00105	0.500523	−	0.865723i	$-0.333141\pi$
$47$	6.86282	1.00105	0.500523	+	0.865723i	$0.333141\pi$
$48$	0	0
$49$	−6.53644	−0.933777
$50$	−1.16074	−0.164153
$51$	0	0
$52$	−15.1860	−2.10592
$53$	11.6635	1.60211	0.801054	−	0.598592i	$-0.204273\pi$
$53$	11.6635	1.60211	0.801054	+	0.598592i	$0.204273\pi$
$54$	0	0
$55$	−2.34211	−0.315810
$56$	2.79447	0.373427
$57$	0	0
$58$	−10.1030	−1.32659
$59$	−2.43364	−0.316833	−0.158417	−	0.987372i	$-0.550639\pi$
$59$	−2.43364	−0.316833	−0.158417	+	0.987372i	$0.550639\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−5.00974	−0.636238
$63$	0	0
$64$	−10.7808	−1.34760
$65$	−9.56978	−1.18699
$66$	0	0
$67$	13.7392	1.67851	0.839255	−	0.543738i	$-0.182991\pi$
$67$	13.7392	1.67851	0.839255	+	0.543738i	$0.182991\pi$
$68$	18.3106	2.22049
$69$	0	0
$70$	3.81268	0.455703
$71$	−11.1807	−1.32691	−0.663454	−	0.748217i	$-0.730910\pi$
$71$	−11.1807	−1.32691	−0.663454	+	0.748217i	$0.730910\pi$
$72$	0	0
$73$	−0.142993	−0.0167360	−0.00836801	−	0.999965i	$-0.502664\pi$
$73$	−0.142993	−0.0167360	−0.00836801	+	0.999965i	$0.502664\pi$
$74$	22.6612	2.63431
$75$	0	0
$76$	−8.83968	−1.01398
$77$	0.680854	0.0775905
$78$	0	0
$79$	6.05595	0.681348	0.340674	−	0.940182i	$-0.389345\pi$
$79$	6.05595	0.681348	0.340674	+	0.940182i	$0.389345\pi$
$80$	5.57439	0.623235
$81$	0	0
$82$	−22.2079	−2.45245
$83$	8.41175	0.923310	0.461655	−	0.887060i	$-0.347256\pi$
$83$	8.41175	0.923310	0.461655	+	0.887060i	$0.347256\pi$
$84$	0	0
$85$	11.5388	1.25156
$86$	−11.5847	−1.24921
$87$	0	0
$88$	4.10437	0.437527
$89$	−13.1648	−1.39546	−0.697731	−	0.716360i	$-0.745807\pi$
$89$	−13.1648	−1.39546	−0.697731	+	0.716360i	$0.745807\pi$
$90$	0	0
$91$	2.78195	0.291627
$92$	−31.7924	−3.31459
$93$	0	0
$94$	−16.4086	−1.69242
$95$	−5.57050	−0.571522
$96$	0	0
$97$	−2.72647	−0.276832	−0.138416	−	0.990374i	$-0.544201\pi$
$97$	−2.72647	−0.276832	−0.138416	+	0.990374i	$0.544201\pi$
$98$	15.6283	1.57869
$99$	0	0
$100$	1.80431	0.180431
$101$	−11.0228	−1.09681	−0.548403	−	0.836214i	$-0.684764\pi$
$101$	−11.0228	−1.09681	−0.548403	+	0.836214i	$0.684764\pi$
$102$	0	0
$103$	15.3232	1.50984	0.754919	−	0.655818i	$-0.227676\pi$
$103$	15.3232	1.50984	0.754919	+	0.655818i	$0.227676\pi$
$104$	16.7703	1.64447
$105$	0	0
$106$	−27.8869	−2.70861
$107$	18.3516	1.77412	0.887059	−	0.461655i	$-0.152744\pi$
$107$	18.3516	1.77412	0.887059	+	0.461655i	$0.152744\pi$
$108$	0	0
$109$	−2.48097	−0.237634	−0.118817	−	0.992916i	$-0.537910\pi$
$109$	−2.48097	−0.237634	−0.118817	+	0.992916i	$0.537910\pi$
$110$	5.59986	0.533925
$111$	0	0
$112$	−1.62048	−0.153121
$113$	−12.1358	−1.14164	−0.570821	−	0.821074i	$-0.693375\pi$
$113$	−12.1358	−1.14164	−0.570821	+	0.821074i	$0.693375\pi$
$114$	0	0
$115$	−20.0346	−1.86824
$116$	15.7047	1.45814
$117$	0	0
$118$	5.81871	0.535656
$119$	−3.35435	−0.307492
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.39095	−0.216466
$123$	0	0
$124$	7.78744	0.699333
$125$	−10.5735	−0.945724
$126$	0	0
$127$	−17.2718	−1.53263	−0.766314	−	0.642466i	$-0.777911\pi$
$127$	−17.2718	−1.53263	−0.766314	+	0.642466i	$0.777911\pi$
$128$	20.7401	1.83319
$129$	0	0
$130$	22.8809	2.00678
$131$	3.80200	0.332183	0.166091	−	0.986110i	$-0.446885\pi$
$131$	3.80200	0.332183	0.166091	+	0.986110i	$0.446885\pi$
$132$	0	0
$133$	1.61935	0.140416
$134$	−32.8497	−2.83778
$135$	0	0
$136$	−20.2209	−1.73393
$137$	−17.3045	−1.47842	−0.739210	−	0.673476i	$-0.764801\pi$
$137$	−17.3045	−1.47842	−0.739210	+	0.673476i	$0.764801\pi$
$138$	0	0
$139$	−16.1217	−1.36742	−0.683710	−	0.729753i	$-0.739635\pi$
$139$	−16.1217	−1.36742	−0.683710	+	0.729753i	$0.739635\pi$
$140$	−5.92666	−0.500894
$141$	0	0
$142$	26.7325	2.24334
$143$	4.08597	0.341686
$144$	0	0
$145$	9.89662	0.821870
$146$	0.341888	0.0282949
$147$	0	0
$148$	−35.2258	−2.89554
$149$	−13.4593	−1.10263	−0.551314	−	0.834298i	$-0.685873\pi$
$149$	−13.4593	−1.10263	−0.551314	+	0.834298i	$0.685873\pi$
$150$	0	0
$151$	−15.3016	−1.24523	−0.622614	−	0.782529i	$-0.713929\pi$
$151$	−15.3016	−1.24523	−0.622614	+	0.782529i	$0.713929\pi$
$152$	9.76189	0.791794
$153$	0	0
$154$	−1.62789	−0.131179
$155$	4.90741	0.394173
$156$	0	0
$157$	−18.1457	−1.44818	−0.724092	−	0.689703i	$-0.757741\pi$
$157$	−18.1457	−1.44818	−0.724092	+	0.689703i	$0.757741\pi$
$158$	−14.4795	−1.15192
$159$	0	0
$160$	5.89769	0.466253
$161$	5.82409	0.459003
$162$	0	0
$163$	−14.4638	−1.13289	−0.566447	−	0.824098i	$-0.691682\pi$
$163$	−14.4638	−1.13289	−0.566447	+	0.824098i	$0.691682\pi$
$164$	34.5212	2.69566
$165$	0	0
$166$	−20.1121	−1.56100
$167$	2.90037	0.224437	0.112219	−	0.993684i	$-0.464204\pi$
$167$	2.90037	0.224437	0.112219	+	0.993684i	$0.464204\pi$
$168$	0	0
$169$	3.69515	0.284243
$170$	−27.5887	−2.11596
$171$	0	0
$172$	18.0080	1.37310
$173$	−2.28013	−0.173355	−0.0866776	−	0.996236i	$-0.527625\pi$
$173$	−2.28013	−0.173355	−0.0866776	+	0.996236i	$0.527625\pi$
$174$	0	0
$175$	−0.330535	−0.0249861
$176$	−2.38007	−0.179405
$177$	0	0
$178$	31.4762	2.35924
$179$	−18.0119	−1.34627	−0.673137	−	0.739518i	$-0.735054\pi$
$179$	−18.0119	−1.34627	−0.673137	+	0.739518i	$0.735054\pi$
$180$	0	0
$181$	−15.6599	−1.16399	−0.581995	−	0.813192i	$-0.697728\pi$
$181$	−15.6599	−1.16399	−0.581995	+	0.813192i	$0.697728\pi$
$182$	−6.65149	−0.493041
$183$	0	0
$184$	35.1092	2.58829
$185$	−22.1983	−1.63205
$186$	0	0
$187$	−4.92668	−0.360274
$188$	25.5066	1.86026
$189$	0	0
$190$	13.3188	0.966246
$191$	14.9404	1.08105	0.540526	−	0.841327i	$-0.318225\pi$
$191$	14.9404	1.08105	0.540526	+	0.841327i	$0.318225\pi$
$192$	0	0
$193$	4.43534	0.319263	0.159632	−	0.987177i	$-0.448969\pi$
$193$	4.43534	0.319263	0.159632	+	0.987177i	$0.448969\pi$
$194$	6.51886	0.468027
$195$	0	0
$196$	−24.2935	−1.73525
$197$	−20.7927	−1.48142	−0.740709	−	0.671826i	$-0.765510\pi$
$197$	−20.7927	−1.48142	−0.740709	+	0.671826i	$0.765510\pi$
$198$	0	0
$199$	23.3453	1.65490	0.827450	−	0.561539i	$-0.189790\pi$
$199$	23.3453	1.65490	0.827450	+	0.561539i	$0.189790\pi$
$200$	−1.99255	−0.140895
$201$	0	0
$202$	26.3548	1.85432
$203$	−2.87696	−0.201923
$204$	0	0
$205$	21.7542	1.51938
$206$	−36.6369	−2.55261
$207$	0	0
$208$	−9.72491	−0.674301
$209$	2.37841	0.164518
$210$	0	0
$211$	−0.664194	−0.0457250	−0.0228625	−	0.999739i	$-0.507278\pi$
$211$	−0.664194	−0.0457250	−0.0228625	+	0.999739i	$0.507278\pi$
$212$	43.3490	2.97722
$213$	0	0
$214$	−43.8778	−2.99942
$215$	11.3481	0.773934
$216$	0	0
$217$	−1.42659	−0.0968433
$218$	5.93187	0.401757
$219$	0	0
$220$	−8.70475	−0.586874
$221$	−20.1303	−1.35411
$222$	0	0
$223$	16.0402	1.07413	0.537064	−	0.843541i	$-0.319533\pi$
$223$	16.0402	1.07413	0.537064	+	0.843541i	$0.319533\pi$
$224$	−1.71446	−0.114552
$225$	0	0
$226$	29.0161	1.93012
$227$	−10.3451	−0.686626	−0.343313	−	0.939221i	$-0.611549\pi$
$227$	−10.3451	−0.686626	−0.343313	+	0.939221i	$0.611549\pi$
$228$	0	0
$229$	−12.8289	−0.847755	−0.423878	−	0.905719i	$-0.639331\pi$
$229$	−12.8289	−0.847755	−0.423878	+	0.905719i	$0.639331\pi$
$230$	47.9018	3.15855
$231$	0	0
$232$	−17.3431	−1.13863
$233$	22.8873	1.49939	0.749697	−	0.661781i	$-0.230199\pi$
$233$	22.8873	1.49939	0.749697	+	0.661781i	$0.230199\pi$
$234$	0	0
$235$	16.0735	1.04852
$236$	−9.04495	−0.588776
$237$	0	0
$238$	8.02007	0.519864
$239$	−6.64045	−0.429535	−0.214768	−	0.976665i	$-0.568899\pi$
$239$	−6.64045	−0.429535	−0.214768	+	0.976665i	$0.568899\pi$
$240$	0	0
$241$	−30.6158	−1.97214	−0.986068	−	0.166340i	$-0.946805\pi$
$241$	−30.6158	−1.97214	−0.986068	+	0.166340i	$0.946805\pi$
$242$	−2.39095	−0.153696
$243$	0	0
$244$	3.71663	0.237933
$245$	−15.3090	−0.978059
$246$	0	0
$247$	9.71813	0.618350
$248$	−8.59987	−0.546092
$249$	0	0
$250$	25.2807	1.59889
$251$	−15.8806	−1.00237	−0.501186	−	0.865340i	$-0.667103\pi$
$251$	−15.8806	−1.00237	−0.501186	+	0.865340i	$0.667103\pi$
$252$	0	0
$253$	8.55411	0.537792
$254$	41.2961	2.59115
$255$	0	0
$256$	−28.0269	−1.75168
$257$	−27.9459	−1.74322	−0.871609	−	0.490202i	$-0.836923\pi$
$257$	−27.9459	−1.74322	−0.871609	+	0.490202i	$0.836923\pi$
$258$	0	0
$259$	6.45306	0.400974
$260$	−35.5673	−2.20579
$261$	0	0
$262$	−9.09039	−0.561606
$263$	4.97640	0.306858	0.153429	−	0.988160i	$-0.450968\pi$
$263$	4.97640	0.306858	0.153429	+	0.988160i	$0.450968\pi$
$264$	0	0
$265$	27.3172	1.67808
$266$	−3.87178	−0.237394
$267$	0	0
$268$	51.0635	3.11920
$269$	−12.5529	−0.765362	−0.382681	−	0.923881i	$-0.624999\pi$
$269$	−12.5529	−0.765362	−0.382681	+	0.923881i	$0.624999\pi$
$270$	0	0
$271$	15.6931	0.953291	0.476645	−	0.879096i	$-0.341853\pi$
$271$	15.6931	0.953291	0.476645	+	0.879096i	$0.341853\pi$
$272$	11.7258	0.710984
$273$	0	0
$274$	41.3740	2.49950
$275$	−0.485471	−0.0292750
$276$	0	0
$277$	28.3012	1.70045	0.850227	−	0.526416i	$-0.176464\pi$
$277$	28.3012	1.70045	0.850227	+	0.526416i	$0.176464\pi$
$278$	38.5460	2.31184
$279$	0	0
$280$	6.54496	0.391136
$281$	0.939387	0.0560392	0.0280196	−	0.999607i	$-0.491080\pi$
$281$	0.939387	0.0560392	0.0280196	+	0.999607i	$0.491080\pi$
$282$	0	0
$283$	−12.0742	−0.717737	−0.358869	−	0.933388i	$-0.616837\pi$
$283$	−12.0742	−0.717737	−0.358869	+	0.933388i	$0.616837\pi$
$284$	−41.5546	−2.46581
$285$	0	0
$286$	−9.76934	−0.577673
$287$	−6.32399	−0.373293
$288$	0	0
$289$	7.27215	0.427774
$290$	−23.6623	−1.38950
$291$	0	0
$292$	−0.531451	−0.0311008
$293$	−6.75821	−0.394819	−0.197410	−	0.980321i	$-0.563253\pi$
$293$	−6.75821	−0.394819	−0.197410	+	0.980321i	$0.563253\pi$
$294$	0	0
$295$	−5.69985	−0.331858
$296$	38.9008	2.26106
$297$	0	0
$298$	32.1805	1.86416
$299$	34.9518	2.02132
$300$	0	0
$301$	−3.29891	−0.190146
$302$	36.5854	2.10525
$303$	0	0
$304$	−5.66080	−0.324669
$305$	2.34211	0.134109
$306$	0	0
$307$	21.5965	1.23258	0.616289	−	0.787520i	$-0.288635\pi$
$307$	21.5965	1.23258	0.616289	+	0.787520i	$0.288635\pi$
$308$	2.53048	0.144187
$309$	0	0
$310$	−11.7334	−0.666410
$311$	17.2548	0.978428	0.489214	−	0.872164i	$-0.337284\pi$
$311$	17.2548	0.978428	0.489214	+	0.872164i	$0.337284\pi$
$312$	0	0
$313$	−0.0253638	−0.00143365	−0.000716823	−	1.00000i	$-0.500228\pi$
$313$	−0.0253638	−0.00143365	−0.000716823		1.00000i	$0.500228\pi$
$314$	43.3854	2.44838
$315$	0	0
$316$	22.5077	1.26616
$317$	32.5423	1.82776	0.913879	−	0.405987i	$-0.133072\pi$
$317$	32.5423	1.82776	0.913879	+	0.405987i	$0.133072\pi$
$318$	0	0
$319$	−4.22552	−0.236584
$320$	−25.2498	−1.41151
$321$	0	0
$322$	−13.9251	−0.776016
$323$	−11.7177	−0.651989
$324$	0	0
$325$	−1.98362	−0.110031
$326$	34.5822	1.91533
$327$	0	0
$328$	−38.1227	−2.10497
$329$	−4.67258	−0.257608
$330$	0	0
$331$	−0.829097	−0.0455713	−0.0227856	−	0.999740i	$-0.507254\pi$
$331$	−0.829097	−0.0455713	−0.0227856	+	0.999740i	$0.507254\pi$
$332$	31.2634	1.71580
$333$	0	0
$334$	−6.93462	−0.379446
$335$	32.1787	1.75811
$336$	0	0
$337$	−34.4197	−1.87496	−0.937480	−	0.348039i	$-0.886848\pi$
$337$	−34.4197	−1.87496	−0.937480	+	0.348039i	$0.886848\pi$
$338$	−8.83492	−0.480556
$339$	0	0
$340$	42.8855	2.32579
$341$	−2.09530	−0.113467
$342$	0	0
$343$	9.21633	0.497635
$344$	−19.8867	−1.07222
$345$	0	0
$346$	5.45167	0.293084
$347$	0.553816	0.0297304	0.0148652	−	0.999890i	$-0.495268\pi$
$347$	0.553816	0.0297304	0.0148652	+	0.999890i	$0.495268\pi$
$348$	0	0
$349$	8.33868	0.446359	0.223180	−	0.974777i	$-0.428356\pi$
$349$	8.33868	0.446359	0.223180	+	0.974777i	$0.428356\pi$
$350$	0.790291	0.0422428
$351$	0	0
$352$	−2.51811	−0.134216
$353$	20.1970	1.07498	0.537490	−	0.843270i	$-0.319372\pi$
$353$	20.1970	1.07498	0.537490	+	0.843270i	$0.319372\pi$
$354$	0	0
$355$	−26.1865	−1.38983
$356$	−48.9285	−2.59321
$357$	0	0
$358$	43.0656	2.27609
$359$	−4.46472	−0.235639	−0.117819	−	0.993035i	$-0.537590\pi$
$359$	−4.46472	−0.235639	−0.117819	+	0.993035i	$0.537590\pi$
$360$	0	0
$361$	−13.3431	−0.702271
$362$	37.4420	1.96791
$363$	0	0
$364$	10.3395	0.541935
$365$	−0.334904	−0.0175297
$366$	0	0
$367$	−13.0961	−0.683612	−0.341806	−	0.939770i	$-0.611039\pi$
$367$	−13.0961	−0.683612	−0.341806	+	0.939770i	$0.611039\pi$
$368$	−20.3594	−1.06131
$369$	0	0
$370$	53.0749	2.75923
$371$	−7.94115	−0.412284
$372$	0	0
$373$	−29.5869	−1.53195	−0.765976	−	0.642869i	$-0.777744\pi$
$373$	−29.5869	−1.53195	−0.765976	+	0.642869i	$0.777744\pi$
$374$	11.7794	0.609100
$375$	0	0
$376$	−28.1676	−1.45263
$377$	−17.2653	−0.889210
$378$	0	0
$379$	−16.5752	−0.851412	−0.425706	−	0.904861i	$-0.639974\pi$
$379$	−16.5752	−0.851412	−0.425706	+	0.904861i	$0.639974\pi$
$380$	−20.7035	−1.06207
$381$	0	0
$382$	−35.7218	−1.82769
$383$	5.22813	0.267145	0.133573	−	0.991039i	$-0.457355\pi$
$383$	5.22813	0.267145	0.133573	+	0.991039i	$0.457355\pi$
$384$	0	0
$385$	1.59463	0.0812700
$386$	−10.6047	−0.539764
$387$	0	0
$388$	−10.1333	−0.514440
$389$	9.66000	0.489781	0.244891	−	0.969551i	$-0.421248\pi$
$389$	9.66000	0.489781	0.244891	+	0.969551i	$0.421248\pi$
$390$	0	0
$391$	−42.1433	−2.13128
$392$	26.8280	1.35502
$393$	0	0
$394$	49.7142	2.50457
$395$	14.1837	0.713659
$396$	0	0
$397$	−22.4867	−1.12857	−0.564287	−	0.825579i	$-0.690849\pi$
$397$	−22.4867	−1.12857	−0.564287	+	0.825579i	$0.690849\pi$
$398$	−55.8173	−2.79787
$399$	0	0
$400$	1.15546	0.0577728
$401$	−33.7633	−1.68606	−0.843030	−	0.537866i	$-0.819231\pi$
$401$	−33.7633	−1.68606	−0.843030	+	0.537866i	$0.819231\pi$
$402$	0	0
$403$	−8.56132	−0.426470
$404$	−40.9675	−2.03821
$405$	0	0
$406$	6.87866	0.341382
$407$	9.47790	0.469802
$408$	0	0
$409$	−16.1810	−0.800098	−0.400049	−	0.916494i	$-0.631007\pi$
$409$	−16.1810	−0.800098	−0.400049	+	0.916494i	$0.631007\pi$
$410$	−52.0133	−2.56875
$411$	0	0
$412$	56.9506	2.80575
$413$	1.65695	0.0815334
$414$	0	0
$415$	19.7012	0.967096
$416$	−10.2889	−0.504456
$417$	0	0
$418$	−5.68666	−0.278144
$419$	5.31843	0.259823	0.129911	−	0.991526i	$-0.458531\pi$
$419$	5.31843	0.259823	0.129911	+	0.991526i	$0.458531\pi$
$420$	0	0
$421$	4.69593	0.228866	0.114433	−	0.993431i	$-0.463495\pi$
$421$	4.69593	0.228866	0.114433	+	0.993431i	$0.463495\pi$
$422$	1.58805	0.0773052
$423$	0	0
$424$	−47.8714	−2.32484
$425$	2.39176	0.116017
$426$	0	0
$427$	−0.680854	−0.0329488
$428$	68.2062	3.29687
$429$	0	0
$430$	−27.1327	−1.30846
$431$	8.03716	0.387136	0.193568	−	0.981087i	$-0.437994\pi$
$431$	8.03716	0.387136	0.193568	+	0.981087i	$0.437994\pi$
$432$	0	0
$433$	−22.0509	−1.05970	−0.529850	−	0.848091i	$-0.677752\pi$
$433$	−22.0509	−1.05970	−0.529850	+	0.848091i	$0.677752\pi$
$434$	3.41090	0.163729
$435$	0	0
$436$	−9.22085	−0.441599
$437$	20.3452	0.973243
$438$	0	0
$439$	38.0798	1.81745	0.908725	−	0.417394i	$-0.137057\pi$
$439$	38.0798	1.81745	0.908725	+	0.417394i	$0.137057\pi$
$440$	9.61288	0.458276
$441$	0	0
$442$	48.1304	2.28933
$443$	−13.2692	−0.630439	−0.315220	−	0.949019i	$-0.602078\pi$
$443$	−13.2692	−0.630439	−0.315220	+	0.949019i	$0.602078\pi$
$444$	0	0
$445$	−30.8333	−1.46164
$446$	−38.3512	−1.81598
$447$	0	0
$448$	7.34016	0.346790
$449$	−5.71516	−0.269715	−0.134858	−	0.990865i	$-0.543058\pi$
$449$	−5.71516	−0.269715	−0.134858	+	0.990865i	$0.543058\pi$
$450$	0	0
$451$	−9.28832	−0.437370
$452$	−45.1044	−2.12153
$453$	0	0
$454$	24.7345	1.16085
$455$	6.51562	0.305457
$456$	0	0
$457$	−0.993366	−0.0464677	−0.0232339	−	0.999730i	$-0.507396\pi$
$457$	−0.993366	−0.0464677	−0.0232339	+	0.999730i	$0.507396\pi$
$458$	30.6731	1.43326
$459$	0	0
$460$	−74.4613	−3.47178
$461$	20.7491	0.966382	0.483191	−	0.875515i	$-0.339478\pi$
$461$	20.7491	0.966382	0.483191	+	0.875515i	$0.339478\pi$
$462$	0	0
$463$	−27.9103	−1.29710	−0.648550	−	0.761172i	$-0.724624\pi$
$463$	−27.9103	−1.29710	−0.648550	+	0.761172i	$0.724624\pi$
$464$	10.0570	0.466886
$465$	0	0
$466$	−54.7222	−2.53496
$467$	8.14215	0.376774	0.188387	−	0.982095i	$-0.439674\pi$
$467$	8.14215	0.376774	0.188387	+	0.982095i	$0.439674\pi$
$468$	0	0
$469$	−9.35438	−0.431945
$470$	−38.4308	−1.77268
$471$	0	0
$472$	9.98857	0.459761
$473$	−4.84525	−0.222785
$474$	0	0
$475$	−1.15465	−0.0529790
$476$	−12.4669	−0.571418
$477$	0	0
$478$	15.8770	0.726196
$479$	−25.8787	−1.18243	−0.591213	−	0.806515i	$-0.701351\pi$
$479$	−25.8787	−1.18243	−0.591213	+	0.806515i	$0.701351\pi$
$480$	0	0
$481$	38.7264	1.76577
$482$	73.2008	3.33420
$483$	0	0
$484$	3.71663	0.168938
$485$	−6.38570	−0.289960
$486$	0	0
$487$	10.1255	0.458830	0.229415	−	0.973329i	$-0.426319\pi$
$487$	10.1255	0.458830	0.229415	+	0.973329i	$0.426319\pi$
$488$	−4.10437	−0.185796
$489$	0	0
$490$	36.6031	1.65356
$491$	2.11058	0.0952493	0.0476246	−	0.998865i	$-0.484835\pi$
$491$	2.11058	0.0952493	0.0476246	+	0.998865i	$0.484835\pi$
$492$	0	0
$493$	20.8178	0.937585
$494$	−23.2355	−1.04542
$495$	0	0
$496$	4.98696	0.223921
$497$	7.61244	0.341464
$498$	0	0
$499$	37.7622	1.69047	0.845233	−	0.534398i	$-0.179461\pi$
$499$	37.7622	1.69047	0.845233	+	0.534398i	$0.179461\pi$
$500$	−39.2978	−1.75745
$501$	0	0
$502$	37.9696	1.69467
$503$	−19.3545	−0.862975	−0.431487	−	0.902119i	$-0.642011\pi$
$503$	−19.3545	−0.862975	−0.431487	+	0.902119i	$0.642011\pi$
$504$	0	0
$505$	−25.8165	−1.14882
$506$	−20.4524	−0.909221
$507$	0	0
$508$	−64.1930	−2.84810
$509$	−3.90614	−0.173137	−0.0865684	−	0.996246i	$-0.527590\pi$
$509$	−3.90614	−0.173137	−0.0865684	+	0.996246i	$0.527590\pi$
$510$	0	0
$511$	0.0973571	0.00430682
$512$	25.5307	1.12831
$513$	0	0
$514$	66.8172	2.94718
$515$	35.8885	1.58144
$516$	0	0
$517$	−6.86282	−0.301827
$518$	−15.4289	−0.677908
$519$	0	0
$520$	39.2779	1.72245
$521$	15.0825	0.660775	0.330388	−	0.943845i	$-0.392821\pi$
$521$	15.0825	0.660775	0.330388	+	0.943845i	$0.392821\pi$
$522$	0	0
$523$	−23.4385	−1.02489	−0.512447	−	0.858719i	$-0.671261\pi$
$523$	−23.4385	−1.02489	−0.512447	+	0.858719i	$0.671261\pi$
$524$	14.1306	0.617300
$525$	0	0
$526$	−11.8983	−0.518791
$527$	10.3229	0.449670
$528$	0	0
$529$	50.1727	2.18142
$530$	−65.3141	−2.83706
$531$	0	0
$532$	6.01853	0.260936
$533$	−37.9518	−1.64387
$534$	0	0
$535$	42.9815	1.85825
$536$	−56.3907	−2.43571
$537$	0	0
$538$	30.0132	1.29396
$539$	6.53644	0.281544
$540$	0	0
$541$	−13.9111	−0.598085	−0.299043	−	0.954240i	$-0.596667\pi$
$541$	−13.9111	−0.598085	−0.299043	+	0.954240i	$0.596667\pi$
$542$	−37.5215	−1.61169
$543$	0	0
$544$	12.4059	0.531899
$545$	−5.81071	−0.248903
$546$	0	0
$547$	−20.3809	−0.871423	−0.435712	−	0.900086i	$-0.643503\pi$
$547$	−20.3809	−0.871423	−0.435712	+	0.900086i	$0.643503\pi$
$548$	−64.3142	−2.74737
$549$	0	0
$550$	1.16074	0.0494939
$551$	−10.0500	−0.428146
$552$	0	0
$553$	−4.12322	−0.175337
$554$	−67.6667	−2.87488
$555$	0	0
$556$	−59.9182	−2.54110
$557$	−2.73158	−0.115741	−0.0578704	−	0.998324i	$-0.518431\pi$
$557$	−2.73158	−0.115741	−0.0578704	+	0.998324i	$0.518431\pi$
$558$	0	0
$559$	−19.7976	−0.837347
$560$	−3.79534	−0.160382
$561$	0	0
$562$	−2.24603	−0.0947429
$563$	−14.3092	−0.603062	−0.301531	−	0.953456i	$-0.597498\pi$
$563$	−14.3092	−0.603062	−0.301531	+	0.953456i	$0.597498\pi$
$564$	0	0
$565$	−28.4234	−1.19578
$566$	28.8688	1.21345
$567$	0	0
$568$	45.8898	1.92549
$569$	−27.4761	−1.15186	−0.575929	−	0.817500i	$-0.695359\pi$
$569$	−27.4761	−1.15186	−0.575929	+	0.817500i	$0.695359\pi$
$570$	0	0
$571$	−36.8322	−1.54138	−0.770689	−	0.637211i	$-0.780088\pi$
$571$	−36.8322	−1.54138	−0.770689	+	0.637211i	$0.780088\pi$
$572$	15.1860	0.634960
$573$	0	0
$574$	15.1203	0.631110
$575$	−4.15277	−0.173182
$576$	0	0
$577$	22.3670	0.931152	0.465576	−	0.885008i	$-0.345847\pi$
$577$	22.3670	0.931152	0.465576	+	0.885008i	$0.345847\pi$
$578$	−17.3873	−0.723218
$579$	0	0
$580$	36.7821	1.52729
$581$	−5.72717	−0.237603
$582$	0	0
$583$	−11.6635	−0.483054
$584$	0.586895	0.0242859
$585$	0	0
$586$	16.1585	0.667503
$587$	22.4116	0.925024	0.462512	−	0.886613i	$-0.346948\pi$
$587$	22.4116	0.925024	0.462512	+	0.886613i	$0.346948\pi$
$588$	0	0
$589$	−4.98348	−0.205341
$590$	13.6281	0.561058
$591$	0	0
$592$	−22.5581	−0.927131
$593$	38.7469	1.59115	0.795573	−	0.605858i	$-0.207170\pi$
$593$	38.7469	1.59115	0.795573	+	0.605858i	$0.207170\pi$
$594$	0	0
$595$	−7.85624	−0.322075
$596$	−50.0232	−2.04903
$597$	0	0
$598$	−83.5680	−3.41735
$599$	8.32006	0.339948	0.169974	−	0.985449i	$-0.445632\pi$
$599$	8.32006	0.339948	0.169974	+	0.985449i	$0.445632\pi$
$600$	0	0
$601$	13.9908	0.570698	0.285349	−	0.958424i	$-0.407890\pi$
$601$	13.9908	0.570698	0.285349	+	0.958424i	$0.407890\pi$
$602$	7.88751	0.321471
$603$	0	0
$604$	−56.8704	−2.31403
$605$	2.34211	0.0952202
$606$	0	0
$607$	−41.0609	−1.66661	−0.833306	−	0.552812i	$-0.813555\pi$
$607$	−41.0609	−1.66661	−0.833306	+	0.552812i	$0.813555\pi$
$608$	−5.98911	−0.242890
$609$	0	0
$610$	−5.59986	−0.226732
$611$	−28.0413	−1.13443
$612$	0	0
$613$	−26.4751	−1.06932	−0.534659	−	0.845068i	$-0.679560\pi$
$613$	−26.4751	−1.06932	−0.534659	+	0.845068i	$0.679560\pi$
$614$	−51.6362	−2.08387
$615$	0	0
$616$	−2.79447	−0.112593
$617$	−26.7282	−1.07604	−0.538019	−	0.842933i	$-0.680827\pi$
$617$	−26.7282	−1.07604	−0.538019	+	0.842933i	$0.680827\pi$
$618$	0	0
$619$	−27.3417	−1.09895	−0.549477	−	0.835509i	$-0.685173\pi$
$619$	−27.3417	−1.09895	−0.549477	+	0.835509i	$0.685173\pi$
$620$	18.2390	0.732497
$621$	0	0
$622$	−41.2552	−1.65418
$623$	8.96327	0.359106
$624$	0	0
$625$	−27.1917	−1.08767
$626$	0.0606435	0.00242380
$627$	0	0
$628$	−67.4408	−2.69118
$629$	−46.6945	−1.86183
$630$	0	0
$631$	−5.54559	−0.220766	−0.110383	−	0.993889i	$-0.535208\pi$
$631$	−5.54559	−0.220766	−0.110383	+	0.993889i	$0.535208\pi$
$632$	−24.8559	−0.988713
$633$	0	0
$634$	−77.8069	−3.09011
$635$	−40.4525	−1.60531
$636$	0	0
$637$	26.7077	1.05820
$638$	10.1030	0.399981
$639$	0	0
$640$	48.5756	1.92012
$641$	19.7959	0.781893	0.390946	−	0.920413i	$-0.372148\pi$
$641$	19.7959	0.781893	0.390946	+	0.920413i	$0.372148\pi$
$642$	0	0
$643$	−30.7766	−1.21371	−0.606855	−	0.794813i	$-0.707569\pi$
$643$	−30.7766	−1.21371	−0.606855	+	0.794813i	$0.707569\pi$
$644$	21.6460	0.852972
$645$	0	0
$646$	28.0164	1.10229
$647$	8.81041	0.346373	0.173186	−	0.984889i	$-0.444594\pi$
$647$	8.81041	0.346373	0.173186	+	0.984889i	$0.444594\pi$
$648$	0	0
$649$	2.43364	0.0955288
$650$	4.74273	0.186025
$651$	0	0
$652$	−53.7567	−2.10527
$653$	46.4924	1.81939	0.909694	−	0.415279i	$-0.136316\pi$
$653$	46.4924	1.81939	0.909694	+	0.415279i	$0.136316\pi$
$654$	0	0
$655$	8.90471	0.347936
$656$	22.1069	0.863128
$657$	0	0
$658$	11.1719	0.435525
$659$	10.0896	0.393034	0.196517	−	0.980500i	$-0.437037\pi$
$659$	10.0896	0.393034	0.196517	+	0.980500i	$0.437037\pi$
$660$	0	0
$661$	−2.33419	−0.0907895	−0.0453948	−	0.998969i	$-0.514455\pi$
$661$	−2.33419	−0.0907895	−0.0453948	+	0.998969i	$0.514455\pi$
$662$	1.98233	0.0770453
$663$	0	0
$664$	−34.5249	−1.33983
$665$	3.79270	0.147074
$666$	0	0
$667$	−36.1455	−1.39956
$668$	10.7796	0.417075
$669$	0	0
$670$	−76.9375	−2.97236
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	12.9252	0.498229	0.249114	−	0.968474i	$-0.419861\pi$
$673$	12.9252	0.498229	0.249114	+	0.968474i	$0.419861\pi$
$674$	82.2957	3.16991
$675$	0	0
$676$	13.7335	0.528212
$677$	23.0322	0.885199	0.442600	−	0.896719i	$-0.354056\pi$
$677$	23.0322	0.885199	0.442600	+	0.896719i	$0.354056\pi$
$678$	0	0
$679$	1.85633	0.0712394
$680$	−47.3595	−1.81616
$681$	0	0
$682$	5.00974	0.191833
$683$	28.2744	1.08189	0.540944	−	0.841058i	$-0.318067\pi$
$683$	28.2744	1.08189	0.540944	+	0.841058i	$0.318067\pi$
$684$	0	0
$685$	−40.5289	−1.54853
$686$	−22.0358	−0.841330
$687$	0	0
$688$	11.5320	0.439655
$689$	−47.6568	−1.81558
$690$	0	0
$691$	−27.6303	−1.05111	−0.525553	−	0.850761i	$-0.676141\pi$
$691$	−27.6303	−1.05111	−0.525553	+	0.850761i	$0.676141\pi$
$692$	−8.47440	−0.322148
$693$	0	0
$694$	−1.32414	−0.0502638
$695$	−37.7587	−1.43227
$696$	0	0
$697$	45.7606	1.73330
$698$	−19.9374	−0.754640
$699$	0	0
$700$	−1.22847	−0.0464320
$701$	24.4630	0.923956	0.461978	−	0.886891i	$-0.347140\pi$
$701$	24.4630	0.923956	0.461978	+	0.886891i	$0.347140\pi$
$702$	0	0
$703$	22.5424	0.850201
$704$	10.7808	0.406317
$705$	0	0
$706$	−48.2901	−1.81742
$707$	7.50488	0.282250
$708$	0	0
$709$	−23.4913	−0.882236	−0.441118	−	0.897449i	$-0.645418\pi$
$709$	−23.4913	−0.882236	−0.441118	+	0.897449i	$0.645418\pi$
$710$	62.6105	2.34973
$711$	0	0
$712$	54.0330	2.02497
$713$	−17.9234	−0.671236
$714$	0	0
$715$	9.56978	0.357890
$716$	−66.9436	−2.50180
$717$	0	0
$718$	10.6749	0.398384
$719$	−19.6963	−0.734550	−0.367275	−	0.930112i	$-0.619709\pi$
$719$	−19.6963	−0.734550	−0.367275	+	0.930112i	$0.619709\pi$
$720$	0	0
$721$	−10.4328	−0.388539
$722$	31.9028	1.18730
$723$	0	0
$724$	−58.2020	−2.16306
$725$	2.05137	0.0761858
$726$	0	0
$727$	−21.6379	−0.802506	−0.401253	−	0.915967i	$-0.631425\pi$
$727$	−21.6379	−0.802506	−0.401253	+	0.915967i	$0.631425\pi$
$728$	−11.4181	−0.423184
$729$	0	0
$730$	0.800739	0.0296367
$731$	23.8710	0.882901
$732$	0	0
$733$	42.1478	1.55677	0.778383	−	0.627790i	$-0.216040\pi$
$733$	42.1478	1.55677	0.778383	+	0.627790i	$0.216040\pi$
$734$	31.3122	1.15575
$735$	0	0
$736$	−21.5402	−0.793982
$737$	−13.7392	−0.506090
$738$	0	0
$739$	19.5967	0.720876	0.360438	−	0.932783i	$-0.382627\pi$
$739$	19.5967	0.720876	0.360438	+	0.932783i	$0.382627\pi$
$740$	−82.5027	−3.03286
$741$	0	0
$742$	18.9869	0.697030
$743$	46.6104	1.70997	0.854985	−	0.518652i	$-0.173566\pi$
$743$	46.6104	1.70997	0.854985	+	0.518652i	$0.173566\pi$
$744$	0	0
$745$	−31.5231	−1.15492
$746$	70.7407	2.59000
$747$	0	0
$748$	−18.3106	−0.669503
$749$	−12.4948	−0.456549
$750$	0	0
$751$	−27.0475	−0.986976	−0.493488	−	0.869753i	$-0.664278\pi$
$751$	−27.0475	−0.986976	−0.493488	+	0.869753i	$0.664278\pi$
$752$	16.3340	0.595640
$753$	0	0
$754$	41.2805	1.50335
$755$	−35.8380	−1.30428
$756$	0	0
$757$	5.95554	0.216458	0.108229	−	0.994126i	$-0.465482\pi$
$757$	5.95554	0.216458	0.108229	+	0.994126i	$0.465482\pi$
$758$	39.6305	1.43944
$759$	0	0
$760$	22.8634	0.829343
$761$	−7.65293	−0.277419	−0.138709	−	0.990333i	$-0.544295\pi$
$761$	−7.65293	−0.277419	−0.138709	+	0.990333i	$0.544295\pi$
$762$	0	0
$763$	1.68918	0.0611524
$764$	55.5281	2.00894
$765$	0	0
$766$	−12.5002	−0.451650
$767$	9.94379	0.359050
$768$	0	0
$769$	−40.5022	−1.46055	−0.730273	−	0.683155i	$-0.760607\pi$
$769$	−40.5022	−1.46055	−0.730273	+	0.683155i	$0.760607\pi$
$770$	−3.81268	−0.137400
$771$	0	0
$772$	16.4845	0.593291
$773$	2.47981	0.0891924	0.0445962	−	0.999005i	$-0.485800\pi$
$773$	2.47981	0.0891924	0.0445962	+	0.999005i	$0.485800\pi$
$774$	0	0
$775$	1.01721	0.0365391
$776$	11.1905	0.401714
$777$	0	0
$778$	−23.0965	−0.828051
$779$	−22.0915	−0.791509
$780$	0	0
$781$	11.1807	0.400078
$782$	100.762	3.60326
$783$	0	0
$784$	−15.5572	−0.555614
$785$	−42.4992	−1.51686
$786$	0	0
$787$	−45.5018	−1.62196	−0.810982	−	0.585072i	$-0.801066\pi$
$787$	−45.5018	−1.62196	−0.810982	+	0.585072i	$0.801066\pi$
$788$	−77.2787	−2.75294
$789$	0	0
$790$	−33.9125	−1.20655
$791$	8.26272	0.293789
$792$	0	0
$793$	−4.08597	−0.145097
$794$	53.7645	1.90803
$795$	0	0
$796$	86.7656	3.07533
$797$	−44.2212	−1.56639	−0.783197	−	0.621774i	$-0.786412\pi$
$797$	−44.2212	−1.56639	−0.783197	+	0.621774i	$0.786412\pi$
$798$	0	0
$799$	33.8109	1.19614
$800$	1.22247	0.0432208
$801$	0	0
$802$	80.7263	2.85055
$803$	0.142993	0.00504610
$804$	0	0
$805$	13.6407	0.480770
$806$	20.4697	0.721013
$807$	0	0
$808$	45.2415	1.59159
$809$	−51.7924	−1.82093	−0.910463	−	0.413592i	$-0.864274\pi$
$809$	−51.7924	−1.82093	−0.910463	+	0.413592i	$0.864274\pi$
$810$	0	0
$811$	43.8409	1.53946	0.769732	−	0.638367i	$-0.220390\pi$
$811$	43.8409	1.53946	0.769732	+	0.638367i	$0.220390\pi$
$812$	−10.6926	−0.375236
$813$	0	0
$814$	−22.6612	−0.794273
$815$	−33.8758	−1.18662
$816$	0	0
$817$	−11.5240	−0.403174
$818$	38.6879	1.35269
$819$	0	0
$820$	80.8525	2.82349
$821$	12.7392	0.444603	0.222301	−	0.974978i	$-0.428643\pi$
$821$	12.7392	0.444603	0.222301	+	0.974978i	$0.428643\pi$
$822$	0	0
$823$	−7.11632	−0.248059	−0.124030	−	0.992279i	$-0.539582\pi$
$823$	−7.11632	−0.248059	−0.124030	+	0.992279i	$0.539582\pi$
$824$	−62.8920	−2.19095
$825$	0	0
$826$	−3.96169	−0.137845
$827$	15.8721	0.551928	0.275964	−	0.961168i	$-0.411003\pi$
$827$	15.8721	0.551928	0.275964	+	0.961168i	$0.411003\pi$
$828$	0	0
$829$	8.79706	0.305535	0.152767	−	0.988262i	$-0.451182\pi$
$829$	8.79706	0.305535	0.152767	+	0.988262i	$0.451182\pi$
$830$	−47.1046	−1.63503
$831$	0	0
$832$	44.0501	1.52716
$833$	−32.2029	−1.11577
$834$	0	0
$835$	6.79297	0.235081
$836$	8.83968	0.305727
$837$	0	0
$838$	−12.7161	−0.439270
$839$	−24.0126	−0.829008	−0.414504	−	0.910047i	$-0.636045\pi$
$839$	−24.0126	−0.829008	−0.414504	+	0.910047i	$0.636045\pi$
$840$	0	0
$841$	−11.1450	−0.384310
$842$	−11.2277	−0.386933
$843$	0	0
$844$	−2.46856	−0.0849714
$845$	8.65445	0.297722
$846$	0	0
$847$	−0.680854	−0.0233944
$848$	27.7600	0.953284
$849$	0	0
$850$	−5.71857	−0.196145
$851$	81.0749	2.77921
$852$	0	0
$853$	−0.428602	−0.0146751	−0.00733753	−	0.999973i	$-0.502336\pi$
$853$	−0.428602	−0.0146751	−0.00733753	+	0.999973i	$0.502336\pi$
$854$	1.62789	0.0557051
$855$	0	0
$856$	−75.3219	−2.57445
$857$	10.7506	0.367234	0.183617	−	0.982998i	$-0.441219\pi$
$857$	10.7506	0.367234	0.183617	+	0.982998i	$0.441219\pi$
$858$	0	0
$859$	−21.0671	−0.718802	−0.359401	−	0.933183i	$-0.617019\pi$
$859$	−21.0671	−0.718802	−0.359401	+	0.933183i	$0.617019\pi$
$860$	42.1767	1.43821
$861$	0	0
$862$	−19.2164	−0.654514
$863$	−52.7185	−1.79456	−0.897279	−	0.441463i	$-0.854460\pi$
$863$	−52.7185	−1.79456	−0.897279	+	0.441463i	$0.854460\pi$
$864$	0	0
$865$	−5.34031	−0.181576
$866$	52.7226	1.79159
$867$	0	0
$868$	−5.30211	−0.179965
$869$	−6.05595	−0.205434
$870$	0	0
$871$	−56.1380	−1.90216
$872$	10.1828	0.344834
$873$	0	0
$874$	−48.6443	−1.64542
$875$	7.19902	0.243371
$876$	0	0
$877$	−8.11996	−0.274192	−0.137096	−	0.990558i	$-0.543777\pi$
$877$	−8.11996	−0.274192	−0.137096	+	0.990558i	$0.543777\pi$
$878$	−91.0469	−3.07268
$879$	0	0
$880$	−5.57439	−0.187913
$881$	11.5045	0.387597	0.193799	−	0.981041i	$-0.437919\pi$
$881$	11.5045	0.387597	0.193799	+	0.981041i	$0.437919\pi$
$882$	0	0
$883$	49.1706	1.65472	0.827361	−	0.561671i	$-0.189841\pi$
$883$	49.1706	1.65472	0.827361	+	0.561671i	$0.189841\pi$
$884$	−74.8167	−2.51636
$885$	0	0
$886$	31.7260	1.06585
$887$	−37.5827	−1.26190	−0.630952	−	0.775822i	$-0.717335\pi$
$887$	−37.5827	−1.26190	−0.630952	+	0.775822i	$0.717335\pi$
$888$	0	0
$889$	11.7596	0.394404
$890$	73.7208	2.47113
$891$	0	0
$892$	59.6153	1.99607
$893$	−16.3226	−0.546216
$894$	0	0
$895$	−42.1859	−1.41012
$896$	−14.1210	−0.471749
$897$	0	0
$898$	13.6647	0.455995
$899$	8.85372	0.295288
$900$	0	0
$901$	57.4624	1.91435
$902$	22.2079	0.739442
$903$	0	0
$904$	49.8099	1.65665
$905$	−36.6772	−1.21919
$906$	0	0
$907$	2.79316	0.0927455	0.0463728	−	0.998924i	$-0.485234\pi$
$907$	2.79316	0.0927455	0.0463728	+	0.998924i	$0.485234\pi$
$908$	−38.4488	−1.27597
$909$	0	0
$910$	−15.5785	−0.516423
$911$	0.569951	0.0188833	0.00944166	−	0.999955i	$-0.496995\pi$
$911$	0.569951	0.0188833	0.00944166	+	0.999955i	$0.496995\pi$
$912$	0	0
$913$	−8.41175	−0.278388
$914$	2.37509	0.0785609
$915$	0	0
$916$	−47.6801	−1.57540
$917$	−2.58861	−0.0854834
$918$	0	0
$919$	46.4852	1.53341	0.766703	−	0.642002i	$-0.221896\pi$
$919$	46.4852	1.53341	0.766703	+	0.642002i	$0.221896\pi$
$920$	82.2296	2.71103
$921$	0	0
$922$	−49.6100	−1.63382
$923$	45.6841	1.50371
$924$	0	0
$925$	−4.60124	−0.151288
$926$	66.7320	2.19295
$927$	0	0
$928$	10.6403	0.349286
$929$	32.9706	1.08173	0.540865	−	0.841110i	$-0.318097\pi$
$929$	32.9706	1.08173	0.540865	+	0.841110i	$0.318097\pi$
$930$	0	0
$931$	15.5464	0.509511
$932$	85.0635	2.78635
$933$	0	0
$934$	−19.4675	−0.636995
$935$	−11.5388	−0.377360
$936$	0	0
$937$	5.63944	0.184232	0.0921162	−	0.995748i	$-0.470637\pi$
$937$	5.63944	0.184232	0.0921162	+	0.995748i	$0.470637\pi$
$938$	22.3658	0.730270
$939$	0	0
$940$	59.7391	1.94848
$941$	39.6392	1.29220	0.646100	−	0.763253i	$-0.276399\pi$
$941$	39.6392	1.29220	0.646100	+	0.763253i	$0.276399\pi$
$942$	0	0
$943$	−79.4533	−2.58735
$944$	−5.79225	−0.188522
$945$	0	0
$946$	11.5847	0.376652
$947$	−17.9550	−0.583458	−0.291729	−	0.956501i	$-0.594231\pi$
$947$	−17.9550	−0.583458	−0.291729	+	0.956501i	$0.594231\pi$
$948$	0	0
$949$	0.584264	0.0189660
$950$	2.76071	0.0895692
$951$	0	0
$952$	13.7675	0.446206
$953$	−3.87276	−0.125451	−0.0627254	−	0.998031i	$-0.519979\pi$
$953$	−3.87276	−0.125451	−0.0627254	+	0.998031i	$0.519979\pi$
$954$	0	0
$955$	34.9921	1.13232
$956$	−24.6801	−0.798211
$957$	0	0
$958$	61.8745	1.99907
$959$	11.7818	0.380454
$960$	0	0
$961$	−26.6097	−0.858378
$962$	−92.5928	−2.98531
$963$	0	0
$964$	−113.788	−3.66485
$965$	10.3881	0.334403
$966$	0	0
$967$	39.2368	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$967$	39.2368	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$968$	−4.10437	−0.131919
$969$	0	0
$970$	15.2679	0.490222
$971$	−1.60009	−0.0513492	−0.0256746	−	0.999670i	$-0.508173\pi$
$971$	−1.60009	−0.0513492	−0.0256746	+	0.999670i	$0.508173\pi$
$972$	0	0
$973$	10.9765	0.351890
$974$	−24.2095	−0.775723
$975$	0	0
$976$	2.38007	0.0761843
$977$	56.8153	1.81768	0.908841	−	0.417142i	$-0.136968\pi$
$977$	56.8153	1.81768	0.908841	+	0.417142i	$0.136968\pi$
$978$	0	0
$979$	13.1648	0.420747
$980$	−56.8980	−1.81754
$981$	0	0
$982$	−5.04629	−0.161034
$983$	13.2440	0.422417	0.211209	−	0.977441i	$-0.432260\pi$
$983$	13.2440	0.422417	0.211209	+	0.977441i	$0.432260\pi$
$984$	0	0
$985$	−48.6987	−1.55167
$986$	−49.7742	−1.58513
$987$	0	0
$988$	36.1187	1.14909
$989$	−41.4468	−1.31793
$990$	0	0
$991$	29.8619	0.948596	0.474298	−	0.880364i	$-0.342702\pi$
$991$	29.8619	0.948596	0.474298	+	0.880364i	$0.342702\pi$
$992$	5.27619	0.167519
$993$	0	0
$994$	−18.2009	−0.577299
$995$	54.6771	1.73338
$996$	0	0
$997$	−2.00081	−0.0633664	−0.0316832	−	0.999498i	$-0.510087\pi$
$997$	−2.00081	−0.0633664	−0.0316832	+	0.999498i	$0.510087\pi$
$998$	−90.2873	−2.85800
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.c.1.1		11
3.2	odd	2		2013.2.a.b.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.b.1.11	✓	11	3.2	odd	2
6039.2.a.c.1.1		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.39095 q^{2} +3.71663 q^{4} +2.34211 q^{5} -0.680854 q^{7} -4.10437 q^{8} +O(q^{10})\)	\(q-2.39095 q^{2} +3.71663 q^{4} +2.34211 q^{5} -0.680854 q^{7} -4.10437 q^{8} -5.59986 q^{10} -1.00000 q^{11} -4.08597 q^{13} +1.62789 q^{14} +2.38007 q^{16} +4.92668 q^{17} -2.37841 q^{19} +8.70475 q^{20} +2.39095 q^{22} -8.55411 q^{23} +0.485471 q^{25} +9.76934 q^{26} -2.53048 q^{28} +4.22552 q^{29} +2.09530 q^{31} +2.51811 q^{32} -11.7794 q^{34} -1.59463 q^{35} -9.47790 q^{37} +5.68666 q^{38} -9.61288 q^{40} +9.28832 q^{41} +4.84525 q^{43} -3.71663 q^{44} +20.4524 q^{46} +6.86282 q^{47} -6.53644 q^{49} -1.16074 q^{50} -15.1860 q^{52} +11.6635 q^{53} -2.34211 q^{55} +2.79447 q^{56} -10.1030 q^{58} -2.43364 q^{59} +1.00000 q^{61} -5.00974 q^{62} -10.7808 q^{64} -9.56978 q^{65} +13.7392 q^{67} +18.3106 q^{68} +3.81268 q^{70} -11.1807 q^{71} -0.142993 q^{73} +22.6612 q^{74} -8.83968 q^{76} +0.680854 q^{77} +6.05595 q^{79} +5.57439 q^{80} -22.2079 q^{82} +8.41175 q^{83} +11.5388 q^{85} -11.5847 q^{86} +4.10437 q^{88} -13.1648 q^{89} +2.78195 q^{91} -31.7924 q^{92} -16.4086 q^{94} -5.57050 q^{95} -2.72647 q^{97} +15.6283 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8}+O(q^{10}) \) 11 * q + 2 * q^2 + 10 * q^4 + q^5 - 11 * q^7 + 3 * q^8	\( 11 q + 2 q^{2} + 10 q^{4} + q^{5} - 11 q^{7} + 3 q^{8} - 8 q^{10} - 11 q^{11} - 13 q^{13} - 5 q^{14} + 4 q^{16} + 13 q^{17} - 12 q^{19} + 7 q^{20} - 2 q^{22} + 3 q^{23} + 12 q^{25} - 12 q^{26} - 13 q^{28} - 2 q^{29} + q^{31} + 23 q^{32} - 14 q^{34} + 4 q^{35} - 14 q^{37} + 8 q^{38} - 34 q^{40} - 3 q^{41} - 21 q^{43} - 10 q^{44} - 12 q^{46} + 16 q^{47} - 18 q^{49} + 13 q^{50} - 33 q^{52} - q^{55} - 16 q^{56} - 17 q^{58} - 3 q^{59} + 11 q^{61} + 21 q^{62} - 7 q^{64} + q^{65} - 24 q^{67} - 2 q^{68} + 4 q^{70} - 7 q^{71} - 42 q^{73} + 16 q^{74} - 13 q^{76} + 11 q^{77} - 11 q^{79} - 42 q^{80} - 38 q^{82} + 34 q^{83} - 14 q^{85} - 42 q^{86} - 3 q^{88} - 29 q^{89} + 9 q^{91} - 42 q^{92} - 33 q^{94} + 31 q^{95} - 45 q^{97} + 33 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 10 * q^4 + q^5 - 11 * q^7 + 3 * q^8 - 8 * q^10 - 11 * q^11 - 13 * q^13 - 5 * q^14 + 4 * q^16 + 13 * q^17 - 12 * q^19 + 7 * q^20 - 2 * q^22 + 3 * q^23 + 12 * q^25 - 12 * q^26 - 13 * q^28 - 2 * q^29 + q^31 + 23 * q^32 - 14 * q^34 + 4 * q^35 - 14 * q^37 + 8 * q^38 - 34 * q^40 - 3 * q^41 - 21 * q^43 - 10 * q^44 - 12 * q^46 + 16 * q^47 - 18 * q^49 + 13 * q^50 - 33 * q^52 - q^55 - 16 * q^56 - 17 * q^58 - 3 * q^59 + 11 * q^61 + 21 * q^62 - 7 * q^64 + q^65 - 24 * q^67 - 2 * q^68 + 4 * q^70 - 7 * q^71 - 42 * q^73 + 16 * q^74 - 13 * q^76 + 11 * q^77 - 11 * q^79 - 42 * q^80 - 38 * q^82 + 34 * q^83 - 14 * q^85 - 42 * q^86 - 3 * q^88 - 29 * q^89 + 9 * q^91 - 42 * q^92 - 33 * q^94 + 31 * q^95 - 45 * q^97 + 33 * q^98

Introduction

L-functions

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Database

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