LMFDB - Embedded newform 6039.2.a.n.1.17

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:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+1.04470 q^{2} -0.908610 q^{4} +2.79878 q^{5} -4.32870 q^{7} -3.03861 q^{8} +O(q^{10})$	\(q+1.04470 q^{2} -0.908610 q^{4} +2.79878 q^{5} -4.32870 q^{7} -3.03861 q^{8} +2.92388 q^{10} -1.00000 q^{11} +2.42009 q^{13} -4.52218 q^{14} -1.35721 q^{16} -1.27789 q^{17} +5.47422 q^{19} -2.54300 q^{20} -1.04470 q^{22} +4.49322 q^{23} +2.83318 q^{25} +2.52826 q^{26} +3.93310 q^{28} -3.83450 q^{29} +0.921333 q^{31} +4.65936 q^{32} -1.33501 q^{34} -12.1151 q^{35} -7.51289 q^{37} +5.71890 q^{38} -8.50442 q^{40} +4.82876 q^{41} -8.17926 q^{43} +0.908610 q^{44} +4.69405 q^{46} +9.90973 q^{47} +11.7377 q^{49} +2.95981 q^{50} -2.19892 q^{52} -9.03740 q^{53} -2.79878 q^{55} +13.1533 q^{56} -4.00589 q^{58} -4.30084 q^{59} -1.00000 q^{61} +0.962513 q^{62} +7.58203 q^{64} +6.77332 q^{65} -10.5758 q^{67} +1.16110 q^{68} -12.6566 q^{70} -4.02364 q^{71} -9.67796 q^{73} -7.84869 q^{74} -4.97393 q^{76} +4.32870 q^{77} -10.5830 q^{79} -3.79853 q^{80} +5.04459 q^{82} +9.35402 q^{83} -3.57653 q^{85} -8.54485 q^{86} +3.03861 q^{88} -11.4151 q^{89} -10.4759 q^{91} -4.08258 q^{92} +10.3527 q^{94} +15.3212 q^{95} -16.2131 q^{97} +12.2623 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.04470	0.738712	0.369356	−	0.929288i	$-0.379578\pi$
$2$	1.04470	0.738712	0.369356	+	0.929288i	$0.379578\pi$
$3$	0	0
$3$	0	0
$4$	−0.908610	−0.454305
$5$	2.79878	1.25165	0.625827	−	0.779962i	$-0.284762\pi$
$5$	2.79878	1.25165	0.625827	+	0.779962i	$0.284762\pi$
$6$	0	0
$7$	−4.32870	−1.63610	−0.818048	−	0.575150i	$-0.804944\pi$
$7$	−4.32870	−1.63610	−0.818048	+	0.575150i	$0.804944\pi$
$8$	−3.03861	−1.07431
$9$	0	0
$10$	2.92388	0.924611
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.42009	0.671213	0.335607	−	0.942002i	$-0.391059\pi$
$13$	2.42009	0.671213	0.335607	+	0.942002i	$0.391059\pi$
$14$	−4.52218	−1.20860
$15$	0	0
$16$	−1.35721	−0.339302
$17$	−1.27789	−0.309933	−0.154967	−	0.987920i	$-0.549527\pi$
$17$	−1.27789	−0.309933	−0.154967	+	0.987920i	$0.549527\pi$
$18$	0	0
$19$	5.47422	1.25587	0.627936	−	0.778265i	$-0.283900\pi$
$19$	5.47422	1.25587	0.627936	+	0.778265i	$0.283900\pi$
$20$	−2.54300	−0.568632
$21$	0	0
$22$	−1.04470	−0.222730
$23$	4.49322	0.936901	0.468450	−	0.883490i	$-0.344813\pi$
$23$	4.49322	0.936901	0.468450	+	0.883490i	$0.344813\pi$
$24$	0	0
$25$	2.83318	0.566636
$26$	2.52826	0.495833
$27$	0	0
$28$	3.93310	0.743286
$29$	−3.83450	−0.712049	−0.356025	−	0.934477i	$-0.615868\pi$
$29$	−3.83450	−0.712049	−0.356025	+	0.934477i	$0.615868\pi$
$30$	0	0
$31$	0.921333	0.165476	0.0827382	−	0.996571i	$-0.473633\pi$
$31$	0.921333	0.165476	0.0827382	+	0.996571i	$0.473633\pi$
$32$	4.65936	0.823666
$33$	0	0
$34$	−1.33501	−0.228952
$35$	−12.1151	−2.04783
$36$	0	0
$37$	−7.51289	−1.23511	−0.617556	−	0.786527i	$-0.711877\pi$
$37$	−7.51289	−1.23511	−0.617556	+	0.786527i	$0.711877\pi$
$38$	5.71890	0.927728
$39$	0	0
$40$	−8.50442	−1.34467
$41$	4.82876	0.754126	0.377063	−	0.926188i	$-0.376934\pi$
$41$	4.82876	0.754126	0.377063	+	0.926188i	$0.376934\pi$
$42$	0	0
$43$	−8.17926	−1.24733	−0.623663	−	0.781693i	$-0.714356\pi$
$43$	−8.17926	−1.24733	−0.623663	+	0.781693i	$0.714356\pi$
$44$	0.908610	0.136978
$45$	0	0
$46$	4.69405	0.692100
$47$	9.90973	1.44548	0.722741	−	0.691119i	$-0.242882\pi$
$47$	9.90973	1.44548	0.722741	+	0.691119i	$0.242882\pi$
$48$	0	0
$49$	11.7377	1.67681
$50$	2.95981	0.418581
$51$	0	0
$52$	−2.19892	−0.304935
$53$	−9.03740	−1.24138	−0.620691	−	0.784055i	$-0.713148\pi$
$53$	−9.03740	−1.24138	−0.620691	+	0.784055i	$0.713148\pi$
$54$	0	0
$55$	−2.79878	−0.377388
$56$	13.1533	1.75768
$57$	0	0
$58$	−4.00589	−0.525999
$59$	−4.30084	−0.559921	−0.279961	−	0.960012i	$-0.590321\pi$
$59$	−4.30084	−0.559921	−0.279961	+	0.960012i	$0.590321\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0.962513	0.122239
$63$	0	0
$64$	7.58203	0.947754
$65$	6.77332	0.840127
$66$	0	0
$67$	−10.5758	−1.29204	−0.646022	−	0.763319i	$-0.723569\pi$
$67$	−10.5758	−1.29204	−0.646022	+	0.763319i	$0.723569\pi$
$68$	1.16110	0.140804
$69$	0	0
$70$	−12.6566	−1.51275
$71$	−4.02364	−0.477518	−0.238759	−	0.971079i	$-0.576741\pi$
$71$	−4.02364	−0.477518	−0.238759	+	0.971079i	$0.576741\pi$
$72$	0	0
$73$	−9.67796	−1.13272	−0.566360	−	0.824158i	$-0.691649\pi$
$73$	−9.67796	−1.13272	−0.566360	+	0.824158i	$0.691649\pi$
$74$	−7.84869	−0.912391
$75$	0	0
$76$	−4.97393	−0.570549
$77$	4.32870	0.493302
$78$	0	0
$79$	−10.5830	−1.19068	−0.595342	−	0.803472i	$-0.702983\pi$
$79$	−10.5830	−1.19068	−0.595342	+	0.803472i	$0.702983\pi$
$80$	−3.79853	−0.424689
$81$	0	0
$82$	5.04459	0.557082
$83$	9.35402	1.02674	0.513368	−	0.858168i	$-0.328397\pi$
$83$	9.35402	1.02674	0.513368	+	0.858168i	$0.328397\pi$
$84$	0	0
$85$	−3.57653	−0.387929
$86$	−8.54485	−0.921415
$87$	0	0
$88$	3.03861	0.323917
$89$	−11.4151	−1.21000	−0.605001	−	0.796224i	$-0.706827\pi$
$89$	−11.4151	−1.21000	−0.605001	+	0.796224i	$0.706827\pi$
$90$	0	0
$91$	−10.4759	−1.09817
$92$	−4.08258	−0.425638
$93$	0	0
$94$	10.3527	1.06780
$95$	15.3212	1.57192
$96$	0	0
$97$	−16.2131	−1.64619	−0.823095	−	0.567904i	$-0.807754\pi$
$97$	−16.2131	−1.64619	−0.823095	+	0.567904i	$0.807754\pi$
$98$	12.2623	1.23868
$99$	0	0
$100$	−2.57425	−0.257425
$101$	−0.825975	−0.0821876	−0.0410938	−	0.999155i	$-0.513084\pi$
$101$	−0.825975	−0.0821876	−0.0410938	+	0.999155i	$0.513084\pi$
$102$	0	0
$103$	3.81263	0.375670	0.187835	−	0.982201i	$-0.439853\pi$
$103$	3.81263	0.375670	0.187835	+	0.982201i	$0.439853\pi$
$104$	−7.35373	−0.721093
$105$	0	0
$106$	−9.44134	−0.917024
$107$	15.4102	1.48976	0.744878	−	0.667200i	$-0.232507\pi$
$107$	15.4102	1.48976	0.744878	+	0.667200i	$0.232507\pi$
$108$	0	0
$109$	5.99961	0.574658	0.287329	−	0.957832i	$-0.407233\pi$
$109$	5.99961	0.574658	0.287329	+	0.957832i	$0.407233\pi$
$110$	−2.92388	−0.278781
$111$	0	0
$112$	5.87496	0.555131
$113$	−18.4691	−1.73743	−0.868714	−	0.495314i	$-0.835053\pi$
$113$	−18.4691	−1.73743	−0.868714	+	0.495314i	$0.835053\pi$
$114$	0	0
$115$	12.5755	1.17267
$116$	3.48407	0.323487
$117$	0	0
$118$	−4.49307	−0.413620
$119$	5.53160	0.507081
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.04470	−0.0945824
$123$	0	0
$124$	−0.837132	−0.0751767
$125$	−6.06446	−0.542421
$126$	0	0
$127$	−13.1572	−1.16751	−0.583756	−	0.811929i	$-0.698418\pi$
$127$	−13.1572	−1.16751	−0.583756	+	0.811929i	$0.698418\pi$
$128$	−1.39779	−0.123548
$129$	0	0
$130$	7.07606	0.620611
$131$	−14.1005	−1.23197	−0.615985	−	0.787758i	$-0.711242\pi$
$131$	−14.1005	−1.23197	−0.615985	+	0.787758i	$0.711242\pi$
$132$	0	0
$133$	−23.6963	−2.05473
$134$	−11.0485	−0.954448
$135$	0	0
$136$	3.88301	0.332965
$137$	−2.39030	−0.204217	−0.102108	−	0.994773i	$-0.532559\pi$
$137$	−2.39030	−0.204217	−0.102108	+	0.994773i	$0.532559\pi$
$138$	0	0
$139$	−1.41844	−0.120310	−0.0601552	−	0.998189i	$-0.519160\pi$
$139$	−1.41844	−0.120310	−0.0601552	+	0.998189i	$0.519160\pi$
$140$	11.0079	0.930337
$141$	0	0
$142$	−4.20348	−0.352748
$143$	−2.42009	−0.202378
$144$	0	0
$145$	−10.7319	−0.891239
$146$	−10.1105	−0.836754
$147$	0	0
$148$	6.82628	0.561117
$149$	−5.53404	−0.453366	−0.226683	−	0.973969i	$-0.572788\pi$
$149$	−5.53404	−0.453366	−0.226683	+	0.973969i	$0.572788\pi$
$150$	0	0
$151$	−3.79810	−0.309085	−0.154542	−	0.987986i	$-0.549390\pi$
$151$	−3.79810	−0.309085	−0.154542	+	0.987986i	$0.549390\pi$
$152$	−16.6340	−1.34920
$153$	0	0
$154$	4.52218	0.364408
$155$	2.57861	0.207119
$156$	0	0
$157$	−17.0924	−1.36413	−0.682063	−	0.731294i	$-0.738917\pi$
$157$	−17.0924	−1.36413	−0.682063	+	0.731294i	$0.738917\pi$
$158$	−11.0561	−0.879573
$159$	0	0
$160$	13.0405	1.03094
$161$	−19.4498	−1.53286
$162$	0	0
$163$	18.8889	1.47949	0.739745	−	0.672888i	$-0.234946\pi$
$163$	18.8889	1.47949	0.739745	+	0.672888i	$0.234946\pi$
$164$	−4.38746	−0.342603
$165$	0	0
$166$	9.77211	0.758463
$167$	2.10168	0.162633	0.0813164	−	0.996688i	$-0.474088\pi$
$167$	2.10168	0.162633	0.0813164	+	0.996688i	$0.474088\pi$
$168$	0	0
$169$	−7.14314	−0.549472
$170$	−3.73639	−0.286568
$171$	0	0
$172$	7.43176	0.566666
$173$	10.7236	0.815297	0.407648	−	0.913139i	$-0.366349\pi$
$173$	10.7236	0.815297	0.407648	+	0.913139i	$0.366349\pi$
$174$	0	0
$175$	−12.2640	−0.927071
$176$	1.35721	0.102304
$177$	0	0
$178$	−11.9254	−0.893843
$179$	−19.6960	−1.47215	−0.736074	−	0.676901i	$-0.763323\pi$
$179$	−19.6960	−1.47215	−0.736074	+	0.676901i	$0.763323\pi$
$180$	0	0
$181$	−26.4204	−1.96381	−0.981906	−	0.189371i	$-0.939355\pi$
$181$	−26.4204	−1.96381	−0.981906	+	0.189371i	$0.939355\pi$
$182$	−10.9441	−0.811231
$183$	0	0
$184$	−13.6532	−1.00652
$185$	−21.0269	−1.54593
$186$	0	0
$187$	1.27789	0.0934485
$188$	−9.00408	−0.656690
$189$	0	0
$190$	16.0060	1.16119
$191$	3.98145	0.288087	0.144044	−	0.989571i	$-0.453989\pi$
$191$	3.98145	0.288087	0.144044	+	0.989571i	$0.453989\pi$
$192$	0	0
$193$	21.1369	1.52147	0.760733	−	0.649065i	$-0.224840\pi$
$193$	21.1369	1.52147	0.760733	+	0.649065i	$0.224840\pi$
$194$	−16.9377	−1.21606
$195$	0	0
$196$	−10.6650	−0.761783
$197$	−0.122002	−0.00869232	−0.00434616	−	0.999991i	$-0.501383\pi$
$197$	−0.122002	−0.00869232	−0.00434616	+	0.999991i	$0.501383\pi$
$198$	0	0
$199$	−17.1579	−1.21629	−0.608145	−	0.793826i	$-0.708086\pi$
$199$	−17.1579	−1.21629	−0.608145	+	0.793826i	$0.708086\pi$
$200$	−8.60894	−0.608744
$201$	0	0
$202$	−0.862893	−0.0607130
$203$	16.5984	1.16498
$204$	0	0
$205$	13.5147	0.943905
$206$	3.98304	0.277512
$207$	0	0
$208$	−3.28458	−0.227744
$209$	−5.47422	−0.378660
$210$	0	0
$211$	3.73045	0.256815	0.128407	−	0.991722i	$-0.459014\pi$
$211$	3.73045	0.256815	0.128407	+	0.991722i	$0.459014\pi$
$212$	8.21147	0.563966
$213$	0	0
$214$	16.0989	1.10050
$215$	−22.8920	−1.56122
$216$	0	0
$217$	−3.98818	−0.270735
$218$	6.26777	0.424507
$219$	0	0
$220$	2.54300	0.171449
$221$	−3.09261	−0.208032
$222$	0	0
$223$	25.2735	1.69244	0.846221	−	0.532832i	$-0.178872\pi$
$223$	25.2735	1.69244	0.846221	+	0.532832i	$0.178872\pi$
$224$	−20.1690	−1.34760
$225$	0	0
$226$	−19.2946	−1.28346
$227$	2.01806	0.133943	0.0669717	−	0.997755i	$-0.478666\pi$
$227$	2.01806	0.133943	0.0669717	+	0.997755i	$0.478666\pi$
$228$	0	0
$229$	16.5927	1.09648	0.548238	−	0.836323i	$-0.315299\pi$
$229$	16.5927	1.09648	0.548238	+	0.836323i	$0.315299\pi$
$230$	13.1376	0.866269
$231$	0	0
$232$	11.6516	0.764963
$233$	23.3233	1.52796	0.763979	−	0.645242i	$-0.223243\pi$
$233$	23.3233	1.52796	0.763979	+	0.645242i	$0.223243\pi$
$234$	0	0
$235$	27.7352	1.80924
$236$	3.90778	0.254375
$237$	0	0
$238$	5.77884	0.374587
$239$	27.9100	1.80535	0.902674	−	0.430325i	$-0.141601\pi$
$239$	27.9100	1.80535	0.902674	+	0.430325i	$0.141601\pi$
$240$	0	0
$241$	16.1898	1.04288	0.521439	−	0.853289i	$-0.325395\pi$
$241$	16.1898	1.04288	0.521439	+	0.853289i	$0.325395\pi$
$242$	1.04470	0.0671556
$243$	0	0
$244$	0.908610	0.0581678
$245$	32.8512	2.09879
$246$	0	0
$247$	13.2481	0.842959
$248$	−2.79958	−0.177773
$249$	0	0
$250$	−6.33552	−0.400693
$251$	−17.0827	−1.07825	−0.539125	−	0.842226i	$-0.681245\pi$
$251$	−17.0827	−1.07825	−0.539125	+	0.842226i	$0.681245\pi$
$252$	0	0
$253$	−4.49322	−0.282486
$254$	−13.7453	−0.862455
$255$	0	0
$256$	−16.6243	−1.03902
$257$	9.63585	0.601068	0.300534	−	0.953771i	$-0.402835\pi$
$257$	9.63585	0.601068	0.300534	+	0.953771i	$0.402835\pi$
$258$	0	0
$259$	32.5211	2.02076
$260$	−6.15430	−0.381674
$261$	0	0
$262$	−14.7308	−0.910071
$263$	−25.4339	−1.56832	−0.784161	−	0.620558i	$-0.786906\pi$
$263$	−25.4339	−1.56832	−0.784161	+	0.620558i	$0.786906\pi$
$264$	0	0
$265$	−25.2937	−1.55378
$266$	−24.7554	−1.51785
$267$	0	0
$268$	9.60930	0.586982
$269$	−21.9935	−1.34097	−0.670484	−	0.741924i	$-0.733913\pi$
$269$	−21.9935	−1.34097	−0.670484	+	0.741924i	$0.733913\pi$
$270$	0	0
$271$	−11.2400	−0.682781	−0.341391	−	0.939922i	$-0.610898\pi$
$271$	−11.2400	−0.682781	−0.341391	+	0.939922i	$0.610898\pi$
$272$	1.73436	0.105161
$273$	0	0
$274$	−2.49713	−0.150857
$275$	−2.83318	−0.170847
$276$	0	0
$277$	−8.99023	−0.540170	−0.270085	−	0.962836i	$-0.587052\pi$
$277$	−8.99023	−0.540170	−0.270085	+	0.962836i	$0.587052\pi$
$278$	−1.48184	−0.0888747
$279$	0	0
$280$	36.8131	2.20000
$281$	−26.6926	−1.59235	−0.796174	−	0.605068i	$-0.793146\pi$
$281$	−26.6926	−1.59235	−0.796174	+	0.605068i	$0.793146\pi$
$282$	0	0
$283$	11.7072	0.695920	0.347960	−	0.937509i	$-0.386874\pi$
$283$	11.7072	0.695920	0.347960	+	0.937509i	$0.386874\pi$
$284$	3.65591	0.216939
$285$	0	0
$286$	−2.52826	−0.149499
$287$	−20.9023	−1.23382
$288$	0	0
$289$	−15.3670	−0.903941
$290$	−11.2116	−0.658369
$291$	0	0
$292$	8.79349	0.514600
$293$	−3.92311	−0.229190	−0.114595	−	0.993412i	$-0.536557\pi$
$293$	−3.92311	−0.229190	−0.114595	+	0.993412i	$0.536557\pi$
$294$	0	0
$295$	−12.0371	−0.700827
$296$	22.8288	1.32689
$297$	0	0
$298$	−5.78139	−0.334907
$299$	10.8740	0.628860
$300$	0	0
$301$	35.4056	2.04075
$302$	−3.96786	−0.228325
$303$	0	0
$304$	−7.42967	−0.426121
$305$	−2.79878	−0.160258
$306$	0	0
$307$	16.9334	0.966438	0.483219	−	0.875499i	$-0.339467\pi$
$307$	16.9334	0.966438	0.483219	+	0.875499i	$0.339467\pi$
$308$	−3.93310	−0.224109
$309$	0	0
$310$	2.69386	0.153001
$311$	4.27067	0.242167	0.121084	−	0.992642i	$-0.461363\pi$
$311$	4.27067	0.242167	0.121084	+	0.992642i	$0.461363\pi$
$312$	0	0
$313$	10.5476	0.596186	0.298093	−	0.954537i	$-0.403650\pi$
$313$	10.5476	0.596186	0.298093	+	0.954537i	$0.403650\pi$
$314$	−17.8564	−1.00770
$315$	0	0
$316$	9.61584	0.540934
$317$	1.90799	0.107163	0.0535816	−	0.998563i	$-0.482936\pi$
$317$	1.90799	0.107163	0.0535816	+	0.998563i	$0.482936\pi$
$318$	0	0
$319$	3.83450	0.214691
$320$	21.2205	1.18626
$321$	0	0
$322$	−20.3191	−1.13234
$323$	−6.99545	−0.389237
$324$	0	0
$325$	6.85656	0.380334
$326$	19.7331	1.09292
$327$	0	0
$328$	−14.6727	−0.810167
$329$	−42.8963	−2.36495
$330$	0	0
$331$	9.78394	0.537774	0.268887	−	0.963172i	$-0.413344\pi$
$331$	9.78394	0.537774	0.268887	+	0.963172i	$0.413344\pi$
$332$	−8.49915	−0.466451
$333$	0	0
$334$	2.19562	0.120139
$335$	−29.5995	−1.61719
$336$	0	0
$337$	−13.9029	−0.757341	−0.378671	−	0.925531i	$-0.623619\pi$
$337$	−13.9029	−0.757341	−0.378671	+	0.925531i	$0.623619\pi$
$338$	−7.46241	−0.405902
$339$	0	0
$340$	3.24967	0.176238
$341$	−0.921333	−0.0498930
$342$	0	0
$343$	−20.5080	−1.10733
$344$	24.8536	1.34002
$345$	0	0
$346$	11.2029	0.602269
$347$	−15.2884	−0.820724	−0.410362	−	0.911923i	$-0.634598\pi$
$347$	−15.2884	−0.820724	−0.410362	+	0.911923i	$0.634598\pi$
$348$	0	0
$349$	26.2711	1.40626	0.703129	−	0.711063i	$-0.251786\pi$
$349$	26.2711	1.40626	0.703129	+	0.711063i	$0.251786\pi$
$350$	−12.8122	−0.684838
$351$	0	0
$352$	−4.65936	−0.248344
$353$	1.07074	0.0569896	0.0284948	−	0.999594i	$-0.490929\pi$
$353$	1.07074	0.0569896	0.0284948	+	0.999594i	$0.490929\pi$
$354$	0	0
$355$	−11.2613	−0.597687
$356$	10.3719	0.549710
$357$	0	0
$358$	−20.5763	−1.08749
$359$	−9.42762	−0.497571	−0.248785	−	0.968559i	$-0.580031\pi$
$359$	−9.42762	−0.497571	−0.248785	+	0.968559i	$0.580031\pi$
$360$	0	0
$361$	10.9671	0.577216
$362$	−27.6013	−1.45069
$363$	0	0
$364$	9.51848	0.498904
$365$	−27.0865	−1.41777
$366$	0	0
$367$	−7.94473	−0.414711	−0.207356	−	0.978266i	$-0.566486\pi$
$367$	−7.94473	−0.414711	−0.207356	+	0.978266i	$0.566486\pi$
$368$	−6.09824	−0.317893
$369$	0	0
$370$	−21.9668	−1.14200
$371$	39.1202	2.03102
$372$	0	0
$373$	12.6905	0.657090	0.328545	−	0.944488i	$-0.393442\pi$
$373$	12.6905	0.657090	0.328545	+	0.944488i	$0.393442\pi$
$374$	1.33501	0.0690315
$375$	0	0
$376$	−30.1118	−1.55290
$377$	−9.27986	−0.477937
$378$	0	0
$379$	−6.89596	−0.354222	−0.177111	−	0.984191i	$-0.556675\pi$
$379$	−6.89596	−0.354222	−0.177111	+	0.984191i	$0.556675\pi$
$380$	−13.9209	−0.714130
$381$	0	0
$382$	4.15940	0.212814
$383$	27.3587	1.39796	0.698981	−	0.715140i	$-0.253637\pi$
$383$	27.3587	1.39796	0.698981	+	0.715140i	$0.253637\pi$
$384$	0	0
$385$	12.1151	0.617443
$386$	22.0816	1.12392
$387$	0	0
$388$	14.7314	0.747872
$389$	5.36933	0.272236	0.136118	−	0.990693i	$-0.456537\pi$
$389$	5.36933	0.272236	0.136118	+	0.990693i	$0.456537\pi$
$390$	0	0
$391$	−5.74183	−0.290377
$392$	−35.6663	−1.80142
$393$	0	0
$394$	−0.127456	−0.00642112
$395$	−29.6196	−1.49032
$396$	0	0
$397$	21.9613	1.10221	0.551103	−	0.834437i	$-0.314207\pi$
$397$	21.9613	1.10221	0.551103	+	0.834437i	$0.314207\pi$
$398$	−17.9248	−0.898488
$399$	0	0
$400$	−3.84522	−0.192261
$401$	8.13964	0.406474	0.203237	−	0.979130i	$-0.434854\pi$
$401$	8.13964	0.406474	0.203237	+	0.979130i	$0.434854\pi$
$402$	0	0
$403$	2.22971	0.111070
$404$	0.750489	0.0373382
$405$	0	0
$406$	17.3403	0.860585
$407$	7.51289	0.372400
$408$	0	0
$409$	−12.2607	−0.606251	−0.303125	−	0.952951i	$-0.598030\pi$
$409$	−12.2607	−0.606251	−0.303125	+	0.952951i	$0.598030\pi$
$410$	14.1187	0.697274
$411$	0	0
$412$	−3.46419	−0.170669
$413$	18.6170	0.916085
$414$	0	0
$415$	26.1799	1.28512
$416$	11.2761	0.552855
$417$	0	0
$418$	−5.71890	−0.279721
$419$	18.8611	0.921427	0.460713	−	0.887549i	$-0.347594\pi$
$419$	18.8611	0.921427	0.460713	+	0.887549i	$0.347594\pi$
$420$	0	0
$421$	3.07757	0.149992	0.0749958	−	0.997184i	$-0.476106\pi$
$421$	3.07757	0.149992	0.0749958	+	0.997184i	$0.476106\pi$
$422$	3.89719	0.189712
$423$	0	0
$424$	27.4612	1.33363
$425$	−3.62049	−0.175619
$426$	0	0
$427$	4.32870	0.209481
$428$	−14.0018	−0.676804
$429$	0	0
$430$	−23.9152	−1.15329
$431$	2.15924	0.104007	0.0520034	−	0.998647i	$-0.483439\pi$
$431$	2.15924	0.104007	0.0520034	+	0.998647i	$0.483439\pi$
$432$	0	0
$433$	−24.2999	−1.16778	−0.583889	−	0.811833i	$-0.698470\pi$
$433$	−24.2999	−1.16778	−0.583889	+	0.811833i	$0.698470\pi$
$434$	−4.16643	−0.199995
$435$	0	0
$436$	−5.45130	−0.261070
$437$	24.5969	1.17663
$438$	0	0
$439$	−40.3487	−1.92574	−0.962869	−	0.269971i	$-0.912986\pi$
$439$	−40.3487	−1.92574	−0.962869	+	0.269971i	$0.912986\pi$
$440$	8.50442	0.405432
$441$	0	0
$442$	−3.23084	−0.153675
$443$	36.8185	1.74930	0.874650	−	0.484755i	$-0.161091\pi$
$443$	36.8185	1.74930	0.874650	+	0.484755i	$0.161091\pi$
$444$	0	0
$445$	−31.9485	−1.51450
$446$	26.4032	1.25023
$447$	0	0
$448$	−32.8204	−1.55062
$449$	23.1714	1.09353	0.546764	−	0.837287i	$-0.315859\pi$
$449$	23.1714	1.09353	0.546764	+	0.837287i	$0.315859\pi$
$450$	0	0
$451$	−4.82876	−0.227378
$452$	16.7812	0.789322
$453$	0	0
$454$	2.10826	0.0989456
$455$	−29.3197	−1.37453
$456$	0	0
$457$	12.7892	0.598252	0.299126	−	0.954214i	$-0.403305\pi$
$457$	12.7892	0.598252	0.299126	+	0.954214i	$0.403305\pi$
$458$	17.3343	0.809979
$459$	0	0
$460$	−11.4263	−0.532752
$461$	10.9121	0.508227	0.254113	−	0.967174i	$-0.418216\pi$
$461$	10.9121	0.508227	0.254113	+	0.967174i	$0.418216\pi$
$462$	0	0
$463$	−23.5247	−1.09328	−0.546642	−	0.837366i	$-0.684094\pi$
$463$	−23.5247	−1.09328	−0.546642	+	0.837366i	$0.684094\pi$
$464$	5.20422	0.241600
$465$	0	0
$466$	24.3657	1.12872
$467$	−12.8256	−0.593499	−0.296749	−	0.954955i	$-0.595903\pi$
$467$	−12.8256	−0.593499	−0.296749	+	0.954955i	$0.595903\pi$
$468$	0	0
$469$	45.7796	2.11391
$470$	28.9748	1.33651
$471$	0	0
$472$	13.0686	0.601530
$473$	8.17926	0.376083
$474$	0	0
$475$	15.5095	0.711623
$476$	−5.02606	−0.230369
$477$	0	0
$478$	29.1575	1.33363
$479$	−0.809863	−0.0370036	−0.0185018	−	0.999829i	$-0.505890\pi$
$479$	−0.809863	−0.0370036	−0.0185018	+	0.999829i	$0.505890\pi$
$480$	0	0
$481$	−18.1819	−0.829023
$482$	16.9134	0.770386
$483$	0	0
$484$	−0.908610	−0.0413004
$485$	−45.3769	−2.06046
$486$	0	0
$487$	10.0691	0.456274	0.228137	−	0.973629i	$-0.426737\pi$
$487$	10.0691	0.456274	0.228137	+	0.973629i	$0.426737\pi$
$488$	3.03861	0.137552
$489$	0	0
$490$	34.3195	1.55040
$491$	27.5136	1.24167	0.620836	−	0.783940i	$-0.286793\pi$
$491$	27.5136	1.24167	0.620836	+	0.783940i	$0.286793\pi$
$492$	0	0
$493$	4.90007	0.220688
$494$	13.8403	0.622704
$495$	0	0
$496$	−1.25044	−0.0561465
$497$	17.4171	0.781265
$498$	0	0
$499$	−24.1975	−1.08323	−0.541615	−	0.840627i	$-0.682187\pi$
$499$	−24.1975	−1.08323	−0.541615	+	0.840627i	$0.682187\pi$
$500$	5.51022	0.246425
$501$	0	0
$502$	−17.8462	−0.796516
$503$	4.83125	0.215415	0.107707	−	0.994183i	$-0.465649\pi$
$503$	4.83125	0.215415	0.107707	+	0.994183i	$0.465649\pi$
$504$	0	0
$505$	−2.31172	−0.102870
$506$	−4.69405	−0.208676
$507$	0	0
$508$	11.9547	0.530406
$509$	−41.1026	−1.82184	−0.910920	−	0.412583i	$-0.864627\pi$
$509$	−41.1026	−1.82184	−0.910920	+	0.412583i	$0.864627\pi$
$510$	0	0
$511$	41.8930	1.85324
$512$	−14.5718	−0.643988
$513$	0	0
$514$	10.0665	0.444016
$515$	10.6707	0.470208
$516$	0	0
$517$	−9.90973	−0.435829
$518$	33.9746	1.49276
$519$	0	0
$520$	−20.5815	−0.902558
$521$	−35.1732	−1.54097	−0.770483	−	0.637461i	$-0.779985\pi$
$521$	−35.1732	−1.54097	−0.770483	+	0.637461i	$0.779985\pi$
$522$	0	0
$523$	24.9263	1.08995	0.544975	−	0.838453i	$-0.316539\pi$
$523$	24.9263	1.08995	0.544975	+	0.838453i	$0.316539\pi$
$524$	12.8119	0.559690
$525$	0	0
$526$	−26.5707	−1.15854
$527$	−1.17736	−0.0512866
$528$	0	0
$529$	−2.81100	−0.122217
$530$	−26.4242	−1.14780
$531$	0	0
$532$	21.5307	0.933473
$533$	11.6861	0.506180
$534$	0	0
$535$	43.1297	1.86466
$536$	32.1359	1.38806
$537$	0	0
$538$	−22.9765	−0.990589
$539$	−11.7377	−0.505577
$540$	0	0
$541$	25.9974	1.11772	0.558858	−	0.829263i	$-0.311240\pi$
$541$	25.9974	1.11772	0.558858	+	0.829263i	$0.311240\pi$
$542$	−11.7424	−0.504379
$543$	0	0
$544$	−5.95414	−0.255281
$545$	16.7916	0.719273
$546$	0	0
$547$	−7.05671	−0.301723	−0.150862	−	0.988555i	$-0.548205\pi$
$547$	−7.05671	−0.301723	−0.150862	+	0.988555i	$0.548205\pi$
$548$	2.17185	0.0927766
$549$	0	0
$550$	−2.95981	−0.126207
$551$	−20.9909	−0.894243
$552$	0	0
$553$	45.8108	1.94807
$554$	−9.39206	−0.399030
$555$	0	0
$556$	1.28881	0.0546576
$557$	−22.6508	−0.959745	−0.479873	−	0.877338i	$-0.659317\pi$
$557$	−22.6508	−0.959745	−0.479873	+	0.877338i	$0.659317\pi$
$558$	0	0
$559$	−19.7946	−0.837222
$560$	16.4427	0.694832
$561$	0	0
$562$	−27.8857	−1.17629
$563$	−41.5219	−1.74994	−0.874971	−	0.484176i	$-0.839120\pi$
$563$	−41.5219	−1.74994	−0.874971	+	0.484176i	$0.839120\pi$
$564$	0	0
$565$	−51.6910	−2.17466
$566$	12.2305	0.514084
$567$	0	0
$568$	12.2263	0.513003
$569$	3.83373	0.160718	0.0803592	−	0.996766i	$-0.474393\pi$
$569$	3.83373	0.160718	0.0803592	+	0.996766i	$0.474393\pi$
$570$	0	0
$571$	22.4937	0.941333	0.470667	−	0.882311i	$-0.344013\pi$
$571$	22.4937	0.941333	0.470667	+	0.882311i	$0.344013\pi$
$572$	2.19892	0.0919415
$573$	0	0
$574$	−21.8365	−0.911440
$575$	12.7301	0.530882
$576$	0	0
$577$	28.8540	1.20121	0.600604	−	0.799547i	$-0.294927\pi$
$577$	28.8540	1.20121	0.600604	+	0.799547i	$0.294927\pi$
$578$	−16.0538	−0.667752
$579$	0	0
$580$	9.75114	0.404894
$581$	−40.4908	−1.67984
$582$	0	0
$583$	9.03740	0.374291
$584$	29.4076	1.21689
$585$	0	0
$586$	−4.09845	−0.169306
$587$	−36.6854	−1.51417	−0.757084	−	0.653318i	$-0.773377\pi$
$587$	−36.6854	−1.51417	−0.757084	+	0.653318i	$0.773377\pi$
$588$	0	0
$589$	5.04358	0.207817
$590$	−12.5751	−0.517709
$591$	0	0
$592$	10.1966	0.419076
$593$	−7.14294	−0.293325	−0.146663	−	0.989187i	$-0.546853\pi$
$593$	−7.14294	−0.293325	−0.146663	+	0.989187i	$0.546853\pi$
$594$	0	0
$595$	15.4817	0.634690
$596$	5.02828	0.205966
$597$	0	0
$598$	11.3600	0.464547
$599$	1.36897	0.0559348	0.0279674	−	0.999609i	$-0.491097\pi$
$599$	1.36897	0.0559348	0.0279674	+	0.999609i	$0.491097\pi$
$600$	0	0
$601$	−44.8728	−1.83040	−0.915201	−	0.402999i	$-0.867968\pi$
$601$	−44.8728	−1.83040	−0.915201	+	0.402999i	$0.867968\pi$
$602$	36.9881	1.50752
$603$	0	0
$604$	3.45099	0.140419
$605$	2.79878	0.113787
$606$	0	0
$607$	−6.57008	−0.266671	−0.133336	−	0.991071i	$-0.542569\pi$
$607$	−6.57008	−0.266671	−0.133336	+	0.991071i	$0.542569\pi$
$608$	25.5063	1.03442
$609$	0	0
$610$	−2.92388	−0.118384
$611$	23.9825	0.970227
$612$	0	0
$613$	19.6075	0.791941	0.395971	−	0.918263i	$-0.370408\pi$
$613$	19.6075	0.791941	0.395971	+	0.918263i	$0.370408\pi$
$614$	17.6902	0.713919
$615$	0	0
$616$	−13.1533	−0.529960
$617$	−9.44772	−0.380351	−0.190175	−	0.981750i	$-0.560906\pi$
$617$	−9.44772	−0.380351	−0.190175	+	0.981750i	$0.560906\pi$
$618$	0	0
$619$	26.4430	1.06283	0.531416	−	0.847111i	$-0.321660\pi$
$619$	26.4430	1.06283	0.531416	+	0.847111i	$0.321660\pi$
$620$	−2.34295	−0.0940951
$621$	0	0
$622$	4.46155	0.178892
$623$	49.4128	1.97968
$624$	0	0
$625$	−31.1390	−1.24556
$626$	11.0190	0.440409
$627$	0	0
$628$	15.5304	0.619729
$629$	9.60063	0.382802
$630$	0	0
$631$	−14.8228	−0.590088	−0.295044	−	0.955484i	$-0.595334\pi$
$631$	−14.8228	−0.590088	−0.295044	+	0.955484i	$0.595334\pi$
$632$	32.1577	1.27917
$633$	0	0
$634$	1.99327	0.0791627
$635$	−36.8241	−1.46132
$636$	0	0
$637$	28.4063	1.12550
$638$	4.00589	0.158595
$639$	0	0
$640$	−3.91211	−0.154640
$641$	−47.7755	−1.88702	−0.943510	−	0.331345i	$-0.892497\pi$
$641$	−47.7755	−1.88702	−0.943510	+	0.331345i	$0.892497\pi$
$642$	0	0
$643$	−20.5426	−0.810121	−0.405060	−	0.914290i	$-0.632750\pi$
$643$	−20.5426	−0.810121	−0.405060	+	0.914290i	$0.632750\pi$
$644$	17.6723	0.696385
$645$	0	0
$646$	−7.30812	−0.287534
$647$	9.72268	0.382238	0.191119	−	0.981567i	$-0.438788\pi$
$647$	9.72268	0.382238	0.191119	+	0.981567i	$0.438788\pi$
$648$	0	0
$649$	4.30084	0.168823
$650$	7.16303	0.280957
$651$	0	0
$652$	−17.1626	−0.672139
$653$	−4.76982	−0.186658	−0.0933288	−	0.995635i	$-0.529751\pi$
$653$	−4.76982	−0.186658	−0.0933288	+	0.995635i	$0.529751\pi$
$654$	0	0
$655$	−39.4644	−1.54200
$656$	−6.55364	−0.255877
$657$	0	0
$658$	−44.8136	−1.74702
$659$	−34.0252	−1.32543	−0.662716	−	0.748871i	$-0.730596\pi$
$659$	−34.0252	−1.32543	−0.662716	+	0.748871i	$0.730596\pi$
$660$	0	0
$661$	−35.3308	−1.37421	−0.687104	−	0.726559i	$-0.741119\pi$
$661$	−35.3308	−1.37421	−0.687104	+	0.726559i	$0.741119\pi$
$662$	10.2212	0.397260
$663$	0	0
$664$	−28.4232	−1.10304
$665$	−66.3207	−2.57181
$666$	0	0
$667$	−17.2293	−0.667119
$668$	−1.90961	−0.0738849
$669$	0	0
$670$	−30.9224	−1.19464
$671$	1.00000	0.0386046
$672$	0	0
$673$	29.7522	1.14686	0.573432	−	0.819253i	$-0.305612\pi$
$673$	29.7522	1.14686	0.573432	+	0.819253i	$0.305612\pi$
$674$	−14.5243	−0.559457
$675$	0	0
$676$	6.49033	0.249628
$677$	−10.9778	−0.421911	−0.210956	−	0.977496i	$-0.567658\pi$
$677$	−10.9778	−0.421911	−0.210956	+	0.977496i	$0.567658\pi$
$678$	0	0
$679$	70.1816	2.69332
$680$	10.8677	0.416757
$681$	0	0
$682$	−0.962513	−0.0368565
$683$	7.69966	0.294619	0.147310	−	0.989090i	$-0.452939\pi$
$683$	7.69966	0.294619	0.147310	+	0.989090i	$0.452939\pi$
$684$	0	0
$685$	−6.68992	−0.255609
$686$	−21.4246	−0.817996
$687$	0	0
$688$	11.1010	0.423221
$689$	−21.8714	−0.833232
$690$	0	0
$691$	18.9627	0.721376	0.360688	−	0.932687i	$-0.382542\pi$
$691$	18.9627	0.721376	0.360688	+	0.932687i	$0.382542\pi$
$692$	−9.74352	−0.370393
$693$	0	0
$694$	−15.9717	−0.606278
$695$	−3.96990	−0.150587
$696$	0	0
$697$	−6.17062	−0.233729
$698$	27.4453	1.03882
$699$	0	0
$700$	11.1432	0.421173
$701$	12.9447	0.488914	0.244457	−	0.969660i	$-0.421390\pi$
$701$	12.9447	0.488914	0.244457	+	0.969660i	$0.421390\pi$
$702$	0	0
$703$	−41.1272	−1.55114
$704$	−7.58203	−0.285759
$705$	0	0
$706$	1.11859	0.0420989
$707$	3.57540	0.134467
$708$	0	0
$709$	−35.8799	−1.34750	−0.673749	−	0.738961i	$-0.735317\pi$
$709$	−35.8799	−1.34750	−0.673749	+	0.738961i	$0.735317\pi$
$710$	−11.7646	−0.441518
$711$	0	0
$712$	34.6862	1.29992
$713$	4.13975	0.155035
$714$	0	0
$715$	−6.77332	−0.253308
$716$	17.8960	0.668804
$717$	0	0
$718$	−9.84900	−0.367562
$719$	22.0081	0.820764	0.410382	−	0.911914i	$-0.365395\pi$
$719$	22.0081	0.820764	0.410382	+	0.911914i	$0.365395\pi$
$720$	0	0
$721$	−16.5037	−0.614632
$722$	11.4573	0.426397
$723$	0	0
$724$	24.0058	0.892169
$725$	−10.8638	−0.403473
$726$	0	0
$727$	17.5655	0.651467	0.325733	−	0.945462i	$-0.394389\pi$
$727$	17.5655	0.651467	0.325733	+	0.945462i	$0.394389\pi$
$728$	31.8321	1.17978
$729$	0	0
$730$	−28.2972	−1.04733
$731$	10.4522	0.386588
$732$	0	0
$733$	−39.6252	−1.46359	−0.731796	−	0.681524i	$-0.761318\pi$
$733$	−39.6252	−1.46359	−0.731796	+	0.681524i	$0.761318\pi$
$734$	−8.29983	−0.306352
$735$	0	0
$736$	20.9355	0.771693
$737$	10.5758	0.389566
$738$	0	0
$739$	24.2721	0.892865	0.446433	−	0.894817i	$-0.352694\pi$
$739$	24.2721	0.892865	0.446433	+	0.894817i	$0.352694\pi$
$740$	19.1053	0.702324
$741$	0	0
$742$	40.8687	1.50034
$743$	−5.37873	−0.197326	−0.0986632	−	0.995121i	$-0.531457\pi$
$743$	−5.37873	−0.197326	−0.0986632	+	0.995121i	$0.531457\pi$
$744$	0	0
$745$	−15.4886	−0.567457
$746$	13.2577	0.485400
$747$	0	0
$748$	−1.16110	−0.0424541
$749$	−66.7060	−2.43739
$750$	0	0
$751$	32.3243	1.17953	0.589766	−	0.807574i	$-0.299220\pi$
$751$	32.3243	1.17953	0.589766	+	0.807574i	$0.299220\pi$
$752$	−13.4496	−0.490456
$753$	0	0
$754$	−9.69464	−0.353058
$755$	−10.6301	−0.386867
$756$	0	0
$757$	20.8951	0.759446	0.379723	−	0.925100i	$-0.376019\pi$
$757$	20.8951	0.759446	0.379723	+	0.925100i	$0.376019\pi$
$758$	−7.20418	−0.261668
$759$	0	0
$760$	−46.5551	−1.68873
$761$	8.09143	0.293314	0.146657	−	0.989187i	$-0.453149\pi$
$761$	8.09143	0.293314	0.146657	+	0.989187i	$0.453149\pi$
$762$	0	0
$763$	−25.9705	−0.940196
$764$	−3.61758	−0.130879
$765$	0	0
$766$	28.5815	1.03269
$767$	−10.4084	−0.375827
$768$	0	0
$769$	2.73230	0.0985294	0.0492647	−	0.998786i	$-0.484312\pi$
$769$	2.73230	0.0985294	0.0492647	+	0.998786i	$0.484312\pi$
$770$	12.6566	0.456112
$771$	0	0
$772$	−19.2052	−0.691209
$773$	−20.4205	−0.734474	−0.367237	−	0.930127i	$-0.619696\pi$
$773$	−20.4205	−0.734474	−0.367237	+	0.930127i	$0.619696\pi$
$774$	0	0
$775$	2.61030	0.0937648
$776$	49.2653	1.76852
$777$	0	0
$778$	5.60932	0.201104
$779$	26.4337	0.947087
$780$	0	0
$781$	4.02364	0.143977
$782$	−5.99847	−0.214505
$783$	0	0
$784$	−15.9305	−0.568946
$785$	−47.8380	−1.70741
$786$	0	0
$787$	32.7385	1.16700	0.583500	−	0.812113i	$-0.301683\pi$
$787$	32.7385	1.16700	0.583500	+	0.812113i	$0.301683\pi$
$788$	0.110853	0.00394896
$789$	0	0
$790$	−30.9435	−1.10092
$791$	79.9473	2.84260
$792$	0	0
$793$	−2.42009	−0.0859401
$794$	22.9429	0.814213
$795$	0	0
$796$	15.5898	0.552566
$797$	−27.2102	−0.963834	−0.481917	−	0.876217i	$-0.660059\pi$
$797$	−27.2102	−0.963834	−0.481917	+	0.876217i	$0.660059\pi$
$798$	0	0
$799$	−12.6635	−0.448003
$800$	13.2008	0.466719
$801$	0	0
$802$	8.50345	0.300267
$803$	9.67796	0.341528
$804$	0	0
$805$	−54.4358	−1.91861
$806$	2.32937	0.0820487
$807$	0	0
$808$	2.50982	0.0882952
$809$	1.42448	0.0500821	0.0250411	−	0.999686i	$-0.492028\pi$
$809$	1.42448	0.0500821	0.0250411	+	0.999686i	$0.492028\pi$
$810$	0	0
$811$	21.9405	0.770434	0.385217	−	0.922826i	$-0.374127\pi$
$811$	21.9405	0.770434	0.385217	+	0.922826i	$0.374127\pi$
$812$	−15.0815	−0.529257
$813$	0	0
$814$	7.84869	0.275096
$815$	52.8658	1.85181
$816$	0	0
$817$	−44.7751	−1.56648
$818$	−12.8087	−0.447845
$819$	0	0
$820$	−12.2795	−0.428820
$821$	−31.4598	−1.09796	−0.548978	−	0.835837i	$-0.684983\pi$
$821$	−31.4598	−1.09796	−0.548978	+	0.835837i	$0.684983\pi$
$822$	0	0
$823$	−32.3429	−1.12740	−0.563701	−	0.825979i	$-0.690623\pi$
$823$	−32.3429	−1.12740	−0.563701	+	0.825979i	$0.690623\pi$
$824$	−11.5851	−0.403586
$825$	0	0
$826$	19.4492	0.676723
$827$	15.6607	0.544576	0.272288	−	0.962216i	$-0.412220\pi$
$827$	15.6607	0.544576	0.272288	+	0.962216i	$0.412220\pi$
$828$	0	0
$829$	7.37509	0.256147	0.128074	−	0.991765i	$-0.459121\pi$
$829$	7.37509	0.256147	0.128074	+	0.991765i	$0.459121\pi$
$830$	27.3500	0.949332
$831$	0	0
$832$	18.3492	0.636145
$833$	−14.9994	−0.519700
$834$	0	0
$835$	5.88214	0.203560
$836$	4.97393	0.172027
$837$	0	0
$838$	19.7042	0.680669
$839$	−32.8520	−1.13418	−0.567088	−	0.823657i	$-0.691930\pi$
$839$	−32.8520	−1.13418	−0.567088	+	0.823657i	$0.691930\pi$
$840$	0	0
$841$	−14.2966	−0.492986
$842$	3.21513	0.110801
$843$	0	0
$844$	−3.38952	−0.116672
$845$	−19.9921	−0.687749
$846$	0	0
$847$	−4.32870	−0.148736
$848$	12.2656	0.421204
$849$	0	0
$850$	−3.78231	−0.129732
$851$	−33.7570	−1.15718
$852$	0	0
$853$	49.2231	1.68537	0.842684	−	0.538409i	$-0.180975\pi$
$853$	49.2231	1.68537	0.842684	+	0.538409i	$0.180975\pi$
$854$	4.52218	0.154746
$855$	0	0
$856$	−46.8255	−1.60046
$857$	12.0223	0.410672	0.205336	−	0.978691i	$-0.434171\pi$
$857$	12.0223	0.410672	0.205336	+	0.978691i	$0.434171\pi$
$858$	0	0
$859$	46.9504	1.60193	0.800964	−	0.598713i	$-0.204321\pi$
$859$	46.9504	1.60193	0.800964	+	0.598713i	$0.204321\pi$
$860$	20.7999	0.709270
$861$	0	0
$862$	2.25575	0.0768311
$863$	2.14569	0.0730401	0.0365201	−	0.999333i	$-0.488373\pi$
$863$	2.14569	0.0730401	0.0365201	+	0.999333i	$0.488373\pi$
$864$	0	0
$865$	30.0129	1.02047
$866$	−25.3860	−0.862652
$867$	0	0
$868$	3.62370	0.122996
$869$	10.5830	0.359005
$870$	0	0
$871$	−25.5945	−0.867237
$872$	−18.2305	−0.617362
$873$	0	0
$874$	25.6963	0.869189
$875$	26.2512	0.887454
$876$	0	0
$877$	34.8679	1.17741	0.588703	−	0.808349i	$-0.299639\pi$
$877$	34.8679	1.17741	0.588703	+	0.808349i	$0.299639\pi$
$878$	−42.1521	−1.42256
$879$	0	0
$880$	3.79853	0.128049
$881$	8.43752	0.284267	0.142134	−	0.989847i	$-0.454604\pi$
$881$	8.43752	0.284267	0.142134	+	0.989847i	$0.454604\pi$
$882$	0	0
$883$	−29.1483	−0.980920	−0.490460	−	0.871464i	$-0.663171\pi$
$883$	−29.1483	−0.980920	−0.490460	+	0.871464i	$0.663171\pi$
$884$	2.80998	0.0945097
$885$	0	0
$886$	38.4642	1.29223
$887$	7.20367	0.241876	0.120938	−	0.992660i	$-0.461410\pi$
$887$	7.20367	0.241876	0.120938	+	0.992660i	$0.461410\pi$
$888$	0	0
$889$	56.9536	1.91016
$890$	−33.3765	−1.11878
$891$	0	0
$892$	−22.9638	−0.768885
$893$	54.2481	1.81534
$894$	0	0
$895$	−55.1248	−1.84262
$896$	6.05063	0.202137
$897$	0	0
$898$	24.2071	0.807802
$899$	−3.53285	−0.117827
$900$	0	0
$901$	11.5488	0.384746
$902$	−5.04459	−0.167967
$903$	0	0
$904$	56.1205	1.86654
$905$	−73.9449	−2.45801
$906$	0	0
$907$	54.3748	1.80549	0.902743	−	0.430179i	$-0.141550\pi$
$907$	54.3748	1.80549	0.902743	+	0.430179i	$0.141550\pi$
$908$	−1.83363	−0.0608511
$909$	0	0
$910$	−30.6302	−1.01538
$911$	52.3016	1.73283	0.866415	−	0.499325i	$-0.166419\pi$
$911$	52.3016	1.73283	0.866415	+	0.499325i	$0.166419\pi$
$912$	0	0
$913$	−9.35402	−0.309573
$914$	13.3608	0.441936
$915$	0	0
$916$	−15.0763	−0.498134
$917$	61.0371	2.01562
$918$	0	0
$919$	15.3827	0.507428	0.253714	−	0.967279i	$-0.418348\pi$
$919$	15.3827	0.507428	0.253714	+	0.967279i	$0.418348\pi$
$920$	−38.2122	−1.25982
$921$	0	0
$922$	11.3998	0.375433
$923$	−9.73758	−0.320516
$924$	0	0
$925$	−21.2854	−0.699858
$926$	−24.5762	−0.807622
$927$	0	0
$928$	−17.8663	−0.586491
$929$	0.723184	0.0237269	0.0118635	−	0.999930i	$-0.496224\pi$
$929$	0.723184	0.0237269	0.0118635	+	0.999930i	$0.496224\pi$
$930$	0	0
$931$	64.2546	2.10586
$932$	−21.1917	−0.694158
$933$	0	0
$934$	−13.3989	−0.438425
$935$	3.57653	0.116965
$936$	0	0
$937$	−12.3757	−0.404297	−0.202148	−	0.979355i	$-0.564792\pi$
$937$	−12.3757	−0.404297	−0.202148	+	0.979355i	$0.564792\pi$
$938$	47.8258	1.56157
$939$	0	0
$940$	−25.2004	−0.821948
$941$	−29.1545	−0.950408	−0.475204	−	0.879876i	$-0.657626\pi$
$941$	−29.1545	−0.950408	−0.475204	+	0.879876i	$0.657626\pi$
$942$	0	0
$943$	21.6967	0.706541
$944$	5.83713	0.189983
$945$	0	0
$946$	8.54485	0.277817
$947$	−17.9883	−0.584540	−0.292270	−	0.956336i	$-0.594411\pi$
$947$	−17.9883	−0.584540	−0.292270	+	0.956336i	$0.594411\pi$
$948$	0	0
$949$	−23.4216	−0.760297
$950$	16.2027	0.525684
$951$	0	0
$952$	−16.8084	−0.544763
$953$	−47.0445	−1.52392	−0.761960	−	0.647625i	$-0.775763\pi$
$953$	−47.0445	−1.52392	−0.761960	+	0.647625i	$0.775763\pi$
$954$	0	0
$955$	11.1432	0.360586
$956$	−25.3593	−0.820178
$957$	0	0
$958$	−0.846061	−0.0273350
$959$	10.3469	0.334118
$960$	0	0
$961$	−30.1511	−0.972618
$962$	−18.9946	−0.612409
$963$	0	0
$964$	−14.7102	−0.473784
$965$	59.1575	1.90435
$966$	0	0
$967$	12.0189	0.386502	0.193251	−	0.981149i	$-0.438097\pi$
$967$	12.0189	0.386502	0.193251	+	0.981149i	$0.438097\pi$
$968$	−3.03861	−0.0976647
$969$	0	0
$970$	−47.4051	−1.52208
$971$	−30.3964	−0.975467	−0.487733	−	0.872993i	$-0.662176\pi$
$971$	−30.3964	−0.975467	−0.487733	+	0.872993i	$0.662176\pi$
$972$	0	0
$973$	6.14000	0.196839
$974$	10.5192	0.337055
$975$	0	0
$976$	1.35721	0.0434432
$977$	−48.5752	−1.55406	−0.777030	−	0.629464i	$-0.783275\pi$
$977$	−48.5752	−1.55406	−0.777030	+	0.629464i	$0.783275\pi$
$978$	0	0
$979$	11.4151	0.364830
$980$	−29.8489	−0.953488
$981$	0	0
$982$	28.7434	0.917238
$983$	−24.6759	−0.787039	−0.393519	−	0.919316i	$-0.628743\pi$
$983$	−24.6759	−0.787039	−0.393519	+	0.919316i	$0.628743\pi$
$984$	0	0
$985$	−0.341458	−0.0108798
$986$	5.11908	0.163025
$987$	0	0
$988$	−12.0374	−0.382960
$989$	−36.7512	−1.16862
$990$	0	0
$991$	2.12597	0.0675336	0.0337668	−	0.999430i	$-0.489250\pi$
$991$	2.12597	0.0675336	0.0337668	+	0.999430i	$0.489250\pi$
$992$	4.29282	0.136297
$993$	0	0
$994$	18.1956	0.577130
$995$	−48.0212	−1.52237
$996$	0	0
$997$	−18.2192	−0.577008	−0.288504	−	0.957479i	$-0.593158\pi$
$997$	−18.2192	−0.577008	−0.288504	+	0.957479i	$0.593158\pi$
$998$	−25.2791	−0.800195
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.n.1.17	✓	25
3.2	odd	2		6039.2.a.o.1.9	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.17	✓	25	1.1	even	1	trivial
6039.2.a.o.1.9	yes	25	3.2	odd	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)

\(f(q)\)	\(=\)	\(q+1.04470 q^{2} -0.908610 q^{4} +2.79878 q^{5} -4.32870 q^{7} -3.03861 q^{8} +O(q^{10})\)	\(q+1.04470 q^{2} -0.908610 q^{4} +2.79878 q^{5} -4.32870 q^{7} -3.03861 q^{8} +2.92388 q^{10} -1.00000 q^{11} +2.42009 q^{13} -4.52218 q^{14} -1.35721 q^{16} -1.27789 q^{17} +5.47422 q^{19} -2.54300 q^{20} -1.04470 q^{22} +4.49322 q^{23} +2.83318 q^{25} +2.52826 q^{26} +3.93310 q^{28} -3.83450 q^{29} +0.921333 q^{31} +4.65936 q^{32} -1.33501 q^{34} -12.1151 q^{35} -7.51289 q^{37} +5.71890 q^{38} -8.50442 q^{40} +4.82876 q^{41} -8.17926 q^{43} +0.908610 q^{44} +4.69405 q^{46} +9.90973 q^{47} +11.7377 q^{49} +2.95981 q^{50} -2.19892 q^{52} -9.03740 q^{53} -2.79878 q^{55} +13.1533 q^{56} -4.00589 q^{58} -4.30084 q^{59} -1.00000 q^{61} +0.962513 q^{62} +7.58203 q^{64} +6.77332 q^{65} -10.5758 q^{67} +1.16110 q^{68} -12.6566 q^{70} -4.02364 q^{71} -9.67796 q^{73} -7.84869 q^{74} -4.97393 q^{76} +4.32870 q^{77} -10.5830 q^{79} -3.79853 q^{80} +5.04459 q^{82} +9.35402 q^{83} -3.57653 q^{85} -8.54485 q^{86} +3.03861 q^{88} -11.4151 q^{89} -10.4759 q^{91} -4.08258 q^{92} +10.3527 q^{94} +15.3212 q^{95} -16.2131 q^{97} +12.2623 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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