LMFDB - Embedded newform 8037.2.a.v.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.531889 q^{2} -1.71709 q^{4} -3.75672 q^{5} -1.03505 q^{7} -1.97708 q^{8} +O(q^{10})$	\(q+0.531889 q^{2} -1.71709 q^{4} -3.75672 q^{5} -1.03505 q^{7} -1.97708 q^{8} -1.99816 q^{10} -5.91108 q^{11} +2.28429 q^{13} -0.550530 q^{14} +2.38260 q^{16} +1.81931 q^{17} -1.00000 q^{19} +6.45064 q^{20} -3.14404 q^{22} +2.52941 q^{23} +9.11292 q^{25} +1.21499 q^{26} +1.77727 q^{28} -2.51429 q^{29} +1.50271 q^{31} +5.22144 q^{32} +0.967672 q^{34} +3.88838 q^{35} +9.33327 q^{37} -0.531889 q^{38} +7.42733 q^{40} +1.35631 q^{41} -2.52496 q^{43} +10.1499 q^{44} +1.34536 q^{46} -1.00000 q^{47} -5.92868 q^{49} +4.84706 q^{50} -3.92234 q^{52} +0.845805 q^{53} +22.2063 q^{55} +2.04637 q^{56} -1.33732 q^{58} -6.50160 q^{59} +14.0178 q^{61} +0.799274 q^{62} -1.98798 q^{64} -8.58143 q^{65} +13.0841 q^{67} -3.12393 q^{68} +2.06819 q^{70} -4.35549 q^{71} +0.142716 q^{73} +4.96426 q^{74} +1.71709 q^{76} +6.11825 q^{77} +3.24900 q^{79} -8.95076 q^{80} +0.721404 q^{82} -2.83757 q^{83} -6.83464 q^{85} -1.34300 q^{86} +11.6867 q^{88} +14.4382 q^{89} -2.36435 q^{91} -4.34323 q^{92} -0.531889 q^{94} +3.75672 q^{95} -5.49993 q^{97} -3.15340 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q - 5 q^{2} + 31 q^{4} - 6 q^{5} - 15 q^{8}+O(q^{10}) $ 34 * q - 5 * q^2 + 31 * q^4 - 6 * q^5 - 15 * q^8	$ 34 q - 5 q^{2} + 31 q^{4} - 6 q^{5} - 15 q^{8} + 4 q^{10} - 6 q^{11} + 2 q^{13} - 12 q^{14} + 21 q^{16} - 4 q^{17} - 34 q^{19} - 20 q^{20} - 8 q^{22} - 26 q^{23} + 32 q^{25} - 29 q^{26} - 4 q^{28} - 14 q^{29} + 2 q^{31} - 35 q^{32} - 18 q^{34} - 50 q^{35} - 10 q^{37} + 5 q^{38} + 17 q^{40} - 18 q^{41} + 6 q^{43} - 6 q^{44} + 18 q^{46} - 34 q^{47} + 28 q^{49} - 41 q^{50} + 10 q^{52} - 40 q^{53} - 8 q^{55} - 76 q^{56} + 4 q^{58} - 62 q^{59} - 2 q^{61} - 50 q^{62} + 11 q^{64} - 32 q^{65} + 20 q^{67} - 28 q^{68} + 22 q^{70} - 52 q^{71} - 8 q^{73} - 10 q^{74} - 31 q^{76} - 36 q^{77} - 12 q^{79} - 92 q^{80} + 10 q^{82} - 82 q^{83} - 4 q^{85} - 40 q^{86} - 16 q^{88} - 58 q^{89} - 100 q^{92} + 5 q^{94} + 6 q^{95} - 6 q^{97} - 45 q^{98}+O(q^{100}) $ 34 * q - 5 * q^2 + 31 * q^4 - 6 * q^5 - 15 * q^8 + 4 * q^10 - 6 * q^11 + 2 * q^13 - 12 * q^14 + 21 * q^16 - 4 * q^17 - 34 * q^19 - 20 * q^20 - 8 * q^22 - 26 * q^23 + 32 * q^25 - 29 * q^26 - 4 * q^28 - 14 * q^29 + 2 * q^31 - 35 * q^32 - 18 * q^34 - 50 * q^35 - 10 * q^37 + 5 * q^38 + 17 * q^40 - 18 * q^41 + 6 * q^43 - 6 * q^44 + 18 * q^46 - 34 * q^47 + 28 * q^49 - 41 * q^50 + 10 * q^52 - 40 * q^53 - 8 * q^55 - 76 * q^56 + 4 * q^58 - 62 * q^59 - 2 * q^61 - 50 * q^62 + 11 * q^64 - 32 * q^65 + 20 * q^67 - 28 * q^68 + 22 * q^70 - 52 * q^71 - 8 * q^73 - 10 * q^74 - 31 * q^76 - 36 * q^77 - 12 * q^79 - 92 * q^80 + 10 * q^82 - 82 * q^83 - 4 * q^85 - 40 * q^86 - 16 * q^88 - 58 * q^89 - 100 * q^92 + 5 * q^94 + 6 * q^95 - 6 * q^97 - 45 * 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 $

$2$	0.531889	0.376102	0.188051	−	0.982159i	$-0.439783\pi$
$2$	0.531889	0.376102	0.188051	+	0.982159i	$0.439783\pi$
$3$	0	0
$3$	0	0
$4$	−1.71709	−0.858547
$5$	−3.75672	−1.68005	−0.840027	−	0.542544i	$-0.817461\pi$
$5$	−3.75672	−1.68005	−0.840027	+	0.542544i	$0.817461\pi$
$6$	0	0
$7$	−1.03505	−0.391211	−0.195606	−	0.980683i	$-0.562667\pi$
$7$	−1.03505	−0.391211	−0.195606	+	0.980683i	$0.562667\pi$
$8$	−1.97708	−0.699004
$9$	0	0
$10$	−1.99816	−0.631872
$11$	−5.91108	−1.78226	−0.891129	−	0.453750i	$-0.850086\pi$
$11$	−5.91108	−1.78226	−0.891129	+	0.453750i	$0.850086\pi$
$12$	0	0
$13$	2.28429	0.633548	0.316774	−	0.948501i	$-0.397400\pi$
$13$	2.28429	0.633548	0.316774	+	0.948501i	$0.397400\pi$
$14$	−0.550530	−0.147135
$15$	0	0
$16$	2.38260	0.595650
$17$	1.81931	0.441248	0.220624	−	0.975359i	$-0.429191\pi$
$17$	1.81931	0.441248	0.220624	+	0.975359i	$0.429191\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	6.45064	1.44241
$21$	0	0
$22$	−3.14404	−0.670311
$23$	2.52941	0.527418	0.263709	−	0.964602i	$-0.415054\pi$
$23$	2.52941	0.527418	0.263709	+	0.964602i	$0.415054\pi$
$24$	0	0
$25$	9.11292	1.82258
$26$	1.21499	0.238279
$27$	0	0
$28$	1.77727	0.335873
$29$	−2.51429	−0.466892	−0.233446	−	0.972370i	$-0.575000\pi$
$29$	−2.51429	−0.466892	−0.233446	+	0.972370i	$0.575000\pi$
$30$	0	0
$31$	1.50271	0.269894	0.134947	−	0.990853i	$-0.456914\pi$
$31$	1.50271	0.269894	0.134947	+	0.990853i	$0.456914\pi$
$32$	5.22144	0.923029
$33$	0	0
$34$	0.967672	0.165954
$35$	3.88838	0.657256
$36$	0	0
$37$	9.33327	1.53438	0.767190	−	0.641419i	$-0.221654\pi$
$37$	9.33327	1.53438	0.767190	+	0.641419i	$0.221654\pi$
$38$	−0.531889	−0.0862838
$39$	0	0
$40$	7.42733	1.17436
$41$	1.35631	0.211819	0.105910	−	0.994376i	$-0.466225\pi$
$41$	1.35631	0.211819	0.105910	+	0.994376i	$0.466225\pi$
$42$	0	0
$43$	−2.52496	−0.385052	−0.192526	−	0.981292i	$-0.561668\pi$
$43$	−2.52496	−0.385052	−0.192526	+	0.981292i	$0.561668\pi$
$44$	10.1499	1.53015
$45$	0	0
$46$	1.34536	0.198363
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.92868	−0.846954
$50$	4.84706	0.685478
$51$	0	0
$52$	−3.92234	−0.543931
$53$	0.845805	0.116180	0.0580901	−	0.998311i	$-0.481499\pi$
$53$	0.845805	0.116180	0.0580901	+	0.998311i	$0.481499\pi$
$54$	0	0
$55$	22.2063	2.99429
$56$	2.04637	0.273458
$57$	0	0
$58$	−1.33732	−0.175599
$59$	−6.50160	−0.846436	−0.423218	−	0.906028i	$-0.639100\pi$
$59$	−6.50160	−0.846436	−0.423218	+	0.906028i	$0.639100\pi$
$60$	0	0
$61$	14.0178	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$61$	14.0178	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$62$	0.799274	0.101508
$63$	0	0
$64$	−1.98798	−0.248497
$65$	−8.58143	−1.06440
$66$	0	0
$67$	13.0841	1.59848	0.799241	−	0.601011i	$-0.205235\pi$
$67$	13.0841	1.59848	0.799241	+	0.601011i	$0.205235\pi$
$68$	−3.12393	−0.378832
$69$	0	0
$70$	2.06819	0.247195
$71$	−4.35549	−0.516901	−0.258451	−	0.966024i	$-0.583212\pi$
$71$	−4.35549	−0.516901	−0.258451	+	0.966024i	$0.583212\pi$
$72$	0	0
$73$	0.142716	0.0167037	0.00835183	−	0.999965i	$-0.497341\pi$
$73$	0.142716	0.0167037	0.00835183	+	0.999965i	$0.497341\pi$
$74$	4.96426	0.577084
$75$	0	0
$76$	1.71709	0.196964
$77$	6.11825	0.697239
$78$	0	0
$79$	3.24900	0.365541	0.182771	−	0.983156i	$-0.441493\pi$
$79$	3.24900	0.365541	0.182771	+	0.983156i	$0.441493\pi$
$80$	−8.95076	−1.00072
$81$	0	0
$82$	0.721404	0.0796657
$83$	−2.83757	−0.311464	−0.155732	−	0.987799i	$-0.549774\pi$
$83$	−2.83757	−0.311464	−0.155732	+	0.987799i	$0.549774\pi$
$84$	0	0
$85$	−6.83464	−0.741321
$86$	−1.34300	−0.144819
$87$	0	0
$88$	11.6867	1.24581
$89$	14.4382	1.53045	0.765223	−	0.643765i	$-0.222629\pi$
$89$	14.4382	1.53045	0.765223	+	0.643765i	$0.222629\pi$
$90$	0	0
$91$	−2.36435	−0.247851
$92$	−4.34323	−0.452813
$93$	0	0
$94$	−0.531889	−0.0548602
$95$	3.75672	0.385431
$96$	0	0
$97$	−5.49993	−0.558433	−0.279216	−	0.960228i	$-0.590075\pi$
$97$	−5.49993	−0.558433	−0.279216	+	0.960228i	$0.590075\pi$
$98$	−3.15340	−0.318541
$99$	0	0
$100$	−15.6477	−1.56477
$101$	−6.41619	−0.638435	−0.319217	−	0.947682i	$-0.603420\pi$
$101$	−6.41619	−0.638435	−0.319217	+	0.947682i	$0.603420\pi$
$102$	0	0
$103$	−13.0964	−1.29042	−0.645212	−	0.764003i	$-0.723231\pi$
$103$	−13.0964	−1.29042	−0.645212	+	0.764003i	$0.723231\pi$
$104$	−4.51623	−0.442853
$105$	0	0
$106$	0.449874	0.0436957
$107$	−3.10910	−0.300568	−0.150284	−	0.988643i	$-0.548019\pi$
$107$	−3.10910	−0.300568	−0.150284	+	0.988643i	$0.548019\pi$
$108$	0	0
$109$	−12.1770	−1.16635	−0.583175	−	0.812347i	$-0.698190\pi$
$109$	−12.1770	−1.16635	−0.583175	+	0.812347i	$0.698190\pi$
$110$	11.8113	1.12616
$111$	0	0
$112$	−2.46610	−0.233025
$113$	7.90641	0.743773	0.371886	−	0.928278i	$-0.378711\pi$
$113$	7.90641	0.743773	0.371886	+	0.928278i	$0.378711\pi$
$114$	0	0
$115$	−9.50226	−0.886090
$116$	4.31727	0.400848
$117$	0	0
$118$	−3.45813	−0.318346
$119$	−1.88307	−0.172621
$120$	0	0
$121$	23.9409	2.17644
$122$	7.45592	0.675027
$123$	0	0
$124$	−2.58029	−0.231717
$125$	−15.4511	−1.38199
$126$	0	0
$127$	16.0043	1.42015	0.710076	−	0.704125i	$-0.248661\pi$
$127$	16.0043	1.42015	0.710076	+	0.704125i	$0.248661\pi$
$128$	−11.5003	−1.01649
$129$	0	0
$130$	−4.56437	−0.400322
$131$	3.64385	0.318365	0.159182	−	0.987249i	$-0.449114\pi$
$131$	3.64385	0.318365	0.159182	+	0.987249i	$0.449114\pi$
$132$	0	0
$133$	1.03505	0.0897500
$134$	6.95931	0.601193
$135$	0	0
$136$	−3.59693	−0.308434
$137$	8.84983	0.756092	0.378046	−	0.925787i	$-0.376596\pi$
$137$	8.84983	0.756092	0.378046	+	0.925787i	$0.376596\pi$
$138$	0	0
$139$	2.45148	0.207932	0.103966	−	0.994581i	$-0.466847\pi$
$139$	2.45148	0.207932	0.103966	+	0.994581i	$0.466847\pi$
$140$	−6.67671	−0.564285
$141$	0	0
$142$	−2.31663	−0.194408
$143$	−13.5026	−1.12915
$144$	0	0
$145$	9.44547	0.784403
$146$	0.0759092	0.00628229
$147$	0	0
$148$	−16.0261	−1.31734
$149$	6.89729	0.565048	0.282524	−	0.959260i	$-0.408828\pi$
$149$	6.89729	0.565048	0.282524	+	0.959260i	$0.408828\pi$
$150$	0	0
$151$	−12.6044	−1.02574	−0.512868	−	0.858468i	$-0.671417\pi$
$151$	−12.6044	−1.02574	−0.512868	+	0.858468i	$0.671417\pi$
$152$	1.97708	0.160362
$153$	0	0
$154$	3.25423	0.262233
$155$	−5.64525	−0.453437
$156$	0	0
$157$	−12.6384	−1.00866	−0.504329	−	0.863512i	$-0.668260\pi$
$157$	−12.6384	−1.00866	−0.504329	+	0.863512i	$0.668260\pi$
$158$	1.72811	0.137481
$159$	0	0
$160$	−19.6155	−1.55074
$161$	−2.61805	−0.206332
$162$	0	0
$163$	3.33733	0.261400	0.130700	−	0.991422i	$-0.458278\pi$
$163$	3.33733	0.261400	0.130700	+	0.991422i	$0.458278\pi$
$164$	−2.32890	−0.181857
$165$	0	0
$166$	−1.50927	−0.117142
$167$	−10.6790	−0.826370	−0.413185	−	0.910647i	$-0.635584\pi$
$167$	−10.6790	−0.826370	−0.413185	+	0.910647i	$0.635584\pi$
$168$	0	0
$169$	−7.78201	−0.598616
$170$	−3.63527	−0.278813
$171$	0	0
$172$	4.33559	0.330586
$173$	−10.8081	−0.821724	−0.410862	−	0.911698i	$-0.634772\pi$
$173$	−10.8081	−0.821724	−0.410862	+	0.911698i	$0.634772\pi$
$174$	0	0
$175$	−9.43230	−0.713015
$176$	−14.0837	−1.06160
$177$	0	0
$178$	7.67952	0.575604
$179$	10.9866	0.821180	0.410590	−	0.911820i	$-0.365323\pi$
$179$	10.9866	0.821180	0.410590	+	0.911820i	$0.365323\pi$
$180$	0	0
$181$	−10.6276	−0.789943	−0.394972	−	0.918693i	$-0.629246\pi$
$181$	−10.6276	−0.789943	−0.394972	+	0.918693i	$0.629246\pi$
$182$	−1.25757	−0.0932174
$183$	0	0
$184$	−5.00084	−0.368667
$185$	−35.0625	−2.57784
$186$	0	0
$187$	−10.7541	−0.786418
$188$	1.71709	0.125232
$189$	0	0
$190$	1.99816	0.144961
$191$	−16.2963	−1.17916	−0.589581	−	0.807710i	$-0.700707\pi$
$191$	−16.2963	−1.17916	−0.589581	+	0.807710i	$0.700707\pi$
$192$	0	0
$193$	8.96357	0.645212	0.322606	−	0.946533i	$-0.395441\pi$
$193$	8.96357	0.645212	0.322606	+	0.946533i	$0.395441\pi$
$194$	−2.92535	−0.210028
$195$	0	0
$196$	10.1801	0.727150
$197$	1.36339	0.0971377	0.0485688	−	0.998820i	$-0.484534\pi$
$197$	1.36339	0.0971377	0.0485688	+	0.998820i	$0.484534\pi$
$198$	0	0
$199$	1.35887	0.0963275	0.0481637	−	0.998839i	$-0.484663\pi$
$199$	1.35887	0.0963275	0.0481637	+	0.998839i	$0.484663\pi$
$200$	−18.0170	−1.27399
$201$	0	0
$202$	−3.41270	−0.240117
$203$	2.60241	0.182653
$204$	0	0
$205$	−5.09525	−0.355868
$206$	−6.96582	−0.485332
$207$	0	0
$208$	5.44255	0.377373
$209$	5.91108	0.408878
$210$	0	0
$211$	−6.35073	−0.437202	−0.218601	−	0.975814i	$-0.570149\pi$
$211$	−6.35073	−0.437202	−0.218601	+	0.975814i	$0.570149\pi$
$212$	−1.45233	−0.0997463
$213$	0	0
$214$	−1.65370	−0.113044
$215$	9.48555	0.646909
$216$	0	0
$217$	−1.55537	−0.105586
$218$	−6.47684	−0.438667
$219$	0	0
$220$	−38.1302	−2.57074
$221$	4.15584	0.279552
$222$	0	0
$223$	−3.36586	−0.225394	−0.112697	−	0.993629i	$-0.535949\pi$
$223$	−3.36586	−0.225394	−0.112697	+	0.993629i	$0.535949\pi$
$224$	−5.40444	−0.361099
$225$	0	0
$226$	4.20533	0.279735
$227$	−8.30118	−0.550969	−0.275484	−	0.961306i	$-0.588838\pi$
$227$	−8.30118	−0.550969	−0.275484	+	0.961306i	$0.588838\pi$
$228$	0	0
$229$	18.1753	1.20106	0.600528	−	0.799604i	$-0.294957\pi$
$229$	18.1753	1.20106	0.600528	+	0.799604i	$0.294957\pi$
$230$	−5.05415	−0.333261
$231$	0	0
$232$	4.97095	0.326359
$233$	18.8792	1.23682	0.618408	−	0.785858i	$-0.287778\pi$
$233$	18.8792	1.23682	0.618408	+	0.785858i	$0.287778\pi$
$234$	0	0
$235$	3.75672	0.245061
$236$	11.1639	0.726705
$237$	0	0
$238$	−1.00159	−0.0649232
$239$	17.8738	1.15616	0.578081	−	0.815979i	$-0.303802\pi$
$239$	17.8738	1.15616	0.578081	+	0.815979i	$0.303802\pi$
$240$	0	0
$241$	−23.7086	−1.52720	−0.763602	−	0.645687i	$-0.776571\pi$
$241$	−23.7086	−1.52720	−0.763602	+	0.645687i	$0.776571\pi$
$242$	12.7339	0.818565
$243$	0	0
$244$	−24.0699	−1.54092
$245$	22.2724	1.42293
$246$	0	0
$247$	−2.28429	−0.145346
$248$	−2.97098	−0.188657
$249$	0	0
$250$	−8.21825	−0.519768
$251$	−20.0297	−1.26426	−0.632132	−	0.774861i	$-0.717820\pi$
$251$	−20.0297	−1.26426	−0.632132	+	0.774861i	$0.717820\pi$
$252$	0	0
$253$	−14.9515	−0.939994
$254$	8.51251	0.534122
$255$	0	0
$256$	−2.14091	−0.133807
$257$	23.9650	1.49490	0.747448	−	0.664320i	$-0.231279\pi$
$257$	23.9650	1.49490	0.747448	+	0.664320i	$0.231279\pi$
$258$	0	0
$259$	−9.66038	−0.600267
$260$	14.7351	0.913834
$261$	0	0
$262$	1.93812	0.119738
$263$	−24.6889	−1.52239	−0.761193	−	0.648526i	$-0.775386\pi$
$263$	−24.6889	−1.52239	−0.761193	+	0.648526i	$0.775386\pi$
$264$	0	0
$265$	−3.17745	−0.195189
$266$	0.550530	0.0337552
$267$	0	0
$268$	−22.4667	−1.37237
$269$	−30.9756	−1.88862	−0.944308	−	0.329064i	$-0.893267\pi$
$269$	−30.9756	−1.88862	−0.944308	+	0.329064i	$0.893267\pi$
$270$	0	0
$271$	−9.53804	−0.579394	−0.289697	−	0.957118i	$-0.593555\pi$
$271$	−9.53804	−0.579394	−0.289697	+	0.957118i	$0.593555\pi$
$272$	4.33470	0.262830
$273$	0	0
$274$	4.70713	0.284368
$275$	−53.8672	−3.24831
$276$	0	0
$277$	7.01087	0.421242	0.210621	−	0.977568i	$-0.432451\pi$
$277$	7.01087	0.421242	0.210621	+	0.977568i	$0.432451\pi$
$278$	1.30391	0.0782036
$279$	0	0
$280$	−7.68764	−0.459424
$281$	−10.3615	−0.618116	−0.309058	−	0.951043i	$-0.600014\pi$
$281$	−10.3615	−0.618116	−0.309058	+	0.951043i	$0.600014\pi$
$282$	0	0
$283$	12.1656	0.723172	0.361586	−	0.932339i	$-0.382235\pi$
$283$	12.1656	0.723172	0.361586	+	0.932339i	$0.382235\pi$
$284$	7.47878	0.443784
$285$	0	0
$286$	−7.18190	−0.424675
$287$	−1.40384	−0.0828660
$288$	0	0
$289$	−13.6901	−0.805300
$290$	5.02394	0.295016
$291$	0	0
$292$	−0.245057	−0.0143409
$293$	12.2434	0.715267	0.357634	−	0.933862i	$-0.383584\pi$
$293$	12.2434	0.715267	0.357634	+	0.933862i	$0.383584\pi$
$294$	0	0
$295$	24.4247	1.42206
$296$	−18.4526	−1.07254
$297$	0	0
$298$	3.66859	0.212516
$299$	5.77790	0.334145
$300$	0	0
$301$	2.61345	0.150637
$302$	−6.70417	−0.385781
$303$	0	0
$304$	−2.38260	−0.136652
$305$	−52.6609	−3.01536
$306$	0	0
$307$	6.95565	0.396980	0.198490	−	0.980103i	$-0.436396\pi$
$307$	6.95565	0.396980	0.198490	+	0.980103i	$0.436396\pi$
$308$	−10.5056	−0.598613
$309$	0	0
$310$	−3.00265	−0.170539
$311$	16.8261	0.954123	0.477061	−	0.878870i	$-0.341702\pi$
$311$	16.8261	0.954123	0.477061	+	0.878870i	$0.341702\pi$
$312$	0	0
$313$	4.10369	0.231954	0.115977	−	0.993252i	$-0.463000\pi$
$313$	4.10369	0.231954	0.115977	+	0.993252i	$0.463000\pi$
$314$	−6.72225	−0.379358
$315$	0	0
$316$	−5.57884	−0.313834
$317$	−4.48281	−0.251780	−0.125890	−	0.992044i	$-0.540179\pi$
$317$	−4.48281	−0.251780	−0.125890	+	0.992044i	$0.540179\pi$
$318$	0	0
$319$	14.8622	0.832121
$320$	7.46826	0.417488
$321$	0	0
$322$	−1.39251	−0.0776018
$323$	−1.81931	−0.101229
$324$	0	0
$325$	20.8166	1.15469
$326$	1.77509	0.0983130
$327$	0	0
$328$	−2.68153	−0.148062
$329$	1.03505	0.0570640
$330$	0	0
$331$	−12.4708	−0.685455	−0.342728	−	0.939435i	$-0.611351\pi$
$331$	−12.4708	−0.685455	−0.342728	+	0.939435i	$0.611351\pi$
$332$	4.87238	0.267407
$333$	0	0
$334$	−5.68007	−0.310799
$335$	−49.1534	−2.68554
$336$	0	0
$337$	10.4696	0.570315	0.285157	−	0.958481i	$-0.407954\pi$
$337$	10.4696	0.570315	0.285157	+	0.958481i	$0.407954\pi$
$338$	−4.13917	−0.225141
$339$	0	0
$340$	11.7357	0.636459
$341$	−8.88263	−0.481022
$342$	0	0
$343$	13.3818	0.722549
$344$	4.99204	0.269153
$345$	0	0
$346$	−5.74870	−0.309052
$347$	−10.1248	−0.543530	−0.271765	−	0.962364i	$-0.587607\pi$
$347$	−10.1248	−0.543530	−0.271765	+	0.962364i	$0.587607\pi$
$348$	0	0
$349$	−11.7417	−0.628519	−0.314259	−	0.949337i	$-0.601756\pi$
$349$	−11.7417	−0.628519	−0.314259	+	0.949337i	$0.601756\pi$
$350$	−5.01694	−0.268166
$351$	0	0
$352$	−30.8644	−1.64508
$353$	25.9738	1.38245	0.691224	−	0.722641i	$-0.257072\pi$
$353$	25.9738	1.38245	0.691224	+	0.722641i	$0.257072\pi$
$354$	0	0
$355$	16.3623	0.868422
$356$	−24.7917	−1.31396
$357$	0	0
$358$	5.84367	0.308848
$359$	−18.3419	−0.968048	−0.484024	−	0.875055i	$-0.660825\pi$
$359$	−18.3419	−0.968048	−0.484024	+	0.875055i	$0.660825\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.65270	−0.297099
$363$	0	0
$364$	4.05981	0.212792
$365$	−0.536144	−0.0280631
$366$	0	0
$367$	27.9476	1.45885	0.729427	−	0.684059i	$-0.239787\pi$
$367$	27.9476	1.45885	0.729427	+	0.684059i	$0.239787\pi$
$368$	6.02656	0.314156
$369$	0	0
$370$	−18.6493	−0.969533
$371$	−0.875448	−0.0454510
$372$	0	0
$373$	7.33393	0.379737	0.189868	−	0.981810i	$-0.439194\pi$
$373$	7.33393	0.379737	0.189868	+	0.981810i	$0.439194\pi$
$374$	−5.71999	−0.295774
$375$	0	0
$376$	1.97708	0.101960
$377$	−5.74337	−0.295798
$378$	0	0
$379$	−2.87140	−0.147494	−0.0737469	−	0.997277i	$-0.523496\pi$
$379$	−2.87140	−0.147494	−0.0737469	+	0.997277i	$0.523496\pi$
$380$	−6.45064	−0.330911
$381$	0	0
$382$	−8.66784	−0.443485
$383$	11.1037	0.567373	0.283687	−	0.958917i	$-0.408442\pi$
$383$	11.1037	0.567373	0.283687	+	0.958917i	$0.408442\pi$
$384$	0	0
$385$	−22.9845	−1.17140
$386$	4.76762	0.242666
$387$	0	0
$388$	9.44389	0.479441
$389$	10.5622	0.535524	0.267762	−	0.963485i	$-0.413716\pi$
$389$	10.5622	0.535524	0.267762	+	0.963485i	$0.413716\pi$
$390$	0	0
$391$	4.60178	0.232722
$392$	11.7215	0.592024
$393$	0	0
$394$	0.725173	0.0365337
$395$	−12.2056	−0.614129
$396$	0	0
$397$	24.9823	1.25383	0.626914	−	0.779088i	$-0.284318\pi$
$397$	24.9823	1.25383	0.626914	+	0.779088i	$0.284318\pi$
$398$	0.722766	0.0362290
$399$	0	0
$400$	21.7124	1.08562
$401$	13.0543	0.651903	0.325951	−	0.945387i	$-0.394315\pi$
$401$	13.0543	0.651903	0.325951	+	0.945387i	$0.394315\pi$
$402$	0	0
$403$	3.43262	0.170991
$404$	11.0172	0.548126
$405$	0	0
$406$	1.38419	0.0686963
$407$	−55.1697	−2.73466
$408$	0	0
$409$	26.4509	1.30791	0.653956	−	0.756533i	$-0.273108\pi$
$409$	26.4509	1.30791	0.653956	+	0.756533i	$0.273108\pi$
$410$	−2.71011	−0.133843
$411$	0	0
$412$	22.4877	1.10789
$413$	6.72946	0.331135
$414$	0	0
$415$	10.6600	0.523277
$416$	11.9273	0.584784
$417$	0	0
$418$	3.14404	0.153780
$419$	−0.198058	−0.00967578	−0.00483789	−	0.999988i	$-0.501540\pi$
$419$	−0.198058	−0.00967578	−0.00483789	+	0.999988i	$0.501540\pi$
$420$	0	0
$421$	22.6584	1.10430	0.552150	−	0.833745i	$-0.313807\pi$
$421$	22.6584	1.10430	0.552150	+	0.833745i	$0.313807\pi$
$422$	−3.37788	−0.164433
$423$	0	0
$424$	−1.67223	−0.0812105
$425$	16.5792	0.804212
$426$	0	0
$427$	−14.5091	−0.702145
$428$	5.33862	0.258052
$429$	0	0
$430$	5.04526	0.243304
$431$	−9.05454	−0.436142	−0.218071	−	0.975933i	$-0.569976\pi$
$431$	−9.05454	−0.436142	−0.218071	+	0.975933i	$0.569976\pi$
$432$	0	0
$433$	0.614648	0.0295381	0.0147691	−	0.999891i	$-0.495299\pi$
$433$	0.614648	0.0295381	0.0147691	+	0.999891i	$0.495299\pi$
$434$	−0.827287	−0.0397110
$435$	0	0
$436$	20.9091	1.00137
$437$	−2.52941	−0.120998
$438$	0	0
$439$	−32.2795	−1.54061	−0.770307	−	0.637673i	$-0.779897\pi$
$439$	−32.2795	−1.54061	−0.770307	+	0.637673i	$0.779897\pi$
$440$	−43.9036	−2.09302
$441$	0	0
$442$	2.21045	0.105140
$443$	−32.1515	−1.52757	−0.763783	−	0.645473i	$-0.776660\pi$
$443$	−32.1515	−1.52757	−0.763783	+	0.645473i	$0.776660\pi$
$444$	0	0
$445$	−54.2402	−2.57123
$446$	−1.79026	−0.0847713
$447$	0	0
$448$	2.05765	0.0972147
$449$	−23.3960	−1.10413	−0.552064	−	0.833802i	$-0.686159\pi$
$449$	−23.3960	−1.10413	−0.552064	+	0.833802i	$0.686159\pi$
$450$	0	0
$451$	−8.01723	−0.377517
$452$	−13.5761	−0.638564
$453$	0	0
$454$	−4.41530	−0.207221
$455$	8.88219	0.416403
$456$	0	0
$457$	−29.8893	−1.39816	−0.699081	−	0.715042i	$-0.746408\pi$
$457$	−29.8893	−1.39816	−0.699081	+	0.715042i	$0.746408\pi$
$458$	9.66722	0.451720
$459$	0	0
$460$	16.3163	0.760750
$461$	25.9941	1.21067	0.605334	−	0.795972i	$-0.293040\pi$
$461$	25.9941	1.21067	0.605334	+	0.795972i	$0.293040\pi$
$462$	0	0
$463$	31.9275	1.48380	0.741898	−	0.670513i	$-0.233926\pi$
$463$	31.9275	1.48380	0.741898	+	0.670513i	$0.233926\pi$
$464$	−5.99055	−0.278104
$465$	0	0
$466$	10.0416	0.465169
$467$	−15.2410	−0.705271	−0.352636	−	0.935761i	$-0.614715\pi$
$467$	−15.2410	−0.705271	−0.352636	+	0.935761i	$0.614715\pi$
$468$	0	0
$469$	−13.5427	−0.625344
$470$	1.99816	0.0921680
$471$	0	0
$472$	12.8542	0.591662
$473$	14.9252	0.686263
$474$	0	0
$475$	−9.11292	−0.418129
$476$	3.23342	0.148203
$477$	0	0
$478$	9.50690	0.434835
$479$	11.9701	0.546929	0.273465	−	0.961882i	$-0.411830\pi$
$479$	11.9701	0.546929	0.273465	+	0.961882i	$0.411830\pi$
$480$	0	0
$481$	21.3199	0.972105
$482$	−12.6103	−0.574385
$483$	0	0
$484$	−41.1087	−1.86858
$485$	20.6617	0.938198
$486$	0	0
$487$	7.81848	0.354289	0.177145	−	0.984185i	$-0.443314\pi$
$487$	7.81848	0.354289	0.177145	+	0.984185i	$0.443314\pi$
$488$	−27.7144	−1.25457
$489$	0	0
$490$	11.8464	0.535167
$491$	−30.1202	−1.35931	−0.679653	−	0.733534i	$-0.737869\pi$
$491$	−30.1202	−1.35931	−0.679653	+	0.733534i	$0.737869\pi$
$492$	0	0
$493$	−4.57428	−0.206015
$494$	−1.21499	−0.0546649
$495$	0	0
$496$	3.58036	0.160763
$497$	4.50813	0.202217
$498$	0	0
$499$	−31.3526	−1.40354	−0.701768	−	0.712406i	$-0.747606\pi$
$499$	−31.3526	−1.40354	−0.701768	+	0.712406i	$0.747606\pi$
$500$	26.5309	1.18650
$501$	0	0
$502$	−10.6536	−0.475492
$503$	24.3284	1.08475	0.542376	−	0.840136i	$-0.317525\pi$
$503$	24.3284	1.08475	0.542376	+	0.840136i	$0.317525\pi$
$504$	0	0
$505$	24.1038	1.07260
$506$	−7.95255	−0.353534
$507$	0	0
$508$	−27.4809	−1.21927
$509$	−27.8740	−1.23549	−0.617747	−	0.786377i	$-0.711954\pi$
$509$	−27.8740	−1.23549	−0.617747	+	0.786377i	$0.711954\pi$
$510$	0	0
$511$	−0.147718	−0.00653466
$512$	21.8618	0.966164
$513$	0	0
$514$	12.7467	0.562234
$515$	49.1994	2.16798
$516$	0	0
$517$	5.91108	0.259969
$518$	−5.13825	−0.225762
$519$	0	0
$520$	16.9662	0.744017
$521$	−2.95089	−0.129281	−0.0646403	−	0.997909i	$-0.520590\pi$
$521$	−2.95089	−0.129281	−0.0646403	+	0.997909i	$0.520590\pi$
$522$	0	0
$523$	−8.16227	−0.356911	−0.178456	−	0.983948i	$-0.557110\pi$
$523$	−8.16227	−0.356911	−0.178456	+	0.983948i	$0.557110\pi$
$524$	−6.25684	−0.273331
$525$	0	0
$526$	−13.1318	−0.572572
$527$	2.73390	0.119090
$528$	0	0
$529$	−16.6021	−0.721831
$530$	−1.69005	−0.0734111
$531$	0	0
$532$	−1.77727	−0.0770546
$533$	3.09820	0.134198
$534$	0	0
$535$	11.6800	0.504971
$536$	−25.8684	−1.11734
$537$	0	0
$538$	−16.4756	−0.710312
$539$	35.0449	1.50949
$540$	0	0
$541$	10.6967	0.459887	0.229943	−	0.973204i	$-0.426146\pi$
$541$	10.6967	0.459887	0.229943	+	0.973204i	$0.426146\pi$
$542$	−5.07318	−0.217912
$543$	0	0
$544$	9.49944	0.407285
$545$	45.7457	1.95953
$546$	0	0
$547$	36.1432	1.54537	0.772687	−	0.634788i	$-0.218912\pi$
$547$	36.1432	1.54537	0.772687	+	0.634788i	$0.218912\pi$
$548$	−15.1960	−0.649141
$549$	0	0
$550$	−28.6514	−1.22170
$551$	2.51429	0.107112
$552$	0	0
$553$	−3.36287	−0.143004
$554$	3.72900	0.158430
$555$	0	0
$556$	−4.20942	−0.178519
$557$	−4.06910	−0.172413	−0.0862067	−	0.996277i	$-0.527475\pi$
$557$	−4.06910	−0.172413	−0.0862067	+	0.996277i	$0.527475\pi$
$558$	0	0
$559$	−5.76774	−0.243949
$560$	9.26445	0.391495
$561$	0	0
$562$	−5.51117	−0.232475
$563$	−22.9162	−0.965805	−0.482902	−	0.875674i	$-0.660417\pi$
$563$	−22.9162	−0.965805	−0.482902	+	0.875674i	$0.660417\pi$
$564$	0	0
$565$	−29.7022	−1.24958
$566$	6.47076	0.271986
$567$	0	0
$568$	8.61115	0.361316
$569$	−8.23045	−0.345039	−0.172519	−	0.985006i	$-0.555191\pi$
$569$	−8.23045	−0.345039	−0.172519	+	0.985006i	$0.555191\pi$
$570$	0	0
$571$	−47.1652	−1.97380	−0.986901	−	0.161325i	$-0.948423\pi$
$571$	−47.1652	−1.97380	−0.986901	+	0.161325i	$0.948423\pi$
$572$	23.1853	0.969426
$573$	0	0
$574$	−0.746687	−0.0311661
$575$	23.0503	0.961262
$576$	0	0
$577$	38.0320	1.58329	0.791647	−	0.610979i	$-0.209224\pi$
$577$	38.0320	1.58329	0.791647	+	0.610979i	$0.209224\pi$
$578$	−7.28161	−0.302875
$579$	0	0
$580$	−16.2188	−0.673447
$581$	2.93702	0.121848
$582$	0	0
$583$	−4.99962	−0.207063
$584$	−0.282162	−0.0116759
$585$	0	0
$586$	6.51213	0.269014
$587$	18.6013	0.767757	0.383879	−	0.923384i	$-0.374588\pi$
$587$	18.6013	0.767757	0.383879	+	0.923384i	$0.374588\pi$
$588$	0	0
$589$	−1.50271	−0.0619180
$590$	12.9912	0.534839
$591$	0	0
$592$	22.2375	0.913954
$593$	44.7621	1.83816	0.919079	−	0.394074i	$-0.128935\pi$
$593$	44.7621	1.83816	0.919079	+	0.394074i	$0.128935\pi$
$594$	0	0
$595$	7.07418	0.290013
$596$	−11.8433	−0.485120
$597$	0	0
$598$	3.07320	0.125673
$599$	−23.5118	−0.960667	−0.480333	−	0.877086i	$-0.659484\pi$
$599$	−23.5118	−0.960667	−0.480333	+	0.877086i	$0.659484\pi$
$600$	0	0
$601$	−1.26831	−0.0517356	−0.0258678	−	0.999665i	$-0.508235\pi$
$601$	−1.26831	−0.0517356	−0.0258678	+	0.999665i	$0.508235\pi$
$602$	1.39006	0.0566548
$603$	0	0
$604$	21.6430	0.880642
$605$	−89.9391	−3.65654
$606$	0	0
$607$	23.3516	0.947814	0.473907	−	0.880575i	$-0.342843\pi$
$607$	23.3516	0.947814	0.473907	+	0.880575i	$0.342843\pi$
$608$	−5.22144	−0.211757
$609$	0	0
$610$	−28.0098	−1.13408
$611$	−2.28429	−0.0924125
$612$	0	0
$613$	−21.2993	−0.860271	−0.430135	−	0.902764i	$-0.641534\pi$
$613$	−21.2993	−0.860271	−0.430135	+	0.902764i	$0.641534\pi$
$614$	3.69964	0.149305
$615$	0	0
$616$	−12.0963	−0.487373
$617$	10.7686	0.433529	0.216765	−	0.976224i	$-0.430450\pi$
$617$	10.7686	0.433529	0.216765	+	0.976224i	$0.430450\pi$
$618$	0	0
$619$	−15.0177	−0.603612	−0.301806	−	0.953369i	$-0.597590\pi$
$619$	−15.0177	−0.603612	−0.301806	+	0.953369i	$0.597590\pi$
$620$	9.69343	0.389297
$621$	0	0
$622$	8.94964	0.358848
$623$	−14.9442	−0.598727
$624$	0	0
$625$	12.4807	0.499227
$626$	2.18270	0.0872384
$627$	0	0
$628$	21.7014	0.865980
$629$	16.9801	0.677043
$630$	0	0
$631$	−8.51835	−0.339110	−0.169555	−	0.985521i	$-0.554233\pi$
$631$	−8.51835	−0.339110	−0.169555	+	0.985521i	$0.554233\pi$
$632$	−6.42354	−0.255515
$633$	0	0
$634$	−2.38436	−0.0946949
$635$	−60.1236	−2.38593
$636$	0	0
$637$	−13.5428	−0.536586
$638$	7.90502	0.312963
$639$	0	0
$640$	43.2032	1.70776
$641$	20.4297	0.806925	0.403463	−	0.914996i	$-0.367807\pi$
$641$	20.4297	0.806925	0.403463	+	0.914996i	$0.367807\pi$
$642$	0	0
$643$	5.98901	0.236183	0.118092	−	0.993003i	$-0.462322\pi$
$643$	5.98901	0.236183	0.118092	+	0.993003i	$0.462322\pi$
$644$	4.49545	0.177145
$645$	0	0
$646$	−0.967672	−0.0380726
$647$	−41.2701	−1.62249	−0.811247	−	0.584703i	$-0.801211\pi$
$647$	−41.2701	−1.62249	−0.811247	+	0.584703i	$0.801211\pi$
$648$	0	0
$649$	38.4315	1.50857
$650$	11.0721	0.434283
$651$	0	0
$652$	−5.73051	−0.224424
$653$	−35.3968	−1.38518	−0.692591	−	0.721330i	$-0.743531\pi$
$653$	−35.3968	−1.38518	−0.692591	+	0.721330i	$0.743531\pi$
$654$	0	0
$655$	−13.6889	−0.534870
$656$	3.23153	0.126170
$657$	0	0
$658$	0.550530	0.0214619
$659$	−14.7699	−0.575353	−0.287677	−	0.957728i	$-0.592883\pi$
$659$	−14.7699	−0.575353	−0.287677	+	0.957728i	$0.592883\pi$
$660$	0	0
$661$	14.4742	0.562981	0.281490	−	0.959564i	$-0.409171\pi$
$661$	14.4742	0.562981	0.281490	+	0.959564i	$0.409171\pi$
$662$	−6.63306	−0.257801
$663$	0	0
$664$	5.61012	0.217715
$665$	−3.88838	−0.150785
$666$	0	0
$667$	−6.35965	−0.246247
$668$	18.3369	0.709477
$669$	0	0
$670$	−26.1441	−1.01004
$671$	−82.8604	−3.19879
$672$	0	0
$673$	2.28788	0.0881913	0.0440956	−	0.999027i	$-0.485959\pi$
$673$	2.28788	0.0881913	0.0440956	+	0.999027i	$0.485959\pi$
$674$	5.56866	0.214497
$675$	0	0
$676$	13.3624	0.513940
$677$	−23.0928	−0.887528	−0.443764	−	0.896144i	$-0.646357\pi$
$677$	−23.0928	−0.887528	−0.443764	+	0.896144i	$0.646357\pi$
$678$	0	0
$679$	5.69268	0.218465
$680$	13.5126	0.518186
$681$	0	0
$682$	−4.72457	−0.180913
$683$	36.2026	1.38525	0.692627	−	0.721296i	$-0.256453\pi$
$683$	36.2026	1.38525	0.692627	+	0.721296i	$0.256453\pi$
$684$	0	0
$685$	−33.2463	−1.27028
$686$	7.11763	0.271752
$687$	0	0
$688$	−6.01596	−0.229357
$689$	1.93207	0.0736058
$690$	0	0
$691$	−4.93955	−0.187909	−0.0939546	−	0.995576i	$-0.529951\pi$
$691$	−4.93955	−0.187909	−0.0939546	+	0.995576i	$0.529951\pi$
$692$	18.5585	0.705489
$693$	0	0
$694$	−5.38529	−0.204423
$695$	−9.20951	−0.349337
$696$	0	0
$697$	2.46754	0.0934649
$698$	−6.24528	−0.236387
$699$	0	0
$700$	16.1961	0.612157
$701$	−2.36233	−0.0892240	−0.0446120	−	0.999004i	$-0.514205\pi$
$701$	−2.36233	−0.0892240	−0.0446120	+	0.999004i	$0.514205\pi$
$702$	0	0
$703$	−9.33327	−0.352011
$704$	11.7511	0.442886
$705$	0	0
$706$	13.8152	0.519942
$707$	6.64106	0.249763
$708$	0	0
$709$	−5.78508	−0.217263	−0.108632	−	0.994082i	$-0.534647\pi$
$709$	−5.78508	−0.217263	−0.108632	+	0.994082i	$0.534647\pi$
$710$	8.70294	0.326615
$711$	0	0
$712$	−28.5455	−1.06979
$713$	3.80096	0.142347
$714$	0	0
$715$	50.7256	1.89703
$716$	−18.8651	−0.705022
$717$	0	0
$718$	−9.75585	−0.364085
$719$	36.4124	1.35795	0.678977	−	0.734159i	$-0.262423\pi$
$719$	36.4124	1.35795	0.678977	+	0.734159i	$0.262423\pi$
$720$	0	0
$721$	13.5554	0.504828
$722$	0.531889	0.0197949
$723$	0	0
$724$	18.2486	0.678203
$725$	−22.9125	−0.850949
$726$	0	0
$727$	−45.1027	−1.67277	−0.836383	−	0.548146i	$-0.815334\pi$
$727$	−45.1027	−1.67277	−0.836383	+	0.548146i	$0.815334\pi$
$728$	4.67451	0.173249
$729$	0	0
$730$	−0.285169	−0.0105546
$731$	−4.59369	−0.169904
$732$	0	0
$733$	−0.952364	−0.0351764	−0.0175882	−	0.999845i	$-0.505599\pi$
$733$	−0.952364	−0.0351764	−0.0175882	+	0.999845i	$0.505599\pi$
$734$	14.8650	0.548678
$735$	0	0
$736$	13.2071	0.486822
$737$	−77.3414	−2.84891
$738$	0	0
$739$	−30.7706	−1.13191	−0.565957	−	0.824435i	$-0.691493\pi$
$739$	−30.7706	−1.13191	−0.565957	+	0.824435i	$0.691493\pi$
$740$	60.2056	2.21320
$741$	0	0
$742$	−0.465641	−0.0170942
$743$	−21.5799	−0.791689	−0.395845	−	0.918318i	$-0.629548\pi$
$743$	−21.5799	−0.791689	−0.395845	+	0.918318i	$0.629548\pi$
$744$	0	0
$745$	−25.9111	−0.949311
$746$	3.90084	0.142820
$747$	0	0
$748$	18.4658	0.675177
$749$	3.21807	0.117586
$750$	0	0
$751$	−6.88939	−0.251397	−0.125699	−	0.992068i	$-0.540117\pi$
$751$	−6.88939	−0.251397	−0.125699	+	0.992068i	$0.540117\pi$
$752$	−2.38260	−0.0868845
$753$	0	0
$754$	−3.05483	−0.111250
$755$	47.3513	1.72329
$756$	0	0
$757$	−31.0828	−1.12972	−0.564861	−	0.825186i	$-0.691070\pi$
$757$	−31.0828	−1.12972	−0.564861	+	0.825186i	$0.691070\pi$
$758$	−1.52726	−0.0554728
$759$	0	0
$760$	−7.42733	−0.269418
$761$	−47.3064	−1.71486	−0.857428	−	0.514604i	$-0.827939\pi$
$761$	−47.3064	−1.71486	−0.857428	+	0.514604i	$0.827939\pi$
$762$	0	0
$763$	12.6038	0.456289
$764$	27.9823	1.01237
$765$	0	0
$766$	5.90594	0.213390
$767$	−14.8515	−0.536258
$768$	0	0
$769$	27.6720	0.997880	0.498940	−	0.866637i	$-0.333723\pi$
$769$	27.6720	0.997880	0.498940	+	0.866637i	$0.333723\pi$
$770$	−12.2252	−0.440566
$771$	0	0
$772$	−15.3913	−0.553945
$773$	−8.28223	−0.297891	−0.148945	−	0.988845i	$-0.547588\pi$
$773$	−8.28223	−0.297891	−0.148945	+	0.988845i	$0.547588\pi$
$774$	0	0
$775$	13.6941	0.491905
$776$	10.8738	0.390347
$777$	0	0
$778$	5.61791	0.201412
$779$	−1.35631	−0.0485947
$780$	0	0
$781$	25.7456	0.921251
$782$	2.44764	0.0875273
$783$	0	0
$784$	−14.1257	−0.504488
$785$	47.4790	1.69460
$786$	0	0
$787$	36.9609	1.31751	0.658757	−	0.752356i	$-0.271083\pi$
$787$	36.9609	1.31751	0.658757	+	0.752356i	$0.271083\pi$
$788$	−2.34107	−0.0833973
$789$	0	0
$790$	−6.49201	−0.230975
$791$	−8.18351	−0.290972
$792$	0	0
$793$	32.0208	1.13709
$794$	13.2878	0.471568
$795$	0	0
$796$	−2.33330	−0.0827017
$797$	−27.1896	−0.963107	−0.481553	−	0.876417i	$-0.659927\pi$
$797$	−27.1896	−0.963107	−0.481553	+	0.876417i	$0.659927\pi$
$798$	0	0
$799$	−1.81931	−0.0643627
$800$	47.5826	1.68230
$801$	0	0
$802$	6.94346	0.245182
$803$	−0.843607	−0.0297702
$804$	0	0
$805$	9.83529	0.346648
$806$	1.82578	0.0643102
$807$	0	0
$808$	12.6853	0.446268
$809$	10.4837	0.368589	0.184294	−	0.982871i	$-0.441000\pi$
$809$	10.4837	0.368589	0.184294	+	0.982871i	$0.441000\pi$
$810$	0	0
$811$	13.4273	0.471497	0.235749	−	0.971814i	$-0.424246\pi$
$811$	13.4273	0.471497	0.235749	+	0.971814i	$0.424246\pi$
$812$	−4.46858	−0.156816
$813$	0	0
$814$	−29.3442	−1.02851
$815$	−12.5374	−0.439166
$816$	0	0
$817$	2.52496	0.0883371
$818$	14.0689	0.491909
$819$	0	0
$820$	8.74903	0.305529
$821$	35.7950	1.24925	0.624626	−	0.780924i	$-0.285251\pi$
$821$	35.7950	1.24925	0.624626	+	0.780924i	$0.285251\pi$
$822$	0	0
$823$	37.4130	1.30414	0.652068	−	0.758160i	$-0.273901\pi$
$823$	37.4130	1.30414	0.652068	+	0.758160i	$0.273901\pi$
$824$	25.8926	0.902012
$825$	0	0
$826$	3.57933	0.124541
$827$	26.2285	0.912055	0.456028	−	0.889966i	$-0.349272\pi$
$827$	26.2285	0.912055	0.456028	+	0.889966i	$0.349272\pi$
$828$	0	0
$829$	25.0339	0.869462	0.434731	−	0.900560i	$-0.356844\pi$
$829$	25.0339	0.869462	0.434731	+	0.900560i	$0.356844\pi$
$830$	5.66992	0.196806
$831$	0	0
$832$	−4.54111	−0.157435
$833$	−10.7861	−0.373717
$834$	0	0
$835$	40.1182	1.38835
$836$	−10.1499	−0.351041
$837$	0	0
$838$	−0.105345	−0.00363908
$839$	−2.02465	−0.0698986	−0.0349493	−	0.999389i	$-0.511127\pi$
$839$	−2.02465	−0.0698986	−0.0349493	+	0.999389i	$0.511127\pi$
$840$	0	0
$841$	−22.6784	−0.782012
$842$	12.0517	0.415330
$843$	0	0
$844$	10.9048	0.375359
$845$	29.2348	1.00571
$846$	0	0
$847$	−24.7799	−0.851449
$848$	2.01522	0.0692028
$849$	0	0
$850$	8.81832	0.302466
$851$	23.6076	0.809259
$852$	0	0
$853$	−20.9796	−0.718329	−0.359164	−	0.933274i	$-0.616938\pi$
$853$	−20.9796	−0.718329	−0.359164	+	0.933274i	$0.616938\pi$
$854$	−7.71723	−0.264078
$855$	0	0
$856$	6.14695	0.210098
$857$	−37.9048	−1.29480	−0.647401	−	0.762149i	$-0.724144\pi$
$857$	−37.9048	−1.29480	−0.647401	+	0.762149i	$0.724144\pi$
$858$	0	0
$859$	11.0094	0.375634	0.187817	−	0.982204i	$-0.439859\pi$
$859$	11.0094	0.375634	0.187817	+	0.982204i	$0.439859\pi$
$860$	−16.2876	−0.555402
$861$	0	0
$862$	−4.81601	−0.164034
$863$	10.9203	0.371732	0.185866	−	0.982575i	$-0.440491\pi$
$863$	10.9203	0.371732	0.185866	+	0.982575i	$0.440491\pi$
$864$	0	0
$865$	40.6029	1.38054
$866$	0.326925	0.0111094
$867$	0	0
$868$	2.67072	0.0906503
$869$	−19.2051	−0.651489
$870$	0	0
$871$	29.8880	1.01272
$872$	24.0750	0.815283
$873$	0	0
$874$	−1.34536	−0.0455076
$875$	15.9926	0.540648
$876$	0	0
$877$	38.1486	1.28819	0.644094	−	0.764946i	$-0.277235\pi$
$877$	38.1486	1.28819	0.644094	+	0.764946i	$0.277235\pi$
$878$	−17.1691	−0.579429
$879$	0	0
$880$	52.9086	1.78355
$881$	−36.0876	−1.21582	−0.607912	−	0.794005i	$-0.707993\pi$
$881$	−36.0876	−1.21582	−0.607912	+	0.794005i	$0.707993\pi$
$882$	0	0
$883$	−34.7014	−1.16780	−0.583898	−	0.811827i	$-0.698473\pi$
$883$	−34.7014	−1.16780	−0.583898	+	0.811827i	$0.698473\pi$
$884$	−7.13597	−0.240009
$885$	0	0
$886$	−17.1011	−0.574521
$887$	−48.1002	−1.61505	−0.807523	−	0.589837i	$-0.799192\pi$
$887$	−48.1002	−1.61505	−0.807523	+	0.589837i	$0.799192\pi$
$888$	0	0
$889$	−16.5652	−0.555579
$890$	−28.8498	−0.967046
$891$	0	0
$892$	5.77949	0.193512
$893$	1.00000	0.0334637
$894$	0	0
$895$	−41.2737	−1.37963
$896$	11.9033	0.397662
$897$	0	0
$898$	−12.4441	−0.415265
$899$	−3.77824	−0.126011
$900$	0	0
$901$	1.53878	0.0512643
$902$	−4.26428	−0.141985
$903$	0	0
$904$	−15.6316	−0.519900
$905$	39.9249	1.32715
$906$	0	0
$907$	36.4039	1.20877	0.604385	−	0.796692i	$-0.293419\pi$
$907$	36.4039	1.20877	0.604385	+	0.796692i	$0.293419\pi$
$908$	14.2539	0.473032
$909$	0	0
$910$	4.72434	0.156610
$911$	−50.0322	−1.65764	−0.828820	−	0.559515i	$-0.810988\pi$
$911$	−50.0322	−1.65764	−0.828820	+	0.559515i	$0.810988\pi$
$912$	0	0
$913$	16.7731	0.555110
$914$	−15.8978	−0.525852
$915$	0	0
$916$	−31.2086	−1.03116
$917$	−3.77156	−0.124548
$918$	0	0
$919$	−32.8435	−1.08341	−0.541703	−	0.840570i	$-0.682220\pi$
$919$	−32.8435	−1.08341	−0.541703	+	0.840570i	$0.682220\pi$
$920$	18.7867	0.619380
$921$	0	0
$922$	13.8260	0.455335
$923$	−9.94920	−0.327482
$924$	0	0
$925$	85.0534	2.79654
$926$	16.9819	0.558059
$927$	0	0
$928$	−13.1282	−0.430955
$929$	−38.5527	−1.26487	−0.632437	−	0.774612i	$-0.717945\pi$
$929$	−38.5527	−1.26487	−0.632437	+	0.774612i	$0.717945\pi$
$930$	0	0
$931$	5.92868	0.194305
$932$	−32.4173	−1.06186
$933$	0	0
$934$	−8.10654	−0.265254
$935$	40.4001	1.32123
$936$	0	0
$937$	20.4773	0.668965	0.334482	−	0.942402i	$-0.391439\pi$
$937$	20.4773	0.668965	0.334482	+	0.942402i	$0.391439\pi$
$938$	−7.20321	−0.235193
$939$	0	0
$940$	−6.45064	−0.210397
$941$	30.7270	1.00167	0.500835	−	0.865543i	$-0.333026\pi$
$941$	30.7270	1.00167	0.500835	+	0.865543i	$0.333026\pi$
$942$	0	0
$943$	3.43065	0.111717
$944$	−15.4907	−0.504180
$945$	0	0
$946$	7.93856	0.258105
$947$	36.3931	1.18262	0.591309	−	0.806445i	$-0.298611\pi$
$947$	36.3931	1.18262	0.591309	+	0.806445i	$0.298611\pi$
$948$	0	0
$949$	0.326005	0.0105826
$950$	−4.84706	−0.157259
$951$	0	0
$952$	3.72299	0.120663
$953$	17.8019	0.576662	0.288331	−	0.957531i	$-0.406900\pi$
$953$	17.8019	0.576662	0.288331	+	0.957531i	$0.406900\pi$
$954$	0	0
$955$	61.2207	1.98106
$956$	−30.6911	−0.992620
$957$	0	0
$958$	6.36678	0.205701
$959$	−9.15999	−0.295792
$960$	0	0
$961$	−28.7419	−0.927157
$962$	11.3398	0.365611
$963$	0	0
$964$	40.7099	1.31118
$965$	−33.6736	−1.08399
$966$	0	0
$967$	5.11254	0.164408	0.0822041	−	0.996616i	$-0.473804\pi$
$967$	5.11254	0.164408	0.0822041	+	0.996616i	$0.473804\pi$
$968$	−47.3331	−1.52134
$969$	0	0
$970$	10.9897	0.352858
$971$	21.6401	0.694464	0.347232	−	0.937779i	$-0.387122\pi$
$971$	21.6401	0.694464	0.347232	+	0.937779i	$0.387122\pi$
$972$	0	0
$973$	−2.53740	−0.0813452
$974$	4.15856	0.133249
$975$	0	0
$976$	33.3989	1.06907
$977$	9.80200	0.313594	0.156797	−	0.987631i	$-0.449883\pi$
$977$	9.80200	0.313594	0.156797	+	0.987631i	$0.449883\pi$
$978$	0	0
$979$	−85.3453	−2.72765
$980$	−38.2437	−1.22165
$981$	0	0
$982$	−16.0206	−0.511238
$983$	38.9298	1.24167	0.620834	−	0.783942i	$-0.286794\pi$
$983$	38.9298	1.24167	0.620834	+	0.783942i	$0.286794\pi$
$984$	0	0
$985$	−5.12188	−0.163197
$986$	−2.43301	−0.0774827
$987$	0	0
$988$	3.92234	0.124786
$989$	−6.38664	−0.203083
$990$	0	0
$991$	50.7479	1.61206	0.806030	−	0.591874i	$-0.201612\pi$
$991$	50.7479	1.61206	0.806030	+	0.591874i	$0.201612\pi$
$992$	7.84631	0.249120
$993$	0	0
$994$	2.39783	0.0760544
$995$	−5.10487	−0.161835
$996$	0	0
$997$	5.16049	0.163434	0.0817171	−	0.996656i	$-0.473960\pi$
$997$	5.16049	0.163434	0.0817171	+	0.996656i	$0.473960\pi$
$998$	−16.6761	−0.527873
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.v.1.22	✓	34
3.2	odd	2		8037.2.a.w.1.13	yes	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.22	✓	34	1.1	even	1	trivial
8037.2.a.w.1.13	yes	34	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.531889 q^{2} -1.71709 q^{4} -3.75672 q^{5} -1.03505 q^{7} -1.97708 q^{8} +O(q^{10})\)	\(q+0.531889 q^{2} -1.71709 q^{4} -3.75672 q^{5} -1.03505 q^{7} -1.97708 q^{8} -1.99816 q^{10} -5.91108 q^{11} +2.28429 q^{13} -0.550530 q^{14} +2.38260 q^{16} +1.81931 q^{17} -1.00000 q^{19} +6.45064 q^{20} -3.14404 q^{22} +2.52941 q^{23} +9.11292 q^{25} +1.21499 q^{26} +1.77727 q^{28} -2.51429 q^{29} +1.50271 q^{31} +5.22144 q^{32} +0.967672 q^{34} +3.88838 q^{35} +9.33327 q^{37} -0.531889 q^{38} +7.42733 q^{40} +1.35631 q^{41} -2.52496 q^{43} +10.1499 q^{44} +1.34536 q^{46} -1.00000 q^{47} -5.92868 q^{49} +4.84706 q^{50} -3.92234 q^{52} +0.845805 q^{53} +22.2063 q^{55} +2.04637 q^{56} -1.33732 q^{58} -6.50160 q^{59} +14.0178 q^{61} +0.799274 q^{62} -1.98798 q^{64} -8.58143 q^{65} +13.0841 q^{67} -3.12393 q^{68} +2.06819 q^{70} -4.35549 q^{71} +0.142716 q^{73} +4.96426 q^{74} +1.71709 q^{76} +6.11825 q^{77} +3.24900 q^{79} -8.95076 q^{80} +0.721404 q^{82} -2.83757 q^{83} -6.83464 q^{85} -1.34300 q^{86} +11.6867 q^{88} +14.4382 q^{89} -2.36435 q^{91} -4.34323 q^{92} -0.531889 q^{94} +3.75672 q^{95} -5.49993 q^{97} -3.15340 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q - 5 q^{2} + 31 q^{4} - 6 q^{5} - 15 q^{8}+O(q^{10}) \) 34 * q - 5 * q^2 + 31 * q^4 - 6 * q^5 - 15 * q^8	\( 34 q - 5 q^{2} + 31 q^{4} - 6 q^{5} - 15 q^{8} + 4 q^{10} - 6 q^{11} + 2 q^{13} - 12 q^{14} + 21 q^{16} - 4 q^{17} - 34 q^{19} - 20 q^{20} - 8 q^{22} - 26 q^{23} + 32 q^{25} - 29 q^{26} - 4 q^{28} - 14 q^{29} + 2 q^{31} - 35 q^{32} - 18 q^{34} - 50 q^{35} - 10 q^{37} + 5 q^{38} + 17 q^{40} - 18 q^{41} + 6 q^{43} - 6 q^{44} + 18 q^{46} - 34 q^{47} + 28 q^{49} - 41 q^{50} + 10 q^{52} - 40 q^{53} - 8 q^{55} - 76 q^{56} + 4 q^{58} - 62 q^{59} - 2 q^{61} - 50 q^{62} + 11 q^{64} - 32 q^{65} + 20 q^{67} - 28 q^{68} + 22 q^{70} - 52 q^{71} - 8 q^{73} - 10 q^{74} - 31 q^{76} - 36 q^{77} - 12 q^{79} - 92 q^{80} + 10 q^{82} - 82 q^{83} - 4 q^{85} - 40 q^{86} - 16 q^{88} - 58 q^{89} - 100 q^{92} + 5 q^{94} + 6 q^{95} - 6 q^{97} - 45 q^{98}+O(q^{100}) \) 34 * q - 5 * q^2 + 31 * q^4 - 6 * q^5 - 15 * q^8 + 4 * q^10 - 6 * q^11 + 2 * q^13 - 12 * q^14 + 21 * q^16 - 4 * q^17 - 34 * q^19 - 20 * q^20 - 8 * q^22 - 26 * q^23 + 32 * q^25 - 29 * q^26 - 4 * q^28 - 14 * q^29 + 2 * q^31 - 35 * q^32 - 18 * q^34 - 50 * q^35 - 10 * q^37 + 5 * q^38 + 17 * q^40 - 18 * q^41 + 6 * q^43 - 6 * q^44 + 18 * q^46 - 34 * q^47 + 28 * q^49 - 41 * q^50 + 10 * q^52 - 40 * q^53 - 8 * q^55 - 76 * q^56 + 4 * q^58 - 62 * q^59 - 2 * q^61 - 50 * q^62 + 11 * q^64 - 32 * q^65 + 20 * q^67 - 28 * q^68 + 22 * q^70 - 52 * q^71 - 8 * q^73 - 10 * q^74 - 31 * q^76 - 36 * q^77 - 12 * q^79 - 92 * q^80 + 10 * q^82 - 82 * q^83 - 4 * q^85 - 40 * q^86 - 16 * q^88 - 58 * q^89 - 100 * q^92 + 5 * q^94 + 6 * q^95 - 6 * q^97 - 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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