LMFDB - Embedded newform 8037.2.a.w.1.20

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:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+0.805016 q^{2} -1.35195 q^{4} +1.59878 q^{5} +1.80711 q^{7} -2.69837 q^{8} +O(q^{10})$	\(q+0.805016 q^{2} -1.35195 q^{4} +1.59878 q^{5} +1.80711 q^{7} -2.69837 q^{8} +1.28705 q^{10} +4.74394 q^{11} +2.50532 q^{13} +1.45475 q^{14} +0.531663 q^{16} -6.69081 q^{17} -1.00000 q^{19} -2.16148 q^{20} +3.81895 q^{22} +6.91332 q^{23} -2.44389 q^{25} +2.01683 q^{26} -2.44312 q^{28} -1.55172 q^{29} +4.96972 q^{31} +5.82474 q^{32} -5.38621 q^{34} +2.88918 q^{35} +1.44202 q^{37} -0.805016 q^{38} -4.31412 q^{40} +9.91176 q^{41} +6.27362 q^{43} -6.41357 q^{44} +5.56533 q^{46} +1.00000 q^{47} -3.73435 q^{49} -1.96737 q^{50} -3.38707 q^{52} +5.98559 q^{53} +7.58455 q^{55} -4.87626 q^{56} -1.24916 q^{58} -14.8645 q^{59} -6.34947 q^{61} +4.00071 q^{62} +3.62569 q^{64} +4.00547 q^{65} -3.95185 q^{67} +9.04563 q^{68} +2.32584 q^{70} -11.8954 q^{71} +12.2279 q^{73} +1.16085 q^{74} +1.35195 q^{76} +8.57283 q^{77} -1.50085 q^{79} +0.850015 q^{80} +7.97913 q^{82} +4.08301 q^{83} -10.6972 q^{85} +5.05036 q^{86} -12.8009 q^{88} +10.6776 q^{89} +4.52740 q^{91} -9.34645 q^{92} +0.805016 q^{94} -1.59878 q^{95} -7.85603 q^{97} -3.00621 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.805016	0.569232	0.284616	−	0.958642i	$-0.408134\pi$
$2$	0.805016	0.569232	0.284616	+	0.958642i	$0.408134\pi$
$3$	0	0
$3$	0	0
$4$	−1.35195	−0.675974
$5$	1.59878	0.714998	0.357499	−	0.933913i	$-0.383629\pi$
$5$	1.59878	0.714998	0.357499	+	0.933913i	$0.383629\pi$
$6$	0	0
$7$	1.80711	0.683024	0.341512	−	0.939877i	$-0.389061\pi$
$7$	1.80711	0.683024	0.341512	+	0.939877i	$0.389061\pi$
$8$	−2.69837	−0.954019
$9$	0	0
$10$	1.28705	0.407000
$11$	4.74394	1.43035	0.715176	−	0.698944i	$-0.246346\pi$
$11$	4.74394	1.43035	0.715176	+	0.698944i	$0.246346\pi$
$12$	0	0
$13$	2.50532	0.694852	0.347426	−	0.937707i	$-0.387056\pi$
$13$	2.50532	0.694852	0.347426	+	0.937707i	$0.387056\pi$
$14$	1.45475	0.388799
$15$	0	0
$16$	0.531663	0.132916
$17$	−6.69081	−1.62276	−0.811380	−	0.584519i	$-0.801283\pi$
$17$	−6.69081	−1.62276	−0.811380	+	0.584519i	$0.801283\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.16148	−0.483321
$21$	0	0
$22$	3.81895	0.814203
$23$	6.91332	1.44153	0.720763	−	0.693181i	$-0.243792\pi$
$23$	6.91332	1.44153	0.720763	+	0.693181i	$0.243792\pi$
$24$	0	0
$25$	−2.44389	−0.488777
$26$	2.01683	0.395532
$27$	0	0
$28$	−2.44312	−0.461707
$29$	−1.55172	−0.288147	−0.144074	−	0.989567i	$-0.546020\pi$
$29$	−1.55172	−0.288147	−0.144074	+	0.989567i	$0.546020\pi$
$30$	0	0
$31$	4.96972	0.892588	0.446294	−	0.894886i	$-0.352744\pi$
$31$	4.96972	0.892588	0.446294	+	0.894886i	$0.352744\pi$
$32$	5.82474	1.02968
$33$	0	0
$34$	−5.38621	−0.923728
$35$	2.88918	0.488361
$36$	0	0
$37$	1.44202	0.237067	0.118534	−	0.992950i	$-0.462181\pi$
$37$	1.44202	0.237067	0.118534	+	0.992950i	$0.462181\pi$
$38$	−0.805016	−0.130591
$39$	0	0
$40$	−4.31412	−0.682122
$41$	9.91176	1.54796	0.773979	−	0.633212i	$-0.218264\pi$
$41$	9.91176	1.54796	0.773979	+	0.633212i	$0.218264\pi$
$42$	0	0
$43$	6.27362	0.956717	0.478359	−	0.878165i	$-0.341232\pi$
$43$	6.27362	0.956717	0.478359	+	0.878165i	$0.341232\pi$
$44$	−6.41357	−0.966882
$45$	0	0
$46$	5.56533	0.820564
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−3.73435	−0.533479
$50$	−1.96737	−0.278228
$51$	0	0
$52$	−3.38707	−0.469702
$53$	5.98559	0.822184	0.411092	−	0.911594i	$-0.365147\pi$
$53$	5.98559	0.822184	0.411092	+	0.911594i	$0.365147\pi$
$54$	0	0
$55$	7.58455	1.02270
$56$	−4.87626	−0.651618
$57$	0	0
$58$	−1.24916	−0.164023
$59$	−14.8645	−1.93519	−0.967594	−	0.252511i	$-0.918744\pi$
$59$	−14.8645	−1.93519	−0.967594	+	0.252511i	$0.918744\pi$
$60$	0	0
$61$	−6.34947	−0.812967	−0.406483	−	0.913658i	$-0.633245\pi$
$61$	−6.34947	−0.812967	−0.406483	+	0.913658i	$0.633245\pi$
$62$	4.00071	0.508090
$63$	0	0
$64$	3.62569	0.453211
$65$	4.00547	0.496818
$66$	0	0
$67$	−3.95185	−0.482795	−0.241397	−	0.970426i	$-0.577606\pi$
$67$	−3.95185	−0.482795	−0.241397	+	0.970426i	$0.577606\pi$
$68$	9.04563	1.09694
$69$	0	0
$70$	2.32584	0.277991
$71$	−11.8954	−1.41172	−0.705861	−	0.708350i	$-0.749440\pi$
$71$	−11.8954	−1.41172	−0.705861	+	0.708350i	$0.749440\pi$
$72$	0	0
$73$	12.2279	1.43117	0.715584	−	0.698527i	$-0.246161\pi$
$73$	12.2279	1.43117	0.715584	+	0.698527i	$0.246161\pi$
$74$	1.16085	0.134946
$75$	0	0
$76$	1.35195	0.155079
$77$	8.57283	0.976965
$78$	0	0
$79$	−1.50085	−0.168859	−0.0844293	−	0.996429i	$-0.526907\pi$
$79$	−1.50085	−0.168859	−0.0844293	+	0.996429i	$0.526907\pi$
$80$	0.850015	0.0950346
$81$	0	0
$82$	7.97913	0.881147
$83$	4.08301	0.448169	0.224084	−	0.974570i	$-0.428061\pi$
$83$	4.08301	0.448169	0.224084	+	0.974570i	$0.428061\pi$
$84$	0	0
$85$	−10.6972	−1.16027
$86$	5.05036	0.544595
$87$	0	0
$88$	−12.8009	−1.36458
$89$	10.6776	1.13182	0.565912	−	0.824465i	$-0.308524\pi$
$89$	10.6776	1.13182	0.565912	+	0.824465i	$0.308524\pi$
$90$	0	0
$91$	4.52740	0.474600
$92$	−9.34645	−0.974435
$93$	0	0
$94$	0.805016	0.0830311
$95$	−1.59878	−0.164032
$96$	0	0
$97$	−7.85603	−0.797659	−0.398830	−	0.917025i	$-0.630584\pi$
$97$	−7.85603	−0.797659	−0.398830	+	0.917025i	$0.630584\pi$
$98$	−3.00621	−0.303673
$99$	0	0
$100$	3.30401	0.330401
$101$	9.64484	0.959697	0.479849	−	0.877351i	$-0.340692\pi$
$101$	9.64484	0.959697	0.479849	+	0.877351i	$0.340692\pi$
$102$	0	0
$103$	0.356072	0.0350848	0.0175424	−	0.999846i	$-0.494416\pi$
$103$	0.356072	0.0350848	0.0175424	+	0.999846i	$0.494416\pi$
$104$	−6.76030	−0.662902
$105$	0	0
$106$	4.81850	0.468014
$107$	4.22679	0.408619	0.204310	−	0.978906i	$-0.434505\pi$
$107$	4.22679	0.408619	0.204310	+	0.978906i	$0.434505\pi$
$108$	0	0
$109$	12.8521	1.23100	0.615502	−	0.788135i	$-0.288953\pi$
$109$	12.8521	1.23100	0.615502	+	0.788135i	$0.288953\pi$
$110$	6.10568	0.582154
$111$	0	0
$112$	0.960775	0.0907847
$113$	2.94570	0.277108	0.138554	−	0.990355i	$-0.455755\pi$
$113$	2.94570	0.277108	0.138554	+	0.990355i	$0.455755\pi$
$114$	0	0
$115$	11.0529	1.03069
$116$	2.09785	0.194780
$117$	0	0
$118$	−11.9661	−1.10157
$119$	−12.0910	−1.10838
$120$	0	0
$121$	11.5050	1.04591
$122$	−5.11143	−0.462767
$123$	0	0
$124$	−6.71881	−0.603367
$125$	−11.9012	−1.06447
$126$	0	0
$127$	17.3019	1.53530	0.767648	−	0.640872i	$-0.221427\pi$
$127$	17.3019	1.53530	0.767648	+	0.640872i	$0.221427\pi$
$128$	−8.73075	−0.771697
$129$	0	0
$130$	3.22447	0.282805
$131$	−11.8794	−1.03791	−0.518954	−	0.854802i	$-0.673678\pi$
$131$	−11.8794	−1.03791	−0.518954	+	0.854802i	$0.673678\pi$
$132$	0	0
$133$	−1.80711	−0.156696
$134$	−3.18130	−0.274823
$135$	0	0
$136$	18.0543	1.54814
$137$	−9.97380	−0.852119	−0.426060	−	0.904695i	$-0.640099\pi$
$137$	−9.97380	−0.852119	−0.426060	+	0.904695i	$0.640099\pi$
$138$	0	0
$139$	6.42313	0.544802	0.272401	−	0.962184i	$-0.412182\pi$
$139$	6.42313	0.544802	0.272401	+	0.962184i	$0.412182\pi$
$140$	−3.90603	−0.330119
$141$	0	0
$142$	−9.57598	−0.803598
$143$	11.8851	0.993883
$144$	0	0
$145$	−2.48087	−0.206025
$146$	9.84366	0.814667
$147$	0	0
$148$	−1.94954	−0.160251
$149$	15.9155	1.30385	0.651924	−	0.758285i	$-0.273962\pi$
$149$	15.9155	1.30385	0.651924	+	0.758285i	$0.273962\pi$
$150$	0	0
$151$	13.2061	1.07470	0.537351	−	0.843359i	$-0.319425\pi$
$151$	13.2061	1.07470	0.537351	+	0.843359i	$0.319425\pi$
$152$	2.69837	0.218867
$153$	0	0
$154$	6.90127	0.556120
$155$	7.94552	0.638199
$156$	0	0
$157$	4.94611	0.394742	0.197371	−	0.980329i	$-0.436760\pi$
$157$	4.94611	0.394742	0.197371	+	0.980329i	$0.436760\pi$
$158$	−1.20821	−0.0961198
$159$	0	0
$160$	9.31251	0.736219
$161$	12.4931	0.984597
$162$	0	0
$163$	−8.47395	−0.663731	−0.331865	−	0.943327i	$-0.607678\pi$
$163$	−8.47395	−0.663731	−0.331865	+	0.943327i	$0.607678\pi$
$164$	−13.4002	−1.04638
$165$	0	0
$166$	3.28689	0.255112
$167$	−10.9444	−0.846904	−0.423452	−	0.905918i	$-0.639182\pi$
$167$	−10.9444	−0.846904	−0.423452	+	0.905918i	$0.639182\pi$
$168$	0	0
$169$	−6.72335	−0.517181
$170$	−8.61139	−0.660464
$171$	0	0
$172$	−8.48161	−0.646716
$173$	8.83201	0.671485	0.335743	−	0.941954i	$-0.391013\pi$
$173$	8.83201	0.671485	0.335743	+	0.941954i	$0.391013\pi$
$174$	0	0
$175$	−4.41637	−0.333846
$176$	2.52218	0.190117
$177$	0	0
$178$	8.59565	0.644271
$179$	8.76233	0.654928	0.327464	−	0.944864i	$-0.393806\pi$
$179$	8.76233	0.654928	0.327464	+	0.944864i	$0.393806\pi$
$180$	0	0
$181$	19.3028	1.43477	0.717385	−	0.696677i	$-0.245339\pi$
$181$	19.3028	1.43477	0.717385	+	0.696677i	$0.245339\pi$
$182$	3.64463	0.270158
$183$	0	0
$184$	−18.6547	−1.37524
$185$	2.30548	0.169503
$186$	0	0
$187$	−31.7408	−2.32112
$188$	−1.35195	−0.0986010
$189$	0	0
$190$	−1.28705	−0.0933723
$191$	8.03074	0.581084	0.290542	−	0.956862i	$-0.406164\pi$
$191$	8.03074	0.581084	0.290542	+	0.956862i	$0.406164\pi$
$192$	0	0
$193$	−6.70667	−0.482757	−0.241378	−	0.970431i	$-0.577599\pi$
$193$	−6.70667	−0.482757	−0.241378	+	0.970431i	$0.577599\pi$
$194$	−6.32423	−0.454054
$195$	0	0
$196$	5.04865	0.360618
$197$	2.04152	0.145452	0.0727262	−	0.997352i	$-0.476830\pi$
$197$	2.04152	0.145452	0.0727262	+	0.997352i	$0.476830\pi$
$198$	0	0
$199$	−24.5253	−1.73855	−0.869276	−	0.494327i	$-0.835415\pi$
$199$	−24.5253	−1.73855	−0.869276	+	0.494327i	$0.835415\pi$
$200$	6.59452	0.466303
$201$	0	0
$202$	7.76425	0.546291
$203$	−2.80413	−0.196811
$204$	0	0
$205$	15.8468	1.10679
$206$	0.286644	0.0199714
$207$	0	0
$208$	1.33199	0.0923568
$209$	−4.74394	−0.328145
$210$	0	0
$211$	−24.4356	−1.68221	−0.841106	−	0.540870i	$-0.818095\pi$
$211$	−24.4356	−1.68221	−0.841106	+	0.540870i	$0.818095\pi$
$212$	−8.09221	−0.555775
$213$	0	0
$214$	3.40264	0.232599
$215$	10.0302	0.684051
$216$	0	0
$217$	8.98084	0.609659
$218$	10.3461	0.700728
$219$	0	0
$220$	−10.2539	−0.691319
$221$	−16.7626	−1.12758
$222$	0	0
$223$	−8.96084	−0.600062	−0.300031	−	0.953929i	$-0.596997\pi$
$223$	−8.96084	−0.600062	−0.300031	+	0.953929i	$0.596997\pi$
$224$	10.5260	0.703295
$225$	0	0
$226$	2.37134	0.157739
$227$	−13.7898	−0.915260	−0.457630	−	0.889143i	$-0.651302\pi$
$227$	−13.7898	−0.915260	−0.457630	+	0.889143i	$0.651302\pi$
$228$	0	0
$229$	−11.8943	−0.785999	−0.393000	−	0.919539i	$-0.628563\pi$
$229$	−11.8943	−0.785999	−0.393000	+	0.919539i	$0.628563\pi$
$230$	8.89777	0.586702
$231$	0	0
$232$	4.18712	0.274898
$233$	−4.05983	−0.265968	−0.132984	−	0.991118i	$-0.542456\pi$
$233$	−4.05983	−0.265968	−0.132984	+	0.991118i	$0.542456\pi$
$234$	0	0
$235$	1.59878	0.104293
$236$	20.0960	1.30814
$237$	0	0
$238$	−9.73348	−0.630928
$239$	26.3475	1.70428	0.852138	−	0.523317i	$-0.175306\pi$
$239$	26.3475	1.70428	0.852138	+	0.523317i	$0.175306\pi$
$240$	0	0
$241$	18.2916	1.17827	0.589134	−	0.808036i	$-0.299469\pi$
$241$	18.2916	1.17827	0.589134	+	0.808036i	$0.299469\pi$
$242$	9.26171	0.595365
$243$	0	0
$244$	8.58416	0.549545
$245$	−5.97042	−0.381436
$246$	0	0
$247$	−2.50532	−0.159410
$248$	−13.4102	−0.851546
$249$	0	0
$250$	−9.58064	−0.605933
$251$	21.5037	1.35730	0.678652	−	0.734460i	$-0.262565\pi$
$251$	21.5037	1.35730	0.678652	+	0.734460i	$0.262565\pi$
$252$	0	0
$253$	32.7964	2.06189
$254$	13.9283	0.873940
$255$	0	0
$256$	−14.2798	−0.892486
$257$	−22.3701	−1.39541	−0.697704	−	0.716386i	$-0.745795\pi$
$257$	−22.3701	−1.39541	−0.697704	+	0.716386i	$0.745795\pi$
$258$	0	0
$259$	2.60590	0.161922
$260$	−5.41520	−0.335836
$261$	0	0
$262$	−9.56311	−0.590811
$263$	−26.0665	−1.60733	−0.803664	−	0.595083i	$-0.797119\pi$
$263$	−26.0665	−1.60733	−0.803664	+	0.595083i	$0.797119\pi$
$264$	0	0
$265$	9.56967	0.587860
$266$	−1.45475	−0.0891967
$267$	0	0
$268$	5.34270	0.326357
$269$	9.58594	0.584465	0.292233	−	0.956347i	$-0.405602\pi$
$269$	9.58594	0.584465	0.292233	+	0.956347i	$0.405602\pi$
$270$	0	0
$271$	6.49767	0.394705	0.197353	−	0.980333i	$-0.436766\pi$
$271$	6.49767	0.394705	0.197353	+	0.980333i	$0.436766\pi$
$272$	−3.55726	−0.215691
$273$	0	0
$274$	−8.02907	−0.485054
$275$	−11.5937	−0.699124
$276$	0	0
$277$	−26.1418	−1.57071	−0.785355	−	0.619045i	$-0.787520\pi$
$277$	−26.1418	−1.57071	−0.785355	+	0.619045i	$0.787520\pi$
$278$	5.17072	0.310119
$279$	0	0
$280$	−7.79609	−0.465905
$281$	11.1156	0.663100	0.331550	−	0.943438i	$-0.392429\pi$
$281$	11.1156	0.663100	0.331550	+	0.943438i	$0.392429\pi$
$282$	0	0
$283$	31.0996	1.84868	0.924339	−	0.381573i	$-0.124617\pi$
$283$	31.0996	1.84868	0.924339	+	0.381573i	$0.124617\pi$
$284$	16.0820	0.954288
$285$	0	0
$286$	9.56771	0.565751
$287$	17.9117	1.05729
$288$	0	0
$289$	27.7669	1.63335
$290$	−1.99714	−0.117276
$291$	0	0
$292$	−16.5315	−0.967433
$293$	17.5657	1.02620	0.513100	−	0.858329i	$-0.328497\pi$
$293$	17.5657	1.02620	0.513100	+	0.858329i	$0.328497\pi$
$294$	0	0
$295$	−23.7651	−1.38366
$296$	−3.89112	−0.226167
$297$	0	0
$298$	12.8122	0.742192
$299$	17.3201	1.00165
$300$	0	0
$301$	11.3371	0.653461
$302$	10.6312	0.611755
$303$	0	0
$304$	−0.531663	−0.0304930
$305$	−10.1514	−0.581270
$306$	0	0
$307$	−1.07652	−0.0614402	−0.0307201	−	0.999528i	$-0.509780\pi$
$307$	−1.07652	−0.0614402	−0.0307201	+	0.999528i	$0.509780\pi$
$308$	−11.5900	−0.660403
$309$	0	0
$310$	6.39627	0.363284
$311$	21.2078	1.20258	0.601291	−	0.799030i	$-0.294653\pi$
$311$	21.2078	1.20258	0.601291	+	0.799030i	$0.294653\pi$
$312$	0	0
$313$	−8.93660	−0.505126	−0.252563	−	0.967580i	$-0.581274\pi$
$313$	−8.93660	−0.505126	−0.252563	+	0.967580i	$0.581274\pi$
$314$	3.98170	0.224700
$315$	0	0
$316$	2.02907	0.114144
$317$	9.40852	0.528435	0.264217	−	0.964463i	$-0.414886\pi$
$317$	9.40852	0.528435	0.264217	+	0.964463i	$0.414886\pi$
$318$	0	0
$319$	−7.36128	−0.412152
$320$	5.79669	0.324045
$321$	0	0
$322$	10.0572	0.560464
$323$	6.69081	0.372287
$324$	0	0
$325$	−6.12273	−0.339628
$326$	−6.82167	−0.377817
$327$	0	0
$328$	−26.7456	−1.47678
$329$	1.80711	0.0996292
$330$	0	0
$331$	−19.1104	−1.05040	−0.525200	−	0.850979i	$-0.676009\pi$
$331$	−19.1104	−1.05040	−0.525200	+	0.850979i	$0.676009\pi$
$332$	−5.52002	−0.302950
$333$	0	0
$334$	−8.81043	−0.482085
$335$	−6.31815	−0.345198
$336$	0	0
$337$	11.6199	0.632976	0.316488	−	0.948597i	$-0.397496\pi$
$337$	11.6199	0.632976	0.316488	+	0.948597i	$0.397496\pi$
$338$	−5.41241	−0.294396
$339$	0	0
$340$	14.4620	0.784313
$341$	23.5761	1.27672
$342$	0	0
$343$	−19.3982	−1.04740
$344$	−16.9286	−0.912727
$345$	0	0
$346$	7.10991	0.382231
$347$	19.4707	1.04524	0.522621	−	0.852565i	$-0.324954\pi$
$347$	19.4707	1.04524	0.522621	+	0.852565i	$0.324954\pi$
$348$	0	0
$349$	20.8116	1.11402	0.557009	−	0.830507i	$-0.311949\pi$
$349$	20.8116	1.11402	0.557009	+	0.830507i	$0.311949\pi$
$350$	−3.55525	−0.190036
$351$	0	0
$352$	27.6323	1.47280
$353$	8.21951	0.437480	0.218740	−	0.975783i	$-0.429805\pi$
$353$	8.21951	0.437480	0.218740	+	0.975783i	$0.429805\pi$
$354$	0	0
$355$	−19.0182	−1.00938
$356$	−14.4356	−0.765084
$357$	0	0
$358$	7.05382	0.372806
$359$	14.4559	0.762953	0.381477	−	0.924379i	$-0.375416\pi$
$359$	14.4559	0.762953	0.381477	+	0.924379i	$0.375416\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	15.5391	0.816717
$363$	0	0
$364$	−6.12081	−0.320818
$365$	19.5498	1.02328
$366$	0	0
$367$	35.4194	1.84888	0.924438	−	0.381332i	$-0.124535\pi$
$367$	35.4194	1.84888	0.924438	+	0.381332i	$0.124535\pi$
$368$	3.67556	0.191602
$369$	0	0
$370$	1.85595	0.0964864
$371$	10.8166	0.561571
$372$	0	0
$373$	33.6235	1.74096	0.870480	−	0.492205i	$-0.163809\pi$
$373$	33.6235	1.74096	0.870480	+	0.492205i	$0.163809\pi$
$374$	−25.5519	−1.32126
$375$	0	0
$376$	−2.69837	−0.139158
$377$	−3.88756	−0.200220
$378$	0	0
$379$	−0.942019	−0.0483883	−0.0241941	−	0.999707i	$-0.507702\pi$
$379$	−0.942019	−0.0483883	−0.0241941	+	0.999707i	$0.507702\pi$
$380$	2.16148	0.110881
$381$	0	0
$382$	6.46488	0.330772
$383$	32.0341	1.63686	0.818432	−	0.574603i	$-0.194843\pi$
$383$	32.0341	1.63686	0.818432	+	0.574603i	$0.194843\pi$
$384$	0	0
$385$	13.7061	0.698528
$386$	−5.39898	−0.274801
$387$	0	0
$388$	10.6210	0.539197
$389$	36.3892	1.84501	0.922503	−	0.385989i	$-0.126140\pi$
$389$	36.3892	1.84501	0.922503	+	0.385989i	$0.126140\pi$
$390$	0	0
$391$	−46.2557	−2.33925
$392$	10.0767	0.508949
$393$	0	0
$394$	1.64346	0.0827962
$395$	−2.39953	−0.120734
$396$	0	0
$397$	−2.16234	−0.108525	−0.0542624	−	0.998527i	$-0.517281\pi$
$397$	−2.16234	−0.108525	−0.0542624	+	0.998527i	$0.517281\pi$
$398$	−19.7433	−0.989640
$399$	0	0
$400$	−1.29933	−0.0649663
$401$	−7.51035	−0.375049	−0.187525	−	0.982260i	$-0.560046\pi$
$401$	−7.51035	−0.375049	−0.187525	+	0.982260i	$0.560046\pi$
$402$	0	0
$403$	12.4508	0.620217
$404$	−13.0393	−0.648731
$405$	0	0
$406$	−2.25737	−0.112031
$407$	6.84088	0.339090
$408$	0	0
$409$	21.7976	1.07782	0.538910	−	0.842363i	$-0.318836\pi$
$409$	21.7976	1.07782	0.538910	+	0.842363i	$0.318836\pi$
$410$	12.7569	0.630019
$411$	0	0
$412$	−0.481391	−0.0237164
$413$	−26.8617	−1.32178
$414$	0	0
$415$	6.52785	0.320440
$416$	14.5929	0.715474
$417$	0	0
$418$	−3.81895	−0.186791
$419$	4.99830	0.244183	0.122091	−	0.992519i	$-0.461040\pi$
$419$	4.99830	0.244183	0.122091	+	0.992519i	$0.461040\pi$
$420$	0	0
$421$	−27.5417	−1.34230	−0.671151	−	0.741321i	$-0.734200\pi$
$421$	−27.5417	−1.34230	−0.671151	+	0.741321i	$0.734200\pi$
$422$	−19.6710	−0.957570
$423$	0	0
$424$	−16.1514	−0.784379
$425$	16.3516	0.793168
$426$	0	0
$427$	−11.4742	−0.555276
$428$	−5.71441	−0.276216
$429$	0	0
$430$	8.07444	0.389384
$431$	19.8570	0.956479	0.478239	−	0.878230i	$-0.341275\pi$
$431$	19.8570	0.956479	0.478239	+	0.878230i	$0.341275\pi$
$432$	0	0
$433$	4.67721	0.224772	0.112386	−	0.993665i	$-0.464151\pi$
$433$	4.67721	0.224772	0.112386	+	0.993665i	$0.464151\pi$
$434$	7.22972	0.347038
$435$	0	0
$436$	−17.3753	−0.832128
$437$	−6.91332	−0.330709
$438$	0	0
$439$	−23.5122	−1.12218	−0.561088	−	0.827756i	$-0.689617\pi$
$439$	−23.5122	−1.12218	−0.561088	+	0.827756i	$0.689617\pi$
$440$	−20.4659	−0.975675
$441$	0	0
$442$	−13.4942	−0.641854
$443$	0.578458	0.0274834	0.0137417	−	0.999906i	$-0.495626\pi$
$443$	0.578458	0.0274834	0.0137417	+	0.999906i	$0.495626\pi$
$444$	0	0
$445$	17.0712	0.809253
$446$	−7.21362	−0.341575
$447$	0	0
$448$	6.55202	0.309554
$449$	−32.8642	−1.55096	−0.775479	−	0.631373i	$-0.782492\pi$
$449$	−32.8642	−1.55096	−0.775479	+	0.631373i	$0.782492\pi$
$450$	0	0
$451$	47.0208	2.21412
$452$	−3.98244	−0.187318
$453$	0	0
$454$	−11.1010	−0.520996
$455$	7.23834	0.339338
$456$	0	0
$457$	13.6749	0.639686	0.319843	−	0.947470i	$-0.396370\pi$
$457$	13.6749	0.639686	0.319843	+	0.947470i	$0.396370\pi$
$458$	−9.57513	−0.447416
$459$	0	0
$460$	−14.9430	−0.696719
$461$	30.1940	1.40627	0.703137	−	0.711054i	$-0.251782\pi$
$461$	30.1940	1.40627	0.703137	+	0.711054i	$0.251782\pi$
$462$	0	0
$463$	−34.6236	−1.60909	−0.804546	−	0.593890i	$-0.797592\pi$
$463$	−34.6236	−1.60909	−0.804546	+	0.593890i	$0.797592\pi$
$464$	−0.824993	−0.0382994
$465$	0	0
$466$	−3.26823	−0.151398
$467$	3.32148	0.153700	0.0768498	−	0.997043i	$-0.475514\pi$
$467$	3.32148	0.153700	0.0768498	+	0.997043i	$0.475514\pi$
$468$	0	0
$469$	−7.14143	−0.329760
$470$	1.28705	0.0593671
$471$	0	0
$472$	40.1099	1.84621
$473$	29.7617	1.36844
$474$	0	0
$475$	2.44389	0.112133
$476$	16.3465	0.749239
$477$	0	0
$478$	21.2101	0.970129
$479$	12.7386	0.582041	0.291020	−	0.956717i	$-0.406005\pi$
$479$	12.7386	0.582041	0.291020	+	0.956717i	$0.406005\pi$
$480$	0	0
$481$	3.61273	0.164727
$482$	14.7251	0.670708
$483$	0	0
$484$	−15.5542	−0.707008
$485$	−12.5601	−0.570325
$486$	0	0
$487$	−30.5854	−1.38596	−0.692979	−	0.720958i	$-0.743702\pi$
$487$	−30.5854	−1.38596	−0.692979	+	0.720958i	$0.743702\pi$
$488$	17.1333	0.775586
$489$	0	0
$490$	−4.80629	−0.217126
$491$	36.5781	1.65074	0.825372	−	0.564589i	$-0.190965\pi$
$491$	36.5781	1.65074	0.825372	+	0.564589i	$0.190965\pi$
$492$	0	0
$493$	10.3823	0.467594
$494$	−2.01683	−0.0907413
$495$	0	0
$496$	2.64222	0.118639
$497$	−21.4963	−0.964240
$498$	0	0
$499$	−38.3273	−1.71576	−0.857882	−	0.513847i	$-0.828220\pi$
$499$	−38.3273	−1.71576	−0.857882	+	0.513847i	$0.828220\pi$
$500$	16.0898	0.719557
$501$	0	0
$502$	17.3109	0.772621
$503$	−6.66193	−0.297041	−0.148520	−	0.988909i	$-0.547451\pi$
$503$	−6.66193	−0.297041	−0.148520	+	0.988909i	$0.547451\pi$
$504$	0	0
$505$	15.4200	0.686182
$506$	26.4016	1.17370
$507$	0	0
$508$	−23.3913	−1.03782
$509$	−18.6331	−0.825900	−0.412950	−	0.910754i	$-0.635502\pi$
$509$	−18.6331	−0.825900	−0.412950	+	0.910754i	$0.635502\pi$
$510$	0	0
$511$	22.0972	0.977522
$512$	5.96606	0.263665
$513$	0	0
$514$	−18.0083	−0.794312
$515$	0.569282	0.0250856
$516$	0	0
$517$	4.74394	0.208638
$518$	2.09779	0.0921715
$519$	0	0
$520$	−10.8083	−0.473974
$521$	11.4272	0.500633	0.250317	−	0.968164i	$-0.419465\pi$
$521$	11.4272	0.500633	0.250317	+	0.968164i	$0.419465\pi$
$522$	0	0
$523$	9.44075	0.412815	0.206408	−	0.978466i	$-0.433823\pi$
$523$	9.44075	0.412815	0.206408	+	0.978466i	$0.433823\pi$
$524$	16.0603	0.701599
$525$	0	0
$526$	−20.9839	−0.914944
$527$	−33.2515	−1.44846
$528$	0	0
$529$	24.7940	1.07800
$530$	7.70374	0.334629
$531$	0	0
$532$	2.44312	0.105923
$533$	24.8322	1.07560
$534$	0	0
$535$	6.75773	0.292162
$536$	10.6636	0.460596
$537$	0	0
$538$	7.71684	0.332697
$539$	−17.7155	−0.763063
$540$	0	0
$541$	−8.89983	−0.382634	−0.191317	−	0.981528i	$-0.561276\pi$
$541$	−8.89983	−0.382634	−0.191317	+	0.981528i	$0.561276\pi$
$542$	5.23073	0.224679
$543$	0	0
$544$	−38.9723	−1.67092
$545$	20.5477	0.880166
$546$	0	0
$547$	4.87600	0.208483	0.104241	−	0.994552i	$-0.466759\pi$
$547$	4.87600	0.208483	0.104241	+	0.994552i	$0.466759\pi$
$548$	13.4841	0.576011
$549$	0	0
$550$	−9.33309	−0.397964
$551$	1.55172	0.0661055
$552$	0	0
$553$	−2.71220	−0.115334
$554$	−21.0446	−0.894099
$555$	0	0
$556$	−8.68374	−0.368272
$557$	3.61228	0.153057	0.0765287	−	0.997067i	$-0.475616\pi$
$557$	3.61228	0.153057	0.0765287	+	0.997067i	$0.475616\pi$
$558$	0	0
$559$	15.7174	0.664777
$560$	1.53607	0.0649109
$561$	0	0
$562$	8.94822	0.377458
$563$	20.9760	0.884032	0.442016	−	0.897007i	$-0.354263\pi$
$563$	20.9760	0.884032	0.442016	+	0.897007i	$0.354263\pi$
$564$	0	0
$565$	4.70954	0.198132
$566$	25.0357	1.05233
$567$	0	0
$568$	32.0982	1.34681
$569$	44.4401	1.86303	0.931514	−	0.363706i	$-0.118489\pi$
$569$	44.4401	1.86303	0.931514	+	0.363706i	$0.118489\pi$
$570$	0	0
$571$	−26.7211	−1.11824	−0.559122	−	0.829085i	$-0.688862\pi$
$571$	−26.7211	−1.11824	−0.559122	+	0.829085i	$0.688862\pi$
$572$	−16.0681	−0.671840
$573$	0	0
$574$	14.4192	0.601844
$575$	−16.8954	−0.704585
$576$	0	0
$577$	−18.0431	−0.751146	−0.375573	−	0.926793i	$-0.622554\pi$
$577$	−18.0431	−0.751146	−0.375573	+	0.926793i	$0.622554\pi$
$578$	22.3528	0.929756
$579$	0	0
$580$	3.35401	0.139268
$581$	7.37845	0.306110
$582$	0	0
$583$	28.3953	1.17601
$584$	−32.9954	−1.36536
$585$	0	0
$586$	14.1407	0.584147
$587$	24.5838	1.01468	0.507340	−	0.861746i	$-0.330629\pi$
$587$	24.5838	1.01468	0.507340	+	0.861746i	$0.330629\pi$
$588$	0	0
$589$	−4.96972	−0.204774
$590$	−19.1313	−0.787622
$591$	0	0
$592$	0.766671	0.0315100
$593$	−9.48863	−0.389652	−0.194826	−	0.980838i	$-0.562414\pi$
$593$	−9.48863	−0.389652	−0.194826	+	0.980838i	$0.562414\pi$
$594$	0	0
$595$	−19.3310	−0.792492
$596$	−21.5169	−0.881367
$597$	0	0
$598$	13.9430	0.570170
$599$	−7.40718	−0.302649	−0.151325	−	0.988484i	$-0.548354\pi$
$599$	−7.40718	−0.302649	−0.151325	+	0.988484i	$0.548354\pi$
$600$	0	0
$601$	−3.22616	−0.131598	−0.0657990	−	0.997833i	$-0.520960\pi$
$601$	−3.22616	−0.131598	−0.0657990	+	0.997833i	$0.520960\pi$
$602$	9.12656	0.371971
$603$	0	0
$604$	−17.8540	−0.726471
$605$	18.3940	0.747823
$606$	0	0
$607$	20.4693	0.830825	0.415413	−	0.909633i	$-0.363637\pi$
$607$	20.4693	0.830825	0.415413	+	0.909633i	$0.363637\pi$
$608$	−5.82474	−0.236225
$609$	0	0
$610$	−8.17208	−0.330878
$611$	2.50532	0.101355
$612$	0	0
$613$	−36.4385	−1.47174	−0.735870	−	0.677123i	$-0.763226\pi$
$613$	−36.4385	−1.47174	−0.735870	+	0.677123i	$0.763226\pi$
$614$	−0.866616	−0.0349738
$615$	0	0
$616$	−23.1327	−0.932043
$617$	−5.16133	−0.207787	−0.103894	−	0.994588i	$-0.533130\pi$
$617$	−5.16133	−0.207787	−0.103894	+	0.994588i	$0.533130\pi$
$618$	0	0
$619$	−29.0219	−1.16649	−0.583245	−	0.812297i	$-0.698217\pi$
$619$	−29.0219	−1.16649	−0.583245	+	0.812297i	$0.698217\pi$
$620$	−10.7419	−0.431406
$621$	0	0
$622$	17.0726	0.684549
$623$	19.2956	0.773063
$624$	0	0
$625$	−6.80798	−0.272319
$626$	−7.19411	−0.287534
$627$	0	0
$628$	−6.68688	−0.266836
$629$	−9.64830	−0.384703
$630$	0	0
$631$	6.77440	0.269684	0.134842	−	0.990867i	$-0.456947\pi$
$631$	6.77440	0.269684	0.134842	+	0.990867i	$0.456947\pi$
$632$	4.04985	0.161094
$633$	0	0
$634$	7.57401	0.300802
$635$	27.6620	1.09773
$636$	0	0
$637$	−9.35576	−0.370689
$638$	−5.92595	−0.234611
$639$	0	0
$640$	−13.9586	−0.551762
$641$	1.18562	0.0468291	0.0234145	−	0.999726i	$-0.492546\pi$
$641$	1.18562	0.0468291	0.0234145	+	0.999726i	$0.492546\pi$
$642$	0	0
$643$	−42.9240	−1.69276	−0.846379	−	0.532582i	$-0.821222\pi$
$643$	−42.9240	−1.69276	−0.846379	+	0.532582i	$0.821222\pi$
$644$	−16.8901	−0.665562
$645$	0	0
$646$	5.38621	0.211918
$647$	−15.9484	−0.626994	−0.313497	−	0.949589i	$-0.601501\pi$
$647$	−15.9484	−0.626994	−0.313497	+	0.949589i	$0.601501\pi$
$648$	0	0
$649$	−70.5162	−2.76800
$650$	−4.92890	−0.193327
$651$	0	0
$652$	11.4563	0.448665
$653$	−11.5505	−0.452007	−0.226004	−	0.974126i	$-0.572566\pi$
$653$	−11.5505	−0.452007	−0.226004	+	0.974126i	$0.572566\pi$
$654$	0	0
$655$	−18.9926	−0.742102
$656$	5.26972	0.205748
$657$	0	0
$658$	1.45475	0.0567122
$659$	−23.8656	−0.929673	−0.464836	−	0.885397i	$-0.653887\pi$
$659$	−23.8656	−0.929673	−0.464836	+	0.885397i	$0.653887\pi$
$660$	0	0
$661$	21.3972	0.832254	0.416127	−	0.909307i	$-0.363387\pi$
$661$	21.3972	0.832254	0.416127	+	0.909307i	$0.363387\pi$
$662$	−15.3841	−0.597922
$663$	0	0
$664$	−11.0175	−0.427561
$665$	−2.88918	−0.112038
$666$	0	0
$667$	−10.7275	−0.415372
$668$	14.7963	0.572486
$669$	0	0
$670$	−5.08622	−0.196498
$671$	−30.1215	−1.16283
$672$	0	0
$673$	−29.5989	−1.14096	−0.570478	−	0.821313i	$-0.693242\pi$
$673$	−29.5989	−1.14096	−0.570478	+	0.821313i	$0.693242\pi$
$674$	9.35420	0.360310
$675$	0	0
$676$	9.08963	0.349601
$677$	−30.5620	−1.17459	−0.587296	−	0.809372i	$-0.699807\pi$
$677$	−30.5620	−1.17459	−0.587296	+	0.809372i	$0.699807\pi$
$678$	0	0
$679$	−14.1967	−0.544820
$680$	28.8650	1.10692
$681$	0	0
$682$	18.9791	0.726748
$683$	−38.9757	−1.49136	−0.745682	−	0.666302i	$-0.767876\pi$
$683$	−38.9757	−1.49136	−0.745682	+	0.666302i	$0.767876\pi$
$684$	0	0
$685$	−15.9460	−0.609264
$686$	−15.6158	−0.596215
$687$	0	0
$688$	3.33545	0.127163
$689$	14.9958	0.571296
$690$	0	0
$691$	−45.0542	−1.71394	−0.856972	−	0.515364i	$-0.827657\pi$
$691$	−45.0542	−1.71394	−0.856972	+	0.515364i	$0.827657\pi$
$692$	−11.9404	−0.453907
$693$	0	0
$694$	15.6742	0.594986
$695$	10.2692	0.389533
$696$	0	0
$697$	−66.3177	−2.51196
$698$	16.7536	0.634135
$699$	0	0
$700$	5.97071	0.225672
$701$	−11.3589	−0.429020	−0.214510	−	0.976722i	$-0.568815\pi$
$701$	−11.3589	−0.429020	−0.214510	+	0.976722i	$0.568815\pi$
$702$	0	0
$703$	−1.44202	−0.0543869
$704$	17.2001	0.648251
$705$	0	0
$706$	6.61684	0.249028
$707$	17.4293	0.655496
$708$	0	0
$709$	−28.1509	−1.05723	−0.528615	−	0.848862i	$-0.677288\pi$
$709$	−28.1509	−1.05723	−0.528615	+	0.848862i	$0.677288\pi$
$710$	−15.3099	−0.574571
$711$	0	0
$712$	−28.8122	−1.07978
$713$	34.3573	1.28669
$714$	0	0
$715$	19.0017	0.710625
$716$	−11.8462	−0.442714
$717$	0	0
$718$	11.6372	0.434298
$719$	25.0341	0.933614	0.466807	−	0.884359i	$-0.345404\pi$
$719$	25.0341	0.933614	0.466807	+	0.884359i	$0.345404\pi$
$720$	0	0
$721$	0.643461	0.0239637
$722$	0.805016	0.0299596
$723$	0	0
$724$	−26.0965	−0.969868
$725$	3.79223	0.140840
$726$	0	0
$727$	26.1148	0.968546	0.484273	−	0.874917i	$-0.339084\pi$
$727$	26.1148	0.968546	0.484273	+	0.874917i	$0.339084\pi$
$728$	−12.2166	−0.452778
$729$	0	0
$730$	15.7379	0.582486
$731$	−41.9756	−1.55252
$732$	0	0
$733$	43.5393	1.60816	0.804080	−	0.594520i	$-0.202658\pi$
$733$	43.5393	1.60816	0.804080	+	0.594520i	$0.202658\pi$
$734$	28.5132	1.05244
$735$	0	0
$736$	40.2683	1.48431
$737$	−18.7473	−0.690567
$738$	0	0
$739$	−10.7235	−0.394471	−0.197235	−	0.980356i	$-0.563196\pi$
$739$	−10.7235	−0.394471	−0.197235	+	0.980356i	$0.563196\pi$
$740$	−3.11690	−0.114579
$741$	0	0
$742$	8.70756	0.319665
$743$	15.7826	0.579007	0.289504	−	0.957177i	$-0.406510\pi$
$743$	15.7826	0.579007	0.289504	+	0.957177i	$0.406510\pi$
$744$	0	0
$745$	25.4454	0.932249
$746$	27.0675	0.991010
$747$	0	0
$748$	42.9120	1.56902
$749$	7.63828	0.279097
$750$	0	0
$751$	−22.1621	−0.808706	−0.404353	−	0.914603i	$-0.632503\pi$
$751$	−22.1621	−0.808706	−0.404353	+	0.914603i	$0.632503\pi$
$752$	0.531663	0.0193878
$753$	0	0
$754$	−3.12955	−0.113972
$755$	21.1138	0.768410
$756$	0	0
$757$	6.07080	0.220647	0.110323	−	0.993896i	$-0.464811\pi$
$757$	6.07080	0.220647	0.110323	+	0.993896i	$0.464811\pi$
$758$	−0.758340	−0.0275442
$759$	0	0
$760$	4.31412	0.156490
$761$	−1.03036	−0.0373504	−0.0186752	−	0.999826i	$-0.505945\pi$
$761$	−1.03036	−0.0373504	−0.0186752	+	0.999826i	$0.505945\pi$
$762$	0	0
$763$	23.2251	0.840805
$764$	−10.8572	−0.392798
$765$	0	0
$766$	25.7880	0.931757
$767$	−37.2403	−1.34467
$768$	0	0
$769$	15.3068	0.551978	0.275989	−	0.961161i	$-0.410995\pi$
$769$	15.3068	0.551978	0.275989	+	0.961161i	$0.410995\pi$
$770$	11.0336	0.397625
$771$	0	0
$772$	9.06708	0.326331
$773$	−22.1744	−0.797557	−0.398778	−	0.917047i	$-0.630566\pi$
$773$	−22.1744	−0.797557	−0.398778	+	0.917047i	$0.630566\pi$
$774$	0	0
$775$	−12.1454	−0.436277
$776$	21.1985	0.760982
$777$	0	0
$778$	29.2939	1.05024
$779$	−9.91176	−0.355126
$780$	0	0
$781$	−56.4310	−2.01926
$782$	−37.2366	−1.33158
$783$	0	0
$784$	−1.98542	−0.0709078
$785$	7.90776	0.282240
$786$	0	0
$787$	−13.5447	−0.482815	−0.241408	−	0.970424i	$-0.577609\pi$
$787$	−13.5447	−0.482815	−0.241408	+	0.970424i	$0.577609\pi$
$788$	−2.76003	−0.0983221
$789$	0	0
$790$	−1.93166	−0.0687255
$791$	5.32321	0.189271
$792$	0	0
$793$	−15.9075	−0.564892
$794$	−1.74072	−0.0617758
$795$	0	0
$796$	33.1570	1.17522
$797$	−7.33025	−0.259651	−0.129825	−	0.991537i	$-0.541442\pi$
$797$	−7.33025	−0.259651	−0.129825	+	0.991537i	$0.541442\pi$
$798$	0	0
$799$	−6.69081	−0.236704
$800$	−14.2350	−0.503284
$801$	0	0
$802$	−6.04595	−0.213490
$803$	58.0085	2.04708
$804$	0	0
$805$	19.9738	0.703985
$806$	10.0231	0.353047
$807$	0	0
$808$	−26.0254	−0.915569
$809$	−36.5243	−1.28413	−0.642063	−	0.766652i	$-0.721921\pi$
$809$	−36.5243	−1.28413	−0.642063	+	0.766652i	$0.721921\pi$
$810$	0	0
$811$	−41.8499	−1.46955	−0.734774	−	0.678312i	$-0.762712\pi$
$811$	−41.8499	−1.46955	−0.734774	+	0.678312i	$0.762712\pi$
$812$	3.79104	0.133040
$813$	0	0
$814$	5.50702	0.193021
$815$	−13.5480	−0.474566
$816$	0	0
$817$	−6.27362	−0.219486
$818$	17.5474	0.613531
$819$	0	0
$820$	−21.4240	−0.748160
$821$	−34.6458	−1.20915	−0.604574	−	0.796549i	$-0.706657\pi$
$821$	−34.6458	−1.20915	−0.604574	+	0.796549i	$0.706657\pi$
$822$	0	0
$823$	−34.5183	−1.20323	−0.601617	−	0.798785i	$-0.705477\pi$
$823$	−34.5183	−1.20323	−0.601617	+	0.798785i	$0.705477\pi$
$824$	−0.960814	−0.0334716
$825$	0	0
$826$	−21.6241	−0.752400
$827$	52.6383	1.83041	0.915207	−	0.402983i	$-0.132027\pi$
$827$	52.6383	1.83041	0.915207	+	0.402983i	$0.132027\pi$
$828$	0	0
$829$	8.86117	0.307761	0.153881	−	0.988089i	$-0.450823\pi$
$829$	8.86117	0.307761	0.153881	+	0.988089i	$0.450823\pi$
$830$	5.25503	0.182405
$831$	0	0
$832$	9.08352	0.314914
$833$	24.9858	0.865708
$834$	0	0
$835$	−17.4978	−0.605535
$836$	6.41357	0.221818
$837$	0	0
$838$	4.02371	0.138997
$839$	40.1920	1.38758	0.693791	−	0.720176i	$-0.255939\pi$
$839$	40.1920	1.38758	0.693791	+	0.720176i	$0.255939\pi$
$840$	0	0
$841$	−26.5922	−0.916971
$842$	−22.1715	−0.764082
$843$	0	0
$844$	33.0356	1.13713
$845$	−10.7492	−0.369784
$846$	0	0
$847$	20.7908	0.714381
$848$	3.18232	0.109281
$849$	0	0
$850$	13.1633	0.451497
$851$	9.96916	0.341738
$852$	0	0
$853$	−8.02002	−0.274600	−0.137300	−	0.990529i	$-0.543842\pi$
$853$	−8.02002	−0.274600	−0.137300	+	0.990529i	$0.543842\pi$
$854$	−9.23692	−0.316081
$855$	0	0
$856$	−11.4055	−0.389831
$857$	−40.5531	−1.38527	−0.692634	−	0.721289i	$-0.743550\pi$
$857$	−40.5531	−1.38527	−0.692634	+	0.721289i	$0.743550\pi$
$858$	0	0
$859$	5.42686	0.185162	0.0925810	−	0.995705i	$-0.470488\pi$
$859$	5.42686	0.185162	0.0925810	+	0.995705i	$0.470488\pi$
$860$	−13.5603	−0.462401
$861$	0	0
$862$	15.9852	0.544459
$863$	8.74148	0.297563	0.148782	−	0.988870i	$-0.452465\pi$
$863$	8.74148	0.297563	0.148782	+	0.988870i	$0.452465\pi$
$864$	0	0
$865$	14.1205	0.480111
$866$	3.76523	0.127948
$867$	0	0
$868$	−12.1416	−0.412114
$869$	−7.11994	−0.241527
$870$	0	0
$871$	−9.90066	−0.335471
$872$	−34.6797	−1.17440
$873$	0	0
$874$	−5.56533	−0.188250
$875$	−21.5067	−0.727060
$876$	0	0
$877$	−14.5540	−0.491453	−0.245726	−	0.969339i	$-0.579026\pi$
$877$	−14.5540	−0.491453	−0.245726	+	0.969339i	$0.579026\pi$
$878$	−18.9277	−0.638779
$879$	0	0
$880$	4.03243	0.135933
$881$	−14.3609	−0.483832	−0.241916	−	0.970297i	$-0.577776\pi$
$881$	−14.3609	−0.483832	−0.241916	+	0.970297i	$0.577776\pi$
$882$	0	0
$883$	−4.35973	−0.146717	−0.0733583	−	0.997306i	$-0.523372\pi$
$883$	−4.35973	−0.146717	−0.0733583	+	0.997306i	$0.523372\pi$
$884$	22.6622	0.762214
$885$	0	0
$886$	0.465668	0.0156444
$887$	54.7989	1.83997	0.919984	−	0.391956i	$-0.128201\pi$
$887$	54.7989	1.83997	0.919984	+	0.391956i	$0.128201\pi$
$888$	0	0
$889$	31.2665	1.04864
$890$	13.7426	0.460653
$891$	0	0
$892$	12.1146	0.405627
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	14.0091	0.468272
$896$	−15.7774	−0.527087
$897$	0	0
$898$	−26.4562	−0.882856
$899$	−7.71162	−0.257197
$900$	0	0
$901$	−40.0485	−1.33421
$902$	37.8525	1.26035
$903$	0	0
$904$	−7.94860	−0.264366
$905$	30.8611	1.02586
$906$	0	0
$907$	42.9844	1.42727	0.713636	−	0.700516i	$-0.247047\pi$
$907$	42.9844	1.42727	0.713636	+	0.700516i	$0.247047\pi$
$908$	18.6431	0.618692
$909$	0	0
$910$	5.82698	0.193162
$911$	29.5997	0.980683	0.490342	−	0.871530i	$-0.336872\pi$
$911$	29.5997	0.980683	0.490342	+	0.871530i	$0.336872\pi$
$912$	0	0
$913$	19.3696	0.641039
$914$	11.0085	0.364130
$915$	0	0
$916$	16.0805	0.531315
$917$	−21.4674	−0.708916
$918$	0	0
$919$	41.9641	1.38427	0.692133	−	0.721770i	$-0.256671\pi$
$919$	41.9641	1.38427	0.692133	+	0.721770i	$0.256671\pi$
$920$	−29.8249	−0.983297
$921$	0	0
$922$	24.3066	0.800497
$923$	−29.8018	−0.980938
$924$	0	0
$925$	−3.52414	−0.115873
$926$	−27.8725	−0.915948
$927$	0	0
$928$	−9.03838	−0.296699
$929$	−59.2080	−1.94255	−0.971276	−	0.237955i	$-0.923523\pi$
$929$	−59.2080	−1.94255	−0.971276	+	0.237955i	$0.923523\pi$
$930$	0	0
$931$	3.73435	0.122388
$932$	5.48868	0.179788
$933$	0	0
$934$	2.67384	0.0874908
$935$	−50.7468	−1.65960
$936$	0	0
$937$	−29.3462	−0.958698	−0.479349	−	0.877624i	$-0.659127\pi$
$937$	−29.3462	−0.958698	−0.479349	+	0.877624i	$0.659127\pi$
$938$	−5.74896	−0.187710
$939$	0	0
$940$	−2.16148	−0.0704996
$941$	4.00175	0.130453	0.0652266	−	0.997870i	$-0.479223\pi$
$941$	4.00175	0.130453	0.0652266	+	0.997870i	$0.479223\pi$
$942$	0	0
$943$	68.5232	2.23142
$944$	−7.90289	−0.257217
$945$	0	0
$946$	23.9586	0.778962
$947$	17.0515	0.554100	0.277050	−	0.960855i	$-0.410643\pi$
$947$	17.0515	0.554100	0.277050	+	0.960855i	$0.410643\pi$
$948$	0	0
$949$	30.6349	0.994450
$950$	1.96737	0.0638299
$951$	0	0
$952$	32.6261	1.05742
$953$	20.4004	0.660835	0.330418	−	0.943835i	$-0.392810\pi$
$953$	20.4004	0.660835	0.330418	+	0.943835i	$0.392810\pi$
$954$	0	0
$955$	12.8394	0.415474
$956$	−35.6204	−1.15205
$957$	0	0
$958$	10.2548	0.331316
$959$	−18.0238	−0.582018
$960$	0	0
$961$	−6.30187	−0.203286
$962$	2.90831	0.0937677
$963$	0	0
$964$	−24.7293	−0.796479
$965$	−10.7225	−0.345170
$966$	0	0
$967$	−0.267352	−0.00859746	−0.00429873	−	0.999991i	$-0.501368\pi$
$967$	−0.267352	−0.00859746	−0.00429873	+	0.999991i	$0.501368\pi$
$968$	−31.0448	−0.997817
$969$	0	0
$970$	−10.1111	−0.324648
$971$	−33.6259	−1.07911	−0.539553	−	0.841952i	$-0.681407\pi$
$971$	−33.6259	−1.07911	−0.539553	+	0.841952i	$0.681407\pi$
$972$	0	0
$973$	11.6073	0.372113
$974$	−24.6218	−0.788932
$975$	0	0
$976$	−3.37578	−0.108056
$977$	−51.3008	−1.64126	−0.820629	−	0.571461i	$-0.806377\pi$
$977$	−51.3008	−1.64126	−0.820629	+	0.571461i	$0.806377\pi$
$978$	0	0
$979$	50.6540	1.61891
$980$	8.07171	0.257841
$981$	0	0
$982$	29.4459	0.939657
$983$	36.9572	1.17875	0.589375	−	0.807859i	$-0.299374\pi$
$983$	36.9572	1.17875	0.589375	+	0.807859i	$0.299374\pi$
$984$	0	0
$985$	3.26395	0.103998
$986$	8.35790	0.266170
$987$	0	0
$988$	3.38707	0.107757
$989$	43.3715	1.37913
$990$	0	0
$991$	23.6879	0.752472	0.376236	−	0.926524i	$-0.377218\pi$
$991$	23.6879	0.752472	0.376236	+	0.926524i	$0.377218\pi$
$992$	28.9474	0.919079
$993$	0	0
$994$	−17.3049	−0.548877
$995$	−39.2107	−1.24306
$996$	0	0
$997$	16.9487	0.536769	0.268385	−	0.963312i	$-0.413510\pi$
$997$	16.9487	0.536769	0.268385	+	0.963312i	$0.413510\pi$
$998$	−30.8541	−0.976668
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.w.1.20	yes	34
3.2	odd	2		8037.2.a.v.1.15	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8037.2.a.v.1.15	✓	34	3.2	odd	2
8037.2.a.w.1.20	yes	34	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.805016 q^{2} -1.35195 q^{4} +1.59878 q^{5} +1.80711 q^{7} -2.69837 q^{8} +O(q^{10})\)	\(q+0.805016 q^{2} -1.35195 q^{4} +1.59878 q^{5} +1.80711 q^{7} -2.69837 q^{8} +1.28705 q^{10} +4.74394 q^{11} +2.50532 q^{13} +1.45475 q^{14} +0.531663 q^{16} -6.69081 q^{17} -1.00000 q^{19} -2.16148 q^{20} +3.81895 q^{22} +6.91332 q^{23} -2.44389 q^{25} +2.01683 q^{26} -2.44312 q^{28} -1.55172 q^{29} +4.96972 q^{31} +5.82474 q^{32} -5.38621 q^{34} +2.88918 q^{35} +1.44202 q^{37} -0.805016 q^{38} -4.31412 q^{40} +9.91176 q^{41} +6.27362 q^{43} -6.41357 q^{44} +5.56533 q^{46} +1.00000 q^{47} -3.73435 q^{49} -1.96737 q^{50} -3.38707 q^{52} +5.98559 q^{53} +7.58455 q^{55} -4.87626 q^{56} -1.24916 q^{58} -14.8645 q^{59} -6.34947 q^{61} +4.00071 q^{62} +3.62569 q^{64} +4.00547 q^{65} -3.95185 q^{67} +9.04563 q^{68} +2.32584 q^{70} -11.8954 q^{71} +12.2279 q^{73} +1.16085 q^{74} +1.35195 q^{76} +8.57283 q^{77} -1.50085 q^{79} +0.850015 q^{80} +7.97913 q^{82} +4.08301 q^{83} -10.6972 q^{85} +5.05036 q^{86} -12.8009 q^{88} +10.6776 q^{89} +4.52740 q^{91} -9.34645 q^{92} +0.805016 q^{94} -1.59878 q^{95} -7.85603 q^{97} -3.00621 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) 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

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Groups

Database

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