LMFDB - Embedded newform 2151.4.a.g.1.50

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.50
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.10283 q^{2} +8.83322 q^{4} +7.42154 q^{5} +5.67497 q^{7} +3.41855 q^{8} +O(q^{10})$	\(q+4.10283 q^{2} +8.83322 q^{4} +7.42154 q^{5} +5.67497 q^{7} +3.41855 q^{8} +30.4493 q^{10} -1.97818 q^{11} +0.731609 q^{13} +23.2834 q^{14} -56.6400 q^{16} -109.522 q^{17} +82.2039 q^{19} +65.5561 q^{20} -8.11612 q^{22} -189.006 q^{23} -69.9207 q^{25} +3.00167 q^{26} +50.1282 q^{28} -180.383 q^{29} -85.3258 q^{31} -259.733 q^{32} -449.352 q^{34} +42.1170 q^{35} +336.056 q^{37} +337.269 q^{38} +25.3709 q^{40} +337.799 q^{41} +46.0679 q^{43} -17.4737 q^{44} -775.460 q^{46} -411.751 q^{47} -310.795 q^{49} -286.873 q^{50} +6.46246 q^{52} -672.423 q^{53} -14.6811 q^{55} +19.4002 q^{56} -740.081 q^{58} -219.015 q^{59} +480.554 q^{61} -350.077 q^{62} -612.519 q^{64} +5.42967 q^{65} -575.008 q^{67} -967.434 q^{68} +172.799 q^{70} +483.841 q^{71} +165.298 q^{73} +1378.78 q^{74} +726.125 q^{76} -11.2261 q^{77} +539.826 q^{79} -420.356 q^{80} +1385.93 q^{82} -97.1930 q^{83} -812.825 q^{85} +189.009 q^{86} -6.76249 q^{88} -68.4685 q^{89} +4.15186 q^{91} -1669.53 q^{92} -1689.35 q^{94} +610.080 q^{95} -96.0144 q^{97} -1275.14 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * 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 $

$2$	4.10283	1.45057	0.725285	−	0.688449i	$-0.241708\pi$
$2$	4.10283	1.45057	0.725285	+	0.688449i	$0.241708\pi$
$3$	0	0
$3$	0	0
$4$	8.83322	1.10415
$5$	7.42154	0.663803	0.331902	−	0.943314i	$-0.392310\pi$
$5$	7.42154	0.663803	0.331902	+	0.943314i	$0.392310\pi$
$6$	0	0
$7$	5.67497	0.306420	0.153210	−	0.988194i	$-0.451039\pi$
$7$	5.67497	0.306420	0.153210	+	0.988194i	$0.451039\pi$
$8$	3.41855	0.151080
$9$	0	0
$10$	30.4493	0.962893
$11$	−1.97818	−0.0542221	−0.0271110	−	0.999632i	$-0.508631\pi$
$11$	−1.97818	−0.0542221	−0.0271110	+	0.999632i	$0.508631\pi$
$12$	0	0
$13$	0.731609	0.0156086	0.00780430	−	0.999970i	$-0.497516\pi$
$13$	0.731609	0.0156086	0.00780430	+	0.999970i	$0.497516\pi$
$14$	23.2834	0.444483
$15$	0	0
$16$	−56.6400	−0.885000
$17$	−109.522	−1.56253	−0.781267	−	0.624197i	$-0.785426\pi$
$17$	−109.522	−1.56253	−0.781267	+	0.624197i	$0.785426\pi$
$18$	0	0
$19$	82.2039	0.992573	0.496286	−	0.868159i	$-0.334697\pi$
$19$	82.2039	0.992573	0.496286	+	0.868159i	$0.334697\pi$
$20$	65.5561	0.732940
$21$	0	0
$22$	−8.11612	−0.0786529
$23$	−189.006	−1.71350	−0.856750	−	0.515732i	$-0.827520\pi$
$23$	−189.006	−1.71350	−0.856750	+	0.515732i	$0.827520\pi$
$24$	0	0
$25$	−69.9207	−0.559365
$26$	3.00167	0.0226414
$27$	0	0
$28$	50.1282	0.338334
$29$	−180.383	−1.15504	−0.577522	−	0.816375i	$-0.695980\pi$
$29$	−180.383	−1.15504	−0.577522	+	0.816375i	$0.695980\pi$
$30$	0	0
$31$	−85.3258	−0.494354	−0.247177	−	0.968970i	$-0.579503\pi$
$31$	−85.3258	−0.494354	−0.247177	+	0.968970i	$0.579503\pi$
$32$	−259.733	−1.43483
$33$	0	0
$34$	−449.352	−2.26656
$35$	42.1170	0.203402
$36$	0	0
$37$	336.056	1.49317	0.746584	−	0.665291i	$-0.231693\pi$
$37$	336.056	1.49317	0.746584	+	0.665291i	$0.231693\pi$
$38$	337.269	1.43980
$39$	0	0
$40$	25.3709	0.100287
$41$	337.799	1.28672	0.643358	−	0.765565i	$-0.277541\pi$
$41$	337.799	1.28672	0.643358	+	0.765565i	$0.277541\pi$
$42$	0	0
$43$	46.0679	0.163379	0.0816894	−	0.996658i	$-0.473968\pi$
$43$	46.0679	0.163379	0.0816894	+	0.996658i	$0.473968\pi$
$44$	−17.4737	−0.0598694
$45$	0	0
$46$	−775.460	−2.48555
$47$	−411.751	−1.27787	−0.638937	−	0.769259i	$-0.720626\pi$
$47$	−411.751	−1.27787	−0.638937	+	0.769259i	$0.720626\pi$
$48$	0	0
$49$	−310.795	−0.906107
$50$	−286.873	−0.811399
$51$	0	0
$52$	6.46246	0.0172343
$53$	−672.423	−1.74272	−0.871362	−	0.490641i	$-0.836763\pi$
$53$	−672.423	−1.74272	−0.871362	+	0.490641i	$0.836763\pi$
$54$	0	0
$55$	−14.6811	−0.0359928
$56$	19.4002	0.0462938
$57$	0	0
$58$	−740.081	−1.67547
$59$	−219.015	−0.483277	−0.241639	−	0.970366i	$-0.577685\pi$
$59$	−219.015	−0.483277	−0.241639	+	0.970366i	$0.577685\pi$
$60$	0	0
$61$	480.554	1.00867	0.504333	−	0.863509i	$-0.331739\pi$
$61$	480.554	1.00867	0.504333	+	0.863509i	$0.331739\pi$
$62$	−350.077	−0.717094
$63$	0	0
$64$	−612.519	−1.19633
$65$	5.42967	0.0103610
$66$	0	0
$67$	−575.008	−1.04848	−0.524241	−	0.851570i	$-0.675651\pi$
$67$	−575.008	−1.04848	−0.524241	+	0.851570i	$0.675651\pi$
$68$	−967.434	−1.72527
$69$	0	0
$70$	172.799	0.295049
$71$	483.841	0.808751	0.404376	−	0.914593i	$-0.367489\pi$
$71$	483.841	0.808751	0.404376	+	0.914593i	$0.367489\pi$
$72$	0	0
$73$	165.298	0.265023	0.132511	−	0.991181i	$-0.457696\pi$
$73$	165.298	0.265023	0.132511	+	0.991181i	$0.457696\pi$
$74$	1378.78	2.16594
$75$	0	0
$76$	726.125	1.09595
$77$	−11.2261	−0.0166147
$78$	0	0
$79$	539.826	0.768799	0.384400	−	0.923167i	$-0.374408\pi$
$79$	539.826	0.768799	0.384400	+	0.923167i	$0.374408\pi$
$80$	−420.356	−0.587466
$81$	0	0
$82$	1385.93	1.86647
$83$	−97.1930	−0.128534	−0.0642670	−	0.997933i	$-0.520471\pi$
$83$	−97.1930	−0.128534	−0.0642670	+	0.997933i	$0.520471\pi$
$84$	0	0
$85$	−812.825	−1.03721
$86$	189.009	0.236992
$87$	0	0
$88$	−6.76249	−0.00819186
$89$	−68.4685	−0.0815465	−0.0407733	−	0.999168i	$-0.512982\pi$
$89$	−68.4685	−0.0815465	−0.0407733	+	0.999168i	$0.512982\pi$
$90$	0	0
$91$	4.15186	0.00478278
$92$	−1669.53	−1.89196
$93$	0	0
$94$	−1689.35	−1.85365
$95$	610.080	0.658873
$96$	0	0
$97$	−96.0144	−0.100503	−0.0502515	−	0.998737i	$-0.516002\pi$
$97$	−96.0144	−0.100503	−0.0502515	+	0.998737i	$0.516002\pi$
$98$	−1275.14	−1.31437
$99$	0	0
$100$	−617.625	−0.617625
$101$	263.049	0.259152	0.129576	−	0.991570i	$-0.458638\pi$
$101$	263.049	0.259152	0.129576	+	0.991570i	$0.458638\pi$
$102$	0	0
$103$	658.943	0.630365	0.315183	−	0.949031i	$-0.397934\pi$
$103$	658.943	0.630365	0.315183	+	0.949031i	$0.397934\pi$
$104$	2.50104	0.00235815
$105$	0	0
$106$	−2758.84	−2.52794
$107$	−410.600	−0.370973	−0.185487	−	0.982647i	$-0.559386\pi$
$107$	−410.600	−0.370973	−0.185487	+	0.982647i	$0.559386\pi$
$108$	0	0
$109$	163.540	0.143709	0.0718547	−	0.997415i	$-0.477108\pi$
$109$	163.540	0.143709	0.0718547	+	0.997415i	$0.477108\pi$
$110$	−60.2342	−0.0522100
$111$	0	0
$112$	−321.430	−0.271181
$113$	329.469	0.274282	0.137141	−	0.990552i	$-0.456209\pi$
$113$	329.469	0.274282	0.137141	+	0.990552i	$0.456209\pi$
$114$	0	0
$115$	−1402.72	−1.13743
$116$	−1593.36	−1.27534
$117$	0	0
$118$	−898.582	−0.701027
$119$	−621.536	−0.478791
$120$	0	0
$121$	−1327.09	−0.997060
$122$	1971.63	1.46314
$123$	0	0
$124$	−753.701	−0.545842
$125$	−1446.61	−1.03511
$126$	0	0
$127$	−13.3598	−0.00933458	−0.00466729	−	0.999989i	$-0.501486\pi$
$127$	−13.3598	−0.00933458	−0.00466729	+	0.999989i	$0.501486\pi$
$128$	−435.201	−0.300521
$129$	0	0
$130$	22.2770	0.0150294
$131$	2588.42	1.72635	0.863174	−	0.504906i	$-0.168473\pi$
$131$	2588.42	1.72635	0.863174	+	0.504906i	$0.168473\pi$
$132$	0	0
$133$	466.505	0.304144
$134$	−2359.16	−1.52090
$135$	0	0
$136$	−374.407	−0.236067
$137$	2002.40	1.24873	0.624367	−	0.781131i	$-0.285357\pi$
$137$	2002.40	1.24873	0.624367	+	0.781131i	$0.285357\pi$
$138$	0	0
$139$	−388.896	−0.237307	−0.118654	−	0.992936i	$-0.537858\pi$
$139$	−388.896	−0.237307	−0.118654	+	0.992936i	$0.537858\pi$
$140$	372.029	0.224587
$141$	0	0
$142$	1985.12	1.17315
$143$	−1.44725	−0.000846331 0
$144$	0	0
$145$	−1338.72	−0.766722
$146$	678.190	0.384434
$147$	0	0
$148$	2968.45	1.64868
$149$	−2677.35	−1.47206	−0.736030	−	0.676949i	$-0.763302\pi$
$149$	−2677.35	−1.47206	−0.736030	+	0.676949i	$0.763302\pi$
$150$	0	0
$151$	−850.590	−0.458411	−0.229205	−	0.973378i	$-0.573613\pi$
$151$	−850.590	−0.458411	−0.229205	+	0.973378i	$0.573613\pi$
$152$	281.018	0.149958
$153$	0	0
$154$	−46.0587	−0.0241008
$155$	−633.249	−0.328153
$156$	0	0
$157$	1988.14	1.01064	0.505320	−	0.862932i	$-0.331374\pi$
$157$	1988.14	1.01064	0.505320	+	0.862932i	$0.331374\pi$
$158$	2214.81	1.11520
$159$	0	0
$160$	−1927.62	−0.952447
$161$	−1072.60	−0.525050
$162$	0	0
$163$	1204.81	0.578946	0.289473	−	0.957186i	$-0.406520\pi$
$163$	1204.81	0.578946	0.289473	+	0.957186i	$0.406520\pi$
$164$	2983.85	1.42073
$165$	0	0
$166$	−398.766	−0.186447
$167$	−3692.79	−1.71112	−0.855559	−	0.517706i	$-0.826786\pi$
$167$	−3692.79	−1.71112	−0.855559	+	0.517706i	$0.826786\pi$
$168$	0	0
$169$	−2196.46	−0.999756
$170$	−3334.88	−1.50455
$171$	0	0
$172$	406.928	0.180395
$173$	451.652	0.198488	0.0992442	−	0.995063i	$-0.468358\pi$
$173$	451.652	0.198488	0.0992442	+	0.995063i	$0.468358\pi$
$174$	0	0
$175$	−396.798	−0.171400
$176$	112.044	0.0479865
$177$	0	0
$178$	−280.914	−0.118289
$179$	−629.199	−0.262729	−0.131365	−	0.991334i	$-0.541936\pi$
$179$	−629.199	−0.262729	−0.131365	+	0.991334i	$0.541936\pi$
$180$	0	0
$181$	−2513.18	−1.03206	−0.516032	−	0.856569i	$-0.672591\pi$
$181$	−2513.18	−1.03206	−0.516032	+	0.856569i	$0.672591\pi$
$182$	17.0344	0.00693776
$183$	0	0
$184$	−646.126	−0.258875
$185$	2494.05	0.991169
$186$	0	0
$187$	216.654	0.0847238
$188$	−3637.09	−1.41097
$189$	0	0
$190$	2503.06	0.955741
$191$	−228.649	−0.0866201	−0.0433100	−	0.999062i	$-0.513790\pi$
$191$	−228.649	−0.0866201	−0.0433100	+	0.999062i	$0.513790\pi$
$192$	0	0
$193$	−4350.14	−1.62243	−0.811217	−	0.584745i	$-0.801195\pi$
$193$	−4350.14	−1.62243	−0.811217	+	0.584745i	$0.801195\pi$
$194$	−393.931	−0.145787
$195$	0	0
$196$	−2745.32	−1.00048
$197$	−584.342	−0.211333	−0.105667	−	0.994402i	$-0.533698\pi$
$197$	−584.342	−0.211333	−0.105667	+	0.994402i	$0.533698\pi$
$198$	0	0
$199$	−2528.13	−0.900576	−0.450288	−	0.892883i	$-0.648679\pi$
$199$	−2528.13	−0.900576	−0.450288	+	0.892883i	$0.648679\pi$
$200$	−239.027	−0.0845089
$201$	0	0
$202$	1079.24	0.375917
$203$	−1023.67	−0.353928
$204$	0	0
$205$	2506.99	0.854126
$206$	2703.53	0.914389
$207$	0	0
$208$	−41.4384	−0.0138136
$209$	−162.614	−0.0538193
$210$	0	0
$211$	−5296.60	−1.72812	−0.864059	−	0.503390i	$-0.832086\pi$
$211$	−5296.60	−1.72812	−0.864059	+	0.503390i	$0.832086\pi$
$212$	−5939.66	−1.92423
$213$	0	0
$214$	−1684.62	−0.538123
$215$	341.895	0.108451
$216$	0	0
$217$	−484.221	−0.151480
$218$	670.978	0.208460
$219$	0	0
$220$	−129.682	−0.0397415
$221$	−80.1276	−0.0243890
$222$	0	0
$223$	3139.10	0.942643	0.471321	−	0.881961i	$-0.343777\pi$
$223$	3139.10	0.942643	0.471321	+	0.881961i	$0.343777\pi$
$224$	−1473.98	−0.439661
$225$	0	0
$226$	1351.76	0.397865
$227$	11.5535	0.00337812	0.00168906	−	0.999999i	$-0.499462\pi$
$227$	11.5535	0.00337812	0.00168906	+	0.999999i	$0.499462\pi$
$228$	0	0
$229$	−1148.68	−0.331470	−0.165735	−	0.986170i	$-0.553000\pi$
$229$	−1148.68	−0.331470	−0.165735	+	0.986170i	$0.553000\pi$
$230$	−5755.11	−1.64992
$231$	0	0
$232$	−616.648	−0.174504
$233$	950.126	0.267145	0.133573	−	0.991039i	$-0.457355\pi$
$233$	950.126	0.267145	0.133573	+	0.991039i	$0.457355\pi$
$234$	0	0
$235$	−3055.83	−0.848257
$236$	−1934.61	−0.533611
$237$	0	0
$238$	−2550.06	−0.694519
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	1914.79	0.511794	0.255897	−	0.966704i	$-0.417629\pi$
$241$	1914.79	0.511794	0.255897	+	0.966704i	$0.417629\pi$
$242$	−5444.81	−1.44630
$243$	0	0
$244$	4244.83	1.11372
$245$	−2306.58	−0.601477
$246$	0	0
$247$	60.1412	0.0154927
$248$	−291.690	−0.0746869
$249$	0	0
$250$	−5935.21	−1.50150
$251$	6780.29	1.70505	0.852526	−	0.522685i	$-0.175070\pi$
$251$	6780.29	1.70505	0.852526	+	0.522685i	$0.175070\pi$
$252$	0	0
$253$	373.887	0.0929095
$254$	−54.8131	−0.0135405
$255$	0	0
$256$	3114.60	0.760400
$257$	2358.47	0.572442	0.286221	−	0.958164i	$-0.407601\pi$
$257$	2358.47	0.572442	0.286221	+	0.958164i	$0.407601\pi$
$258$	0	0
$259$	1907.11	0.457536
$260$	47.9615	0.0114402
$261$	0	0
$262$	10619.9	2.50419
$263$	−3356.03	−0.786849	−0.393425	−	0.919357i	$-0.628710\pi$
$263$	−3356.03	−0.786849	−0.393425	+	0.919357i	$0.628710\pi$
$264$	0	0
$265$	−4990.42	−1.15683
$266$	1913.99	0.441181
$267$	0	0
$268$	−5079.17	−1.15768
$269$	7072.46	1.60303	0.801516	−	0.597973i	$-0.204027\pi$
$269$	7072.46	1.60303	0.801516	+	0.597973i	$0.204027\pi$
$270$	0	0
$271$	−2169.26	−0.486248	−0.243124	−	0.969995i	$-0.578172\pi$
$271$	−2169.26	−0.486248	−0.243124	+	0.969995i	$0.578172\pi$
$272$	6203.35	1.38284
$273$	0	0
$274$	8215.51	1.81138
$275$	138.315	0.0303299
$276$	0	0
$277$	3570.50	0.774479	0.387240	−	0.921979i	$-0.373429\pi$
$277$	3570.50	0.774479	0.387240	+	0.921979i	$0.373429\pi$
$278$	−1595.57	−0.344231
$279$	0	0
$280$	143.979	0.0307300
$281$	−2408.14	−0.511238	−0.255619	−	0.966778i	$-0.582279\pi$
$281$	−2408.14	−0.511238	−0.255619	+	0.966778i	$0.582279\pi$
$282$	0	0
$283$	4218.51	0.886093	0.443046	−	0.896499i	$-0.353898\pi$
$283$	4218.51	0.886093	0.443046	+	0.896499i	$0.353898\pi$
$284$	4273.87	0.892984
$285$	0	0
$286$	−5.93783	−0.00122766
$287$	1917.00	0.394275
$288$	0	0
$289$	7082.14	1.44151
$290$	−5492.54	−1.11218
$291$	0	0
$292$	1460.11	0.292626
$293$	7019.97	1.39970	0.699848	−	0.714292i	$-0.253251\pi$
$293$	7019.97	1.39970	0.699848	+	0.714292i	$0.253251\pi$
$294$	0	0
$295$	−1625.43	−0.320801
$296$	1148.82	0.225588
$297$	0	0
$298$	−10984.7	−2.13533
$299$	−138.279	−0.0267453
$300$	0	0
$301$	261.434	0.0500624
$302$	−3489.82	−0.664956
$303$	0	0
$304$	−4656.03	−0.878427
$305$	3566.45	0.669555
$306$	0	0
$307$	3707.37	0.689221	0.344610	−	0.938746i	$-0.388011\pi$
$307$	3707.37	0.689221	0.344610	+	0.938746i	$0.388011\pi$
$308$	−99.1625	−0.0183452
$309$	0	0
$310$	−2598.11	−0.476009
$311$	−7560.97	−1.37860	−0.689298	−	0.724478i	$-0.742081\pi$
$311$	−7560.97	−1.37860	−0.689298	+	0.724478i	$0.742081\pi$
$312$	0	0
$313$	−629.159	−0.113617	−0.0568086	−	0.998385i	$-0.518092\pi$
$313$	−629.159	−0.113617	−0.0568086	+	0.998385i	$0.518092\pi$
$314$	8156.99	1.46600
$315$	0	0
$316$	4768.40	0.848871
$317$	−10835.3	−1.91979	−0.959893	−	0.280366i	$-0.909544\pi$
$317$	−10835.3	−1.91979	−0.959893	+	0.280366i	$0.909544\pi$
$318$	0	0
$319$	356.829	0.0626289
$320$	−4545.84	−0.794125
$321$	0	0
$322$	−4400.71	−0.761621
$323$	−9003.17	−1.55093
$324$	0	0
$325$	−51.1546	−0.00873092
$326$	4943.14	0.839801
$327$	0	0
$328$	1154.78	0.194397
$329$	−2336.68	−0.391566
$330$	0	0
$331$	−4714.96	−0.782953	−0.391476	−	0.920188i	$-0.628036\pi$
$331$	−4714.96	−0.782953	−0.391476	+	0.920188i	$0.628036\pi$
$332$	−858.527	−0.141921
$333$	0	0
$334$	−15150.9	−2.48210
$335$	−4267.44	−0.695986
$336$	0	0
$337$	−2761.14	−0.446318	−0.223159	−	0.974782i	$-0.571637\pi$
$337$	−2761.14	−0.446318	−0.223159	+	0.974782i	$0.571637\pi$
$338$	−9011.72	−1.45022
$339$	0	0
$340$	−7179.86	−1.14524
$341$	168.789	0.0268049
$342$	0	0
$343$	−3710.26	−0.584068
$344$	157.485	0.0246832
$345$	0	0
$346$	1853.05	0.287921
$347$	−2930.86	−0.453420	−0.226710	−	0.973962i	$-0.572797\pi$
$347$	−2930.86	−0.453420	−0.226710	+	0.973962i	$0.572797\pi$
$348$	0	0
$349$	11425.2	1.75236	0.876181	−	0.481982i	$-0.160083\pi$
$349$	11425.2	1.75236	0.876181	+	0.481982i	$0.160083\pi$
$350$	−1627.99	−0.248628
$351$	0	0
$352$	513.797	0.0777997
$353$	3672.52	0.553735	0.276868	−	0.960908i	$-0.410704\pi$
$353$	3672.52	0.553735	0.276868	+	0.960908i	$0.410704\pi$
$354$	0	0
$355$	3590.85	0.536852
$356$	−604.797	−0.0900398
$357$	0	0
$358$	−2581.50	−0.381107
$359$	194.274	0.0285609	0.0142805	−	0.999898i	$-0.495454\pi$
$359$	194.274	0.0285609	0.0142805	+	0.999898i	$0.495454\pi$
$360$	0	0
$361$	−101.511	−0.0147997
$362$	−10311.2	−1.49708
$363$	0	0
$364$	36.6743	0.00528092
$365$	1226.77	0.175923
$366$	0	0
$367$	−8736.56	−1.24263	−0.621315	−	0.783561i	$-0.713401\pi$
$367$	−8736.56	−1.24263	−0.621315	+	0.783561i	$0.713401\pi$
$368$	10705.3	1.51645
$369$	0	0
$370$	10232.7	1.43776
$371$	−3815.98	−0.534005
$372$	0	0
$373$	8401.37	1.16624	0.583118	−	0.812387i	$-0.301832\pi$
$373$	8401.37	1.16624	0.583118	+	0.812387i	$0.301832\pi$
$374$	888.897	0.122898
$375$	0	0
$376$	−1407.59	−0.193061
$377$	−131.970	−0.0180286
$378$	0	0
$379$	8294.84	1.12421	0.562107	−	0.827064i	$-0.309991\pi$
$379$	8294.84	1.12421	0.562107	+	0.827064i	$0.309991\pi$
$380$	5388.97	0.727496
$381$	0	0
$382$	−938.106	−0.125648
$383$	13231.8	1.76531	0.882656	−	0.470020i	$-0.155753\pi$
$383$	13231.8	1.76531	0.882656	+	0.470020i	$0.155753\pi$
$384$	0	0
$385$	−83.3149	−0.0110289
$386$	−17847.9	−2.35345
$387$	0	0
$388$	−848.116	−0.110971
$389$	−1343.03	−0.175050	−0.0875250	−	0.996162i	$-0.527896\pi$
$389$	−1343.03	−0.175050	−0.0875250	+	0.996162i	$0.527896\pi$
$390$	0	0
$391$	20700.4	2.67740
$392$	−1062.47	−0.136895
$393$	0	0
$394$	−2397.46	−0.306553
$395$	4006.34	0.510331
$396$	0	0
$397$	−5223.00	−0.660290	−0.330145	−	0.943930i	$-0.607098\pi$
$397$	−5223.00	−0.660290	−0.330145	+	0.943930i	$0.607098\pi$
$398$	−10372.5	−1.30635
$399$	0	0
$400$	3960.31	0.495039
$401$	2065.04	0.257165	0.128582	−	0.991699i	$-0.458957\pi$
$401$	2065.04	0.257165	0.128582	+	0.991699i	$0.458957\pi$
$402$	0	0
$403$	−62.4251	−0.00771617
$404$	2323.57	0.286143
$405$	0	0
$406$	−4199.94	−0.513397
$407$	−664.777	−0.0809626
$408$	0	0
$409$	6781.26	0.819833	0.409917	−	0.912123i	$-0.365558\pi$
$409$	6781.26	0.819833	0.409917	+	0.912123i	$0.365558\pi$
$410$	10285.8	1.23897
$411$	0	0
$412$	5820.59	0.696019
$413$	−1242.90	−0.148086
$414$	0	0
$415$	−721.322	−0.0853213
$416$	−190.023	−0.0223958
$417$	0	0
$418$	−667.177	−0.0780687
$419$	−12444.2	−1.45092	−0.725462	−	0.688262i	$-0.758374\pi$
$419$	−12444.2	−1.45092	−0.725462	+	0.688262i	$0.758374\pi$
$420$	0	0
$421$	1394.38	0.161420	0.0807099	−	0.996738i	$-0.474281\pi$
$421$	1394.38	0.161420	0.0807099	+	0.996738i	$0.474281\pi$
$422$	−21731.1	−2.50676
$423$	0	0
$424$	−2298.71	−0.263291
$425$	7657.88	0.874027
$426$	0	0
$427$	2727.13	0.309075
$428$	−3626.91	−0.409611
$429$	0	0
$430$	1402.74	0.157316
$431$	−10732.9	−1.19950	−0.599750	−	0.800187i	$-0.704734\pi$
$431$	−10732.9	−1.19950	−0.599750	+	0.800187i	$0.704734\pi$
$432$	0	0
$433$	4781.67	0.530698	0.265349	−	0.964152i	$-0.414513\pi$
$433$	4781.67	0.530698	0.265349	+	0.964152i	$0.414513\pi$
$434$	−1986.68	−0.219732
$435$	0	0
$436$	1444.59	0.158677
$437$	−15537.0	−1.70077
$438$	0	0
$439$	7803.24	0.848356	0.424178	−	0.905579i	$-0.360563\pi$
$439$	7803.24	0.848356	0.424178	+	0.905579i	$0.360563\pi$
$440$	−50.1881	−0.00543778
$441$	0	0
$442$	−328.750	−0.0353779
$443$	−5285.63	−0.566880	−0.283440	−	0.958990i	$-0.591476\pi$
$443$	−5285.63	−0.566880	−0.283440	+	0.958990i	$0.591476\pi$
$444$	0	0
$445$	−508.142	−0.0541308
$446$	12879.2	1.36737
$447$	0	0
$448$	−3476.03	−0.366578
$449$	−12053.2	−1.26687	−0.633436	−	0.773795i	$-0.718356\pi$
$449$	−12053.2	−1.26687	−0.633436	+	0.773795i	$0.718356\pi$
$450$	0	0
$451$	−668.227	−0.0697684
$452$	2910.27	0.302849
$453$	0	0
$454$	47.4021	0.00490020
$455$	30.8132	0.00317483
$456$	0	0
$457$	3222.88	0.329891	0.164945	−	0.986303i	$-0.447255\pi$
$457$	3222.88	0.329891	0.164945	+	0.986303i	$0.447255\pi$
$458$	−4712.83	−0.480821
$459$	0	0
$460$	−12390.5	−1.25589
$461$	−11082.0	−1.11961	−0.559805	−	0.828624i	$-0.689124\pi$
$461$	−11082.0	−1.11961	−0.559805	+	0.828624i	$0.689124\pi$
$462$	0	0
$463$	9696.33	0.973276	0.486638	−	0.873604i	$-0.338223\pi$
$463$	9696.33	0.973276	0.486638	+	0.873604i	$0.338223\pi$
$464$	10216.9	1.02221
$465$	0	0
$466$	3898.21	0.387513
$467$	−13395.3	−1.32732	−0.663662	−	0.748033i	$-0.730999\pi$
$467$	−13395.3	−1.32732	−0.663662	+	0.748033i	$0.730999\pi$
$468$	0	0
$469$	−3263.15	−0.321276
$470$	−12537.6	−1.23046
$471$	0	0
$472$	−748.714	−0.0730135
$473$	−91.1304	−0.00885873
$474$	0	0
$475$	−5747.76	−0.555211
$476$	−5490.16	−0.528658
$477$	0	0
$478$	980.576	0.0938295
$479$	−12902.1	−1.23071	−0.615356	−	0.788249i	$-0.710988\pi$
$479$	−12902.1	−1.23071	−0.615356	+	0.788249i	$0.710988\pi$
$480$	0	0
$481$	245.861	0.0233063
$482$	7856.05	0.742392
$483$	0	0
$484$	−11722.4	−1.10091
$485$	−712.575	−0.0667142
$486$	0	0
$487$	15869.1	1.47658	0.738292	−	0.674481i	$-0.235633\pi$
$487$	15869.1	1.47658	0.738292	+	0.674481i	$0.235633\pi$
$488$	1642.80	0.152389
$489$	0	0
$490$	−9463.49	−0.872484
$491$	−15691.2	−1.44223	−0.721114	−	0.692816i	$-0.756370\pi$
$491$	−15691.2	−1.44223	−0.721114	+	0.692816i	$0.756370\pi$
$492$	0	0
$493$	19756.0	1.80480
$494$	246.749	0.0224732
$495$	0	0
$496$	4832.85	0.437503
$497$	2745.78	0.247817
$498$	0	0
$499$	−15943.3	−1.43030	−0.715149	−	0.698972i	$-0.753641\pi$
$499$	−15943.3	−1.43030	−0.715149	+	0.698972i	$0.753641\pi$
$500$	−12778.2	−1.14292
$501$	0	0
$502$	27818.4	2.47330
$503$	1777.64	0.157576	0.0787882	−	0.996891i	$-0.474895\pi$
$503$	1777.64	0.157576	0.0787882	+	0.996891i	$0.474895\pi$
$504$	0	0
$505$	1952.23	0.172026
$506$	1534.00	0.134772
$507$	0	0
$508$	−118.010	−0.0103068
$509$	−9604.37	−0.836358	−0.418179	−	0.908365i	$-0.637331\pi$
$509$	−9604.37	−0.836358	−0.418179	+	0.908365i	$0.637331\pi$
$510$	0	0
$511$	938.061	0.0812082
$512$	16260.3	1.40353
$513$	0	0
$514$	9676.42	0.830367
$515$	4890.38	0.418438
$516$	0	0
$517$	814.517	0.0692890
$518$	7824.53	0.663688
$519$	0	0
$520$	18.5616	0.00156535
$521$	−3976.84	−0.334412	−0.167206	−	0.985922i	$-0.553474\pi$
$521$	−3976.84	−0.334412	−0.167206	+	0.985922i	$0.553474\pi$
$522$	0	0
$523$	5611.16	0.469137	0.234569	−	0.972100i	$-0.424632\pi$
$523$	5611.16	0.469137	0.234569	+	0.972100i	$0.424632\pi$
$524$	22864.1	1.90615
$525$	0	0
$526$	−13769.2	−1.14138
$527$	9345.08	0.772444
$528$	0	0
$529$	23556.3	1.93608
$530$	−20474.8	−1.67806
$531$	0	0
$532$	4120.74	0.335821
$533$	247.137	0.0200839
$534$	0	0
$535$	−3047.28	−0.246253
$536$	−1965.69	−0.158405
$537$	0	0
$538$	29017.1	2.32531
$539$	614.807	0.0491310
$540$	0	0
$541$	−8065.22	−0.640944	−0.320472	−	0.947258i	$-0.603842\pi$
$541$	−8065.22	−0.640944	−0.320472	+	0.947258i	$0.603842\pi$
$542$	−8900.12	−0.705337
$543$	0	0
$544$	28446.5	2.24198
$545$	1213.72	0.0953947
$546$	0	0
$547$	−7739.19	−0.604943	−0.302472	−	0.953158i	$-0.597812\pi$
$547$	−7739.19	−0.604943	−0.302472	+	0.953158i	$0.597812\pi$
$548$	17687.6	1.37879
$549$	0	0
$550$	567.485	0.0439957
$551$	−14828.2	−1.14647
$552$	0	0
$553$	3063.49	0.235575
$554$	14649.2	1.12344
$555$	0	0
$556$	−3435.20	−0.262024
$557$	12486.6	0.949865	0.474933	−	0.880022i	$-0.342472\pi$
$557$	12486.6	0.949865	0.474933	+	0.880022i	$0.342472\pi$
$558$	0	0
$559$	33.7037	0.00255012
$560$	−2385.51	−0.180011
$561$	0	0
$562$	−9880.20	−0.741586
$563$	11511.2	0.861703	0.430852	−	0.902423i	$-0.358213\pi$
$563$	11511.2	0.861703	0.430852	+	0.902423i	$0.358213\pi$
$564$	0	0
$565$	2445.17	0.182069
$566$	17307.8	1.28534
$567$	0	0
$568$	1654.03	0.122186
$569$	−15667.4	−1.15432	−0.577162	−	0.816629i	$-0.695840\pi$
$569$	−15667.4	−1.15432	−0.577162	+	0.816629i	$0.695840\pi$
$570$	0	0
$571$	−3951.30	−0.289591	−0.144796	−	0.989462i	$-0.546252\pi$
$571$	−3951.30	−0.289591	−0.144796	+	0.989462i	$0.546252\pi$
$572$	−12.7839	−0.000934478 0
$573$	0	0
$574$	7865.13	0.571923
$575$	13215.4	0.958473
$576$	0	0
$577$	1864.58	0.134530	0.0672648	−	0.997735i	$-0.478573\pi$
$577$	1864.58	0.134530	0.0672648	+	0.997735i	$0.478573\pi$
$578$	29056.8	2.09101
$579$	0	0
$580$	−11825.2	−0.846578
$581$	−551.567	−0.0393853
$582$	0	0
$583$	1330.17	0.0944941
$584$	565.079	0.0400396
$585$	0	0
$586$	28801.7	2.03036
$587$	22511.7	1.58289	0.791447	−	0.611238i	$-0.209328\pi$
$587$	22511.7	1.58289	0.791447	+	0.611238i	$0.209328\pi$
$588$	0	0
$589$	−7014.12	−0.490682
$590$	−6668.87	−0.465344
$591$	0	0
$592$	−19034.2	−1.32145
$593$	−6682.50	−0.462761	−0.231380	−	0.972863i	$-0.574324\pi$
$593$	−6682.50	−0.462761	−0.231380	+	0.972863i	$0.574324\pi$
$594$	0	0
$595$	−4612.76	−0.317823
$596$	−23649.6	−1.62538
$597$	0	0
$598$	−567.334	−0.0387960
$599$	14057.5	0.958890	0.479445	−	0.877572i	$-0.340838\pi$
$599$	14057.5	0.958890	0.479445	+	0.877572i	$0.340838\pi$
$600$	0	0
$601$	−17801.8	−1.20824	−0.604118	−	0.796895i	$-0.706474\pi$
$601$	−17801.8	−1.20824	−0.604118	+	0.796895i	$0.706474\pi$
$602$	1072.62	0.0726191
$603$	0	0
$604$	−7513.44	−0.506155
$605$	−9849.03	−0.661851
$606$	0	0
$607$	19937.3	1.33316	0.666581	−	0.745433i	$-0.267757\pi$
$607$	19937.3	1.33316	0.666581	+	0.745433i	$0.267757\pi$
$608$	−21351.1	−1.42418
$609$	0	0
$610$	14632.5	0.971237
$611$	−301.241	−0.0199458
$612$	0	0
$613$	27224.1	1.79375	0.896877	−	0.442281i	$-0.145831\pi$
$613$	27224.1	1.79375	0.896877	+	0.442281i	$0.145831\pi$
$614$	15210.7	0.999762
$615$	0	0
$616$	−38.3769	−0.00251015
$617$	15111.7	0.986018	0.493009	−	0.870024i	$-0.335897\pi$
$617$	15111.7	0.986018	0.493009	+	0.870024i	$0.335897\pi$
$618$	0	0
$619$	12209.5	0.792796	0.396398	−	0.918079i	$-0.370260\pi$
$619$	12209.5	0.792796	0.396398	+	0.918079i	$0.370260\pi$
$620$	−5593.63	−0.362331
$621$	0	0
$622$	−31021.4	−1.99975
$623$	−388.556	−0.0249874
$624$	0	0
$625$	−1996.01	−0.127745
$626$	−2581.33	−0.164810
$627$	0	0
$628$	17561.6	1.11590
$629$	−36805.6	−2.33312
$630$	0	0
$631$	3761.08	0.237284	0.118642	−	0.992937i	$-0.462146\pi$
$631$	3761.08	0.237284	0.118642	+	0.992937i	$0.462146\pi$
$632$	1845.42	0.116150
$633$	0	0
$634$	−44455.5	−2.78478
$635$	−99.1505	−0.00619633
$636$	0	0
$637$	−227.380	−0.0141431
$638$	1464.01	0.0908476
$639$	0	0
$640$	−3229.86	−0.199487
$641$	29899.9	1.84239	0.921197	−	0.389097i	$-0.127213\pi$
$641$	29899.9	1.84239	0.921197	+	0.389097i	$0.127213\pi$
$642$	0	0
$643$	21028.7	1.28972	0.644860	−	0.764300i	$-0.276915\pi$
$643$	21028.7	1.28972	0.644860	+	0.764300i	$0.276915\pi$
$644$	−9474.54	−0.579735
$645$	0	0
$646$	−36938.5	−2.24973
$647$	6910.94	0.419934	0.209967	−	0.977709i	$-0.432664\pi$
$647$	6910.94	0.419934	0.209967	+	0.977709i	$0.432664\pi$
$648$	0	0
$649$	433.251	0.0262043
$650$	−209.879	−0.0126648
$651$	0	0
$652$	10642.4	0.639244
$653$	24024.8	1.43976	0.719879	−	0.694099i	$-0.244197\pi$
$653$	24024.8	1.43976	0.719879	+	0.694099i	$0.244197\pi$
$654$	0	0
$655$	19210.1	1.14596
$656$	−19133.0	−1.13874
$657$	0	0
$658$	−9586.98	−0.567993
$659$	−7052.74	−0.416898	−0.208449	−	0.978033i	$-0.566841\pi$
$659$	−7052.74	−0.416898	−0.208449	+	0.978033i	$0.566841\pi$
$660$	0	0
$661$	4552.03	0.267857	0.133929	−	0.990991i	$-0.457241\pi$
$661$	4552.03	0.267857	0.133929	+	0.990991i	$0.457241\pi$
$662$	−19344.7	−1.13573
$663$	0	0
$664$	−332.259	−0.0194189
$665$	3462.19	0.201891
$666$	0	0
$667$	34093.5	1.97917
$668$	−32619.2	−1.88933
$669$	0	0
$670$	−17508.6	−1.00958
$671$	−950.620	−0.0546919
$672$	0	0
$673$	−7578.33	−0.434061	−0.217031	−	0.976165i	$-0.569637\pi$
$673$	−7578.33	−0.434061	−0.217031	+	0.976165i	$0.569637\pi$
$674$	−11328.5	−0.647415
$675$	0	0
$676$	−19401.8	−1.10388
$677$	−14482.8	−0.822187	−0.411093	−	0.911593i	$-0.634853\pi$
$677$	−14482.8	−0.822187	−0.411093	+	0.911593i	$0.634853\pi$
$678$	0	0
$679$	−544.879	−0.0307961
$680$	−2778.68	−0.156702
$681$	0	0
$682$	692.514	0.0388823
$683$	−12786.8	−0.716357	−0.358179	−	0.933653i	$-0.616602\pi$
$683$	−12786.8	−0.716357	−0.358179	+	0.933653i	$0.616602\pi$
$684$	0	0
$685$	14860.9	0.828914
$686$	−15222.6	−0.847232
$687$	0	0
$688$	−2609.29	−0.144590
$689$	−491.951	−0.0272015
$690$	0	0
$691$	−20525.4	−1.12999	−0.564994	−	0.825095i	$-0.691122\pi$
$691$	−20525.4	−1.12999	−0.564994	+	0.825095i	$0.691122\pi$
$692$	3989.54	0.219161
$693$	0	0
$694$	−12024.8	−0.657718
$695$	−2886.21	−0.157525
$696$	0	0
$697$	−36996.6	−2.01054
$698$	46875.5	2.54192
$699$	0	0
$700$	−3505.00	−0.189252
$701$	26106.0	1.40658	0.703288	−	0.710905i	$-0.251714\pi$
$701$	26106.0	1.40658	0.703288	+	0.710905i	$0.251714\pi$
$702$	0	0
$703$	27625.1	1.48208
$704$	1211.67	0.0648673
$705$	0	0
$706$	15067.7	0.803231
$707$	1492.79	0.0794091
$708$	0	0
$709$	11015.3	0.583480	0.291740	−	0.956498i	$-0.405766\pi$
$709$	11015.3	0.583480	0.291740	+	0.956498i	$0.405766\pi$
$710$	14732.6	0.778741
$711$	0	0
$712$	−234.063	−0.0123200
$713$	16127.1	0.847075
$714$	0	0
$715$	−10.7408	−0.000561797 0
$716$	−5557.85	−0.290093
$717$	0	0
$718$	797.072	0.0414296
$719$	22222.9	1.15267	0.576337	−	0.817212i	$-0.304482\pi$
$719$	22222.9	1.15267	0.576337	+	0.817212i	$0.304482\pi$
$720$	0	0
$721$	3739.48	0.193156
$722$	−416.483	−0.0214680
$723$	0	0
$724$	−22199.5	−1.13956
$725$	12612.5	0.646092
$726$	0	0
$727$	−1110.63	−0.0566591	−0.0283295	−	0.999599i	$-0.509019\pi$
$727$	−1110.63	−0.0566591	−0.0283295	+	0.999599i	$0.509019\pi$
$728$	14.1933	0.000722582 0
$729$	0	0
$730$	5033.21	0.255189
$731$	−5045.46	−0.255285
$732$	0	0
$733$	−12745.1	−0.642225	−0.321113	−	0.947041i	$-0.604057\pi$
$733$	−12745.1	−0.642225	−0.321113	+	0.947041i	$0.604057\pi$
$734$	−35844.6	−1.80252
$735$	0	0
$736$	49091.1	2.45859
$737$	1137.47	0.0568509
$738$	0	0
$739$	−33895.1	−1.68721	−0.843607	−	0.536961i	$-0.819572\pi$
$739$	−33895.1	−1.68721	−0.843607	+	0.536961i	$0.819572\pi$
$740$	22030.5	1.09440
$741$	0	0
$742$	−15656.3	−0.774611
$743$	22181.2	1.09522	0.547611	−	0.836733i	$-0.315537\pi$
$743$	22181.2	1.09522	0.547611	+	0.836733i	$0.315537\pi$
$744$	0	0
$745$	−19870.1	−0.977158
$746$	34469.4	1.69171
$747$	0	0
$748$	1913.76	0.0935479
$749$	−2330.14	−0.113673
$750$	0	0
$751$	−11657.1	−0.566410	−0.283205	−	0.959059i	$-0.591398\pi$
$751$	−11657.1	−0.566410	−0.283205	+	0.959059i	$0.591398\pi$
$752$	23321.6	1.13092
$753$	0	0
$754$	−541.450	−0.0261518
$755$	−6312.69	−0.304294
$756$	0	0
$757$	−15371.4	−0.738021	−0.369010	−	0.929425i	$-0.620303\pi$
$757$	−15371.4	−0.738021	−0.369010	+	0.929425i	$0.620303\pi$
$758$	34032.3	1.63075
$759$	0	0
$760$	2085.59	0.0995424
$761$	6164.19	0.293629	0.146815	−	0.989164i	$-0.453098\pi$
$761$	6164.19	0.293629	0.146815	+	0.989164i	$0.453098\pi$
$762$	0	0
$763$	928.086	0.0440354
$764$	−2019.70	−0.0956417
$765$	0	0
$766$	54287.9	2.56071
$767$	−160.234	−0.00754328
$768$	0	0
$769$	15507.5	0.727197	0.363599	−	0.931556i	$-0.381548\pi$
$769$	15507.5	0.727197	0.363599	+	0.931556i	$0.381548\pi$
$770$	−341.827	−0.0159982
$771$	0	0
$772$	−38425.7	−1.79141
$773$	−31731.2	−1.47644	−0.738222	−	0.674558i	$-0.764334\pi$
$773$	−31731.2	−1.47644	−0.738222	+	0.674558i	$0.764334\pi$
$774$	0	0
$775$	5966.04	0.276524
$776$	−328.230	−0.0151840
$777$	0	0
$778$	−5510.23	−0.253922
$779$	27768.4	1.27716
$780$	0	0
$781$	−957.122	−0.0438522
$782$	84930.2	3.88376
$783$	0	0
$784$	17603.4	0.801905
$785$	14755.0	0.670867
$786$	0	0
$787$	42805.3	1.93881	0.969405	−	0.245467i	$-0.0789414\pi$
$787$	42805.3	1.93881	0.969405	+	0.245467i	$0.0789414\pi$
$788$	−5161.62	−0.233344
$789$	0	0
$790$	16437.3	0.740271
$791$	1869.73	0.0840453
$792$	0	0
$793$	351.578	0.0157439
$794$	−21429.1	−0.957796
$795$	0	0
$796$	−22331.6	−0.994373
$797$	644.298	0.0286351	0.0143176	−	0.999897i	$-0.495442\pi$
$797$	644.298	0.0286351	0.0143176	+	0.999897i	$0.495442\pi$
$798$	0	0
$799$	45096.0	1.99672
$800$	18160.7	0.802597
$801$	0	0
$802$	8472.50	0.373036
$803$	−326.989	−0.0143701
$804$	0	0
$805$	−7960.38	−0.348530
$806$	−256.120	−0.0111928
$807$	0	0
$808$	899.244	0.0391526
$809$	23877.9	1.03771	0.518853	−	0.854864i	$-0.326359\pi$
$809$	23877.9	1.03771	0.518853	+	0.854864i	$0.326359\pi$
$810$	0	0
$811$	−5572.29	−0.241270	−0.120635	−	0.992697i	$-0.538493\pi$
$811$	−5572.29	−0.241270	−0.120635	+	0.992697i	$0.538493\pi$
$812$	−9042.28	−0.390791
$813$	0	0
$814$	−2727.47	−0.117442
$815$	8941.57	0.384306
$816$	0	0
$817$	3786.96	0.162165
$818$	27822.4	1.18923
$819$	0	0
$820$	22144.8	0.943086
$821$	−33978.6	−1.44441	−0.722206	−	0.691678i	$-0.756872\pi$
$821$	−33978.6	−1.44441	−0.722206	+	0.691678i	$0.756872\pi$
$822$	0	0
$823$	35936.7	1.52208	0.761042	−	0.648702i	$-0.224688\pi$
$823$	35936.7	1.52208	0.761042	+	0.648702i	$0.224688\pi$
$824$	2252.63	0.0952355
$825$	0	0
$826$	−5099.43	−0.214808
$827$	−42648.5	−1.79327	−0.896634	−	0.442773i	$-0.853995\pi$
$827$	−42648.5	−1.79327	−0.896634	+	0.442773i	$0.853995\pi$
$828$	0	0
$829$	4856.64	0.203472	0.101736	−	0.994811i	$-0.467560\pi$
$829$	4856.64	0.203472	0.101736	+	0.994811i	$0.467560\pi$
$830$	−2959.46	−0.123764
$831$	0	0
$832$	−448.125	−0.0186730
$833$	34039.0	1.41582
$834$	0	0
$835$	−27406.2	−1.13585
$836$	−1436.40	−0.0594247
$837$	0	0
$838$	−51056.3	−2.10467
$839$	−2091.03	−0.0860435	−0.0430217	−	0.999074i	$-0.513698\pi$
$839$	−2091.03	−0.0860435	−0.0430217	+	0.999074i	$0.513698\pi$
$840$	0	0
$841$	8149.04	0.334128
$842$	5720.89	0.234151
$843$	0	0
$844$	−46786.0	−1.90811
$845$	−16301.2	−0.663641
$846$	0	0
$847$	−7531.18	−0.305519
$848$	38086.0	1.54231
$849$	0	0
$850$	31419.0	1.26784
$851$	−63516.6	−2.55854
$852$	0	0
$853$	−23427.5	−0.940377	−0.470188	−	0.882566i	$-0.655814\pi$
$853$	−23427.5	−0.940377	−0.470188	+	0.882566i	$0.655814\pi$
$854$	11188.9	0.448334
$855$	0	0
$856$	−1403.65	−0.0560466
$857$	13818.2	0.550781	0.275390	−	0.961332i	$-0.411193\pi$
$857$	13818.2	0.550781	0.275390	+	0.961332i	$0.411193\pi$
$858$	0	0
$859$	4369.03	0.173538	0.0867692	−	0.996228i	$-0.472346\pi$
$859$	4369.03	0.173538	0.0867692	+	0.996228i	$0.472346\pi$
$860$	3020.03	0.119747
$861$	0	0
$862$	−44035.2	−1.73996
$863$	47079.7	1.85702	0.928511	−	0.371304i	$-0.121089\pi$
$863$	47079.7	1.85702	0.928511	+	0.371304i	$0.121089\pi$
$864$	0	0
$865$	3351.96	0.131757
$866$	19618.4	0.769815
$867$	0	0
$868$	−4277.23	−0.167257
$869$	−1067.87	−0.0416859
$870$	0	0
$871$	−420.681	−0.0163654
$872$	559.070	0.0217116
$873$	0	0
$874$	−63745.9	−2.46709
$875$	−8209.48	−0.317178
$876$	0	0
$877$	−47259.9	−1.81967	−0.909837	−	0.414966i	$-0.863794\pi$
$877$	−47259.9	−1.81967	−0.909837	+	0.414966i	$0.863794\pi$
$878$	32015.3	1.23060
$879$	0	0
$880$	831.539	0.0318536
$881$	−46130.0	−1.76409	−0.882043	−	0.471169i	$-0.843832\pi$
$881$	−46130.0	−1.76409	−0.882043	+	0.471169i	$0.843832\pi$
$882$	0	0
$883$	−45009.5	−1.71539	−0.857695	−	0.514158i	$-0.828104\pi$
$883$	−45009.5	−1.71539	−0.857695	+	0.514158i	$0.828104\pi$
$884$	−707.784	−0.0269291
$885$	0	0
$886$	−21686.0	−0.822299
$887$	−4655.29	−0.176223	−0.0881113	−	0.996111i	$-0.528083\pi$
$887$	−4655.29	−0.176223	−0.0881113	+	0.996111i	$0.528083\pi$
$888$	0	0
$889$	−75.8166	−0.00286030
$890$	−2084.82	−0.0785205
$891$	0	0
$892$	27728.3	1.04082
$893$	−33847.6	−1.26838
$894$	0	0
$895$	−4669.63	−0.174400
$896$	−2469.75	−0.0920855
$897$	0	0
$898$	−49452.3	−1.83769
$899$	15391.3	0.571000
$900$	0	0
$901$	73645.3	2.72306
$902$	−2741.62	−0.101204
$903$	0	0
$904$	1126.31	0.0414384
$905$	−18651.7	−0.685087
$906$	0	0
$907$	−11431.6	−0.418500	−0.209250	−	0.977862i	$-0.567102\pi$
$907$	−11431.6	−0.418500	−0.209250	+	0.977862i	$0.567102\pi$
$908$	102.055	0.00372996
$909$	0	0
$910$	126.421	0.00460531
$911$	37817.7	1.37536	0.687682	−	0.726012i	$-0.258628\pi$
$911$	37817.7	1.37536	0.687682	+	0.726012i	$0.258628\pi$
$912$	0	0
$913$	192.265	0.00696938
$914$	13222.9	0.478529
$915$	0	0
$916$	−10146.5	−0.365994
$917$	14689.2	0.528987
$918$	0	0
$919$	23292.2	0.836058	0.418029	−	0.908434i	$-0.362721\pi$
$919$	23292.2	0.836058	0.418029	+	0.908434i	$0.362721\pi$
$920$	−4795.25	−0.171842
$921$	0	0
$922$	−45467.6	−1.62407
$923$	353.982	0.0126235
$924$	0	0
$925$	−23497.2	−0.835227
$926$	39782.4	1.41180
$927$	0	0
$928$	46851.4	1.65730
$929$	−19457.9	−0.687183	−0.343591	−	0.939119i	$-0.611644\pi$
$929$	−19457.9	−0.687183	−0.343591	+	0.939119i	$0.611644\pi$
$930$	0	0
$931$	−25548.6	−0.899377
$932$	8392.67	0.294969
$933$	0	0
$934$	−54958.6	−1.92538
$935$	1607.91	0.0562399
$936$	0	0
$937$	10504.7	0.366247	0.183124	−	0.983090i	$-0.441379\pi$
$937$	10504.7	0.366247	0.183124	+	0.983090i	$0.441379\pi$
$938$	−13388.2	−0.466033
$939$	0	0
$940$	−26992.8	−0.936605
$941$	14020.6	0.485717	0.242858	−	0.970062i	$-0.421915\pi$
$941$	14020.6	0.485717	0.242858	+	0.970062i	$0.421915\pi$
$942$	0	0
$943$	−63846.1	−2.20479
$944$	12405.0	0.427700
$945$	0	0
$946$	−373.893	−0.0128502
$947$	4803.37	0.164824	0.0824121	−	0.996598i	$-0.473738\pi$
$947$	4803.37	0.164824	0.0824121	+	0.996598i	$0.473738\pi$
$948$	0	0
$949$	120.934	0.00413664
$950$	−23582.1	−0.805372
$951$	0	0
$952$	−2124.75	−0.0723356
$953$	−28669.0	−0.974482	−0.487241	−	0.873268i	$-0.661997\pi$
$953$	−28669.0	−0.974482	−0.487241	+	0.873268i	$0.661997\pi$
$954$	0	0
$955$	−1696.93	−0.0574987
$956$	2111.14	0.0714217
$957$	0	0
$958$	−52935.0	−1.78523
$959$	11363.6	0.382636
$960$	0	0
$961$	−22510.5	−0.755615
$962$	1008.73	0.0338074
$963$	0	0
$964$	16913.7	0.565098
$965$	−32284.7	−1.07698
$966$	0	0
$967$	37377.7	1.24300	0.621502	−	0.783413i	$-0.286523\pi$
$967$	37377.7	1.24300	0.621502	+	0.783413i	$0.286523\pi$
$968$	−4536.71	−0.150636
$969$	0	0
$970$	−2923.58	−0.0967736
$971$	15760.4	0.520880	0.260440	−	0.965490i	$-0.416132\pi$
$971$	15760.4	0.520880	0.260440	+	0.965490i	$0.416132\pi$
$972$	0	0
$973$	−2206.97	−0.0727156
$974$	65108.1	2.14189
$975$	0	0
$976$	−27218.6	−0.892669
$977$	6093.82	0.199548	0.0997741	−	0.995010i	$-0.468188\pi$
$977$	6093.82	0.199548	0.0997741	+	0.995010i	$0.468188\pi$
$978$	0	0
$979$	135.443	0.00442162
$980$	−20374.5	−0.664122
$981$	0	0
$982$	−64378.3	−2.09205
$983$	23957.8	0.777351	0.388676	−	0.921375i	$-0.372933\pi$
$983$	23957.8	0.777351	0.388676	+	0.921375i	$0.372933\pi$
$984$	0	0
$985$	−4336.72	−0.140284
$986$	81055.4	2.61798
$987$	0	0
$988$	531.240	0.0171063
$989$	−8707.11	−0.279949
$990$	0	0
$991$	15949.9	0.511267	0.255633	−	0.966774i	$-0.417716\pi$
$991$	15949.9	0.511267	0.255633	+	0.966774i	$0.417716\pi$
$992$	22161.9	0.709316
$993$	0	0
$994$	11265.5	0.359476
$995$	−18762.7	−0.597805
$996$	0	0
$997$	−2544.36	−0.0808232	−0.0404116	−	0.999183i	$-0.512867\pi$
$997$	−2544.36	−0.0808232	−0.0404116	+	0.999183i	$0.512867\pi$
$998$	−65412.6	−2.07475
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.50	✓	59
3.2	odd	2		2151.4.a.h.1.10	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.50	✓	59	1.1	even	1	trivial
2151.4.a.h.1.10	yes	59	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+4.10283 q^{2} +8.83322 q^{4} +7.42154 q^{5} +5.67497 q^{7} +3.41855 q^{8} +O(q^{10})\)	\(q+4.10283 q^{2} +8.83322 q^{4} +7.42154 q^{5} +5.67497 q^{7} +3.41855 q^{8} +30.4493 q^{10} -1.97818 q^{11} +0.731609 q^{13} +23.2834 q^{14} -56.6400 q^{16} -109.522 q^{17} +82.2039 q^{19} +65.5561 q^{20} -8.11612 q^{22} -189.006 q^{23} -69.9207 q^{25} +3.00167 q^{26} +50.1282 q^{28} -180.383 q^{29} -85.3258 q^{31} -259.733 q^{32} -449.352 q^{34} +42.1170 q^{35} +336.056 q^{37} +337.269 q^{38} +25.3709 q^{40} +337.799 q^{41} +46.0679 q^{43} -17.4737 q^{44} -775.460 q^{46} -411.751 q^{47} -310.795 q^{49} -286.873 q^{50} +6.46246 q^{52} -672.423 q^{53} -14.6811 q^{55} +19.4002 q^{56} -740.081 q^{58} -219.015 q^{59} +480.554 q^{61} -350.077 q^{62} -612.519 q^{64} +5.42967 q^{65} -575.008 q^{67} -967.434 q^{68} +172.799 q^{70} +483.841 q^{71} +165.298 q^{73} +1378.78 q^{74} +726.125 q^{76} -11.2261 q^{77} +539.826 q^{79} -420.356 q^{80} +1385.93 q^{82} -97.1930 q^{83} -812.825 q^{85} +189.009 q^{86} -6.76249 q^{88} -68.4685 q^{89} +4.15186 q^{91} -1669.53 q^{92} -1689.35 q^{94} +610.080 q^{95} -96.0144 q^{97} -1275.14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

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