LMFDB - Embedded newform 531.4.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 $ x^8 - 2x^7 - 45x^6 + 47x^5 + 654x^4 - 157x^3 - 2898x^2 + 96*x + 2432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.04902$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.04902 q^{2} -3.80150 q^{4} -16.1855 q^{5} -1.13960 q^{7} +24.1816 q^{8} +O(q^{10})$	\(q-2.04902 q^{2} -3.80150 q^{4} -16.1855 q^{5} -1.13960 q^{7} +24.1816 q^{8} +33.1645 q^{10} -36.3258 q^{11} +78.1440 q^{13} +2.33506 q^{14} -19.1366 q^{16} +43.6811 q^{17} -18.4272 q^{19} +61.5292 q^{20} +74.4325 q^{22} +4.45159 q^{23} +136.971 q^{25} -160.119 q^{26} +4.33218 q^{28} +161.921 q^{29} +245.150 q^{31} -154.241 q^{32} -89.5037 q^{34} +18.4450 q^{35} -173.192 q^{37} +37.7579 q^{38} -391.391 q^{40} -128.039 q^{41} -229.176 q^{43} +138.092 q^{44} -9.12142 q^{46} +138.962 q^{47} -341.701 q^{49} -280.657 q^{50} -297.064 q^{52} +168.587 q^{53} +587.952 q^{55} -27.5572 q^{56} -331.780 q^{58} -59.0000 q^{59} -878.522 q^{61} -502.318 q^{62} +469.137 q^{64} -1264.80 q^{65} +361.058 q^{67} -166.054 q^{68} -37.7942 q^{70} +761.904 q^{71} -404.740 q^{73} +354.874 q^{74} +70.0511 q^{76} +41.3968 q^{77} -493.825 q^{79} +309.736 q^{80} +262.356 q^{82} -398.428 q^{83} -707.002 q^{85} +469.588 q^{86} -878.415 q^{88} +1327.94 q^{89} -89.0527 q^{91} -16.9227 q^{92} -284.736 q^{94} +298.254 q^{95} -1462.81 q^{97} +700.154 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) $ 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8	$ 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) $ 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8 + 21 * q^10 - 67 * q^11 + 33 * q^13 - 79 * q^14 - 30 * q^16 - 139 * q^17 + 64 * q^19 - 117 * q^20 - 84 * q^22 - 226 * q^23 + 96 * q^25 - 24 * q^26 + 34 * q^28 - 456 * q^29 + 124 * q^31 - 174 * q^32 - 114 * q^34 - 556 * q^35 + 127 * q^37 - 237 * q^38 - 188 * q^40 - 425 * q^41 - 115 * q^43 - 510 * q^44 - 711 * q^46 - 420 * q^47 + 171 * q^49 + 137 * q^50 - 922 * q^52 - 98 * q^53 - 616 * q^55 + 412 * q^56 - 1548 * q^58 - 472 * q^59 - 1254 * q^61 + 766 * q^62 - 2019 * q^64 + 734 * q^65 - 1010 * q^67 + 503 * q^68 - 2956 * q^70 + 17 * q^71 - 1180 * q^73 + 1228 * q^74 - 2008 * q^76 - 441 * q^77 - 873 * q^79 + 865 * q^80 - 3645 * q^82 - 759 * q^83 - 850 * q^85 + 1226 * q^86 - 3047 * q^88 - 988 * q^89 - 2111 * q^91 + 1062 * q^92 - 2240 * q^94 - 1822 * q^95 - 668 * q^97 + 1368 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.04902	−0.724440	−0.362220	−	0.932093i	$-0.617981\pi$
$2$	−2.04902	−0.724440	−0.362220	+	0.932093i	$0.617981\pi$
$3$	0	0
$3$	0	0
$4$	−3.80150	−0.475187
$5$	−16.1855	−1.44768	−0.723838	−	0.689970i	$-0.757624\pi$
$5$	−16.1855	−1.44768	−0.723838	+	0.689970i	$0.757624\pi$
$6$	0	0
$7$	−1.13960	−0.0615325	−0.0307662	−	0.999527i	$-0.509795\pi$
$7$	−1.13960	−0.0615325	−0.0307662	+	0.999527i	$0.509795\pi$
$8$	24.1816	1.06868
$9$	0	0
$10$	33.1645	1.04875
$11$	−36.3258	−0.995695	−0.497847	−	0.867265i	$-0.665876\pi$
$11$	−36.3258	−0.995695	−0.497847	+	0.867265i	$0.665876\pi$
$12$	0	0
$13$	78.1440	1.66717	0.833586	−	0.552389i	$-0.186284\pi$
$13$	78.1440	1.66717	0.833586	+	0.552389i	$0.186284\pi$
$14$	2.33506	0.0445766
$15$	0	0
$16$	−19.1366	−0.299010
$17$	43.6811	0.623190	0.311595	−	0.950215i	$-0.399137\pi$
$17$	43.6811	0.623190	0.311595	+	0.950215i	$0.399137\pi$
$18$	0	0
$19$	−18.4272	−0.222500	−0.111250	−	0.993792i	$-0.535485\pi$
$19$	−18.4272	−0.222500	−0.111250	+	0.993792i	$0.535485\pi$
$20$	61.5292	0.687917
$21$	0	0
$22$	74.4325	0.721321
$23$	4.45159	0.0403574	0.0201787	−	0.999796i	$-0.493576\pi$
$23$	4.45159	0.0403574	0.0201787	+	0.999796i	$0.493576\pi$
$24$	0	0
$25$	136.971	1.09577
$26$	−160.119	−1.20777
$27$	0	0
$28$	4.33218	0.0292394
$29$	161.921	1.03682	0.518412	−	0.855131i	$-0.326523\pi$
$29$	161.921	1.03682	0.518412	+	0.855131i	$0.326523\pi$
$30$	0	0
$31$	245.150	1.42033	0.710164	−	0.704036i	$-0.248621\pi$
$31$	245.150	1.42033	0.710164	+	0.704036i	$0.248621\pi$
$32$	−154.241	−0.852069
$33$	0	0
$34$	−89.5037	−0.451464
$35$	18.4450	0.0890791
$36$	0	0
$37$	−173.192	−0.769528	−0.384764	−	0.923015i	$-0.625717\pi$
$37$	−173.192	−0.769528	−0.384764	+	0.923015i	$0.625717\pi$
$38$	37.7579	0.161188
$39$	0	0
$40$	−391.391	−1.54711
$41$	−128.039	−0.487716	−0.243858	−	0.969811i	$-0.578413\pi$
$41$	−128.039	−0.487716	−0.243858	+	0.969811i	$0.578413\pi$
$42$	0	0
$43$	−229.176	−0.812769	−0.406384	−	0.913702i	$-0.633211\pi$
$43$	−229.176	−0.812769	−0.406384	+	0.913702i	$0.633211\pi$
$44$	138.092	0.473141
$45$	0	0
$46$	−9.12142	−0.0292365
$47$	138.962	0.431269	0.215635	−	0.976474i	$-0.430818\pi$
$47$	138.962	0.431269	0.215635	+	0.976474i	$0.430818\pi$
$48$	0	0
$49$	−341.701	−0.996214
$50$	−280.657	−0.793817
$51$	0	0
$52$	−297.064	−0.792219
$53$	168.587	0.436929	0.218464	−	0.975845i	$-0.429895\pi$
$53$	168.587	0.436929	0.218464	+	0.975845i	$0.429895\pi$
$54$	0	0
$55$	587.952	1.44144
$56$	−27.5572	−0.0657588
$57$	0	0
$58$	−331.780	−0.751117
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	−878.522	−1.84399	−0.921994	−	0.387205i	$-0.873441\pi$
$61$	−878.522	−1.84399	−0.921994	+	0.387205i	$0.873441\pi$
$62$	−502.318	−1.02894
$63$	0	0
$64$	469.137	0.916283
$65$	−1264.80	−2.41353
$66$	0	0
$67$	361.058	0.658362	0.329181	−	0.944267i	$-0.393227\pi$
$67$	361.058	0.658362	0.329181	+	0.944267i	$0.393227\pi$
$68$	−166.054	−0.296132
$69$	0	0
$70$	−37.7942	−0.0645324
$71$	761.904	1.27354	0.636771	−	0.771053i	$-0.280270\pi$
$71$	761.904	1.27354	0.636771	+	0.771053i	$0.280270\pi$
$72$	0	0
$73$	−404.740	−0.648921	−0.324460	−	0.945899i	$-0.605183\pi$
$73$	−404.740	−0.648921	−0.324460	+	0.945899i	$0.605183\pi$
$74$	354.874	0.557477
$75$	0	0
$76$	70.0511	0.105729
$77$	41.3968	0.0612676
$78$	0	0
$79$	−493.825	−0.703286	−0.351643	−	0.936134i	$-0.614377\pi$
$79$	−493.825	−0.703286	−0.351643	+	0.936134i	$0.614377\pi$
$80$	309.736	0.432870
$81$	0	0
$82$	262.356	0.353321
$83$	−398.428	−0.526906	−0.263453	−	0.964672i	$-0.584861\pi$
$83$	−398.428	−0.526906	−0.263453	+	0.964672i	$0.584861\pi$
$84$	0	0
$85$	−707.002	−0.902177
$86$	469.588	0.588802
$87$	0	0
$88$	−878.415	−1.06408
$89$	1327.94	1.58159	0.790796	−	0.612080i	$-0.209667\pi$
$89$	1327.94	1.58159	0.790796	+	0.612080i	$0.209667\pi$
$90$	0	0
$91$	−89.0527	−0.102585
$92$	−16.9227	−0.0191773
$93$	0	0
$94$	−284.736	−0.312429
$95$	298.254	0.322108
$96$	0	0
$97$	−1462.81	−1.53120	−0.765598	−	0.643320i	$-0.777557\pi$
$97$	−1462.81	−1.53120	−0.765598	+	0.643320i	$0.777557\pi$
$98$	700.154	0.721697
$99$	0	0
$100$	−520.694	−0.520694
$101$	−228.118	−0.224738	−0.112369	−	0.993667i	$-0.535844\pi$
$101$	−228.118	−0.224738	−0.112369	+	0.993667i	$0.535844\pi$
$102$	0	0
$103$	764.688	0.731524	0.365762	−	0.930708i	$-0.380808\pi$
$103$	764.688	0.731524	0.365762	+	0.930708i	$0.380808\pi$
$104$	1889.64	1.78168
$105$	0	0
$106$	−345.439	−0.316529
$107$	−1557.21	−1.40692	−0.703462	−	0.710733i	$-0.748363\pi$
$107$	−1557.21	−1.40692	−0.703462	+	0.710733i	$0.748363\pi$
$108$	0	0
$109$	1589.34	1.39662	0.698310	−	0.715796i	$-0.253936\pi$
$109$	1589.34	1.39662	0.698310	+	0.715796i	$0.253936\pi$
$110$	−1204.73	−1.04424
$111$	0	0
$112$	21.8081	0.0183988
$113$	−1409.48	−1.17339	−0.586694	−	0.809809i	$-0.699571\pi$
$113$	−1409.48	−1.17339	−0.586694	+	0.809809i	$0.699571\pi$
$114$	0	0
$115$	−72.0513	−0.0584245
$116$	−615.541	−0.492686
$117$	0	0
$118$	120.892	0.0943140
$119$	−49.7789	−0.0383464
$120$	0	0
$121$	−11.4358	−0.00859188
$122$	1800.11	1.33586
$123$	0	0
$124$	−931.935	−0.674921
$125$	−193.755	−0.138640
$126$	0	0
$127$	−1270.35	−0.887603	−0.443802	−	0.896125i	$-0.646370\pi$
$127$	−1270.35	−0.887603	−0.443802	+	0.896125i	$0.646370\pi$
$128$	272.655	0.188278
$129$	0	0
$130$	2591.61	1.74845
$131$	−2502.45	−1.66901	−0.834504	−	0.551002i	$-0.814246\pi$
$131$	−2502.45	−1.66901	−0.834504	+	0.551002i	$0.814246\pi$
$132$	0	0
$133$	20.9996	0.0136910
$134$	−739.817	−0.476944
$135$	0	0
$136$	1056.28	0.665993
$137$	−458.017	−0.285628	−0.142814	−	0.989750i	$-0.545615\pi$
$137$	−458.017	−0.285628	−0.142814	+	0.989750i	$0.545615\pi$
$138$	0	0
$139$	286.953	0.175101	0.0875505	−	0.996160i	$-0.472096\pi$
$139$	286.953	0.175101	0.0875505	+	0.996160i	$0.472096\pi$
$140$	−70.1185	−0.0423293
$141$	0	0
$142$	−1561.16	−0.922604
$143$	−2838.64	−1.65999
$144$	0	0
$145$	−2620.77	−1.50099
$146$	829.322	0.470104
$147$	0	0
$148$	658.388	0.365670
$149$	−2607.82	−1.43383	−0.716915	−	0.697160i	$-0.754447\pi$
$149$	−2607.82	−1.43383	−0.716915	+	0.697160i	$0.754447\pi$
$150$	0	0
$151$	−20.5789	−0.0110907	−0.00554533	−	0.999985i	$-0.501765\pi$
$151$	−20.5789	−0.0110907	−0.00554533	+	0.999985i	$0.501765\pi$
$152$	−445.600	−0.237782
$153$	0	0
$154$	−84.8231	−0.0443847
$155$	−3967.87	−2.05617
$156$	0	0
$157$	−1903.56	−0.967649	−0.483825	−	0.875165i	$-0.660753\pi$
$157$	−1903.56	−0.967649	−0.483825	+	0.875165i	$0.660753\pi$
$158$	1011.86	0.509488
$159$	0	0
$160$	2496.47	1.23352
$161$	−5.07302	−0.00248329
$162$	0	0
$163$	−195.798	−0.0940862	−0.0470431	−	0.998893i	$-0.514980\pi$
$163$	−195.798	−0.0940862	−0.0470431	+	0.998893i	$0.514980\pi$
$164$	486.741	0.231757
$165$	0	0
$166$	816.390	0.381712
$167$	2098.12	0.972198	0.486099	−	0.873904i	$-0.338419\pi$
$167$	2098.12	0.972198	0.486099	+	0.873904i	$0.338419\pi$
$168$	0	0
$169$	3909.48	1.77946
$170$	1448.66	0.653573
$171$	0	0
$172$	871.213	0.386217
$173$	−1996.80	−0.877537	−0.438769	−	0.898600i	$-0.644585\pi$
$173$	−1996.80	−0.877537	−0.438769	+	0.898600i	$0.644585\pi$
$174$	0	0
$175$	−156.092	−0.0674253
$176$	695.154	0.297723
$177$	0	0
$178$	−2720.99	−1.14577
$179$	2474.76	1.03336	0.516682	−	0.856177i	$-0.327167\pi$
$179$	2474.76	1.03336	0.516682	+	0.856177i	$0.327167\pi$
$180$	0	0
$181$	618.103	0.253830	0.126915	−	0.991914i	$-0.459492\pi$
$181$	618.103	0.253830	0.126915	+	0.991914i	$0.459492\pi$
$182$	182.471	0.0743168
$183$	0	0
$184$	107.646	0.0431294
$185$	2803.20	1.11403
$186$	0	0
$187$	−1586.75	−0.620507
$188$	−528.263	−0.204934
$189$	0	0
$190$	−611.131	−0.233348
$191$	−552.196	−0.209191	−0.104596	−	0.994515i	$-0.533355\pi$
$191$	−552.196	−0.209191	−0.104596	+	0.994515i	$0.533355\pi$
$192$	0	0
$193$	4002.75	1.49287	0.746436	−	0.665457i	$-0.231763\pi$
$193$	4002.75	1.49287	0.746436	+	0.665457i	$0.231763\pi$
$194$	2997.34	1.10926
$195$	0	0
$196$	1298.98	0.473388
$197$	−1380.23	−0.499173	−0.249586	−	0.968353i	$-0.580295\pi$
$197$	−1380.23	−0.499173	−0.249586	+	0.968353i	$0.580295\pi$
$198$	0	0
$199$	2132.94	0.759800	0.379900	−	0.925028i	$-0.375958\pi$
$199$	2132.94	0.759800	0.379900	+	0.925028i	$0.375958\pi$
$200$	3312.17	1.17103
$201$	0	0
$202$	467.419	0.162809
$203$	−184.524	−0.0637984
$204$	0	0
$205$	2072.38	0.706055
$206$	−1566.86	−0.529945
$207$	0	0
$208$	−1495.41	−0.498501
$209$	669.385	0.221542
$210$	0	0
$211$	6000.66	1.95783	0.978916	−	0.204265i	$-0.0654806\pi$
$211$	6000.66	1.95783	0.978916	+	0.204265i	$0.0654806\pi$
$212$	−640.884	−0.207623
$213$	0	0
$214$	3190.76	1.01923
$215$	3709.34	1.17663
$216$	0	0
$217$	−279.372	−0.0873963
$218$	−3256.60	−1.01177
$219$	0	0
$220$	−2235.10	−0.684956
$221$	3413.42	1.03897
$222$	0	0
$223$	−4919.66	−1.47733	−0.738666	−	0.674071i	$-0.764544\pi$
$223$	−4919.66	−1.47733	−0.738666	+	0.674071i	$0.764544\pi$
$224$	175.773	0.0524299
$225$	0	0
$226$	2888.06	0.850048
$227$	−1021.59	−0.298701	−0.149350	−	0.988784i	$-0.547718\pi$
$227$	−1021.59	−0.298701	−0.149350	+	0.988784i	$0.547718\pi$
$228$	0	0
$229$	1745.58	0.503716	0.251858	−	0.967764i	$-0.418958\pi$
$229$	1745.58	0.503716	0.251858	+	0.967764i	$0.418958\pi$
$230$	147.635	0.0423250
$231$	0	0
$232$	3915.50	1.10804
$233$	1961.28	0.551450	0.275725	−	0.961237i	$-0.411082\pi$
$233$	1961.28	0.551450	0.275725	+	0.961237i	$0.411082\pi$
$234$	0	0
$235$	−2249.17	−0.624339
$236$	224.288	0.0618641
$237$	0	0
$238$	101.998	0.0277797
$239$	−1822.98	−0.493385	−0.246692	−	0.969094i	$-0.579344\pi$
$239$	−1822.98	−0.493385	−0.246692	+	0.969094i	$0.579344\pi$
$240$	0	0
$241$	−6154.85	−1.64510	−0.822549	−	0.568695i	$-0.807449\pi$
$241$	−6154.85	−1.64510	−0.822549	+	0.568695i	$0.807449\pi$
$242$	23.4322	0.00622430
$243$	0	0
$244$	3339.70	0.876239
$245$	5530.61	1.44220
$246$	0	0
$247$	−1439.98	−0.370946
$248$	5928.10	1.51788
$249$	0	0
$250$	397.009	0.100436
$251$	−4896.77	−1.23140	−0.615700	−	0.787981i	$-0.711127\pi$
$251$	−4896.77	−1.23140	−0.615700	+	0.787981i	$0.711127\pi$
$252$	0	0
$253$	−161.708	−0.0401837
$254$	2602.98	0.643015
$255$	0	0
$256$	−4311.77	−1.05268
$257$	6238.24	1.51413	0.757063	−	0.653341i	$-0.226633\pi$
$257$	6238.24	1.51413	0.757063	+	0.653341i	$0.226633\pi$
$258$	0	0
$259$	197.369	0.0473510
$260$	4808.14	1.14688
$261$	0	0
$262$	5127.58	1.20910
$263$	5359.90	1.25667	0.628337	−	0.777941i	$-0.283736\pi$
$263$	5359.90	1.25667	0.628337	+	0.777941i	$0.283736\pi$
$264$	0	0
$265$	−2728.67	−0.632532
$266$	−43.0288	−0.00991829
$267$	0	0
$268$	−1372.56	−0.312845
$269$	7201.10	1.63219	0.816094	−	0.577919i	$-0.196135\pi$
$269$	7201.10	1.63219	0.816094	+	0.577919i	$0.196135\pi$
$270$	0	0
$271$	−1135.65	−0.254560	−0.127280	−	0.991867i	$-0.540625\pi$
$271$	−1135.65	−0.254560	−0.127280	+	0.991867i	$0.540625\pi$
$272$	−835.910	−0.186340
$273$	0	0
$274$	938.489	0.206920
$275$	−4975.58	−1.09105
$276$	0	0
$277$	1044.58	0.226580	0.113290	−	0.993562i	$-0.463861\pi$
$277$	1044.58	0.226580	0.113290	+	0.993562i	$0.463861\pi$
$278$	−587.974	−0.126850
$279$	0	0
$280$	446.028	0.0951974
$281$	−6506.31	−1.38126	−0.690630	−	0.723209i	$-0.742667\pi$
$281$	−6506.31	−1.38126	−0.690630	+	0.723209i	$0.742667\pi$
$282$	0	0
$283$	−8390.09	−1.76233	−0.881164	−	0.472811i	$-0.843239\pi$
$283$	−8390.09	−1.76233	−0.881164	+	0.472811i	$0.843239\pi$
$284$	−2896.38	−0.605170
$285$	0	0
$286$	5816.45	1.20257
$287$	145.913	0.0300104
$288$	0	0
$289$	−3004.96	−0.611634
$290$	5370.02	1.08737
$291$	0	0
$292$	1538.62	0.308359
$293$	−3643.33	−0.726436	−0.363218	−	0.931704i	$-0.618322\pi$
$293$	−3643.33	−0.726436	−0.363218	+	0.931704i	$0.618322\pi$
$294$	0	0
$295$	954.945	0.188471
$296$	−4188.05	−0.822383
$297$	0	0
$298$	5343.48	1.03872
$299$	347.865	0.0672828
$300$	0	0
$301$	261.169	0.0500117
$302$	42.1667	0.00803451
$303$	0	0
$304$	352.636	0.0665297
$305$	14219.3	2.66950
$306$	0	0
$307$	−521.575	−0.0969637	−0.0484819	−	0.998824i	$-0.515438\pi$
$307$	−521.575	−0.0969637	−0.0484819	+	0.998824i	$0.515438\pi$
$308$	−157.370	−0.0291136
$309$	0	0
$310$	8130.27	1.48957
$311$	−5344.76	−0.974513	−0.487256	−	0.873259i	$-0.662002\pi$
$311$	−5344.76	−0.974513	−0.487256	+	0.873259i	$0.662002\pi$
$312$	0	0
$313$	4368.71	0.788927	0.394463	−	0.918912i	$-0.370931\pi$
$313$	4368.71	0.788927	0.394463	+	0.918912i	$0.370931\pi$
$314$	3900.45	0.701004
$315$	0	0
$316$	1877.27	0.334193
$317$	−5679.69	−1.00632	−0.503160	−	0.864193i	$-0.667829\pi$
$317$	−5679.69	−1.00632	−0.503160	+	0.864193i	$0.667829\pi$
$318$	0	0
$319$	−5881.90	−1.03236
$320$	−7593.22	−1.32648
$321$	0	0
$322$	10.3948	0.00179900
$323$	−804.923	−0.138660
$324$	0	0
$325$	10703.5	1.82683
$326$	401.194	0.0681598
$327$	0	0
$328$	−3096.19	−0.521215
$329$	−158.361	−0.0265371
$330$	0	0
$331$	3620.41	0.601196	0.300598	−	0.953751i	$-0.402814\pi$
$331$	3620.41	0.601196	0.300598	+	0.953751i	$0.402814\pi$
$332$	1514.62	0.250379
$333$	0	0
$334$	−4299.09	−0.704299
$335$	−5843.91	−0.953096
$336$	0	0
$337$	−4951.57	−0.800383	−0.400191	−	0.916432i	$-0.631056\pi$
$337$	−4951.57	−0.800383	−0.400191	+	0.916432i	$0.631056\pi$
$338$	−8010.63	−1.28911
$339$	0	0
$340$	2687.66	0.428703
$341$	−8905.26	−1.41421
$342$	0	0
$343$	780.284	0.122832
$344$	−5541.84	−0.868593
$345$	0	0
$346$	4091.49	0.635723
$347$	8903.74	1.37746	0.688729	−	0.725019i	$-0.258169\pi$
$347$	8903.74	1.37746	0.688729	+	0.725019i	$0.258169\pi$
$348$	0	0
$349$	−10124.5	−1.55287	−0.776433	−	0.630200i	$-0.782973\pi$
$349$	−10124.5	−1.55287	−0.776433	+	0.630200i	$0.782973\pi$
$350$	319.836	0.0488455
$351$	0	0
$352$	5602.93	0.848401
$353$	−10278.4	−1.54975	−0.774877	−	0.632112i	$-0.782188\pi$
$353$	−10278.4	−1.54975	−0.774877	+	0.632112i	$0.782188\pi$
$354$	0	0
$355$	−12331.8	−1.84368
$356$	−5048.17	−0.751552
$357$	0	0
$358$	−5070.84	−0.748610
$359$	−11976.1	−1.76065	−0.880324	−	0.474374i	$-0.842675\pi$
$359$	−11976.1	−1.76065	−0.880324	+	0.474374i	$0.842675\pi$
$360$	0	0
$361$	−6519.44	−0.950494
$362$	−1266.51	−0.183885
$363$	0	0
$364$	338.534	0.0487472
$365$	6550.92	0.939427
$366$	0	0
$367$	11740.3	1.66987	0.834933	−	0.550352i	$-0.185507\pi$
$367$	11740.3	1.66987	0.834933	+	0.550352i	$0.185507\pi$
$368$	−85.1885	−0.0120673
$369$	0	0
$370$	−5743.82	−0.807046
$371$	−192.122	−0.0268853
$372$	0	0
$373$	−4055.41	−0.562952	−0.281476	−	0.959568i	$-0.590824\pi$
$373$	−4055.41	−0.562952	−0.281476	+	0.959568i	$0.590824\pi$
$374$	3251.29	0.449520
$375$	0	0
$376$	3360.31	0.460891
$377$	12653.1	1.72857
$378$	0	0
$379$	−1798.03	−0.243691	−0.121845	−	0.992549i	$-0.538881\pi$
$379$	−1798.03	−0.243691	−0.121845	+	0.992549i	$0.538881\pi$
$380$	−1133.81	−0.153062
$381$	0	0
$382$	1131.46	0.151546
$383$	−7620.06	−1.01662	−0.508312	−	0.861173i	$-0.669730\pi$
$383$	−7620.06	−1.01662	−0.508312	+	0.861173i	$0.669730\pi$
$384$	0	0
$385$	−670.028	−0.0886956
$386$	−8201.73	−1.08150
$387$	0	0
$388$	5560.87	0.727604
$389$	978.747	0.127569	0.0637846	−	0.997964i	$-0.479683\pi$
$389$	978.747	0.127569	0.0637846	+	0.997964i	$0.479683\pi$
$390$	0	0
$391$	194.451	0.0251504
$392$	−8262.87	−1.06464
$393$	0	0
$394$	2828.12	0.361621
$395$	7992.80	1.01813
$396$	0	0
$397$	−2554.43	−0.322930	−0.161465	−	0.986878i	$-0.551622\pi$
$397$	−2554.43	−0.322930	−0.161465	+	0.986878i	$0.551622\pi$
$398$	−4370.45	−0.550429
$399$	0	0
$400$	−2621.16	−0.327645
$401$	12779.6	1.59148	0.795740	−	0.605639i	$-0.207082\pi$
$401$	12779.6	1.59148	0.795740	+	0.605639i	$0.207082\pi$
$402$	0	0
$403$	19157.0	2.36793
$404$	867.190	0.106793
$405$	0	0
$406$	378.095	0.0462181
$407$	6291.33	0.766215
$408$	0	0
$409$	−14112.8	−1.70620	−0.853099	−	0.521749i	$-0.825280\pi$
$409$	−14112.8	−1.70620	−0.853099	+	0.521749i	$0.825280\pi$
$410$	−4246.36	−0.511495
$411$	0	0
$412$	−2906.96	−0.347611
$413$	67.2362	0.00801085
$414$	0	0
$415$	6448.77	0.762789
$416$	−12053.0	−1.42055
$417$	0	0
$418$	−1371.59	−0.160494
$419$	−3577.00	−0.417060	−0.208530	−	0.978016i	$-0.566868\pi$
$419$	−3577.00	−0.417060	−0.208530	+	0.978016i	$0.566868\pi$
$420$	0	0
$421$	−10454.2	−1.21023	−0.605116	−	0.796137i	$-0.706873\pi$
$421$	−10454.2	−1.21023	−0.605116	+	0.796137i	$0.706873\pi$
$422$	−12295.5	−1.41833
$423$	0	0
$424$	4076.70	0.466939
$425$	5983.04	0.682871
$426$	0	0
$427$	1001.16	0.113465
$428$	5919.72	0.668552
$429$	0	0
$430$	−7600.52	−0.852395
$431$	12521.0	1.39934	0.699670	−	0.714466i	$-0.253330\pi$
$431$	12521.0	1.39934	0.699670	+	0.714466i	$0.253330\pi$
$432$	0	0
$433$	13582.0	1.50741	0.753707	−	0.657210i	$-0.228264\pi$
$433$	13582.0	1.50741	0.753707	+	0.657210i	$0.228264\pi$
$434$	572.440	0.0633133
$435$	0	0
$436$	−6041.89	−0.663656
$437$	−82.0306	−0.00897953
$438$	0	0
$439$	1232.27	0.133970	0.0669851	−	0.997754i	$-0.478662\pi$
$439$	1232.27	0.133970	0.0669851	+	0.997754i	$0.478662\pi$
$440$	14217.6	1.54045
$441$	0	0
$442$	−6994.18	−0.752668
$443$	−12543.1	−1.34524	−0.672622	−	0.739986i	$-0.734832\pi$
$443$	−12543.1	−1.34524	−0.672622	+	0.739986i	$0.734832\pi$
$444$	0	0
$445$	−21493.4	−2.28963
$446$	10080.5	1.07024
$447$	0	0
$448$	−534.627	−0.0563812
$449$	1370.68	0.144068	0.0720340	−	0.997402i	$-0.477051\pi$
$449$	1370.68	0.144068	0.0720340	+	0.997402i	$0.477051\pi$
$450$	0	0
$451$	4651.13	0.485617
$452$	5358.14	0.557579
$453$	0	0
$454$	2093.25	0.216391
$455$	1441.36	0.148510
$456$	0	0
$457$	−7052.45	−0.721881	−0.360941	−	0.932589i	$-0.617544\pi$
$457$	−7052.45	−0.721881	−0.360941	+	0.932589i	$0.617544\pi$
$458$	−3576.73	−0.364912
$459$	0	0
$460$	273.903	0.0277626
$461$	−5082.44	−0.513476	−0.256738	−	0.966481i	$-0.582648\pi$
$461$	−5082.44	−0.513476	−0.256738	+	0.966481i	$0.582648\pi$
$462$	0	0
$463$	−5699.92	−0.572133	−0.286067	−	0.958210i	$-0.592348\pi$
$463$	−5699.92	−0.572133	−0.286067	+	0.958210i	$0.592348\pi$
$464$	−3098.62	−0.310021
$465$	0	0
$466$	−4018.71	−0.399492
$467$	−16957.5	−1.68030	−0.840149	−	0.542355i	$-0.817533\pi$
$467$	−16957.5	−1.68030	−0.840149	+	0.542355i	$0.817533\pi$
$468$	0	0
$469$	−411.461	−0.0405107
$470$	4608.60	0.452296
$471$	0	0
$472$	−1426.71	−0.139131
$473$	8325.02	0.809270
$474$	0	0
$475$	−2524.00	−0.243808
$476$	189.234	0.0182217
$477$	0	0
$478$	3735.34	0.357427
$479$	−15755.7	−1.50291	−0.751456	−	0.659783i	$-0.770648\pi$
$479$	−15755.7	−1.50291	−0.751456	+	0.659783i	$0.770648\pi$
$480$	0	0
$481$	−13533.9	−1.28294
$482$	12611.4	1.19177
$483$	0	0
$484$	43.4731	0.00408275
$485$	23676.4	2.21668
$486$	0	0
$487$	−6390.04	−0.594580	−0.297290	−	0.954787i	$-0.596083\pi$
$487$	−6390.04	−0.594580	−0.297290	+	0.954787i	$0.596083\pi$
$488$	−21244.0	−1.97064
$489$	0	0
$490$	−11332.4	−1.04478
$491$	−15901.1	−1.46152	−0.730760	−	0.682635i	$-0.760834\pi$
$491$	−15901.1	−1.46152	−0.730760	+	0.682635i	$0.760834\pi$
$492$	0	0
$493$	7072.88	0.646139
$494$	2950.55	0.268728
$495$	0	0
$496$	−4691.34	−0.424692
$497$	−868.264	−0.0783641
$498$	0	0
$499$	3902.49	0.350100	0.175050	−	0.984560i	$-0.443991\pi$
$499$	3902.49	0.350100	0.175050	+	0.984560i	$0.443991\pi$
$500$	736.559	0.0658799
$501$	0	0
$502$	10033.6	0.892075
$503$	3985.69	0.353306	0.176653	−	0.984273i	$-0.443473\pi$
$503$	3985.69	0.353306	0.176653	+	0.984273i	$0.443473\pi$
$504$	0	0
$505$	3692.21	0.325349
$506$	331.343	0.0291107
$507$	0	0
$508$	4829.24	0.421778
$509$	4706.40	0.409838	0.204919	−	0.978779i	$-0.434307\pi$
$509$	4706.40	0.409838	0.204919	+	0.978779i	$0.434307\pi$
$510$	0	0
$511$	461.240	0.0399297
$512$	6653.69	0.574325
$513$	0	0
$514$	−12782.3	−1.09689
$515$	−12376.9	−1.05901
$516$	0	0
$517$	−5047.90	−0.429413
$518$	−404.414	−0.0343029
$519$	0	0
$520$	−30584.9	−2.57930
$521$	−3269.28	−0.274913	−0.137456	−	0.990508i	$-0.543893\pi$
$521$	−3269.28	−0.274913	−0.137456	+	0.990508i	$0.543893\pi$
$522$	0	0
$523$	6507.07	0.544043	0.272021	−	0.962291i	$-0.412308\pi$
$523$	6507.07	0.544043	0.272021	+	0.962291i	$0.412308\pi$
$524$	9513.06	0.793091
$525$	0	0
$526$	−10982.6	−0.910385
$527$	10708.4	0.885134
$528$	0	0
$529$	−12147.2	−0.998371
$530$	5591.11	0.458231
$531$	0	0
$532$	−79.8301	−0.00650578
$533$	−10005.5	−0.813107
$534$	0	0
$535$	25204.2	2.03677
$536$	8730.95	0.703581
$537$	0	0
$538$	−14755.2	−1.18242
$539$	12412.6	0.991925
$540$	0	0
$541$	17118.2	1.36039	0.680193	−	0.733033i	$-0.261896\pi$
$541$	17118.2	1.36039	0.680193	+	0.733033i	$0.261896\pi$
$542$	2326.97	0.184413
$543$	0	0
$544$	−6737.42	−0.531001
$545$	−25724.3	−2.02185
$546$	0	0
$547$	18217.3	1.42398	0.711988	−	0.702192i	$-0.247795\pi$
$547$	18217.3	1.42398	0.711988	+	0.702192i	$0.247795\pi$
$548$	1741.15	0.135727
$549$	0	0
$550$	10195.1	0.790400
$551$	−2983.75	−0.230694
$552$	0	0
$553$	562.761	0.0432749
$554$	−2140.37	−0.164144
$555$	0	0
$556$	−1090.85	−0.0832057
$557$	−11664.8	−0.887347	−0.443673	−	0.896189i	$-0.646325\pi$
$557$	−11664.8	−0.887347	−0.443673	+	0.896189i	$0.646325\pi$
$558$	0	0
$559$	−17908.8	−1.35503
$560$	−352.975	−0.0266355
$561$	0	0
$562$	13331.6	1.00064
$563$	−11698.6	−0.875733	−0.437867	−	0.899040i	$-0.644266\pi$
$563$	−11698.6	−0.875733	−0.437867	+	0.899040i	$0.644266\pi$
$564$	0	0
$565$	22813.2	1.69869
$566$	17191.5	1.27670
$567$	0	0
$568$	18424.0	1.36101
$569$	−3094.45	−0.227990	−0.113995	−	0.993481i	$-0.536365\pi$
$569$	−3094.45	−0.227990	−0.113995	+	0.993481i	$0.536365\pi$
$570$	0	0
$571$	20401.7	1.49525	0.747623	−	0.664123i	$-0.231195\pi$
$571$	20401.7	1.49525	0.747623	+	0.664123i	$0.231195\pi$
$572$	10791.1	0.788808
$573$	0	0
$574$	−298.980	−0.0217407
$575$	609.739	0.0442224
$576$	0	0
$577$	−20917.5	−1.50920	−0.754599	−	0.656187i	$-0.772168\pi$
$577$	−20917.5	−1.50920	−0.754599	+	0.656187i	$0.772168\pi$
$578$	6157.24	0.443092
$579$	0	0
$580$	9962.85	0.713250
$581$	454.048	0.0324218
$582$	0	0
$583$	−6124.07	−0.435048
$584$	−9787.24	−0.693491
$585$	0	0
$586$	7465.28	0.526259
$587$	18046.9	1.26895	0.634475	−	0.772943i	$-0.281216\pi$
$587$	18046.9	1.26895	0.634475	+	0.772943i	$0.281216\pi$
$588$	0	0
$589$	−4517.43	−0.316023
$590$	−1956.71	−0.136536
$591$	0	0
$592$	3314.31	0.230097
$593$	−5854.76	−0.405441	−0.202720	−	0.979237i	$-0.564978\pi$
$593$	−5854.76	−0.405441	−0.202720	+	0.979237i	$0.564978\pi$
$594$	0	0
$595$	805.697	0.0555132
$596$	9913.61	0.681338
$597$	0	0
$598$	−712.784	−0.0487423
$599$	5850.74	0.399090	0.199545	−	0.979889i	$-0.436054\pi$
$599$	5850.74	0.399090	0.199545	+	0.979889i	$0.436054\pi$
$600$	0	0
$601$	−5057.35	−0.343251	−0.171625	−	0.985162i	$-0.554902\pi$
$601$	−5057.35	−0.343251	−0.171625	+	0.985162i	$0.554902\pi$
$602$	−535.141	−0.0362305
$603$	0	0
$604$	78.2307	0.00527014
$605$	185.094	0.0124383
$606$	0	0
$607$	7535.66	0.503893	0.251946	−	0.967741i	$-0.418929\pi$
$607$	7535.66	0.503893	0.251946	+	0.967741i	$0.418929\pi$
$608$	2842.24	0.189585
$609$	0	0
$610$	−29135.8	−1.93389
$611$	10859.0	0.719000
$612$	0	0
$613$	−9172.53	−0.604364	−0.302182	−	0.953250i	$-0.597715\pi$
$613$	−9172.53	−0.604364	−0.302182	+	0.953250i	$0.597715\pi$
$614$	1068.72	0.0702444
$615$	0	0
$616$	1001.04	0.0654757
$617$	−6795.93	−0.443426	−0.221713	−	0.975112i	$-0.571165\pi$
$617$	−6795.93	−0.443426	−0.221713	+	0.975112i	$0.571165\pi$
$618$	0	0
$619$	−18841.8	−1.22345	−0.611726	−	0.791070i	$-0.709524\pi$
$619$	−18841.8	−1.22345	−0.611726	+	0.791070i	$0.709524\pi$
$620$	15083.9	0.977068
$621$	0	0
$622$	10951.5	0.705976
$623$	−1513.32	−0.0973192
$624$	0	0
$625$	−13985.3	−0.895062
$626$	−8951.59	−0.571530
$627$	0	0
$628$	7236.39	0.459815
$629$	−7565.21	−0.479562
$630$	0	0
$631$	4816.45	0.303867	0.151933	−	0.988391i	$-0.451450\pi$
$631$	4816.45	0.303867	0.151933	+	0.988391i	$0.451450\pi$
$632$	−11941.4	−0.751591
$633$	0	0
$634$	11637.8	0.729018
$635$	20561.3	1.28496
$636$	0	0
$637$	−26701.9	−1.66086
$638$	12052.2	0.747883
$639$	0	0
$640$	−4413.07	−0.272565
$641$	4109.33	0.253212	0.126606	−	0.991953i	$-0.459592\pi$
$641$	4109.33	0.253212	0.126606	+	0.991953i	$0.459592\pi$
$642$	0	0
$643$	−20097.1	−1.23259	−0.616293	−	0.787517i	$-0.711366\pi$
$643$	−20097.1	−1.23259	−0.616293	+	0.787517i	$0.711366\pi$
$644$	19.2851	0.00118003
$645$	0	0
$646$	1649.31	0.100451
$647$	11607.1	0.705290	0.352645	−	0.935757i	$-0.385282\pi$
$647$	11607.1	0.705290	0.352645	+	0.935757i	$0.385282\pi$
$648$	0	0
$649$	2143.22	0.129628
$650$	−21931.6	−1.32343
$651$	0	0
$652$	744.324	0.0447085
$653$	−28649.4	−1.71691	−0.858453	−	0.512893i	$-0.828574\pi$
$653$	−28649.4	−1.71691	−0.858453	+	0.512893i	$0.828574\pi$
$654$	0	0
$655$	40503.4	2.41618
$656$	2450.24	0.145832
$657$	0	0
$658$	324.485	0.0192245
$659$	2146.45	0.126880	0.0634398	−	0.997986i	$-0.479793\pi$
$659$	2146.45	0.126880	0.0634398	+	0.997986i	$0.479793\pi$
$660$	0	0
$661$	7235.64	0.425769	0.212885	−	0.977077i	$-0.431714\pi$
$661$	7235.64	0.425769	0.212885	+	0.977077i	$0.431714\pi$
$662$	−7418.32	−0.435530
$663$	0	0
$664$	−9634.62	−0.563096
$665$	−339.890	−0.0198201
$666$	0	0
$667$	720.805	0.0418436
$668$	−7975.98	−0.461976
$669$	0	0
$670$	11974.3	0.690460
$671$	31913.0	1.83605
$672$	0	0
$673$	16869.8	0.966247	0.483124	−	0.875552i	$-0.339502\pi$
$673$	16869.8	0.966247	0.483124	+	0.875552i	$0.339502\pi$
$674$	10145.9	0.579829
$675$	0	0
$676$	−14861.9	−0.845578
$677$	−269.724	−0.0153121	−0.00765607	−	0.999971i	$-0.502437\pi$
$677$	−269.724	−0.0153121	−0.00765607	+	0.999971i	$0.502437\pi$
$678$	0	0
$679$	1667.02	0.0942183
$680$	−17096.4	−0.964143
$681$	0	0
$682$	18247.1	1.02451
$683$	2809.90	0.157420	0.0787099	−	0.996898i	$-0.474920\pi$
$683$	2809.90	0.157420	0.0787099	+	0.996898i	$0.474920\pi$
$684$	0	0
$685$	7413.25	0.413497
$686$	−1598.82	−0.0889844
$687$	0	0
$688$	4385.67	0.243026
$689$	13174.1	0.728436
$690$	0	0
$691$	24053.0	1.32420	0.662098	−	0.749418i	$-0.269666\pi$
$691$	24053.0	1.32420	0.662098	+	0.749418i	$0.269666\pi$
$692$	7590.83	0.416994
$693$	0	0
$694$	−18244.0	−0.997885
$695$	−4644.48	−0.253490
$696$	0	0
$697$	−5592.90	−0.303940
$698$	20745.3	1.12496
$699$	0	0
$700$	593.382	0.0320396
$701$	−32736.0	−1.76379	−0.881897	−	0.471441i	$-0.843734\pi$
$701$	−32736.0	−1.76379	−0.881897	+	0.471441i	$0.843734\pi$
$702$	0	0
$703$	3191.45	0.171220
$704$	−17041.8	−0.912338
$705$	0	0
$706$	21060.7	1.12270
$707$	259.963	0.0138287
$708$	0	0
$709$	−20629.3	−1.09273	−0.546367	−	0.837546i	$-0.683989\pi$
$709$	−20629.3	−1.09273	−0.546367	+	0.837546i	$0.683989\pi$
$710$	25268.2	1.33563
$711$	0	0
$712$	32111.7	1.69022
$713$	1091.31	0.0573208
$714$	0	0
$715$	45944.9	2.40314
$716$	−9407.79	−0.491041
$717$	0	0
$718$	24539.2	1.27548
$719$	−6316.05	−0.327606	−0.163803	−	0.986493i	$-0.552376\pi$
$719$	−6316.05	−0.327606	−0.163803	+	0.986493i	$0.552376\pi$
$720$	0	0
$721$	−871.437	−0.0450125
$722$	13358.5	0.688575
$723$	0	0
$724$	−2349.72	−0.120617
$725$	22178.4	1.13612
$726$	0	0
$727$	36426.5	1.85830	0.929150	−	0.369703i	$-0.120541\pi$
$727$	36426.5	1.85830	0.929150	+	0.369703i	$0.120541\pi$
$728$	−2153.43	−0.109631
$729$	0	0
$730$	−13423.0	−0.680558
$731$	−10010.7	−0.506509
$732$	0	0
$733$	−14013.9	−0.706161	−0.353081	−	0.935593i	$-0.614866\pi$
$733$	−14013.9	−0.706161	−0.353081	+	0.935593i	$0.614866\pi$
$734$	−24056.2	−1.20972
$735$	0	0
$736$	−686.618	−0.0343873
$737$	−13115.7	−0.655528
$738$	0	0
$739$	−23407.6	−1.16517	−0.582587	−	0.812768i	$-0.697960\pi$
$739$	−23407.6	−1.16517	−0.582587	+	0.812768i	$0.697960\pi$
$740$	−10656.3	−0.529372
$741$	0	0
$742$	393.662	0.0194768
$743$	−9526.11	−0.470362	−0.235181	−	0.971952i	$-0.575568\pi$
$743$	−9526.11	−0.470362	−0.235181	+	0.971952i	$0.575568\pi$
$744$	0	0
$745$	42208.9	2.07572
$746$	8309.64	0.407825
$747$	0	0
$748$	6032.03	0.294857
$749$	1774.59	0.0865715
$750$	0	0
$751$	−17166.3	−0.834098	−0.417049	−	0.908884i	$-0.636936\pi$
$751$	−17166.3	−0.834098	−0.417049	+	0.908884i	$0.636936\pi$
$752$	−2659.26	−0.128954
$753$	0	0
$754$	−25926.6	−1.25224
$755$	333.080	0.0160557
$756$	0	0
$757$	9502.71	0.456251	0.228125	−	0.973632i	$-0.426740\pi$
$757$	9502.71	0.456251	0.228125	+	0.973632i	$0.426740\pi$
$758$	3684.21	0.176539
$759$	0	0
$760$	7212.26	0.344232
$761$	−23314.4	−1.11057	−0.555287	−	0.831659i	$-0.687392\pi$
$761$	−23314.4	−1.11057	−0.555287	+	0.831659i	$0.687392\pi$
$762$	0	0
$763$	−1811.21	−0.0859375
$764$	2099.17	0.0994049
$765$	0	0
$766$	15613.7	0.736483
$767$	−4610.50	−0.217047
$768$	0	0
$769$	36770.4	1.72428	0.862142	−	0.506667i	$-0.169123\pi$
$769$	36770.4	1.72428	0.862142	+	0.506667i	$0.169123\pi$
$770$	1372.90	0.0642546
$771$	0	0
$772$	−15216.4	−0.709393
$773$	−30333.2	−1.41140	−0.705698	−	0.708513i	$-0.749366\pi$
$773$	−30333.2	−1.41140	−0.705698	+	0.708513i	$0.749366\pi$
$774$	0	0
$775$	33578.4	1.55635
$776$	−35373.1	−1.63636
$777$	0	0
$778$	−2005.48	−0.0924162
$779$	2359.41	0.108517
$780$	0	0
$781$	−27676.8	−1.26806
$782$	−398.434	−0.0182199
$783$	0	0
$784$	6539.02	0.297878
$785$	30810.2	1.40084
$786$	0	0
$787$	−10130.6	−0.458851	−0.229426	−	0.973326i	$-0.573685\pi$
$787$	−10130.6	−0.458851	−0.229426	+	0.973326i	$0.573685\pi$
$788$	5246.93	0.237201
$789$	0	0
$790$	−16377.5	−0.737574
$791$	1606.24	0.0722014
$792$	0	0
$793$	−68651.2	−3.07424
$794$	5234.09	0.233943
$795$	0	0
$796$	−8108.37	−0.361047
$797$	24869.8	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$797$	24869.8	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$798$	0	0
$799$	6070.01	0.268763
$800$	−21126.5	−0.933670
$801$	0	0
$802$	−26185.7	−1.15293
$803$	14702.5	0.646127
$804$	0	0
$805$	82.1095	0.00359501
$806$	−39253.1	−1.71542
$807$	0	0
$808$	−5516.25	−0.240174
$809$	28593.0	1.24262	0.621308	−	0.783567i	$-0.286602\pi$
$809$	28593.0	1.24262	0.621308	+	0.783567i	$0.286602\pi$
$810$	0	0
$811$	−28737.4	−1.24428	−0.622138	−	0.782908i	$-0.713736\pi$
$811$	−28737.4	−1.24428	−0.622138	+	0.782908i	$0.713736\pi$
$812$	701.469	0.0303162
$813$	0	0
$814$	−12891.1	−0.555077
$815$	3169.08	0.136206
$816$	0	0
$817$	4223.09	0.180841
$818$	28917.6	1.23604
$819$	0	0
$820$	−7878.15	−0.335508
$821$	−28082.4	−1.19377	−0.596884	−	0.802328i	$-0.703595\pi$
$821$	−28082.4	−1.19377	−0.596884	+	0.802328i	$0.703595\pi$
$822$	0	0
$823$	16565.4	0.701621	0.350810	−	0.936447i	$-0.385906\pi$
$823$	16565.4	0.701621	0.350810	+	0.936447i	$0.385906\pi$
$824$	18491.3	0.781768
$825$	0	0
$826$	−137.769	−0.00580338
$827$	25792.9	1.08453	0.542265	−	0.840207i	$-0.317567\pi$
$827$	25792.9	1.08453	0.542265	+	0.840207i	$0.317567\pi$
$828$	0	0
$829$	34787.4	1.45744	0.728720	−	0.684812i	$-0.240116\pi$
$829$	34787.4	1.45744	0.728720	+	0.684812i	$0.240116\pi$
$830$	−13213.7	−0.552595
$831$	0	0
$832$	36660.2	1.52760
$833$	−14925.9	−0.620830
$834$	0	0
$835$	−33959.1	−1.40743
$836$	−2544.66	−0.105274
$837$	0	0
$838$	7329.37	0.302135
$839$	40608.0	1.67097	0.835485	−	0.549513i	$-0.185187\pi$
$839$	40608.0	1.67097	0.835485	+	0.549513i	$0.185187\pi$
$840$	0	0
$841$	1829.31	0.0750056
$842$	21421.0	0.876740
$843$	0	0
$844$	−22811.5	−0.930336
$845$	−63277.0	−2.57609
$846$	0	0
$847$	13.0322	0.000528680 0
$848$	−3226.19	−0.130646
$849$	0	0
$850$	−12259.4	−0.494699
$851$	−770.979	−0.0310562
$852$	0	0
$853$	18149.1	0.728505	0.364252	−	0.931300i	$-0.381325\pi$
$853$	18149.1	0.728505	0.364252	+	0.931300i	$0.381325\pi$
$854$	−2051.40	−0.0821986
$855$	0	0
$856$	−37655.7	−1.50356
$857$	4806.87	0.191598	0.0957990	−	0.995401i	$-0.469459\pi$
$857$	4806.87	0.191598	0.0957990	+	0.995401i	$0.469459\pi$
$858$	0	0
$859$	15908.4	0.631885	0.315942	−	0.948778i	$-0.397679\pi$
$859$	15908.4	0.631885	0.315942	+	0.948778i	$0.397679\pi$
$860$	−14101.0	−0.559118
$861$	0	0
$862$	−25655.8	−1.01374
$863$	−8799.82	−0.347102	−0.173551	−	0.984825i	$-0.555524\pi$
$863$	−8799.82	−0.347102	−0.173551	+	0.984825i	$0.555524\pi$
$864$	0	0
$865$	32319.2	1.27039
$866$	−27829.9	−1.09203
$867$	0	0
$868$	1062.03	0.0415296
$869$	17938.6	0.700258
$870$	0	0
$871$	28214.5	1.09760
$872$	38432.8	1.49255
$873$	0	0
$874$	168.083	0.00650513
$875$	220.803	0.00853085
$876$	0	0
$877$	8805.63	0.339048	0.169524	−	0.985526i	$-0.445777\pi$
$877$	8805.63	0.339048	0.169524	+	0.985526i	$0.445777\pi$
$878$	−2524.95	−0.0970533
$879$	0	0
$880$	−11251.4	−0.431006
$881$	37620.6	1.43867	0.719335	−	0.694663i	$-0.244447\pi$
$881$	37620.6	1.43867	0.719335	+	0.694663i	$0.244447\pi$
$882$	0	0
$883$	15013.0	0.572173	0.286086	−	0.958204i	$-0.407646\pi$
$883$	15013.0	0.572173	0.286086	+	0.958204i	$0.407646\pi$
$884$	−12976.1	−0.493703
$885$	0	0
$886$	25701.2	0.974548
$887$	−32943.4	−1.24705	−0.623525	−	0.781804i	$-0.714300\pi$
$887$	−32943.4	−1.24705	−0.623525	+	0.781804i	$0.714300\pi$
$888$	0	0
$889$	1447.69	0.0546164
$890$	44040.6	1.65870
$891$	0	0
$892$	18702.1	0.702009
$893$	−2560.68	−0.0959574
$894$	0	0
$895$	−40055.3	−1.49598
$896$	−310.717	−0.0115852
$897$	0	0
$898$	−2808.56	−0.104369
$899$	39694.8	1.47263
$900$	0	0
$901$	7364.08	0.272290
$902$	−9530.28	−0.351800
$903$	0	0
$904$	−34083.4	−1.25398
$905$	−10004.3	−0.367464
$906$	0	0
$907$	46052.7	1.68595	0.842973	−	0.537955i	$-0.180803\pi$
$907$	46052.7	1.68595	0.842973	+	0.537955i	$0.180803\pi$
$908$	3883.55	0.141939
$909$	0	0
$910$	−2953.39	−0.107587
$911$	20657.9	0.751290	0.375645	−	0.926764i	$-0.377421\pi$
$911$	20657.9	0.751290	0.375645	+	0.926764i	$0.377421\pi$
$912$	0	0
$913$	14473.2	0.524638
$914$	14450.7	0.522960
$915$	0	0
$916$	−6635.80	−0.239359
$917$	2851.79	0.102698
$918$	0	0
$919$	−47631.7	−1.70971	−0.854855	−	0.518867i	$-0.826354\pi$
$919$	−47631.7	−1.70971	−0.854855	+	0.518867i	$0.826354\pi$
$920$	−1742.31	−0.0624374
$921$	0	0
$922$	10414.0	0.371983
$923$	59538.3	2.12321
$924$	0	0
$925$	−23722.2	−0.843224
$926$	11679.3	0.414476
$927$	0	0
$928$	−24974.8	−0.883447
$929$	27546.0	0.972827	0.486413	−	0.873729i	$-0.338305\pi$
$929$	27546.0	0.972827	0.486413	+	0.873729i	$0.338305\pi$
$930$	0	0
$931$	6296.61	0.221658
$932$	−7455.81	−0.262042
$933$	0	0
$934$	34746.3	1.21727
$935$	25682.4	0.898293
$936$	0	0
$937$	−8405.09	−0.293044	−0.146522	−	0.989207i	$-0.546808\pi$
$937$	−8405.09	−0.293044	−0.146522	+	0.989207i	$0.546808\pi$
$938$	843.094	0.0293475
$939$	0	0
$940$	8550.21	0.296678
$941$	12903.8	0.447027	0.223514	−	0.974701i	$-0.428247\pi$
$941$	12903.8	0.447027	0.223514	+	0.974701i	$0.428247\pi$
$942$	0	0
$943$	−569.978	−0.0196830
$944$	1129.06	0.0389278
$945$	0	0
$946$	−17058.2	−0.586267
$947$	26496.9	0.909223	0.454612	−	0.890690i	$-0.349778\pi$
$947$	26496.9	0.909223	0.454612	+	0.890690i	$0.349778\pi$
$948$	0	0
$949$	−31628.0	−1.08186
$950$	5171.73	0.176624
$951$	0	0
$952$	−1203.73	−0.0409802
$953$	−3951.58	−0.134317	−0.0671585	−	0.997742i	$-0.521393\pi$
$953$	−3951.58	−0.134317	−0.0671585	+	0.997742i	$0.521393\pi$
$954$	0	0
$955$	8937.57	0.302841
$956$	6930.06	0.234450
$957$	0	0
$958$	32283.8	1.08877
$959$	521.955	0.0175754
$960$	0	0
$961$	30307.3	1.01733
$962$	27731.3	0.929410
$963$	0	0
$964$	23397.6	0.781729
$965$	−64786.6	−2.16120
$966$	0	0
$967$	46296.1	1.53959	0.769793	−	0.638293i	$-0.220359\pi$
$967$	46296.1	1.53959	0.769793	+	0.638293i	$0.220359\pi$
$968$	−276.535	−0.00918200
$969$	0	0
$970$	−48513.4	−1.60585
$971$	49931.5	1.65023	0.825117	−	0.564961i	$-0.191109\pi$
$971$	49931.5	1.65023	0.825117	+	0.564961i	$0.191109\pi$
$972$	0	0
$973$	−327.011	−0.0107744
$974$	13093.4	0.430737
$975$	0	0
$976$	16812.0	0.551371
$977$	13238.2	0.433497	0.216748	−	0.976227i	$-0.430455\pi$
$977$	13238.2	0.433497	0.216748	+	0.976227i	$0.430455\pi$
$978$	0	0
$979$	−48238.6	−1.57478
$980$	−21024.6	−0.685313
$981$	0	0
$982$	32581.7	1.05878
$983$	−30187.5	−0.979483	−0.489742	−	0.871868i	$-0.662909\pi$
$983$	−30187.5	−0.979483	−0.489742	+	0.871868i	$0.662909\pi$
$984$	0	0
$985$	22339.7	0.722641
$986$	−14492.5	−0.468089
$987$	0	0
$988$	5474.07	0.176269
$989$	−1020.20	−0.0328013
$990$	0	0
$991$	−57834.2	−1.85385	−0.926924	−	0.375250i	$-0.877557\pi$
$991$	−57834.2	−1.85385	−0.926924	+	0.375250i	$0.877557\pi$
$992$	−37812.1	−1.21022
$993$	0	0
$994$	1779.10	0.0567701
$995$	−34522.7	−1.09994
$996$	0	0
$997$	22267.8	0.707351	0.353675	−	0.935368i	$-0.384932\pi$
$997$	22267.8	0.707351	0.353675	+	0.935368i	$0.384932\pi$
$998$	−7996.31	−0.253626
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.e.1.4		8
3.2	odd	2		177.4.a.d.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.d.1.5	✓	8	3.2	odd	2
531.4.a.e.1.4		8	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-2.04902 q^{2} -3.80150 q^{4} -16.1855 q^{5} -1.13960 q^{7} +24.1816 q^{8} +O(q^{10})\)	\(q-2.04902 q^{2} -3.80150 q^{4} -16.1855 q^{5} -1.13960 q^{7} +24.1816 q^{8} +33.1645 q^{10} -36.3258 q^{11} +78.1440 q^{13} +2.33506 q^{14} -19.1366 q^{16} +43.6811 q^{17} -18.4272 q^{19} +61.5292 q^{20} +74.4325 q^{22} +4.45159 q^{23} +136.971 q^{25} -160.119 q^{26} +4.33218 q^{28} +161.921 q^{29} +245.150 q^{31} -154.241 q^{32} -89.5037 q^{34} +18.4450 q^{35} -173.192 q^{37} +37.7579 q^{38} -391.391 q^{40} -128.039 q^{41} -229.176 q^{43} +138.092 q^{44} -9.12142 q^{46} +138.962 q^{47} -341.701 q^{49} -280.657 q^{50} -297.064 q^{52} +168.587 q^{53} +587.952 q^{55} -27.5572 q^{56} -331.780 q^{58} -59.0000 q^{59} -878.522 q^{61} -502.318 q^{62} +469.137 q^{64} -1264.80 q^{65} +361.058 q^{67} -166.054 q^{68} -37.7942 q^{70} +761.904 q^{71} -404.740 q^{73} +354.874 q^{74} +70.0511 q^{76} +41.3968 q^{77} -493.825 q^{79} +309.736 q^{80} +262.356 q^{82} -398.428 q^{83} -707.002 q^{85} +469.588 q^{86} -878.415 q^{88} +1327.94 q^{89} -89.0527 q^{91} -16.9227 q^{92} -284.736 q^{94} +298.254 q^{95} -1462.81 q^{97} +700.154 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) \) 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) \) 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8 + 21 * q^10 - 67 * q^11 + 33 * q^13 - 79 * q^14 - 30 * q^16 - 139 * q^17 + 64 * q^19 - 117 * q^20 - 84 * q^22 - 226 * q^23 + 96 * q^25 - 24 * q^26 + 34 * q^28 - 456 * q^29 + 124 * q^31 - 174 * q^32 - 114 * q^34 - 556 * q^35 + 127 * q^37 - 237 * q^38 - 188 * q^40 - 425 * q^41 - 115 * q^43 - 510 * q^44 - 711 * q^46 - 420 * q^47 + 171 * q^49 + 137 * q^50 - 922 * q^52 - 98 * q^53 - 616 * q^55 + 412 * q^56 - 1548 * q^58 - 472 * q^59 - 1254 * q^61 + 766 * q^62 - 2019 * q^64 + 734 * q^65 - 1010 * q^67 + 503 * q^68 - 2956 * q^70 + 17 * q^71 - 1180 * q^73 + 1228 * q^74 - 2008 * q^76 - 441 * q^77 - 873 * q^79 + 865 * q^80 - 3645 * q^82 - 759 * q^83 - 850 * q^85 + 1226 * q^86 - 3047 * q^88 - 988 * q^89 - 2111 * q^91 + 1062 * q^92 - 2240 * q^94 - 1822 * q^95 - 668 * q^97 + 1368 * q^98

Introduction

L-functions

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