LMFDB - Embedded newform 531.4.a.d.1.1

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 $ x^7 - x^6 - 34x^5 + 25x^4 + 315x^3 - 146x^2 - 736*x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.44426$ 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)

$f(q)$	$=$	$q-3.44426 q^{2} +3.86292 q^{4} +20.9057 q^{5} -30.0572 q^{7} +14.2492 q^{8} +O(q^{10})$	\(q-3.44426 q^{2} +3.86292 q^{4} +20.9057 q^{5} -30.0572 q^{7} +14.2492 q^{8} -72.0046 q^{10} -51.2171 q^{11} -22.3086 q^{13} +103.525 q^{14} -79.9812 q^{16} +89.1524 q^{17} +96.6448 q^{19} +80.7571 q^{20} +176.405 q^{22} -76.1937 q^{23} +312.048 q^{25} +76.8365 q^{26} -116.108 q^{28} +71.5601 q^{29} -129.288 q^{31} +161.483 q^{32} -307.064 q^{34} -628.366 q^{35} -108.382 q^{37} -332.870 q^{38} +297.889 q^{40} +357.190 q^{41} -237.327 q^{43} -197.848 q^{44} +262.431 q^{46} -97.2732 q^{47} +560.433 q^{49} -1074.77 q^{50} -86.1763 q^{52} +705.515 q^{53} -1070.73 q^{55} -428.290 q^{56} -246.471 q^{58} +59.0000 q^{59} -549.755 q^{61} +445.302 q^{62} +83.6615 q^{64} -466.376 q^{65} +652.131 q^{67} +344.389 q^{68} +2164.25 q^{70} +37.3589 q^{71} -600.093 q^{73} +373.297 q^{74} +373.331 q^{76} +1539.44 q^{77} +1221.13 q^{79} -1672.06 q^{80} -1230.26 q^{82} +577.079 q^{83} +1863.79 q^{85} +817.417 q^{86} -729.802 q^{88} +375.999 q^{89} +670.532 q^{91} -294.330 q^{92} +335.034 q^{94} +2020.43 q^{95} +322.400 q^{97} -1930.28 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) $ 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8	$ 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8} - 79 q^{10} + 131 q^{11} - 123 q^{13} + 117 q^{14} + 202 q^{16} + 235 q^{17} - 80 q^{19} - 61 q^{20} + 688 q^{22} + 274 q^{23} + 193 q^{25} + 180 q^{26} - 118 q^{28} + 406 q^{29} - 346 q^{31} + 854 q^{32} + 178 q^{34} + 424 q^{35} - 157 q^{37} + 129 q^{38} - 590 q^{40} + 825 q^{41} - 815 q^{43} + 1690 q^{44} + 1457 q^{46} + 1196 q^{47} + 914 q^{49} - 713 q^{50} + 1030 q^{52} + 900 q^{53} - 1044 q^{55} - 2172 q^{56} + 1242 q^{58} + 413 q^{59} + 420 q^{61} - 646 q^{62} + 3541 q^{64} - 190 q^{65} + 1316 q^{67} + 611 q^{68} + 4658 q^{70} + 173 q^{71} - 418 q^{73} - 660 q^{74} + 1540 q^{76} + 753 q^{77} + 2635 q^{79} - 6155 q^{80} - 125 q^{82} - 457 q^{83} + 1270 q^{85} - 3482 q^{86} + 7685 q^{88} - 592 q^{89} + 3179 q^{91} + 3500 q^{92} + 2064 q^{94} + 2250 q^{95} - 1906 q^{97} - 2994 q^{98}+O(q^{100}) $ 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8 - 79 * q^10 + 131 * q^11 - 123 * q^13 + 117 * q^14 + 202 * q^16 + 235 * q^17 - 80 * q^19 - 61 * q^20 + 688 * q^22 + 274 * q^23 + 193 * q^25 + 180 * q^26 - 118 * q^28 + 406 * q^29 - 346 * q^31 + 854 * q^32 + 178 * q^34 + 424 * q^35 - 157 * q^37 + 129 * q^38 - 590 * q^40 + 825 * q^41 - 815 * q^43 + 1690 * q^44 + 1457 * q^46 + 1196 * q^47 + 914 * q^49 - 713 * q^50 + 1030 * q^52 + 900 * q^53 - 1044 * q^55 - 2172 * q^56 + 1242 * q^58 + 413 * q^59 + 420 * q^61 - 646 * q^62 + 3541 * q^64 - 190 * q^65 + 1316 * q^67 + 611 * q^68 + 4658 * q^70 + 173 * q^71 - 418 * q^73 - 660 * q^74 + 1540 * q^76 + 753 * q^77 + 2635 * q^79 - 6155 * q^80 - 125 * q^82 - 457 * q^83 + 1270 * q^85 - 3482 * q^86 + 7685 * q^88 - 592 * q^89 + 3179 * q^91 + 3500 * q^92 + 2064 * q^94 + 2250 * q^95 - 1906 * q^97 - 2994 * 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 $

$2$	−3.44426	−1.21773	−0.608865	−	0.793274i	$-0.708375\pi$
$2$	−3.44426	−1.21773	−0.608865	+	0.793274i	$0.708375\pi$
$3$	0	0
$3$	0	0
$4$	3.86292	0.482865
$5$	20.9057	1.86986	0.934931	−	0.354830i	$-0.115461\pi$
$5$	20.9057	1.86986	0.934931	+	0.354830i	$0.115461\pi$
$6$	0	0
$7$	−30.0572	−1.62293	−0.811467	−	0.584398i	$-0.801331\pi$
$7$	−30.0572	−1.62293	−0.811467	+	0.584398i	$0.801331\pi$
$8$	14.2492	0.629730
$9$	0	0
$10$	−72.0046	−2.27699
$11$	−51.2171	−1.40387	−0.701934	−	0.712242i	$-0.747680\pi$
$11$	−51.2171	−1.40387	−0.701934	+	0.712242i	$0.747680\pi$
$12$	0	0
$13$	−22.3086	−0.475945	−0.237973	−	0.971272i	$-0.576483\pi$
$13$	−22.3086	−0.475945	−0.237973	+	0.971272i	$0.576483\pi$
$14$	103.525	1.97629
$15$	0	0
$16$	−79.9812	−1.24971
$17$	89.1524	1.27192	0.635960	−	0.771722i	$-0.280605\pi$
$17$	89.1524	1.27192	0.635960	+	0.771722i	$0.280605\pi$
$18$	0	0
$19$	96.6448	1.16694	0.583469	−	0.812135i	$-0.301695\pi$
$19$	96.6448	1.16694	0.583469	+	0.812135i	$0.301695\pi$
$20$	80.7571	0.902891
$21$	0	0
$22$	176.405	1.70953
$23$	−76.1937	−0.690760	−0.345380	−	0.938463i	$-0.612250\pi$
$23$	−76.1937	−0.690760	−0.345380	+	0.938463i	$0.612250\pi$
$24$	0	0
$25$	312.048	2.49638
$26$	76.8365	0.579572
$27$	0	0
$28$	−116.108	−0.783658
$29$	71.5601	0.458220	0.229110	−	0.973401i	$-0.426419\pi$
$29$	71.5601	0.458220	0.229110	+	0.973401i	$0.426419\pi$
$30$	0	0
$31$	−129.288	−0.749059	−0.374530	−	0.927215i	$-0.622196\pi$
$31$	−129.288	−0.749059	−0.374530	+	0.927215i	$0.622196\pi$
$32$	161.483	0.892074
$33$	0	0
$34$	−307.064	−1.54885
$35$	−628.366	−3.03466
$36$	0	0
$37$	−108.382	−0.481567	−0.240783	−	0.970579i	$-0.577404\pi$
$37$	−108.382	−0.481567	−0.240783	+	0.970579i	$0.577404\pi$
$38$	−332.870	−1.42102
$39$	0	0
$40$	297.889	1.17751
$41$	357.190	1.36058	0.680290	−	0.732943i	$-0.261854\pi$
$41$	357.190	1.36058	0.680290	+	0.732943i	$0.261854\pi$
$42$	0	0
$43$	−237.327	−0.841676	−0.420838	−	0.907136i	$-0.638264\pi$
$43$	−237.327	−0.841676	−0.420838	+	0.907136i	$0.638264\pi$
$44$	−197.848	−0.677879
$45$	0	0
$46$	262.431	0.841159
$47$	−97.2732	−0.301889	−0.150944	−	0.988542i	$-0.548231\pi$
$47$	−97.2732	−0.301889	−0.150944	+	0.988542i	$0.548231\pi$
$48$	0	0
$49$	560.433	1.63392
$50$	−1074.77	−3.03992
$51$	0	0
$52$	−86.1763	−0.229817
$53$	705.515	1.82849	0.914245	−	0.405162i	$-0.132785\pi$
$53$	705.515	1.82849	0.914245	+	0.405162i	$0.132785\pi$
$54$	0	0
$55$	−1070.73	−2.62504
$56$	−428.290	−1.02201
$57$	0	0
$58$	−246.471	−0.557988
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	−549.755	−1.15392	−0.576958	−	0.816774i	$-0.695760\pi$
$61$	−549.755	−1.15392	−0.576958	+	0.816774i	$0.695760\pi$
$62$	445.302	0.912152
$63$	0	0
$64$	83.6615	0.163401
$65$	−466.376	−0.889951
$66$	0	0
$67$	652.131	1.18911	0.594555	−	0.804055i	$-0.297328\pi$
$67$	652.131	1.18911	0.594555	+	0.804055i	$0.297328\pi$
$68$	344.389	0.614166
$69$	0	0
$70$	2164.25	3.69540
$71$	37.3589	0.0624463	0.0312232	−	0.999512i	$-0.490060\pi$
$71$	37.3589	0.0624463	0.0312232	+	0.999512i	$0.490060\pi$
$72$	0	0
$73$	−600.093	−0.962131	−0.481066	−	0.876685i	$-0.659750\pi$
$73$	−600.093	−0.962131	−0.481066	+	0.876685i	$0.659750\pi$
$74$	373.297	0.586418
$75$	0	0
$76$	373.331	0.563474
$77$	1539.44	2.27839
$78$	0	0
$79$	1221.13	1.73909	0.869546	−	0.493851i	$-0.164411\pi$
$79$	1221.13	1.73909	0.869546	+	0.493851i	$0.164411\pi$
$80$	−1672.06	−2.33678
$81$	0	0
$82$	−1230.26	−1.65682
$83$	577.079	0.763164	0.381582	−	0.924335i	$-0.375379\pi$
$83$	577.079	0.763164	0.381582	+	0.924335i	$0.375379\pi$
$84$	0	0
$85$	1863.79	2.37831
$86$	817.417	1.02493
$87$	0	0
$88$	−729.802	−0.884058
$89$	375.999	0.447818	0.223909	−	0.974610i	$-0.428118\pi$
$89$	375.999	0.447818	0.223909	+	0.974610i	$0.428118\pi$
$90$	0	0
$91$	670.532	0.772427
$92$	−294.330	−0.333544
$93$	0	0
$94$	335.034	0.367619
$95$	2020.43	2.18201
$96$	0	0
$97$	322.400	0.337472	0.168736	−	0.985661i	$-0.446032\pi$
$97$	322.400	0.337472	0.168736	+	0.985661i	$0.446032\pi$
$98$	−1930.28	−1.98967
$99$	0	0
$100$	1205.42	1.20542
$101$	−657.772	−0.648027	−0.324014	−	0.946052i	$-0.605032\pi$
$101$	−657.772	−0.648027	−0.324014	+	0.946052i	$0.605032\pi$
$102$	0	0
$103$	1003.17	0.959667	0.479834	−	0.877359i	$-0.340697\pi$
$103$	1003.17	0.959667	0.479834	+	0.877359i	$0.340697\pi$
$104$	−317.879	−0.299717
$105$	0	0
$106$	−2429.98	−2.22661
$107$	882.607	0.797428	0.398714	−	0.917075i	$-0.369457\pi$
$107$	882.607	0.797428	0.398714	+	0.917075i	$0.369457\pi$
$108$	0	0
$109$	608.066	0.534332	0.267166	−	0.963651i	$-0.413913\pi$
$109$	608.066	0.534332	0.267166	+	0.963651i	$0.413913\pi$
$110$	3687.87	3.19659
$111$	0	0
$112$	2404.01	2.02819
$113$	−373.130	−0.310629	−0.155315	−	0.987865i	$-0.549639\pi$
$113$	−373.130	−0.310629	−0.155315	+	0.987865i	$0.549639\pi$
$114$	0	0
$115$	−1592.88	−1.29163
$116$	276.431	0.221258
$117$	0	0
$118$	−203.211	−0.158535
$119$	−2679.67	−2.06424
$120$	0	0
$121$	1292.20	0.970846
$122$	1893.50	1.40516
$123$	0	0
$124$	−499.430	−0.361695
$125$	3910.37	2.79803
$126$	0	0
$127$	626.337	0.437625	0.218813	−	0.975767i	$-0.429782\pi$
$127$	626.337	0.437625	0.218813	+	0.975767i	$0.429782\pi$
$128$	−1580.01	−1.09105
$129$	0	0
$130$	1606.32	1.08372
$131$	−1191.46	−0.794641	−0.397320	−	0.917680i	$-0.630060\pi$
$131$	−1191.46	−0.794641	−0.397320	+	0.917680i	$0.630060\pi$
$132$	0	0
$133$	−2904.87	−1.89387
$134$	−2246.11	−1.44802
$135$	0	0
$136$	1270.35	0.800966
$137$	487.325	0.303905	0.151952	−	0.988388i	$-0.451444\pi$
$137$	487.325	0.303905	0.151952	+	0.988388i	$0.451444\pi$
$138$	0	0
$139$	552.716	0.337272	0.168636	−	0.985678i	$-0.446064\pi$
$139$	552.716	0.337272	0.168636	+	0.985678i	$0.446064\pi$
$140$	−2427.33	−1.46533
$141$	0	0
$142$	−128.674	−0.0760427
$143$	1142.58	0.668164
$144$	0	0
$145$	1496.01	0.856807
$146$	2066.88	1.17162
$147$	0	0
$148$	−418.673	−0.232532
$149$	3509.24	1.92945	0.964725	−	0.263260i	$-0.0847978\pi$
$149$	3509.24	1.92945	0.964725	+	0.263260i	$0.0847978\pi$
$150$	0	0
$151$	3175.51	1.71138	0.855692	−	0.517485i	$-0.173132\pi$
$151$	3175.51	1.71138	0.855692	+	0.517485i	$0.173132\pi$
$152$	1377.11	0.734857
$153$	0	0
$154$	−5302.24	−2.77446
$155$	−2702.86	−1.40064
$156$	0	0
$157$	719.754	0.365877	0.182938	−	0.983124i	$-0.441439\pi$
$157$	719.754	0.365877	0.182938	+	0.983124i	$0.441439\pi$
$158$	−4205.90	−2.11774
$159$	0	0
$160$	3375.91	1.66806
$161$	2290.17	1.12106
$162$	0	0
$163$	−1106.29	−0.531602	−0.265801	−	0.964028i	$-0.585636\pi$
$163$	−1106.29	−0.531602	−0.265801	+	0.964028i	$0.585636\pi$
$164$	1379.80	0.656977
$165$	0	0
$166$	−1987.61	−0.929327
$167$	821.206	0.380520	0.190260	−	0.981734i	$-0.439067\pi$
$167$	821.206	0.380520	0.190260	+	0.981734i	$0.439067\pi$
$168$	0	0
$169$	−1699.33	−0.773476
$170$	−6419.38	−2.89614
$171$	0	0
$172$	−916.777	−0.406416
$173$	4010.92	1.76269	0.881343	−	0.472476i	$-0.156640\pi$
$173$	4010.92	1.76269	0.881343	+	0.472476i	$0.156640\pi$
$174$	0	0
$175$	−9379.28	−4.05147
$176$	4096.41	1.75442
$177$	0	0
$178$	−1295.04	−0.545321
$179$	716.188	0.299053	0.149526	−	0.988758i	$-0.452225\pi$
$179$	716.188	0.299053	0.149526	+	0.988758i	$0.452225\pi$
$180$	0	0
$181$	1786.44	0.733619	0.366809	−	0.930296i	$-0.380450\pi$
$181$	1786.44	0.733619	0.366809	+	0.930296i	$0.380450\pi$
$182$	−2309.49	−0.940608
$183$	0	0
$184$	−1085.70	−0.434993
$185$	−2265.81	−0.900463
$186$	0	0
$187$	−4566.13	−1.78561
$188$	−375.759	−0.145771
$189$	0	0
$190$	−6958.87	−2.65710
$191$	−853.280	−0.323252	−0.161626	−	0.986852i	$-0.551674\pi$
$191$	−853.280	−0.323252	−0.161626	+	0.986852i	$0.551674\pi$
$192$	0	0
$193$	−3478.84	−1.29747	−0.648737	−	0.761013i	$-0.724702\pi$
$193$	−3478.84	−1.29747	−0.648737	+	0.761013i	$0.724702\pi$
$194$	−1110.43	−0.410949
$195$	0	0
$196$	2164.91	0.788961
$197$	337.299	0.121988	0.0609938	−	0.998138i	$-0.480573\pi$
$197$	337.299	0.121988	0.0609938	+	0.998138i	$0.480573\pi$
$198$	0	0
$199$	−2613.64	−0.931035	−0.465518	−	0.885039i	$-0.654132\pi$
$199$	−2613.64	−0.931035	−0.465518	+	0.885039i	$0.654132\pi$
$200$	4446.42	1.57205
$201$	0	0
$202$	2265.54	0.789122
$203$	−2150.89	−0.743660
$204$	0	0
$205$	7467.31	2.54410
$206$	−3455.19	−1.16862
$207$	0	0
$208$	1784.27	0.594792
$209$	−4949.87	−1.63823
$210$	0	0
$211$	4988.17	1.62749	0.813743	−	0.581224i	$-0.197426\pi$
$211$	4988.17	1.62749	0.813743	+	0.581224i	$0.197426\pi$
$212$	2725.35	0.882914
$213$	0	0
$214$	−3039.93	−0.971052
$215$	−4961.49	−1.57382
$216$	0	0
$217$	3886.04	1.21567
$218$	−2094.34	−0.650672
$219$	0	0
$220$	−4136.15	−1.26754
$221$	−1988.86	−0.605364
$222$	0	0
$223$	1752.62	0.526296	0.263148	−	0.964755i	$-0.415239\pi$
$223$	1752.62	0.526296	0.263148	+	0.964755i	$0.415239\pi$
$224$	−4853.71	−1.44778
$225$	0	0
$226$	1285.16	0.378262
$227$	2983.27	0.872276	0.436138	−	0.899880i	$-0.356346\pi$
$227$	2983.27	0.872276	0.436138	+	0.899880i	$0.356346\pi$
$228$	0	0
$229$	1672.34	0.482583	0.241292	−	0.970453i	$-0.422429\pi$
$229$	1672.34	0.482583	0.241292	+	0.970453i	$0.422429\pi$
$230$	5486.30	1.57285
$231$	0	0
$232$	1019.67	0.288555
$233$	−508.127	−0.142869	−0.0714345	−	0.997445i	$-0.522758\pi$
$233$	−508.127	−0.142869	−0.0714345	+	0.997445i	$0.522758\pi$
$234$	0	0
$235$	−2033.56	−0.564490
$236$	227.912	0.0628637
$237$	0	0
$238$	9229.47	2.51369
$239$	2497.08	0.675828	0.337914	−	0.941177i	$-0.390279\pi$
$239$	2497.08	0.675828	0.337914	+	0.941177i	$0.390279\pi$
$240$	0	0
$241$	−4476.15	−1.19641	−0.598204	−	0.801344i	$-0.704119\pi$
$241$	−4476.15	−1.19641	−0.598204	+	0.801344i	$0.704119\pi$
$242$	−4450.66	−1.18223
$243$	0	0
$244$	−2123.66	−0.557186
$245$	11716.2	3.05520
$246$	0	0
$247$	−2156.01	−0.555399
$248$	−1842.25	−0.471705
$249$	0	0
$250$	−13468.3	−3.40724
$251$	5524.04	1.38914	0.694571	−	0.719425i	$-0.255594\pi$
$251$	5524.04	1.38914	0.694571	+	0.719425i	$0.255594\pi$
$252$	0	0
$253$	3902.43	0.969737
$254$	−2157.27	−0.532909
$255$	0	0
$256$	4772.68	1.16521
$257$	−7709.96	−1.87134	−0.935669	−	0.352879i	$-0.885203\pi$
$257$	−7709.96	−1.87134	−0.935669	+	0.352879i	$0.885203\pi$
$258$	0	0
$259$	3257.67	0.781551
$260$	−1801.57	−0.429727
$261$	0	0
$262$	4103.68	0.967657
$263$	2619.57	0.614181	0.307091	−	0.951680i	$-0.400644\pi$
$263$	2619.57	0.614181	0.307091	+	0.951680i	$0.400644\pi$
$264$	0	0
$265$	14749.3	3.41902
$266$	10005.1	2.30622
$267$	0	0
$268$	2519.13	0.574180
$269$	−3075.82	−0.697160	−0.348580	−	0.937279i	$-0.613336\pi$
$269$	−3075.82	−0.697160	−0.348580	+	0.937279i	$0.613336\pi$
$270$	0	0
$271$	−4028.15	−0.902926	−0.451463	−	0.892290i	$-0.649098\pi$
$271$	−4028.15	−0.902926	−0.451463	+	0.892290i	$0.649098\pi$
$272$	−7130.51	−1.58953
$273$	0	0
$274$	−1678.47	−0.370074
$275$	−15982.2	−3.50459
$276$	0	0
$277$	−3905.75	−0.847197	−0.423599	−	0.905850i	$-0.639233\pi$
$277$	−3905.75	−0.847197	−0.423599	+	0.905850i	$0.639233\pi$
$278$	−1903.70	−0.410706
$279$	0	0
$280$	−8953.69	−1.91102
$281$	3502.01	0.743461	0.371730	−	0.928341i	$-0.378765\pi$
$281$	3502.01	0.743461	0.371730	+	0.928341i	$0.378765\pi$
$282$	0	0
$283$	−6544.11	−1.37458	−0.687291	−	0.726382i	$-0.741200\pi$
$283$	−6544.11	−1.37458	−0.687291	+	0.726382i	$0.741200\pi$
$284$	144.315	0.0301532
$285$	0	0
$286$	−3935.35	−0.813643
$287$	−10736.1	−2.20813
$288$	0	0
$289$	3035.14	0.617778
$290$	−5152.65	−1.04336
$291$	0	0
$292$	−2318.11	−0.464580
$293$	−6261.31	−1.24843	−0.624215	−	0.781253i	$-0.714581\pi$
$293$	−6261.31	−1.24843	−0.624215	+	0.781253i	$0.714581\pi$
$294$	0	0
$295$	1233.44	0.243435
$296$	−1544.36	−0.303257
$297$	0	0
$298$	−12086.7	−2.34955
$299$	1699.77	0.328764
$300$	0	0
$301$	7133.38	1.36598
$302$	−10937.3	−2.08400
$303$	0	0
$304$	−7729.77	−1.45833
$305$	−11493.0	−2.15766
$306$	0	0
$307$	−3929.77	−0.730566	−0.365283	−	0.930896i	$-0.619028\pi$
$307$	−3929.77	−0.730566	−0.365283	+	0.930896i	$0.619028\pi$
$308$	5946.74	1.10015
$309$	0	0
$310$	9309.35	1.70560
$311$	6515.56	1.18798	0.593992	−	0.804471i	$-0.297551\pi$
$311$	6515.56	1.18798	0.593992	+	0.804471i	$0.297551\pi$
$312$	0	0
$313$	−61.8440	−0.0111681	−0.00558407	−	0.999984i	$-0.501777\pi$
$313$	−61.8440	−0.0111681	−0.00558407	+	0.999984i	$0.501777\pi$
$314$	−2479.02	−0.445539
$315$	0	0
$316$	4717.15	0.839747
$317$	−4490.24	−0.795574	−0.397787	−	0.917478i	$-0.630222\pi$
$317$	−4490.24	−0.795574	−0.397787	+	0.917478i	$0.630222\pi$
$318$	0	0
$319$	−3665.10	−0.643280
$320$	1749.00	0.305538
$321$	0	0
$322$	−7887.93	−1.36515
$323$	8616.11	1.48425
$324$	0	0
$325$	−6961.35	−1.18814
$326$	3810.34	0.647348
$327$	0	0
$328$	5089.67	0.856798
$329$	2923.76	0.489945
$330$	0	0
$331$	7049.54	1.17063	0.585314	−	0.810807i	$-0.300971\pi$
$331$	7049.54	1.17063	0.585314	+	0.810807i	$0.300971\pi$
$332$	2229.21	0.368505
$333$	0	0
$334$	−2828.45	−0.463371
$335$	13633.2	2.22347
$336$	0	0
$337$	4495.19	0.726613	0.363307	−	0.931670i	$-0.381648\pi$
$337$	4495.19	0.726613	0.363307	+	0.931670i	$0.381648\pi$
$338$	5852.92	0.941885
$339$	0	0
$340$	7199.68	1.14840
$341$	6621.77	1.05158
$342$	0	0
$343$	−6535.42	−1.02880
$344$	−3381.72	−0.530029
$345$	0	0
$346$	−13814.7	−2.14648
$347$	−9082.61	−1.40513	−0.702565	−	0.711620i	$-0.747962\pi$
$347$	−9082.61	−1.40513	−0.702565	+	0.711620i	$0.747962\pi$
$348$	0	0
$349$	12020.9	1.84374	0.921870	−	0.387498i	$-0.126661\pi$
$349$	12020.9	1.84374	0.921870	+	0.387498i	$0.126661\pi$
$350$	32304.7	4.93359
$351$	0	0
$352$	−8270.68	−1.25235
$353$	5538.86	0.835138	0.417569	−	0.908645i	$-0.362882\pi$
$353$	5538.86	0.835138	0.417569	+	0.908645i	$0.362882\pi$
$354$	0	0
$355$	781.014	0.116766
$356$	1452.45	0.216236
$357$	0	0
$358$	−2466.74	−0.364165
$359$	10116.0	1.48719	0.743595	−	0.668630i	$-0.233119\pi$
$359$	10116.0	1.48719	0.743595	+	0.668630i	$0.233119\pi$
$360$	0	0
$361$	2481.22	0.361747
$362$	−6152.96	−0.893349
$363$	0	0
$364$	2590.21	0.372978
$365$	−12545.4	−1.79905
$366$	0	0
$367$	−5324.58	−0.757332	−0.378666	−	0.925533i	$-0.623617\pi$
$367$	−5324.58	−0.757332	−0.378666	+	0.925533i	$0.623617\pi$
$368$	6094.07	0.863248
$369$	0	0
$370$	7804.04	1.09652
$371$	−21205.8	−2.96752
$372$	0	0
$373$	−7097.68	−0.985266	−0.492633	−	0.870237i	$-0.663965\pi$
$373$	−7097.68	−0.985266	−0.492633	+	0.870237i	$0.663965\pi$
$374$	15726.9	2.17439
$375$	0	0
$376$	−1386.06	−0.190108
$377$	−1596.40	−0.218087
$378$	0	0
$379$	−8367.60	−1.13408	−0.567038	−	0.823692i	$-0.691911\pi$
$379$	−8367.60	−1.13408	−0.567038	+	0.823692i	$0.691911\pi$
$380$	7804.75	1.05362
$381$	0	0
$382$	2938.92	0.393634
$383$	10657.5	1.42186	0.710928	−	0.703265i	$-0.248275\pi$
$383$	10657.5	1.42186	0.710928	+	0.703265i	$0.248275\pi$
$384$	0	0
$385$	32183.1	4.26027
$386$	11982.0	1.57997
$387$	0	0
$388$	1245.41	0.162953
$389$	−1280.56	−0.166907	−0.0834534	−	0.996512i	$-0.526595\pi$
$389$	−1280.56	−0.166907	−0.0834534	+	0.996512i	$0.526595\pi$
$390$	0	0
$391$	−6792.85	−0.878591
$392$	7985.70	1.02893
$393$	0	0
$394$	−1161.75	−0.148548
$395$	25528.7	3.25186
$396$	0	0
$397$	831.103	0.105068	0.0525338	−	0.998619i	$-0.483270\pi$
$397$	831.103	0.105068	0.0525338	+	0.998619i	$0.483270\pi$
$398$	9002.05	1.13375
$399$	0	0
$400$	−24958.0	−3.11975
$401$	−4713.15	−0.586941	−0.293471	−	0.955968i	$-0.594810\pi$
$401$	−4713.15	−0.586941	−0.293471	+	0.955968i	$0.594810\pi$
$402$	0	0
$403$	2884.23	0.356511
$404$	−2540.92	−0.312910
$405$	0	0
$406$	7408.23	0.905577
$407$	5551.04	0.676056
$408$	0	0
$409$	10700.4	1.29364	0.646820	−	0.762643i	$-0.276098\pi$
$409$	10700.4	1.29364	0.646820	+	0.762643i	$0.276098\pi$
$410$	−25719.4	−3.09802
$411$	0	0
$412$	3875.19	0.463390
$413$	−1773.37	−0.211288
$414$	0	0
$415$	12064.2	1.42701
$416$	−3602.45	−0.424578
$417$	0	0
$418$	17048.6	1.99492
$419$	−3995.59	−0.465864	−0.232932	−	0.972493i	$-0.574832\pi$
$419$	−3995.59	−0.465864	−0.232932	+	0.972493i	$0.574832\pi$
$420$	0	0
$421$	5484.63	0.634928	0.317464	−	0.948270i	$-0.397169\pi$
$421$	5484.63	0.634928	0.317464	+	0.948270i	$0.397169\pi$
$422$	−17180.5	−1.98184
$423$	0	0
$424$	10053.0	1.15146
$425$	27819.8	3.17520
$426$	0	0
$427$	16524.1	1.87273
$428$	3409.44	0.385050
$429$	0	0
$430$	17088.7	1.91648
$431$	9116.24	1.01883	0.509413	−	0.860522i	$-0.329863\pi$
$431$	9116.24	1.01883	0.509413	+	0.860522i	$0.329863\pi$
$432$	0	0
$433$	−13162.3	−1.46083	−0.730415	−	0.683004i	$-0.760673\pi$
$433$	−13162.3	−1.46083	−0.730415	+	0.683004i	$0.760673\pi$
$434$	−13384.5	−1.48036
$435$	0	0
$436$	2348.91	0.258010
$437$	−7363.73	−0.806075
$438$	0	0
$439$	−2279.58	−0.247833	−0.123916	−	0.992293i	$-0.539545\pi$
$439$	−2279.58	−0.247833	−0.123916	+	0.992293i	$0.539545\pi$
$440$	−15257.0	−1.65307
$441$	0	0
$442$	6850.16	0.737169
$443$	−13194.2	−1.41507	−0.707536	−	0.706678i	$-0.750193\pi$
$443$	−13194.2	−1.41507	−0.707536	+	0.706678i	$0.750193\pi$
$444$	0	0
$445$	7860.51	0.837357
$446$	−6036.47	−0.640886
$447$	0	0
$448$	−2514.63	−0.265190
$449$	−3287.96	−0.345587	−0.172794	−	0.984958i	$-0.555279\pi$
$449$	−3287.96	−0.345587	−0.172794	+	0.984958i	$0.555279\pi$
$450$	0	0
$451$	−18294.3	−1.91007
$452$	−1441.37	−0.149992
$453$	0	0
$454$	−10275.2	−1.06220
$455$	14017.9	1.44433
$456$	0	0
$457$	−14476.7	−1.48182	−0.740908	−	0.671606i	$-0.765605\pi$
$457$	−14476.7	−1.48182	−0.740908	+	0.671606i	$0.765605\pi$
$458$	−5759.99	−0.587656
$459$	0	0
$460$	−6153.18	−0.623682
$461$	4765.61	0.481468	0.240734	−	0.970591i	$-0.422612\pi$
$461$	4765.61	0.481468	0.240734	+	0.970591i	$0.422612\pi$
$462$	0	0
$463$	6286.30	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$463$	6286.30	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$464$	−5723.46	−0.572640
$465$	0	0
$466$	1750.12	0.173976
$467$	−5623.77	−0.557253	−0.278626	−	0.960400i	$-0.589879\pi$
$467$	−5623.77	−0.557253	−0.278626	+	0.960400i	$0.589879\pi$
$468$	0	0
$469$	−19601.2	−1.92985
$470$	7004.12	0.687396
$471$	0	0
$472$	840.701	0.0819839
$473$	12155.2	1.18160
$474$	0	0
$475$	30157.8	2.91313
$476$	−10351.3	−0.996750
$477$	0	0
$478$	−8600.61	−0.822976
$479$	−10680.0	−1.01875	−0.509376	−	0.860544i	$-0.670124\pi$
$479$	−10680.0	−1.01875	−0.509376	+	0.860544i	$0.670124\pi$
$480$	0	0
$481$	2417.86	0.229199
$482$	15417.0	1.45690
$483$	0	0
$484$	4991.65	0.468788
$485$	6739.99	0.631025
$486$	0	0
$487$	8071.47	0.751034	0.375517	−	0.926816i	$-0.377465\pi$
$487$	8071.47	0.751034	0.375517	+	0.926816i	$0.377465\pi$
$488$	−7833.55	−0.726656
$489$	0	0
$490$	−40353.8	−3.72040
$491$	11495.5	1.05659	0.528295	−	0.849061i	$-0.322832\pi$
$491$	11495.5	1.05659	0.528295	+	0.849061i	$0.322832\pi$
$492$	0	0
$493$	6379.75	0.582818
$494$	7425.85	0.676326
$495$	0	0
$496$	10340.6	0.936104
$497$	−1122.90	−0.101346
$498$	0	0
$499$	−340.307	−0.0305295	−0.0152648	−	0.999883i	$-0.504859\pi$
$499$	−340.307	−0.0305295	−0.0152648	+	0.999883i	$0.504859\pi$
$500$	15105.4	1.35107
$501$	0	0
$502$	−19026.2	−1.69160
$503$	6818.69	0.604434	0.302217	−	0.953239i	$-0.402273\pi$
$503$	6818.69	0.604434	0.302217	+	0.953239i	$0.402273\pi$
$504$	0	0
$505$	−13751.2	−1.21172
$506$	−13441.0	−1.18088
$507$	0	0
$508$	2419.49	0.211314
$509$	−19771.2	−1.72170	−0.860849	−	0.508860i	$-0.830067\pi$
$509$	−19771.2	−1.72170	−0.860849	+	0.508860i	$0.830067\pi$
$510$	0	0
$511$	18037.1	1.56148
$512$	−3798.25	−0.327853
$513$	0	0
$514$	26555.1	2.27878
$515$	20972.1	1.79445
$516$	0	0
$517$	4982.06	0.423812
$518$	−11220.3	−0.951718
$519$	0	0
$520$	−6645.47	−0.560429
$521$	−20826.6	−1.75130	−0.875652	−	0.482943i	$-0.839568\pi$
$521$	−20826.6	−1.75130	−0.875652	+	0.482943i	$0.839568\pi$
$522$	0	0
$523$	5693.85	0.476051	0.238026	−	0.971259i	$-0.423500\pi$
$523$	5693.85	0.476051	0.238026	+	0.971259i	$0.423500\pi$
$524$	−4602.50	−0.383704
$525$	0	0
$526$	−9022.49	−0.747907
$527$	−11526.3	−0.952743
$528$	0	0
$529$	−6361.52	−0.522850
$530$	−50800.4	−4.16345
$531$	0	0
$532$	−11221.3	−0.914482
$533$	−7968.41	−0.647561
$534$	0	0
$535$	18451.5	1.49108
$536$	9292.32	0.748819
$537$	0	0
$538$	10593.9	0.848952
$539$	−28703.8	−2.29380
$540$	0	0
$541$	1565.72	0.124428	0.0622140	−	0.998063i	$-0.480184\pi$
$541$	1565.72	0.124428	0.0622140	+	0.998063i	$0.480184\pi$
$542$	13874.0	1.09952
$543$	0	0
$544$	14396.6	1.13465
$545$	12712.0	0.999127
$546$	0	0
$547$	2599.10	0.203162	0.101581	−	0.994827i	$-0.467610\pi$
$547$	2599.10	0.203162	0.101581	+	0.994827i	$0.467610\pi$
$548$	1882.50	0.146745
$549$	0	0
$550$	55046.9	4.26765
$551$	6915.91	0.534714
$552$	0	0
$553$	−36703.8	−2.82243
$554$	13452.4	1.03166
$555$	0	0
$556$	2135.10	0.162857
$557$	−14567.3	−1.10814	−0.554072	−	0.832469i	$-0.686927\pi$
$557$	−14567.3	−1.10814	−0.554072	+	0.832469i	$0.686927\pi$
$558$	0	0
$559$	5294.43	0.400591
$560$	50257.5	3.79244
$561$	0	0
$562$	−12061.8	−0.905334
$563$	−11318.1	−0.847251	−0.423626	−	0.905837i	$-0.639243\pi$
$563$	−11318.1	−0.847251	−0.423626	+	0.905837i	$0.639243\pi$
$564$	0	0
$565$	−7800.54	−0.580834
$566$	22539.6	1.67387
$567$	0	0
$568$	532.334	0.0393243
$569$	16128.1	1.18827	0.594134	−	0.804366i	$-0.297495\pi$
$569$	16128.1	1.18827	0.594134	+	0.804366i	$0.297495\pi$
$570$	0	0
$571$	−13301.5	−0.974867	−0.487433	−	0.873160i	$-0.662067\pi$
$571$	−13301.5	−0.974867	−0.487433	+	0.873160i	$0.662067\pi$
$572$	4413.70	0.322633
$573$	0	0
$574$	36978.0	2.68891
$575$	−23776.1	−1.72440
$576$	0	0
$577$	−19302.8	−1.39270	−0.696348	−	0.717704i	$-0.745193\pi$
$577$	−19302.8	−1.39270	−0.696348	+	0.717704i	$0.745193\pi$
$578$	−10453.8	−0.752287
$579$	0	0
$580$	5778.98	0.413723
$581$	−17345.3	−1.23857
$582$	0	0
$583$	−36134.5	−2.56696
$584$	−8550.83	−0.605883
$585$	0	0
$586$	21565.6	1.52025
$587$	−17933.2	−1.26096	−0.630480	−	0.776205i	$-0.717142\pi$
$587$	−17933.2	−1.26096	−0.630480	+	0.776205i	$0.717142\pi$
$588$	0	0
$589$	−12495.0	−0.874107
$590$	−4248.27	−0.296438
$591$	0	0
$592$	8668.56	0.601817
$593$	26500.4	1.83515	0.917573	−	0.397567i	$-0.130145\pi$
$593$	26500.4	1.83515	0.917573	+	0.397567i	$0.130145\pi$
$594$	0	0
$595$	−56020.3	−3.85985
$596$	13555.9	0.931664
$597$	0	0
$598$	−5854.46	−0.400346
$599$	−14006.2	−0.955388	−0.477694	−	0.878526i	$-0.658527\pi$
$599$	−14006.2	−0.955388	−0.477694	+	0.878526i	$0.658527\pi$
$600$	0	0
$601$	8209.99	0.557225	0.278613	−	0.960404i	$-0.410125\pi$
$601$	8209.99	0.557225	0.278613	+	0.960404i	$0.410125\pi$
$602$	−24569.2	−1.66340
$603$	0	0
$604$	12266.7	0.826368
$605$	27014.2	1.81535
$606$	0	0
$607$	150.423	0.0100585	0.00502924	−	0.999987i	$-0.498399\pi$
$607$	150.423	0.0100585	0.00502924	+	0.999987i	$0.498399\pi$
$608$	15606.5	1.04100
$609$	0	0
$610$	39584.9	2.62745
$611$	2170.03	0.143682
$612$	0	0
$613$	1790.10	0.117947	0.0589734	−	0.998260i	$-0.481217\pi$
$613$	1790.10	0.117947	0.0589734	+	0.998260i	$0.481217\pi$
$614$	13535.1	0.889632
$615$	0	0
$616$	21935.8	1.43477
$617$	−12232.0	−0.798122	−0.399061	−	0.916924i	$-0.630664\pi$
$617$	−12232.0	−0.798122	−0.399061	+	0.916924i	$0.630664\pi$
$618$	0	0
$619$	−5439.47	−0.353200	−0.176600	−	0.984283i	$-0.556510\pi$
$619$	−5439.47	−0.353200	−0.176600	+	0.984283i	$0.556510\pi$
$620$	−10440.9	−0.676319
$621$	0	0
$622$	−22441.3	−1.44664
$623$	−11301.5	−0.726779
$624$	0	0
$625$	42742.9	2.73555
$626$	213.007	0.0135998
$627$	0	0
$628$	2780.35	0.176669
$629$	−9662.56	−0.612514
$630$	0	0
$631$	12962.2	0.817776	0.408888	−	0.912585i	$-0.365917\pi$
$631$	12962.2	0.817776	0.408888	+	0.912585i	$0.365917\pi$
$632$	17400.1	1.09516
$633$	0	0
$634$	15465.5	0.968794
$635$	13094.0	0.818299
$636$	0	0
$637$	−12502.5	−0.777654
$638$	12623.6	0.783341
$639$	0	0
$640$	−33031.3	−2.04012
$641$	13358.5	0.823131	0.411566	−	0.911380i	$-0.364982\pi$
$641$	13358.5	0.823131	0.411566	+	0.911380i	$0.364982\pi$
$642$	0	0
$643$	−23023.9	−1.41209	−0.706046	−	0.708166i	$-0.749523\pi$
$643$	−23023.9	−1.41209	−0.706046	+	0.708166i	$0.749523\pi$
$644$	8846.74	0.541320
$645$	0	0
$646$	−29676.1	−1.80742
$647$	−29405.1	−1.78676	−0.893379	−	0.449303i	$-0.851672\pi$
$647$	−29405.1	−1.78676	−0.893379	+	0.449303i	$0.851672\pi$
$648$	0	0
$649$	−3021.81	−0.182768
$650$	23976.7	1.44683
$651$	0	0
$652$	−4273.50	−0.256692
$653$	−16468.8	−0.986944	−0.493472	−	0.869762i	$-0.664272\pi$
$653$	−16468.8	−0.986944	−0.493472	+	0.869762i	$0.664272\pi$
$654$	0	0
$655$	−24908.2	−1.48587
$656$	−28568.5	−1.70033
$657$	0	0
$658$	−10070.2	−0.596621
$659$	377.335	0.0223048	0.0111524	−	0.999938i	$-0.496450\pi$
$659$	377.335	0.0223048	0.0111524	+	0.999938i	$0.496450\pi$
$660$	0	0
$661$	−15474.3	−0.910559	−0.455280	−	0.890348i	$-0.650461\pi$
$661$	−15474.3	−0.910559	−0.455280	+	0.890348i	$0.650461\pi$
$662$	−24280.4	−1.42551
$663$	0	0
$664$	8222.89	0.480588
$665$	−60728.3	−3.54127
$666$	0	0
$667$	−5452.43	−0.316520
$668$	3172.26	0.183740
$669$	0	0
$670$	−46956.4	−2.70759
$671$	28156.9	1.61995
$672$	0	0
$673$	15634.5	0.895490	0.447745	−	0.894161i	$-0.352227\pi$
$673$	15634.5	0.895490	0.447745	+	0.894161i	$0.352227\pi$
$674$	−15482.6	−0.884818
$675$	0	0
$676$	−6564.37	−0.373485
$677$	26000.2	1.47603	0.738014	−	0.674786i	$-0.235764\pi$
$677$	26000.2	1.47603	0.738014	+	0.674786i	$0.235764\pi$
$678$	0	0
$679$	−9690.42	−0.547694
$680$	26557.5	1.49770
$681$	0	0
$682$	−22807.1	−1.28054
$683$	19549.4	1.09522	0.547612	−	0.836733i	$-0.315537\pi$
$683$	19549.4	1.09522	0.547612	+	0.836733i	$0.315537\pi$
$684$	0	0
$685$	10187.9	0.568260
$686$	22509.7	1.25280
$687$	0	0
$688$	18981.7	1.05185
$689$	−15739.0	−0.870261
$690$	0	0
$691$	−28184.4	−1.55164	−0.775821	−	0.630952i	$-0.782664\pi$
$691$	−28184.4	−1.55164	−0.775821	+	0.630952i	$0.782664\pi$
$692$	15493.9	0.851140
$693$	0	0
$694$	31282.9	1.71107
$695$	11554.9	0.630652
$696$	0	0
$697$	31844.4	1.73055
$698$	−41403.2	−2.24518
$699$	0	0
$700$	−36231.4	−1.95631
$701$	−2313.19	−0.124633	−0.0623167	−	0.998056i	$-0.519849\pi$
$701$	−2313.19	−0.124633	−0.0623167	+	0.998056i	$0.519849\pi$
$702$	0	0
$703$	−10474.6	−0.561959
$704$	−4284.90	−0.229394
$705$	0	0
$706$	−19077.3	−1.01697
$707$	19770.8	1.05171
$708$	0	0
$709$	−16383.6	−0.867841	−0.433920	−	0.900951i	$-0.642870\pi$
$709$	−16383.6	−0.867841	−0.433920	+	0.900951i	$0.642870\pi$
$710$	−2690.02	−0.142189
$711$	0	0
$712$	5357.67	0.282004
$713$	9850.95	0.517421
$714$	0	0
$715$	23886.5	1.24937
$716$	2766.58	0.144402
$717$	0	0
$718$	−34842.1	−1.81100
$719$	30186.8	1.56575	0.782876	−	0.622177i	$-0.213752\pi$
$719$	30186.8	1.56575	0.782876	+	0.622177i	$0.213752\pi$
$720$	0	0
$721$	−30152.6	−1.55748
$722$	−8545.96	−0.440510
$723$	0	0
$724$	6900.88	0.354239
$725$	22330.2	1.14389
$726$	0	0
$727$	16487.2	0.841095	0.420547	−	0.907271i	$-0.361838\pi$
$727$	16487.2	0.841095	0.420547	+	0.907271i	$0.361838\pi$
$728$	9554.53	0.486421
$729$	0	0
$730$	43209.5	2.19076
$731$	−21158.3	−1.07054
$732$	0	0
$733$	−13859.4	−0.698372	−0.349186	−	0.937053i	$-0.613542\pi$
$733$	−13859.4	−0.698372	−0.349186	+	0.937053i	$0.613542\pi$
$734$	18339.2	0.922226
$735$	0	0
$736$	−12304.0	−0.616210
$737$	−33400.3	−1.66935
$738$	0	0
$739$	36238.9	1.80388	0.901942	−	0.431856i	$-0.142141\pi$
$739$	36238.9	1.80388	0.901942	+	0.431856i	$0.142141\pi$
$740$	−8752.65	−0.434803
$741$	0	0
$742$	73038.2	3.61364
$743$	−11663.4	−0.575893	−0.287946	−	0.957647i	$-0.592972\pi$
$743$	−11663.4	−0.575893	−0.287946	+	0.957647i	$0.592972\pi$
$744$	0	0
$745$	73363.1	3.60780
$746$	24446.3	1.19979
$747$	0	0
$748$	−17638.6	−0.862207
$749$	−26528.6	−1.29417
$750$	0	0
$751$	4619.42	0.224454	0.112227	−	0.993683i	$-0.464202\pi$
$751$	4619.42	0.224454	0.112227	+	0.993683i	$0.464202\pi$
$752$	7780.03	0.377272
$753$	0	0
$754$	5498.43	0.265571
$755$	66386.2	3.20005
$756$	0	0
$757$	−17625.2	−0.846233	−0.423116	−	0.906075i	$-0.639064\pi$
$757$	−17625.2	−0.846233	−0.423116	+	0.906075i	$0.639064\pi$
$758$	28820.2	1.38100
$759$	0	0
$760$	28789.4	1.37408
$761$	3247.66	0.154701	0.0773506	−	0.997004i	$-0.475354\pi$
$761$	3247.66	0.154701	0.0773506	+	0.997004i	$0.475354\pi$
$762$	0	0
$763$	−18276.7	−0.867185
$764$	−3296.16	−0.156087
$765$	0	0
$766$	−36707.1	−1.73144
$767$	−1316.21	−0.0619628
$768$	0	0
$769$	−3993.05	−0.187247	−0.0936236	−	0.995608i	$-0.529845\pi$
$769$	−3993.05	−0.187247	−0.0936236	+	0.995608i	$0.529845\pi$
$770$	−110847.	−5.18785
$771$	0	0
$772$	−13438.5	−0.626505
$773$	−9486.90	−0.441423	−0.220712	−	0.975339i	$-0.570838\pi$
$773$	−9486.90	−0.441423	−0.220712	+	0.975339i	$0.570838\pi$
$774$	0	0
$775$	−40344.1	−1.86994
$776$	4593.93	0.212516
$777$	0	0
$778$	4410.57	0.203247
$779$	34520.6	1.58771
$780$	0	0
$781$	−1913.42	−0.0876664
$782$	23396.3	1.06989
$783$	0	0
$784$	−44824.1	−2.04191
$785$	15047.0	0.684139
$786$	0	0
$787$	9310.51	0.421708	0.210854	−	0.977518i	$-0.432376\pi$
$787$	9310.51	0.421708	0.210854	+	0.977518i	$0.432376\pi$
$788$	1302.96	0.0589036
$789$	0	0
$790$	−87927.3	−3.95989
$791$	11215.2	0.504131
$792$	0	0
$793$	12264.2	0.549200
$794$	−2862.53	−0.127944
$795$	0	0
$796$	−10096.3	−0.449565
$797$	−1402.21	−0.0623196	−0.0311598	−	0.999514i	$-0.509920\pi$
$797$	−1402.21	−0.0623196	−0.0311598	+	0.999514i	$0.509920\pi$
$798$	0	0
$799$	−8672.14	−0.383978
$800$	50390.3	2.22696
$801$	0	0
$802$	16233.3	0.714735
$803$	30735.1	1.35071
$804$	0	0
$805$	47877.5	2.09623
$806$	−9934.05	−0.434134
$807$	0	0
$808$	−9372.71	−0.408082
$809$	−33015.3	−1.43480	−0.717402	−	0.696659i	$-0.754669\pi$
$809$	−33015.3	−1.43480	−0.717402	+	0.696659i	$0.754669\pi$
$810$	0	0
$811$	35812.1	1.55060	0.775298	−	0.631596i	$-0.217600\pi$
$811$	35812.1	1.55060	0.775298	+	0.631596i	$0.217600\pi$
$812$	−8308.73	−0.359088
$813$	0	0
$814$	−19119.2	−0.823254
$815$	−23127.7	−0.994023
$816$	0	0
$817$	−22936.4	−0.982184
$818$	−36854.8	−1.57530
$819$	0	0
$820$	28845.6	1.22846
$821$	1940.92	0.0825076	0.0412538	−	0.999149i	$-0.486865\pi$
$821$	1940.92	0.0825076	0.0412538	+	0.999149i	$0.486865\pi$
$822$	0	0
$823$	−1861.48	−0.0788424	−0.0394212	−	0.999223i	$-0.512551\pi$
$823$	−1861.48	−0.0788424	−0.0394212	+	0.999223i	$0.512551\pi$
$824$	14294.4	0.604332
$825$	0	0
$826$	6107.95	0.257292
$827$	7336.67	0.308490	0.154245	−	0.988033i	$-0.450706\pi$
$827$	7336.67	0.308490	0.154245	+	0.988033i	$0.450706\pi$
$828$	0	0
$829$	30106.6	1.26133	0.630667	−	0.776053i	$-0.282781\pi$
$829$	30106.6	1.26133	0.630667	+	0.776053i	$0.282781\pi$
$830$	−41552.3	−1.73771
$831$	0	0
$832$	−1866.37	−0.0777701
$833$	49963.9	2.07821
$834$	0	0
$835$	17167.9	0.711520
$836$	−19121.0	−0.791044
$837$	0	0
$838$	13761.8	0.567297
$839$	27636.8	1.13722	0.568611	−	0.822607i	$-0.307481\pi$
$839$	27636.8	1.13722	0.568611	+	0.822607i	$0.307481\pi$
$840$	0	0
$841$	−19268.2	−0.790035
$842$	−18890.5	−0.773170
$843$	0	0
$844$	19268.9	0.785857
$845$	−35525.6	−1.44629
$846$	0	0
$847$	−38839.7	−1.57562
$848$	−56428.0	−2.28508
$849$	0	0
$850$	−95818.7	−3.86653
$851$	8258.07	0.332647
$852$	0	0
$853$	−23055.8	−0.925457	−0.462728	−	0.886500i	$-0.653129\pi$
$853$	−23055.8	−0.925457	−0.462728	+	0.886500i	$0.653129\pi$
$854$	−56913.1	−2.28048
$855$	0	0
$856$	12576.4	0.502165
$857$	44339.3	1.76733	0.883665	−	0.468119i	$-0.155068\pi$
$857$	44339.3	1.76733	0.883665	+	0.468119i	$0.155068\pi$
$858$	0	0
$859$	−5864.49	−0.232938	−0.116469	−	0.993194i	$-0.537158\pi$
$859$	−5864.49	−0.232938	−0.116469	+	0.993194i	$0.537158\pi$
$860$	−19165.8	−0.759942
$861$	0	0
$862$	−31398.7	−1.24065
$863$	−8819.67	−0.347886	−0.173943	−	0.984756i	$-0.555651\pi$
$863$	−8819.67	−0.347886	−0.173943	+	0.984756i	$0.555651\pi$
$864$	0	0
$865$	83851.1	3.29598
$866$	45334.3	1.77890
$867$	0	0
$868$	15011.5	0.587007
$869$	−62543.0	−2.44146
$870$	0	0
$871$	−14548.1	−0.565951
$872$	8664.44	0.336485
$873$	0	0
$874$	25362.6	0.981582
$875$	−117535.	−4.54102
$876$	0	0
$877$	−16591.0	−0.638814	−0.319407	−	0.947618i	$-0.603484\pi$
$877$	−16591.0	−0.638814	−0.319407	+	0.947618i	$0.603484\pi$
$878$	7851.47	0.301793
$879$	0	0
$880$	85638.3	3.28053
$881$	28452.1	1.08806	0.544028	−	0.839067i	$-0.316899\pi$
$881$	28452.1	1.08806	0.544028	+	0.839067i	$0.316899\pi$
$882$	0	0
$883$	−6177.75	−0.235445	−0.117722	−	0.993047i	$-0.537559\pi$
$883$	−6177.75	−0.235445	−0.117722	+	0.993047i	$0.537559\pi$
$884$	−7682.82	−0.292309
$885$	0	0
$886$	45444.3	1.72317
$887$	13155.6	0.497996	0.248998	−	0.968504i	$-0.419899\pi$
$887$	13155.6	0.497996	0.248998	+	0.968504i	$0.419899\pi$
$888$	0	0
$889$	−18825.9	−0.710237
$890$	−27073.6	−1.01967
$891$	0	0
$892$	6770.23	0.254130
$893$	−9400.95	−0.352285
$894$	0	0
$895$	14972.4	0.559187
$896$	47490.7	1.77071
$897$	0	0
$898$	11324.6	0.420832
$899$	−9251.87	−0.343234
$900$	0	0
$901$	62898.4	2.32569
$902$	63010.2	2.32595
$903$	0	0
$904$	−5316.79	−0.195613
$905$	37346.8	1.37177
$906$	0	0
$907$	11767.5	0.430796	0.215398	−	0.976526i	$-0.430895\pi$
$907$	11767.5	0.430796	0.215398	+	0.976526i	$0.430895\pi$
$908$	11524.1	0.421192
$909$	0	0
$910$	−48281.4	−1.75881
$911$	−8944.57	−0.325298	−0.162649	−	0.986684i	$-0.552004\pi$
$911$	−8944.57	−0.325298	−0.162649	+	0.986684i	$0.552004\pi$
$912$	0	0
$913$	−29556.3	−1.07138
$914$	49861.4	1.80445
$915$	0	0
$916$	6460.14	0.233023
$917$	35811.8	1.28965
$918$	0	0
$919$	38367.6	1.37718	0.688591	−	0.725150i	$-0.258230\pi$
$919$	38367.6	1.37718	0.688591	+	0.725150i	$0.258230\pi$
$920$	−22697.3	−0.813376
$921$	0	0
$922$	−16414.0	−0.586298
$923$	−833.424	−0.0297210
$924$	0	0
$925$	−33820.5	−1.20218
$926$	−21651.7	−0.768378
$927$	0	0
$928$	11555.7	0.408766
$929$	47966.0	1.69399	0.846993	−	0.531604i	$-0.178410\pi$
$929$	47966.0	1.69399	0.846993	+	0.531604i	$0.178410\pi$
$930$	0	0
$931$	54162.9	1.90668
$932$	−1962.85	−0.0689865
$933$	0	0
$934$	19369.7	0.678583
$935$	−95458.1	−3.33884
$936$	0	0
$937$	72.7929	0.00253793	0.00126896	−	0.999999i	$-0.499596\pi$
$937$	72.7929	0.00253793	0.00126896	+	0.999999i	$0.499596\pi$
$938$	67511.6	2.35003
$939$	0	0
$940$	−7855.50	−0.272573
$941$	1145.95	0.0396992	0.0198496	−	0.999803i	$-0.493681\pi$
$941$	1145.95	0.0396992	0.0198496	+	0.999803i	$0.493681\pi$
$942$	0	0
$943$	−27215.7	−0.939835
$944$	−4718.89	−0.162698
$945$	0	0
$946$	−41865.7	−1.43887
$947$	52363.7	1.79682	0.898412	−	0.439153i	$-0.144721\pi$
$947$	52363.7	1.79682	0.898412	+	0.439153i	$0.144721\pi$
$948$	0	0
$949$	13387.2	0.457922
$950$	−103871.	−3.54740
$951$	0	0
$952$	−38183.0	−1.29992
$953$	18197.6	0.618549	0.309274	−	0.950973i	$-0.399914\pi$
$953$	18197.6	0.618549	0.309274	+	0.950973i	$0.399914\pi$
$954$	0	0
$955$	−17838.4	−0.604437
$956$	9646.04	0.326334
$957$	0	0
$958$	36784.7	1.24056
$959$	−14647.6	−0.493217
$960$	0	0
$961$	−13075.6	−0.438910
$962$	−8327.73	−0.279103
$963$	0	0
$964$	−17291.0	−0.577704
$965$	−72727.5	−2.42610
$966$	0	0
$967$	−15190.1	−0.505149	−0.252575	−	0.967577i	$-0.581277\pi$
$967$	−15190.1	−0.505149	−0.252575	+	0.967577i	$0.581277\pi$
$968$	18412.7	0.611371
$969$	0	0
$970$	−23214.3	−0.768418
$971$	33558.7	1.10911	0.554557	−	0.832146i	$-0.312888\pi$
$971$	33558.7	1.10911	0.554557	+	0.832146i	$0.312888\pi$
$972$	0	0
$973$	−16613.1	−0.547370
$974$	−27800.2	−0.914556
$975$	0	0
$976$	43970.0	1.44206
$977$	−13843.4	−0.453317	−0.226658	−	0.973974i	$-0.572780\pi$
$977$	−13843.4	−0.453317	−0.226658	+	0.973974i	$0.572780\pi$
$978$	0	0
$979$	−19257.6	−0.628677
$980$	45258.9	1.47525
$981$	0	0
$982$	−39593.5	−1.28664
$983$	10695.3	0.347027	0.173514	−	0.984831i	$-0.444488\pi$
$983$	10695.3	0.347027	0.173514	+	0.984831i	$0.444488\pi$
$984$	0	0
$985$	7051.47	0.228100
$986$	−21973.5	−0.709715
$987$	0	0
$988$	−8328.49	−0.268183
$989$	18082.8	0.581396
$990$	0	0
$991$	14749.0	0.472773	0.236386	−	0.971659i	$-0.424037\pi$
$991$	14749.0	0.472773	0.236386	+	0.971659i	$0.424037\pi$
$992$	−20877.8	−0.668216
$993$	0	0
$994$	3867.57	0.123412
$995$	−54640.0	−1.74091
$996$	0	0
$997$	30262.1	0.961294	0.480647	−	0.876914i	$-0.340402\pi$
$997$	30262.1	0.961294	0.480647	+	0.876914i	$0.340402\pi$
$998$	1172.11	0.0371767
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.d.1.1		7
3.2	odd	2		177.4.a.a.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.7	✓	7	3.2	odd	2
531.4.a.d.1.1		7	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)

\(f(q)\)	\(=\)	\(q-3.44426 q^{2} +3.86292 q^{4} +20.9057 q^{5} -30.0572 q^{7} +14.2492 q^{8} +O(q^{10})\)	\(q-3.44426 q^{2} +3.86292 q^{4} +20.9057 q^{5} -30.0572 q^{7} +14.2492 q^{8} -72.0046 q^{10} -51.2171 q^{11} -22.3086 q^{13} +103.525 q^{14} -79.9812 q^{16} +89.1524 q^{17} +96.6448 q^{19} +80.7571 q^{20} +176.405 q^{22} -76.1937 q^{23} +312.048 q^{25} +76.8365 q^{26} -116.108 q^{28} +71.5601 q^{29} -129.288 q^{31} +161.483 q^{32} -307.064 q^{34} -628.366 q^{35} -108.382 q^{37} -332.870 q^{38} +297.889 q^{40} +357.190 q^{41} -237.327 q^{43} -197.848 q^{44} +262.431 q^{46} -97.2732 q^{47} +560.433 q^{49} -1074.77 q^{50} -86.1763 q^{52} +705.515 q^{53} -1070.73 q^{55} -428.290 q^{56} -246.471 q^{58} +59.0000 q^{59} -549.755 q^{61} +445.302 q^{62} +83.6615 q^{64} -466.376 q^{65} +652.131 q^{67} +344.389 q^{68} +2164.25 q^{70} +37.3589 q^{71} -600.093 q^{73} +373.297 q^{74} +373.331 q^{76} +1539.44 q^{77} +1221.13 q^{79} -1672.06 q^{80} -1230.26 q^{82} +577.079 q^{83} +1863.79 q^{85} +817.417 q^{86} -729.802 q^{88} +375.999 q^{89} +670.532 q^{91} -294.330 q^{92} +335.034 q^{94} +2020.43 q^{95} +322.400 q^{97} -1930.28 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) \) 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8} - 79 q^{10} + 131 q^{11} - 123 q^{13} + 117 q^{14} + 202 q^{16} + 235 q^{17} - 80 q^{19} - 61 q^{20} + 688 q^{22} + 274 q^{23} + 193 q^{25} + 180 q^{26} - 118 q^{28} + 406 q^{29} - 346 q^{31} + 854 q^{32} + 178 q^{34} + 424 q^{35} - 157 q^{37} + 129 q^{38} - 590 q^{40} + 825 q^{41} - 815 q^{43} + 1690 q^{44} + 1457 q^{46} + 1196 q^{47} + 914 q^{49} - 713 q^{50} + 1030 q^{52} + 900 q^{53} - 1044 q^{55} - 2172 q^{56} + 1242 q^{58} + 413 q^{59} + 420 q^{61} - 646 q^{62} + 3541 q^{64} - 190 q^{65} + 1316 q^{67} + 611 q^{68} + 4658 q^{70} + 173 q^{71} - 418 q^{73} - 660 q^{74} + 1540 q^{76} + 753 q^{77} + 2635 q^{79} - 6155 q^{80} - 125 q^{82} - 457 q^{83} + 1270 q^{85} - 3482 q^{86} + 7685 q^{88} - 592 q^{89} + 3179 q^{91} + 3500 q^{92} + 2064 q^{94} + 2250 q^{95} - 1906 q^{97} - 2994 q^{98}+O(q^{100}) \) 7 * q + 8 * q^2 + 22 * q^4 + 28 * q^5 - 59 * q^7 + 117 * q^8 - 79 * q^10 + 131 * q^11 - 123 * q^13 + 117 * q^14 + 202 * q^16 + 235 * q^17 - 80 * q^19 - 61 * q^20 + 688 * q^22 + 274 * q^23 + 193 * q^25 + 180 * q^26 - 118 * q^28 + 406 * q^29 - 346 * q^31 + 854 * q^32 + 178 * q^34 + 424 * q^35 - 157 * q^37 + 129 * q^38 - 590 * q^40 + 825 * q^41 - 815 * q^43 + 1690 * q^44 + 1457 * q^46 + 1196 * q^47 + 914 * q^49 - 713 * q^50 + 1030 * q^52 + 900 * q^53 - 1044 * q^55 - 2172 * q^56 + 1242 * q^58 + 413 * q^59 + 420 * q^61 - 646 * q^62 + 3541 * q^64 - 190 * q^65 + 1316 * q^67 + 611 * q^68 + 4658 * q^70 + 173 * q^71 - 418 * q^73 - 660 * q^74 + 1540 * q^76 + 753 * q^77 + 2635 * q^79 - 6155 * q^80 - 125 * q^82 - 457 * q^83 + 1270 * q^85 - 3482 * q^86 + 7685 * q^88 - 592 * q^89 + 3179 * q^91 + 3500 * q^92 + 2064 * q^94 + 2250 * q^95 - 1906 * q^97 - 2994 * q^98

Introduction

L-functions

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