LMFDB - Embedded newform 531.4.a.d.1.2

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			1.2
Root			$-2.68175$ 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)

$f(q)$	$=$	$q-1.68175 q^{2} -5.17172 q^{4} +9.78761 q^{5} +6.87552 q^{7} +22.1515 q^{8} +O(q^{10})$	\(q-1.68175 q^{2} -5.17172 q^{4} +9.78761 q^{5} +6.87552 q^{7} +22.1515 q^{8} -16.4603 q^{10} +37.8468 q^{11} -11.3334 q^{13} -11.5629 q^{14} +4.12039 q^{16} +60.3439 q^{17} -92.5767 q^{19} -50.6188 q^{20} -63.6489 q^{22} +183.718 q^{23} -29.2026 q^{25} +19.0599 q^{26} -35.5582 q^{28} -265.559 q^{29} +177.376 q^{31} -184.142 q^{32} -101.483 q^{34} +67.2949 q^{35} -70.0886 q^{37} +155.691 q^{38} +216.811 q^{40} -208.760 q^{41} +393.127 q^{43} -195.733 q^{44} -308.968 q^{46} +134.917 q^{47} -295.727 q^{49} +49.1115 q^{50} +58.6131 q^{52} +650.432 q^{53} +370.430 q^{55} +152.303 q^{56} +446.604 q^{58} +59.0000 q^{59} +22.5322 q^{61} -298.302 q^{62} +276.717 q^{64} -110.927 q^{65} -209.476 q^{67} -312.082 q^{68} -113.173 q^{70} -209.160 q^{71} +865.282 q^{73} +117.872 q^{74} +478.781 q^{76} +260.217 q^{77} +843.604 q^{79} +40.3287 q^{80} +351.082 q^{82} +845.384 q^{83} +590.623 q^{85} -661.141 q^{86} +838.365 q^{88} +974.766 q^{89} -77.9229 q^{91} -950.139 q^{92} -226.897 q^{94} -906.105 q^{95} +338.796 q^{97} +497.339 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.68175	−0.594588	−0.297294	−	0.954786i	$-0.596084\pi$
$2$	−1.68175	−0.594588	−0.297294	+	0.954786i	$0.596084\pi$
$3$	0	0
$3$	0	0
$4$	−5.17172	−0.646465
$5$	9.78761	0.875431	0.437715	−	0.899114i	$-0.355788\pi$
$5$	9.78761	0.875431	0.437715	+	0.899114i	$0.355788\pi$
$6$	0	0
$7$	6.87552	0.371243	0.185622	−	0.982621i	$-0.440570\pi$
$7$	6.87552	0.371243	0.185622	+	0.982621i	$0.440570\pi$
$8$	22.1515	0.978969
$9$	0	0
$10$	−16.4603	−0.520521
$11$	37.8468	1.03739	0.518693	−	0.854961i	$-0.326419\pi$
$11$	37.8468	1.03739	0.518693	+	0.854961i	$0.326419\pi$
$12$	0	0
$13$	−11.3334	−0.241794	−0.120897	−	0.992665i	$-0.538577\pi$
$13$	−11.3334	−0.241794	−0.120897	+	0.992665i	$0.538577\pi$
$14$	−11.5629	−0.220737
$15$	0	0
$16$	4.12039	0.0643810
$17$	60.3439	0.860915	0.430457	−	0.902611i	$-0.358352\pi$
$17$	60.3439	0.860915	0.430457	+	0.902611i	$0.358352\pi$
$18$	0	0
$19$	−92.5767	−1.11782	−0.558909	−	0.829229i	$-0.688780\pi$
$19$	−92.5767	−1.11782	−0.558909	+	0.829229i	$0.688780\pi$
$20$	−50.6188	−0.565935
$21$	0	0
$22$	−63.6489	−0.616818
$23$	183.718	1.66556	0.832781	−	0.553603i	$-0.186748\pi$
$23$	183.718	1.66556	0.832781	+	0.553603i	$0.186748\pi$
$24$	0	0
$25$	−29.2026	−0.233621
$26$	19.0599	0.143768
$27$	0	0
$28$	−35.5582	−0.239995
$29$	−265.559	−1.70045	−0.850225	−	0.526419i	$-0.823534\pi$
$29$	−265.559	−1.70045	−0.850225	+	0.526419i	$0.823534\pi$
$30$	0	0
$31$	177.376	1.02767	0.513833	−	0.857890i	$-0.328225\pi$
$31$	177.376	1.02767	0.513833	+	0.857890i	$0.328225\pi$
$32$	−184.142	−1.01725
$33$	0	0
$34$	−101.483	−0.511890
$35$	67.2949	0.324998
$36$	0	0
$37$	−70.0886	−0.311419	−0.155709	−	0.987803i	$-0.549766\pi$
$37$	−70.0886	−0.311419	−0.155709	+	0.987803i	$0.549766\pi$
$38$	155.691	0.664642
$39$	0	0
$40$	216.811	0.857019
$41$	−208.760	−0.795190	−0.397595	−	0.917561i	$-0.630155\pi$
$41$	−208.760	−0.795190	−0.397595	+	0.917561i	$0.630155\pi$
$42$	0	0
$43$	393.127	1.39421	0.697107	−	0.716967i	$-0.254470\pi$
$43$	393.127	1.39421	0.697107	+	0.716967i	$0.254470\pi$
$44$	−195.733	−0.670633
$45$	0	0
$46$	−308.968	−0.990323
$47$	134.917	0.418717	0.209358	−	0.977839i	$-0.432862\pi$
$47$	134.917	0.418717	0.209358	+	0.977839i	$0.432862\pi$
$48$	0	0
$49$	−295.727	−0.862179
$50$	49.1115	0.138908
$51$	0	0
$52$	58.6131	0.156311
$53$	650.432	1.68573	0.842865	−	0.538126i	$-0.180867\pi$
$53$	650.432	1.68573	0.842865	+	0.538126i	$0.180867\pi$
$54$	0	0
$55$	370.430	0.908160
$56$	152.303	0.363435
$57$	0	0
$58$	446.604	1.01107
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	22.5322	0.0472943	0.0236472	−	0.999720i	$-0.492472\pi$
$61$	22.5322	0.0472943	0.0236472	+	0.999720i	$0.492472\pi$
$62$	−298.302	−0.611038
$63$	0	0
$64$	276.717	0.540463
$65$	−110.927	−0.211674
$66$	0	0
$67$	−209.476	−0.381964	−0.190982	−	0.981594i	$-0.561167\pi$
$67$	−209.476	−0.381964	−0.190982	+	0.981594i	$0.561167\pi$
$68$	−312.082	−0.556551
$69$	0	0
$70$	−113.173	−0.193240
$71$	−209.160	−0.349616	−0.174808	−	0.984603i	$-0.555930\pi$
$71$	−209.160	−0.349616	−0.174808	+	0.984603i	$0.555930\pi$
$72$	0	0
$73$	865.282	1.38731	0.693655	−	0.720308i	$-0.255999\pi$
$73$	865.282	1.38731	0.693655	+	0.720308i	$0.255999\pi$
$74$	117.872	0.185166
$75$	0	0
$76$	478.781	0.722630
$77$	260.217	0.385122
$78$	0	0
$79$	843.604	1.20143	0.600715	−	0.799464i	$-0.294883\pi$
$79$	843.604	1.20143	0.600715	+	0.799464i	$0.294883\pi$
$80$	40.3287	0.0563611
$81$	0	0
$82$	351.082	0.472811
$83$	845.384	1.11799	0.558994	−	0.829172i	$-0.311188\pi$
$83$	845.384	1.11799	0.558994	+	0.829172i	$0.311188\pi$
$84$	0	0
$85$	590.623	0.753671
$86$	−661.141	−0.828984
$87$	0	0
$88$	838.365	1.01557
$89$	974.766	1.16096	0.580478	−	0.814276i	$-0.302866\pi$
$89$	974.766	1.16096	0.580478	+	0.814276i	$0.302866\pi$
$90$	0	0
$91$	−77.9229	−0.0897642
$92$	−950.139	−1.07673
$93$	0	0
$94$	−226.897	−0.248964
$95$	−906.105	−0.978573
$96$	0	0
$97$	338.796	0.354634	0.177317	−	0.984154i	$-0.443258\pi$
$97$	338.796	0.354634	0.177317	+	0.984154i	$0.443258\pi$
$98$	497.339	0.512641
$99$	0	0
$100$	151.028	0.151028
$101$	683.879	0.673747	0.336874	−	0.941550i	$-0.390631\pi$
$101$	683.879	0.673747	0.336874	+	0.941550i	$0.390631\pi$
$102$	0	0
$103$	−1235.54	−1.18196	−0.590979	−	0.806687i	$-0.701258\pi$
$103$	−1235.54	−1.18196	−0.590979	+	0.806687i	$0.701258\pi$
$104$	−251.052	−0.236708
$105$	0	0
$106$	−1093.86	−1.00231
$107$	−431.478	−0.389837	−0.194918	−	0.980819i	$-0.562444\pi$
$107$	−431.478	−0.389837	−0.194918	+	0.980819i	$0.562444\pi$
$108$	0	0
$109$	427.874	0.375989	0.187995	−	0.982170i	$-0.439801\pi$
$109$	427.874	0.375989	0.187995	+	0.982170i	$0.439801\pi$
$110$	−622.971	−0.539981
$111$	0	0
$112$	28.3298	0.0239010
$113$	1934.70	1.61063	0.805317	−	0.592845i	$-0.201995\pi$
$113$	1934.70	1.61063	0.805317	+	0.592845i	$0.201995\pi$
$114$	0	0
$115$	1798.16	1.45808
$116$	1373.40	1.09928
$117$	0	0
$118$	−99.2233	−0.0774088
$119$	414.896	0.319609
$120$	0	0
$121$	101.382	0.0761698
$122$	−37.8935	−0.0281207
$123$	0	0
$124$	−917.337	−0.664349
$125$	−1509.28	−1.07995
$126$	0	0
$127$	−1334.46	−0.932395	−0.466198	−	0.884681i	$-0.654376\pi$
$127$	−1334.46	−0.932395	−0.466198	+	0.884681i	$0.654376\pi$
$128$	1007.76	0.695896
$129$	0	0
$130$	186.551	0.125859
$131$	739.638	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$131$	739.638	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$132$	0	0
$133$	−636.513	−0.414982
$134$	352.287	0.227111
$135$	0	0
$136$	1336.71	0.842809
$137$	1262.30	0.787196	0.393598	−	0.919283i	$-0.371230\pi$
$137$	1262.30	0.787196	0.393598	+	0.919283i	$0.371230\pi$
$138$	0	0
$139$	2630.69	1.60527	0.802634	−	0.596471i	$-0.203431\pi$
$139$	2630.69	1.60527	0.802634	+	0.596471i	$0.203431\pi$
$140$	−348.030	−0.210099
$141$	0	0
$142$	351.755	0.207877
$143$	−428.933	−0.250833
$144$	0	0
$145$	−2599.19	−1.48863
$146$	−1455.19	−0.824878
$147$	0	0
$148$	362.478	0.201321
$149$	434.666	0.238988	0.119494	−	0.992835i	$-0.461873\pi$
$149$	434.666	0.238988	0.119494	+	0.992835i	$0.461873\pi$
$150$	0	0
$151$	−1585.40	−0.854425	−0.427213	−	0.904151i	$-0.640504\pi$
$151$	−1585.40	−0.854425	−0.427213	+	0.904151i	$0.640504\pi$
$152$	−2050.72	−1.09431
$153$	0	0
$154$	−437.619	−0.228989
$155$	1736.09	0.899650
$156$	0	0
$157$	3744.07	1.90324	0.951622	−	0.307270i	$-0.0994155\pi$
$157$	3744.07	1.90324	0.951622	+	0.307270i	$0.0994155\pi$
$158$	−1418.73	−0.714356
$159$	0	0
$160$	−1802.31	−0.890531
$161$	1263.16	0.618328
$162$	0	0
$163$	−1331.85	−0.639993	−0.319996	−	0.947419i	$-0.603682\pi$
$163$	−1331.85	−0.639993	−0.319996	+	0.947419i	$0.603682\pi$
$164$	1079.65	0.514062
$165$	0	0
$166$	−1421.73	−0.664743
$167$	−16.5584	−0.00767260	−0.00383630	−	0.999993i	$-0.501221\pi$
$167$	−16.5584	−0.00767260	−0.00383630	+	0.999993i	$0.501221\pi$
$168$	0	0
$169$	−2068.55	−0.941536
$170$	−993.280	−0.448124
$171$	0	0
$172$	−2033.14	−0.901310
$173$	2397.07	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$173$	2397.07	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$174$	0	0
$175$	−200.783	−0.0867302
$176$	155.944	0.0667880
$177$	0	0
$178$	−1639.31	−0.690291
$179$	−1223.00	−0.510680	−0.255340	−	0.966851i	$-0.582187\pi$
$179$	−1223.00	−0.510680	−0.255340	+	0.966851i	$0.582187\pi$
$180$	0	0
$181$	−4266.97	−1.75227	−0.876137	−	0.482063i	$-0.839888\pi$
$181$	−4266.97	−1.75227	−0.876137	+	0.482063i	$0.839888\pi$
$182$	131.047	0.0533728
$183$	0	0
$184$	4069.64	1.63053
$185$	−686.000	−0.272626
$186$	0	0
$187$	2283.83	0.893101
$188$	−697.753	−0.270686
$189$	0	0
$190$	1523.84	0.581848
$191$	3080.45	1.16698	0.583491	−	0.812120i	$-0.301687\pi$
$191$	3080.45	1.16698	0.583491	+	0.812120i	$0.301687\pi$
$192$	0	0
$193$	3469.71	1.29407	0.647033	−	0.762462i	$-0.276009\pi$
$193$	3469.71	1.29407	0.647033	+	0.762462i	$0.276009\pi$
$194$	−569.770	−0.210862
$195$	0	0
$196$	1529.42	0.557368
$197$	1487.74	0.538055	0.269027	−	0.963133i	$-0.413298\pi$
$197$	1487.74	0.538055	0.269027	+	0.963133i	$0.413298\pi$
$198$	0	0
$199$	−3027.73	−1.07854	−0.539271	−	0.842132i	$-0.681300\pi$
$199$	−3027.73	−1.07854	−0.539271	+	0.842132i	$0.681300\pi$
$200$	−646.883	−0.228708
$201$	0	0
$202$	−1150.11	−0.400602
$203$	−1825.86	−0.631280
$204$	0	0
$205$	−2043.26	−0.696134
$206$	2077.87	0.702779
$207$	0	0
$208$	−46.6980	−0.0155669
$209$	−3503.74	−1.15961
$210$	0	0
$211$	−1655.90	−0.540269	−0.270135	−	0.962823i	$-0.587068\pi$
$211$	−1655.90	−0.540269	−0.270135	+	0.962823i	$0.587068\pi$
$212$	−3363.85	−1.08976
$213$	0	0
$214$	725.638	0.231792
$215$	3847.77	1.22054
$216$	0	0
$217$	1219.55	0.381514
$218$	−719.576	−0.223559
$219$	0	0
$220$	−1915.76	−0.587093
$221$	−683.901	−0.208164
$222$	0	0
$223$	−1225.20	−0.367916	−0.183958	−	0.982934i	$-0.558891\pi$
$223$	−1225.20	−0.367916	−0.183958	+	0.982934i	$0.558891\pi$
$224$	−1266.07	−0.377647
$225$	0	0
$226$	−3253.69	−0.957664
$227$	657.677	0.192298	0.0961488	−	0.995367i	$-0.469348\pi$
$227$	657.677	0.192298	0.0961488	+	0.995367i	$0.469348\pi$
$228$	0	0
$229$	1407.97	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$229$	1407.97	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$230$	−3024.06	−0.866959
$231$	0	0
$232$	−5882.54	−1.66469
$233$	−6393.48	−1.79764	−0.898821	−	0.438315i	$-0.855575\pi$
$233$	−6393.48	−1.79764	−0.898821	+	0.438315i	$0.855575\pi$
$234$	0	0
$235$	1320.52	0.366558
$236$	−305.131	−0.0841625
$237$	0	0
$238$	−697.751	−0.190036
$239$	−1941.60	−0.525489	−0.262744	−	0.964865i	$-0.584628\pi$
$239$	−1941.60	−0.525489	−0.262744	+	0.964865i	$0.584628\pi$
$240$	0	0
$241$	−3701.61	−0.989385	−0.494692	−	0.869068i	$-0.664719\pi$
$241$	−3701.61	−0.989385	−0.494692	+	0.869068i	$0.664719\pi$
$242$	−170.499	−0.0452897
$243$	0	0
$244$	−116.530	−0.0305741
$245$	−2894.46	−0.754778
$246$	0	0
$247$	1049.21	0.270282
$248$	3929.15	1.00605
$249$	0	0
$250$	2538.22	0.642126
$251$	239.067	0.0601186	0.0300593	−	0.999548i	$-0.490430\pi$
$251$	239.067	0.0601186	0.0300593	+	0.999548i	$0.490430\pi$
$252$	0	0
$253$	6953.15	1.72783
$254$	2244.23	0.554392
$255$	0	0
$256$	−3908.55	−0.954235
$257$	−6633.12	−1.60997	−0.804985	−	0.593295i	$-0.797827\pi$
$257$	−6633.12	−1.60997	−0.804985	+	0.593295i	$0.797827\pi$
$258$	0	0
$259$	−481.895	−0.115612
$260$	573.682	0.136839
$261$	0	0
$262$	−1243.89	−0.293311
$263$	2010.45	0.471368	0.235684	−	0.971830i	$-0.424267\pi$
$263$	2010.45	0.471368	0.235684	+	0.971830i	$0.424267\pi$
$264$	0	0
$265$	6366.17	1.47574
$266$	1070.46	0.246744
$267$	0	0
$268$	1083.35	0.246926
$269$	−5923.22	−1.34255	−0.671273	−	0.741210i	$-0.734252\pi$
$269$	−5923.22	−1.34255	−0.671273	+	0.741210i	$0.734252\pi$
$270$	0	0
$271$	−1908.94	−0.427896	−0.213948	−	0.976845i	$-0.568632\pi$
$271$	−1908.94	−0.427896	−0.213948	+	0.976845i	$0.568632\pi$
$272$	248.640	0.0554266
$273$	0	0
$274$	−2122.88	−0.468057
$275$	−1105.23	−0.242355
$276$	0	0
$277$	7685.54	1.66707	0.833536	−	0.552464i	$-0.186312\pi$
$277$	7685.54	1.66707	0.833536	+	0.552464i	$0.186312\pi$
$278$	−4424.17	−0.954474
$279$	0	0
$280$	1490.69	0.318162
$281$	−1223.20	−0.259680	−0.129840	−	0.991535i	$-0.541446\pi$
$281$	−1223.20	−0.259680	−0.129840	+	0.991535i	$0.541446\pi$
$282$	0	0
$283$	−1543.59	−0.324229	−0.162115	−	0.986772i	$-0.551831\pi$
$283$	−1543.59	−0.324229	−0.162115	+	0.986772i	$0.551831\pi$
$284$	1081.72	0.226014
$285$	0	0
$286$	721.358	0.149143
$287$	−1435.33	−0.295209
$288$	0	0
$289$	−1271.61	−0.258826
$290$	4371.18	0.885120
$291$	0	0
$292$	−4474.99	−0.896846
$293$	3987.52	0.795063	0.397532	−	0.917589i	$-0.369867\pi$
$293$	3987.52	0.795063	0.397532	+	0.917589i	$0.369867\pi$
$294$	0	0
$295$	577.469	0.113971
$296$	−1552.57	−0.304869
$297$	0	0
$298$	−731.000	−0.142100
$299$	−2082.15	−0.402722
$300$	0	0
$301$	2702.95	0.517593
$302$	2666.25	0.508031
$303$	0	0
$304$	−381.452	−0.0719663
$305$	220.537	0.0414029
$306$	0	0
$307$	−895.829	−0.166540	−0.0832698	−	0.996527i	$-0.526536\pi$
$307$	−895.829	−0.166540	−0.0832698	+	0.996527i	$0.526536\pi$
$308$	−1345.77	−0.248968
$309$	0	0
$310$	−2919.66	−0.534921
$311$	−4166.91	−0.759754	−0.379877	−	0.925037i	$-0.624034\pi$
$311$	−4166.91	−0.759754	−0.379877	+	0.925037i	$0.624034\pi$
$312$	0	0
$313$	−3619.78	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$313$	−3619.78	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$314$	−6296.59	−1.13165
$315$	0	0
$316$	−4362.88	−0.776681
$317$	−865.480	−0.153344	−0.0766722	−	0.997056i	$-0.524430\pi$
$317$	−865.480	−0.153344	−0.0766722	+	0.997056i	$0.524430\pi$
$318$	0	0
$319$	−10050.6	−1.76402
$320$	2708.40	0.473138
$321$	0	0
$322$	−2124.32	−0.367651
$323$	−5586.44	−0.962347
$324$	0	0
$325$	330.965	0.0564881
$326$	2239.84	0.380532
$327$	0	0
$328$	−4624.35	−0.778467
$329$	927.626	0.155446
$330$	0	0
$331$	8140.89	1.35185	0.675927	−	0.736968i	$-0.263743\pi$
$331$	8140.89	1.35185	0.675927	+	0.736968i	$0.263743\pi$
$332$	−4372.09	−0.722739
$333$	0	0
$334$	27.8470	0.00456204
$335$	−2050.27	−0.334383
$336$	0	0
$337$	7396.34	1.19556	0.597781	−	0.801659i	$-0.296049\pi$
$337$	7396.34	1.19556	0.597781	+	0.801659i	$0.296049\pi$
$338$	3478.79	0.559826
$339$	0	0
$340$	−3054.53	−0.487222
$341$	6713.11	1.06609
$342$	0	0
$343$	−4391.58	−0.691321
$344$	8708.36	1.36489
$345$	0	0
$346$	−4031.27	−0.626365
$347$	1341.98	0.207612	0.103806	−	0.994598i	$-0.466898\pi$
$347$	1341.98	0.207612	0.103806	+	0.994598i	$0.466898\pi$
$348$	0	0
$349$	2193.32	0.336407	0.168203	−	0.985752i	$-0.446203\pi$
$349$	2193.32	0.336407	0.168203	+	0.985752i	$0.446203\pi$
$350$	337.667	0.0515688
$351$	0	0
$352$	−6969.18	−1.05528
$353$	636.290	0.0959386	0.0479693	−	0.998849i	$-0.484725\pi$
$353$	636.290	0.0959386	0.0479693	+	0.998849i	$0.484725\pi$
$354$	0	0
$355$	−2047.18	−0.306064
$356$	−5041.22	−0.750516
$357$	0	0
$358$	2056.79	0.303644
$359$	7124.07	1.04734	0.523668	−	0.851922i	$-0.324563\pi$
$359$	7124.07	1.04734	0.523668	+	0.851922i	$0.324563\pi$
$360$	0	0
$361$	1711.45	0.249519
$362$	7175.98	1.04188
$363$	0	0
$364$	402.995	0.0580294
$365$	8469.04	1.21449
$366$	0	0
$367$	−8714.69	−1.23952	−0.619759	−	0.784792i	$-0.712770\pi$
$367$	−8714.69	−1.23952	−0.619759	+	0.784792i	$0.712770\pi$
$368$	756.990	0.107231
$369$	0	0
$370$	1153.68	0.162100
$371$	4472.05	0.625815
$372$	0	0
$373$	9123.29	1.26645	0.633225	−	0.773967i	$-0.281731\pi$
$373$	9123.29	1.26645	0.633225	+	0.773967i	$0.281731\pi$
$374$	−3840.82	−0.531027
$375$	0	0
$376$	2988.62	0.409911
$377$	3009.68	0.411158
$378$	0	0
$379$	8539.81	1.15742	0.578708	−	0.815535i	$-0.303557\pi$
$379$	8539.81	1.15742	0.578708	+	0.815535i	$0.303557\pi$
$380$	4686.12	0.632613
$381$	0	0
$382$	−5180.54	−0.693873
$383$	8691.08	1.15951	0.579757	−	0.814790i	$-0.303148\pi$
$383$	8691.08	1.15951	0.579757	+	0.814790i	$0.303148\pi$
$384$	0	0
$385$	2546.90	0.337148
$386$	−5835.18	−0.769437
$387$	0	0
$388$	−1752.16	−0.229259
$389$	10079.9	1.31381	0.656904	−	0.753974i	$-0.271866\pi$
$389$	10079.9	1.31381	0.656904	+	0.753974i	$0.271866\pi$
$390$	0	0
$391$	11086.3	1.43391
$392$	−6550.81	−0.844046
$393$	0	0
$394$	−2502.00	−0.319921
$395$	8256.87	1.05177
$396$	0	0
$397$	−1829.51	−0.231286	−0.115643	−	0.993291i	$-0.536893\pi$
$397$	−1829.51	−0.231286	−0.115643	+	0.993291i	$0.536893\pi$
$398$	5091.88	0.641288
$399$	0	0
$400$	−120.326	−0.0150408
$401$	10919.6	1.35985	0.679926	−	0.733281i	$-0.262012\pi$
$401$	10919.6	1.35985	0.679926	+	0.733281i	$0.262012\pi$
$402$	0	0
$403$	−2010.27	−0.248483
$404$	−3536.83	−0.435554
$405$	0	0
$406$	3070.63	0.375352
$407$	−2652.63	−0.323061
$408$	0	0
$409$	6441.50	0.778757	0.389378	−	0.921078i	$-0.372690\pi$
$409$	6441.50	0.778757	0.389378	+	0.921078i	$0.372690\pi$
$410$	3436.25	0.413913
$411$	0	0
$412$	6389.88	0.764094
$413$	405.656	0.0483317
$414$	0	0
$415$	8274.29	0.978721
$416$	2086.95	0.245964
$417$	0	0
$418$	5892.41	0.689491
$419$	−6784.29	−0.791012	−0.395506	−	0.918463i	$-0.629431\pi$
$419$	−6784.29	−0.791012	−0.395506	+	0.918463i	$0.629431\pi$
$420$	0	0
$421$	827.842	0.0958350	0.0479175	−	0.998851i	$-0.484742\pi$
$421$	827.842	0.0958350	0.0479175	+	0.998851i	$0.484742\pi$
$422$	2784.81	0.321238
$423$	0	0
$424$	14408.1	1.65028
$425$	−1762.20	−0.201128
$426$	0	0
$427$	154.921	0.0175577
$428$	2231.48	0.252016
$429$	0	0
$430$	−6470.99	−0.725718
$431$	−1719.44	−0.192163	−0.0960815	−	0.995373i	$-0.530631\pi$
$431$	−1719.44	−0.192163	−0.0960815	+	0.995373i	$0.530631\pi$
$432$	0	0
$433$	11944.5	1.32567	0.662836	−	0.748765i	$-0.269353\pi$
$433$	11944.5	1.32567	0.662836	+	0.748765i	$0.269353\pi$
$434$	−2050.98	−0.226844
$435$	0	0
$436$	−2212.84	−0.243064
$437$	−17008.0	−1.86180
$438$	0	0
$439$	316.306	0.0343882	0.0171941	−	0.999852i	$-0.494527\pi$
$439$	316.306	0.0343882	0.0171941	+	0.999852i	$0.494527\pi$
$440$	8205.59	0.889060
$441$	0	0
$442$	1150.15	0.123772
$443$	9043.41	0.969899	0.484950	−	0.874542i	$-0.338838\pi$
$443$	9043.41	0.969899	0.484950	+	0.874542i	$0.338838\pi$
$444$	0	0
$445$	9540.63	1.01634
$446$	2060.47	0.218758
$447$	0	0
$448$	1902.57	0.200643
$449$	−9058.55	−0.952114	−0.476057	−	0.879414i	$-0.657934\pi$
$449$	−9058.55	−0.952114	−0.476057	+	0.879414i	$0.657934\pi$
$450$	0	0
$451$	−7900.90	−0.824920
$452$	−10005.7	−1.04122
$453$	0	0
$454$	−1106.05	−0.114338
$455$	−762.680	−0.0785824
$456$	0	0
$457$	14344.7	1.46831	0.734155	−	0.678982i	$-0.237579\pi$
$457$	14344.7	1.46831	0.734155	+	0.678982i	$0.237579\pi$
$458$	−2367.85	−0.241578
$459$	0	0
$460$	−9299.59	−0.942599
$461$	10742.1	1.08527	0.542634	−	0.839969i	$-0.317427\pi$
$461$	10742.1	1.08527	0.542634	+	0.839969i	$0.317427\pi$
$462$	0	0
$463$	−17574.2	−1.76402	−0.882009	−	0.471232i	$-0.843809\pi$
$463$	−17574.2	−1.76402	−0.882009	+	0.471232i	$0.843809\pi$
$464$	−1094.21	−0.109477
$465$	0	0
$466$	10752.2	1.06886
$467$	−8958.29	−0.887667	−0.443833	−	0.896109i	$-0.646382\pi$
$467$	−8958.29	−0.887667	−0.443833	+	0.896109i	$0.646382\pi$
$468$	0	0
$469$	−1440.26	−0.141802
$470$	−2220.78	−0.217951
$471$	0	0
$472$	1306.94	0.127451
$473$	14878.6	1.44634
$474$	0	0
$475$	2703.49	0.261146
$476$	−2145.72	−0.206616
$477$	0	0
$478$	3265.29	0.312450
$479$	−6909.71	−0.659108	−0.329554	−	0.944137i	$-0.606898\pi$
$479$	−6909.71	−0.659108	−0.329554	+	0.944137i	$0.606898\pi$
$480$	0	0
$481$	794.342	0.0752991
$482$	6225.19	0.588277
$483$	0	0
$484$	−524.319	−0.0492411
$485$	3316.01	0.310458
$486$	0	0
$487$	−15008.4	−1.39650	−0.698248	−	0.715856i	$-0.746037\pi$
$487$	−15008.4	−1.39650	−0.698248	+	0.715856i	$0.746037\pi$
$488$	499.123	0.0462997
$489$	0	0
$490$	4867.76	0.448782
$491$	−10939.9	−1.00552	−0.502761	−	0.864425i	$-0.667682\pi$
$491$	−10939.9	−1.00552	−0.502761	+	0.864425i	$0.667682\pi$
$492$	0	0
$493$	−16024.9	−1.46394
$494$	−1764.51	−0.160706
$495$	0	0
$496$	730.857	0.0661622
$497$	−1438.08	−0.129792
$498$	0	0
$499$	−1960.94	−0.175919	−0.0879596	−	0.996124i	$-0.528035\pi$
$499$	−1960.94	−0.175919	−0.0879596	+	0.996124i	$0.528035\pi$
$500$	7805.55	0.698149
$501$	0	0
$502$	−402.051	−0.0357458
$503$	7812.26	0.692508	0.346254	−	0.938141i	$-0.387453\pi$
$503$	7812.26	0.692508	0.346254	+	0.938141i	$0.387453\pi$
$504$	0	0
$505$	6693.54	0.589819
$506$	−11693.5	−1.02735
$507$	0	0
$508$	6901.45	0.602761
$509$	−11915.7	−1.03763	−0.518816	−	0.854886i	$-0.673627\pi$
$509$	−11915.7	−1.03763	−0.518816	+	0.854886i	$0.673627\pi$
$510$	0	0
$511$	5949.26	0.515029
$512$	−1488.92	−0.128519
$513$	0	0
$514$	11155.2	0.957270
$515$	−12093.0	−1.03472
$516$	0	0
$517$	5106.19	0.434371
$518$	810.428	0.0687416
$519$	0	0
$520$	−2457.20	−0.207222
$521$	−12795.0	−1.07593	−0.537963	−	0.842969i	$-0.680806\pi$
$521$	−12795.0	−1.07593	−0.537963	+	0.842969i	$0.680806\pi$
$522$	0	0
$523$	−18908.5	−1.58090	−0.790452	−	0.612524i	$-0.790154\pi$
$523$	−18908.5	−1.58090	−0.790452	+	0.612524i	$0.790154\pi$
$524$	−3825.20	−0.318902
$525$	0	0
$526$	−3381.08	−0.280270
$527$	10703.5	0.884732
$528$	0	0
$529$	21585.4	1.77409
$530$	−10706.3	−0.877457
$531$	0	0
$532$	3291.86	0.268271
$533$	2365.96	0.192272
$534$	0	0
$535$	−4223.14	−0.341275
$536$	−4640.22	−0.373931
$537$	0	0
$538$	9961.37	0.798263
$539$	−11192.3	−0.894412
$540$	0	0
$541$	−13497.9	−1.07268	−0.536340	−	0.844002i	$-0.680194\pi$
$541$	−13497.9	−1.07268	−0.536340	+	0.844002i	$0.680194\pi$
$542$	3210.36	0.254422
$543$	0	0
$544$	−11111.8	−0.875765
$545$	4187.86	0.329153
$546$	0	0
$547$	7859.46	0.614344	0.307172	−	0.951654i	$-0.400617\pi$
$547$	7859.46	0.614344	0.307172	+	0.951654i	$0.400617\pi$
$548$	−6528.27	−0.508894
$549$	0	0
$550$	1858.72	0.144102
$551$	24584.6	1.90080
$552$	0	0
$553$	5800.22	0.446022
$554$	−12925.2	−0.991222
$555$	0	0
$556$	−13605.2	−1.03775
$557$	−20367.3	−1.54936	−0.774678	−	0.632356i	$-0.782088\pi$
$557$	−20367.3	−1.54936	−0.774678	+	0.632356i	$0.782088\pi$
$558$	0	0
$559$	−4455.46	−0.337112
$560$	277.281	0.0209237
$561$	0	0
$562$	2057.12	0.154403
$563$	−3392.89	−0.253985	−0.126992	−	0.991904i	$-0.540532\pi$
$563$	−3392.89	−0.253985	−0.126992	+	0.991904i	$0.540532\pi$
$564$	0	0
$565$	18936.1	1.41000
$566$	2595.93	0.192783
$567$	0	0
$568$	−4633.21	−0.342263
$569$	−8441.27	−0.621927	−0.310964	−	0.950422i	$-0.600652\pi$
$569$	−8441.27	−0.621927	−0.310964	+	0.950422i	$0.600652\pi$
$570$	0	0
$571$	1840.67	0.134903	0.0674514	−	0.997723i	$-0.478513\pi$
$571$	1840.67	0.134903	0.0674514	+	0.997723i	$0.478513\pi$
$572$	2218.32	0.162155
$573$	0	0
$574$	2413.87	0.175528
$575$	−5365.06	−0.389110
$576$	0	0
$577$	−14007.9	−1.01067	−0.505335	−	0.862923i	$-0.668631\pi$
$577$	−14007.9	−1.01067	−0.505335	+	0.862923i	$0.668631\pi$
$578$	2138.53	0.153895
$579$	0	0
$580$	13442.3	0.962344
$581$	5812.45	0.415045
$582$	0	0
$583$	24616.8	1.74875
$584$	19167.3	1.35813
$585$	0	0
$586$	−6706.01	−0.472735
$587$	26870.9	1.88940	0.944702	−	0.327931i	$-0.106351\pi$
$587$	26870.9	1.88940	0.944702	+	0.327931i	$0.106351\pi$
$588$	0	0
$589$	−16420.9	−1.14874
$590$	−971.159	−0.0677661
$591$	0	0
$592$	−288.792	−0.0200495
$593$	7529.46	0.521413	0.260707	−	0.965418i	$-0.416044\pi$
$593$	7529.46	0.521413	0.260707	+	0.965418i	$0.416044\pi$
$594$	0	0
$595$	4060.84	0.279795
$596$	−2247.97	−0.154497
$597$	0	0
$598$	3501.66	0.239454
$599$	−3363.55	−0.229434	−0.114717	−	0.993398i	$-0.536596\pi$
$599$	−3363.55	−0.229434	−0.114717	+	0.993398i	$0.536596\pi$
$600$	0	0
$601$	13338.7	0.905316	0.452658	−	0.891684i	$-0.350476\pi$
$601$	13338.7	0.905316	0.452658	+	0.891684i	$0.350476\pi$
$602$	−4545.68	−0.307755
$603$	0	0
$604$	8199.25	0.552356
$605$	992.288	0.0666814
$606$	0	0
$607$	−7189.25	−0.480729	−0.240365	−	0.970683i	$-0.577267\pi$
$607$	−7189.25	−0.480729	−0.240365	+	0.970683i	$0.577267\pi$
$608$	17047.2	1.13710
$609$	0	0
$610$	−370.887	−0.0246177
$611$	−1529.07	−0.101243
$612$	0	0
$613$	−22156.6	−1.45986	−0.729932	−	0.683520i	$-0.760448\pi$
$613$	−22156.6	−1.45986	−0.729932	+	0.683520i	$0.760448\pi$
$614$	1506.56	0.0990225
$615$	0	0
$616$	5764.20	0.377023
$617$	15462.4	1.00890	0.504452	−	0.863440i	$-0.331695\pi$
$617$	15462.4	1.00890	0.504452	+	0.863440i	$0.331695\pi$
$618$	0	0
$619$	−18599.5	−1.20772	−0.603858	−	0.797092i	$-0.706371\pi$
$619$	−18599.5	−1.20772	−0.603858	+	0.797092i	$0.706371\pi$
$620$	−8978.54	−0.581592
$621$	0	0
$622$	7007.69	0.451741
$623$	6702.02	0.430997
$624$	0	0
$625$	−11121.9	−0.711800
$626$	6087.57	0.388671
$627$	0	0
$628$	−19363.3	−1.23038
$629$	−4229.42	−0.268105
$630$	0	0
$631$	−14886.6	−0.939187	−0.469594	−	0.882883i	$-0.655600\pi$
$631$	−14886.6	−0.939187	−0.469594	+	0.882883i	$0.655600\pi$
$632$	18687.1	1.17616
$633$	0	0
$634$	1455.52	0.0911768
$635$	−13061.2	−0.816248
$636$	0	0
$637$	3351.59	0.208469
$638$	16902.5	1.04887
$639$	0	0
$640$	9863.61	0.609208
$641$	−30295.2	−1.86675	−0.933376	−	0.358899i	$-0.883152\pi$
$641$	−30295.2	−1.86675	−0.933376	+	0.358899i	$0.883152\pi$
$642$	0	0
$643$	12393.0	0.760080	0.380040	−	0.924970i	$-0.375910\pi$
$643$	12393.0	0.760080	0.380040	+	0.924970i	$0.375910\pi$
$644$	−6532.70	−0.399727
$645$	0	0
$646$	9395.00	0.572200
$647$	18759.0	1.13986	0.569931	−	0.821692i	$-0.306970\pi$
$647$	18759.0	1.13986	0.569931	+	0.821692i	$0.306970\pi$
$648$	0	0
$649$	2232.96	0.135056
$650$	−556.601	−0.0335872
$651$	0	0
$652$	6887.97	0.413733
$653$	−17228.9	−1.03249	−0.516247	−	0.856440i	$-0.672671\pi$
$653$	−17228.9	−1.03249	−0.516247	+	0.856440i	$0.672671\pi$
$654$	0	0
$655$	7239.29	0.431851
$656$	−860.171	−0.0511952
$657$	0	0
$658$	−1560.03	−0.0924262
$659$	−10789.4	−0.637776	−0.318888	−	0.947792i	$-0.603309\pi$
$659$	−10789.4	−0.637776	−0.318888	+	0.947792i	$0.603309\pi$
$660$	0	0
$661$	91.3456	0.00537509	0.00268754	−	0.999996i	$-0.499145\pi$
$661$	91.3456	0.00537509	0.00268754	+	0.999996i	$0.499145\pi$
$662$	−13690.9	−0.803797
$663$	0	0
$664$	18726.6	1.09448
$665$	−6229.94	−0.363288
$666$	0	0
$667$	−48788.0	−2.83220
$668$	85.6351	0.00496006
$669$	0	0
$670$	3448.05	0.198820
$671$	852.773	0.0490625
$672$	0	0
$673$	−25294.9	−1.44881	−0.724404	−	0.689376i	$-0.757885\pi$
$673$	−25294.9	−1.44881	−0.724404	+	0.689376i	$0.757885\pi$
$674$	−12438.8	−0.710868
$675$	0	0
$676$	10698.0	0.608670
$677$	−1529.96	−0.0868557	−0.0434278	−	0.999057i	$-0.513828\pi$
$677$	−1529.96	−0.0868557	−0.0434278	+	0.999057i	$0.513828\pi$
$678$	0	0
$679$	2329.40	0.131656
$680$	13083.2	0.737820
$681$	0	0
$682$	−11289.8	−0.633882
$683$	−7588.88	−0.425154	−0.212577	−	0.977144i	$-0.568186\pi$
$683$	−7588.88	−0.425154	−0.212577	+	0.977144i	$0.568186\pi$
$684$	0	0
$685$	12354.9	0.689135
$686$	7385.54	0.411051
$687$	0	0
$688$	1619.83	0.0897610
$689$	−7371.60	−0.407599
$690$	0	0
$691$	−7781.95	−0.428421	−0.214211	−	0.976787i	$-0.568718\pi$
$691$	−7781.95	−0.428421	−0.214211	+	0.976787i	$0.568718\pi$
$692$	−12396.9	−0.681013
$693$	0	0
$694$	−2256.87	−0.123443
$695$	25748.2	1.40530
$696$	0	0
$697$	−12597.4	−0.684591
$698$	−3688.62	−0.200024
$699$	0	0
$700$	1038.39	0.0560680
$701$	−16501.9	−0.889111	−0.444555	−	0.895751i	$-0.646638\pi$
$701$	−16501.9	−0.889111	−0.444555	+	0.895751i	$0.646638\pi$
$702$	0	0
$703$	6488.57	0.348110
$704$	10472.9	0.560669
$705$	0	0
$706$	−1070.08	−0.0570440
$707$	4702.02	0.250124
$708$	0	0
$709$	−26435.2	−1.40027	−0.700136	−	0.714009i	$-0.746877\pi$
$709$	−26435.2	−1.40027	−0.700136	+	0.714009i	$0.746877\pi$
$710$	3442.84	0.181982
$711$	0	0
$712$	21592.6	1.13654
$713$	32587.2	1.71164
$714$	0	0
$715$	−4198.23	−0.219587
$716$	6325.03	0.330136
$717$	0	0
$718$	−11980.9	−0.622734
$719$	9576.02	0.496697	0.248348	−	0.968671i	$-0.420112\pi$
$719$	9576.02	0.496697	0.248348	+	0.968671i	$0.420112\pi$
$720$	0	0
$721$	−8495.00	−0.438794
$722$	−2878.23	−0.148361
$723$	0	0
$724$	22067.6	1.13278
$725$	7755.02	0.397261
$726$	0	0
$727$	−11408.0	−0.581979	−0.290990	−	0.956726i	$-0.593985\pi$
$727$	−11408.0	−0.581979	−0.290990	+	0.956726i	$0.593985\pi$
$728$	−1726.11	−0.0878764
$729$	0	0
$730$	−14242.8	−0.722124
$731$	23722.8	1.20030
$732$	0	0
$733$	−25829.0	−1.30152	−0.650762	−	0.759282i	$-0.725550\pi$
$733$	−25829.0	−1.30152	−0.650762	+	0.759282i	$0.725550\pi$
$734$	14655.9	0.737003
$735$	0	0
$736$	−33830.2	−1.69429
$737$	−7928.01	−0.396244
$738$	0	0
$739$	37906.8	1.88691	0.943454	−	0.331503i	$-0.107556\pi$
$739$	37906.8	1.88691	0.943454	+	0.331503i	$0.107556\pi$
$740$	3547.80	0.176243
$741$	0	0
$742$	−7520.88	−0.372102
$743$	−17868.7	−0.882286	−0.441143	−	0.897437i	$-0.645427\pi$
$743$	−17868.7	−0.882286	−0.441143	+	0.897437i	$0.645427\pi$
$744$	0	0
$745$	4254.35	0.209218
$746$	−15343.1	−0.753017
$747$	0	0
$748$	−11811.3	−0.577358
$749$	−2966.63	−0.144724
$750$	0	0
$751$	−12.6803	−0.000616125 0	−0.000308063	−	1.00000i	$-0.500098\pi$
$751$	−12.6803	−0.000616125 0	−0.000308063		1.00000i	$0.500098\pi$
$752$	555.911	0.0269574
$753$	0	0
$754$	−5061.54	−0.244470
$755$	−15517.3	−0.747990
$756$	0	0
$757$	39700.2	1.90611	0.953056	−	0.302793i	$-0.0979192\pi$
$757$	39700.2	1.90611	0.953056	+	0.302793i	$0.0979192\pi$
$758$	−14361.8	−0.688186
$759$	0	0
$760$	−20071.6	−0.957992
$761$	9323.77	0.444134	0.222067	−	0.975031i	$-0.428720\pi$
$761$	9323.77	0.444134	0.222067	+	0.975031i	$0.428720\pi$
$762$	0	0
$763$	2941.85	0.139583
$764$	−15931.2	−0.754412
$765$	0	0
$766$	−14616.2	−0.689433
$767$	−668.670	−0.0314789
$768$	0	0
$769$	20630.0	0.967408	0.483704	−	0.875232i	$-0.339291\pi$
$769$	20630.0	0.967408	0.483704	+	0.875232i	$0.339291\pi$
$770$	−4283.25	−0.200464
$771$	0	0
$772$	−17944.3	−0.836568
$773$	−9335.33	−0.434370	−0.217185	−	0.976130i	$-0.569688\pi$
$773$	−9335.33	−0.434370	−0.217185	+	0.976130i	$0.569688\pi$
$774$	0	0
$775$	−5179.84	−0.240084
$776$	7504.85	0.347176
$777$	0	0
$778$	−16951.9	−0.781175
$779$	19326.3	0.888879
$780$	0	0
$781$	−7916.04	−0.362686
$782$	−18644.3	−0.852584
$783$	0	0
$784$	−1218.51	−0.0555080
$785$	36645.5	1.66616
$786$	0	0
$787$	−14965.2	−0.677831	−0.338916	−	0.940817i	$-0.610060\pi$
$787$	−14965.2	−0.677831	−0.338916	+	0.940817i	$0.610060\pi$
$788$	−7694.15	−0.347833
$789$	0	0
$790$	−13886.0	−0.625369
$791$	13302.1	0.597937
$792$	0	0
$793$	−255.366	−0.0114355
$794$	3076.78	0.137520
$795$	0	0
$796$	15658.5	0.697239
$797$	−35050.2	−1.55777	−0.778885	−	0.627167i	$-0.784214\pi$
$797$	−35050.2	−1.55777	−0.778885	+	0.627167i	$0.784214\pi$
$798$	0	0
$799$	8141.43	0.360480
$800$	5377.43	0.237651
$801$	0	0
$802$	−18364.1	−0.808552
$803$	32748.2	1.43918
$804$	0	0
$805$	12363.3	0.541303
$806$	3380.77	0.147745
$807$	0	0
$808$	15149.0	0.659577
$809$	14925.1	0.648628	0.324314	−	0.945950i	$-0.394867\pi$
$809$	14925.1	0.648628	0.324314	+	0.945950i	$0.394867\pi$
$810$	0	0
$811$	−28928.7	−1.25256	−0.626280	−	0.779598i	$-0.715423\pi$
$811$	−28928.7	−1.25256	−0.626280	+	0.779598i	$0.715423\pi$
$812$	9442.81	0.408100
$813$	0	0
$814$	4461.06	0.192089
$815$	−13035.7	−0.560269
$816$	0	0
$817$	−36394.4	−1.55848
$818$	−10833.0	−0.463040
$819$	0	0
$820$	10567.2	0.450026
$821$	1009.83	0.0429274	0.0214637	−	0.999770i	$-0.493167\pi$
$821$	1009.83	0.0429274	0.0214637	+	0.999770i	$0.493167\pi$
$822$	0	0
$823$	4148.74	0.175718	0.0878590	−	0.996133i	$-0.471998\pi$
$823$	4148.74	0.175718	0.0878590	+	0.996133i	$0.471998\pi$
$824$	−27369.2	−1.15710
$825$	0	0
$826$	−682.211	−0.0287375
$827$	−46387.4	−1.95048	−0.975241	−	0.221147i	$-0.929020\pi$
$827$	−46387.4	−1.95048	−0.975241	+	0.221147i	$0.929020\pi$
$828$	0	0
$829$	16193.2	0.678423	0.339211	−	0.940710i	$-0.389840\pi$
$829$	16193.2	0.678423	0.339211	+	0.940710i	$0.389840\pi$
$830$	−13915.3	−0.581936
$831$	0	0
$832$	−3136.15	−0.130681
$833$	−17845.3	−0.742262
$834$	0	0
$835$	−162.067	−0.00671683
$836$	18120.3	0.749647
$837$	0	0
$838$	11409.5	0.470327
$839$	26644.1	1.09637	0.548186	−	0.836357i	$-0.315319\pi$
$839$	26644.1	1.09637	0.548186	+	0.836357i	$0.315319\pi$
$840$	0	0
$841$	46132.5	1.89153
$842$	−1392.22	−0.0569824
$843$	0	0
$844$	8563.84	0.349265
$845$	−20246.2	−0.824249
$846$	0	0
$847$	697.054	0.0282775
$848$	2680.03	0.108529
$849$	0	0
$850$	2963.58	0.119588
$851$	−12876.6	−0.518687
$852$	0	0
$853$	−28697.2	−1.15190	−0.575952	−	0.817484i	$-0.695368\pi$
$853$	−28697.2	−1.15190	−0.575952	+	0.817484i	$0.695368\pi$
$854$	−260.538	−0.0104396
$855$	0	0
$856$	−9557.89	−0.381638
$857$	−19383.8	−0.772623	−0.386311	−	0.922368i	$-0.626251\pi$
$857$	−19383.8	−0.772623	−0.386311	+	0.922368i	$0.626251\pi$
$858$	0	0
$859$	−18991.9	−0.754362	−0.377181	−	0.926140i	$-0.623107\pi$
$859$	−18991.9	−0.754362	−0.377181	+	0.926140i	$0.623107\pi$
$860$	−19899.6	−0.789035
$861$	0	0
$862$	2891.66	0.114258
$863$	−34623.5	−1.36570	−0.682849	−	0.730560i	$-0.739259\pi$
$863$	−34623.5	−1.36570	−0.682849	+	0.730560i	$0.739259\pi$
$864$	0	0
$865$	23461.6	0.922216
$866$	−20087.7	−0.788229
$867$	0	0
$868$	−6307.17	−0.246635
$869$	31927.7	1.24635
$870$	0	0
$871$	2374.08	0.0923565
$872$	9478.06	0.368082
$873$	0	0
$874$	28603.3	1.10700
$875$	−10377.1	−0.400924
$876$	0	0
$877$	−33705.2	−1.29777	−0.648884	−	0.760887i	$-0.724764\pi$
$877$	−33705.2	−1.29777	−0.648884	+	0.760887i	$0.724764\pi$
$878$	−531.947	−0.0204469
$879$	0	0
$880$	1526.31	0.0584683
$881$	24525.5	0.937896	0.468948	−	0.883226i	$-0.344633\pi$
$881$	24525.5	0.937896	0.468948	+	0.883226i	$0.344633\pi$
$882$	0	0
$883$	3141.61	0.119732	0.0598662	−	0.998206i	$-0.480933\pi$
$883$	3141.61	0.119732	0.0598662	+	0.998206i	$0.480933\pi$
$884$	3536.94	0.134570
$885$	0	0
$886$	−15208.8	−0.576691
$887$	5590.87	0.211638	0.105819	−	0.994385i	$-0.466254\pi$
$887$	5590.87	0.211638	0.105819	+	0.994385i	$0.466254\pi$
$888$	0	0
$889$	−9175.11	−0.346145
$890$	−16045.0	−0.604302
$891$	0	0
$892$	6336.37	0.237845
$893$	−12490.2	−0.468050
$894$	0	0
$895$	−11970.3	−0.447065
$896$	6928.90	0.258346
$897$	0	0
$898$	15234.2	0.566116
$899$	−47103.7	−1.74749
$900$	0	0
$901$	39249.6	1.45127
$902$	13287.3	0.490488
$903$	0	0
$904$	42856.7	1.57676
$905$	−41763.5	−1.53399
$906$	0	0
$907$	−27732.8	−1.01527	−0.507637	−	0.861571i	$-0.669481\pi$
$907$	−27732.8	−1.01527	−0.507637	+	0.861571i	$0.669481\pi$
$908$	−3401.32	−0.124314
$909$	0	0
$910$	1282.64	0.0467242
$911$	−7724.27	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$911$	−7724.27	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$912$	0	0
$913$	31995.1	1.15978
$914$	−24124.2	−0.873040
$915$	0	0
$916$	−7281.62	−0.262655
$917$	5085.39	0.183135
$918$	0	0
$919$	40052.6	1.43767	0.718833	−	0.695183i	$-0.244677\pi$
$919$	40052.6	1.43767	0.718833	+	0.695183i	$0.244677\pi$
$920$	39832.1	1.42742
$921$	0	0
$922$	−18065.5	−0.645288
$923$	2370.49	0.0845349
$924$	0	0
$925$	2046.77	0.0727540
$926$	29555.3	1.04887
$927$	0	0
$928$	48900.5	1.72978
$929$	−14184.7	−0.500952	−0.250476	−	0.968123i	$-0.580587\pi$
$929$	−14184.7	−0.500952	−0.250476	+	0.968123i	$0.580587\pi$
$930$	0	0
$931$	27377.5	0.963759
$932$	33065.3	1.16211
$933$	0	0
$934$	15065.6	0.527796
$935$	22353.2	0.781848
$936$	0	0
$937$	−30556.2	−1.06534	−0.532672	−	0.846322i	$-0.678812\pi$
$937$	−30556.2	−1.06534	−0.532672	+	0.846322i	$0.678812\pi$
$938$	2422.15	0.0843135
$939$	0	0
$940$	−6829.34	−0.236967
$941$	9474.36	0.328220	0.164110	−	0.986442i	$-0.447525\pi$
$941$	9474.36	0.328220	0.164110	+	0.986442i	$0.447525\pi$
$942$	0	0
$943$	−38353.0	−1.32444
$944$	243.103	0.00838170
$945$	0	0
$946$	−25022.1	−0.859976
$947$	−55248.5	−1.89581	−0.947906	−	0.318549i	$-0.896804\pi$
$947$	−55248.5	−1.89581	−0.947906	+	0.318549i	$0.896804\pi$
$948$	0	0
$949$	−9806.58	−0.335443
$950$	−4546.59	−0.155274
$951$	0	0
$952$	9190.58	0.312887
$953$	−6103.29	−0.207455	−0.103728	−	0.994606i	$-0.533077\pi$
$953$	−6103.29	−0.207455	−0.103728	+	0.994606i	$0.533077\pi$
$954$	0	0
$955$	30150.2	1.02161
$956$	10041.4	0.339710
$957$	0	0
$958$	11620.4	0.391898
$959$	8678.99	0.292241
$960$	0	0
$961$	1671.17	0.0560963
$962$	−1335.88	−0.0447720
$963$	0	0
$964$	19143.7	0.639602
$965$	33960.1	1.13287
$966$	0	0
$967$	−22070.6	−0.733965	−0.366982	−	0.930228i	$-0.619609\pi$
$967$	−22070.6	−0.733965	−0.366982	+	0.930228i	$0.619609\pi$
$968$	2245.77	0.0745678
$969$	0	0
$970$	−5576.69	−0.184595
$971$	46982.7	1.55278	0.776389	−	0.630254i	$-0.217049\pi$
$971$	46982.7	1.55278	0.776389	+	0.630254i	$0.217049\pi$
$972$	0	0
$973$	18087.4	0.595945
$974$	25240.3	0.830341
$975$	0	0
$976$	92.8414	0.00304486
$977$	38521.7	1.26143	0.630715	−	0.776014i	$-0.282762\pi$
$977$	38521.7	1.26143	0.630715	+	0.776014i	$0.282762\pi$
$978$	0	0
$979$	36891.8	1.20436
$980$	14969.3	0.487937
$981$	0	0
$982$	18398.2	0.597872
$983$	17793.1	0.577327	0.288664	−	0.957431i	$-0.406789\pi$
$983$	17793.1	0.577327	0.288664	+	0.957431i	$0.406789\pi$
$984$	0	0
$985$	14561.4	0.471030
$986$	26949.8	0.870443
$987$	0	0
$988$	−5426.21	−0.174727
$989$	72224.5	2.32215
$990$	0	0
$991$	−52191.4	−1.67297	−0.836485	−	0.547990i	$-0.815393\pi$
$991$	−52191.4	−1.67297	−0.836485	+	0.547990i	$0.815393\pi$
$992$	−32662.3	−1.04539
$993$	0	0
$994$	2418.50	0.0771731
$995$	−29634.2	−0.944189
$996$	0	0
$997$	6732.09	0.213849	0.106924	−	0.994267i	$-0.465900\pi$
$997$	6732.09	0.213849	0.106924	+	0.994267i	$0.465900\pi$
$998$	3297.81	0.104600
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.d.1.2		7
3.2	odd	2		177.4.a.a.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.6	✓	7	3.2	odd	2
531.4.a.d.1.2		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)

\(f(q)\)	\(=\)	\(q-1.68175 q^{2} -5.17172 q^{4} +9.78761 q^{5} +6.87552 q^{7} +22.1515 q^{8} +O(q^{10})\)	\(q-1.68175 q^{2} -5.17172 q^{4} +9.78761 q^{5} +6.87552 q^{7} +22.1515 q^{8} -16.4603 q^{10} +37.8468 q^{11} -11.3334 q^{13} -11.5629 q^{14} +4.12039 q^{16} +60.3439 q^{17} -92.5767 q^{19} -50.6188 q^{20} -63.6489 q^{22} +183.718 q^{23} -29.2026 q^{25} +19.0599 q^{26} -35.5582 q^{28} -265.559 q^{29} +177.376 q^{31} -184.142 q^{32} -101.483 q^{34} +67.2949 q^{35} -70.0886 q^{37} +155.691 q^{38} +216.811 q^{40} -208.760 q^{41} +393.127 q^{43} -195.733 q^{44} -308.968 q^{46} +134.917 q^{47} -295.727 q^{49} +49.1115 q^{50} +58.6131 q^{52} +650.432 q^{53} +370.430 q^{55} +152.303 q^{56} +446.604 q^{58} +59.0000 q^{59} +22.5322 q^{61} -298.302 q^{62} +276.717 q^{64} -110.927 q^{65} -209.476 q^{67} -312.082 q^{68} -113.173 q^{70} -209.160 q^{71} +865.282 q^{73} +117.872 q^{74} +478.781 q^{76} +260.217 q^{77} +843.604 q^{79} +40.3287 q^{80} +351.082 q^{82} +845.384 q^{83} +590.623 q^{85} -661.141 q^{86} +838.365 q^{88} +974.766 q^{89} -77.9229 q^{91} -950.139 q^{92} -226.897 q^{94} -906.105 q^{95} +338.796 q^{97} +497.339 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 8 q^{2} + 22 q^{4} + 28 q^{5} - 59 q^{7} + 117 q^{8}+O(q^{10}) \) 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

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