LMFDB - Embedded newform 531.8.a.d.1.5

Newspace parameters

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$165.876448532$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
Defining polynomial:	$x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{10}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$12.7630$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

$f(q)$	$=$	$q-10.7630 q^{2} -12.1577 q^{4} -451.863 q^{5} -504.672 q^{7} +1508.52 q^{8} +O(q^{10})$	\(q-10.7630 q^{2} -12.1577 q^{4} -451.863 q^{5} -504.672 q^{7} +1508.52 q^{8} +4863.40 q^{10} +8070.72 q^{11} -4674.69 q^{13} +5431.79 q^{14} -14680.0 q^{16} +13499.0 q^{17} -11364.6 q^{19} +5493.61 q^{20} -86865.2 q^{22} +66475.4 q^{23} +126055. q^{25} +50313.7 q^{26} +6135.65 q^{28} +174542. q^{29} +62482.4 q^{31} -35089.3 q^{32} -145289. q^{34} +228043. q^{35} +277403. q^{37} +122318. q^{38} -681643. q^{40} -529842. q^{41} +675111. q^{43} -98121.3 q^{44} -715475. q^{46} -175110. q^{47} -568849. q^{49} -1.35673e6 q^{50} +56833.4 q^{52} +313480. q^{53} -3.64686e6 q^{55} -761307. q^{56} -1.87860e6 q^{58} +205379. q^{59} -3.01895e6 q^{61} -672499. q^{62} +2.25671e6 q^{64} +2.11232e6 q^{65} -885587. q^{67} -164116. q^{68} -2.45442e6 q^{70} +1.15026e6 q^{71} +33498.4 q^{73} -2.98569e6 q^{74} +138168. q^{76} -4.07307e6 q^{77} -694337. q^{79} +6.63335e6 q^{80} +5.70270e6 q^{82} +672731. q^{83} -6.09968e6 q^{85} -7.26622e6 q^{86} +1.21748e7 q^{88} -7.67580e6 q^{89} +2.35918e6 q^{91} -808188. q^{92} +1.88471e6 q^{94} +5.13525e6 q^{95} +4.89911e6 q^{97} +6.12253e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) $	$ 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) $

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 $

$2$	−10.7630	−0.951324	−0.475662	−	0.879628i	$-0.657792\pi$
$2$	−10.7630	−0.951324	−0.475662	+	0.879628i	$0.657792\pi$
$3$	0	0
$3$	0	0
$4$	−12.1577	−0.0949820
$5$	−451.863	−1.61663	−0.808317	−	0.588748i	$-0.799621\pi$
$5$	−451.863	−1.61663	−0.808317	+	0.588748i	$0.799621\pi$
$6$	0	0
$7$	−504.672	−0.556117	−0.278058	−	0.960564i	$-0.589691\pi$
$7$	−504.672	−0.556117	−0.278058	+	0.960564i	$0.589691\pi$
$8$	1508.52	1.04168
$9$	0	0
$10$	4863.40	1.53794
$11$	8070.72	1.82826	0.914129	−	0.405422i	$-0.132876\pi$
$11$	8070.72	1.82826	0.914129	+	0.405422i	$0.132876\pi$
$12$	0	0
$13$	−4674.69	−0.590134	−0.295067	−	0.955477i	$-0.595342\pi$
$13$	−4674.69	−0.590134	−0.295067	+	0.955477i	$0.595342\pi$
$14$	5431.79	0.529048
$15$	0	0
$16$	−14680.0	−0.895996
$17$	13499.0	0.666391	0.333196	−	0.942858i	$-0.391873\pi$
$17$	13499.0	0.666391	0.333196	+	0.942858i	$0.391873\pi$
$18$	0	0
$19$	−11364.6	−0.380117	−0.190059	−	0.981773i	$-0.560868\pi$
$19$	−11364.6	−0.380117	−0.190059	+	0.981773i	$0.560868\pi$
$20$	5493.61	0.153551
$21$	0	0
$22$	−86865.2	−1.73927
$23$	66475.4	1.13924	0.569618	−	0.821910i	$-0.307091\pi$
$23$	66475.4	1.13924	0.569618	+	0.821910i	$0.307091\pi$
$24$	0	0
$25$	126055.	1.61350
$26$	50313.7	0.561409
$27$	0	0
$28$	6135.65	0.0528211
$29$	174542.	1.32894	0.664472	−	0.747313i	$-0.268657\pi$
$29$	174542.	1.32894	0.664472	+	0.747313i	$0.268657\pi$
$30$	0	0
$31$	62482.4	0.376697	0.188348	−	0.982102i	$-0.439687\pi$
$31$	62482.4	0.376697	0.188348	+	0.982102i	$0.439687\pi$
$32$	−35089.3	−0.189300
$33$	0	0
$34$	−145289.	−0.633954
$35$	228043.	0.899038
$36$	0	0
$37$	277403.	0.900336	0.450168	−	0.892944i	$-0.351364\pi$
$37$	277403.	0.900336	0.450168	+	0.892944i	$0.351364\pi$
$38$	122318.	0.361615
$39$	0	0
$40$	−681643.	−1.68402
$41$	−529842.	−1.20061	−0.600307	−	0.799770i	$-0.704955\pi$
$41$	−529842.	−1.20061	−0.600307	+	0.799770i	$0.704955\pi$
$42$	0	0
$43$	675111.	1.29490	0.647449	−	0.762109i	$-0.275836\pi$
$43$	675111.	1.29490	0.647449	+	0.762109i	$0.275836\pi$
$44$	−98121.3	−0.173652
$45$	0	0
$46$	−715475.	−1.08378
$47$	−175110.	−0.246019	−0.123010	−	0.992405i	$-0.539255\pi$
$47$	−175110.	−0.246019	−0.123010	+	0.992405i	$0.539255\pi$
$48$	0	0
$49$	−568849.	−0.690734
$50$	−1.35673e6	−1.53497
$51$	0	0
$52$	56833.4	0.0560521
$53$	313480.	0.289231	0.144615	−	0.989488i	$-0.453805\pi$
$53$	313480.	0.289231	0.144615	+	0.989488i	$0.453805\pi$
$54$	0	0
$55$	−3.64686e6	−2.95563
$56$	−761307.	−0.579298
$57$	0	0
$58$	−1.87860e6	−1.26426
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	0	0
$61$	−3.01895e6	−1.70295	−0.851474	−	0.524396i	$-0.824291\pi$
$61$	−3.01895e6	−1.70295	−0.851474	+	0.524396i	$0.824291\pi$
$62$	−672499.	−0.358361
$63$	0	0
$64$	2.25671e6	1.07608
$65$	2.11232e6	0.954031
$66$	0	0
$67$	−885587.	−0.359724	−0.179862	−	0.983692i	$-0.557565\pi$
$67$	−885587.	−0.359724	−0.179862	+	0.983692i	$0.557565\pi$
$68$	−164116.	−0.0632952
$69$	0	0
$70$	−2.45442e6	−0.855276
$71$	1.15026e6	0.381411	0.190706	−	0.981647i	$-0.438922\pi$
$71$	1.15026e6	0.381411	0.190706	+	0.981647i	$0.438922\pi$
$72$	0	0
$73$	33498.4	0.0100785	0.00503923	−	0.999987i	$-0.498396\pi$
$73$	33498.4	0.0100785	0.00503923	+	0.999987i	$0.498396\pi$
$74$	−2.98569e6	−0.856511
$75$	0	0
$76$	138168.	0.0361043
$77$	−4.07307e6	−1.01673
$78$	0	0
$79$	−694337.	−0.158444	−0.0792219	−	0.996857i	$-0.525244\pi$
$79$	−694337.	−0.158444	−0.0792219	+	0.996857i	$0.525244\pi$
$80$	6.63335e6	1.44850
$81$	0	0
$82$	5.70270e6	1.14217
$83$	672731.	0.129142	0.0645711	−	0.997913i	$-0.479432\pi$
$83$	672731.	0.129142	0.0645711	+	0.997913i	$0.479432\pi$
$84$	0	0
$85$	−6.09968e6	−1.07731
$86$	−7.26622e6	−1.23187
$87$	0	0
$88$	1.21748e7	1.90447
$89$	−7.67580e6	−1.15414	−0.577070	−	0.816695i	$-0.695804\pi$
$89$	−7.67580e6	−1.15414	−0.577070	+	0.816695i	$0.695804\pi$
$90$	0	0
$91$	2.35918e6	0.328184
$92$	−808188.	−0.108207
$93$	0	0
$94$	1.88471e6	0.234044
$95$	5.13525e6	0.614510
$96$	0	0
$97$	4.89911e6	0.545025	0.272512	−	0.962152i	$-0.412145\pi$
$97$	4.89911e6	0.545025	0.272512	+	0.962152i	$0.412145\pi$
$98$	6.12253e6	0.657112
$99$	0	0
$100$	−1.53254e6	−0.153254
$101$	1.84644e7	1.78324	0.891621	−	0.452782i	$-0.149568\pi$
$101$	1.84644e7	1.78324	0.891621	+	0.452782i	$0.149568\pi$
$102$	0	0
$103$	−5.21469e6	−0.470217	−0.235109	−	0.971969i	$-0.575545\pi$
$103$	−5.21469e6	−0.470217	−0.235109	+	0.971969i	$0.575545\pi$
$104$	−7.05185e6	−0.614733
$105$	0	0
$106$	−3.37399e6	−0.275152
$107$	2.16312e6	0.170701	0.0853506	−	0.996351i	$-0.472799\pi$
$107$	2.16312e6	0.170701	0.0853506	+	0.996351i	$0.472799\pi$
$108$	0	0
$109$	−1.59770e7	−1.18169	−0.590845	−	0.806785i	$-0.701206\pi$
$109$	−1.59770e7	−1.18169	−0.590845	+	0.806785i	$0.701206\pi$
$110$	3.92511e7	2.81176
$111$	0	0
$112$	7.40859e6	0.498279
$113$	−9.29263e6	−0.605849	−0.302924	−	0.953015i	$-0.597963\pi$
$113$	−9.29263e6	−0.605849	−0.302924	+	0.953015i	$0.597963\pi$
$114$	0	0
$115$	−3.00378e7	−1.84173
$116$	−2.12203e6	−0.126226
$117$	0	0
$118$	−2.21050e6	−0.123852
$119$	−6.81255e6	−0.370591
$120$	0	0
$121$	4.56493e7	2.34253
$122$	3.24930e7	1.62006
$123$	0	0
$124$	−759642.	−0.0357794
$125$	−2.16578e7	−0.991813
$126$	0	0
$127$	8.17874e6	0.354302	0.177151	−	0.984184i	$-0.443312\pi$
$127$	8.17874e6	0.354302	0.177151	+	0.984184i	$0.443312\pi$
$128$	−1.97975e7	−0.834403
$129$	0	0
$130$	−2.27349e7	−0.907593
$131$	−4.40912e6	−0.171357	−0.0856787	−	0.996323i	$-0.527306\pi$
$131$	−4.40912e6	−0.171357	−0.0856787	+	0.996323i	$0.527306\pi$
$132$	0	0
$133$	5.73541e6	0.211390
$134$	9.53158e6	0.342214
$135$	0	0
$136$	2.03634e7	0.694168
$137$	8.21867e6	0.273073	0.136537	−	0.990635i	$-0.456403\pi$
$137$	8.21867e6	0.273073	0.136537	+	0.990635i	$0.456403\pi$
$138$	0	0
$139$	−5.56805e7	−1.75854	−0.879268	−	0.476328i	$-0.841968\pi$
$139$	−5.56805e7	−1.75854	−0.879268	+	0.476328i	$0.841968\pi$
$140$	−2.77247e6	−0.0853924
$141$	0	0
$142$	−1.23803e7	−0.362846
$143$	−3.77281e7	−1.07892
$144$	0	0
$145$	−7.88690e7	−2.14842
$146$	−360544.	−0.00958789
$147$	0	0
$148$	−3.37258e6	−0.0855157
$149$	−2.85493e7	−0.707039	−0.353520	−	0.935427i	$-0.615015\pi$
$149$	−2.85493e7	−0.707039	−0.353520	+	0.935427i	$0.615015\pi$
$150$	0	0
$151$	−1.41537e7	−0.334541	−0.167271	−	0.985911i	$-0.553495\pi$
$151$	−1.41537e7	−0.334541	−0.167271	+	0.985911i	$0.553495\pi$
$152$	−1.71437e7	−0.395962
$153$	0	0
$154$	4.38384e7	0.967236
$155$	−2.82335e7	−0.608981
$156$	0	0
$157$	6.04002e7	1.24563	0.622816	−	0.782368i	$-0.285989\pi$
$157$	6.04002e7	1.24563	0.622816	+	0.782368i	$0.285989\pi$
$158$	7.47315e6	0.150732
$159$	0	0
$160$	1.58556e7	0.306029
$161$	−3.35483e7	−0.633548
$162$	0	0
$163$	7.29006e7	1.31848	0.659241	−	0.751931i	$-0.270878\pi$
$163$	7.29006e7	1.31848	0.659241	+	0.751931i	$0.270878\pi$
$164$	6.44166e6	0.114037
$165$	0	0
$166$	−7.24061e6	−0.122856
$167$	9.50243e7	1.57880	0.789400	−	0.613879i	$-0.210392\pi$
$167$	9.50243e7	1.57880	0.789400	+	0.613879i	$0.210392\pi$
$168$	0	0
$169$	−4.08958e7	−0.651742
$170$	6.56509e7	1.02487
$171$	0	0
$172$	−8.20779e6	−0.122992
$173$	−5.86358e7	−0.860996	−0.430498	−	0.902592i	$-0.641662\pi$
$173$	−5.86358e7	−0.860996	−0.430498	+	0.902592i	$0.641662\pi$
$174$	0	0
$175$	−6.36165e7	−0.897298
$176$	−1.18478e8	−1.63811
$177$	0	0
$178$	8.26147e7	1.09796
$179$	3.55324e7	0.463061	0.231531	−	0.972828i	$-0.425627\pi$
$179$	3.55324e7	0.463061	0.231531	+	0.972828i	$0.425627\pi$
$180$	0	0
$181$	1.28354e8	1.60892	0.804459	−	0.594008i	$-0.202455\pi$
$181$	1.28354e8	1.60892	0.804459	+	0.594008i	$0.202455\pi$
$182$	−2.53919e7	−0.312209
$183$	0	0
$184$	1.00279e8	1.18672
$185$	−1.25348e8	−1.45551
$186$	0	0
$187$	1.08946e8	1.21834
$188$	2.12894e6	0.0233674
$189$	0	0
$190$	−5.52708e7	−0.584599
$191$	8.71206e7	0.904699	0.452350	−	0.891841i	$-0.350586\pi$
$191$	8.71206e7	0.904699	0.452350	+	0.891841i	$0.350586\pi$
$192$	0	0
$193$	3.34478e7	0.334902	0.167451	−	0.985880i	$-0.446446\pi$
$193$	3.34478e7	0.334902	0.167451	+	0.985880i	$0.446446\pi$
$194$	−5.27292e7	−0.518495
$195$	0	0
$196$	6.91589e6	0.0656073
$197$	−1.43873e8	−1.34074	−0.670372	−	0.742025i	$-0.733866\pi$
$197$	−1.43873e8	−1.34074	−0.670372	+	0.742025i	$0.733866\pi$
$198$	0	0
$199$	−3.39107e7	−0.305036	−0.152518	−	0.988301i	$-0.548738\pi$
$199$	−3.39107e7	−0.305036	−0.152518	+	0.988301i	$0.548738\pi$
$200$	1.90156e8	1.68076
$201$	0	0
$202$	−1.98732e8	−1.69644
$203$	−8.80864e7	−0.739048
$204$	0	0
$205$	2.39416e8	1.94095
$206$	5.61258e7	0.447329
$207$	0	0
$208$	6.86244e7	0.528758
$209$	−9.17207e7	−0.694953
$210$	0	0
$211$	−1.13428e8	−0.831249	−0.415624	−	0.909536i	$-0.636437\pi$
$211$	−1.13428e8	−0.831249	−0.415624	+	0.909536i	$0.636437\pi$
$212$	−3.81120e6	−0.0274717
$213$	0	0
$214$	−2.32816e7	−0.162392
$215$	−3.05058e8	−2.09338
$216$	0	0
$217$	−3.15331e7	−0.209487
$218$	1.71961e8	1.12417
$219$	0	0
$220$	4.43374e7	0.280731
$221$	−6.31034e7	−0.393260
$222$	0	0
$223$	−1.58906e8	−0.959565	−0.479782	−	0.877387i	$-0.659284\pi$
$223$	−1.58906e8	−0.959565	−0.479782	+	0.877387i	$0.659284\pi$
$224$	1.77086e7	0.105273
$225$	0	0
$226$	1.00017e8	0.576358
$227$	2.46698e7	0.139983	0.0699915	−	0.997548i	$-0.477703\pi$
$227$	2.46698e7	0.139983	0.0699915	+	0.997548i	$0.477703\pi$
$228$	0	0
$229$	2.79166e6	0.0153617	0.00768084	−	0.999971i	$-0.497555\pi$
$229$	2.79166e6	0.0153617	0.00768084	+	0.999971i	$0.497555\pi$
$230$	3.23297e8	1.75208
$231$	0	0
$232$	2.63300e8	1.38434
$233$	−1.72078e8	−0.891207	−0.445604	−	0.895230i	$-0.647011\pi$
$233$	−1.72078e8	−0.891207	−0.445604	+	0.895230i	$0.647011\pi$
$234$	0	0
$235$	7.91259e7	0.397723
$236$	−2.49694e6	−0.0123656
$237$	0	0
$238$	7.33235e7	0.352553
$239$	3.26143e8	1.54531	0.772656	−	0.634825i	$-0.218928\pi$
$239$	3.26143e8	1.54531	0.772656	+	0.634825i	$0.218928\pi$
$240$	0	0
$241$	4.12780e8	1.89958	0.949792	−	0.312882i	$-0.101294\pi$
$241$	4.12780e8	1.89958	0.949792	+	0.312882i	$0.101294\pi$
$242$	−4.91324e8	−2.22851
$243$	0	0
$244$	3.67035e7	0.161750
$245$	2.57042e8	1.11666
$246$	0	0
$247$	5.31261e7	0.224320
$248$	9.42559e7	0.392399
$249$	0	0
$250$	2.33103e8	0.943536
$251$	−4.66684e8	−1.86279	−0.931397	−	0.364005i	$-0.881409\pi$
$251$	−4.66684e8	−1.86279	−0.931397	+	0.364005i	$0.881409\pi$
$252$	0	0
$253$	5.36504e8	2.08282
$254$	−8.80278e7	−0.337056
$255$	0	0
$256$	−7.57777e7	−0.282294
$257$	9.36406e7	0.344111	0.172055	−	0.985087i	$-0.444959\pi$
$257$	9.36406e7	0.344111	0.172055	+	0.985087i	$0.444959\pi$
$258$	0	0
$259$	−1.39997e8	−0.500692
$260$	−2.56809e7	−0.0906158
$261$	0	0
$262$	4.74554e7	0.163016
$263$	5.00867e8	1.69776	0.848882	−	0.528582i	$-0.177276\pi$
$263$	5.00867e8	1.69776	0.848882	+	0.528582i	$0.177276\pi$
$264$	0	0
$265$	−1.41650e8	−0.467581
$266$	−6.17303e7	−0.201100
$267$	0	0
$268$	1.07667e7	0.0341673
$269$	−7.84305e7	−0.245670	−0.122835	−	0.992427i	$-0.539199\pi$
$269$	−7.84305e7	−0.245670	−0.122835	+	0.992427i	$0.539199\pi$
$270$	0	0
$271$	3.97558e8	1.21341	0.606705	−	0.794927i	$-0.292491\pi$
$271$	3.97558e8	1.21341	0.606705	+	0.794927i	$0.292491\pi$
$272$	−1.98165e8	−0.597084
$273$	0	0
$274$	−8.84576e7	−0.259781
$275$	1.01735e9	2.94990
$276$	0	0
$277$	−9.64793e7	−0.272744	−0.136372	−	0.990658i	$-0.543544\pi$
$277$	−9.64793e7	−0.272744	−0.136372	+	0.990658i	$0.543544\pi$
$278$	5.99289e8	1.67294
$279$	0	0
$280$	3.44006e8	0.936512
$281$	5.37086e8	1.44402	0.722008	−	0.691885i	$-0.243219\pi$
$281$	5.37086e8	1.44402	0.722008	+	0.691885i	$0.243219\pi$
$282$	0	0
$283$	−6.18129e8	−1.62116	−0.810582	−	0.585626i	$-0.800849\pi$
$283$	−6.18129e8	−1.62116	−0.810582	+	0.585626i	$0.800849\pi$
$284$	−1.39846e7	−0.0362272
$285$	0	0
$286$	4.06067e8	1.02640
$287$	2.67397e8	0.667682
$288$	0	0
$289$	−2.28117e8	−0.555923
$290$	8.48867e8	2.04384
$291$	0	0
$292$	−407264.	−0.000957273 0
$293$	−3.39223e8	−0.787858	−0.393929	−	0.919141i	$-0.628884\pi$
$293$	−3.39223e8	−0.787858	−0.393929	+	0.919141i	$0.628884\pi$
$294$	0	0
$295$	−9.28031e7	−0.210468
$296$	4.18467e8	0.937864
$297$	0	0
$298$	3.07276e8	0.672623
$299$	−3.10752e8	−0.672302
$300$	0	0
$301$	−3.40710e8	−0.720115
$302$	1.52336e8	0.318257
$303$	0	0
$304$	1.66833e8	0.340584
$305$	1.36415e9	2.75304
$306$	0	0
$307$	−9.62816e7	−0.189915	−0.0949575	−	0.995481i	$-0.530272\pi$
$307$	−9.62816e7	−0.189915	−0.0949575	+	0.995481i	$0.530272\pi$
$308$	4.95191e7	0.0965707
$309$	0	0
$310$	3.03877e8	0.579338
$311$	7.91196e8	1.49150	0.745749	−	0.666227i	$-0.232092\pi$
$311$	7.91196e8	1.49150	0.745749	+	0.666227i	$0.232092\pi$
$312$	0	0
$313$	−8.83580e7	−0.162870	−0.0814349	−	0.996679i	$-0.525950\pi$
$313$	−8.83580e7	−0.162870	−0.0814349	+	0.996679i	$0.525950\pi$
$314$	−6.50088e8	−1.18500
$315$	0	0
$316$	8.44154e6	0.0150493
$317$	3.65056e8	0.643654	0.321827	−	0.946799i	$-0.395703\pi$
$317$	3.65056e8	0.643654	0.321827	+	0.946799i	$0.395703\pi$
$318$	0	0
$319$	1.40868e9	2.42965
$320$	−1.01972e9	−1.73963
$321$	0	0
$322$	3.61080e8	0.602710
$323$	−1.53411e8	−0.253307
$324$	0	0
$325$	−5.89268e8	−0.952184
$326$	−7.84630e8	−1.25430
$327$	0	0
$328$	−7.99277e8	−1.25066
$329$	8.83733e7	0.136816
$330$	0	0
$331$	8.84610e8	1.34077	0.670384	−	0.742014i	$-0.266129\pi$
$331$	8.84610e8	1.34077	0.670384	+	0.742014i	$0.266129\pi$
$332$	−8.17886e6	−0.0122662
$333$	0	0
$334$	−1.02275e9	−1.50195
$335$	4.00164e8	0.581542
$336$	0	0
$337$	−1.01312e9	−1.44197	−0.720983	−	0.692953i	$-0.756309\pi$
$337$	−1.01312e9	−1.44197	−0.720983	+	0.692953i	$0.756309\pi$
$338$	4.40162e8	0.620018
$339$	0	0
$340$	7.41581e7	0.102325
$341$	5.04278e8	0.688699
$342$	0	0
$343$	7.02701e8	0.940246
$344$	1.01842e9	1.34887
$345$	0	0
$346$	6.31097e8	0.819086
$347$	4.59045e8	0.589796	0.294898	−	0.955529i	$-0.404714\pi$
$347$	4.59045e8	0.589796	0.294898	+	0.955529i	$0.404714\pi$
$348$	0	0
$349$	6.65833e8	0.838448	0.419224	−	0.907883i	$-0.362302\pi$
$349$	6.65833e8	0.838448	0.419224	+	0.907883i	$0.362302\pi$
$350$	6.84705e8	0.853621
$351$	0	0
$352$	−2.83196e8	−0.346089
$353$	7.46486e7	0.0903255	0.0451627	−	0.998980i	$-0.485619\pi$
$353$	7.46486e7	0.0903255	0.0451627	+	0.998980i	$0.485619\pi$
$354$	0	0
$355$	−5.19762e8	−0.616603
$356$	9.33201e7	0.109623
$357$	0	0
$358$	−3.82435e8	−0.440521
$359$	1.55696e9	1.77601	0.888005	−	0.459833i	$-0.152091\pi$
$359$	1.55696e9	1.77601	0.888005	+	0.459833i	$0.152091\pi$
$360$	0	0
$361$	−7.64717e8	−0.855511
$362$	−1.38147e9	−1.53060
$363$	0	0
$364$	−2.86822e7	−0.0311715
$365$	−1.51367e7	−0.0162932
$366$	0	0
$367$	−4.78302e7	−0.0505092	−0.0252546	−	0.999681i	$-0.508040\pi$
$367$	−4.78302e7	−0.0505092	−0.0252546	+	0.999681i	$0.508040\pi$
$368$	−9.75859e8	−1.02075
$369$	0	0
$370$	1.34912e9	1.38467
$371$	−1.58205e8	−0.160846
$372$	0	0
$373$	−1.46088e9	−1.45759	−0.728793	−	0.684735i	$-0.759918\pi$
$373$	−1.46088e9	−1.45759	−0.728793	+	0.684735i	$0.759918\pi$
$374$	−1.17259e9	−1.15903
$375$	0	0
$376$	−2.64157e8	−0.256274
$377$	−8.15928e8	−0.784255
$378$	0	0
$379$	8.99699e8	0.848907	0.424453	−	0.905450i	$-0.360466\pi$
$379$	8.99699e8	0.848907	0.424453	+	0.905450i	$0.360466\pi$
$380$	−6.24329e7	−0.0583674
$381$	0	0
$382$	−9.37680e8	−0.860662
$383$	−7.51806e8	−0.683770	−0.341885	−	0.939742i	$-0.611065\pi$
$383$	−7.51806e8	−0.683770	−0.341885	+	0.939742i	$0.611065\pi$
$384$	0	0
$385$	1.84047e9	1.64367
$386$	−3.59999e8	−0.318600
$387$	0	0
$388$	−5.95619e7	−0.0517675
$389$	−1.98944e9	−1.71359	−0.856794	−	0.515660i	$-0.827547\pi$
$389$	−1.98944e9	−1.71359	−0.856794	+	0.515660i	$0.827547\pi$
$390$	0	0
$391$	8.97349e8	0.759176
$392$	−8.58119e8	−0.719526
$393$	0	0
$394$	1.54850e9	1.27548
$395$	3.13745e8	0.256146
$396$	0	0
$397$	−2.39171e9	−1.91841	−0.959207	−	0.282704i	$-0.908769\pi$
$397$	−2.39171e9	−1.91841	−0.959207	+	0.282704i	$0.908769\pi$
$398$	3.64981e8	0.290188
$399$	0	0
$400$	−1.85049e9	−1.44569
$401$	−1.23670e9	−0.957767	−0.478883	−	0.877879i	$-0.658958\pi$
$401$	−1.23670e9	−0.957767	−0.478883	+	0.877879i	$0.658958\pi$
$402$	0	0
$403$	−2.92086e8	−0.222302
$404$	−2.24485e8	−0.169376
$405$	0	0
$406$	9.48075e8	0.703075
$407$	2.23884e9	1.64605
$408$	0	0
$409$	1.08001e9	0.780545	0.390272	−	0.920699i	$-0.372381\pi$
$409$	1.08001e9	0.780545	0.390272	+	0.920699i	$0.372381\pi$
$410$	−2.57684e9	−1.84648
$411$	0	0
$412$	6.33986e7	0.0446622
$413$	−1.03649e8	−0.0724003
$414$	0	0
$415$	−3.03982e8	−0.208776
$416$	1.64032e8	0.111712
$417$	0	0
$418$	9.87190e8	0.661125
$419$	1.95040e9	1.29531	0.647656	−	0.761933i	$-0.275749\pi$
$419$	1.95040e9	1.29531	0.647656	+	0.761933i	$0.275749\pi$
$420$	0	0
$421$	4.11047e8	0.268475	0.134238	−	0.990949i	$-0.457141\pi$
$421$	4.11047e8	0.268475	0.134238	+	0.990949i	$0.457141\pi$
$422$	1.22082e9	0.790787
$423$	0	0
$424$	4.72891e8	0.301287
$425$	1.70161e9	1.07523
$426$	0	0
$427$	1.52358e9	0.947039
$428$	−2.62985e7	−0.0162135
$429$	0	0
$430$	3.28334e9	1.99148
$431$	2.36621e9	1.42358	0.711792	−	0.702390i	$-0.247884\pi$
$431$	2.36621e9	1.42358	0.711792	+	0.702390i	$0.247884\pi$
$432$	0	0
$433$	−2.22967e8	−0.131987	−0.0659937	−	0.997820i	$-0.521022\pi$
$433$	−2.22967e8	−0.131987	−0.0659937	+	0.997820i	$0.521022\pi$
$434$	3.39391e8	0.199291
$435$	0	0
$436$	1.94244e8	0.112239
$437$	−7.55468e8	−0.433043
$438$	0	0
$439$	−6.34602e8	−0.357994	−0.178997	−	0.983850i	$-0.557285\pi$
$439$	−6.34602e8	−0.357994	−0.178997	+	0.983850i	$0.557285\pi$
$440$	−5.50135e9	−3.07882
$441$	0	0
$442$	6.79182e8	0.374118
$443$	1.44884e9	0.791788	0.395894	−	0.918296i	$-0.370435\pi$
$443$	1.44884e9	0.791788	0.395894	+	0.918296i	$0.370435\pi$
$444$	0	0
$445$	3.46841e9	1.86582
$446$	1.71031e9	0.912857
$447$	0	0
$448$	−1.13890e9	−0.598427
$449$	−3.21927e9	−1.67840	−0.839199	−	0.543825i	$-0.816976\pi$
$449$	−3.21927e9	−1.67840	−0.839199	+	0.543825i	$0.816976\pi$
$450$	0	0
$451$	−4.27621e9	−2.19503
$452$	1.12977e8	0.0575447
$453$	0	0
$454$	−2.65521e8	−0.133169
$455$	−1.06603e9	−0.530553
$456$	0	0
$457$	−1.03131e9	−0.505454	−0.252727	−	0.967538i	$-0.581328\pi$
$457$	−1.03131e9	−0.505454	−0.252727	+	0.967538i	$0.581328\pi$
$458$	−3.00467e7	−0.0146139
$459$	0	0
$460$	3.65190e8	0.174931
$461$	−3.21305e9	−1.52744	−0.763720	−	0.645547i	$-0.776629\pi$
$461$	−3.21305e9	−1.52744	−0.763720	+	0.645547i	$0.776629\pi$
$462$	0	0
$463$	−1.66945e9	−0.781698	−0.390849	−	0.920455i	$-0.627819\pi$
$463$	−1.66945e9	−0.781698	−0.390849	+	0.920455i	$0.627819\pi$
$464$	−2.56228e9	−1.19073
$465$	0	0
$466$	1.85207e9	0.847827
$467$	−3.79498e8	−0.172425	−0.0862125	−	0.996277i	$-0.527476\pi$
$467$	−3.79498e8	−0.172425	−0.0862125	+	0.996277i	$0.527476\pi$
$468$	0	0
$469$	4.46931e8	0.200049
$470$	−8.51632e8	−0.378364
$471$	0	0
$472$	3.09818e8	0.135616
$473$	5.44863e9	2.36741
$474$	0	0
$475$	−1.43257e9	−0.613321
$476$	8.28249e7	0.0351995
$477$	0	0
$478$	−3.51028e9	−1.47009
$479$	−1.60692e9	−0.668068	−0.334034	−	0.942561i	$-0.608410\pi$
$479$	−1.60692e9	−0.668068	−0.334034	+	0.942561i	$0.608410\pi$
$480$	0	0
$481$	−1.29677e9	−0.531319
$482$	−4.44275e9	−1.80712
$483$	0	0
$484$	−5.54990e8	−0.222498
$485$	−2.21373e9	−0.881105
$486$	0	0
$487$	4.45492e9	1.74779	0.873893	−	0.486118i	$-0.161587\pi$
$487$	4.45492e9	1.74779	0.873893	+	0.486118i	$0.161587\pi$
$488$	−4.55414e9	−1.77393
$489$	0	0
$490$	−2.76654e9	−1.06231
$491$	3.60524e9	1.37451	0.687256	−	0.726415i	$-0.258815\pi$
$491$	3.60524e9	1.37451	0.687256	+	0.726415i	$0.258815\pi$
$492$	0	0
$493$	2.35613e9	0.885596
$494$	−5.71796e8	−0.213401
$495$	0	0
$496$	−9.17242e8	−0.337519
$497$	−5.80507e8	−0.212109
$498$	0	0
$499$	−2.51714e9	−0.906893	−0.453446	−	0.891284i	$-0.649806\pi$
$499$	−2.51714e9	−0.906893	−0.453446	+	0.891284i	$0.649806\pi$
$500$	2.63309e8	0.0942044
$501$	0	0
$502$	5.02292e9	1.77212
$503$	−3.07735e9	−1.07817	−0.539087	−	0.842250i	$-0.681230\pi$
$503$	−3.07735e9	−1.07817	−0.539087	+	0.842250i	$0.681230\pi$
$504$	0	0
$505$	−8.34338e9	−2.88285
$506$	−5.77440e9	−1.98143
$507$	0	0
$508$	−9.94346e7	−0.0336523
$509$	5.06424e9	1.70217	0.851083	−	0.525031i	$-0.175946\pi$
$509$	5.06424e9	1.70217	0.851083	+	0.525031i	$0.175946\pi$
$510$	0	0
$511$	−1.69057e7	−0.00560481
$512$	3.34968e9	1.10296
$513$	0	0
$514$	−1.00785e9	−0.327361
$515$	2.35633e9	0.760169
$516$	0	0
$517$	−1.41327e9	−0.449787
$518$	1.50679e9	0.476320
$519$	0	0
$520$	3.18647e9	0.993798
$521$	3.99436e9	1.23741	0.618707	−	0.785622i	$-0.287657\pi$
$521$	3.99436e9	1.23741	0.618707	+	0.785622i	$0.287657\pi$
$522$	0	0
$523$	6.31987e9	1.93175	0.965877	−	0.259001i	$-0.0833931\pi$
$523$	6.31987e9	1.93175	0.965877	+	0.259001i	$0.0833931\pi$
$524$	5.36048e7	0.0162759
$525$	0	0
$526$	−5.39084e9	−1.61513
$527$	8.43448e8	0.251027
$528$	0	0
$529$	1.01415e9	0.297857
$530$	1.52458e9	0.444821
$531$	0	0
$532$	−6.97294e7	−0.0200782
$533$	2.47685e9	0.708523
$534$	0	0
$535$	−9.77432e8	−0.275961
$536$	−1.33592e9	−0.374718
$537$	0	0
$538$	8.44148e8	0.233712
$539$	−4.59102e9	−1.26284
$540$	0	0
$541$	−4.36436e9	−1.18503	−0.592516	−	0.805559i	$-0.701865\pi$
$541$	−4.36436e9	−1.18503	−0.592516	+	0.805559i	$0.701865\pi$
$542$	−4.27892e9	−1.15435
$543$	0	0
$544$	−4.73670e8	−0.126148
$545$	7.21943e9	1.91036
$546$	0	0
$547$	−2.77236e9	−0.724258	−0.362129	−	0.932128i	$-0.617950\pi$
$547$	−2.77236e9	−0.724258	−0.362129	+	0.932128i	$0.617950\pi$
$548$	−9.99201e7	−0.0259371
$549$	0	0
$550$	−1.09498e10	−2.80632
$551$	−1.98360e9	−0.505154
$552$	0	0
$553$	3.50412e8	0.0881133
$554$	1.03841e9	0.259468
$555$	0	0
$556$	6.76946e8	0.167029
$557$	−6.99835e9	−1.71594	−0.857970	−	0.513699i	$-0.828275\pi$
$557$	−6.99835e9	−1.71594	−0.857970	+	0.513699i	$0.828275\pi$
$558$	0	0
$559$	−3.15593e9	−0.764163
$560$	−3.34767e9	−0.805534
$561$	0	0
$562$	−5.78066e9	−1.37373
$563$	−1.13060e9	−0.267012	−0.133506	−	0.991048i	$-0.542624\pi$
$563$	−1.13060e9	−0.267012	−0.133506	+	0.991048i	$0.542624\pi$
$564$	0	0
$565$	4.19899e9	0.979435
$566$	6.65293e9	1.54225
$567$	0	0
$568$	1.73520e9	0.397310
$569$	4.34076e9	0.987810	0.493905	−	0.869516i	$-0.335569\pi$
$569$	4.34076e9	0.987810	0.493905	+	0.869516i	$0.335569\pi$
$570$	0	0
$571$	1.60830e9	0.361526	0.180763	−	0.983527i	$-0.442143\pi$
$571$	1.60830e9	0.361526	0.180763	+	0.983527i	$0.442143\pi$
$572$	4.58686e8	0.102478
$573$	0	0
$574$	−2.87799e9	−0.635182
$575$	8.37956e9	1.83816
$576$	0	0
$577$	4.44207e9	0.962654	0.481327	−	0.876541i	$-0.340155\pi$
$577$	4.44207e9	0.962654	0.481327	+	0.876541i	$0.340155\pi$
$578$	2.45522e9	0.528863
$579$	0	0
$580$	9.58865e8	0.204061
$581$	−3.39509e8	−0.0718181
$582$	0	0
$583$	2.53001e9	0.528789
$584$	5.05330e7	0.0104986
$585$	0	0
$586$	3.65106e9	0.749509
$587$	−5.12698e8	−0.104623	−0.0523116	−	0.998631i	$-0.516659\pi$
$587$	−5.12698e8	−0.104623	−0.0523116	+	0.998631i	$0.516659\pi$
$588$	0	0
$589$	−7.10089e8	−0.143189
$590$	9.98841e8	0.200223
$591$	0	0
$592$	−4.07227e9	−0.806698
$593$	−4.79247e9	−0.943774	−0.471887	−	0.881659i	$-0.656427\pi$
$593$	−4.79247e9	−0.943774	−0.471887	+	0.881659i	$0.656427\pi$
$594$	0	0
$595$	3.07834e9	0.599111
$596$	3.47094e8	0.0671560
$597$	0	0
$598$	3.34462e9	0.639577
$599$	−6.08958e9	−1.15769	−0.578846	−	0.815437i	$-0.696497\pi$
$599$	−6.08958e9	−1.15769	−0.578846	+	0.815437i	$0.696497\pi$
$600$	0	0
$601$	−5.35978e9	−1.00713	−0.503566	−	0.863957i	$-0.667979\pi$
$601$	−5.35978e9	−1.00713	−0.503566	+	0.863957i	$0.667979\pi$
$602$	3.66706e9	0.685063
$603$	0	0
$604$	1.72076e8	0.0317754
$605$	−2.06272e10	−3.78701
$606$	0	0
$607$	9.02137e9	1.63724	0.818619	−	0.574337i	$-0.194740\pi$
$607$	9.02137e9	1.63724	0.818619	+	0.574337i	$0.194740\pi$
$608$	3.98777e8	0.0719561
$609$	0	0
$610$	−1.46824e10	−2.61904
$611$	8.18586e8	0.145184
$612$	0	0
$613$	7.72450e9	1.35444	0.677219	−	0.735782i	$-0.263185\pi$
$613$	7.72450e9	1.35444	0.677219	+	0.735782i	$0.263185\pi$
$614$	1.03628e9	0.180671
$615$	0	0
$616$	−6.14429e9	−1.05911
$617$	4.39185e9	0.752748	0.376374	−	0.926468i	$-0.377171\pi$
$617$	4.39185e9	0.752748	0.376374	+	0.926468i	$0.377171\pi$
$618$	0	0
$619$	3.33323e9	0.564870	0.282435	−	0.959286i	$-0.408858\pi$
$619$	3.33323e9	0.564870	0.282435	+	0.959286i	$0.408858\pi$
$620$	3.43254e8	0.0578422
$621$	0	0
$622$	−8.51565e9	−1.41890
$623$	3.87376e9	0.641837
$624$	0	0
$625$	−6.16859e7	−0.0101066
$626$	9.50998e8	0.154942
$627$	0	0
$628$	−7.34327e8	−0.118313
$629$	3.74465e9	0.599976
$630$	0	0
$631$	−7.87439e9	−1.24771	−0.623856	−	0.781540i	$-0.714435\pi$
$631$	−7.87439e9	−1.24771	−0.623856	+	0.781540i	$0.714435\pi$
$632$	−1.04742e9	−0.165048
$633$	0	0
$634$	−3.92910e9	−0.612323
$635$	−3.69567e9	−0.572776
$636$	0	0
$637$	2.65919e9	0.407626
$638$	−1.51616e10	−2.31139
$639$	0	0
$640$	8.94577e9	1.34892
$641$	−2.66936e8	−0.0400316	−0.0200158	−	0.999800i	$-0.506372\pi$
$641$	−2.66936e8	−0.0400316	−0.0200158	+	0.999800i	$0.506372\pi$
$642$	0	0
$643$	−9.18242e9	−1.36213	−0.681065	−	0.732223i	$-0.738483\pi$
$643$	−9.18242e9	−1.36213	−0.681065	+	0.732223i	$0.738483\pi$
$644$	4.07870e8	0.0601757
$645$	0	0
$646$	1.65116e9	0.240977
$647$	−7.79548e9	−1.13156	−0.565781	−	0.824556i	$-0.691425\pi$
$647$	−7.79548e9	−1.13156	−0.565781	+	0.824556i	$0.691425\pi$
$648$	0	0
$649$	1.65756e9	0.238019
$650$	6.34229e9	0.905836
$651$	0	0
$652$	−8.86303e8	−0.125232
$653$	7.64228e8	0.107406	0.0537028	−	0.998557i	$-0.482898\pi$
$653$	7.64228e8	0.107406	0.0537028	+	0.998557i	$0.482898\pi$
$654$	0	0
$655$	1.99232e9	0.277022
$656$	7.77809e9	1.07575
$657$	0	0
$658$	−9.51162e8	−0.130156
$659$	4.42497e9	0.602299	0.301149	−	0.953577i	$-0.402630\pi$
$659$	4.42497e9	0.602299	0.301149	+	0.953577i	$0.402630\pi$
$660$	0	0
$661$	−1.21787e9	−0.164019	−0.0820097	−	0.996632i	$-0.526134\pi$
$661$	−1.21787e9	−0.164019	−0.0820097	+	0.996632i	$0.526134\pi$
$662$	−9.52106e9	−1.27551
$663$	0	0
$664$	1.01483e9	0.134525
$665$	−2.59162e9	−0.341740
$666$	0	0
$667$	1.16027e10	1.51398
$668$	−1.15528e9	−0.149958
$669$	0	0
$670$	−4.30697e9	−0.553235
$671$	−2.43651e10	−3.11343
$672$	0	0
$673$	−6.63758e9	−0.839377	−0.419689	−	0.907668i	$-0.637861\pi$
$673$	−6.63758e9	−0.839377	−0.419689	+	0.907668i	$0.637861\pi$
$674$	1.09042e10	1.37178
$675$	0	0
$676$	4.97199e8	0.0619037
$677$	−1.51749e9	−0.187961	−0.0939803	−	0.995574i	$-0.529959\pi$
$677$	−1.51749e9	−0.187961	−0.0939803	+	0.995574i	$0.529959\pi$
$678$	0	0
$679$	−2.47244e9	−0.303098
$680$	−9.20148e9	−1.12222
$681$	0	0
$682$	−5.42755e9	−0.655176
$683$	1.25832e10	1.51119	0.755595	−	0.655039i	$-0.227348\pi$
$683$	1.25832e10	1.51119	0.755595	+	0.655039i	$0.227348\pi$
$684$	0	0
$685$	−3.71371e9	−0.441460
$686$	−7.56318e9	−0.894479
$687$	0	0
$688$	−9.91063e9	−1.16022
$689$	−1.46542e9	−0.170685
$690$	0	0
$691$	−5.99235e9	−0.690914	−0.345457	−	0.938435i	$-0.612276\pi$
$691$	−5.99235e9	−0.690914	−0.345457	+	0.938435i	$0.612276\pi$
$692$	7.12876e8	0.0817791
$693$	0	0
$694$	−4.94070e9	−0.561088
$695$	2.51599e10	2.84291
$696$	0	0
$697$	−7.15232e9	−0.800078
$698$	−7.16636e9	−0.797636
$699$	0	0
$700$	7.73430e8	0.0852271
$701$	1.16633e10	1.27881	0.639406	−	0.768869i	$-0.279180\pi$
$701$	1.16633e10	1.27881	0.639406	+	0.768869i	$0.279180\pi$
$702$	0	0
$703$	−3.15258e9	−0.342233
$704$	1.82132e10	1.96736
$705$	0	0
$706$	−8.03443e8	−0.0859288
$707$	−9.31847e9	−0.991691
$708$	0	0
$709$	−5.32106e9	−0.560707	−0.280353	−	0.959897i	$-0.590452\pi$
$709$	−5.32106e9	−0.560707	−0.280353	+	0.959897i	$0.590452\pi$
$710$	5.59420e9	0.586589
$711$	0	0
$712$	−1.15791e10	−1.20225
$713$	4.15354e9	0.429146
$714$	0	0
$715$	1.70479e10	1.74422
$716$	−4.31992e8	−0.0439825
$717$	0	0
$718$	−1.67575e10	−1.68956
$719$	1.68037e9	0.168599	0.0842993	−	0.996440i	$-0.473135\pi$
$719$	1.68037e9	0.168599	0.0842993	+	0.996440i	$0.473135\pi$
$720$	0	0
$721$	2.63171e9	0.261496
$722$	8.23065e9	0.813868
$723$	0	0
$724$	−1.56049e9	−0.152818
$725$	2.20019e10	2.14426
$726$	0	0
$727$	6.43210e9	0.620844	0.310422	−	0.950599i	$-0.399530\pi$
$727$	6.43210e9	0.620844	0.310422	+	0.950599i	$0.399530\pi$
$728$	3.55887e9	0.341863
$729$	0	0
$730$	1.62916e8	0.0155001
$731$	9.11330e9	0.862908
$732$	0	0
$733$	9.99078e9	0.936991	0.468495	−	0.883466i	$-0.344796\pi$
$733$	9.99078e9	0.936991	0.468495	+	0.883466i	$0.344796\pi$
$734$	5.14796e8	0.0480507
$735$	0	0
$736$	−2.33258e9	−0.215657
$737$	−7.14732e9	−0.657669
$738$	0	0
$739$	5.40356e9	0.492521	0.246260	−	0.969204i	$-0.420798\pi$
$739$	5.40356e9	0.492521	0.246260	+	0.969204i	$0.420798\pi$
$740$	1.52394e9	0.138248
$741$	0	0
$742$	1.70276e9	0.153017
$743$	1.37163e10	1.22680	0.613402	−	0.789771i	$-0.289801\pi$
$743$	1.37163e10	1.22680	0.613402	+	0.789771i	$0.289801\pi$
$744$	0	0
$745$	1.29004e10	1.14302
$746$	1.57235e10	1.38664
$747$	0	0
$748$	−1.32454e9	−0.115720
$749$	−1.09166e9	−0.0949299
$750$	0	0
$751$	1.22637e9	0.105653	0.0528264	−	0.998604i	$-0.483177\pi$
$751$	1.22637e9	0.105653	0.0528264	+	0.998604i	$0.483177\pi$
$752$	2.57062e9	0.220432
$753$	0	0
$754$	8.78184e9	0.746081
$755$	6.39552e9	0.540831
$756$	0	0
$757$	−1.17911e10	−0.987912	−0.493956	−	0.869487i	$-0.664450\pi$
$757$	−1.17911e10	−0.987912	−0.493956	+	0.869487i	$0.664450\pi$
$758$	−9.68347e9	−0.807586
$759$	0	0
$760$	7.74662e9	0.640125
$761$	1.03626e10	0.852359	0.426179	−	0.904639i	$-0.359859\pi$
$761$	1.03626e10	0.852359	0.426179	+	0.904639i	$0.359859\pi$
$762$	0	0
$763$	8.06317e9	0.657158
$764$	−1.05919e9	−0.0859301
$765$	0	0
$766$	8.09169e9	0.650487
$767$	−9.60082e8	−0.0768289
$768$	0	0
$769$	7.77986e9	0.616921	0.308460	−	0.951237i	$-0.400186\pi$
$769$	7.77986e9	0.616921	0.308460	+	0.951237i	$0.400186\pi$
$770$	−1.98090e10	−1.56367
$771$	0	0
$772$	−4.06648e8	−0.0318096
$773$	−8.05208e9	−0.627018	−0.313509	−	0.949585i	$-0.601505\pi$
$773$	−8.05208e9	−0.627018	−0.313509	+	0.949585i	$0.601505\pi$
$774$	0	0
$775$	7.87623e9	0.607802
$776$	7.39040e9	0.567743
$777$	0	0
$778$	2.14123e10	1.63018
$779$	6.02146e9	0.456374
$780$	0	0
$781$	9.28346e9	0.697319
$782$	−9.65817e9	−0.722223
$783$	0	0
$784$	8.35071e9	0.618895
$785$	−2.72926e10	−2.01373
$786$	0	0
$787$	2.48655e10	1.81838	0.909192	−	0.416377i	$-0.136700\pi$
$787$	2.48655e10	1.81838	0.909192	+	0.416377i	$0.136700\pi$
$788$	1.74916e9	0.127347
$789$	0	0
$790$	−3.37684e9	−0.243678
$791$	4.68973e9	0.336923
$792$	0	0
$793$	1.41126e10	1.00497
$794$	2.57420e10	1.82503
$795$	0	0
$796$	4.12276e8	0.0289729
$797$	1.33668e9	0.0935241	0.0467621	−	0.998906i	$-0.485110\pi$
$797$	1.33668e9	0.0935241	0.0467621	+	0.998906i	$0.485110\pi$
$798$	0	0
$799$	−2.36381e9	−0.163945
$800$	−4.42319e9	−0.305436
$801$	0	0
$802$	1.33106e10	0.911147
$803$	2.70356e8	0.0184261
$804$	0	0
$805$	1.51592e10	1.02422
$806$	3.14372e9	0.211481
$807$	0	0
$808$	2.78539e10	1.85757
$809$	5.48115e9	0.363959	0.181979	−	0.983302i	$-0.441750\pi$
$809$	5.48115e9	0.363959	0.181979	+	0.983302i	$0.441750\pi$
$810$	0	0
$811$	1.52888e10	1.00647	0.503233	−	0.864151i	$-0.332144\pi$
$811$	1.52888e10	1.00647	0.503233	+	0.864151i	$0.332144\pi$
$812$	1.07093e9	0.0701963
$813$	0	0
$814$	−2.40966e10	−1.56592
$815$	−3.29411e10	−2.13150
$816$	0	0
$817$	−7.67238e9	−0.492213
$818$	−1.16242e10	−0.742551
$819$	0	0
$820$	−2.91075e9	−0.184356
$821$	−5.00980e9	−0.315951	−0.157975	−	0.987443i	$-0.550497\pi$
$821$	−5.00980e9	−0.315951	−0.157975	+	0.987443i	$0.550497\pi$
$822$	0	0
$823$	−1.43300e10	−0.896077	−0.448038	−	0.894014i	$-0.647877\pi$
$823$	−1.43300e10	−0.896077	−0.448038	+	0.894014i	$0.647877\pi$
$824$	−7.86646e9	−0.489817
$825$	0	0
$826$	1.11558e9	0.0688761
$827$	−2.07445e10	−1.27536	−0.637682	−	0.770300i	$-0.720107\pi$
$827$	−2.07445e10	−1.27536	−0.637682	+	0.770300i	$0.720107\pi$
$828$	0	0
$829$	−3.94434e9	−0.240455	−0.120227	−	0.992746i	$-0.538362\pi$
$829$	−3.94434e9	−0.240455	−0.120227	+	0.992746i	$0.538362\pi$
$830$	3.27176e9	0.198613
$831$	0	0
$832$	−1.05494e10	−0.635033
$833$	−7.67887e9	−0.460299
$834$	0	0
$835$	−4.29380e10	−2.55234
$836$	1.11511e9	0.0660080
$837$	0	0
$838$	−2.09922e10	−1.23226
$839$	1.52467e10	0.891268	0.445634	−	0.895215i	$-0.352978\pi$
$839$	1.52467e10	0.891268	0.445634	+	0.895215i	$0.352978\pi$
$840$	0	0
$841$	1.32150e10	0.766091
$842$	−4.42410e9	−0.255407
$843$	0	0
$844$	1.37902e9	0.0789537
$845$	1.84793e10	1.05363
$846$	0	0
$847$	−2.30379e10	−1.30272
$848$	−4.60189e9	−0.259150
$849$	0	0
$850$	−1.83145e10	−1.02289
$851$	1.84404e10	1.02569
$852$	0	0
$853$	−2.14258e10	−1.18200	−0.590998	−	0.806673i	$-0.701266\pi$
$853$	−2.14258e10	−1.18200	−0.590998	+	0.806673i	$0.701266\pi$
$854$	−1.63983e10	−0.900941
$855$	0	0
$856$	3.26310e9	0.177817
$857$	−2.49225e10	−1.35257	−0.676283	−	0.736642i	$-0.736410\pi$
$857$	−2.49225e10	−1.35257	−0.676283	+	0.736642i	$0.736410\pi$
$858$	0	0
$859$	−8.80692e9	−0.474076	−0.237038	−	0.971500i	$-0.576177\pi$
$859$	−8.80692e9	−0.474076	−0.237038	+	0.971500i	$0.576177\pi$
$860$	3.70880e9	0.198833
$861$	0	0
$862$	−2.54675e10	−1.35429
$863$	−5.86301e9	−0.310515	−0.155257	−	0.987874i	$-0.549621\pi$
$863$	−5.86301e9	−0.310515	−0.155257	+	0.987874i	$0.549621\pi$
$864$	0	0
$865$	2.64953e10	1.39192
$866$	2.39979e9	0.125563
$867$	0	0
$868$	3.83370e8	0.0198975
$869$	−5.60380e9	−0.289676
$870$	0	0
$871$	4.13984e9	0.212285
$872$	−2.41017e10	−1.23095
$873$	0	0
$874$	8.13111e9	0.411964
$875$	1.09301e10	0.551564
$876$	0	0
$877$	3.56534e10	1.78485	0.892427	−	0.451193i	$-0.149001\pi$
$877$	3.56534e10	1.78485	0.892427	+	0.451193i	$0.149001\pi$
$878$	6.83022e9	0.340568
$879$	0	0
$880$	5.35359e10	2.64823
$881$	−3.92292e10	−1.93283	−0.966414	−	0.256989i	$-0.917269\pi$
$881$	−3.92292e10	−1.93283	−0.966414	+	0.256989i	$0.917269\pi$
$882$	0	0
$883$	2.18077e10	1.06598	0.532988	−	0.846123i	$-0.321069\pi$
$883$	2.18077e10	1.06598	0.532988	+	0.846123i	$0.321069\pi$
$884$	7.67192e8	0.0373526
$885$	0	0
$886$	−1.55939e10	−0.753247
$887$	−4.76501e9	−0.229262	−0.114631	−	0.993408i	$-0.536569\pi$
$887$	−4.76501e9	−0.229262	−0.114631	+	0.993408i	$0.536569\pi$
$888$	0	0
$889$	−4.12758e9	−0.197033
$890$	−3.73305e10	−1.77500
$891$	0	0
$892$	1.93194e9	0.0911414
$893$	1.99006e9	0.0935162
$894$	0	0
$895$	−1.60558e10	−0.748601
$896$	9.99126e9	0.464026
$897$	0	0
$898$	3.46490e10	1.59670
$899$	1.09058e10	0.500609
$900$	0	0
$901$	4.23166e9	0.192741
$902$	4.60249e10	2.08819
$903$	0	0
$904$	−1.40181e10	−0.631102
$905$	−5.79984e10	−2.60103
$906$	0	0
$907$	−8.85912e9	−0.394244	−0.197122	−	0.980379i	$-0.563159\pi$
$907$	−8.85912e9	−0.394244	−0.197122	+	0.980379i	$0.563159\pi$
$908$	−2.99928e8	−0.0132959
$909$	0	0
$910$	1.14737e10	0.504728
$911$	5.66060e9	0.248055	0.124028	−	0.992279i	$-0.460419\pi$
$911$	5.66060e9	0.248055	0.124028	+	0.992279i	$0.460419\pi$
$912$	0	0
$913$	5.42942e9	0.236105
$914$	1.11000e10	0.480851
$915$	0	0
$916$	−3.39402e7	−0.00145908
$917$	2.22516e9	0.0952947
$918$	0	0
$919$	3.15687e10	1.34169	0.670846	−	0.741597i	$-0.265931\pi$
$919$	3.15687e10	1.34169	0.670846	+	0.741597i	$0.265931\pi$
$920$	−4.53125e10	−1.91849
$921$	0	0
$922$	3.45821e10	1.45309
$923$	−5.37713e9	−0.225084
$924$	0	0
$925$	3.49680e10	1.45270
$926$	1.79683e10	0.743649
$927$	0	0
$928$	−6.12456e9	−0.251569
$929$	−2.88328e10	−1.17986	−0.589932	−	0.807453i	$-0.700845\pi$
$929$	−2.88328e10	−1.17986	−0.589932	+	0.807453i	$0.700845\pi$
$930$	0	0
$931$	6.46476e9	0.262560
$932$	2.09207e9	0.0846487
$933$	0	0
$934$	4.08454e9	0.164032
$935$	−4.92288e10	−1.96960
$936$	0	0
$937$	−4.19782e10	−1.66700	−0.833500	−	0.552520i	$-0.813666\pi$
$937$	−4.19782e10	−1.66700	−0.833500	+	0.552520i	$0.813666\pi$
$938$	−4.81032e9	−0.190311
$939$	0	0
$940$	−9.61988e8	−0.0377766
$941$	4.83827e10	1.89289	0.946447	−	0.322860i	$-0.104644\pi$
$941$	4.83827e10	1.89289	0.946447	+	0.322860i	$0.104644\pi$
$942$	0	0
$943$	−3.52215e10	−1.36778
$944$	−3.01496e9	−0.116649
$945$	0	0
$946$	−5.86436e10	−2.25217
$947$	7.76682e9	0.297179	0.148590	−	0.988899i	$-0.452527\pi$
$947$	7.76682e9	0.297179	0.148590	+	0.988899i	$0.452527\pi$
$948$	0	0
$949$	−1.56595e8	−0.00594765
$950$	1.54187e10	0.583467
$951$	0	0
$952$	−1.02769e10	−0.386039
$953$	4.85606e9	0.181743	0.0908717	−	0.995863i	$-0.471035\pi$
$953$	4.85606e9	0.181743	0.0908717	+	0.995863i	$0.471035\pi$
$954$	0	0
$955$	−3.93666e10	−1.46257
$956$	−3.96515e9	−0.146777
$957$	0	0
$958$	1.72953e10	0.635549
$959$	−4.14773e9	−0.151861
$960$	0	0
$961$	−2.36086e10	−0.858100
$962$	1.39571e10	0.505457
$963$	0	0
$964$	−5.01845e9	−0.180426
$965$	−1.51138e10	−0.541413
$966$	0	0
$967$	−2.84618e10	−1.01221	−0.506103	−	0.862473i	$-0.668915\pi$
$967$	−2.84618e10	−1.01221	−0.506103	+	0.862473i	$0.668915\pi$
$968$	6.88628e10	2.44017
$969$	0	0
$970$	2.38264e10	0.838217
$971$	3.54110e10	1.24128	0.620642	−	0.784094i	$-0.286872\pi$
$971$	3.54110e10	1.24128	0.620642	+	0.784094i	$0.286872\pi$
$972$	0	0
$973$	2.81004e10	0.977952
$974$	−4.79483e10	−1.66271
$975$	0	0
$976$	4.43182e10	1.52584
$977$	−4.07835e10	−1.39911	−0.699557	−	0.714577i	$-0.746619\pi$
$977$	−4.07835e10	−1.39911	−0.699557	+	0.714577i	$0.746619\pi$
$978$	0	0
$979$	−6.19492e10	−2.11007
$980$	−3.12504e9	−0.106063
$981$	0	0
$982$	−3.88032e10	−1.30761
$983$	4.57225e10	1.53530	0.767649	−	0.640871i	$-0.221427\pi$
$983$	4.57225e10	1.53530	0.767649	+	0.640871i	$0.221427\pi$
$984$	0	0
$985$	6.50107e10	2.16749
$986$	−2.53591e10	−0.842489
$987$	0	0
$988$	−6.45891e8	−0.0213064
$989$	4.48782e10	1.47519
$990$	0	0
$991$	7.18635e9	0.234558	0.117279	−	0.993099i	$-0.462583\pi$
$991$	7.18635e9	0.234558	0.117279	+	0.993099i	$0.462583\pi$
$992$	−2.19247e9	−0.0713086
$993$	0	0
$994$	6.24800e9	0.201785
$995$	1.53230e10	0.493131
$996$	0	0
$997$	5.76916e10	1.84365	0.921827	−	0.387602i	$-0.126696\pi$
$997$	5.76916e10	1.84365	0.921827	+	0.387602i	$0.126696\pi$
$998$	2.70920e10	0.862749
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.d.1.5		17
3.2	odd	2		177.8.a.b.1.13	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.b.1.13	✓	17	3.2	odd	2
531.8.a.d.1.5		17	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-10.7630 q^{2} -12.1577 q^{4} -451.863 q^{5} -504.672 q^{7} +1508.52 q^{8} +O(q^{10})\)	\(q-10.7630 q^{2} -12.1577 q^{4} -451.863 q^{5} -504.672 q^{7} +1508.52 q^{8} +4863.40 q^{10} +8070.72 q^{11} -4674.69 q^{13} +5431.79 q^{14} -14680.0 q^{16} +13499.0 q^{17} -11364.6 q^{19} +5493.61 q^{20} -86865.2 q^{22} +66475.4 q^{23} +126055. q^{25} +50313.7 q^{26} +6135.65 q^{28} +174542. q^{29} +62482.4 q^{31} -35089.3 q^{32} -145289. q^{34} +228043. q^{35} +277403. q^{37} +122318. q^{38} -681643. q^{40} -529842. q^{41} +675111. q^{43} -98121.3 q^{44} -715475. q^{46} -175110. q^{47} -568849. q^{49} -1.35673e6 q^{50} +56833.4 q^{52} +313480. q^{53} -3.64686e6 q^{55} -761307. q^{56} -1.87860e6 q^{58} +205379. q^{59} -3.01895e6 q^{61} -672499. q^{62} +2.25671e6 q^{64} +2.11232e6 q^{65} -885587. q^{67} -164116. q^{68} -2.45442e6 q^{70} +1.15026e6 q^{71} +33498.4 q^{73} -2.98569e6 q^{74} +138168. q^{76} -4.07307e6 q^{77} -694337. q^{79} +6.63335e6 q^{80} +5.70270e6 q^{82} +672731. q^{83} -6.09968e6 q^{85} -7.26622e6 q^{86} +1.21748e7 q^{88} -7.67580e6 q^{89} +2.35918e6 q^{91} -808188. q^{92} +1.88471e6 q^{94} +5.13525e6 q^{95} +4.89911e6 q^{97} +6.12253e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \)	\( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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