Properties

Label 531.8.a.d.1.16
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 1.16
Root \(-19.6388\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+21.6388 q^{2} +340.238 q^{4} +399.107 q^{5} -1780.57 q^{7} +4592.58 q^{8} +O(q^{10})\) \(q+21.6388 q^{2} +340.238 q^{4} +399.107 q^{5} -1780.57 q^{7} +4592.58 q^{8} +8636.21 q^{10} +204.067 q^{11} +7466.22 q^{13} -38529.4 q^{14} +55827.6 q^{16} -2070.83 q^{17} +38611.6 q^{19} +135792. q^{20} +4415.78 q^{22} +47604.4 q^{23} +81161.6 q^{25} +161560. q^{26} -605818. q^{28} -223646. q^{29} +273826. q^{31} +620192. q^{32} -44810.3 q^{34} -710639. q^{35} -545039. q^{37} +835510. q^{38} +1.83293e6 q^{40} +521804. q^{41} +327902. q^{43} +69431.5 q^{44} +1.03010e6 q^{46} +24173.5 q^{47} +2.34689e6 q^{49} +1.75624e6 q^{50} +2.54029e6 q^{52} +1.06061e6 q^{53} +81444.8 q^{55} -8.17742e6 q^{56} -4.83943e6 q^{58} +205379. q^{59} +1.08933e6 q^{61} +5.92527e6 q^{62} +6.27428e6 q^{64} +2.97982e6 q^{65} -55751.2 q^{67} -704575. q^{68} -1.53774e7 q^{70} +851859. q^{71} +3.44512e6 q^{73} -1.17940e7 q^{74} +1.31372e7 q^{76} -363356. q^{77} -8.05252e6 q^{79} +2.22812e7 q^{80} +1.12912e7 q^{82} -2.05020e6 q^{83} -826483. q^{85} +7.09542e6 q^{86} +937197. q^{88} +1.68320e6 q^{89} -1.32941e7 q^{91} +1.61969e7 q^{92} +523086. q^{94} +1.54102e7 q^{95} +1.14133e7 q^{97} +5.07839e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.6388 1.91262 0.956309 0.292356i \(-0.0944393\pi\)
0.956309 + 0.292356i \(0.0944393\pi\)
\(3\) 0 0
\(4\) 340.238 2.65811
\(5\) 399.107 1.42789 0.713945 0.700202i \(-0.246907\pi\)
0.713945 + 0.700202i \(0.246907\pi\)
\(6\) 0 0
\(7\) −1780.57 −1.96208 −0.981039 0.193812i \(-0.937915\pi\)
−0.981039 + 0.193812i \(0.937915\pi\)
\(8\) 4592.58 3.17133
\(9\) 0 0
\(10\) 8636.21 2.73101
\(11\) 204.067 0.0462274 0.0231137 0.999733i \(-0.492642\pi\)
0.0231137 + 0.999733i \(0.492642\pi\)
\(12\) 0 0
\(13\) 7466.22 0.942539 0.471269 0.881989i \(-0.343796\pi\)
0.471269 + 0.881989i \(0.343796\pi\)
\(14\) −38529.4 −3.75271
\(15\) 0 0
\(16\) 55827.6 3.40744
\(17\) −2070.83 −0.102229 −0.0511144 0.998693i \(-0.516277\pi\)
−0.0511144 + 0.998693i \(0.516277\pi\)
\(18\) 0 0
\(19\) 38611.6 1.29146 0.645729 0.763566i \(-0.276553\pi\)
0.645729 + 0.763566i \(0.276553\pi\)
\(20\) 135792. 3.79549
\(21\) 0 0
\(22\) 4415.78 0.0884154
\(23\) 47604.4 0.815831 0.407915 0.913020i \(-0.366256\pi\)
0.407915 + 0.913020i \(0.366256\pi\)
\(24\) 0 0
\(25\) 81161.6 1.03887
\(26\) 161560. 1.80272
\(27\) 0 0
\(28\) −605818. −5.21542
\(29\) −223646. −1.70282 −0.851408 0.524505i \(-0.824251\pi\)
−0.851408 + 0.524505i \(0.824251\pi\)
\(30\) 0 0
\(31\) 273826. 1.65085 0.825427 0.564508i \(-0.190934\pi\)
0.825427 + 0.564508i \(0.190934\pi\)
\(32\) 620192. 3.34581
\(33\) 0 0
\(34\) −44810.3 −0.195525
\(35\) −710639. −2.80163
\(36\) 0 0
\(37\) −545039. −1.76897 −0.884487 0.466565i \(-0.845491\pi\)
−0.884487 + 0.466565i \(0.845491\pi\)
\(38\) 835510. 2.47007
\(39\) 0 0
\(40\) 1.83293e6 4.52832
\(41\) 521804. 1.18240 0.591200 0.806525i \(-0.298654\pi\)
0.591200 + 0.806525i \(0.298654\pi\)
\(42\) 0 0
\(43\) 327902. 0.628934 0.314467 0.949268i \(-0.398174\pi\)
0.314467 + 0.949268i \(0.398174\pi\)
\(44\) 69431.5 0.122878
\(45\) 0 0
\(46\) 1.03010e6 1.56037
\(47\) 24173.5 0.0339623 0.0169811 0.999856i \(-0.494594\pi\)
0.0169811 + 0.999856i \(0.494594\pi\)
\(48\) 0 0
\(49\) 2.34689e6 2.84975
\(50\) 1.75624e6 1.98696
\(51\) 0 0
\(52\) 2.54029e6 2.50537
\(53\) 1.06061e6 0.978570 0.489285 0.872124i \(-0.337258\pi\)
0.489285 + 0.872124i \(0.337258\pi\)
\(54\) 0 0
\(55\) 81444.8 0.0660076
\(56\) −8.17742e6 −6.22240
\(57\) 0 0
\(58\) −4.83943e6 −3.25684
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 1.08933e6 0.614475 0.307238 0.951633i \(-0.400595\pi\)
0.307238 + 0.951633i \(0.400595\pi\)
\(62\) 5.92527e6 3.15746
\(63\) 0 0
\(64\) 6.27428e6 2.99181
\(65\) 2.97982e6 1.34584
\(66\) 0 0
\(67\) −55751.2 −0.0226461 −0.0113230 0.999936i \(-0.503604\pi\)
−0.0113230 + 0.999936i \(0.503604\pi\)
\(68\) −704575. −0.271735
\(69\) 0 0
\(70\) −1.53774e7 −5.35845
\(71\) 851859. 0.282464 0.141232 0.989976i \(-0.454894\pi\)
0.141232 + 0.989976i \(0.454894\pi\)
\(72\) 0 0
\(73\) 3.44512e6 1.03651 0.518256 0.855225i \(-0.326581\pi\)
0.518256 + 0.855225i \(0.326581\pi\)
\(74\) −1.17940e7 −3.38337
\(75\) 0 0
\(76\) 1.31372e7 3.43284
\(77\) −363356. −0.0907017
\(78\) 0 0
\(79\) −8.05252e6 −1.83754 −0.918771 0.394791i \(-0.870817\pi\)
−0.918771 + 0.394791i \(0.870817\pi\)
\(80\) 2.22812e7 4.86545
\(81\) 0 0
\(82\) 1.12912e7 2.26148
\(83\) −2.05020e6 −0.393571 −0.196785 0.980447i \(-0.563050\pi\)
−0.196785 + 0.980447i \(0.563050\pi\)
\(84\) 0 0
\(85\) −826483. −0.145971
\(86\) 7.09542e6 1.20291
\(87\) 0 0
\(88\) 937197. 0.146602
\(89\) 1.68320e6 0.253087 0.126544 0.991961i \(-0.459612\pi\)
0.126544 + 0.991961i \(0.459612\pi\)
\(90\) 0 0
\(91\) −1.32941e7 −1.84933
\(92\) 1.61969e7 2.16857
\(93\) 0 0
\(94\) 523086. 0.0649569
\(95\) 1.54102e7 1.84406
\(96\) 0 0
\(97\) 1.14133e7 1.26972 0.634861 0.772627i \(-0.281058\pi\)
0.634861 + 0.772627i \(0.281058\pi\)
\(98\) 5.07839e7 5.45048
\(99\) 0 0
\(100\) 2.76143e7 2.76143
\(101\) −7.19660e6 −0.695029 −0.347514 0.937675i \(-0.612974\pi\)
−0.347514 + 0.937675i \(0.612974\pi\)
\(102\) 0 0
\(103\) −2.77089e6 −0.249856 −0.124928 0.992166i \(-0.539870\pi\)
−0.124928 + 0.992166i \(0.539870\pi\)
\(104\) 3.42893e7 2.98911
\(105\) 0 0
\(106\) 2.29504e7 1.87163
\(107\) 937525. 0.0739843 0.0369922 0.999316i \(-0.488222\pi\)
0.0369922 + 0.999316i \(0.488222\pi\)
\(108\) 0 0
\(109\) −1.52436e7 −1.12744 −0.563722 0.825965i \(-0.690631\pi\)
−0.563722 + 0.825965i \(0.690631\pi\)
\(110\) 1.76237e6 0.126247
\(111\) 0 0
\(112\) −9.94049e7 −6.68567
\(113\) 1.00374e7 0.654402 0.327201 0.944955i \(-0.393895\pi\)
0.327201 + 0.944955i \(0.393895\pi\)
\(114\) 0 0
\(115\) 1.89993e7 1.16492
\(116\) −7.60928e7 −4.52627
\(117\) 0 0
\(118\) 4.44416e6 0.249002
\(119\) 3.68726e6 0.200581
\(120\) 0 0
\(121\) −1.94455e7 −0.997863
\(122\) 2.35718e7 1.17526
\(123\) 0 0
\(124\) 9.31661e7 4.38816
\(125\) 1.21192e6 0.0554993
\(126\) 0 0
\(127\) 9.06146e6 0.392541 0.196270 0.980550i \(-0.437117\pi\)
0.196270 + 0.980550i \(0.437117\pi\)
\(128\) 5.63835e7 2.37639
\(129\) 0 0
\(130\) 6.44799e7 2.57408
\(131\) −2.09570e7 −0.814478 −0.407239 0.913322i \(-0.633508\pi\)
−0.407239 + 0.913322i \(0.633508\pi\)
\(132\) 0 0
\(133\) −6.87507e7 −2.53394
\(134\) −1.20639e6 −0.0433133
\(135\) 0 0
\(136\) −9.51046e6 −0.324202
\(137\) −2.19828e7 −0.730399 −0.365199 0.930929i \(-0.618999\pi\)
−0.365199 + 0.930929i \(0.618999\pi\)
\(138\) 0 0
\(139\) −6.44663e6 −0.203601 −0.101801 0.994805i \(-0.532460\pi\)
−0.101801 + 0.994805i \(0.532460\pi\)
\(140\) −2.41786e8 −7.44704
\(141\) 0 0
\(142\) 1.84332e7 0.540247
\(143\) 1.52361e6 0.0435711
\(144\) 0 0
\(145\) −8.92586e7 −2.43143
\(146\) 7.45484e7 1.98245
\(147\) 0 0
\(148\) −1.85443e8 −4.70213
\(149\) 3.38724e7 0.838869 0.419434 0.907786i \(-0.362228\pi\)
0.419434 + 0.907786i \(0.362228\pi\)
\(150\) 0 0
\(151\) −6.27255e7 −1.48260 −0.741301 0.671172i \(-0.765791\pi\)
−0.741301 + 0.671172i \(0.765791\pi\)
\(152\) 1.77327e8 4.09565
\(153\) 0 0
\(154\) −7.86260e6 −0.173478
\(155\) 1.09286e8 2.35724
\(156\) 0 0
\(157\) 4.72239e7 0.973898 0.486949 0.873430i \(-0.338110\pi\)
0.486949 + 0.873430i \(0.338110\pi\)
\(158\) −1.74247e8 −3.51452
\(159\) 0 0
\(160\) 2.47523e8 4.77744
\(161\) −8.47631e7 −1.60072
\(162\) 0 0
\(163\) −2.85412e7 −0.516198 −0.258099 0.966119i \(-0.583096\pi\)
−0.258099 + 0.966119i \(0.583096\pi\)
\(164\) 1.77538e8 3.14295
\(165\) 0 0
\(166\) −4.43639e7 −0.752751
\(167\) 6.05397e7 1.00585 0.502924 0.864331i \(-0.332258\pi\)
0.502924 + 0.864331i \(0.332258\pi\)
\(168\) 0 0
\(169\) −7.00401e6 −0.111620
\(170\) −1.78841e7 −0.279188
\(171\) 0 0
\(172\) 1.11565e8 1.67178
\(173\) 8.84147e7 1.29826 0.649132 0.760676i \(-0.275132\pi\)
0.649132 + 0.760676i \(0.275132\pi\)
\(174\) 0 0
\(175\) −1.44514e8 −2.03834
\(176\) 1.13926e7 0.157517
\(177\) 0 0
\(178\) 3.64224e7 0.484059
\(179\) −9.25607e7 −1.20626 −0.603130 0.797643i \(-0.706080\pi\)
−0.603130 + 0.797643i \(0.706080\pi\)
\(180\) 0 0
\(181\) 7.46139e6 0.0935287 0.0467644 0.998906i \(-0.485109\pi\)
0.0467644 + 0.998906i \(0.485109\pi\)
\(182\) −2.87669e8 −3.53707
\(183\) 0 0
\(184\) 2.18627e8 2.58727
\(185\) −2.17529e8 −2.52590
\(186\) 0 0
\(187\) −422589. −0.00472577
\(188\) 8.22475e6 0.0902755
\(189\) 0 0
\(190\) 3.33458e8 3.52698
\(191\) −6.59363e7 −0.684712 −0.342356 0.939570i \(-0.611225\pi\)
−0.342356 + 0.939570i \(0.611225\pi\)
\(192\) 0 0
\(193\) 1.34481e7 0.134651 0.0673257 0.997731i \(-0.478553\pi\)
0.0673257 + 0.997731i \(0.478553\pi\)
\(194\) 2.46969e8 2.42849
\(195\) 0 0
\(196\) 7.98502e8 7.57495
\(197\) −1.59146e7 −0.148308 −0.0741539 0.997247i \(-0.523626\pi\)
−0.0741539 + 0.997247i \(0.523626\pi\)
\(198\) 0 0
\(199\) 3.59450e7 0.323335 0.161668 0.986845i \(-0.448313\pi\)
0.161668 + 0.986845i \(0.448313\pi\)
\(200\) 3.72741e8 3.29460
\(201\) 0 0
\(202\) −1.55726e8 −1.32932
\(203\) 3.98217e8 3.34105
\(204\) 0 0
\(205\) 2.08256e8 1.68834
\(206\) −5.99588e7 −0.477878
\(207\) 0 0
\(208\) 4.16821e8 3.21165
\(209\) 7.87937e6 0.0597007
\(210\) 0 0
\(211\) −2.03281e8 −1.48973 −0.744866 0.667214i \(-0.767486\pi\)
−0.744866 + 0.667214i \(0.767486\pi\)
\(212\) 3.60861e8 2.60115
\(213\) 0 0
\(214\) 2.02869e7 0.141504
\(215\) 1.30868e8 0.898048
\(216\) 0 0
\(217\) −4.87567e8 −3.23910
\(218\) −3.29853e8 −2.15637
\(219\) 0 0
\(220\) 2.77106e7 0.175455
\(221\) −1.54613e7 −0.0963546
\(222\) 0 0
\(223\) −1.24671e8 −0.752830 −0.376415 0.926451i \(-0.622843\pi\)
−0.376415 + 0.926451i \(0.622843\pi\)
\(224\) −1.10429e9 −6.56473
\(225\) 0 0
\(226\) 2.17196e8 1.25162
\(227\) 1.63994e7 0.0930547 0.0465273 0.998917i \(-0.485185\pi\)
0.0465273 + 0.998917i \(0.485185\pi\)
\(228\) 0 0
\(229\) −1.34148e8 −0.738179 −0.369089 0.929394i \(-0.620330\pi\)
−0.369089 + 0.929394i \(0.620330\pi\)
\(230\) 4.11122e8 2.22804
\(231\) 0 0
\(232\) −1.02711e9 −5.40020
\(233\) 6.54300e7 0.338869 0.169434 0.985541i \(-0.445806\pi\)
0.169434 + 0.985541i \(0.445806\pi\)
\(234\) 0 0
\(235\) 9.64781e6 0.0484944
\(236\) 6.98778e7 0.346057
\(237\) 0 0
\(238\) 7.97879e7 0.383634
\(239\) 1.59121e8 0.753936 0.376968 0.926226i \(-0.376967\pi\)
0.376968 + 0.926226i \(0.376967\pi\)
\(240\) 0 0
\(241\) 1.38094e8 0.635501 0.317750 0.948174i \(-0.397073\pi\)
0.317750 + 0.948174i \(0.397073\pi\)
\(242\) −4.20778e8 −1.90853
\(243\) 0 0
\(244\) 3.70631e8 1.63334
\(245\) 9.36661e8 4.06912
\(246\) 0 0
\(247\) 2.88283e8 1.21725
\(248\) 1.25757e9 5.23541
\(249\) 0 0
\(250\) 2.62244e7 0.106149
\(251\) −1.39109e8 −0.555262 −0.277631 0.960688i \(-0.589549\pi\)
−0.277631 + 0.960688i \(0.589549\pi\)
\(252\) 0 0
\(253\) 9.71452e6 0.0377137
\(254\) 1.96079e8 0.750781
\(255\) 0 0
\(256\) 4.16963e8 1.55331
\(257\) −4.09751e8 −1.50575 −0.752877 0.658162i \(-0.771334\pi\)
−0.752877 + 0.658162i \(0.771334\pi\)
\(258\) 0 0
\(259\) 9.70480e8 3.47086
\(260\) 1.01385e9 3.57740
\(261\) 0 0
\(262\) −4.53484e8 −1.55779
\(263\) −1.19894e8 −0.406400 −0.203200 0.979137i \(-0.565134\pi\)
−0.203200 + 0.979137i \(0.565134\pi\)
\(264\) 0 0
\(265\) 4.23299e8 1.39729
\(266\) −1.48768e9 −4.84647
\(267\) 0 0
\(268\) −1.89687e7 −0.0601958
\(269\) 2.45594e8 0.769282 0.384641 0.923066i \(-0.374325\pi\)
0.384641 + 0.923066i \(0.374325\pi\)
\(270\) 0 0
\(271\) 2.52445e8 0.770503 0.385252 0.922812i \(-0.374115\pi\)
0.385252 + 0.922812i \(0.374115\pi\)
\(272\) −1.15609e8 −0.348339
\(273\) 0 0
\(274\) −4.75681e8 −1.39697
\(275\) 1.65624e7 0.0480241
\(276\) 0 0
\(277\) 3.24074e8 0.916147 0.458074 0.888914i \(-0.348540\pi\)
0.458074 + 0.888914i \(0.348540\pi\)
\(278\) −1.39497e8 −0.389412
\(279\) 0 0
\(280\) −3.26367e9 −8.88491
\(281\) 1.78256e8 0.479260 0.239630 0.970864i \(-0.422974\pi\)
0.239630 + 0.970864i \(0.422974\pi\)
\(282\) 0 0
\(283\) −4.64602e8 −1.21851 −0.609254 0.792975i \(-0.708531\pi\)
−0.609254 + 0.792975i \(0.708531\pi\)
\(284\) 2.89835e8 0.750822
\(285\) 0 0
\(286\) 3.29692e7 0.0833349
\(287\) −9.29110e8 −2.31996
\(288\) 0 0
\(289\) −4.06050e8 −0.989549
\(290\) −1.93145e9 −4.65040
\(291\) 0 0
\(292\) 1.17216e9 2.75517
\(293\) 7.63943e8 1.77429 0.887145 0.461492i \(-0.152686\pi\)
0.887145 + 0.461492i \(0.152686\pi\)
\(294\) 0 0
\(295\) 8.19682e7 0.185895
\(296\) −2.50314e9 −5.61001
\(297\) 0 0
\(298\) 7.32959e8 1.60444
\(299\) 3.55425e8 0.768952
\(300\) 0 0
\(301\) −5.83853e8 −1.23402
\(302\) −1.35731e9 −2.83565
\(303\) 0 0
\(304\) 2.15559e9 4.40057
\(305\) 4.34759e8 0.877402
\(306\) 0 0
\(307\) −5.34126e8 −1.05356 −0.526780 0.850001i \(-0.676601\pi\)
−0.526780 + 0.850001i \(0.676601\pi\)
\(308\) −1.23628e8 −0.241095
\(309\) 0 0
\(310\) 2.36482e9 4.50850
\(311\) −7.32126e8 −1.38014 −0.690072 0.723741i \(-0.742421\pi\)
−0.690072 + 0.723741i \(0.742421\pi\)
\(312\) 0 0
\(313\) 5.31534e8 0.979774 0.489887 0.871786i \(-0.337038\pi\)
0.489887 + 0.871786i \(0.337038\pi\)
\(314\) 1.02187e9 1.86270
\(315\) 0 0
\(316\) −2.73978e9 −4.88439
\(317\) −1.07205e9 −1.89020 −0.945102 0.326775i \(-0.894038\pi\)
−0.945102 + 0.326775i \(0.894038\pi\)
\(318\) 0 0
\(319\) −4.56388e7 −0.0787167
\(320\) 2.50411e9 4.27197
\(321\) 0 0
\(322\) −1.83417e9 −3.06157
\(323\) −7.99581e7 −0.132024
\(324\) 0 0
\(325\) 6.05970e8 0.979174
\(326\) −6.17598e8 −0.987289
\(327\) 0 0
\(328\) 2.39643e9 3.74979
\(329\) −4.30426e7 −0.0666366
\(330\) 0 0
\(331\) −4.36057e7 −0.0660914 −0.0330457 0.999454i \(-0.510521\pi\)
−0.0330457 + 0.999454i \(0.510521\pi\)
\(332\) −6.97556e8 −1.04615
\(333\) 0 0
\(334\) 1.31001e9 1.92380
\(335\) −2.22507e7 −0.0323361
\(336\) 0 0
\(337\) −6.67140e8 −0.949538 −0.474769 0.880110i \(-0.657468\pi\)
−0.474769 + 0.880110i \(0.657468\pi\)
\(338\) −1.51559e8 −0.213487
\(339\) 0 0
\(340\) −2.81201e8 −0.388008
\(341\) 5.58790e7 0.0763147
\(342\) 0 0
\(343\) −2.71243e9 −3.62935
\(344\) 1.50592e9 1.99456
\(345\) 0 0
\(346\) 1.91319e9 2.48308
\(347\) −3.15540e7 −0.0405417 −0.0202708 0.999795i \(-0.506453\pi\)
−0.0202708 + 0.999795i \(0.506453\pi\)
\(348\) 0 0
\(349\) 7.24543e8 0.912379 0.456189 0.889883i \(-0.349214\pi\)
0.456189 + 0.889883i \(0.349214\pi\)
\(350\) −3.12711e9 −3.89857
\(351\) 0 0
\(352\) 1.26561e8 0.154668
\(353\) −9.26501e8 −1.12107 −0.560537 0.828129i \(-0.689405\pi\)
−0.560537 + 0.828129i \(0.689405\pi\)
\(354\) 0 0
\(355\) 3.39983e8 0.403328
\(356\) 5.72688e8 0.672734
\(357\) 0 0
\(358\) −2.00290e9 −2.30712
\(359\) 1.08624e9 1.23906 0.619532 0.784971i \(-0.287322\pi\)
0.619532 + 0.784971i \(0.287322\pi\)
\(360\) 0 0
\(361\) 5.96986e8 0.667866
\(362\) 1.61456e8 0.178885
\(363\) 0 0
\(364\) −4.52318e9 −4.91574
\(365\) 1.37497e9 1.48003
\(366\) 0 0
\(367\) 8.58587e8 0.906678 0.453339 0.891338i \(-0.350233\pi\)
0.453339 + 0.891338i \(0.350233\pi\)
\(368\) 2.65764e9 2.77990
\(369\) 0 0
\(370\) −4.70707e9 −4.83108
\(371\) −1.88850e9 −1.92003
\(372\) 0 0
\(373\) −1.80912e8 −0.180504 −0.0902518 0.995919i \(-0.528767\pi\)
−0.0902518 + 0.995919i \(0.528767\pi\)
\(374\) −9.14432e6 −0.00903859
\(375\) 0 0
\(376\) 1.11019e8 0.107706
\(377\) −1.66979e9 −1.60497
\(378\) 0 0
\(379\) −1.73658e9 −1.63854 −0.819270 0.573408i \(-0.805621\pi\)
−0.819270 + 0.573408i \(0.805621\pi\)
\(380\) 5.24313e9 4.90172
\(381\) 0 0
\(382\) −1.42678e9 −1.30959
\(383\) −1.24471e9 −1.13207 −0.566035 0.824381i \(-0.691523\pi\)
−0.566035 + 0.824381i \(0.691523\pi\)
\(384\) 0 0
\(385\) −1.45018e8 −0.129512
\(386\) 2.91001e8 0.257537
\(387\) 0 0
\(388\) 3.88322e9 3.37506
\(389\) 1.41664e9 1.22021 0.610106 0.792320i \(-0.291127\pi\)
0.610106 + 0.792320i \(0.291127\pi\)
\(390\) 0 0
\(391\) −9.85807e7 −0.0834014
\(392\) 1.07783e10 9.03750
\(393\) 0 0
\(394\) −3.44373e8 −0.283656
\(395\) −3.21382e9 −2.62381
\(396\) 0 0
\(397\) −1.99168e9 −1.59755 −0.798773 0.601633i \(-0.794517\pi\)
−0.798773 + 0.601633i \(0.794517\pi\)
\(398\) 7.77808e8 0.618417
\(399\) 0 0
\(400\) 4.53105e9 3.53988
\(401\) 9.22799e8 0.714664 0.357332 0.933977i \(-0.383687\pi\)
0.357332 + 0.933977i \(0.383687\pi\)
\(402\) 0 0
\(403\) 2.04445e9 1.55599
\(404\) −2.44856e9 −1.84746
\(405\) 0 0
\(406\) 8.61694e9 6.39017
\(407\) −1.11225e8 −0.0817750
\(408\) 0 0
\(409\) −3.58871e7 −0.0259362 −0.0129681 0.999916i \(-0.504128\pi\)
−0.0129681 + 0.999916i \(0.504128\pi\)
\(410\) 4.50641e9 3.22914
\(411\) 0 0
\(412\) −9.42763e8 −0.664144
\(413\) −3.65692e8 −0.255441
\(414\) 0 0
\(415\) −8.18249e8 −0.561975
\(416\) 4.63049e9 3.15355
\(417\) 0 0
\(418\) 1.70500e8 0.114185
\(419\) 2.26805e8 0.150627 0.0753135 0.997160i \(-0.476004\pi\)
0.0753135 + 0.997160i \(0.476004\pi\)
\(420\) 0 0
\(421\) 7.22132e8 0.471660 0.235830 0.971794i \(-0.424219\pi\)
0.235830 + 0.971794i \(0.424219\pi\)
\(422\) −4.39876e9 −2.84929
\(423\) 0 0
\(424\) 4.87096e9 3.10337
\(425\) −1.68072e8 −0.106202
\(426\) 0 0
\(427\) −1.93963e9 −1.20565
\(428\) 3.18982e8 0.196659
\(429\) 0 0
\(430\) 2.83183e9 1.71762
\(431\) −2.37550e8 −0.142917 −0.0714586 0.997444i \(-0.522765\pi\)
−0.0714586 + 0.997444i \(0.522765\pi\)
\(432\) 0 0
\(433\) −2.62934e8 −0.155646 −0.0778231 0.996967i \(-0.524797\pi\)
−0.0778231 + 0.996967i \(0.524797\pi\)
\(434\) −1.05504e10 −6.19517
\(435\) 0 0
\(436\) −5.18645e9 −2.99687
\(437\) 1.83809e9 1.05361
\(438\) 0 0
\(439\) 9.64067e8 0.543853 0.271926 0.962318i \(-0.412339\pi\)
0.271926 + 0.962318i \(0.412339\pi\)
\(440\) 3.74042e8 0.209332
\(441\) 0 0
\(442\) −3.34564e8 −0.184290
\(443\) −2.26896e9 −1.23998 −0.619989 0.784611i \(-0.712863\pi\)
−0.619989 + 0.784611i \(0.712863\pi\)
\(444\) 0 0
\(445\) 6.71776e8 0.361380
\(446\) −2.69772e9 −1.43988
\(447\) 0 0
\(448\) −1.11718e10 −5.87016
\(449\) 6.29447e8 0.328168 0.164084 0.986446i \(-0.447533\pi\)
0.164084 + 0.986446i \(0.447533\pi\)
\(450\) 0 0
\(451\) 1.06483e8 0.0546592
\(452\) 3.41509e9 1.73947
\(453\) 0 0
\(454\) 3.54864e8 0.177978
\(455\) −5.30579e9 −2.64064
\(456\) 0 0
\(457\) −2.73973e9 −1.34277 −0.671383 0.741111i \(-0.734299\pi\)
−0.671383 + 0.741111i \(0.734299\pi\)
\(458\) −2.90281e9 −1.41185
\(459\) 0 0
\(460\) 6.46428e9 3.09648
\(461\) −1.50755e9 −0.716668 −0.358334 0.933593i \(-0.616655\pi\)
−0.358334 + 0.933593i \(0.616655\pi\)
\(462\) 0 0
\(463\) −7.07432e8 −0.331246 −0.165623 0.986189i \(-0.552964\pi\)
−0.165623 + 0.986189i \(0.552964\pi\)
\(464\) −1.24856e10 −5.80225
\(465\) 0 0
\(466\) 1.41583e9 0.648126
\(467\) 3.62529e9 1.64715 0.823575 0.567207i \(-0.191976\pi\)
0.823575 + 0.567207i \(0.191976\pi\)
\(468\) 0 0
\(469\) 9.92690e7 0.0444333
\(470\) 2.08767e8 0.0927513
\(471\) 0 0
\(472\) 9.43220e8 0.412873
\(473\) 6.69142e7 0.0290740
\(474\) 0 0
\(475\) 3.13378e9 1.34166
\(476\) 1.25455e9 0.533166
\(477\) 0 0
\(478\) 3.44319e9 1.44199
\(479\) −1.79818e9 −0.747580 −0.373790 0.927513i \(-0.621942\pi\)
−0.373790 + 0.927513i \(0.621942\pi\)
\(480\) 0 0
\(481\) −4.06938e9 −1.66733
\(482\) 2.98820e9 1.21547
\(483\) 0 0
\(484\) −6.61611e9 −2.65243
\(485\) 4.55511e9 1.81302
\(486\) 0 0
\(487\) −1.58236e9 −0.620802 −0.310401 0.950606i \(-0.600463\pi\)
−0.310401 + 0.950606i \(0.600463\pi\)
\(488\) 5.00283e9 1.94871
\(489\) 0 0
\(490\) 2.02682e10 7.78268
\(491\) 2.99676e8 0.114253 0.0571264 0.998367i \(-0.481806\pi\)
0.0571264 + 0.998367i \(0.481806\pi\)
\(492\) 0 0
\(493\) 4.63132e8 0.174077
\(494\) 6.23810e9 2.32814
\(495\) 0 0
\(496\) 1.52870e10 5.62520
\(497\) −1.51680e9 −0.554217
\(498\) 0 0
\(499\) 1.35009e9 0.486419 0.243210 0.969974i \(-0.421800\pi\)
0.243210 + 0.969974i \(0.421800\pi\)
\(500\) 4.12340e8 0.147523
\(501\) 0 0
\(502\) −3.01016e9 −1.06200
\(503\) −1.94191e9 −0.680365 −0.340183 0.940359i \(-0.610489\pi\)
−0.340183 + 0.940359i \(0.610489\pi\)
\(504\) 0 0
\(505\) −2.87222e9 −0.992424
\(506\) 2.10211e8 0.0721320
\(507\) 0 0
\(508\) 3.08305e9 1.04342
\(509\) 1.85092e9 0.622120 0.311060 0.950390i \(-0.399316\pi\)
0.311060 + 0.950390i \(0.399316\pi\)
\(510\) 0 0
\(511\) −6.13429e9 −2.03372
\(512\) 1.80551e9 0.594504
\(513\) 0 0
\(514\) −8.86652e9 −2.87993
\(515\) −1.10588e9 −0.356766
\(516\) 0 0
\(517\) 4.93302e6 0.00156999
\(518\) 2.10000e10 6.63844
\(519\) 0 0
\(520\) 1.36851e10 4.26811
\(521\) −2.35474e9 −0.729475 −0.364738 0.931110i \(-0.618841\pi\)
−0.364738 + 0.931110i \(0.618841\pi\)
\(522\) 0 0
\(523\) 1.86850e9 0.571133 0.285566 0.958359i \(-0.407818\pi\)
0.285566 + 0.958359i \(0.407818\pi\)
\(524\) −7.13037e9 −2.16497
\(525\) 0 0
\(526\) −2.59437e9 −0.777288
\(527\) −5.67047e8 −0.168765
\(528\) 0 0
\(529\) −1.13864e9 −0.334420
\(530\) 9.15968e9 2.67248
\(531\) 0 0
\(532\) −2.33916e10 −6.73550
\(533\) 3.89591e9 1.11446
\(534\) 0 0
\(535\) 3.74173e8 0.105641
\(536\) −2.56042e8 −0.0718183
\(537\) 0 0
\(538\) 5.31437e9 1.47134
\(539\) 4.78924e8 0.131736
\(540\) 0 0
\(541\) −1.58014e8 −0.0429046 −0.0214523 0.999770i \(-0.506829\pi\)
−0.0214523 + 0.999770i \(0.506829\pi\)
\(542\) 5.46261e9 1.47368
\(543\) 0 0
\(544\) −1.28431e9 −0.342038
\(545\) −6.08383e9 −1.60986
\(546\) 0 0
\(547\) −4.19427e8 −0.109572 −0.0547862 0.998498i \(-0.517448\pi\)
−0.0547862 + 0.998498i \(0.517448\pi\)
\(548\) −7.47937e9 −1.94148
\(549\) 0 0
\(550\) 3.58391e8 0.0918519
\(551\) −8.63532e9 −2.19912
\(552\) 0 0
\(553\) 1.43381e10 3.60540
\(554\) 7.01258e9 1.75224
\(555\) 0 0
\(556\) −2.19339e9 −0.541195
\(557\) 7.37086e9 1.80728 0.903639 0.428295i \(-0.140886\pi\)
0.903639 + 0.428295i \(0.140886\pi\)
\(558\) 0 0
\(559\) 2.44819e9 0.592795
\(560\) −3.96732e10 −9.54640
\(561\) 0 0
\(562\) 3.85724e9 0.916642
\(563\) 5.40369e9 1.27618 0.638089 0.769963i \(-0.279725\pi\)
0.638089 + 0.769963i \(0.279725\pi\)
\(564\) 0 0
\(565\) 4.00598e9 0.934414
\(566\) −1.00534e10 −2.33054
\(567\) 0 0
\(568\) 3.91224e9 0.895789
\(569\) 2.39098e9 0.544104 0.272052 0.962283i \(-0.412298\pi\)
0.272052 + 0.962283i \(0.412298\pi\)
\(570\) 0 0
\(571\) −4.97857e9 −1.11912 −0.559562 0.828789i \(-0.689031\pi\)
−0.559562 + 0.828789i \(0.689031\pi\)
\(572\) 5.18391e8 0.115817
\(573\) 0 0
\(574\) −2.01048e10 −4.43720
\(575\) 3.86365e9 0.847541
\(576\) 0 0
\(577\) 3.57830e9 0.775464 0.387732 0.921772i \(-0.373259\pi\)
0.387732 + 0.921772i \(0.373259\pi\)
\(578\) −8.78645e9 −1.89263
\(579\) 0 0
\(580\) −3.03692e10 −6.46302
\(581\) 3.65053e9 0.772216
\(582\) 0 0
\(583\) 2.16437e8 0.0452367
\(584\) 1.58220e10 3.28713
\(585\) 0 0
\(586\) 1.65308e10 3.39354
\(587\) 2.26324e9 0.461845 0.230923 0.972972i \(-0.425826\pi\)
0.230923 + 0.972972i \(0.425826\pi\)
\(588\) 0 0
\(589\) 1.05729e10 2.13201
\(590\) 1.77370e9 0.355547
\(591\) 0 0
\(592\) −3.04282e10 −6.02768
\(593\) −2.71803e9 −0.535257 −0.267628 0.963522i \(-0.586240\pi\)
−0.267628 + 0.963522i \(0.586240\pi\)
\(594\) 0 0
\(595\) 1.47161e9 0.286407
\(596\) 1.15247e10 2.22981
\(597\) 0 0
\(598\) 7.69099e9 1.47071
\(599\) −5.48589e9 −1.04293 −0.521463 0.853274i \(-0.674614\pi\)
−0.521463 + 0.853274i \(0.674614\pi\)
\(600\) 0 0
\(601\) 5.20845e9 0.978695 0.489348 0.872089i \(-0.337235\pi\)
0.489348 + 0.872089i \(0.337235\pi\)
\(602\) −1.26339e10 −2.36020
\(603\) 0 0
\(604\) −2.13416e10 −3.94092
\(605\) −7.76085e9 −1.42484
\(606\) 0 0
\(607\) −8.43683e9 −1.53115 −0.765577 0.643345i \(-0.777546\pi\)
−0.765577 + 0.643345i \(0.777546\pi\)
\(608\) 2.39466e10 4.32097
\(609\) 0 0
\(610\) 9.40766e9 1.67814
\(611\) 1.80485e8 0.0320108
\(612\) 0 0
\(613\) −6.65621e9 −1.16712 −0.583560 0.812070i \(-0.698341\pi\)
−0.583560 + 0.812070i \(0.698341\pi\)
\(614\) −1.15579e10 −2.01506
\(615\) 0 0
\(616\) −1.66874e9 −0.287645
\(617\) −1.08431e10 −1.85846 −0.929231 0.369498i \(-0.879530\pi\)
−0.929231 + 0.369498i \(0.879530\pi\)
\(618\) 0 0
\(619\) −1.05882e10 −1.79434 −0.897172 0.441681i \(-0.854382\pi\)
−0.897172 + 0.441681i \(0.854382\pi\)
\(620\) 3.71833e10 6.26580
\(621\) 0 0
\(622\) −1.58423e10 −2.63969
\(623\) −2.99705e9 −0.496576
\(624\) 0 0
\(625\) −5.85706e9 −0.959621
\(626\) 1.15018e10 1.87394
\(627\) 0 0
\(628\) 1.60674e10 2.58873
\(629\) 1.12868e9 0.180840
\(630\) 0 0
\(631\) 9.06740e9 1.43675 0.718373 0.695658i \(-0.244887\pi\)
0.718373 + 0.695658i \(0.244887\pi\)
\(632\) −3.69819e10 −5.82746
\(633\) 0 0
\(634\) −2.31980e10 −3.61524
\(635\) 3.61649e9 0.560505
\(636\) 0 0
\(637\) 1.75224e10 2.68600
\(638\) −9.87569e8 −0.150555
\(639\) 0 0
\(640\) 2.25030e10 3.39322
\(641\) 6.94011e9 1.04079 0.520395 0.853926i \(-0.325785\pi\)
0.520395 + 0.853926i \(0.325785\pi\)
\(642\) 0 0
\(643\) −4.50631e9 −0.668471 −0.334235 0.942490i \(-0.608478\pi\)
−0.334235 + 0.942490i \(0.608478\pi\)
\(644\) −2.88396e10 −4.25490
\(645\) 0 0
\(646\) −1.73020e9 −0.252512
\(647\) −5.05823e9 −0.734233 −0.367116 0.930175i \(-0.619655\pi\)
−0.367116 + 0.930175i \(0.619655\pi\)
\(648\) 0 0
\(649\) 4.19112e7 0.00601829
\(650\) 1.31125e10 1.87279
\(651\) 0 0
\(652\) −9.71082e9 −1.37211
\(653\) 3.31489e9 0.465879 0.232940 0.972491i \(-0.425166\pi\)
0.232940 + 0.972491i \(0.425166\pi\)
\(654\) 0 0
\(655\) −8.36409e9 −1.16298
\(656\) 2.91311e10 4.02896
\(657\) 0 0
\(658\) −9.31391e8 −0.127450
\(659\) 9.88658e9 1.34570 0.672849 0.739780i \(-0.265071\pi\)
0.672849 + 0.739780i \(0.265071\pi\)
\(660\) 0 0
\(661\) −1.76683e9 −0.237952 −0.118976 0.992897i \(-0.537961\pi\)
−0.118976 + 0.992897i \(0.537961\pi\)
\(662\) −9.43576e8 −0.126408
\(663\) 0 0
\(664\) −9.41571e9 −1.24814
\(665\) −2.74389e10 −3.61819
\(666\) 0 0
\(667\) −1.06465e10 −1.38921
\(668\) 2.05979e10 2.67366
\(669\) 0 0
\(670\) −4.81479e8 −0.0618466
\(671\) 2.22296e8 0.0284056
\(672\) 0 0
\(673\) −8.78145e9 −1.11049 −0.555244 0.831688i \(-0.687375\pi\)
−0.555244 + 0.831688i \(0.687375\pi\)
\(674\) −1.44361e10 −1.81610
\(675\) 0 0
\(676\) −2.38303e9 −0.296699
\(677\) −1.26024e9 −0.156096 −0.0780482 0.996950i \(-0.524869\pi\)
−0.0780482 + 0.996950i \(0.524869\pi\)
\(678\) 0 0
\(679\) −2.03221e10 −2.49129
\(680\) −3.79569e9 −0.462924
\(681\) 0 0
\(682\) 1.20915e9 0.145961
\(683\) −1.20509e10 −1.44726 −0.723630 0.690188i \(-0.757528\pi\)
−0.723630 + 0.690188i \(0.757528\pi\)
\(684\) 0 0
\(685\) −8.77348e9 −1.04293
\(686\) −5.86937e10 −6.94156
\(687\) 0 0
\(688\) 1.83060e10 2.14306
\(689\) 7.91878e9 0.922340
\(690\) 0 0
\(691\) −1.63179e9 −0.188144 −0.0940720 0.995565i \(-0.529988\pi\)
−0.0940720 + 0.995565i \(0.529988\pi\)
\(692\) 3.00820e10 3.45093
\(693\) 0 0
\(694\) −6.82792e8 −0.0775408
\(695\) −2.57290e9 −0.290720
\(696\) 0 0
\(697\) −1.08057e9 −0.120875
\(698\) 1.56782e10 1.74503
\(699\) 0 0
\(700\) −4.91692e10 −5.41813
\(701\) 4.92646e9 0.540159 0.270080 0.962838i \(-0.412950\pi\)
0.270080 + 0.962838i \(0.412950\pi\)
\(702\) 0 0
\(703\) −2.10448e10 −2.28456
\(704\) 1.28038e9 0.138304
\(705\) 0 0
\(706\) −2.00484e10 −2.14419
\(707\) 1.28141e10 1.36370
\(708\) 0 0
\(709\) 1.23998e10 1.30663 0.653315 0.757087i \(-0.273378\pi\)
0.653315 + 0.757087i \(0.273378\pi\)
\(710\) 7.35683e9 0.771413
\(711\) 0 0
\(712\) 7.73022e9 0.802624
\(713\) 1.30353e10 1.34682
\(714\) 0 0
\(715\) 6.08085e8 0.0622147
\(716\) −3.14927e10 −3.20637
\(717\) 0 0
\(718\) 2.35049e10 2.36986
\(719\) 7.30887e9 0.733329 0.366665 0.930353i \(-0.380500\pi\)
0.366665 + 0.930353i \(0.380500\pi\)
\(720\) 0 0
\(721\) 4.93377e9 0.490236
\(722\) 1.29181e10 1.27737
\(723\) 0 0
\(724\) 2.53865e9 0.248610
\(725\) −1.81514e10 −1.76900
\(726\) 0 0
\(727\) −1.04087e9 −0.100467 −0.0502337 0.998737i \(-0.515997\pi\)
−0.0502337 + 0.998737i \(0.515997\pi\)
\(728\) −6.10545e10 −5.86486
\(729\) 0 0
\(730\) 2.97528e10 2.83073
\(731\) −6.79030e8 −0.0642951
\(732\) 0 0
\(733\) 1.03455e10 0.970262 0.485131 0.874441i \(-0.338772\pi\)
0.485131 + 0.874441i \(0.338772\pi\)
\(734\) 1.85788e10 1.73413
\(735\) 0 0
\(736\) 2.95239e10 2.72961
\(737\) −1.13770e7 −0.00104687
\(738\) 0 0
\(739\) −8.36230e9 −0.762203 −0.381101 0.924533i \(-0.624455\pi\)
−0.381101 + 0.924533i \(0.624455\pi\)
\(740\) −7.40117e10 −6.71412
\(741\) 0 0
\(742\) −4.08649e10 −3.67228
\(743\) 1.24177e10 1.11066 0.555330 0.831630i \(-0.312592\pi\)
0.555330 + 0.831630i \(0.312592\pi\)
\(744\) 0 0
\(745\) 1.35187e10 1.19781
\(746\) −3.91472e9 −0.345235
\(747\) 0 0
\(748\) −1.43781e8 −0.0125616
\(749\) −1.66933e9 −0.145163
\(750\) 0 0
\(751\) −4.11718e9 −0.354699 −0.177349 0.984148i \(-0.556752\pi\)
−0.177349 + 0.984148i \(0.556752\pi\)
\(752\) 1.34955e9 0.115725
\(753\) 0 0
\(754\) −3.61322e10 −3.06970
\(755\) −2.50342e10 −2.11699
\(756\) 0 0
\(757\) 7.02287e9 0.588409 0.294204 0.955743i \(-0.404945\pi\)
0.294204 + 0.955743i \(0.404945\pi\)
\(758\) −3.75775e10 −3.13390
\(759\) 0 0
\(760\) 7.07725e10 5.84813
\(761\) 3.65462e9 0.300605 0.150302 0.988640i \(-0.451975\pi\)
0.150302 + 0.988640i \(0.451975\pi\)
\(762\) 0 0
\(763\) 2.71423e10 2.21213
\(764\) −2.24340e10 −1.82004
\(765\) 0 0
\(766\) −2.69341e10 −2.16522
\(767\) 1.53341e9 0.122708
\(768\) 0 0
\(769\) 1.51627e10 1.20236 0.601179 0.799114i \(-0.294698\pi\)
0.601179 + 0.799114i \(0.294698\pi\)
\(770\) −3.13802e9 −0.247707
\(771\) 0 0
\(772\) 4.57556e9 0.357918
\(773\) −1.27681e10 −0.994253 −0.497126 0.867678i \(-0.665612\pi\)
−0.497126 + 0.867678i \(0.665612\pi\)
\(774\) 0 0
\(775\) 2.22242e10 1.71502
\(776\) 5.24163e10 4.02671
\(777\) 0 0
\(778\) 3.06543e10 2.33380
\(779\) 2.01477e10 1.52702
\(780\) 0 0
\(781\) 1.73837e8 0.0130576
\(782\) −2.13317e9 −0.159515
\(783\) 0 0
\(784\) 1.31021e11 9.71036
\(785\) 1.88474e10 1.39062
\(786\) 0 0
\(787\) −3.81227e9 −0.278787 −0.139394 0.990237i \(-0.544515\pi\)
−0.139394 + 0.990237i \(0.544515\pi\)
\(788\) −5.41476e9 −0.394219
\(789\) 0 0
\(790\) −6.95433e10 −5.01834
\(791\) −1.78722e10 −1.28399
\(792\) 0 0
\(793\) 8.13317e9 0.579167
\(794\) −4.30976e10 −3.05550
\(795\) 0 0
\(796\) 1.22299e10 0.859461
\(797\) 9.63179e9 0.673911 0.336956 0.941521i \(-0.390603\pi\)
0.336956 + 0.941521i \(0.390603\pi\)
\(798\) 0 0
\(799\) −5.00592e7 −0.00347192
\(800\) 5.03357e10 3.47585
\(801\) 0 0
\(802\) 1.99683e10 1.36688
\(803\) 7.03037e8 0.0479153
\(804\) 0 0
\(805\) −3.38296e10 −2.28566
\(806\) 4.42394e10 2.97603
\(807\) 0 0
\(808\) −3.30510e10 −2.20417
\(809\) −1.38532e9 −0.0919878 −0.0459939 0.998942i \(-0.514645\pi\)
−0.0459939 + 0.998942i \(0.514645\pi\)
\(810\) 0 0
\(811\) 1.41931e10 0.934339 0.467170 0.884168i \(-0.345274\pi\)
0.467170 + 0.884168i \(0.345274\pi\)
\(812\) 1.35489e11 8.88090
\(813\) 0 0
\(814\) −2.40677e9 −0.156404
\(815\) −1.13910e10 −0.737073
\(816\) 0 0
\(817\) 1.26608e10 0.812242
\(818\) −7.76554e8 −0.0496061
\(819\) 0 0
\(820\) 7.08566e10 4.48778
\(821\) 1.94897e10 1.22915 0.614573 0.788860i \(-0.289328\pi\)
0.614573 + 0.788860i \(0.289328\pi\)
\(822\) 0 0
\(823\) 7.94309e9 0.496696 0.248348 0.968671i \(-0.420112\pi\)
0.248348 + 0.968671i \(0.420112\pi\)
\(824\) −1.27255e10 −0.792376
\(825\) 0 0
\(826\) −7.91314e9 −0.488561
\(827\) −1.74454e10 −1.07254 −0.536268 0.844048i \(-0.680166\pi\)
−0.536268 + 0.844048i \(0.680166\pi\)
\(828\) 0 0
\(829\) 2.55780e10 1.55929 0.779644 0.626223i \(-0.215400\pi\)
0.779644 + 0.626223i \(0.215400\pi\)
\(830\) −1.77059e10 −1.07484
\(831\) 0 0
\(832\) 4.68452e10 2.81990
\(833\) −4.86001e9 −0.291326
\(834\) 0 0
\(835\) 2.41618e10 1.43624
\(836\) 2.68086e9 0.158691
\(837\) 0 0
\(838\) 4.90778e9 0.288092
\(839\) 7.34489e9 0.429357 0.214678 0.976685i \(-0.431130\pi\)
0.214678 + 0.976685i \(0.431130\pi\)
\(840\) 0 0
\(841\) 3.27675e10 1.89958
\(842\) 1.56261e10 0.902106
\(843\) 0 0
\(844\) −6.91639e10 −3.95987
\(845\) −2.79535e9 −0.159382
\(846\) 0 0
\(847\) 3.46241e10 1.95788
\(848\) 5.92115e10 3.33442
\(849\) 0 0
\(850\) −3.63687e9 −0.203124
\(851\) −2.59463e10 −1.44318
\(852\) 0 0
\(853\) −3.22700e10 −1.78023 −0.890116 0.455734i \(-0.849377\pi\)
−0.890116 + 0.455734i \(0.849377\pi\)
\(854\) −4.19712e10 −2.30594
\(855\) 0 0
\(856\) 4.30566e9 0.234629
\(857\) 7.06953e9 0.383670 0.191835 0.981427i \(-0.438556\pi\)
0.191835 + 0.981427i \(0.438556\pi\)
\(858\) 0 0
\(859\) −2.05380e10 −1.10556 −0.552779 0.833328i \(-0.686432\pi\)
−0.552779 + 0.833328i \(0.686432\pi\)
\(860\) 4.45264e10 2.38711
\(861\) 0 0
\(862\) −5.14030e9 −0.273346
\(863\) 3.50261e10 1.85504 0.927520 0.373773i \(-0.121936\pi\)
0.927520 + 0.373773i \(0.121936\pi\)
\(864\) 0 0
\(865\) 3.52869e10 1.85378
\(866\) −5.68957e9 −0.297692
\(867\) 0 0
\(868\) −1.65889e11 −8.60990
\(869\) −1.64326e9 −0.0849447
\(870\) 0 0
\(871\) −4.16251e8 −0.0213448
\(872\) −7.00075e10 −3.57550
\(873\) 0 0
\(874\) 3.97740e10 2.01516
\(875\) −2.15790e9 −0.108894
\(876\) 0 0
\(877\) 2.05838e10 1.03045 0.515225 0.857055i \(-0.327708\pi\)
0.515225 + 0.857055i \(0.327708\pi\)
\(878\) 2.08613e10 1.04018
\(879\) 0 0
\(880\) 4.54686e9 0.224917
\(881\) −3.80144e9 −0.187298 −0.0936488 0.995605i \(-0.529853\pi\)
−0.0936488 + 0.995605i \(0.529853\pi\)
\(882\) 0 0
\(883\) −2.46327e10 −1.20406 −0.602032 0.798472i \(-0.705642\pi\)
−0.602032 + 0.798472i \(0.705642\pi\)
\(884\) −5.26052e9 −0.256121
\(885\) 0 0
\(886\) −4.90976e10 −2.37161
\(887\) 2.28817e10 1.10092 0.550461 0.834861i \(-0.314452\pi\)
0.550461 + 0.834861i \(0.314452\pi\)
\(888\) 0 0
\(889\) −1.61346e10 −0.770196
\(890\) 1.45364e10 0.691183
\(891\) 0 0
\(892\) −4.24177e10 −2.00111
\(893\) 9.33378e8 0.0438609
\(894\) 0 0
\(895\) −3.69416e10 −1.72241
\(896\) −1.00395e11 −4.66265
\(897\) 0 0
\(898\) 1.36205e10 0.627661
\(899\) −6.12400e10 −2.81110
\(900\) 0 0
\(901\) −2.19635e9 −0.100038
\(902\) 2.30417e9 0.104542
\(903\) 0 0
\(904\) 4.60974e10 2.07533
\(905\) 2.97790e9 0.133549
\(906\) 0 0
\(907\) 2.43988e10 1.08578 0.542892 0.839802i \(-0.317329\pi\)
0.542892 + 0.839802i \(0.317329\pi\)
\(908\) 5.57971e9 0.247350
\(909\) 0 0
\(910\) −1.14811e11 −5.05055
\(911\) −7.15550e9 −0.313563 −0.156782 0.987633i \(-0.550112\pi\)
−0.156782 + 0.987633i \(0.550112\pi\)
\(912\) 0 0
\(913\) −4.18379e8 −0.0181937
\(914\) −5.92844e10 −2.56820
\(915\) 0 0
\(916\) −4.56424e10 −1.96216
\(917\) 3.73154e10 1.59807
\(918\) 0 0
\(919\) −3.31022e10 −1.40687 −0.703433 0.710761i \(-0.748351\pi\)
−0.703433 + 0.710761i \(0.748351\pi\)
\(920\) 8.72558e10 3.69434
\(921\) 0 0
\(922\) −3.26216e10 −1.37071
\(923\) 6.36017e9 0.266234
\(924\) 0 0
\(925\) −4.42362e10 −1.83773
\(926\) −1.53080e10 −0.633548
\(927\) 0 0
\(928\) −1.38703e11 −5.69729
\(929\) −4.50956e10 −1.84535 −0.922676 0.385577i \(-0.874003\pi\)
−0.922676 + 0.385577i \(0.874003\pi\)
\(930\) 0 0
\(931\) 9.06172e10 3.68033
\(932\) 2.22618e10 0.900750
\(933\) 0 0
\(934\) 7.84469e10 3.15037
\(935\) −1.68658e8 −0.00674787
\(936\) 0 0
\(937\) 2.71853e10 1.07956 0.539779 0.841807i \(-0.318508\pi\)
0.539779 + 0.841807i \(0.318508\pi\)
\(938\) 2.14806e9 0.0849840
\(939\) 0 0
\(940\) 3.28256e9 0.128903
\(941\) 4.59907e10 1.79931 0.899655 0.436601i \(-0.143818\pi\)
0.899655 + 0.436601i \(0.143818\pi\)
\(942\) 0 0
\(943\) 2.48402e10 0.964638
\(944\) 1.14658e10 0.443611
\(945\) 0 0
\(946\) 1.44794e9 0.0556074
\(947\) −4.47468e10 −1.71213 −0.856065 0.516868i \(-0.827098\pi\)
−0.856065 + 0.516868i \(0.827098\pi\)
\(948\) 0 0
\(949\) 2.57221e10 0.976954
\(950\) 6.78113e10 2.56608
\(951\) 0 0
\(952\) 1.69340e10 0.636109
\(953\) 6.72893e9 0.251838 0.125919 0.992041i \(-0.459812\pi\)
0.125919 + 0.992041i \(0.459812\pi\)
\(954\) 0 0
\(955\) −2.63156e10 −0.977692
\(956\) 5.41390e10 2.00405
\(957\) 0 0
\(958\) −3.89104e10 −1.42984
\(959\) 3.91419e10 1.43310
\(960\) 0 0
\(961\) 4.74681e10 1.72532
\(962\) −8.80566e10 −3.18896
\(963\) 0 0
\(964\) 4.69849e10 1.68923
\(965\) 5.36724e9 0.192267
\(966\) 0 0
\(967\) −1.86690e9 −0.0663941 −0.0331970 0.999449i \(-0.510569\pi\)
−0.0331970 + 0.999449i \(0.510569\pi\)
\(968\) −8.93052e10 −3.16456
\(969\) 0 0
\(970\) 9.85672e10 3.46762
\(971\) −2.50323e10 −0.877471 −0.438735 0.898616i \(-0.644573\pi\)
−0.438735 + 0.898616i \(0.644573\pi\)
\(972\) 0 0
\(973\) 1.14787e10 0.399482
\(974\) −3.42403e10 −1.18736
\(975\) 0 0
\(976\) 6.08145e10 2.09379
\(977\) −3.21353e10 −1.10243 −0.551216 0.834363i \(-0.685836\pi\)
−0.551216 + 0.834363i \(0.685836\pi\)
\(978\) 0 0
\(979\) 3.43486e8 0.0116996
\(980\) 3.18688e11 10.8162
\(981\) 0 0
\(982\) 6.48464e9 0.218522
\(983\) 2.52657e10 0.848387 0.424194 0.905572i \(-0.360558\pi\)
0.424194 + 0.905572i \(0.360558\pi\)
\(984\) 0 0
\(985\) −6.35163e9 −0.211767
\(986\) 1.00216e10 0.332942
\(987\) 0 0
\(988\) 9.80849e10 3.23559
\(989\) 1.56096e10 0.513104
\(990\) 0 0
\(991\) −5.79422e10 −1.89120 −0.945600 0.325331i \(-0.894524\pi\)
−0.945600 + 0.325331i \(0.894524\pi\)
\(992\) 1.69825e11 5.52344
\(993\) 0 0
\(994\) −3.28217e10 −1.06001
\(995\) 1.43459e10 0.461687
\(996\) 0 0
\(997\) −4.76882e10 −1.52398 −0.761988 0.647591i \(-0.775777\pi\)
−0.761988 + 0.647591i \(0.775777\pi\)
\(998\) 2.92143e10 0.930334
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 531.8.a.d.1.16 17
3.2 odd 2 177.8.a.b.1.2 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.2 17 3.2 odd 2
531.8.a.d.1.16 17 1.1 even 1 trivial