Properties

Label 1521.4.a.a.1.1
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{2} +17.0000 q^{4} -7.00000 q^{5} +13.0000 q^{7} -45.0000 q^{8} +O(q^{10})\) \(q-5.00000 q^{2} +17.0000 q^{4} -7.00000 q^{5} +13.0000 q^{7} -45.0000 q^{8} +35.0000 q^{10} -26.0000 q^{11} -65.0000 q^{14} +89.0000 q^{16} -77.0000 q^{17} +126.000 q^{19} -119.000 q^{20} +130.000 q^{22} +96.0000 q^{23} -76.0000 q^{25} +221.000 q^{28} +82.0000 q^{29} -196.000 q^{31} -85.0000 q^{32} +385.000 q^{34} -91.0000 q^{35} +131.000 q^{37} -630.000 q^{38} +315.000 q^{40} +336.000 q^{41} -201.000 q^{43} -442.000 q^{44} -480.000 q^{46} -105.000 q^{47} -174.000 q^{49} +380.000 q^{50} +432.000 q^{53} +182.000 q^{55} -585.000 q^{56} -410.000 q^{58} -294.000 q^{59} -56.0000 q^{61} +980.000 q^{62} -287.000 q^{64} -478.000 q^{67} -1309.00 q^{68} +455.000 q^{70} +9.00000 q^{71} -98.0000 q^{73} -655.000 q^{74} +2142.00 q^{76} -338.000 q^{77} +1304.00 q^{79} -623.000 q^{80} -1680.00 q^{82} -308.000 q^{83} +539.000 q^{85} +1005.00 q^{86} +1170.00 q^{88} -1190.00 q^{89} +1632.00 q^{92} +525.000 q^{94} -882.000 q^{95} -70.0000 q^{97} +870.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 −1.76777 −0.883883 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 0 0
\(4\) 17.0000 2.12500
\(5\) −7.00000 −0.626099 −0.313050 0.949737i \(-0.601351\pi\)
−0.313050 + 0.949737i \(0.601351\pi\)
\(6\) 0 0
\(7\) 13.0000 0.701934 0.350967 0.936388i \(-0.385853\pi\)
0.350967 + 0.936388i \(0.385853\pi\)
\(8\) −45.0000 −1.98874
\(9\) 0 0
\(10\) 35.0000 1.10680
\(11\) −26.0000 −0.712663 −0.356332 0.934360i \(-0.615973\pi\)
−0.356332 + 0.934360i \(0.615973\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −65.0000 −1.24086
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) −77.0000 −1.09854 −0.549272 0.835644i \(-0.685095\pi\)
−0.549272 + 0.835644i \(0.685095\pi\)
\(18\) 0 0
\(19\) 126.000 1.52139 0.760694 0.649110i \(-0.224859\pi\)
0.760694 + 0.649110i \(0.224859\pi\)
\(20\) −119.000 −1.33046
\(21\) 0 0
\(22\) 130.000 1.25982
\(23\) 96.0000 0.870321 0.435161 0.900353i \(-0.356692\pi\)
0.435161 + 0.900353i \(0.356692\pi\)
\(24\) 0 0
\(25\) −76.0000 −0.608000
\(26\) 0 0
\(27\) 0 0
\(28\) 221.000 1.49161
\(29\) 82.0000 0.525070 0.262535 0.964923i \(-0.415442\pi\)
0.262535 + 0.964923i \(0.415442\pi\)
\(30\) 0 0
\(31\) −196.000 −1.13557 −0.567785 0.823177i \(-0.692199\pi\)
−0.567785 + 0.823177i \(0.692199\pi\)
\(32\) −85.0000 −0.469563
\(33\) 0 0
\(34\) 385.000 1.94197
\(35\) −91.0000 −0.439480
\(36\) 0 0
\(37\) 131.000 0.582061 0.291031 0.956714i \(-0.406002\pi\)
0.291031 + 0.956714i \(0.406002\pi\)
\(38\) −630.000 −2.68946
\(39\) 0 0
\(40\) 315.000 1.24515
\(41\) 336.000 1.27986 0.639932 0.768432i \(-0.278963\pi\)
0.639932 + 0.768432i \(0.278963\pi\)
\(42\) 0 0
\(43\) −201.000 −0.712842 −0.356421 0.934325i \(-0.616003\pi\)
−0.356421 + 0.934325i \(0.616003\pi\)
\(44\) −442.000 −1.51441
\(45\) 0 0
\(46\) −480.000 −1.53852
\(47\) −105.000 −0.325869 −0.162934 0.986637i \(-0.552096\pi\)
−0.162934 + 0.986637i \(0.552096\pi\)
\(48\) 0 0
\(49\) −174.000 −0.507289
\(50\) 380.000 1.07480
\(51\) 0 0
\(52\) 0 0
\(53\) 432.000 1.11962 0.559809 0.828622i \(-0.310874\pi\)
0.559809 + 0.828622i \(0.310874\pi\)
\(54\) 0 0
\(55\) 182.000 0.446198
\(56\) −585.000 −1.39596
\(57\) 0 0
\(58\) −410.000 −0.928201
\(59\) −294.000 −0.648738 −0.324369 0.945931i \(-0.605152\pi\)
−0.324369 + 0.945931i \(0.605152\pi\)
\(60\) 0 0
\(61\) −56.0000 −0.117542 −0.0587710 0.998271i \(-0.518718\pi\)
−0.0587710 + 0.998271i \(0.518718\pi\)
\(62\) 980.000 2.00742
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) 0 0
\(67\) −478.000 −0.871597 −0.435798 0.900044i \(-0.643534\pi\)
−0.435798 + 0.900044i \(0.643534\pi\)
\(68\) −1309.00 −2.33441
\(69\) 0 0
\(70\) 455.000 0.776899
\(71\) 9.00000 0.0150437 0.00752186 0.999972i \(-0.497606\pi\)
0.00752186 + 0.999972i \(0.497606\pi\)
\(72\) 0 0
\(73\) −98.0000 −0.157124 −0.0785619 0.996909i \(-0.525033\pi\)
−0.0785619 + 0.996909i \(0.525033\pi\)
\(74\) −655.000 −1.02895
\(75\) 0 0
\(76\) 2142.00 3.23295
\(77\) −338.000 −0.500243
\(78\) 0 0
\(79\) 1304.00 1.85711 0.928554 0.371198i \(-0.121053\pi\)
0.928554 + 0.371198i \(0.121053\pi\)
\(80\) −623.000 −0.870669
\(81\) 0 0
\(82\) −1680.00 −2.26250
\(83\) −308.000 −0.407318 −0.203659 0.979042i \(-0.565283\pi\)
−0.203659 + 0.979042i \(0.565283\pi\)
\(84\) 0 0
\(85\) 539.000 0.687797
\(86\) 1005.00 1.26014
\(87\) 0 0
\(88\) 1170.00 1.41730
\(89\) −1190.00 −1.41730 −0.708650 0.705560i \(-0.750696\pi\)
−0.708650 + 0.705560i \(0.750696\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1632.00 1.84943
\(93\) 0 0
\(94\) 525.000 0.576060
\(95\) −882.000 −0.952540
\(96\) 0 0
\(97\) −70.0000 −0.0732724 −0.0366362 0.999329i \(-0.511664\pi\)
−0.0366362 + 0.999329i \(0.511664\pi\)
\(98\) 870.000 0.896768
\(99\) 0 0
\(100\) −1292.00 −1.29200
\(101\) −420.000 −0.413778 −0.206889 0.978364i \(-0.566334\pi\)
−0.206889 + 0.978364i \(0.566334\pi\)
\(102\) 0 0
\(103\) 588.000 0.562499 0.281249 0.959635i \(-0.409251\pi\)
0.281249 + 0.959635i \(0.409251\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2160.00 −1.97922
\(107\) 684.000 0.617989 0.308994 0.951064i \(-0.400008\pi\)
0.308994 + 0.951064i \(0.400008\pi\)
\(108\) 0 0
\(109\) −373.000 −0.327770 −0.163885 0.986479i \(-0.552403\pi\)
−0.163885 + 0.986479i \(0.552403\pi\)
\(110\) −910.000 −0.788774
\(111\) 0 0
\(112\) 1157.00 0.976127
\(113\) 1734.00 1.44355 0.721774 0.692128i \(-0.243327\pi\)
0.721774 + 0.692128i \(0.243327\pi\)
\(114\) 0 0
\(115\) −672.000 −0.544907
\(116\) 1394.00 1.11577
\(117\) 0 0
\(118\) 1470.00 1.14682
\(119\) −1001.00 −0.771105
\(120\) 0 0
\(121\) −655.000 −0.492111
\(122\) 280.000 0.207787
\(123\) 0 0
\(124\) −3332.00 −2.41308
\(125\) 1407.00 1.00677
\(126\) 0 0
\(127\) 1892.00 1.32195 0.660976 0.750407i \(-0.270143\pi\)
0.660976 + 0.750407i \(0.270143\pi\)
\(128\) 2115.00 1.46048
\(129\) 0 0
\(130\) 0 0
\(131\) −1435.00 −0.957073 −0.478536 0.878068i \(-0.658833\pi\)
−0.478536 + 0.878068i \(0.658833\pi\)
\(132\) 0 0
\(133\) 1638.00 1.06791
\(134\) 2390.00 1.54078
\(135\) 0 0
\(136\) 3465.00 2.18472
\(137\) −1776.00 −1.10755 −0.553773 0.832667i \(-0.686813\pi\)
−0.553773 + 0.832667i \(0.686813\pi\)
\(138\) 0 0
\(139\) −1869.00 −1.14048 −0.570239 0.821479i \(-0.693150\pi\)
−0.570239 + 0.821479i \(0.693150\pi\)
\(140\) −1547.00 −0.933895
\(141\) 0 0
\(142\) −45.0000 −0.0265938
\(143\) 0 0
\(144\) 0 0
\(145\) −574.000 −0.328746
\(146\) 490.000 0.277758
\(147\) 0 0
\(148\) 2227.00 1.23688
\(149\) 2466.00 1.35586 0.677928 0.735128i \(-0.262878\pi\)
0.677928 + 0.735128i \(0.262878\pi\)
\(150\) 0 0
\(151\) 3323.00 1.79087 0.895437 0.445189i \(-0.146863\pi\)
0.895437 + 0.445189i \(0.146863\pi\)
\(152\) −5670.00 −3.02564
\(153\) 0 0
\(154\) 1690.00 0.884312
\(155\) 1372.00 0.710979
\(156\) 0 0
\(157\) −2730.00 −1.38776 −0.693878 0.720092i \(-0.744099\pi\)
−0.693878 + 0.720092i \(0.744099\pi\)
\(158\) −6520.00 −3.28293
\(159\) 0 0
\(160\) 595.000 0.293993
\(161\) 1248.00 0.610908
\(162\) 0 0
\(163\) 544.000 0.261407 0.130704 0.991421i \(-0.458276\pi\)
0.130704 + 0.991421i \(0.458276\pi\)
\(164\) 5712.00 2.71971
\(165\) 0 0
\(166\) 1540.00 0.720043
\(167\) 1624.00 0.752508 0.376254 0.926516i \(-0.377212\pi\)
0.376254 + 0.926516i \(0.377212\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −2695.00 −1.21587
\(171\) 0 0
\(172\) −3417.00 −1.51479
\(173\) 336.000 0.147662 0.0738312 0.997271i \(-0.476477\pi\)
0.0738312 + 0.997271i \(0.476477\pi\)
\(174\) 0 0
\(175\) −988.000 −0.426776
\(176\) −2314.00 −0.991047
\(177\) 0 0
\(178\) 5950.00 2.50546
\(179\) 3029.00 1.26479 0.632397 0.774645i \(-0.282071\pi\)
0.632397 + 0.774645i \(0.282071\pi\)
\(180\) 0 0
\(181\) −28.0000 −0.0114985 −0.00574924 0.999983i \(-0.501830\pi\)
−0.00574924 + 0.999983i \(0.501830\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4320.00 −1.73084
\(185\) −917.000 −0.364428
\(186\) 0 0
\(187\) 2002.00 0.782892
\(188\) −1785.00 −0.692471
\(189\) 0 0
\(190\) 4410.00 1.68387
\(191\) −422.000 −0.159868 −0.0799342 0.996800i \(-0.525471\pi\)
−0.0799342 + 0.996800i \(0.525471\pi\)
\(192\) 0 0
\(193\) −492.000 −0.183497 −0.0917485 0.995782i \(-0.529246\pi\)
−0.0917485 + 0.995782i \(0.529246\pi\)
\(194\) 350.000 0.129529
\(195\) 0 0
\(196\) −2958.00 −1.07799
\(197\) 2991.00 1.08173 0.540863 0.841111i \(-0.318098\pi\)
0.540863 + 0.841111i \(0.318098\pi\)
\(198\) 0 0
\(199\) −70.0000 −0.0249355 −0.0124678 0.999922i \(-0.503969\pi\)
−0.0124678 + 0.999922i \(0.503969\pi\)
\(200\) 3420.00 1.20915
\(201\) 0 0
\(202\) 2100.00 0.731463
\(203\) 1066.00 0.368564
\(204\) 0 0
\(205\) −2352.00 −0.801321
\(206\) −2940.00 −0.994367
\(207\) 0 0
\(208\) 0 0
\(209\) −3276.00 −1.08424
\(210\) 0 0
\(211\) 2851.00 0.930194 0.465097 0.885260i \(-0.346019\pi\)
0.465097 + 0.885260i \(0.346019\pi\)
\(212\) 7344.00 2.37919
\(213\) 0 0
\(214\) −3420.00 −1.09246
\(215\) 1407.00 0.446310
\(216\) 0 0
\(217\) −2548.00 −0.797095
\(218\) 1865.00 0.579421
\(219\) 0 0
\(220\) 3094.00 0.948170
\(221\) 0 0
\(222\) 0 0
\(223\) −217.000 −0.0651632 −0.0325816 0.999469i \(-0.510373\pi\)
−0.0325816 + 0.999469i \(0.510373\pi\)
\(224\) −1105.00 −0.329602
\(225\) 0 0
\(226\) −8670.00 −2.55186
\(227\) −2576.00 −0.753194 −0.376597 0.926377i \(-0.622906\pi\)
−0.376597 + 0.926377i \(0.622906\pi\)
\(228\) 0 0
\(229\) −455.000 −0.131298 −0.0656490 0.997843i \(-0.520912\pi\)
−0.0656490 + 0.997843i \(0.520912\pi\)
\(230\) 3360.00 0.963269
\(231\) 0 0
\(232\) −3690.00 −1.04423
\(233\) −3061.00 −0.860656 −0.430328 0.902673i \(-0.641602\pi\)
−0.430328 + 0.902673i \(0.641602\pi\)
\(234\) 0 0
\(235\) 735.000 0.204026
\(236\) −4998.00 −1.37857
\(237\) 0 0
\(238\) 5005.00 1.36313
\(239\) −3477.00 −0.941039 −0.470520 0.882389i \(-0.655934\pi\)
−0.470520 + 0.882389i \(0.655934\pi\)
\(240\) 0 0
\(241\) 1610.00 0.430329 0.215164 0.976578i \(-0.430971\pi\)
0.215164 + 0.976578i \(0.430971\pi\)
\(242\) 3275.00 0.869938
\(243\) 0 0
\(244\) −952.000 −0.249777
\(245\) 1218.00 0.317613
\(246\) 0 0
\(247\) 0 0
\(248\) 8820.00 2.25835
\(249\) 0 0
\(250\) −7035.00 −1.77973
\(251\) −1008.00 −0.253484 −0.126742 0.991936i \(-0.540452\pi\)
−0.126742 + 0.991936i \(0.540452\pi\)
\(252\) 0 0
\(253\) −2496.00 −0.620246
\(254\) −9460.00 −2.33690
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) −6041.00 −1.46625 −0.733127 0.680092i \(-0.761940\pi\)
−0.733127 + 0.680092i \(0.761940\pi\)
\(258\) 0 0
\(259\) 1703.00 0.408569
\(260\) 0 0
\(261\) 0 0
\(262\) 7175.00 1.69188
\(263\) 3708.00 0.869373 0.434686 0.900582i \(-0.356859\pi\)
0.434686 + 0.900582i \(0.356859\pi\)
\(264\) 0 0
\(265\) −3024.00 −0.700992
\(266\) −8190.00 −1.88782
\(267\) 0 0
\(268\) −8126.00 −1.85214
\(269\) −8344.00 −1.89124 −0.945618 0.325278i \(-0.894542\pi\)
−0.945618 + 0.325278i \(0.894542\pi\)
\(270\) 0 0
\(271\) 1617.00 0.362457 0.181228 0.983441i \(-0.441993\pi\)
0.181228 + 0.983441i \(0.441993\pi\)
\(272\) −6853.00 −1.52766
\(273\) 0 0
\(274\) 8880.00 1.95788
\(275\) 1976.00 0.433299
\(276\) 0 0
\(277\) −3820.00 −0.828598 −0.414299 0.910141i \(-0.635973\pi\)
−0.414299 + 0.910141i \(0.635973\pi\)
\(278\) 9345.00 2.01610
\(279\) 0 0
\(280\) 4095.00 0.874011
\(281\) −6214.00 −1.31920 −0.659602 0.751615i \(-0.729275\pi\)
−0.659602 + 0.751615i \(0.729275\pi\)
\(282\) 0 0
\(283\) −5292.00 −1.11158 −0.555789 0.831323i \(-0.687584\pi\)
−0.555789 + 0.831323i \(0.687584\pi\)
\(284\) 153.000 0.0319679
\(285\) 0 0
\(286\) 0 0
\(287\) 4368.00 0.898379
\(288\) 0 0
\(289\) 1016.00 0.206798
\(290\) 2870.00 0.581146
\(291\) 0 0
\(292\) −1666.00 −0.333888
\(293\) −903.000 −0.180047 −0.0900236 0.995940i \(-0.528694\pi\)
−0.0900236 + 0.995940i \(0.528694\pi\)
\(294\) 0 0
\(295\) 2058.00 0.406174
\(296\) −5895.00 −1.15757
\(297\) 0 0
\(298\) −12330.0 −2.39684
\(299\) 0 0
\(300\) 0 0
\(301\) −2613.00 −0.500368
\(302\) −16615.0 −3.16585
\(303\) 0 0
\(304\) 11214.0 2.11568
\(305\) 392.000 0.0735930
\(306\) 0 0
\(307\) −2114.00 −0.393004 −0.196502 0.980503i \(-0.562958\pi\)
−0.196502 + 0.980503i \(0.562958\pi\)
\(308\) −5746.00 −1.06302
\(309\) 0 0
\(310\) −6860.00 −1.25684
\(311\) −3402.00 −0.620288 −0.310144 0.950690i \(-0.600377\pi\)
−0.310144 + 0.950690i \(0.600377\pi\)
\(312\) 0 0
\(313\) −10689.0 −1.93028 −0.965141 0.261732i \(-0.915706\pi\)
−0.965141 + 0.261732i \(0.915706\pi\)
\(314\) 13650.0 2.45323
\(315\) 0 0
\(316\) 22168.0 3.94635
\(317\) −7054.00 −1.24982 −0.624909 0.780698i \(-0.714864\pi\)
−0.624909 + 0.780698i \(0.714864\pi\)
\(318\) 0 0
\(319\) −2132.00 −0.374198
\(320\) 2009.00 0.350958
\(321\) 0 0
\(322\) −6240.00 −1.07994
\(323\) −9702.00 −1.67131
\(324\) 0 0
\(325\) 0 0
\(326\) −2720.00 −0.462107
\(327\) 0 0
\(328\) −15120.0 −2.54531
\(329\) −1365.00 −0.228738
\(330\) 0 0
\(331\) −9704.00 −1.61142 −0.805710 0.592310i \(-0.798216\pi\)
−0.805710 + 0.592310i \(0.798216\pi\)
\(332\) −5236.00 −0.865551
\(333\) 0 0
\(334\) −8120.00 −1.33026
\(335\) 3346.00 0.545706
\(336\) 0 0
\(337\) −10449.0 −1.68900 −0.844500 0.535555i \(-0.820103\pi\)
−0.844500 + 0.535555i \(0.820103\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 9163.00 1.46157
\(341\) 5096.00 0.809278
\(342\) 0 0
\(343\) −6721.00 −1.05802
\(344\) 9045.00 1.41766
\(345\) 0 0
\(346\) −1680.00 −0.261033
\(347\) 621.000 0.0960721 0.0480361 0.998846i \(-0.484704\pi\)
0.0480361 + 0.998846i \(0.484704\pi\)
\(348\) 0 0
\(349\) −12481.0 −1.91431 −0.957153 0.289584i \(-0.906483\pi\)
−0.957153 + 0.289584i \(0.906483\pi\)
\(350\) 4940.00 0.754440
\(351\) 0 0
\(352\) 2210.00 0.334640
\(353\) −1400.00 −0.211089 −0.105545 0.994415i \(-0.533659\pi\)
−0.105545 + 0.994415i \(0.533659\pi\)
\(354\) 0 0
\(355\) −63.0000 −0.00941885
\(356\) −20230.0 −3.01176
\(357\) 0 0
\(358\) −15145.0 −2.23586
\(359\) −4968.00 −0.730365 −0.365182 0.930936i \(-0.618993\pi\)
−0.365182 + 0.930936i \(0.618993\pi\)
\(360\) 0 0
\(361\) 9017.00 1.31462
\(362\) 140.000 0.0203266
\(363\) 0 0
\(364\) 0 0
\(365\) 686.000 0.0983750
\(366\) 0 0
\(367\) 8722.00 1.24056 0.620279 0.784381i \(-0.287019\pi\)
0.620279 + 0.784381i \(0.287019\pi\)
\(368\) 8544.00 1.21029
\(369\) 0 0
\(370\) 4585.00 0.644224
\(371\) 5616.00 0.785898
\(372\) 0 0
\(373\) 10012.0 1.38982 0.694908 0.719098i \(-0.255445\pi\)
0.694908 + 0.719098i \(0.255445\pi\)
\(374\) −10010.0 −1.38397
\(375\) 0 0
\(376\) 4725.00 0.648067
\(377\) 0 0
\(378\) 0 0
\(379\) 3372.00 0.457013 0.228507 0.973542i \(-0.426616\pi\)
0.228507 + 0.973542i \(0.426616\pi\)
\(380\) −14994.0 −2.02415
\(381\) 0 0
\(382\) 2110.00 0.282610
\(383\) −847.000 −0.113002 −0.0565009 0.998403i \(-0.517994\pi\)
−0.0565009 + 0.998403i \(0.517994\pi\)
\(384\) 0 0
\(385\) 2366.00 0.313201
\(386\) 2460.00 0.324380
\(387\) 0 0
\(388\) −1190.00 −0.155704
\(389\) −11314.0 −1.47466 −0.737330 0.675533i \(-0.763914\pi\)
−0.737330 + 0.675533i \(0.763914\pi\)
\(390\) 0 0
\(391\) −7392.00 −0.956086
\(392\) 7830.00 1.00886
\(393\) 0 0
\(394\) −14955.0 −1.91224
\(395\) −9128.00 −1.16273
\(396\) 0 0
\(397\) −1862.00 −0.235393 −0.117697 0.993050i \(-0.537551\pi\)
−0.117697 + 0.993050i \(0.537551\pi\)
\(398\) 350.000 0.0440802
\(399\) 0 0
\(400\) −6764.00 −0.845500
\(401\) 6820.00 0.849313 0.424657 0.905355i \(-0.360395\pi\)
0.424657 + 0.905355i \(0.360395\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −7140.00 −0.879278
\(405\) 0 0
\(406\) −5330.00 −0.651536
\(407\) −3406.00 −0.414814
\(408\) 0 0
\(409\) 12992.0 1.57069 0.785346 0.619057i \(-0.212485\pi\)
0.785346 + 0.619057i \(0.212485\pi\)
\(410\) 11760.0 1.41655
\(411\) 0 0
\(412\) 9996.00 1.19531
\(413\) −3822.00 −0.455371
\(414\) 0 0
\(415\) 2156.00 0.255021
\(416\) 0 0
\(417\) 0 0
\(418\) 16380.0 1.91668
\(419\) 7343.00 0.856155 0.428078 0.903742i \(-0.359191\pi\)
0.428078 + 0.903742i \(0.359191\pi\)
\(420\) 0 0
\(421\) 5059.00 0.585655 0.292827 0.956165i \(-0.405404\pi\)
0.292827 + 0.956165i \(0.405404\pi\)
\(422\) −14255.0 −1.64437
\(423\) 0 0
\(424\) −19440.0 −2.22663
\(425\) 5852.00 0.667915
\(426\) 0 0
\(427\) −728.000 −0.0825068
\(428\) 11628.0 1.31323
\(429\) 0 0
\(430\) −7035.00 −0.788972
\(431\) 3243.00 0.362436 0.181218 0.983443i \(-0.441996\pi\)
0.181218 + 0.983443i \(0.441996\pi\)
\(432\) 0 0
\(433\) 11599.0 1.28733 0.643663 0.765309i \(-0.277414\pi\)
0.643663 + 0.765309i \(0.277414\pi\)
\(434\) 12740.0 1.40908
\(435\) 0 0
\(436\) −6341.00 −0.696511
\(437\) 12096.0 1.32410
\(438\) 0 0
\(439\) −17374.0 −1.88887 −0.944437 0.328692i \(-0.893392\pi\)
−0.944437 + 0.328692i \(0.893392\pi\)
\(440\) −8190.00 −0.887370
\(441\) 0 0
\(442\) 0 0
\(443\) −989.000 −0.106070 −0.0530348 0.998593i \(-0.516889\pi\)
−0.0530348 + 0.998593i \(0.516889\pi\)
\(444\) 0 0
\(445\) 8330.00 0.887370
\(446\) 1085.00 0.115193
\(447\) 0 0
\(448\) −3731.00 −0.393467
\(449\) −14474.0 −1.52131 −0.760657 0.649154i \(-0.775123\pi\)
−0.760657 + 0.649154i \(0.775123\pi\)
\(450\) 0 0
\(451\) −8736.00 −0.912111
\(452\) 29478.0 3.06754
\(453\) 0 0
\(454\) 12880.0 1.33147
\(455\) 0 0
\(456\) 0 0
\(457\) 1594.00 0.163160 0.0815801 0.996667i \(-0.474003\pi\)
0.0815801 + 0.996667i \(0.474003\pi\)
\(458\) 2275.00 0.232104
\(459\) 0 0
\(460\) −11424.0 −1.15793
\(461\) −5915.00 −0.597590 −0.298795 0.954317i \(-0.596585\pi\)
−0.298795 + 0.954317i \(0.596585\pi\)
\(462\) 0 0
\(463\) 11072.0 1.11136 0.555680 0.831396i \(-0.312458\pi\)
0.555680 + 0.831396i \(0.312458\pi\)
\(464\) 7298.00 0.730175
\(465\) 0 0
\(466\) 15305.0 1.52144
\(467\) −1260.00 −0.124852 −0.0624260 0.998050i \(-0.519884\pi\)
−0.0624260 + 0.998050i \(0.519884\pi\)
\(468\) 0 0
\(469\) −6214.00 −0.611804
\(470\) −3675.00 −0.360670
\(471\) 0 0
\(472\) 13230.0 1.29017
\(473\) 5226.00 0.508016
\(474\) 0 0
\(475\) −9576.00 −0.925004
\(476\) −17017.0 −1.63860
\(477\) 0 0
\(478\) 17385.0 1.66354
\(479\) −12033.0 −1.14781 −0.573906 0.818921i \(-0.694572\pi\)
−0.573906 + 0.818921i \(0.694572\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −8050.00 −0.760721
\(483\) 0 0
\(484\) −11135.0 −1.04574
\(485\) 490.000 0.0458758
\(486\) 0 0
\(487\) 2280.00 0.212149 0.106075 0.994358i \(-0.466172\pi\)
0.106075 + 0.994358i \(0.466172\pi\)
\(488\) 2520.00 0.233760
\(489\) 0 0
\(490\) −6090.00 −0.561466
\(491\) −16767.0 −1.54111 −0.770554 0.637375i \(-0.780020\pi\)
−0.770554 + 0.637375i \(0.780020\pi\)
\(492\) 0 0
\(493\) −6314.00 −0.576812
\(494\) 0 0
\(495\) 0 0
\(496\) −17444.0 −1.57915
\(497\) 117.000 0.0105597
\(498\) 0 0
\(499\) −12840.0 −1.15190 −0.575949 0.817485i \(-0.695367\pi\)
−0.575949 + 0.817485i \(0.695367\pi\)
\(500\) 23919.0 2.13938
\(501\) 0 0
\(502\) 5040.00 0.448100
\(503\) 2198.00 0.194839 0.0974195 0.995243i \(-0.468941\pi\)
0.0974195 + 0.995243i \(0.468941\pi\)
\(504\) 0 0
\(505\) 2940.00 0.259066
\(506\) 12480.0 1.09645
\(507\) 0 0
\(508\) 32164.0 2.80915
\(509\) −17066.0 −1.48612 −0.743062 0.669223i \(-0.766627\pi\)
−0.743062 + 0.669223i \(0.766627\pi\)
\(510\) 0 0
\(511\) −1274.00 −0.110290
\(512\) 24475.0 2.11260
\(513\) 0 0
\(514\) 30205.0 2.59200
\(515\) −4116.00 −0.352180
\(516\) 0 0
\(517\) 2730.00 0.232235
\(518\) −8515.00 −0.722254
\(519\) 0 0
\(520\) 0 0
\(521\) −2583.00 −0.217204 −0.108602 0.994085i \(-0.534637\pi\)
−0.108602 + 0.994085i \(0.534637\pi\)
\(522\) 0 0
\(523\) 18620.0 1.55678 0.778390 0.627781i \(-0.216037\pi\)
0.778390 + 0.627781i \(0.216037\pi\)
\(524\) −24395.0 −2.03378
\(525\) 0 0
\(526\) −18540.0 −1.53685
\(527\) 15092.0 1.24747
\(528\) 0 0
\(529\) −2951.00 −0.242541
\(530\) 15120.0 1.23919
\(531\) 0 0
\(532\) 27846.0 2.26932
\(533\) 0 0
\(534\) 0 0
\(535\) −4788.00 −0.386922
\(536\) 21510.0 1.73338
\(537\) 0 0
\(538\) 41720.0 3.34327
\(539\) 4524.00 0.361526
\(540\) 0 0
\(541\) 16833.0 1.33772 0.668861 0.743388i \(-0.266782\pi\)
0.668861 + 0.743388i \(0.266782\pi\)
\(542\) −8085.00 −0.640739
\(543\) 0 0
\(544\) 6545.00 0.515836
\(545\) 2611.00 0.205216
\(546\) 0 0
\(547\) −8615.00 −0.673402 −0.336701 0.941612i \(-0.609311\pi\)
−0.336701 + 0.941612i \(0.609311\pi\)
\(548\) −30192.0 −2.35354
\(549\) 0 0
\(550\) −9880.00 −0.765972
\(551\) 10332.0 0.798835
\(552\) 0 0
\(553\) 16952.0 1.30357
\(554\) 19100.0 1.46477
\(555\) 0 0
\(556\) −31773.0 −2.42352
\(557\) 8535.00 0.649263 0.324632 0.945841i \(-0.394760\pi\)
0.324632 + 0.945841i \(0.394760\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −8099.00 −0.611152
\(561\) 0 0
\(562\) 31070.0 2.33204
\(563\) 4641.00 0.347415 0.173708 0.984797i \(-0.444425\pi\)
0.173708 + 0.984797i \(0.444425\pi\)
\(564\) 0 0
\(565\) −12138.0 −0.903804
\(566\) 26460.0 1.96501
\(567\) 0 0
\(568\) −405.000 −0.0299180
\(569\) 4793.00 0.353134 0.176567 0.984289i \(-0.443501\pi\)
0.176567 + 0.984289i \(0.443501\pi\)
\(570\) 0 0
\(571\) −5563.00 −0.407713 −0.203857 0.979001i \(-0.565348\pi\)
−0.203857 + 0.979001i \(0.565348\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −21840.0 −1.58813
\(575\) −7296.00 −0.529155
\(576\) 0 0
\(577\) −24038.0 −1.73434 −0.867171 0.498011i \(-0.834064\pi\)
−0.867171 + 0.498011i \(0.834064\pi\)
\(578\) −5080.00 −0.365571
\(579\) 0 0
\(580\) −9758.00 −0.698584
\(581\) −4004.00 −0.285910
\(582\) 0 0
\(583\) −11232.0 −0.797911
\(584\) 4410.00 0.312478
\(585\) 0 0
\(586\) 4515.00 0.318281
\(587\) −21224.0 −1.49235 −0.746174 0.665751i \(-0.768111\pi\)
−0.746174 + 0.665751i \(0.768111\pi\)
\(588\) 0 0
\(589\) −24696.0 −1.72764
\(590\) −10290.0 −0.718021
\(591\) 0 0
\(592\) 11659.0 0.809429
\(593\) 4354.00 0.301513 0.150757 0.988571i \(-0.451829\pi\)
0.150757 + 0.988571i \(0.451829\pi\)
\(594\) 0 0
\(595\) 7007.00 0.482788
\(596\) 41922.0 2.88119
\(597\) 0 0
\(598\) 0 0
\(599\) −7310.00 −0.498629 −0.249314 0.968423i \(-0.580205\pi\)
−0.249314 + 0.968423i \(0.580205\pi\)
\(600\) 0 0
\(601\) −7595.00 −0.515485 −0.257743 0.966214i \(-0.582979\pi\)
−0.257743 + 0.966214i \(0.582979\pi\)
\(602\) 13065.0 0.884534
\(603\) 0 0
\(604\) 56491.0 3.80561
\(605\) 4585.00 0.308110
\(606\) 0 0
\(607\) −826.000 −0.0552328 −0.0276164 0.999619i \(-0.508792\pi\)
−0.0276164 + 0.999619i \(0.508792\pi\)
\(608\) −10710.0 −0.714388
\(609\) 0 0
\(610\) −1960.00 −0.130095
\(611\) 0 0
\(612\) 0 0
\(613\) −14590.0 −0.961312 −0.480656 0.876909i \(-0.659602\pi\)
−0.480656 + 0.876909i \(0.659602\pi\)
\(614\) 10570.0 0.694740
\(615\) 0 0
\(616\) 15210.0 0.994851
\(617\) 4888.00 0.318936 0.159468 0.987203i \(-0.449022\pi\)
0.159468 + 0.987203i \(0.449022\pi\)
\(618\) 0 0
\(619\) 11004.0 0.714520 0.357260 0.934005i \(-0.383711\pi\)
0.357260 + 0.934005i \(0.383711\pi\)
\(620\) 23324.0 1.51083
\(621\) 0 0
\(622\) 17010.0 1.09653
\(623\) −15470.0 −0.994851
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) 53445.0 3.41229
\(627\) 0 0
\(628\) −46410.0 −2.94898
\(629\) −10087.0 −0.639420
\(630\) 0 0
\(631\) 4975.00 0.313869 0.156935 0.987609i \(-0.449839\pi\)
0.156935 + 0.987609i \(0.449839\pi\)
\(632\) −58680.0 −3.69330
\(633\) 0 0
\(634\) 35270.0 2.20939
\(635\) −13244.0 −0.827673
\(636\) 0 0
\(637\) 0 0
\(638\) 10660.0 0.661494
\(639\) 0 0
\(640\) −14805.0 −0.914405
\(641\) −3950.00 −0.243394 −0.121697 0.992567i \(-0.538834\pi\)
−0.121697 + 0.992567i \(0.538834\pi\)
\(642\) 0 0
\(643\) 3682.00 0.225823 0.112911 0.993605i \(-0.463982\pi\)
0.112911 + 0.993605i \(0.463982\pi\)
\(644\) 21216.0 1.29818
\(645\) 0 0
\(646\) 48510.0 2.95449
\(647\) −10402.0 −0.632063 −0.316032 0.948749i \(-0.602351\pi\)
−0.316032 + 0.948749i \(0.602351\pi\)
\(648\) 0 0
\(649\) 7644.00 0.462332
\(650\) 0 0
\(651\) 0 0
\(652\) 9248.00 0.555490
\(653\) 31680.0 1.89852 0.949260 0.314491i \(-0.101834\pi\)
0.949260 + 0.314491i \(0.101834\pi\)
\(654\) 0 0
\(655\) 10045.0 0.599222
\(656\) 29904.0 1.77981
\(657\) 0 0
\(658\) 6825.00 0.404356
\(659\) −21940.0 −1.29691 −0.648453 0.761255i \(-0.724584\pi\)
−0.648453 + 0.761255i \(0.724584\pi\)
\(660\) 0 0
\(661\) 31374.0 1.84615 0.923077 0.384616i \(-0.125666\pi\)
0.923077 + 0.384616i \(0.125666\pi\)
\(662\) 48520.0 2.84862
\(663\) 0 0
\(664\) 13860.0 0.810049
\(665\) −11466.0 −0.668620
\(666\) 0 0
\(667\) 7872.00 0.456979
\(668\) 27608.0 1.59908
\(669\) 0 0
\(670\) −16730.0 −0.964681
\(671\) 1456.00 0.0837679
\(672\) 0 0
\(673\) 18013.0 1.03172 0.515862 0.856672i \(-0.327472\pi\)
0.515862 + 0.856672i \(0.327472\pi\)
\(674\) 52245.0 2.98576
\(675\) 0 0
\(676\) 0 0
\(677\) 10640.0 0.604030 0.302015 0.953303i \(-0.402341\pi\)
0.302015 + 0.953303i \(0.402341\pi\)
\(678\) 0 0
\(679\) −910.000 −0.0514324
\(680\) −24255.0 −1.36785
\(681\) 0 0
\(682\) −25480.0 −1.43062
\(683\) −9336.00 −0.523034 −0.261517 0.965199i \(-0.584223\pi\)
−0.261517 + 0.965199i \(0.584223\pi\)
\(684\) 0 0
\(685\) 12432.0 0.693434
\(686\) 33605.0 1.87033
\(687\) 0 0
\(688\) −17889.0 −0.991296
\(689\) 0 0
\(690\) 0 0
\(691\) −4200.00 −0.231224 −0.115612 0.993294i \(-0.536883\pi\)
−0.115612 + 0.993294i \(0.536883\pi\)
\(692\) 5712.00 0.313783
\(693\) 0 0
\(694\) −3105.00 −0.169833
\(695\) 13083.0 0.714052
\(696\) 0 0
\(697\) −25872.0 −1.40599
\(698\) 62405.0 3.38405
\(699\) 0 0
\(700\) −16796.0 −0.906899
\(701\) −9872.00 −0.531898 −0.265949 0.963987i \(-0.585685\pi\)
−0.265949 + 0.963987i \(0.585685\pi\)
\(702\) 0 0
\(703\) 16506.0 0.885541
\(704\) 7462.00 0.399481
\(705\) 0 0
\(706\) 7000.00 0.373156
\(707\) −5460.00 −0.290445
\(708\) 0 0
\(709\) −28450.0 −1.50700 −0.753499 0.657449i \(-0.771636\pi\)
−0.753499 + 0.657449i \(0.771636\pi\)
\(710\) 315.000 0.0166503
\(711\) 0 0
\(712\) 53550.0 2.81864
\(713\) −18816.0 −0.988310
\(714\) 0 0
\(715\) 0 0
\(716\) 51493.0 2.68769
\(717\) 0 0
\(718\) 24840.0 1.29111
\(719\) −32718.0 −1.69705 −0.848523 0.529159i \(-0.822507\pi\)
−0.848523 + 0.529159i \(0.822507\pi\)
\(720\) 0 0
\(721\) 7644.00 0.394837
\(722\) −45085.0 −2.32395
\(723\) 0 0
\(724\) −476.000 −0.0244343
\(725\) −6232.00 −0.319242
\(726\) 0 0
\(727\) −22834.0 −1.16488 −0.582439 0.812874i \(-0.697901\pi\)
−0.582439 + 0.812874i \(0.697901\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −3430.00 −0.173904
\(731\) 15477.0 0.783088
\(732\) 0 0
\(733\) −7875.00 −0.396821 −0.198410 0.980119i \(-0.563578\pi\)
−0.198410 + 0.980119i \(0.563578\pi\)
\(734\) −43610.0 −2.19302
\(735\) 0 0
\(736\) −8160.00 −0.408671
\(737\) 12428.0 0.621155
\(738\) 0 0
\(739\) 2140.00 0.106524 0.0532620 0.998581i \(-0.483038\pi\)
0.0532620 + 0.998581i \(0.483038\pi\)
\(740\) −15589.0 −0.774410
\(741\) 0 0
\(742\) −28080.0 −1.38928
\(743\) 31971.0 1.57860 0.789302 0.614006i \(-0.210443\pi\)
0.789302 + 0.614006i \(0.210443\pi\)
\(744\) 0 0
\(745\) −17262.0 −0.848900
\(746\) −50060.0 −2.45687
\(747\) 0 0
\(748\) 34034.0 1.66364
\(749\) 8892.00 0.433787
\(750\) 0 0
\(751\) −7432.00 −0.361115 −0.180558 0.983564i \(-0.557790\pi\)
−0.180558 + 0.983564i \(0.557790\pi\)
\(752\) −9345.00 −0.453161
\(753\) 0 0
\(754\) 0 0
\(755\) −23261.0 −1.12126
\(756\) 0 0
\(757\) 20176.0 0.968704 0.484352 0.874873i \(-0.339055\pi\)
0.484352 + 0.874873i \(0.339055\pi\)
\(758\) −16860.0 −0.807893
\(759\) 0 0
\(760\) 39690.0 1.89435
\(761\) −9478.00 −0.451481 −0.225741 0.974187i \(-0.572480\pi\)
−0.225741 + 0.974187i \(0.572480\pi\)
\(762\) 0 0
\(763\) −4849.00 −0.230073
\(764\) −7174.00 −0.339720
\(765\) 0 0
\(766\) 4235.00 0.199761
\(767\) 0 0
\(768\) 0 0
\(769\) 12096.0 0.567221 0.283610 0.958940i \(-0.408468\pi\)
0.283610 + 0.958940i \(0.408468\pi\)
\(770\) −11830.0 −0.553667
\(771\) 0 0
\(772\) −8364.00 −0.389931
\(773\) 17941.0 0.834790 0.417395 0.908725i \(-0.362943\pi\)
0.417395 + 0.908725i \(0.362943\pi\)
\(774\) 0 0
\(775\) 14896.0 0.690426
\(776\) 3150.00 0.145720
\(777\) 0 0
\(778\) 56570.0 2.60685
\(779\) 42336.0 1.94717
\(780\) 0 0
\(781\) −234.000 −0.0107211
\(782\) 36960.0 1.69014
\(783\) 0 0
\(784\) −15486.0 −0.705448
\(785\) 19110.0 0.868873
\(786\) 0 0
\(787\) −6664.00 −0.301837 −0.150919 0.988546i \(-0.548223\pi\)
−0.150919 + 0.988546i \(0.548223\pi\)
\(788\) 50847.0 2.29867
\(789\) 0 0
\(790\) 45640.0 2.05544
\(791\) 22542.0 1.01328
\(792\) 0 0
\(793\) 0 0
\(794\) 9310.00 0.416120
\(795\) 0 0
\(796\) −1190.00 −0.0529880
\(797\) 1442.00 0.0640882 0.0320441 0.999486i \(-0.489798\pi\)
0.0320441 + 0.999486i \(0.489798\pi\)
\(798\) 0 0
\(799\) 8085.00 0.357981
\(800\) 6460.00 0.285494
\(801\) 0 0
\(802\) −34100.0 −1.50139
\(803\) 2548.00 0.111976
\(804\) 0 0
\(805\) −8736.00 −0.382489
\(806\) 0 0
\(807\) 0 0
\(808\) 18900.0 0.822896
\(809\) −30207.0 −1.31276 −0.656379 0.754431i \(-0.727913\pi\)
−0.656379 + 0.754431i \(0.727913\pi\)
\(810\) 0 0
\(811\) −21140.0 −0.915322 −0.457661 0.889127i \(-0.651313\pi\)
−0.457661 + 0.889127i \(0.651313\pi\)
\(812\) 18122.0 0.783199
\(813\) 0 0
\(814\) 17030.0 0.733294
\(815\) −3808.00 −0.163667
\(816\) 0 0
\(817\) −25326.0 −1.08451
\(818\) −64960.0 −2.77662
\(819\) 0 0
\(820\) −39984.0 −1.70281
\(821\) 569.000 0.0241879 0.0120939 0.999927i \(-0.496150\pi\)
0.0120939 + 0.999927i \(0.496150\pi\)
\(822\) 0 0
\(823\) −8538.00 −0.361623 −0.180812 0.983518i \(-0.557872\pi\)
−0.180812 + 0.983518i \(0.557872\pi\)
\(824\) −26460.0 −1.11866
\(825\) 0 0
\(826\) 19110.0 0.804990
\(827\) −32702.0 −1.37504 −0.687521 0.726164i \(-0.741301\pi\)
−0.687521 + 0.726164i \(0.741301\pi\)
\(828\) 0 0
\(829\) −21154.0 −0.886259 −0.443130 0.896458i \(-0.646132\pi\)
−0.443130 + 0.896458i \(0.646132\pi\)
\(830\) −10780.0 −0.450818
\(831\) 0 0
\(832\) 0 0
\(833\) 13398.0 0.557279
\(834\) 0 0
\(835\) −11368.0 −0.471145
\(836\) −55692.0 −2.30400
\(837\) 0 0
\(838\) −36715.0 −1.51348
\(839\) −2184.00 −0.0898690 −0.0449345 0.998990i \(-0.514308\pi\)
−0.0449345 + 0.998990i \(0.514308\pi\)
\(840\) 0 0
\(841\) −17665.0 −0.724302
\(842\) −25295.0 −1.03530
\(843\) 0 0
\(844\) 48467.0 1.97666
\(845\) 0 0
\(846\) 0 0
\(847\) −8515.00 −0.345430
\(848\) 38448.0 1.55697
\(849\) 0 0
\(850\) −29260.0 −1.18072
\(851\) 12576.0 0.506580
\(852\) 0 0
\(853\) −36687.0 −1.47261 −0.736307 0.676648i \(-0.763432\pi\)
−0.736307 + 0.676648i \(0.763432\pi\)
\(854\) 3640.00 0.145853
\(855\) 0 0
\(856\) −30780.0 −1.22902
\(857\) −36806.0 −1.46706 −0.733529 0.679658i \(-0.762128\pi\)
−0.733529 + 0.679658i \(0.762128\pi\)
\(858\) 0 0
\(859\) 4900.00 0.194628 0.0973142 0.995254i \(-0.468975\pi\)
0.0973142 + 0.995254i \(0.468975\pi\)
\(860\) 23919.0 0.948408
\(861\) 0 0
\(862\) −16215.0 −0.640702
\(863\) −13697.0 −0.540268 −0.270134 0.962823i \(-0.587068\pi\)
−0.270134 + 0.962823i \(0.587068\pi\)
\(864\) 0 0
\(865\) −2352.00 −0.0924513
\(866\) −57995.0 −2.27569
\(867\) 0 0
\(868\) −43316.0 −1.69383
\(869\) −33904.0 −1.32349
\(870\) 0 0
\(871\) 0 0
\(872\) 16785.0 0.651848
\(873\) 0 0
\(874\) −60480.0 −2.34069
\(875\) 18291.0 0.706684
\(876\) 0 0
\(877\) −6239.00 −0.240224 −0.120112 0.992760i \(-0.538325\pi\)
−0.120112 + 0.992760i \(0.538325\pi\)
\(878\) 86870.0 3.33909
\(879\) 0 0
\(880\) 16198.0 0.620494
\(881\) −133.000 −0.00508613 −0.00254307 0.999997i \(-0.500809\pi\)
−0.00254307 + 0.999997i \(0.500809\pi\)
\(882\) 0 0
\(883\) −26003.0 −0.991020 −0.495510 0.868602i \(-0.665019\pi\)
−0.495510 + 0.868602i \(0.665019\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 4945.00 0.187506
\(887\) 31248.0 1.18287 0.591435 0.806353i \(-0.298562\pi\)
0.591435 + 0.806353i \(0.298562\pi\)
\(888\) 0 0
\(889\) 24596.0 0.927923
\(890\) −41650.0 −1.56866
\(891\) 0 0
\(892\) −3689.00 −0.138472
\(893\) −13230.0 −0.495773
\(894\) 0 0
\(895\) −21203.0 −0.791886
\(896\) 27495.0 1.02516
\(897\) 0 0
\(898\) 72370.0 2.68933
\(899\) −16072.0 −0.596253
\(900\) 0 0
\(901\) −33264.0 −1.22995
\(902\) 43680.0 1.61240
\(903\) 0 0
\(904\) −78030.0 −2.87084
\(905\) 196.000 0.00719918
\(906\) 0 0
\(907\) −38253.0 −1.40041 −0.700204 0.713943i \(-0.746908\pi\)
−0.700204 + 0.713943i \(0.746908\pi\)
\(908\) −43792.0 −1.60054
\(909\) 0 0
\(910\) 0 0
\(911\) −36374.0 −1.32286 −0.661429 0.750007i \(-0.730050\pi\)
−0.661429 + 0.750007i \(0.730050\pi\)
\(912\) 0 0
\(913\) 8008.00 0.290281
\(914\) −7970.00 −0.288429
\(915\) 0 0
\(916\) −7735.00 −0.279008
\(917\) −18655.0 −0.671802
\(918\) 0 0
\(919\) −27648.0 −0.992408 −0.496204 0.868206i \(-0.665273\pi\)
−0.496204 + 0.868206i \(0.665273\pi\)
\(920\) 30240.0 1.08368
\(921\) 0 0
\(922\) 29575.0 1.05640
\(923\) 0 0
\(924\) 0 0
\(925\) −9956.00 −0.353893
\(926\) −55360.0 −1.96462
\(927\) 0 0
\(928\) −6970.00 −0.246553
\(929\) 756.000 0.0266992 0.0133496 0.999911i \(-0.495751\pi\)
0.0133496 + 0.999911i \(0.495751\pi\)
\(930\) 0 0
\(931\) −21924.0 −0.771783
\(932\) −52037.0 −1.82889
\(933\) 0 0
\(934\) 6300.00 0.220709
\(935\) −14014.0 −0.490168
\(936\) 0 0
\(937\) 20846.0 0.726797 0.363399 0.931634i \(-0.381616\pi\)
0.363399 + 0.931634i \(0.381616\pi\)
\(938\) 31070.0 1.08153
\(939\) 0 0
\(940\) 12495.0 0.433555
\(941\) −41321.0 −1.43148 −0.715742 0.698365i \(-0.753911\pi\)
−0.715742 + 0.698365i \(0.753911\pi\)
\(942\) 0 0
\(943\) 32256.0 1.11389
\(944\) −26166.0 −0.902151
\(945\) 0 0
\(946\) −26130.0 −0.898055
\(947\) 54966.0 1.88612 0.943060 0.332624i \(-0.107934\pi\)
0.943060 + 0.332624i \(0.107934\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 47880.0 1.63519
\(951\) 0 0
\(952\) 45045.0 1.53353
\(953\) 44553.0 1.51439 0.757195 0.653189i \(-0.226569\pi\)
0.757195 + 0.653189i \(0.226569\pi\)
\(954\) 0 0
\(955\) 2954.00 0.100093
\(956\) −59109.0 −1.99971
\(957\) 0 0
\(958\) 60165.0 2.02906
\(959\) −23088.0 −0.777425
\(960\) 0 0
\(961\) 8625.00 0.289517
\(962\) 0 0
\(963\) 0 0
\(964\) 27370.0 0.914448
\(965\) 3444.00 0.114887
\(966\) 0 0
\(967\) 27907.0 0.928054 0.464027 0.885821i \(-0.346404\pi\)
0.464027 + 0.885821i \(0.346404\pi\)
\(968\) 29475.0 0.978680
\(969\) 0 0
\(970\) −2450.00 −0.0810977
\(971\) 16443.0 0.543441 0.271720 0.962376i \(-0.412407\pi\)
0.271720 + 0.962376i \(0.412407\pi\)
\(972\) 0 0
\(973\) −24297.0 −0.800541
\(974\) −11400.0 −0.375030
\(975\) 0 0
\(976\) −4984.00 −0.163457
\(977\) −45414.0 −1.48713 −0.743563 0.668666i \(-0.766866\pi\)
−0.743563 + 0.668666i \(0.766866\pi\)
\(978\) 0 0
\(979\) 30940.0 1.01006
\(980\) 20706.0 0.674927
\(981\) 0 0
\(982\) 83835.0 2.72432
\(983\) −8981.00 −0.291403 −0.145702 0.989329i \(-0.546544\pi\)
−0.145702 + 0.989329i \(0.546544\pi\)
\(984\) 0 0
\(985\) −20937.0 −0.677267
\(986\) 31570.0 1.01967
\(987\) 0 0
\(988\) 0 0
\(989\) −19296.0 −0.620402
\(990\) 0 0
\(991\) −17414.0 −0.558198 −0.279099 0.960262i \(-0.590036\pi\)
−0.279099 + 0.960262i \(0.590036\pi\)
\(992\) 16660.0 0.533221
\(993\) 0 0
\(994\) −585.000 −0.0186671
\(995\) 490.000 0.0156121
\(996\) 0 0
\(997\) −23702.0 −0.752909 −0.376454 0.926435i \(-0.622857\pi\)
−0.376454 + 0.926435i \(0.622857\pi\)
\(998\) 64200.0 2.03629
\(999\) 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 1521.4.a.a.1.1 1
3.2 odd 2 169.4.a.e.1.1 1
13.12 even 2 117.4.a.b.1.1 1
39.2 even 12 169.4.e.e.147.2 4
39.5 even 4 169.4.b.a.168.1 2
39.8 even 4 169.4.b.a.168.2 2
39.11 even 12 169.4.e.e.147.1 4
39.17 odd 6 169.4.c.e.146.1 2
39.20 even 12 169.4.e.e.23.2 4
39.23 odd 6 169.4.c.e.22.1 2
39.29 odd 6 169.4.c.a.22.1 2
39.32 even 12 169.4.e.e.23.1 4
39.35 odd 6 169.4.c.a.146.1 2
39.38 odd 2 13.4.a.a.1.1 1
52.51 odd 2 1872.4.a.k.1.1 1
156.155 even 2 208.4.a.g.1.1 1
195.38 even 4 325.4.b.b.274.2 2
195.77 even 4 325.4.b.b.274.1 2
195.194 odd 2 325.4.a.d.1.1 1
273.272 even 2 637.4.a.a.1.1 1
312.77 odd 2 832.4.a.r.1.1 1
312.155 even 2 832.4.a.a.1.1 1
429.428 even 2 1573.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.a.1.1 1 39.38 odd 2
117.4.a.b.1.1 1 13.12 even 2
169.4.a.e.1.1 1 3.2 odd 2
169.4.b.a.168.1 2 39.5 even 4
169.4.b.a.168.2 2 39.8 even 4
169.4.c.a.22.1 2 39.29 odd 6
169.4.c.a.146.1 2 39.35 odd 6
169.4.c.e.22.1 2 39.23 odd 6
169.4.c.e.146.1 2 39.17 odd 6
169.4.e.e.23.1 4 39.32 even 12
169.4.e.e.23.2 4 39.20 even 12
169.4.e.e.147.1 4 39.11 even 12
169.4.e.e.147.2 4 39.2 even 12
208.4.a.g.1.1 1 156.155 even 2
325.4.a.d.1.1 1 195.194 odd 2
325.4.b.b.274.1 2 195.77 even 4
325.4.b.b.274.2 2 195.38 even 4
637.4.a.a.1.1 1 273.272 even 2
832.4.a.a.1.1 1 312.155 even 2
832.4.a.r.1.1 1 312.77 odd 2
1521.4.a.a.1.1 1 1.1 even 1 trivial
1573.4.a.a.1.1 1 429.428 even 2
1872.4.a.k.1.1 1 52.51 odd 2