Properties

Label 2352.4.a.h.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -4.00000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -4.00000 q^{5} +9.00000 q^{9} +26.0000 q^{11} -2.00000 q^{13} +12.0000 q^{15} +36.0000 q^{17} -76.0000 q^{19} +114.000 q^{23} -109.000 q^{25} -27.0000 q^{27} +6.00000 q^{29} -256.000 q^{31} -78.0000 q^{33} -86.0000 q^{37} +6.00000 q^{39} -160.000 q^{41} +220.000 q^{43} -36.0000 q^{45} +308.000 q^{47} -108.000 q^{51} +258.000 q^{53} -104.000 q^{55} +228.000 q^{57} +264.000 q^{59} -606.000 q^{61} +8.00000 q^{65} +520.000 q^{67} -342.000 q^{69} +286.000 q^{71} +530.000 q^{73} +327.000 q^{75} +44.0000 q^{79} +81.0000 q^{81} +1012.00 q^{83} -144.000 q^{85} -18.0000 q^{87} -768.000 q^{89} +768.000 q^{93} +304.000 q^{95} -222.000 q^{97} +234.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −4.00000 −0.357771 −0.178885 0.983870i \(-0.557249\pi\)
−0.178885 + 0.983870i \(0.557249\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 26.0000 0.712663 0.356332 0.934360i \(-0.384027\pi\)
0.356332 + 0.934360i \(0.384027\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.0426692 −0.0213346 0.999772i \(-0.506792\pi\)
−0.0213346 + 0.999772i \(0.506792\pi\)
\(14\) 0 0
\(15\) 12.0000 0.206559
\(16\) 0 0
\(17\) 36.0000 0.513605 0.256802 0.966464i \(-0.417331\pi\)
0.256802 + 0.966464i \(0.417331\pi\)
\(18\) 0 0
\(19\) −76.0000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 114.000 1.03351 0.516753 0.856134i \(-0.327141\pi\)
0.516753 + 0.856134i \(0.327141\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 6.00000 0.0384197 0.0192099 0.999815i \(-0.493885\pi\)
0.0192099 + 0.999815i \(0.493885\pi\)
\(30\) 0 0
\(31\) −256.000 −1.48319 −0.741596 0.670847i \(-0.765931\pi\)
−0.741596 + 0.670847i \(0.765931\pi\)
\(32\) 0 0
\(33\) −78.0000 −0.411456
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −86.0000 −0.382117 −0.191058 0.981579i \(-0.561192\pi\)
−0.191058 + 0.981579i \(0.561192\pi\)
\(38\) 0 0
\(39\) 6.00000 0.0246351
\(40\) 0 0
\(41\) −160.000 −0.609459 −0.304729 0.952439i \(-0.598566\pi\)
−0.304729 + 0.952439i \(0.598566\pi\)
\(42\) 0 0
\(43\) 220.000 0.780225 0.390113 0.920767i \(-0.372436\pi\)
0.390113 + 0.920767i \(0.372436\pi\)
\(44\) 0 0
\(45\) −36.0000 −0.119257
\(46\) 0 0
\(47\) 308.000 0.955881 0.477941 0.878392i \(-0.341383\pi\)
0.477941 + 0.878392i \(0.341383\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −108.000 −0.296530
\(52\) 0 0
\(53\) 258.000 0.668661 0.334330 0.942456i \(-0.391490\pi\)
0.334330 + 0.942456i \(0.391490\pi\)
\(54\) 0 0
\(55\) −104.000 −0.254970
\(56\) 0 0
\(57\) 228.000 0.529813
\(58\) 0 0
\(59\) 264.000 0.582540 0.291270 0.956641i \(-0.405922\pi\)
0.291270 + 0.956641i \(0.405922\pi\)
\(60\) 0 0
\(61\) −606.000 −1.27197 −0.635986 0.771700i \(-0.719407\pi\)
−0.635986 + 0.771700i \(0.719407\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.00000 0.0152658
\(66\) 0 0
\(67\) 520.000 0.948181 0.474090 0.880476i \(-0.342777\pi\)
0.474090 + 0.880476i \(0.342777\pi\)
\(68\) 0 0
\(69\) −342.000 −0.596695
\(70\) 0 0
\(71\) 286.000 0.478056 0.239028 0.971013i \(-0.423171\pi\)
0.239028 + 0.971013i \(0.423171\pi\)
\(72\) 0 0
\(73\) 530.000 0.849751 0.424875 0.905252i \(-0.360318\pi\)
0.424875 + 0.905252i \(0.360318\pi\)
\(74\) 0 0
\(75\) 327.000 0.503449
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 44.0000 0.0626631 0.0313316 0.999509i \(-0.490025\pi\)
0.0313316 + 0.999509i \(0.490025\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1012.00 1.33833 0.669165 0.743114i \(-0.266652\pi\)
0.669165 + 0.743114i \(0.266652\pi\)
\(84\) 0 0
\(85\) −144.000 −0.183753
\(86\) 0 0
\(87\) −18.0000 −0.0221816
\(88\) 0 0
\(89\) −768.000 −0.914695 −0.457347 0.889288i \(-0.651200\pi\)
−0.457347 + 0.889288i \(0.651200\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 768.000 0.856321
\(94\) 0 0
\(95\) 304.000 0.328313
\(96\) 0 0
\(97\) −222.000 −0.232378 −0.116189 0.993227i \(-0.537068\pi\)
−0.116189 + 0.993227i \(0.537068\pi\)
\(98\) 0 0
\(99\) 234.000 0.237554
\(100\) 0 0
\(101\) 320.000 0.315259 0.157630 0.987498i \(-0.449615\pi\)
0.157630 + 0.987498i \(0.449615\pi\)
\(102\) 0 0
\(103\) −592.000 −0.566325 −0.283163 0.959072i \(-0.591384\pi\)
−0.283163 + 0.959072i \(0.591384\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1782.00 −1.61002 −0.805011 0.593259i \(-0.797841\pi\)
−0.805011 + 0.593259i \(0.797841\pi\)
\(108\) 0 0
\(109\) 230.000 0.202110 0.101055 0.994881i \(-0.467778\pi\)
0.101055 + 0.994881i \(0.467778\pi\)
\(110\) 0 0
\(111\) 258.000 0.220615
\(112\) 0 0
\(113\) −1718.00 −1.43023 −0.715114 0.699007i \(-0.753625\pi\)
−0.715114 + 0.699007i \(0.753625\pi\)
\(114\) 0 0
\(115\) −456.000 −0.369758
\(116\) 0 0
\(117\) −18.0000 −0.0142231
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −655.000 −0.492111
\(122\) 0 0
\(123\) 480.000 0.351871
\(124\) 0 0
\(125\) 936.000 0.669747
\(126\) 0 0
\(127\) 2444.00 1.70764 0.853819 0.520571i \(-0.174281\pi\)
0.853819 + 0.520571i \(0.174281\pi\)
\(128\) 0 0
\(129\) −660.000 −0.450463
\(130\) 0 0
\(131\) 1996.00 1.33123 0.665616 0.746295i \(-0.268169\pi\)
0.665616 + 0.746295i \(0.268169\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 108.000 0.0688530
\(136\) 0 0
\(137\) −1746.00 −1.08884 −0.544419 0.838813i \(-0.683250\pi\)
−0.544419 + 0.838813i \(0.683250\pi\)
\(138\) 0 0
\(139\) 1804.00 1.10081 0.550407 0.834896i \(-0.314472\pi\)
0.550407 + 0.834896i \(0.314472\pi\)
\(140\) 0 0
\(141\) −924.000 −0.551878
\(142\) 0 0
\(143\) −52.0000 −0.0304088
\(144\) 0 0
\(145\) −24.0000 −0.0137455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2814.00 −1.54719 −0.773597 0.633678i \(-0.781544\pi\)
−0.773597 + 0.633678i \(0.781544\pi\)
\(150\) 0 0
\(151\) 792.000 0.426835 0.213417 0.976961i \(-0.431541\pi\)
0.213417 + 0.976961i \(0.431541\pi\)
\(152\) 0 0
\(153\) 324.000 0.171202
\(154\) 0 0
\(155\) 1024.00 0.530643
\(156\) 0 0
\(157\) 2778.00 1.41216 0.706078 0.708134i \(-0.250463\pi\)
0.706078 + 0.708134i \(0.250463\pi\)
\(158\) 0 0
\(159\) −774.000 −0.386052
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2880.00 −1.38392 −0.691960 0.721936i \(-0.743253\pi\)
−0.691960 + 0.721936i \(0.743253\pi\)
\(164\) 0 0
\(165\) 312.000 0.147207
\(166\) 0 0
\(167\) −1060.00 −0.491169 −0.245585 0.969375i \(-0.578980\pi\)
−0.245585 + 0.969375i \(0.578980\pi\)
\(168\) 0 0
\(169\) −2193.00 −0.998179
\(170\) 0 0
\(171\) −684.000 −0.305888
\(172\) 0 0
\(173\) 1440.00 0.632839 0.316420 0.948619i \(-0.397519\pi\)
0.316420 + 0.948619i \(0.397519\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −792.000 −0.336330
\(178\) 0 0
\(179\) −1210.00 −0.505249 −0.252625 0.967564i \(-0.581294\pi\)
−0.252625 + 0.967564i \(0.581294\pi\)
\(180\) 0 0
\(181\) −1618.00 −0.664447 −0.332224 0.943201i \(-0.607799\pi\)
−0.332224 + 0.943201i \(0.607799\pi\)
\(182\) 0 0
\(183\) 1818.00 0.734374
\(184\) 0 0
\(185\) 344.000 0.136710
\(186\) 0 0
\(187\) 936.000 0.366027
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4018.00 −1.52216 −0.761079 0.648659i \(-0.775330\pi\)
−0.761079 + 0.648659i \(0.775330\pi\)
\(192\) 0 0
\(193\) 3382.00 1.26136 0.630678 0.776045i \(-0.282777\pi\)
0.630678 + 0.776045i \(0.282777\pi\)
\(194\) 0 0
\(195\) −24.0000 −0.00881372
\(196\) 0 0
\(197\) −4302.00 −1.55586 −0.777931 0.628350i \(-0.783731\pi\)
−0.777931 + 0.628350i \(0.783731\pi\)
\(198\) 0 0
\(199\) −2640.00 −0.940425 −0.470213 0.882553i \(-0.655823\pi\)
−0.470213 + 0.882553i \(0.655823\pi\)
\(200\) 0 0
\(201\) −1560.00 −0.547432
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 640.000 0.218047
\(206\) 0 0
\(207\) 1026.00 0.344502
\(208\) 0 0
\(209\) −1976.00 −0.653985
\(210\) 0 0
\(211\) −3396.00 −1.10801 −0.554005 0.832513i \(-0.686901\pi\)
−0.554005 + 0.832513i \(0.686901\pi\)
\(212\) 0 0
\(213\) −858.000 −0.276006
\(214\) 0 0
\(215\) −880.000 −0.279142
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1590.00 −0.490604
\(220\) 0 0
\(221\) −72.0000 −0.0219151
\(222\) 0 0
\(223\) 3480.00 1.04501 0.522507 0.852635i \(-0.324997\pi\)
0.522507 + 0.852635i \(0.324997\pi\)
\(224\) 0 0
\(225\) −981.000 −0.290667
\(226\) 0 0
\(227\) −1504.00 −0.439753 −0.219877 0.975528i \(-0.570565\pi\)
−0.219877 + 0.975528i \(0.570565\pi\)
\(228\) 0 0
\(229\) −5122.00 −1.47804 −0.739020 0.673683i \(-0.764711\pi\)
−0.739020 + 0.673683i \(0.764711\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1630.00 0.458304 0.229152 0.973391i \(-0.426405\pi\)
0.229152 + 0.973391i \(0.426405\pi\)
\(234\) 0 0
\(235\) −1232.00 −0.341986
\(236\) 0 0
\(237\) −132.000 −0.0361786
\(238\) 0 0
\(239\) 2522.00 0.682572 0.341286 0.939960i \(-0.389138\pi\)
0.341286 + 0.939960i \(0.389138\pi\)
\(240\) 0 0
\(241\) −3022.00 −0.807735 −0.403867 0.914817i \(-0.632334\pi\)
−0.403867 + 0.914817i \(0.632334\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 152.000 0.0391560
\(248\) 0 0
\(249\) −3036.00 −0.772685
\(250\) 0 0
\(251\) 3480.00 0.875122 0.437561 0.899189i \(-0.355842\pi\)
0.437561 + 0.899189i \(0.355842\pi\)
\(252\) 0 0
\(253\) 2964.00 0.736542
\(254\) 0 0
\(255\) 432.000 0.106090
\(256\) 0 0
\(257\) −3496.00 −0.848539 −0.424269 0.905536i \(-0.639469\pi\)
−0.424269 + 0.905536i \(0.639469\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 54.0000 0.0128066
\(262\) 0 0
\(263\) 5058.00 1.18589 0.592946 0.805242i \(-0.297965\pi\)
0.592946 + 0.805242i \(0.297965\pi\)
\(264\) 0 0
\(265\) −1032.00 −0.239227
\(266\) 0 0
\(267\) 2304.00 0.528099
\(268\) 0 0
\(269\) −3596.00 −0.815063 −0.407532 0.913191i \(-0.633610\pi\)
−0.407532 + 0.913191i \(0.633610\pi\)
\(270\) 0 0
\(271\) −2424.00 −0.543349 −0.271674 0.962389i \(-0.587577\pi\)
−0.271674 + 0.962389i \(0.587577\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2834.00 −0.621442
\(276\) 0 0
\(277\) 3634.00 0.788252 0.394126 0.919056i \(-0.371047\pi\)
0.394126 + 0.919056i \(0.371047\pi\)
\(278\) 0 0
\(279\) −2304.00 −0.494397
\(280\) 0 0
\(281\) −2970.00 −0.630517 −0.315259 0.949006i \(-0.602091\pi\)
−0.315259 + 0.949006i \(0.602091\pi\)
\(282\) 0 0
\(283\) −7028.00 −1.47622 −0.738112 0.674679i \(-0.764282\pi\)
−0.738112 + 0.674679i \(0.764282\pi\)
\(284\) 0 0
\(285\) −912.000 −0.189552
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3617.00 −0.736210
\(290\) 0 0
\(291\) 666.000 0.134164
\(292\) 0 0
\(293\) 528.000 0.105277 0.0526384 0.998614i \(-0.483237\pi\)
0.0526384 + 0.998614i \(0.483237\pi\)
\(294\) 0 0
\(295\) −1056.00 −0.208416
\(296\) 0 0
\(297\) −702.000 −0.137152
\(298\) 0 0
\(299\) −228.000 −0.0440989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −960.000 −0.182015
\(304\) 0 0
\(305\) 2424.00 0.455075
\(306\) 0 0
\(307\) 4300.00 0.799394 0.399697 0.916647i \(-0.369115\pi\)
0.399697 + 0.916647i \(0.369115\pi\)
\(308\) 0 0
\(309\) 1776.00 0.326968
\(310\) 0 0
\(311\) −4580.00 −0.835074 −0.417537 0.908660i \(-0.637107\pi\)
−0.417537 + 0.908660i \(0.637107\pi\)
\(312\) 0 0
\(313\) −2266.00 −0.409207 −0.204604 0.978845i \(-0.565591\pi\)
−0.204604 + 0.978845i \(0.565591\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7926.00 −1.40432 −0.702159 0.712021i \(-0.747780\pi\)
−0.702159 + 0.712021i \(0.747780\pi\)
\(318\) 0 0
\(319\) 156.000 0.0273803
\(320\) 0 0
\(321\) 5346.00 0.929547
\(322\) 0 0
\(323\) −2736.00 −0.471316
\(324\) 0 0
\(325\) 218.000 0.0372076
\(326\) 0 0
\(327\) −690.000 −0.116688
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4132.00 0.686149 0.343074 0.939308i \(-0.388532\pi\)
0.343074 + 0.939308i \(0.388532\pi\)
\(332\) 0 0
\(333\) −774.000 −0.127372
\(334\) 0 0
\(335\) −2080.00 −0.339231
\(336\) 0 0
\(337\) −3622.00 −0.585469 −0.292734 0.956194i \(-0.594565\pi\)
−0.292734 + 0.956194i \(0.594565\pi\)
\(338\) 0 0
\(339\) 5154.00 0.825743
\(340\) 0 0
\(341\) −6656.00 −1.05702
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1368.00 0.213480
\(346\) 0 0
\(347\) 10254.0 1.58635 0.793175 0.608994i \(-0.208426\pi\)
0.793175 + 0.608994i \(0.208426\pi\)
\(348\) 0 0
\(349\) 8178.00 1.25432 0.627161 0.778890i \(-0.284217\pi\)
0.627161 + 0.778890i \(0.284217\pi\)
\(350\) 0 0
\(351\) 54.0000 0.00821170
\(352\) 0 0
\(353\) −9932.00 −1.49753 −0.748763 0.662837i \(-0.769352\pi\)
−0.748763 + 0.662837i \(0.769352\pi\)
\(354\) 0 0
\(355\) −1144.00 −0.171034
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6546.00 −0.962353 −0.481176 0.876624i \(-0.659790\pi\)
−0.481176 + 0.876624i \(0.659790\pi\)
\(360\) 0 0
\(361\) −1083.00 −0.157895
\(362\) 0 0
\(363\) 1965.00 0.284121
\(364\) 0 0
\(365\) −2120.00 −0.304016
\(366\) 0 0
\(367\) −12176.0 −1.73183 −0.865916 0.500190i \(-0.833263\pi\)
−0.865916 + 0.500190i \(0.833263\pi\)
\(368\) 0 0
\(369\) −1440.00 −0.203153
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −8210.00 −1.13967 −0.569836 0.821758i \(-0.692993\pi\)
−0.569836 + 0.821758i \(0.692993\pi\)
\(374\) 0 0
\(375\) −2808.00 −0.386679
\(376\) 0 0
\(377\) −12.0000 −0.00163934
\(378\) 0 0
\(379\) −6908.00 −0.936254 −0.468127 0.883661i \(-0.655071\pi\)
−0.468127 + 0.883661i \(0.655071\pi\)
\(380\) 0 0
\(381\) −7332.00 −0.985905
\(382\) 0 0
\(383\) −10248.0 −1.36723 −0.683614 0.729844i \(-0.739593\pi\)
−0.683614 + 0.729844i \(0.739593\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1980.00 0.260075
\(388\) 0 0
\(389\) −274.000 −0.0357130 −0.0178565 0.999841i \(-0.505684\pi\)
−0.0178565 + 0.999841i \(0.505684\pi\)
\(390\) 0 0
\(391\) 4104.00 0.530814
\(392\) 0 0
\(393\) −5988.00 −0.768587
\(394\) 0 0
\(395\) −176.000 −0.0224190
\(396\) 0 0
\(397\) 10010.0 1.26546 0.632730 0.774373i \(-0.281934\pi\)
0.632730 + 0.774373i \(0.281934\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1998.00 0.248816 0.124408 0.992231i \(-0.460297\pi\)
0.124408 + 0.992231i \(0.460297\pi\)
\(402\) 0 0
\(403\) 512.000 0.0632867
\(404\) 0 0
\(405\) −324.000 −0.0397523
\(406\) 0 0
\(407\) −2236.00 −0.272320
\(408\) 0 0
\(409\) 12842.0 1.55256 0.776279 0.630390i \(-0.217105\pi\)
0.776279 + 0.630390i \(0.217105\pi\)
\(410\) 0 0
\(411\) 5238.00 0.628641
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4048.00 −0.478816
\(416\) 0 0
\(417\) −5412.00 −0.635556
\(418\) 0 0
\(419\) −10400.0 −1.21259 −0.606293 0.795242i \(-0.707344\pi\)
−0.606293 + 0.795242i \(0.707344\pi\)
\(420\) 0 0
\(421\) 15586.0 1.80431 0.902156 0.431410i \(-0.141984\pi\)
0.902156 + 0.431410i \(0.141984\pi\)
\(422\) 0 0
\(423\) 2772.00 0.318627
\(424\) 0 0
\(425\) −3924.00 −0.447863
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 156.000 0.0175565
\(430\) 0 0
\(431\) −8066.00 −0.901451 −0.450726 0.892663i \(-0.648835\pi\)
−0.450726 + 0.892663i \(0.648835\pi\)
\(432\) 0 0
\(433\) 5222.00 0.579569 0.289784 0.957092i \(-0.406416\pi\)
0.289784 + 0.957092i \(0.406416\pi\)
\(434\) 0 0
\(435\) 72.0000 0.00793594
\(436\) 0 0
\(437\) −8664.00 −0.948410
\(438\) 0 0
\(439\) −10920.0 −1.18721 −0.593603 0.804758i \(-0.702295\pi\)
−0.593603 + 0.804758i \(0.702295\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1650.00 −0.176961 −0.0884807 0.996078i \(-0.528201\pi\)
−0.0884807 + 0.996078i \(0.528201\pi\)
\(444\) 0 0
\(445\) 3072.00 0.327251
\(446\) 0 0
\(447\) 8442.00 0.893273
\(448\) 0 0
\(449\) 11858.0 1.24636 0.623178 0.782080i \(-0.285841\pi\)
0.623178 + 0.782080i \(0.285841\pi\)
\(450\) 0 0
\(451\) −4160.00 −0.434339
\(452\) 0 0
\(453\) −2376.00 −0.246433
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17894.0 −1.83161 −0.915805 0.401623i \(-0.868446\pi\)
−0.915805 + 0.401623i \(0.868446\pi\)
\(458\) 0 0
\(459\) −972.000 −0.0988433
\(460\) 0 0
\(461\) 2088.00 0.210950 0.105475 0.994422i \(-0.466364\pi\)
0.105475 + 0.994422i \(0.466364\pi\)
\(462\) 0 0
\(463\) −13532.0 −1.35828 −0.679142 0.734007i \(-0.737648\pi\)
−0.679142 + 0.734007i \(0.737648\pi\)
\(464\) 0 0
\(465\) −3072.00 −0.306367
\(466\) 0 0
\(467\) 6344.00 0.628620 0.314310 0.949320i \(-0.398227\pi\)
0.314310 + 0.949320i \(0.398227\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −8334.00 −0.815309
\(472\) 0 0
\(473\) 5720.00 0.556038
\(474\) 0 0
\(475\) 8284.00 0.800202
\(476\) 0 0
\(477\) 2322.00 0.222887
\(478\) 0 0
\(479\) 9948.00 0.948926 0.474463 0.880275i \(-0.342642\pi\)
0.474463 + 0.880275i \(0.342642\pi\)
\(480\) 0 0
\(481\) 172.000 0.0163046
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 888.000 0.0831382
\(486\) 0 0
\(487\) −20144.0 −1.87436 −0.937178 0.348850i \(-0.886572\pi\)
−0.937178 + 0.348850i \(0.886572\pi\)
\(488\) 0 0
\(489\) 8640.00 0.799007
\(490\) 0 0
\(491\) −10530.0 −0.967846 −0.483923 0.875111i \(-0.660788\pi\)
−0.483923 + 0.875111i \(0.660788\pi\)
\(492\) 0 0
\(493\) 216.000 0.0197326
\(494\) 0 0
\(495\) −936.000 −0.0849900
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5548.00 0.497721 0.248860 0.968539i \(-0.419944\pi\)
0.248860 + 0.968539i \(0.419944\pi\)
\(500\) 0 0
\(501\) 3180.00 0.283577
\(502\) 0 0
\(503\) −2152.00 −0.190761 −0.0953807 0.995441i \(-0.530407\pi\)
−0.0953807 + 0.995441i \(0.530407\pi\)
\(504\) 0 0
\(505\) −1280.00 −0.112791
\(506\) 0 0
\(507\) 6579.00 0.576299
\(508\) 0 0
\(509\) 19780.0 1.72246 0.861231 0.508214i \(-0.169694\pi\)
0.861231 + 0.508214i \(0.169694\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2052.00 0.176604
\(514\) 0 0
\(515\) 2368.00 0.202615
\(516\) 0 0
\(517\) 8008.00 0.681221
\(518\) 0 0
\(519\) −4320.00 −0.365370
\(520\) 0 0
\(521\) −23052.0 −1.93844 −0.969219 0.246199i \(-0.920818\pi\)
−0.969219 + 0.246199i \(0.920818\pi\)
\(522\) 0 0
\(523\) −13364.0 −1.11734 −0.558668 0.829391i \(-0.688687\pi\)
−0.558668 + 0.829391i \(0.688687\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9216.00 −0.761775
\(528\) 0 0
\(529\) 829.000 0.0681351
\(530\) 0 0
\(531\) 2376.00 0.194180
\(532\) 0 0
\(533\) 320.000 0.0260051
\(534\) 0 0
\(535\) 7128.00 0.576019
\(536\) 0 0
\(537\) 3630.00 0.291706
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9766.00 −0.776106 −0.388053 0.921637i \(-0.626852\pi\)
−0.388053 + 0.921637i \(0.626852\pi\)
\(542\) 0 0
\(543\) 4854.00 0.383619
\(544\) 0 0
\(545\) −920.000 −0.0723091
\(546\) 0 0
\(547\) −3768.00 −0.294530 −0.147265 0.989097i \(-0.547047\pi\)
−0.147265 + 0.989097i \(0.547047\pi\)
\(548\) 0 0
\(549\) −5454.00 −0.423991
\(550\) 0 0
\(551\) −456.000 −0.0352564
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1032.00 −0.0789297
\(556\) 0 0
\(557\) 1618.00 0.123082 0.0615412 0.998105i \(-0.480398\pi\)
0.0615412 + 0.998105i \(0.480398\pi\)
\(558\) 0 0
\(559\) −440.000 −0.0332916
\(560\) 0 0
\(561\) −2808.00 −0.211326
\(562\) 0 0
\(563\) −5768.00 −0.431780 −0.215890 0.976418i \(-0.569265\pi\)
−0.215890 + 0.976418i \(0.569265\pi\)
\(564\) 0 0
\(565\) 6872.00 0.511694
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8114.00 −0.597815 −0.298907 0.954282i \(-0.596622\pi\)
−0.298907 + 0.954282i \(0.596622\pi\)
\(570\) 0 0
\(571\) −7816.00 −0.572836 −0.286418 0.958105i \(-0.592465\pi\)
−0.286418 + 0.958105i \(0.592465\pi\)
\(572\) 0 0
\(573\) 12054.0 0.878819
\(574\) 0 0
\(575\) −12426.0 −0.901217
\(576\) 0 0
\(577\) 18278.0 1.31876 0.659379 0.751811i \(-0.270819\pi\)
0.659379 + 0.751811i \(0.270819\pi\)
\(578\) 0 0
\(579\) −10146.0 −0.728244
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6708.00 0.476530
\(584\) 0 0
\(585\) 72.0000 0.00508860
\(586\) 0 0
\(587\) −24960.0 −1.75504 −0.877521 0.479539i \(-0.840804\pi\)
−0.877521 + 0.479539i \(0.840804\pi\)
\(588\) 0 0
\(589\) 19456.0 1.36107
\(590\) 0 0
\(591\) 12906.0 0.898277
\(592\) 0 0
\(593\) −10892.0 −0.754268 −0.377134 0.926159i \(-0.623090\pi\)
−0.377134 + 0.926159i \(0.623090\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7920.00 0.542955
\(598\) 0 0
\(599\) 20022.0 1.36574 0.682869 0.730541i \(-0.260732\pi\)
0.682869 + 0.730541i \(0.260732\pi\)
\(600\) 0 0
\(601\) −10370.0 −0.703829 −0.351914 0.936032i \(-0.614469\pi\)
−0.351914 + 0.936032i \(0.614469\pi\)
\(602\) 0 0
\(603\) 4680.00 0.316060
\(604\) 0 0
\(605\) 2620.00 0.176063
\(606\) 0 0
\(607\) −8992.00 −0.601275 −0.300638 0.953738i \(-0.597199\pi\)
−0.300638 + 0.953738i \(0.597199\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −616.000 −0.0407867
\(612\) 0 0
\(613\) 12318.0 0.811614 0.405807 0.913959i \(-0.366991\pi\)
0.405807 + 0.913959i \(0.366991\pi\)
\(614\) 0 0
\(615\) −1920.00 −0.125889
\(616\) 0 0
\(617\) −26026.0 −1.69816 −0.849082 0.528261i \(-0.822844\pi\)
−0.849082 + 0.528261i \(0.822844\pi\)
\(618\) 0 0
\(619\) −18332.0 −1.19035 −0.595174 0.803597i \(-0.702917\pi\)
−0.595174 + 0.803597i \(0.702917\pi\)
\(620\) 0 0
\(621\) −3078.00 −0.198898
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) 0 0
\(627\) 5928.00 0.377578
\(628\) 0 0
\(629\) −3096.00 −0.196257
\(630\) 0 0
\(631\) −10572.0 −0.666980 −0.333490 0.942754i \(-0.608226\pi\)
−0.333490 + 0.942754i \(0.608226\pi\)
\(632\) 0 0
\(633\) 10188.0 0.639710
\(634\) 0 0
\(635\) −9776.00 −0.610943
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2574.00 0.159352
\(640\) 0 0
\(641\) 24822.0 1.52950 0.764750 0.644327i \(-0.222862\pi\)
0.764750 + 0.644327i \(0.222862\pi\)
\(642\) 0 0
\(643\) −6620.00 −0.406014 −0.203007 0.979177i \(-0.565071\pi\)
−0.203007 + 0.979177i \(0.565071\pi\)
\(644\) 0 0
\(645\) 2640.00 0.161163
\(646\) 0 0
\(647\) 15956.0 0.969544 0.484772 0.874641i \(-0.338903\pi\)
0.484772 + 0.874641i \(0.338903\pi\)
\(648\) 0 0
\(649\) 6864.00 0.415155
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30802.0 −1.84590 −0.922952 0.384915i \(-0.874231\pi\)
−0.922952 + 0.384915i \(0.874231\pi\)
\(654\) 0 0
\(655\) −7984.00 −0.476276
\(656\) 0 0
\(657\) 4770.00 0.283250
\(658\) 0 0
\(659\) 12714.0 0.751543 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(660\) 0 0
\(661\) −5822.00 −0.342586 −0.171293 0.985220i \(-0.554795\pi\)
−0.171293 + 0.985220i \(0.554795\pi\)
\(662\) 0 0
\(663\) 216.000 0.0126527
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 684.000 0.0397070
\(668\) 0 0
\(669\) −10440.0 −0.603339
\(670\) 0 0
\(671\) −15756.0 −0.906488
\(672\) 0 0
\(673\) 28510.0 1.63296 0.816478 0.577376i \(-0.195923\pi\)
0.816478 + 0.577376i \(0.195923\pi\)
\(674\) 0 0
\(675\) 2943.00 0.167816
\(676\) 0 0
\(677\) 21864.0 1.24121 0.620607 0.784122i \(-0.286886\pi\)
0.620607 + 0.784122i \(0.286886\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 4512.00 0.253892
\(682\) 0 0
\(683\) 17862.0 1.00069 0.500344 0.865826i \(-0.333207\pi\)
0.500344 + 0.865826i \(0.333207\pi\)
\(684\) 0 0
\(685\) 6984.00 0.389555
\(686\) 0 0
\(687\) 15366.0 0.853347
\(688\) 0 0
\(689\) −516.000 −0.0285313
\(690\) 0 0
\(691\) −6068.00 −0.334063 −0.167032 0.985952i \(-0.553418\pi\)
−0.167032 + 0.985952i \(0.553418\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7216.00 −0.393840
\(696\) 0 0
\(697\) −5760.00 −0.313021
\(698\) 0 0
\(699\) −4890.00 −0.264602
\(700\) 0 0
\(701\) 7734.00 0.416703 0.208352 0.978054i \(-0.433190\pi\)
0.208352 + 0.978054i \(0.433190\pi\)
\(702\) 0 0
\(703\) 6536.00 0.350654
\(704\) 0 0
\(705\) 3696.00 0.197446
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 19902.0 1.05421 0.527105 0.849800i \(-0.323277\pi\)
0.527105 + 0.849800i \(0.323277\pi\)
\(710\) 0 0
\(711\) 396.000 0.0208877
\(712\) 0 0
\(713\) −29184.0 −1.53289
\(714\) 0 0
\(715\) 208.000 0.0108794
\(716\) 0 0
\(717\) −7566.00 −0.394083
\(718\) 0 0
\(719\) 12160.0 0.630725 0.315363 0.948971i \(-0.397874\pi\)
0.315363 + 0.948971i \(0.397874\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 9066.00 0.466346
\(724\) 0 0
\(725\) −654.000 −0.0335020
\(726\) 0 0
\(727\) 3088.00 0.157534 0.0787672 0.996893i \(-0.474902\pi\)
0.0787672 + 0.996893i \(0.474902\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 7920.00 0.400727
\(732\) 0 0
\(733\) −10178.0 −0.512869 −0.256435 0.966562i \(-0.582548\pi\)
−0.256435 + 0.966562i \(0.582548\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13520.0 0.675734
\(738\) 0 0
\(739\) 5840.00 0.290701 0.145350 0.989380i \(-0.453569\pi\)
0.145350 + 0.989380i \(0.453569\pi\)
\(740\) 0 0
\(741\) −456.000 −0.0226067
\(742\) 0 0
\(743\) 14414.0 0.711707 0.355854 0.934542i \(-0.384190\pi\)
0.355854 + 0.934542i \(0.384190\pi\)
\(744\) 0 0
\(745\) 11256.0 0.553541
\(746\) 0 0
\(747\) 9108.00 0.446110
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12608.0 0.612613 0.306307 0.951933i \(-0.400907\pi\)
0.306307 + 0.951933i \(0.400907\pi\)
\(752\) 0 0
\(753\) −10440.0 −0.505252
\(754\) 0 0
\(755\) −3168.00 −0.152709
\(756\) 0 0
\(757\) 22606.0 1.08538 0.542688 0.839935i \(-0.317407\pi\)
0.542688 + 0.839935i \(0.317407\pi\)
\(758\) 0 0
\(759\) −8892.00 −0.425243
\(760\) 0 0
\(761\) −5952.00 −0.283521 −0.141761 0.989901i \(-0.545276\pi\)
−0.141761 + 0.989901i \(0.545276\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1296.00 −0.0612510
\(766\) 0 0
\(767\) −528.000 −0.0248566
\(768\) 0 0
\(769\) −14138.0 −0.662977 −0.331489 0.943459i \(-0.607551\pi\)
−0.331489 + 0.943459i \(0.607551\pi\)
\(770\) 0 0
\(771\) 10488.0 0.489904
\(772\) 0 0
\(773\) −12616.0 −0.587019 −0.293510 0.955956i \(-0.594823\pi\)
−0.293510 + 0.955956i \(0.594823\pi\)
\(774\) 0 0
\(775\) 27904.0 1.29334
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12160.0 0.559278
\(780\) 0 0
\(781\) 7436.00 0.340693
\(782\) 0 0
\(783\) −162.000 −0.00739388
\(784\) 0 0
\(785\) −11112.0 −0.505228
\(786\) 0 0
\(787\) −12292.0 −0.556750 −0.278375 0.960472i \(-0.589796\pi\)
−0.278375 + 0.960472i \(0.589796\pi\)
\(788\) 0 0
\(789\) −15174.0 −0.684675
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1212.00 0.0542741
\(794\) 0 0
\(795\) 3096.00 0.138118
\(796\) 0 0
\(797\) 20784.0 0.923723 0.461861 0.886952i \(-0.347182\pi\)
0.461861 + 0.886952i \(0.347182\pi\)
\(798\) 0 0
\(799\) 11088.0 0.490945
\(800\) 0 0
\(801\) −6912.00 −0.304898
\(802\) 0 0
\(803\) 13780.0 0.605586
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10788.0 0.470577
\(808\) 0 0
\(809\) −13574.0 −0.589909 −0.294955 0.955511i \(-0.595305\pi\)
−0.294955 + 0.955511i \(0.595305\pi\)
\(810\) 0 0
\(811\) 32308.0 1.39887 0.699437 0.714694i \(-0.253434\pi\)
0.699437 + 0.714694i \(0.253434\pi\)
\(812\) 0 0
\(813\) 7272.00 0.313703
\(814\) 0 0
\(815\) 11520.0 0.495126
\(816\) 0 0
\(817\) −16720.0 −0.715984
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2390.00 −0.101598 −0.0507988 0.998709i \(-0.516177\pi\)
−0.0507988 + 0.998709i \(0.516177\pi\)
\(822\) 0 0
\(823\) −28020.0 −1.18677 −0.593387 0.804917i \(-0.702210\pi\)
−0.593387 + 0.804917i \(0.702210\pi\)
\(824\) 0 0
\(825\) 8502.00 0.358790
\(826\) 0 0
\(827\) −34006.0 −1.42987 −0.714936 0.699190i \(-0.753544\pi\)
−0.714936 + 0.699190i \(0.753544\pi\)
\(828\) 0 0
\(829\) 6310.00 0.264361 0.132181 0.991226i \(-0.457802\pi\)
0.132181 + 0.991226i \(0.457802\pi\)
\(830\) 0 0
\(831\) −10902.0 −0.455098
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 4240.00 0.175726
\(836\) 0 0
\(837\) 6912.00 0.285440
\(838\) 0 0
\(839\) −16524.0 −0.679943 −0.339971 0.940436i \(-0.610417\pi\)
−0.339971 + 0.940436i \(0.610417\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 0 0
\(843\) 8910.00 0.364029
\(844\) 0 0
\(845\) 8772.00 0.357119
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 21084.0 0.852298
\(850\) 0 0
\(851\) −9804.00 −0.394920
\(852\) 0 0
\(853\) −36662.0 −1.47161 −0.735805 0.677194i \(-0.763196\pi\)
−0.735805 + 0.677194i \(0.763196\pi\)
\(854\) 0 0
\(855\) 2736.00 0.109438
\(856\) 0 0
\(857\) 42000.0 1.67409 0.837044 0.547136i \(-0.184282\pi\)
0.837044 + 0.547136i \(0.184282\pi\)
\(858\) 0 0
\(859\) 28388.0 1.12757 0.563787 0.825920i \(-0.309344\pi\)
0.563787 + 0.825920i \(0.309344\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23918.0 −0.943428 −0.471714 0.881752i \(-0.656364\pi\)
−0.471714 + 0.881752i \(0.656364\pi\)
\(864\) 0 0
\(865\) −5760.00 −0.226411
\(866\) 0 0
\(867\) 10851.0 0.425051
\(868\) 0 0
\(869\) 1144.00 0.0446577
\(870\) 0 0
\(871\) −1040.00 −0.0404582
\(872\) 0 0
\(873\) −1998.00 −0.0774594
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2346.00 −0.0903293 −0.0451646 0.998980i \(-0.514381\pi\)
−0.0451646 + 0.998980i \(0.514381\pi\)
\(878\) 0 0
\(879\) −1584.00 −0.0607816
\(880\) 0 0
\(881\) −51876.0 −1.98382 −0.991911 0.126937i \(-0.959485\pi\)
−0.991911 + 0.126937i \(0.959485\pi\)
\(882\) 0 0
\(883\) 11372.0 0.433407 0.216703 0.976237i \(-0.430470\pi\)
0.216703 + 0.976237i \(0.430470\pi\)
\(884\) 0 0
\(885\) 3168.00 0.120329
\(886\) 0 0
\(887\) 38764.0 1.46738 0.733691 0.679483i \(-0.237796\pi\)
0.733691 + 0.679483i \(0.237796\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2106.00 0.0791848
\(892\) 0 0
\(893\) −23408.0 −0.877177
\(894\) 0 0
\(895\) 4840.00 0.180764
\(896\) 0 0
\(897\) 684.000 0.0254605
\(898\) 0 0
\(899\) −1536.00 −0.0569838
\(900\) 0 0
\(901\) 9288.00 0.343427
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6472.00 0.237720
\(906\) 0 0
\(907\) 11212.0 0.410461 0.205231 0.978714i \(-0.434206\pi\)
0.205231 + 0.978714i \(0.434206\pi\)
\(908\) 0 0
\(909\) 2880.00 0.105086
\(910\) 0 0
\(911\) −1458.00 −0.0530249 −0.0265125 0.999648i \(-0.508440\pi\)
−0.0265125 + 0.999648i \(0.508440\pi\)
\(912\) 0 0
\(913\) 26312.0 0.953779
\(914\) 0 0
\(915\) −7272.00 −0.262738
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 38608.0 1.38581 0.692906 0.721028i \(-0.256330\pi\)
0.692906 + 0.721028i \(0.256330\pi\)
\(920\) 0 0
\(921\) −12900.0 −0.461530
\(922\) 0 0
\(923\) −572.000 −0.0203983
\(924\) 0 0
\(925\) 9374.00 0.333206
\(926\) 0 0
\(927\) −5328.00 −0.188775
\(928\) 0 0
\(929\) −7376.00 −0.260494 −0.130247 0.991482i \(-0.541577\pi\)
−0.130247 + 0.991482i \(0.541577\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 13740.0 0.482130
\(934\) 0 0
\(935\) −3744.00 −0.130954
\(936\) 0 0
\(937\) 24878.0 0.867373 0.433687 0.901064i \(-0.357212\pi\)
0.433687 + 0.901064i \(0.357212\pi\)
\(938\) 0 0
\(939\) 6798.00 0.236256
\(940\) 0 0
\(941\) 26148.0 0.905845 0.452923 0.891550i \(-0.350381\pi\)
0.452923 + 0.891550i \(0.350381\pi\)
\(942\) 0 0
\(943\) −18240.0 −0.629879
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18350.0 −0.629667 −0.314834 0.949147i \(-0.601949\pi\)
−0.314834 + 0.949147i \(0.601949\pi\)
\(948\) 0 0
\(949\) −1060.00 −0.0362582
\(950\) 0 0
\(951\) 23778.0 0.810783
\(952\) 0 0
\(953\) −57510.0 −1.95481 −0.977404 0.211381i \(-0.932204\pi\)
−0.977404 + 0.211381i \(0.932204\pi\)
\(954\) 0 0
\(955\) 16072.0 0.544584
\(956\) 0 0
\(957\) −468.000 −0.0158080
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 35745.0 1.19986
\(962\) 0 0
\(963\) −16038.0 −0.536674
\(964\) 0 0
\(965\) −13528.0 −0.451276
\(966\) 0 0
\(967\) 33364.0 1.10953 0.554764 0.832008i \(-0.312808\pi\)
0.554764 + 0.832008i \(0.312808\pi\)
\(968\) 0 0
\(969\) 8208.00 0.272114
\(970\) 0 0
\(971\) 26892.0 0.888780 0.444390 0.895833i \(-0.353420\pi\)
0.444390 + 0.895833i \(0.353420\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −654.000 −0.0214818
\(976\) 0 0
\(977\) −31970.0 −1.04689 −0.523445 0.852060i \(-0.675353\pi\)
−0.523445 + 0.852060i \(0.675353\pi\)
\(978\) 0 0
\(979\) −19968.0 −0.651869
\(980\) 0 0
\(981\) 2070.00 0.0673700
\(982\) 0 0
\(983\) −19728.0 −0.640107 −0.320054 0.947399i \(-0.603701\pi\)
−0.320054 + 0.947399i \(0.603701\pi\)
\(984\) 0 0
\(985\) 17208.0 0.556642
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25080.0 0.806368
\(990\) 0 0
\(991\) 21400.0 0.685967 0.342984 0.939341i \(-0.388562\pi\)
0.342984 + 0.939341i \(0.388562\pi\)
\(992\) 0 0
\(993\) −12396.0 −0.396148
\(994\) 0 0
\(995\) 10560.0 0.336457
\(996\) 0 0
\(997\) 4754.00 0.151014 0.0755069 0.997145i \(-0.475943\pi\)
0.0755069 + 0.997145i \(0.475943\pi\)
\(998\) 0 0
\(999\) 2322.00 0.0735384
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 2352.4.a.h.1.1 1
4.3 odd 2 1176.4.a.j.1.1 1
7.6 odd 2 336.4.a.j.1.1 1
21.20 even 2 1008.4.a.i.1.1 1
28.27 even 2 168.4.a.c.1.1 1
56.13 odd 2 1344.4.a.e.1.1 1
56.27 even 2 1344.4.a.s.1.1 1
84.83 odd 2 504.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.a.c.1.1 1 28.27 even 2
336.4.a.j.1.1 1 7.6 odd 2
504.4.a.b.1.1 1 84.83 odd 2
1008.4.a.i.1.1 1 21.20 even 2
1176.4.a.j.1.1 1 4.3 odd 2
1344.4.a.e.1.1 1 56.13 odd 2
1344.4.a.s.1.1 1 56.27 even 2
2352.4.a.h.1.1 1 1.1 even 1 trivial