Properties

Label 28.4.a.b.1.1
Level $28$
Weight $4$
Character 28.1
Self dual yes
Analytic conductor $1.652$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 28.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.65205348016\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 28.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{3} +6.00000 q^{5} +7.00000 q^{7} -11.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{3} +6.00000 q^{5} +7.00000 q^{7} -11.0000 q^{9} -12.0000 q^{11} -82.0000 q^{13} +24.0000 q^{15} -30.0000 q^{17} +68.0000 q^{19} +28.0000 q^{21} +216.000 q^{23} -89.0000 q^{25} -152.000 q^{27} +246.000 q^{29} -112.000 q^{31} -48.0000 q^{33} +42.0000 q^{35} +110.000 q^{37} -328.000 q^{39} -246.000 q^{41} -172.000 q^{43} -66.0000 q^{45} +192.000 q^{47} +49.0000 q^{49} -120.000 q^{51} +558.000 q^{53} -72.0000 q^{55} +272.000 q^{57} +540.000 q^{59} +110.000 q^{61} -77.0000 q^{63} -492.000 q^{65} +140.000 q^{67} +864.000 q^{69} -840.000 q^{71} -550.000 q^{73} -356.000 q^{75} -84.0000 q^{77} -208.000 q^{79} -311.000 q^{81} +516.000 q^{83} -180.000 q^{85} +984.000 q^{87} -1398.00 q^{89} -574.000 q^{91} -448.000 q^{93} +408.000 q^{95} +1586.00 q^{97} +132.000 q^{99} +O(q^{100})\)

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
\(3\) 4.00000 0.769800 0.384900 0.922958i \(-0.374236\pi\)
0.384900 + 0.922958i \(0.374236\pi\)
\(4\) 0 0
\(5\) 6.00000 0.536656 0.268328 0.963328i \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) −11.0000 −0.407407
\(10\) 0 0
\(11\) −12.0000 −0.328921 −0.164461 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) 0 0
\(13\) −82.0000 −1.74944 −0.874720 0.484629i \(-0.838954\pi\)
−0.874720 + 0.484629i \(0.838954\pi\)
\(14\) 0 0
\(15\) 24.0000 0.413118
\(16\) 0 0
\(17\) −30.0000 −0.428004 −0.214002 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 0 0
\(19\) 68.0000 0.821067 0.410533 0.911846i \(-0.365343\pi\)
0.410533 + 0.911846i \(0.365343\pi\)
\(20\) 0 0
\(21\) 28.0000 0.290957
\(22\) 0 0
\(23\) 216.000 1.95822 0.979111 0.203326i \(-0.0651750\pi\)
0.979111 + 0.203326i \(0.0651750\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) 0 0
\(27\) −152.000 −1.08342
\(28\) 0 0
\(29\) 246.000 1.57521 0.787604 0.616181i \(-0.211321\pi\)
0.787604 + 0.616181i \(0.211321\pi\)
\(30\) 0 0
\(31\) −112.000 −0.648897 −0.324448 0.945903i \(-0.605179\pi\)
−0.324448 + 0.945903i \(0.605179\pi\)
\(32\) 0 0
\(33\) −48.0000 −0.253204
\(34\) 0 0
\(35\) 42.0000 0.202837
\(36\) 0 0
\(37\) 110.000 0.488754 0.244377 0.969680i \(-0.421417\pi\)
0.244377 + 0.969680i \(0.421417\pi\)
\(38\) 0 0
\(39\) −328.000 −1.34672
\(40\) 0 0
\(41\) −246.000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −172.000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −66.0000 −0.218638
\(46\) 0 0
\(47\) 192.000 0.595874 0.297937 0.954586i \(-0.403701\pi\)
0.297937 + 0.954586i \(0.403701\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −120.000 −0.329478
\(52\) 0 0
\(53\) 558.000 1.44617 0.723087 0.690757i \(-0.242723\pi\)
0.723087 + 0.690757i \(0.242723\pi\)
\(54\) 0 0
\(55\) −72.0000 −0.176518
\(56\) 0 0
\(57\) 272.000 0.632058
\(58\) 0 0
\(59\) 540.000 1.19156 0.595780 0.803148i \(-0.296843\pi\)
0.595780 + 0.803148i \(0.296843\pi\)
\(60\) 0 0
\(61\) 110.000 0.230886 0.115443 0.993314i \(-0.463171\pi\)
0.115443 + 0.993314i \(0.463171\pi\)
\(62\) 0 0
\(63\) −77.0000 −0.153986
\(64\) 0 0
\(65\) −492.000 −0.938848
\(66\) 0 0
\(67\) 140.000 0.255279 0.127640 0.991821i \(-0.459260\pi\)
0.127640 + 0.991821i \(0.459260\pi\)
\(68\) 0 0
\(69\) 864.000 1.50744
\(70\) 0 0
\(71\) −840.000 −1.40408 −0.702040 0.712138i \(-0.747727\pi\)
−0.702040 + 0.712138i \(0.747727\pi\)
\(72\) 0 0
\(73\) −550.000 −0.881817 −0.440908 0.897552i \(-0.645344\pi\)
−0.440908 + 0.897552i \(0.645344\pi\)
\(74\) 0 0
\(75\) −356.000 −0.548098
\(76\) 0 0
\(77\) −84.0000 −0.124321
\(78\) 0 0
\(79\) −208.000 −0.296226 −0.148113 0.988970i \(-0.547320\pi\)
−0.148113 + 0.988970i \(0.547320\pi\)
\(80\) 0 0
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) 516.000 0.682390 0.341195 0.939993i \(-0.389168\pi\)
0.341195 + 0.939993i \(0.389168\pi\)
\(84\) 0 0
\(85\) −180.000 −0.229691
\(86\) 0 0
\(87\) 984.000 1.21260
\(88\) 0 0
\(89\) −1398.00 −1.66503 −0.832515 0.554002i \(-0.813100\pi\)
−0.832515 + 0.554002i \(0.813100\pi\)
\(90\) 0 0
\(91\) −574.000 −0.661226
\(92\) 0 0
\(93\) −448.000 −0.499521
\(94\) 0 0
\(95\) 408.000 0.440631
\(96\) 0 0
\(97\) 1586.00 1.66014 0.830072 0.557657i \(-0.188299\pi\)
0.830072 + 0.557657i \(0.188299\pi\)
\(98\) 0 0
\(99\) 132.000 0.134005
\(100\) 0 0
\(101\) −1242.00 −1.22360 −0.611800 0.791012i \(-0.709554\pi\)
−0.611800 + 0.791012i \(0.709554\pi\)
\(102\) 0 0
\(103\) 680.000 0.650509 0.325254 0.945627i \(-0.394550\pi\)
0.325254 + 0.945627i \(0.394550\pi\)
\(104\) 0 0
\(105\) 168.000 0.156144
\(106\) 0 0
\(107\) 996.000 0.899878 0.449939 0.893059i \(-0.351446\pi\)
0.449939 + 0.893059i \(0.351446\pi\)
\(108\) 0 0
\(109\) 1382.00 1.21442 0.607209 0.794542i \(-0.292289\pi\)
0.607209 + 0.794542i \(0.292289\pi\)
\(110\) 0 0
\(111\) 440.000 0.376243
\(112\) 0 0
\(113\) −750.000 −0.624372 −0.312186 0.950021i \(-0.601061\pi\)
−0.312186 + 0.950021i \(0.601061\pi\)
\(114\) 0 0
\(115\) 1296.00 1.05089
\(116\) 0 0
\(117\) 902.000 0.712734
\(118\) 0 0
\(119\) −210.000 −0.161770
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) 0 0
\(123\) −984.000 −0.721336
\(124\) 0 0
\(125\) −1284.00 −0.918756
\(126\) 0 0
\(127\) 176.000 0.122972 0.0614861 0.998108i \(-0.480416\pi\)
0.0614861 + 0.998108i \(0.480416\pi\)
\(128\) 0 0
\(129\) −688.000 −0.469574
\(130\) 0 0
\(131\) −1548.00 −1.03244 −0.516219 0.856457i \(-0.672661\pi\)
−0.516219 + 0.856457i \(0.672661\pi\)
\(132\) 0 0
\(133\) 476.000 0.310334
\(134\) 0 0
\(135\) −912.000 −0.581426
\(136\) 0 0
\(137\) 378.000 0.235728 0.117864 0.993030i \(-0.462395\pi\)
0.117864 + 0.993030i \(0.462395\pi\)
\(138\) 0 0
\(139\) −2500.00 −1.52552 −0.762760 0.646682i \(-0.776156\pi\)
−0.762760 + 0.646682i \(0.776156\pi\)
\(140\) 0 0
\(141\) 768.000 0.458704
\(142\) 0 0
\(143\) 984.000 0.575428
\(144\) 0 0
\(145\) 1476.00 0.845346
\(146\) 0 0
\(147\) 196.000 0.109971
\(148\) 0 0
\(149\) 846.000 0.465148 0.232574 0.972579i \(-0.425285\pi\)
0.232574 + 0.972579i \(0.425285\pi\)
\(150\) 0 0
\(151\) −2536.00 −1.36673 −0.683367 0.730075i \(-0.739485\pi\)
−0.683367 + 0.730075i \(0.739485\pi\)
\(152\) 0 0
\(153\) 330.000 0.174372
\(154\) 0 0
\(155\) −672.000 −0.348234
\(156\) 0 0
\(157\) −1186.00 −0.602886 −0.301443 0.953484i \(-0.597468\pi\)
−0.301443 + 0.953484i \(0.597468\pi\)
\(158\) 0 0
\(159\) 2232.00 1.11326
\(160\) 0 0
\(161\) 1512.00 0.740138
\(162\) 0 0
\(163\) 2108.00 1.01295 0.506476 0.862254i \(-0.330948\pi\)
0.506476 + 0.862254i \(0.330948\pi\)
\(164\) 0 0
\(165\) −288.000 −0.135883
\(166\) 0 0
\(167\) −1944.00 −0.900786 −0.450393 0.892830i \(-0.648716\pi\)
−0.450393 + 0.892830i \(0.648716\pi\)
\(168\) 0 0
\(169\) 4527.00 2.06054
\(170\) 0 0
\(171\) −748.000 −0.334509
\(172\) 0 0
\(173\) −1362.00 −0.598560 −0.299280 0.954165i \(-0.596747\pi\)
−0.299280 + 0.954165i \(0.596747\pi\)
\(174\) 0 0
\(175\) −623.000 −0.269111
\(176\) 0 0
\(177\) 2160.00 0.917263
\(178\) 0 0
\(179\) 1596.00 0.666428 0.333214 0.942851i \(-0.391867\pi\)
0.333214 + 0.942851i \(0.391867\pi\)
\(180\) 0 0
\(181\) −1690.00 −0.694015 −0.347007 0.937862i \(-0.612802\pi\)
−0.347007 + 0.937862i \(0.612802\pi\)
\(182\) 0 0
\(183\) 440.000 0.177736
\(184\) 0 0
\(185\) 660.000 0.262293
\(186\) 0 0
\(187\) 360.000 0.140780
\(188\) 0 0
\(189\) −1064.00 −0.409495
\(190\) 0 0
\(191\) 3552.00 1.34562 0.672811 0.739815i \(-0.265087\pi\)
0.672811 + 0.739815i \(0.265087\pi\)
\(192\) 0 0
\(193\) −2686.00 −1.00177 −0.500887 0.865512i \(-0.666993\pi\)
−0.500887 + 0.865512i \(0.666993\pi\)
\(194\) 0 0
\(195\) −1968.00 −0.722725
\(196\) 0 0
\(197\) −1410.00 −0.509941 −0.254970 0.966949i \(-0.582066\pi\)
−0.254970 + 0.966949i \(0.582066\pi\)
\(198\) 0 0
\(199\) −2968.00 −1.05727 −0.528633 0.848850i \(-0.677295\pi\)
−0.528633 + 0.848850i \(0.677295\pi\)
\(200\) 0 0
\(201\) 560.000 0.196514
\(202\) 0 0
\(203\) 1722.00 0.595373
\(204\) 0 0
\(205\) −1476.00 −0.502870
\(206\) 0 0
\(207\) −2376.00 −0.797794
\(208\) 0 0
\(209\) −816.000 −0.270067
\(210\) 0 0
\(211\) −1348.00 −0.439811 −0.219906 0.975521i \(-0.570575\pi\)
−0.219906 + 0.975521i \(0.570575\pi\)
\(212\) 0 0
\(213\) −3360.00 −1.08086
\(214\) 0 0
\(215\) −1032.00 −0.327357
\(216\) 0 0
\(217\) −784.000 −0.245260
\(218\) 0 0
\(219\) −2200.00 −0.678823
\(220\) 0 0
\(221\) 2460.00 0.748767
\(222\) 0 0
\(223\) 3872.00 1.16273 0.581364 0.813644i \(-0.302519\pi\)
0.581364 + 0.813644i \(0.302519\pi\)
\(224\) 0 0
\(225\) 979.000 0.290074
\(226\) 0 0
\(227\) 5364.00 1.56838 0.784188 0.620524i \(-0.213080\pi\)
0.784188 + 0.620524i \(0.213080\pi\)
\(228\) 0 0
\(229\) −874.000 −0.252208 −0.126104 0.992017i \(-0.540247\pi\)
−0.126104 + 0.992017i \(0.540247\pi\)
\(230\) 0 0
\(231\) −336.000 −0.0957021
\(232\) 0 0
\(233\) 378.000 0.106282 0.0531408 0.998587i \(-0.483077\pi\)
0.0531408 + 0.998587i \(0.483077\pi\)
\(234\) 0 0
\(235\) 1152.00 0.319780
\(236\) 0 0
\(237\) −832.000 −0.228035
\(238\) 0 0
\(239\) 1920.00 0.519642 0.259821 0.965657i \(-0.416336\pi\)
0.259821 + 0.965657i \(0.416336\pi\)
\(240\) 0 0
\(241\) 4322.00 1.15521 0.577603 0.816318i \(-0.303988\pi\)
0.577603 + 0.816318i \(0.303988\pi\)
\(242\) 0 0
\(243\) 2860.00 0.755017
\(244\) 0 0
\(245\) 294.000 0.0766652
\(246\) 0 0
\(247\) −5576.00 −1.43641
\(248\) 0 0
\(249\) 2064.00 0.525304
\(250\) 0 0
\(251\) 5292.00 1.33079 0.665395 0.746492i \(-0.268263\pi\)
0.665395 + 0.746492i \(0.268263\pi\)
\(252\) 0 0
\(253\) −2592.00 −0.644101
\(254\) 0 0
\(255\) −720.000 −0.176816
\(256\) 0 0
\(257\) −5118.00 −1.24223 −0.621113 0.783721i \(-0.713319\pi\)
−0.621113 + 0.783721i \(0.713319\pi\)
\(258\) 0 0
\(259\) 770.000 0.184732
\(260\) 0 0
\(261\) −2706.00 −0.641752
\(262\) 0 0
\(263\) 3768.00 0.883440 0.441720 0.897153i \(-0.354368\pi\)
0.441720 + 0.897153i \(0.354368\pi\)
\(264\) 0 0
\(265\) 3348.00 0.776098
\(266\) 0 0
\(267\) −5592.00 −1.28174
\(268\) 0 0
\(269\) 3918.00 0.888047 0.444024 0.896015i \(-0.353551\pi\)
0.444024 + 0.896015i \(0.353551\pi\)
\(270\) 0 0
\(271\) 4880.00 1.09387 0.546935 0.837175i \(-0.315794\pi\)
0.546935 + 0.837175i \(0.315794\pi\)
\(272\) 0 0
\(273\) −2296.00 −0.509012
\(274\) 0 0
\(275\) 1068.00 0.234192
\(276\) 0 0
\(277\) −3538.00 −0.767429 −0.383714 0.923452i \(-0.625355\pi\)
−0.383714 + 0.923452i \(0.625355\pi\)
\(278\) 0 0
\(279\) 1232.00 0.264365
\(280\) 0 0
\(281\) −5430.00 −1.15276 −0.576382 0.817180i \(-0.695536\pi\)
−0.576382 + 0.817180i \(0.695536\pi\)
\(282\) 0 0
\(283\) −6436.00 −1.35187 −0.675937 0.736959i \(-0.736261\pi\)
−0.675937 + 0.736959i \(0.736261\pi\)
\(284\) 0 0
\(285\) 1632.00 0.339198
\(286\) 0 0
\(287\) −1722.00 −0.354169
\(288\) 0 0
\(289\) −4013.00 −0.816813
\(290\) 0 0
\(291\) 6344.00 1.27798
\(292\) 0 0
\(293\) 1350.00 0.269174 0.134587 0.990902i \(-0.457029\pi\)
0.134587 + 0.990902i \(0.457029\pi\)
\(294\) 0 0
\(295\) 3240.00 0.639458
\(296\) 0 0
\(297\) 1824.00 0.356361
\(298\) 0 0
\(299\) −17712.0 −3.42579
\(300\) 0 0
\(301\) −1204.00 −0.230556
\(302\) 0 0
\(303\) −4968.00 −0.941928
\(304\) 0 0
\(305\) 660.000 0.123907
\(306\) 0 0
\(307\) 3332.00 0.619437 0.309719 0.950828i \(-0.399765\pi\)
0.309719 + 0.950828i \(0.399765\pi\)
\(308\) 0 0
\(309\) 2720.00 0.500762
\(310\) 0 0
\(311\) −4728.00 −0.862059 −0.431029 0.902338i \(-0.641849\pi\)
−0.431029 + 0.902338i \(0.641849\pi\)
\(312\) 0 0
\(313\) 5114.00 0.923516 0.461758 0.887006i \(-0.347219\pi\)
0.461758 + 0.887006i \(0.347219\pi\)
\(314\) 0 0
\(315\) −462.000 −0.0826373
\(316\) 0 0
\(317\) 7206.00 1.27675 0.638374 0.769726i \(-0.279607\pi\)
0.638374 + 0.769726i \(0.279607\pi\)
\(318\) 0 0
\(319\) −2952.00 −0.518120
\(320\) 0 0
\(321\) 3984.00 0.692726
\(322\) 0 0
\(323\) −2040.00 −0.351420
\(324\) 0 0
\(325\) 7298.00 1.24560
\(326\) 0 0
\(327\) 5528.00 0.934860
\(328\) 0 0
\(329\) 1344.00 0.225219
\(330\) 0 0
\(331\) 6260.00 1.03952 0.519759 0.854313i \(-0.326022\pi\)
0.519759 + 0.854313i \(0.326022\pi\)
\(332\) 0 0
\(333\) −1210.00 −0.199122
\(334\) 0 0
\(335\) 840.000 0.136997
\(336\) 0 0
\(337\) −5326.00 −0.860907 −0.430454 0.902613i \(-0.641646\pi\)
−0.430454 + 0.902613i \(0.641646\pi\)
\(338\) 0 0
\(339\) −3000.00 −0.480642
\(340\) 0 0
\(341\) 1344.00 0.213436
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 5184.00 0.808977
\(346\) 0 0
\(347\) 36.0000 0.00556940 0.00278470 0.999996i \(-0.499114\pi\)
0.00278470 + 0.999996i \(0.499114\pi\)
\(348\) 0 0
\(349\) 3134.00 0.480685 0.240343 0.970688i \(-0.422740\pi\)
0.240343 + 0.970688i \(0.422740\pi\)
\(350\) 0 0
\(351\) 12464.0 1.89538
\(352\) 0 0
\(353\) 1218.00 0.183648 0.0918238 0.995775i \(-0.470730\pi\)
0.0918238 + 0.995775i \(0.470730\pi\)
\(354\) 0 0
\(355\) −5040.00 −0.753508
\(356\) 0 0
\(357\) −840.000 −0.124531
\(358\) 0 0
\(359\) −10008.0 −1.47131 −0.735657 0.677354i \(-0.763127\pi\)
−0.735657 + 0.677354i \(0.763127\pi\)
\(360\) 0 0
\(361\) −2235.00 −0.325849
\(362\) 0 0
\(363\) −4748.00 −0.686516
\(364\) 0 0
\(365\) −3300.00 −0.473233
\(366\) 0 0
\(367\) −1072.00 −0.152474 −0.0762370 0.997090i \(-0.524291\pi\)
−0.0762370 + 0.997090i \(0.524291\pi\)
\(368\) 0 0
\(369\) 2706.00 0.381758
\(370\) 0 0
\(371\) 3906.00 0.546602
\(372\) 0 0
\(373\) −274.000 −0.0380353 −0.0190177 0.999819i \(-0.506054\pi\)
−0.0190177 + 0.999819i \(0.506054\pi\)
\(374\) 0 0
\(375\) −5136.00 −0.707258
\(376\) 0 0
\(377\) −20172.0 −2.75573
\(378\) 0 0
\(379\) 7652.00 1.03709 0.518545 0.855051i \(-0.326474\pi\)
0.518545 + 0.855051i \(0.326474\pi\)
\(380\) 0 0
\(381\) 704.000 0.0946641
\(382\) 0 0
\(383\) 2160.00 0.288175 0.144087 0.989565i \(-0.453975\pi\)
0.144087 + 0.989565i \(0.453975\pi\)
\(384\) 0 0
\(385\) −504.000 −0.0667175
\(386\) 0 0
\(387\) 1892.00 0.248516
\(388\) 0 0
\(389\) −1074.00 −0.139984 −0.0699922 0.997548i \(-0.522297\pi\)
−0.0699922 + 0.997548i \(0.522297\pi\)
\(390\) 0 0
\(391\) −6480.00 −0.838127
\(392\) 0 0
\(393\) −6192.00 −0.794771
\(394\) 0 0
\(395\) −1248.00 −0.158971
\(396\) 0 0
\(397\) 6926.00 0.875582 0.437791 0.899077i \(-0.355761\pi\)
0.437791 + 0.899077i \(0.355761\pi\)
\(398\) 0 0
\(399\) 1904.00 0.238895
\(400\) 0 0
\(401\) 1938.00 0.241344 0.120672 0.992692i \(-0.461495\pi\)
0.120672 + 0.992692i \(0.461495\pi\)
\(402\) 0 0
\(403\) 9184.00 1.13521
\(404\) 0 0
\(405\) −1866.00 −0.228944
\(406\) 0 0
\(407\) −1320.00 −0.160762
\(408\) 0 0
\(409\) −9574.00 −1.15747 −0.578733 0.815517i \(-0.696453\pi\)
−0.578733 + 0.815517i \(0.696453\pi\)
\(410\) 0 0
\(411\) 1512.00 0.181463
\(412\) 0 0
\(413\) 3780.00 0.450367
\(414\) 0 0
\(415\) 3096.00 0.366209
\(416\) 0 0
\(417\) −10000.0 −1.17435
\(418\) 0 0
\(419\) −5052.00 −0.589037 −0.294518 0.955646i \(-0.595159\pi\)
−0.294518 + 0.955646i \(0.595159\pi\)
\(420\) 0 0
\(421\) 3422.00 0.396147 0.198074 0.980187i \(-0.436531\pi\)
0.198074 + 0.980187i \(0.436531\pi\)
\(422\) 0 0
\(423\) −2112.00 −0.242763
\(424\) 0 0
\(425\) 2670.00 0.304739
\(426\) 0 0
\(427\) 770.000 0.0872668
\(428\) 0 0
\(429\) 3936.00 0.442965
\(430\) 0 0
\(431\) 2208.00 0.246765 0.123382 0.992359i \(-0.460626\pi\)
0.123382 + 0.992359i \(0.460626\pi\)
\(432\) 0 0
\(433\) −6814.00 −0.756259 −0.378129 0.925753i \(-0.623433\pi\)
−0.378129 + 0.925753i \(0.623433\pi\)
\(434\) 0 0
\(435\) 5904.00 0.650747
\(436\) 0 0
\(437\) 14688.0 1.60783
\(438\) 0 0
\(439\) 12584.0 1.36811 0.684056 0.729429i \(-0.260214\pi\)
0.684056 + 0.729429i \(0.260214\pi\)
\(440\) 0 0
\(441\) −539.000 −0.0582011
\(442\) 0 0
\(443\) 6996.00 0.750316 0.375158 0.926961i \(-0.377588\pi\)
0.375158 + 0.926961i \(0.377588\pi\)
\(444\) 0 0
\(445\) −8388.00 −0.893549
\(446\) 0 0
\(447\) 3384.00 0.358071
\(448\) 0 0
\(449\) 9474.00 0.995781 0.497891 0.867240i \(-0.334108\pi\)
0.497891 + 0.867240i \(0.334108\pi\)
\(450\) 0 0
\(451\) 2952.00 0.308213
\(452\) 0 0
\(453\) −10144.0 −1.05211
\(454\) 0 0
\(455\) −3444.00 −0.354851
\(456\) 0 0
\(457\) 5786.00 0.592249 0.296124 0.955149i \(-0.404306\pi\)
0.296124 + 0.955149i \(0.404306\pi\)
\(458\) 0 0
\(459\) 4560.00 0.463709
\(460\) 0 0
\(461\) 3438.00 0.347340 0.173670 0.984804i \(-0.444437\pi\)
0.173670 + 0.984804i \(0.444437\pi\)
\(462\) 0 0
\(463\) 9392.00 0.942728 0.471364 0.881939i \(-0.343762\pi\)
0.471364 + 0.881939i \(0.343762\pi\)
\(464\) 0 0
\(465\) −2688.00 −0.268071
\(466\) 0 0
\(467\) −4956.00 −0.491084 −0.245542 0.969386i \(-0.578966\pi\)
−0.245542 + 0.969386i \(0.578966\pi\)
\(468\) 0 0
\(469\) 980.000 0.0964866
\(470\) 0 0
\(471\) −4744.00 −0.464102
\(472\) 0 0
\(473\) 2064.00 0.200640
\(474\) 0 0
\(475\) −6052.00 −0.584600
\(476\) 0 0
\(477\) −6138.00 −0.589182
\(478\) 0 0
\(479\) −20592.0 −1.96424 −0.982122 0.188248i \(-0.939719\pi\)
−0.982122 + 0.188248i \(0.939719\pi\)
\(480\) 0 0
\(481\) −9020.00 −0.855045
\(482\) 0 0
\(483\) 6048.00 0.569759
\(484\) 0 0
\(485\) 9516.00 0.890926
\(486\) 0 0
\(487\) −13432.0 −1.24982 −0.624910 0.780697i \(-0.714864\pi\)
−0.624910 + 0.780697i \(0.714864\pi\)
\(488\) 0 0
\(489\) 8432.00 0.779771
\(490\) 0 0
\(491\) −14172.0 −1.30259 −0.651297 0.758823i \(-0.725775\pi\)
−0.651297 + 0.758823i \(0.725775\pi\)
\(492\) 0 0
\(493\) −7380.00 −0.674196
\(494\) 0 0
\(495\) 792.000 0.0719147
\(496\) 0 0
\(497\) −5880.00 −0.530692
\(498\) 0 0
\(499\) −5956.00 −0.534323 −0.267162 0.963652i \(-0.586086\pi\)
−0.267162 + 0.963652i \(0.586086\pi\)
\(500\) 0 0
\(501\) −7776.00 −0.693425
\(502\) 0 0
\(503\) 16968.0 1.50411 0.752053 0.659102i \(-0.229064\pi\)
0.752053 + 0.659102i \(0.229064\pi\)
\(504\) 0 0
\(505\) −7452.00 −0.656653
\(506\) 0 0
\(507\) 18108.0 1.58620
\(508\) 0 0
\(509\) 5214.00 0.454040 0.227020 0.973890i \(-0.427102\pi\)
0.227020 + 0.973890i \(0.427102\pi\)
\(510\) 0 0
\(511\) −3850.00 −0.333295
\(512\) 0 0
\(513\) −10336.0 −0.889562
\(514\) 0 0
\(515\) 4080.00 0.349100
\(516\) 0 0
\(517\) −2304.00 −0.195996
\(518\) 0 0
\(519\) −5448.00 −0.460772
\(520\) 0 0
\(521\) −1398.00 −0.117558 −0.0587788 0.998271i \(-0.518721\pi\)
−0.0587788 + 0.998271i \(0.518721\pi\)
\(522\) 0 0
\(523\) −18580.0 −1.55344 −0.776718 0.629849i \(-0.783117\pi\)
−0.776718 + 0.629849i \(0.783117\pi\)
\(524\) 0 0
\(525\) −2492.00 −0.207162
\(526\) 0 0
\(527\) 3360.00 0.277730
\(528\) 0 0
\(529\) 34489.0 2.83463
\(530\) 0 0
\(531\) −5940.00 −0.485450
\(532\) 0 0
\(533\) 20172.0 1.63930
\(534\) 0 0
\(535\) 5976.00 0.482925
\(536\) 0 0
\(537\) 6384.00 0.513017
\(538\) 0 0
\(539\) −588.000 −0.0469888
\(540\) 0 0
\(541\) −18970.0 −1.50755 −0.753774 0.657133i \(-0.771769\pi\)
−0.753774 + 0.657133i \(0.771769\pi\)
\(542\) 0 0
\(543\) −6760.00 −0.534253
\(544\) 0 0
\(545\) 8292.00 0.651725
\(546\) 0 0
\(547\) −16036.0 −1.25347 −0.626737 0.779231i \(-0.715610\pi\)
−0.626737 + 0.779231i \(0.715610\pi\)
\(548\) 0 0
\(549\) −1210.00 −0.0940647
\(550\) 0 0
\(551\) 16728.0 1.29335
\(552\) 0 0
\(553\) −1456.00 −0.111963
\(554\) 0 0
\(555\) 2640.00 0.201913
\(556\) 0 0
\(557\) 8310.00 0.632147 0.316074 0.948735i \(-0.397635\pi\)
0.316074 + 0.948735i \(0.397635\pi\)
\(558\) 0 0
\(559\) 14104.0 1.06715
\(560\) 0 0
\(561\) 1440.00 0.108372
\(562\) 0 0
\(563\) 7092.00 0.530892 0.265446 0.964126i \(-0.414481\pi\)
0.265446 + 0.964126i \(0.414481\pi\)
\(564\) 0 0
\(565\) −4500.00 −0.335073
\(566\) 0 0
\(567\) −2177.00 −0.161244
\(568\) 0 0
\(569\) −7158.00 −0.527380 −0.263690 0.964608i \(-0.584940\pi\)
−0.263690 + 0.964608i \(0.584940\pi\)
\(570\) 0 0
\(571\) 6500.00 0.476386 0.238193 0.971218i \(-0.423445\pi\)
0.238193 + 0.971218i \(0.423445\pi\)
\(572\) 0 0
\(573\) 14208.0 1.03586
\(574\) 0 0
\(575\) −19224.0 −1.39425
\(576\) 0 0
\(577\) 21794.0 1.57244 0.786218 0.617949i \(-0.212036\pi\)
0.786218 + 0.617949i \(0.212036\pi\)
\(578\) 0 0
\(579\) −10744.0 −0.771166
\(580\) 0 0
\(581\) 3612.00 0.257919
\(582\) 0 0
\(583\) −6696.00 −0.475678
\(584\) 0 0
\(585\) 5412.00 0.382493
\(586\) 0 0
\(587\) 9756.00 0.685985 0.342993 0.939338i \(-0.388559\pi\)
0.342993 + 0.939338i \(0.388559\pi\)
\(588\) 0 0
\(589\) −7616.00 −0.532787
\(590\) 0 0
\(591\) −5640.00 −0.392553
\(592\) 0 0
\(593\) 5586.00 0.386829 0.193414 0.981117i \(-0.438044\pi\)
0.193414 + 0.981117i \(0.438044\pi\)
\(594\) 0 0
\(595\) −1260.00 −0.0868151
\(596\) 0 0
\(597\) −11872.0 −0.813884
\(598\) 0 0
\(599\) −24.0000 −0.00163708 −0.000818542 1.00000i \(-0.500261\pi\)
−0.000818542 1.00000i \(0.500261\pi\)
\(600\) 0 0
\(601\) 4298.00 0.291712 0.145856 0.989306i \(-0.453406\pi\)
0.145856 + 0.989306i \(0.453406\pi\)
\(602\) 0 0
\(603\) −1540.00 −0.104003
\(604\) 0 0
\(605\) −7122.00 −0.478596
\(606\) 0 0
\(607\) 8480.00 0.567039 0.283519 0.958966i \(-0.408498\pi\)
0.283519 + 0.958966i \(0.408498\pi\)
\(608\) 0 0
\(609\) 6888.00 0.458318
\(610\) 0 0
\(611\) −15744.0 −1.04245
\(612\) 0 0
\(613\) −1906.00 −0.125583 −0.0627917 0.998027i \(-0.520000\pi\)
−0.0627917 + 0.998027i \(0.520000\pi\)
\(614\) 0 0
\(615\) −5904.00 −0.387109
\(616\) 0 0
\(617\) 7482.00 0.488191 0.244096 0.969751i \(-0.421509\pi\)
0.244096 + 0.969751i \(0.421509\pi\)
\(618\) 0 0
\(619\) −7348.00 −0.477126 −0.238563 0.971127i \(-0.576676\pi\)
−0.238563 + 0.971127i \(0.576676\pi\)
\(620\) 0 0
\(621\) −32832.0 −2.12158
\(622\) 0 0
\(623\) −9786.00 −0.629322
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) −3264.00 −0.207897
\(628\) 0 0
\(629\) −3300.00 −0.209189
\(630\) 0 0
\(631\) 4520.00 0.285164 0.142582 0.989783i \(-0.454460\pi\)
0.142582 + 0.989783i \(0.454460\pi\)
\(632\) 0 0
\(633\) −5392.00 −0.338567
\(634\) 0 0
\(635\) 1056.00 0.0659938
\(636\) 0 0
\(637\) −4018.00 −0.249920
\(638\) 0 0
\(639\) 9240.00 0.572032
\(640\) 0 0
\(641\) −19806.0 −1.22042 −0.610211 0.792239i \(-0.708915\pi\)
−0.610211 + 0.792239i \(0.708915\pi\)
\(642\) 0 0
\(643\) −5020.00 −0.307884 −0.153942 0.988080i \(-0.549197\pi\)
−0.153942 + 0.988080i \(0.549197\pi\)
\(644\) 0 0
\(645\) −4128.00 −0.252000
\(646\) 0 0
\(647\) 28392.0 1.72520 0.862600 0.505886i \(-0.168834\pi\)
0.862600 + 0.505886i \(0.168834\pi\)
\(648\) 0 0
\(649\) −6480.00 −0.391930
\(650\) 0 0
\(651\) −3136.00 −0.188801
\(652\) 0 0
\(653\) −17562.0 −1.05246 −0.526228 0.850343i \(-0.676394\pi\)
−0.526228 + 0.850343i \(0.676394\pi\)
\(654\) 0 0
\(655\) −9288.00 −0.554064
\(656\) 0 0
\(657\) 6050.00 0.359259
\(658\) 0 0
\(659\) 4716.00 0.278770 0.139385 0.990238i \(-0.455487\pi\)
0.139385 + 0.990238i \(0.455487\pi\)
\(660\) 0 0
\(661\) −22762.0 −1.33939 −0.669697 0.742635i \(-0.733576\pi\)
−0.669697 + 0.742635i \(0.733576\pi\)
\(662\) 0 0
\(663\) 9840.00 0.576401
\(664\) 0 0
\(665\) 2856.00 0.166543
\(666\) 0 0
\(667\) 53136.0 3.08461
\(668\) 0 0
\(669\) 15488.0 0.895068
\(670\) 0 0
\(671\) −1320.00 −0.0759434
\(672\) 0 0
\(673\) 4802.00 0.275042 0.137521 0.990499i \(-0.456086\pi\)
0.137521 + 0.990499i \(0.456086\pi\)
\(674\) 0 0
\(675\) 13528.0 0.771397
\(676\) 0 0
\(677\) 21558.0 1.22384 0.611921 0.790919i \(-0.290397\pi\)
0.611921 + 0.790919i \(0.290397\pi\)
\(678\) 0 0
\(679\) 11102.0 0.627475
\(680\) 0 0
\(681\) 21456.0 1.20734
\(682\) 0 0
\(683\) 3780.00 0.211768 0.105884 0.994378i \(-0.466233\pi\)
0.105884 + 0.994378i \(0.466233\pi\)
\(684\) 0 0
\(685\) 2268.00 0.126505
\(686\) 0 0
\(687\) −3496.00 −0.194150
\(688\) 0 0
\(689\) −45756.0 −2.52999
\(690\) 0 0
\(691\) −5500.00 −0.302793 −0.151396 0.988473i \(-0.548377\pi\)
−0.151396 + 0.988473i \(0.548377\pi\)
\(692\) 0 0
\(693\) 924.000 0.0506491
\(694\) 0 0
\(695\) −15000.0 −0.818680
\(696\) 0 0
\(697\) 7380.00 0.401058
\(698\) 0 0
\(699\) 1512.00 0.0818156
\(700\) 0 0
\(701\) 10230.0 0.551187 0.275593 0.961274i \(-0.411126\pi\)
0.275593 + 0.961274i \(0.411126\pi\)
\(702\) 0 0
\(703\) 7480.00 0.401299
\(704\) 0 0
\(705\) 4608.00 0.246166
\(706\) 0 0
\(707\) −8694.00 −0.462477
\(708\) 0 0
\(709\) 10190.0 0.539765 0.269883 0.962893i \(-0.413015\pi\)
0.269883 + 0.962893i \(0.413015\pi\)
\(710\) 0 0
\(711\) 2288.00 0.120685
\(712\) 0 0
\(713\) −24192.0 −1.27068
\(714\) 0 0
\(715\) 5904.00 0.308807
\(716\) 0 0
\(717\) 7680.00 0.400021
\(718\) 0 0
\(719\) 9408.00 0.487982 0.243991 0.969777i \(-0.421543\pi\)
0.243991 + 0.969777i \(0.421543\pi\)
\(720\) 0 0
\(721\) 4760.00 0.245869
\(722\) 0 0
\(723\) 17288.0 0.889278
\(724\) 0 0
\(725\) −21894.0 −1.12155
\(726\) 0 0
\(727\) −33064.0 −1.68676 −0.843381 0.537316i \(-0.819438\pi\)
−0.843381 + 0.537316i \(0.819438\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 0 0
\(731\) 5160.00 0.261080
\(732\) 0 0
\(733\) −6322.00 −0.318565 −0.159283 0.987233i \(-0.550918\pi\)
−0.159283 + 0.987233i \(0.550918\pi\)
\(734\) 0 0
\(735\) 1176.00 0.0590169
\(736\) 0 0
\(737\) −1680.00 −0.0839669
\(738\) 0 0
\(739\) −20740.0 −1.03239 −0.516193 0.856472i \(-0.672651\pi\)
−0.516193 + 0.856472i \(0.672651\pi\)
\(740\) 0 0
\(741\) −22304.0 −1.10575
\(742\) 0 0
\(743\) 32040.0 1.58201 0.791005 0.611810i \(-0.209558\pi\)
0.791005 + 0.611810i \(0.209558\pi\)
\(744\) 0 0
\(745\) 5076.00 0.249624
\(746\) 0 0
\(747\) −5676.00 −0.278011
\(748\) 0 0
\(749\) 6972.00 0.340122
\(750\) 0 0
\(751\) −12832.0 −0.623497 −0.311749 0.950165i \(-0.600915\pi\)
−0.311749 + 0.950165i \(0.600915\pi\)
\(752\) 0 0
\(753\) 21168.0 1.02444
\(754\) 0 0
\(755\) −15216.0 −0.733466
\(756\) 0 0
\(757\) −19906.0 −0.955741 −0.477870 0.878430i \(-0.658591\pi\)
−0.477870 + 0.878430i \(0.658591\pi\)
\(758\) 0 0
\(759\) −10368.0 −0.495829
\(760\) 0 0
\(761\) 10842.0 0.516455 0.258227 0.966084i \(-0.416862\pi\)
0.258227 + 0.966084i \(0.416862\pi\)
\(762\) 0 0
\(763\) 9674.00 0.459007
\(764\) 0 0
\(765\) 1980.00 0.0935778
\(766\) 0 0
\(767\) −44280.0 −2.08456
\(768\) 0 0
\(769\) 28274.0 1.32586 0.662930 0.748681i \(-0.269313\pi\)
0.662930 + 0.748681i \(0.269313\pi\)
\(770\) 0 0
\(771\) −20472.0 −0.956266
\(772\) 0 0
\(773\) −32346.0 −1.50505 −0.752526 0.658563i \(-0.771165\pi\)
−0.752526 + 0.658563i \(0.771165\pi\)
\(774\) 0 0
\(775\) 9968.00 0.462014
\(776\) 0 0
\(777\) 3080.00 0.142206
\(778\) 0 0
\(779\) −16728.0 −0.769375
\(780\) 0 0
\(781\) 10080.0 0.461832
\(782\) 0 0
\(783\) −37392.0 −1.70662
\(784\) 0 0
\(785\) −7116.00 −0.323543
\(786\) 0 0
\(787\) 30116.0 1.36407 0.682033 0.731322i \(-0.261096\pi\)
0.682033 + 0.731322i \(0.261096\pi\)
\(788\) 0 0
\(789\) 15072.0 0.680073
\(790\) 0 0
\(791\) −5250.00 −0.235991
\(792\) 0 0
\(793\) −9020.00 −0.403921
\(794\) 0 0
\(795\) 13392.0 0.597441
\(796\) 0 0
\(797\) −6594.00 −0.293063 −0.146532 0.989206i \(-0.546811\pi\)
−0.146532 + 0.989206i \(0.546811\pi\)
\(798\) 0 0
\(799\) −5760.00 −0.255036
\(800\) 0 0
\(801\) 15378.0 0.678346
\(802\) 0 0
\(803\) 6600.00 0.290048
\(804\) 0 0
\(805\) 9072.00 0.397200
\(806\) 0 0
\(807\) 15672.0 0.683619
\(808\) 0 0
\(809\) −43014.0 −1.86933 −0.934667 0.355524i \(-0.884303\pi\)
−0.934667 + 0.355524i \(0.884303\pi\)
\(810\) 0 0
\(811\) −14164.0 −0.613274 −0.306637 0.951827i \(-0.599204\pi\)
−0.306637 + 0.951827i \(0.599204\pi\)
\(812\) 0 0
\(813\) 19520.0 0.842062
\(814\) 0 0
\(815\) 12648.0 0.543608
\(816\) 0 0
\(817\) −11696.0 −0.500846
\(818\) 0 0
\(819\) 6314.00 0.269388
\(820\) 0 0
\(821\) 34830.0 1.48060 0.740302 0.672275i \(-0.234683\pi\)
0.740302 + 0.672275i \(0.234683\pi\)
\(822\) 0 0
\(823\) 31016.0 1.31367 0.656835 0.754035i \(-0.271895\pi\)
0.656835 + 0.754035i \(0.271895\pi\)
\(824\) 0 0
\(825\) 4272.00 0.180281
\(826\) 0 0
\(827\) 9876.00 0.415263 0.207631 0.978207i \(-0.433425\pi\)
0.207631 + 0.978207i \(0.433425\pi\)
\(828\) 0 0
\(829\) −3154.00 −0.132139 −0.0660693 0.997815i \(-0.521046\pi\)
−0.0660693 + 0.997815i \(0.521046\pi\)
\(830\) 0 0
\(831\) −14152.0 −0.590767
\(832\) 0 0
\(833\) −1470.00 −0.0611434
\(834\) 0 0
\(835\) −11664.0 −0.483412
\(836\) 0 0
\(837\) 17024.0 0.703029
\(838\) 0 0
\(839\) 36936.0 1.51987 0.759936 0.649998i \(-0.225230\pi\)
0.759936 + 0.649998i \(0.225230\pi\)
\(840\) 0 0
\(841\) 36127.0 1.48128
\(842\) 0 0
\(843\) −21720.0 −0.887398
\(844\) 0 0
\(845\) 27162.0 1.10580
\(846\) 0 0
\(847\) −8309.00 −0.337073
\(848\) 0 0
\(849\) −25744.0 −1.04067
\(850\) 0 0
\(851\) 23760.0 0.957088
\(852\) 0 0
\(853\) 9638.00 0.386869 0.193434 0.981113i \(-0.438037\pi\)
0.193434 + 0.981113i \(0.438037\pi\)
\(854\) 0 0
\(855\) −4488.00 −0.179516
\(856\) 0 0
\(857\) 10266.0 0.409195 0.204597 0.978846i \(-0.434411\pi\)
0.204597 + 0.978846i \(0.434411\pi\)
\(858\) 0 0
\(859\) −4084.00 −0.162217 −0.0811084 0.996705i \(-0.525846\pi\)
−0.0811084 + 0.996705i \(0.525846\pi\)
\(860\) 0 0
\(861\) −6888.00 −0.272639
\(862\) 0 0
\(863\) −192.000 −0.00757330 −0.00378665 0.999993i \(-0.501205\pi\)
−0.00378665 + 0.999993i \(0.501205\pi\)
\(864\) 0 0
\(865\) −8172.00 −0.321221
\(866\) 0 0
\(867\) −16052.0 −0.628783
\(868\) 0 0
\(869\) 2496.00 0.0974350
\(870\) 0 0
\(871\) −11480.0 −0.446596
\(872\) 0 0
\(873\) −17446.0 −0.676355
\(874\) 0 0
\(875\) −8988.00 −0.347257
\(876\) 0 0
\(877\) 19910.0 0.766605 0.383303 0.923623i \(-0.374787\pi\)
0.383303 + 0.923623i \(0.374787\pi\)
\(878\) 0 0
\(879\) 5400.00 0.207210
\(880\) 0 0
\(881\) 14802.0 0.566052 0.283026 0.959112i \(-0.408662\pi\)
0.283026 + 0.959112i \(0.408662\pi\)
\(882\) 0 0
\(883\) −32548.0 −1.24046 −0.620231 0.784419i \(-0.712961\pi\)
−0.620231 + 0.784419i \(0.712961\pi\)
\(884\) 0 0
\(885\) 12960.0 0.492255
\(886\) 0 0
\(887\) 1464.00 0.0554186 0.0277093 0.999616i \(-0.491179\pi\)
0.0277093 + 0.999616i \(0.491179\pi\)
\(888\) 0 0
\(889\) 1232.00 0.0464791
\(890\) 0 0
\(891\) 3732.00 0.140322
\(892\) 0 0
\(893\) 13056.0 0.489252
\(894\) 0 0
\(895\) 9576.00 0.357643
\(896\) 0 0
\(897\) −70848.0 −2.63717
\(898\) 0 0
\(899\) −27552.0 −1.02215
\(900\) 0 0
\(901\) −16740.0 −0.618968
\(902\) 0 0
\(903\) −4816.00 −0.177482
\(904\) 0 0
\(905\) −10140.0 −0.372448
\(906\) 0 0
\(907\) −49564.0 −1.81449 −0.907247 0.420599i \(-0.861820\pi\)
−0.907247 + 0.420599i \(0.861820\pi\)
\(908\) 0 0
\(909\) 13662.0 0.498504
\(910\) 0 0
\(911\) 8448.00 0.307239 0.153619 0.988130i \(-0.450907\pi\)
0.153619 + 0.988130i \(0.450907\pi\)
\(912\) 0 0
\(913\) −6192.00 −0.224453
\(914\) 0 0
\(915\) 2640.00 0.0953833
\(916\) 0 0
\(917\) −10836.0 −0.390225
\(918\) 0 0
\(919\) 14600.0 0.524058 0.262029 0.965060i \(-0.415608\pi\)
0.262029 + 0.965060i \(0.415608\pi\)
\(920\) 0 0
\(921\) 13328.0 0.476843
\(922\) 0 0
\(923\) 68880.0 2.45635
\(924\) 0 0
\(925\) −9790.00 −0.347993
\(926\) 0 0
\(927\) −7480.00 −0.265022
\(928\) 0 0
\(929\) −21102.0 −0.745247 −0.372623 0.927983i \(-0.621542\pi\)
−0.372623 + 0.927983i \(0.621542\pi\)
\(930\) 0 0
\(931\) 3332.00 0.117295
\(932\) 0 0
\(933\) −18912.0 −0.663613
\(934\) 0 0
\(935\) 2160.00 0.0755503
\(936\) 0 0
\(937\) −20806.0 −0.725403 −0.362701 0.931905i \(-0.618146\pi\)
−0.362701 + 0.931905i \(0.618146\pi\)
\(938\) 0 0
\(939\) 20456.0 0.710923
\(940\) 0 0
\(941\) 24510.0 0.849100 0.424550 0.905404i \(-0.360432\pi\)
0.424550 + 0.905404i \(0.360432\pi\)
\(942\) 0 0
\(943\) −53136.0 −1.83494
\(944\) 0 0
\(945\) −6384.00 −0.219758
\(946\) 0 0
\(947\) −44148.0 −1.51491 −0.757454 0.652889i \(-0.773557\pi\)
−0.757454 + 0.652889i \(0.773557\pi\)
\(948\) 0 0
\(949\) 45100.0 1.54268
\(950\) 0 0
\(951\) 28824.0 0.982841
\(952\) 0 0
\(953\) 27114.0 0.921625 0.460812 0.887498i \(-0.347558\pi\)
0.460812 + 0.887498i \(0.347558\pi\)
\(954\) 0 0
\(955\) 21312.0 0.722136
\(956\) 0 0
\(957\) −11808.0 −0.398849
\(958\) 0 0
\(959\) 2646.00 0.0890968
\(960\) 0 0
\(961\) −17247.0 −0.578933
\(962\) 0 0
\(963\) −10956.0 −0.366617
\(964\) 0 0
\(965\) −16116.0 −0.537609
\(966\) 0 0
\(967\) −10264.0 −0.341332 −0.170666 0.985329i \(-0.554592\pi\)
−0.170666 + 0.985329i \(0.554592\pi\)
\(968\) 0 0
\(969\) −8160.00 −0.270523
\(970\) 0 0
\(971\) 51468.0 1.70102 0.850508 0.525962i \(-0.176295\pi\)
0.850508 + 0.525962i \(0.176295\pi\)
\(972\) 0 0
\(973\) −17500.0 −0.576592
\(974\) 0 0
\(975\) 29192.0 0.958864
\(976\) 0 0
\(977\) −23790.0 −0.779027 −0.389514 0.921021i \(-0.627357\pi\)
−0.389514 + 0.921021i \(0.627357\pi\)
\(978\) 0 0
\(979\) 16776.0 0.547664
\(980\) 0 0
\(981\) −15202.0 −0.494763
\(982\) 0 0
\(983\) 26424.0 0.857370 0.428685 0.903454i \(-0.358977\pi\)
0.428685 + 0.903454i \(0.358977\pi\)
\(984\) 0 0
\(985\) −8460.00 −0.273663
\(986\) 0 0
\(987\) 5376.00 0.173374
\(988\) 0 0
\(989\) −37152.0 −1.19450
\(990\) 0 0
\(991\) 39488.0 1.26577 0.632885 0.774246i \(-0.281871\pi\)
0.632885 + 0.774246i \(0.281871\pi\)
\(992\) 0 0
\(993\) 25040.0 0.800222
\(994\) 0 0
\(995\) −17808.0 −0.567388
\(996\) 0 0
\(997\) 30854.0 0.980096 0.490048 0.871695i \(-0.336979\pi\)
0.490048 + 0.871695i \(0.336979\pi\)
\(998\) 0 0
\(999\) −16720.0 −0.529527
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 28.4.a.b.1.1 1
3.2 odd 2 252.4.a.c.1.1 1
4.3 odd 2 112.4.a.c.1.1 1
5.2 odd 4 700.4.e.f.449.1 2
5.3 odd 4 700.4.e.f.449.2 2
5.4 even 2 700.4.a.e.1.1 1
7.2 even 3 196.4.e.c.165.1 2
7.3 odd 6 196.4.e.d.177.1 2
7.4 even 3 196.4.e.c.177.1 2
7.5 odd 6 196.4.e.d.165.1 2
7.6 odd 2 196.4.a.b.1.1 1
8.3 odd 2 448.4.a.m.1.1 1
8.5 even 2 448.4.a.d.1.1 1
12.11 even 2 1008.4.a.f.1.1 1
21.2 odd 6 1764.4.k.k.361.1 2
21.5 even 6 1764.4.k.e.361.1 2
21.11 odd 6 1764.4.k.k.1549.1 2
21.17 even 6 1764.4.k.e.1549.1 2
21.20 even 2 1764.4.a.k.1.1 1
28.27 even 2 784.4.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.4.a.b.1.1 1 1.1 even 1 trivial
112.4.a.c.1.1 1 4.3 odd 2
196.4.a.b.1.1 1 7.6 odd 2
196.4.e.c.165.1 2 7.2 even 3
196.4.e.c.177.1 2 7.4 even 3
196.4.e.d.165.1 2 7.5 odd 6
196.4.e.d.177.1 2 7.3 odd 6
252.4.a.c.1.1 1 3.2 odd 2
448.4.a.d.1.1 1 8.5 even 2
448.4.a.m.1.1 1 8.3 odd 2
700.4.a.e.1.1 1 5.4 even 2
700.4.e.f.449.1 2 5.2 odd 4
700.4.e.f.449.2 2 5.3 odd 4
784.4.a.n.1.1 1 28.27 even 2
1008.4.a.f.1.1 1 12.11 even 2
1764.4.a.k.1.1 1 21.20 even 2
1764.4.k.e.361.1 2 21.5 even 6
1764.4.k.e.1549.1 2 21.17 even 6
1764.4.k.k.361.1 2 21.2 odd 6
1764.4.k.k.1549.1 2 21.11 odd 6