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Properties

Label 2352.4.a.c.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

\(f(q)\) \(=\) \(q-3.00000 q^{3} -7.00000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -7.00000 q^{5} +9.00000 q^{9} -7.00000 q^{11} +52.0000 q^{13} +21.0000 q^{15} -72.0000 q^{17} +20.0000 q^{19} +48.0000 q^{23} -76.0000 q^{25} -27.0000 q^{27} -243.000 q^{29} +95.0000 q^{31} +21.0000 q^{33} +352.000 q^{37} -156.000 q^{39} +296.000 q^{41} -158.000 q^{43} -63.0000 q^{45} -142.000 q^{47} +216.000 q^{51} -375.000 q^{53} +49.0000 q^{55} -60.0000 q^{57} +279.000 q^{59} -246.000 q^{61} -364.000 q^{65} +730.000 q^{67} -144.000 q^{69} -338.000 q^{71} +542.000 q^{73} +228.000 q^{75} +305.000 q^{79} +81.0000 q^{81} +1123.00 q^{83} +504.000 q^{85} +729.000 q^{87} +426.000 q^{89} -285.000 q^{93} -140.000 q^{95} +369.000 q^{97} -63.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
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
\(5\) −7.00000 −0.626099 −0.313050 − 0.949737i \(-0.601351\pi\)
−0.313050 + 0.949737i \(0.601351\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −7.00000 −0.191871 −0.0959354 − 0.995388i \(-0.530584\pi\)
−0.0959354 + 0.995388i \(0.530584\pi\)
\(12\) 0 0
\(13\) 52.0000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 21.0000 0.361478
\(16\) 0 0
\(17\) −72.0000 −1.02721 −0.513605 − 0.858027i \(-0.671690\pi\)
−0.513605 + 0.858027i \(0.671690\pi\)
\(18\) 0 0
\(19\) 20.0000 0.241490 0.120745 − 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 48.0000 0.435161 0.217580 − 0.976042i \(-0.430184\pi\)
0.217580 + 0.976042i \(0.430184\pi\)
\(24\) 0 0
\(25\) −76.0000 −0.608000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −243.000 −1.55600 −0.777999 − 0.628265i \(-0.783765\pi\)
−0.777999 + 0.628265i \(0.783765\pi\)
\(30\) 0 0
\(31\) 95.0000 0.550403 0.275202 − 0.961387i \(-0.411255\pi\)
0.275202 + 0.961387i \(0.411255\pi\)
\(32\) 0 0
\(33\) 21.0000 0.110777
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 352.000 1.56401 0.782006 − 0.623271i \(-0.214197\pi\)
0.782006 + 0.623271i \(0.214197\pi\)
\(38\) 0 0
\(39\) −156.000 −0.640513
\(40\) 0 0
\(41\) 296.000 1.12750 0.563749 − 0.825946i \(-0.309358\pi\)
0.563749 + 0.825946i \(0.309358\pi\)
\(42\) 0 0
\(43\) −158.000 −0.560344 −0.280172 − 0.959950i \(-0.590391\pi\)
−0.280172 + 0.959950i \(0.590391\pi\)
\(44\) 0 0
\(45\) −63.0000 −0.208700
\(46\) 0 0
\(47\) −142.000 −0.440698 −0.220349 − 0.975421i \(-0.570720\pi\)
−0.220349 + 0.975421i \(0.570720\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 216.000 0.593060
\(52\) 0 0
\(53\) −375.000 −0.971891 −0.485945 − 0.873989i \(-0.661525\pi\)
−0.485945 + 0.873989i \(0.661525\pi\)
\(54\) 0 0
\(55\) 49.0000 0.120130
\(56\) 0 0
\(57\) −60.0000 −0.139424
\(58\) 0 0
\(59\) 279.000 0.615639 0.307820 − 0.951445i \(-0.400401\pi\)
0.307820 + 0.951445i \(0.400401\pi\)
\(60\) 0 0
\(61\) −246.000 −0.516345 −0.258173 − 0.966099i \(-0.583120\pi\)
−0.258173 + 0.966099i \(0.583120\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −364.000 −0.694595
\(66\) 0 0
\(67\) 730.000 1.33110 0.665550 − 0.746353i \(-0.268197\pi\)
0.665550 + 0.746353i \(0.268197\pi\)
\(68\) 0 0
\(69\) −144.000 −0.251240
\(70\) 0 0
\(71\) −338.000 −0.564975 −0.282487 − 0.959271i \(-0.591160\pi\)
−0.282487 + 0.959271i \(0.591160\pi\)
\(72\) 0 0
\(73\) 542.000 0.868990 0.434495 − 0.900674i \(-0.356927\pi\)
0.434495 + 0.900674i \(0.356927\pi\)
\(74\) 0 0
\(75\) 228.000 0.351029
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 305.000 0.434369 0.217185 − 0.976131i \(-0.430313\pi\)
0.217185 + 0.976131i \(0.430313\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1123.00 1.48512 0.742562 − 0.669778i \(-0.233611\pi\)
0.742562 + 0.669778i \(0.233611\pi\)
\(84\) 0 0
\(85\) 504.000 0.643135
\(86\) 0 0
\(87\) 729.000 0.898356
\(88\) 0 0
\(89\) 426.000 0.507370 0.253685 − 0.967287i \(-0.418357\pi\)
0.253685 + 0.967287i \(0.418357\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −285.000 −0.317776
\(94\) 0 0
\(95\) −140.000 −0.151197
\(96\) 0 0
\(97\) 369.000 0.386250 0.193125 − 0.981174i \(-0.438138\pi\)
0.193125 + 0.981174i \(0.438138\pi\)
\(98\) 0 0
\(99\) −63.0000 −0.0639570
\(100\) 0 0
\(101\) −1270.00 −1.25119 −0.625593 − 0.780150i \(-0.715143\pi\)
−0.625593 + 0.780150i \(0.715143\pi\)
\(102\) 0 0
\(103\) 1832.00 1.75255 0.876273 − 0.481814i \(-0.160022\pi\)
0.876273 + 0.481814i \(0.160022\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 999.000 0.902589 0.451294 − 0.892375i \(-0.350962\pi\)
0.451294 + 0.892375i \(0.350962\pi\)
\(108\) 0 0
\(109\) −1666.00 −1.46398 −0.731990 − 0.681315i \(-0.761408\pi\)
−0.731990 + 0.681315i \(0.761408\pi\)
\(110\) 0 0
\(111\) −1056.00 −0.902983
\(112\) 0 0
\(113\) −1832.00 −1.52513 −0.762567 − 0.646910i \(-0.776061\pi\)
−0.762567 + 0.646910i \(0.776061\pi\)
\(114\) 0 0
\(115\) −336.000 −0.272454
\(116\) 0 0
\(117\) 468.000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1282.00 −0.963186
\(122\) 0 0
\(123\) −888.000 −0.650961
\(124\) 0 0
\(125\) 1407.00 1.00677
\(126\) 0 0
\(127\) 1931.00 1.34920 0.674601 − 0.738183i \(-0.264316\pi\)
0.674601 + 0.738183i \(0.264316\pi\)
\(128\) 0 0
\(129\) 474.000 0.323515
\(130\) 0 0
\(131\) 979.000 0.652944 0.326472 − 0.945207i \(-0.394140\pi\)
0.326472 + 0.945207i \(0.394140\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 189.000 0.120493
\(136\) 0 0
\(137\) −486.000 −0.303079 −0.151539 − 0.988451i \(-0.548423\pi\)
−0.151539 + 0.988451i \(0.548423\pi\)
\(138\) 0 0
\(139\) −2630.00 −1.60485 −0.802423 − 0.596755i \(-0.796456\pi\)
−0.802423 + 0.596755i \(0.796456\pi\)
\(140\) 0 0
\(141\) 426.000 0.254437
\(142\) 0 0
\(143\) −364.000 −0.212862
\(144\) 0 0
\(145\) 1701.00 0.974209
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1878.00 −1.03256 −0.516281 − 0.856419i \(-0.672684\pi\)
−0.516281 + 0.856419i \(0.672684\pi\)
\(150\) 0 0
\(151\) −2103.00 −1.13338 −0.566688 − 0.823933i \(-0.691775\pi\)
−0.566688 + 0.823933i \(0.691775\pi\)
\(152\) 0 0
\(153\) −648.000 −0.342403
\(154\) 0 0
\(155\) −665.000 −0.344607
\(156\) 0 0
\(157\) −1404.00 −0.713703 −0.356852 − 0.934161i \(-0.616150\pi\)
−0.356852 + 0.934161i \(0.616150\pi\)
\(158\) 0 0
\(159\) 1125.00 0.561121
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1536.00 0.738091 0.369045 − 0.929411i \(-0.379685\pi\)
0.369045 + 0.929411i \(0.379685\pi\)
\(164\) 0 0
\(165\) −147.000 −0.0693572
\(166\) 0 0
\(167\) −3634.00 −1.68388 −0.841938 − 0.539574i \(-0.818585\pi\)
−0.841938 + 0.539574i \(0.818585\pi\)
\(168\) 0 0
\(169\) 507.000 0.230769
\(170\) 0 0
\(171\) 180.000 0.0804967
\(172\) 0 0
\(173\) −858.000 −0.377067 −0.188533 − 0.982067i \(-0.560373\pi\)
−0.188533 + 0.982067i \(0.560373\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −837.000 −0.355439
\(178\) 0 0
\(179\) 2420.00 1.01050 0.505249 − 0.862973i \(-0.331400\pi\)
0.505249 + 0.862973i \(0.331400\pi\)
\(180\) 0 0
\(181\) 2672.00 1.09728 0.548641 − 0.836058i \(-0.315145\pi\)
0.548641 + 0.836058i \(0.315145\pi\)
\(182\) 0 0
\(183\) 738.000 0.298112
\(184\) 0 0
\(185\) −2464.00 −0.979226
\(186\) 0 0
\(187\) 504.000 0.197092
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1712.00 0.648565 0.324283 − 0.945960i \(-0.394877\pi\)
0.324283 + 0.945960i \(0.394877\pi\)
\(192\) 0 0
\(193\) −3395.00 −1.26620 −0.633102 − 0.774068i \(-0.718219\pi\)
−0.633102 + 0.774068i \(0.718219\pi\)
\(194\) 0 0
\(195\) 1092.00 0.401024
\(196\) 0 0
\(197\) 510.000 0.184447 0.0922233 − 0.995738i \(-0.470603\pi\)
0.0922233 + 0.995738i \(0.470603\pi\)
\(198\) 0 0
\(199\) 276.000 0.0983172 0.0491586 − 0.998791i \(-0.484346\pi\)
0.0491586 + 0.998791i \(0.484346\pi\)
\(200\) 0 0
\(201\) −2190.00 −0.768511
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2072.00 −0.705926
\(206\) 0 0
\(207\) 432.000 0.145054
\(208\) 0 0
\(209\) −140.000 −0.0463349
\(210\) 0 0
\(211\) −3198.00 −1.04341 −0.521705 − 0.853126i \(-0.674704\pi\)
−0.521705 + 0.853126i \(0.674704\pi\)
\(212\) 0 0
\(213\) 1014.00 0.326188
\(214\) 0 0
\(215\) 1106.00 0.350831
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1626.00 −0.501712
\(220\) 0 0
\(221\) −3744.00 −1.13959
\(222\) 0 0
\(223\) −5091.00 −1.52878 −0.764391 − 0.644752i \(-0.776960\pi\)
−0.764391 + 0.644752i \(0.776960\pi\)
\(224\) 0 0
\(225\) −684.000 −0.202667
\(226\) 0 0
\(227\) −3895.00 −1.13886 −0.569428 − 0.822041i \(-0.692835\pi\)
−0.569428 + 0.822041i \(0.692835\pi\)
\(228\) 0 0
\(229\) 6428.00 1.85491 0.927455 − 0.373936i \(-0.121992\pi\)
0.927455 + 0.373936i \(0.121992\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3824.00 −1.07519 −0.537593 − 0.843204i \(-0.680666\pi\)
−0.537593 + 0.843204i \(0.680666\pi\)
\(234\) 0 0
\(235\) 994.000 0.275921
\(236\) 0 0
\(237\) −915.000 −0.250783
\(238\) 0 0
\(239\) −568.000 −0.153727 −0.0768637 − 0.997042i \(-0.524491\pi\)
−0.0768637 + 0.997042i \(0.524491\pi\)
\(240\) 0 0
\(241\) −3535.00 −0.944852 −0.472426 − 0.881370i \(-0.656622\pi\)
−0.472426 + 0.881370i \(0.656622\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1040.00 0.267909
\(248\) 0 0
\(249\) −3369.00 −0.857437
\(250\) 0 0
\(251\) 4335.00 1.09013 0.545065 − 0.838394i \(-0.316505\pi\)
0.545065 + 0.838394i \(0.316505\pi\)
\(252\) 0 0
\(253\) −336.000 −0.0834946
\(254\) 0 0
\(255\) −1512.00 −0.371314
\(256\) 0 0
\(257\) −3574.00 −0.867471 −0.433735 − 0.901040i \(-0.642805\pi\)
−0.433735 + 0.901040i \(0.642805\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2187.00 −0.518666
\(262\) 0 0
\(263\) −5466.00 −1.28155 −0.640776 − 0.767728i \(-0.721387\pi\)
−0.640776 + 0.767728i \(0.721387\pi\)
\(264\) 0 0
\(265\) 2625.00 0.608500
\(266\) 0 0
\(267\) −1278.00 −0.292930
\(268\) 0 0
\(269\) 5815.00 1.31802 0.659009 − 0.752135i \(-0.270976\pi\)
0.659009 + 0.752135i \(0.270976\pi\)
\(270\) 0 0
\(271\) −3237.00 −0.725586 −0.362793 − 0.931870i \(-0.618177\pi\)
−0.362793 + 0.931870i \(0.618177\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 532.000 0.116657
\(276\) 0 0
\(277\) −3176.00 −0.688907 −0.344454 − 0.938803i \(-0.611936\pi\)
−0.344454 + 0.938803i \(0.611936\pi\)
\(278\) 0 0
\(279\) 855.000 0.183468
\(280\) 0 0
\(281\) −3282.00 −0.696753 −0.348377 − 0.937355i \(-0.613267\pi\)
−0.348377 + 0.937355i \(0.613267\pi\)
\(282\) 0 0
\(283\) 2182.00 0.458327 0.229163 − 0.973388i \(-0.426401\pi\)
0.229163 + 0.973388i \(0.426401\pi\)
\(284\) 0 0
\(285\) 420.000 0.0872935
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 271.000 0.0551598
\(290\) 0 0
\(291\) −1107.00 −0.223002
\(292\) 0 0
\(293\) −3021.00 −0.602351 −0.301175 − 0.953569i \(-0.597379\pi\)
−0.301175 + 0.953569i \(0.597379\pi\)
\(294\) 0 0
\(295\) −1953.00 −0.385451
\(296\) 0 0
\(297\) 189.000 0.0369256
\(298\) 0 0
\(299\) 2496.00 0.482767
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3810.00 0.722372
\(304\) 0 0
\(305\) 1722.00 0.323283
\(306\) 0 0
\(307\) −1304.00 −0.242421 −0.121210 − 0.992627i \(-0.538678\pi\)
−0.121210 + 0.992627i \(0.538678\pi\)
\(308\) 0 0
\(309\) −5496.00 −1.01183
\(310\) 0 0
\(311\) 6208.00 1.13191 0.565954 − 0.824437i \(-0.308508\pi\)
0.565954 + 0.824437i \(0.308508\pi\)
\(312\) 0 0
\(313\) −3337.00 −0.602615 −0.301307 − 0.953527i \(-0.597423\pi\)
−0.301307 + 0.953527i \(0.597423\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8637.00 1.53029 0.765146 − 0.643857i \(-0.222667\pi\)
0.765146 + 0.643857i \(0.222667\pi\)
\(318\) 0 0
\(319\) 1701.00 0.298551
\(320\) 0 0
\(321\) −2997.00 −0.521110
\(322\) 0 0
\(323\) −1440.00 −0.248061
\(324\) 0 0
\(325\) −3952.00 −0.674515
\(326\) 0 0
\(327\) 4998.00 0.845229
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1940.00 −0.322151 −0.161076 − 0.986942i \(-0.551496\pi\)
−0.161076 + 0.986942i \(0.551496\pi\)
\(332\) 0 0
\(333\) 3168.00 0.521337
\(334\) 0 0
\(335\) −5110.00 −0.833400
\(336\) 0 0
\(337\) −5527.00 −0.893397 −0.446699 − 0.894684i \(-0.647400\pi\)
−0.446699 + 0.894684i \(0.647400\pi\)
\(338\) 0 0
\(339\) 5496.00 0.880536
\(340\) 0 0
\(341\) −665.000 −0.105606
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1008.00 0.157301
\(346\) 0 0
\(347\) −9564.00 −1.47960 −0.739802 − 0.672825i \(-0.765081\pi\)
−0.739802 + 0.672825i \(0.765081\pi\)
\(348\) 0 0
\(349\) 918.000 0.140801 0.0704003 − 0.997519i \(-0.477572\pi\)
0.0704003 + 0.997519i \(0.477572\pi\)
\(350\) 0 0
\(351\) −1404.00 −0.213504
\(352\) 0 0
\(353\) −11480.0 −1.73093 −0.865466 − 0.500968i \(-0.832977\pi\)
−0.865466 + 0.500968i \(0.832977\pi\)
\(354\) 0 0
\(355\) 2366.00 0.353730
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3654.00 −0.537189 −0.268594 − 0.963253i \(-0.586559\pi\)
−0.268594 + 0.963253i \(0.586559\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) 3846.00 0.556095
\(364\) 0 0
\(365\) −3794.00 −0.544074
\(366\) 0 0
\(367\) −10067.0 −1.43186 −0.715931 − 0.698171i \(-0.753997\pi\)
−0.715931 + 0.698171i \(0.753997\pi\)
\(368\) 0 0
\(369\) 2664.00 0.375833
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 8800.00 1.22157 0.610786 − 0.791795i \(-0.290853\pi\)
0.610786 + 0.791795i \(0.290853\pi\)
\(374\) 0 0
\(375\) −4221.00 −0.581257
\(376\) 0 0
\(377\) −12636.0 −1.72623
\(378\) 0 0
\(379\) −1136.00 −0.153964 −0.0769821 − 0.997032i \(-0.524528\pi\)
−0.0769821 + 0.997032i \(0.524528\pi\)
\(380\) 0 0
\(381\) −5793.00 −0.778962
\(382\) 0 0
\(383\) 4290.00 0.572347 0.286173 − 0.958178i \(-0.407617\pi\)
0.286173 + 0.958178i \(0.407617\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1422.00 −0.186781
\(388\) 0 0
\(389\) −862.000 −0.112353 −0.0561763 − 0.998421i \(-0.517891\pi\)
−0.0561763 + 0.998421i \(0.517891\pi\)
\(390\) 0 0
\(391\) −3456.00 −0.447001
\(392\) 0 0
\(393\) −2937.00 −0.376977
\(394\) 0 0
\(395\) −2135.00 −0.271958
\(396\) 0 0
\(397\) 9140.00 1.15547 0.577737 − 0.816223i \(-0.303936\pi\)
0.577737 + 0.816223i \(0.303936\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14232.0 −1.77235 −0.886175 − 0.463351i \(-0.846647\pi\)
−0.886175 + 0.463351i \(0.846647\pi\)
\(402\) 0 0
\(403\) 4940.00 0.610618
\(404\) 0 0
\(405\) −567.000 −0.0695666
\(406\) 0 0
\(407\) −2464.00 −0.300088
\(408\) 0 0
\(409\) 10001.0 1.20909 0.604545 − 0.796571i \(-0.293355\pi\)
0.604545 + 0.796571i \(0.293355\pi\)
\(410\) 0 0
\(411\) 1458.00 0.174983
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −7861.00 −0.929834
\(416\) 0 0
\(417\) 7890.00 0.926559
\(418\) 0 0
\(419\) −10256.0 −1.19580 −0.597898 − 0.801572i \(-0.703997\pi\)
−0.597898 + 0.801572i \(0.703997\pi\)
\(420\) 0 0
\(421\) −4502.00 −0.521174 −0.260587 − 0.965450i \(-0.583916\pi\)
−0.260587 + 0.965450i \(0.583916\pi\)
\(422\) 0 0
\(423\) −1278.00 −0.146899
\(424\) 0 0
\(425\) 5472.00 0.624544
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1092.00 0.122896
\(430\) 0 0
\(431\) −12356.0 −1.38090 −0.690450 − 0.723380i \(-0.742587\pi\)
−0.690450 + 0.723380i \(0.742587\pi\)
\(432\) 0 0
\(433\) −862.000 −0.0956699 −0.0478350 − 0.998855i \(-0.515232\pi\)
−0.0478350 + 0.998855i \(0.515232\pi\)
\(434\) 0 0
\(435\) −5103.00 −0.562460
\(436\) 0 0
\(437\) 960.000 0.105087
\(438\) 0 0
\(439\) −1365.00 −0.148401 −0.0742003 − 0.997243i \(-0.523640\pi\)
−0.0742003 + 0.997243i \(0.523640\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8931.00 −0.957843 −0.478922 − 0.877858i \(-0.658972\pi\)
−0.478922 + 0.877858i \(0.658972\pi\)
\(444\) 0 0
\(445\) −2982.00 −0.317664
\(446\) 0 0
\(447\) 5634.00 0.596150
\(448\) 0 0
\(449\) 11228.0 1.18014 0.590069 − 0.807353i \(-0.299100\pi\)
0.590069 + 0.807353i \(0.299100\pi\)
\(450\) 0 0
\(451\) −2072.00 −0.216334
\(452\) 0 0
\(453\) 6309.00 0.654355
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12151.0 1.24376 0.621882 − 0.783111i \(-0.286368\pi\)
0.621882 + 0.783111i \(0.286368\pi\)
\(458\) 0 0
\(459\) 1944.00 0.197687
\(460\) 0 0
\(461\) −18534.0 −1.87248 −0.936241 − 0.351358i \(-0.885720\pi\)
−0.936241 + 0.351358i \(0.885720\pi\)
\(462\) 0 0
\(463\) −17096.0 −1.71602 −0.858011 − 0.513631i \(-0.828300\pi\)
−0.858011 + 0.513631i \(0.828300\pi\)
\(464\) 0 0
\(465\) 1995.00 0.198959
\(466\) 0 0
\(467\) −2140.00 −0.212050 −0.106025 − 0.994363i \(-0.533812\pi\)
−0.106025 + 0.994363i \(0.533812\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4212.00 0.412057
\(472\) 0 0
\(473\) 1106.00 0.107514
\(474\) 0 0
\(475\) −1520.00 −0.146826
\(476\) 0 0
\(477\) −3375.00 −0.323964
\(478\) 0 0
\(479\) −13266.0 −1.26543 −0.632713 − 0.774386i \(-0.718059\pi\)
−0.632713 + 0.774386i \(0.718059\pi\)
\(480\) 0 0
\(481\) 18304.0 1.73512
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2583.00 −0.241831
\(486\) 0 0
\(487\) 19183.0 1.78494 0.892469 − 0.451109i \(-0.148971\pi\)
0.892469 + 0.451109i \(0.148971\pi\)
\(488\) 0 0
\(489\) −4608.00 −0.426137
\(490\) 0 0
\(491\) 21693.0 1.99387 0.996936 − 0.0782185i \(-0.0249232\pi\)
0.996936 + 0.0782185i \(0.0249232\pi\)
\(492\) 0 0
\(493\) 17496.0 1.59834
\(494\) 0 0
\(495\) 441.000 0.0400434
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −19658.0 −1.76355 −0.881776 − 0.471667i \(-0.843652\pi\)
−0.881776 + 0.471667i \(0.843652\pi\)
\(500\) 0 0
\(501\) 10902.0 0.972187
\(502\) 0 0
\(503\) 19436.0 1.72288 0.861440 − 0.507860i \(-0.169563\pi\)
0.861440 + 0.507860i \(0.169563\pi\)
\(504\) 0 0
\(505\) 8890.00 0.783366
\(506\) 0 0
\(507\) −1521.00 −0.133235
\(508\) 0 0
\(509\) −3089.00 −0.268993 −0.134497 − 0.990914i \(-0.542942\pi\)
−0.134497 + 0.990914i \(0.542942\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −540.000 −0.0464748
\(514\) 0 0
\(515\) −12824.0 −1.09727
\(516\) 0 0
\(517\) 994.000 0.0845572
\(518\) 0 0
\(519\) 2574.00 0.217700
\(520\) 0 0
\(521\) −1950.00 −0.163975 −0.0819876 − 0.996633i \(-0.526127\pi\)
−0.0819876 + 0.996633i \(0.526127\pi\)
\(522\) 0 0
\(523\) 13132.0 1.09794 0.548970 − 0.835842i \(-0.315020\pi\)
0.548970 + 0.835842i \(0.315020\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6840.00 −0.565380
\(528\) 0 0
\(529\) −9863.00 −0.810635
\(530\) 0 0
\(531\) 2511.00 0.205213
\(532\) 0 0
\(533\) 15392.0 1.25085
\(534\) 0 0
\(535\) −6993.00 −0.565110
\(536\) 0 0
\(537\) −7260.00 −0.583412
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2930.00 0.232848 0.116424 − 0.993200i \(-0.462857\pi\)
0.116424 + 0.993200i \(0.462857\pi\)
\(542\) 0 0
\(543\) −8016.00 −0.633517
\(544\) 0 0
\(545\) 11662.0 0.916597
\(546\) 0 0
\(547\) −19824.0 −1.54957 −0.774783 − 0.632227i \(-0.782141\pi\)
−0.774783 + 0.632227i \(0.782141\pi\)
\(548\) 0 0
\(549\) −2214.00 −0.172115
\(550\) 0 0
\(551\) −4860.00 −0.375759
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 7392.00 0.565357
\(556\) 0 0
\(557\) −19169.0 −1.45820 −0.729099 − 0.684408i \(-0.760061\pi\)
−0.729099 + 0.684408i \(0.760061\pi\)
\(558\) 0 0
\(559\) −8216.00 −0.621645
\(560\) 0 0
\(561\) −1512.00 −0.113791
\(562\) 0 0
\(563\) 7267.00 0.543992 0.271996 − 0.962298i \(-0.412316\pi\)
0.271996 + 0.962298i \(0.412316\pi\)
\(564\) 0 0
\(565\) 12824.0 0.954884
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4892.00 −0.360428 −0.180214 − 0.983627i \(-0.557679\pi\)
−0.180214 + 0.983627i \(0.557679\pi\)
\(570\) 0 0
\(571\) −22126.0 −1.62162 −0.810809 − 0.585310i \(-0.800973\pi\)
−0.810809 + 0.585310i \(0.800973\pi\)
\(572\) 0 0
\(573\) −5136.00 −0.374449
\(574\) 0 0
\(575\) −3648.00 −0.264578
\(576\) 0 0
\(577\) 1097.00 0.0791485 0.0395743 − 0.999217i \(-0.487400\pi\)
0.0395743 + 0.999217i \(0.487400\pi\)
\(578\) 0 0
\(579\) 10185.0 0.731043
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2625.00 0.186478
\(584\) 0 0
\(585\) −3276.00 −0.231532
\(586\) 0 0
\(587\) 9969.00 0.700962 0.350481 − 0.936570i \(-0.386018\pi\)
0.350481 + 0.936570i \(0.386018\pi\)
\(588\) 0 0
\(589\) 1900.00 0.132917
\(590\) 0 0
\(591\) −1530.00 −0.106490
\(592\) 0 0
\(593\) 6424.00 0.444860 0.222430 − 0.974949i \(-0.428601\pi\)
0.222430 + 0.974949i \(0.428601\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −828.000 −0.0567635
\(598\) 0 0
\(599\) −17046.0 −1.16274 −0.581370 − 0.813640i \(-0.697483\pi\)
−0.581370 + 0.813640i \(0.697483\pi\)
\(600\) 0 0
\(601\) 18877.0 1.28121 0.640606 − 0.767869i \(-0.278683\pi\)
0.640606 + 0.767869i \(0.278683\pi\)
\(602\) 0 0
\(603\) 6570.00 0.443700
\(604\) 0 0
\(605\) 8974.00 0.603050
\(606\) 0 0
\(607\) −13783.0 −0.921639 −0.460819 − 0.887494i \(-0.652444\pi\)
−0.460819 + 0.887494i \(0.652444\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7384.00 −0.488911
\(612\) 0 0
\(613\) 18192.0 1.19864 0.599321 − 0.800509i \(-0.295437\pi\)
0.599321 + 0.800509i \(0.295437\pi\)
\(614\) 0 0
\(615\) 6216.00 0.407566
\(616\) 0 0
\(617\) −13054.0 −0.851757 −0.425879 − 0.904780i \(-0.640035\pi\)
−0.425879 + 0.904780i \(0.640035\pi\)
\(618\) 0 0
\(619\) 7246.00 0.470503 0.235251 − 0.971935i \(-0.424409\pi\)
0.235251 + 0.971935i \(0.424409\pi\)
\(620\) 0 0
\(621\) −1296.00 −0.0837467
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) 0 0
\(627\) 420.000 0.0267515
\(628\) 0 0
\(629\) −25344.0 −1.60657
\(630\) 0 0
\(631\) −2817.00 −0.177723 −0.0888613 − 0.996044i \(-0.528323\pi\)
−0.0888613 + 0.996044i \(0.528323\pi\)
\(632\) 0 0
\(633\) 9594.00 0.602413
\(634\) 0 0
\(635\) −13517.0 −0.844734
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3042.00 −0.188325
\(640\) 0 0
\(641\) 15786.0 0.972714 0.486357 − 0.873760i \(-0.338326\pi\)
0.486357 + 0.873760i \(0.338326\pi\)
\(642\) 0 0
\(643\) −17426.0 −1.06876 −0.534381 − 0.845244i \(-0.679455\pi\)
−0.534381 + 0.845244i \(0.679455\pi\)
\(644\) 0 0
\(645\) −3318.00 −0.202552
\(646\) 0 0
\(647\) −25834.0 −1.56977 −0.784884 − 0.619643i \(-0.787277\pi\)
−0.784884 + 0.619643i \(0.787277\pi\)
\(648\) 0 0
\(649\) −1953.00 −0.118123
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27983.0 1.67697 0.838483 − 0.544927i \(-0.183443\pi\)
0.838483 + 0.544927i \(0.183443\pi\)
\(654\) 0 0
\(655\) −6853.00 −0.408807
\(656\) 0 0
\(657\) 4878.00 0.289663
\(658\) 0 0
\(659\) 28296.0 1.67262 0.836309 − 0.548258i \(-0.184709\pi\)
0.836309 + 0.548258i \(0.184709\pi\)
\(660\) 0 0
\(661\) −21254.0 −1.25066 −0.625329 − 0.780361i \(-0.715035\pi\)
−0.625329 + 0.780361i \(0.715035\pi\)
\(662\) 0 0
\(663\) 11232.0 0.657941
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −11664.0 −0.677109
\(668\) 0 0
\(669\) 15273.0 0.882643
\(670\) 0 0
\(671\) 1722.00 0.0990716
\(672\) 0 0
\(673\) −28259.0 −1.61858 −0.809290 − 0.587409i \(-0.800148\pi\)
−0.809290 + 0.587409i \(0.800148\pi\)
\(674\) 0 0
\(675\) 2052.00 0.117010
\(676\) 0 0
\(677\) −31779.0 −1.80409 −0.902043 − 0.431646i \(-0.857933\pi\)
−0.902043 + 0.431646i \(0.857933\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 11685.0 0.657519
\(682\) 0 0
\(683\) 9747.00 0.546059 0.273030 − 0.962006i \(-0.411974\pi\)
0.273030 + 0.962006i \(0.411974\pi\)
\(684\) 0 0
\(685\) 3402.00 0.189757
\(686\) 0 0
\(687\) −19284.0 −1.07093
\(688\) 0 0
\(689\) −19500.0 −1.07822
\(690\) 0 0
\(691\) −24512.0 −1.34947 −0.674733 − 0.738062i \(-0.735741\pi\)
−0.674733 + 0.738062i \(0.735741\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18410.0 1.00479
\(696\) 0 0
\(697\) −21312.0 −1.15818
\(698\) 0 0
\(699\) 11472.0 0.620760
\(700\) 0 0
\(701\) 14325.0 0.771823 0.385911 − 0.922536i \(-0.373887\pi\)
0.385911 + 0.922536i \(0.373887\pi\)
\(702\) 0 0
\(703\) 7040.00 0.377694
\(704\) 0 0
\(705\) −2982.00 −0.159303
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 12078.0 0.639773 0.319886 − 0.947456i \(-0.396355\pi\)
0.319886 + 0.947456i \(0.396355\pi\)
\(710\) 0 0
\(711\) 2745.00 0.144790
\(712\) 0 0
\(713\) 4560.00 0.239514
\(714\) 0 0
\(715\) 2548.00 0.133272
\(716\) 0 0
\(717\) 1704.00 0.0887546
\(718\) 0 0
\(719\) −10658.0 −0.552818 −0.276409 − 0.961040i \(-0.589144\pi\)
−0.276409 + 0.961040i \(0.589144\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 10605.0 0.545511
\(724\) 0 0
\(725\) 18468.0 0.946047
\(726\) 0 0
\(727\) 19099.0 0.974337 0.487168 − 0.873308i \(-0.338030\pi\)
0.487168 + 0.873308i \(0.338030\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 11376.0 0.575590
\(732\) 0 0
\(733\) −1406.00 −0.0708483 −0.0354241 − 0.999372i \(-0.511278\pi\)
−0.0354241 + 0.999372i \(0.511278\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5110.00 −0.255399
\(738\) 0 0
\(739\) −6730.00 −0.335003 −0.167501 − 0.985872i \(-0.553570\pi\)
−0.167501 + 0.985872i \(0.553570\pi\)
\(740\) 0 0
\(741\) −3120.00 −0.154678
\(742\) 0 0
\(743\) 1166.00 0.0575725 0.0287863 − 0.999586i \(-0.490836\pi\)
0.0287863 + 0.999586i \(0.490836\pi\)
\(744\) 0 0
\(745\) 13146.0 0.646486
\(746\) 0 0
\(747\) 10107.0 0.495041
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16967.0 0.824414 0.412207 − 0.911090i \(-0.364758\pi\)
0.412207 + 0.911090i \(0.364758\pi\)
\(752\) 0 0
\(753\) −13005.0 −0.629387
\(754\) 0 0
\(755\) 14721.0 0.709605
\(756\) 0 0
\(757\) 11878.0 0.570295 0.285147 − 0.958484i \(-0.407957\pi\)
0.285147 + 0.958484i \(0.407957\pi\)
\(758\) 0 0
\(759\) 1008.00 0.0482056
\(760\) 0 0
\(761\) −27324.0 −1.30157 −0.650785 − 0.759262i \(-0.725560\pi\)
−0.650785 + 0.759262i \(0.725560\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4536.00 0.214378
\(766\) 0 0
\(767\) 14508.0 0.682990
\(768\) 0 0
\(769\) 11971.0 0.561359 0.280680 − 0.959802i \(-0.409440\pi\)
0.280680 + 0.959802i \(0.409440\pi\)
\(770\) 0 0
\(771\) 10722.0 0.500834
\(772\) 0 0
\(773\) 338.000 0.0157271 0.00786353 − 0.999969i \(-0.497497\pi\)
0.00786353 + 0.999969i \(0.497497\pi\)
\(774\) 0 0
\(775\) −7220.00 −0.334645
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5920.00 0.272280
\(780\) 0 0
\(781\) 2366.00 0.108402
\(782\) 0 0
\(783\) 6561.00 0.299452
\(784\) 0 0
\(785\) 9828.00 0.446849
\(786\) 0 0
\(787\) 14114.0 0.639275 0.319638 − 0.947540i \(-0.396439\pi\)
0.319638 + 0.947540i \(0.396439\pi\)
\(788\) 0 0
\(789\) 16398.0 0.739904
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12792.0 −0.572834
\(794\) 0 0
\(795\) −7875.00 −0.351318
\(796\) 0 0
\(797\) −1563.00 −0.0694659 −0.0347329 − 0.999397i \(-0.511058\pi\)
−0.0347329 + 0.999397i \(0.511058\pi\)
\(798\) 0 0
\(799\) 10224.0 0.452690
\(800\) 0 0
\(801\) 3834.00 0.169123
\(802\) 0 0
\(803\) −3794.00 −0.166734
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −17445.0 −0.760958
\(808\) 0 0
\(809\) −31568.0 −1.37191 −0.685953 − 0.727646i \(-0.740614\pi\)
−0.685953 + 0.727646i \(0.740614\pi\)
\(810\) 0 0
\(811\) 2626.00 0.113701 0.0568504 − 0.998383i \(-0.481894\pi\)
0.0568504 + 0.998383i \(0.481894\pi\)
\(812\) 0 0
\(813\) 9711.00 0.418917
\(814\) 0 0
\(815\) −10752.0 −0.462118
\(816\) 0 0
\(817\) −3160.00 −0.135318
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 361.000 0.0153459 0.00767295 − 0.999971i \(-0.497558\pi\)
0.00767295 + 0.999971i \(0.497558\pi\)
\(822\) 0 0
\(823\) 29544.0 1.25132 0.625662 − 0.780095i \(-0.284829\pi\)
0.625662 + 0.780095i \(0.284829\pi\)
\(824\) 0 0
\(825\) −1596.00 −0.0673522
\(826\) 0 0
\(827\) −24163.0 −1.01600 −0.507999 − 0.861358i \(-0.669615\pi\)
−0.507999 + 0.861358i \(0.669615\pi\)
\(828\) 0 0
\(829\) 39196.0 1.64214 0.821070 − 0.570828i \(-0.193378\pi\)
0.821070 + 0.570828i \(0.193378\pi\)
\(830\) 0 0
\(831\) 9528.00 0.397741
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 25438.0 1.05427
\(836\) 0 0
\(837\) −2565.00 −0.105925
\(838\) 0 0
\(839\) 6528.00 0.268619 0.134310 − 0.990939i \(-0.457118\pi\)
0.134310 + 0.990939i \(0.457118\pi\)
\(840\) 0 0
\(841\) 34660.0 1.42113
\(842\) 0 0
\(843\) 9846.00 0.402271
\(844\) 0 0
\(845\) −3549.00 −0.144484
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −6546.00 −0.264615
\(850\) 0 0
\(851\) 16896.0 0.680596
\(852\) 0 0
\(853\) −1046.00 −0.0419864 −0.0209932 − 0.999780i \(-0.506683\pi\)
−0.0209932 + 0.999780i \(0.506683\pi\)
\(854\) 0 0
\(855\) −1260.00 −0.0503989
\(856\) 0 0
\(857\) −32910.0 −1.31177 −0.655883 − 0.754862i \(-0.727704\pi\)
−0.655883 + 0.754862i \(0.727704\pi\)
\(858\) 0 0
\(859\) 22682.0 0.900931 0.450466 − 0.892794i \(-0.351258\pi\)
0.450466 + 0.892794i \(0.351258\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7450.00 0.293860 0.146930 − 0.989147i \(-0.453061\pi\)
0.146930 + 0.989147i \(0.453061\pi\)
\(864\) 0 0
\(865\) 6006.00 0.236081
\(866\) 0 0
\(867\) −813.000 −0.0318465
\(868\) 0 0
\(869\) −2135.00 −0.0833428
\(870\) 0 0
\(871\) 37960.0 1.47672
\(872\) 0 0
\(873\) 3321.00 0.128750
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10080.0 0.388116 0.194058 − 0.980990i \(-0.437835\pi\)
0.194058 + 0.980990i \(0.437835\pi\)
\(878\) 0 0
\(879\) 9063.00 0.347767
\(880\) 0 0
\(881\) 32958.0 1.26037 0.630183 − 0.776446i \(-0.282980\pi\)
0.630183 + 0.776446i \(0.282980\pi\)
\(882\) 0 0
\(883\) 19784.0 0.754003 0.377001 − 0.926213i \(-0.376955\pi\)
0.377001 + 0.926213i \(0.376955\pi\)
\(884\) 0 0
\(885\) 5859.00 0.222540
\(886\) 0 0
\(887\) −11480.0 −0.434567 −0.217283 − 0.976109i \(-0.569720\pi\)
−0.217283 + 0.976109i \(0.569720\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −567.000 −0.0213190
\(892\) 0 0
\(893\) −2840.00 −0.106424
\(894\) 0 0
\(895\) −16940.0 −0.632672
\(896\) 0 0
\(897\) −7488.00 −0.278726
\(898\) 0 0
\(899\) −23085.0 −0.856427
\(900\) 0 0
\(901\) 27000.0 0.998336
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18704.0 −0.687008
\(906\) 0 0
\(907\) 45352.0 1.66030 0.830148 − 0.557543i \(-0.188256\pi\)
0.830148 + 0.557543i \(0.188256\pi\)
\(908\) 0 0
\(909\) −11430.0 −0.417062
\(910\) 0 0
\(911\) 39906.0 1.45131 0.725656 − 0.688058i \(-0.241537\pi\)
0.725656 + 0.688058i \(0.241537\pi\)
\(912\) 0 0
\(913\) −7861.00 −0.284952
\(914\) 0 0
\(915\) −5166.00 −0.186648
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −17528.0 −0.629157 −0.314579 − 0.949231i \(-0.601863\pi\)
−0.314579 + 0.949231i \(0.601863\pi\)
\(920\) 0 0
\(921\) 3912.00 0.139962
\(922\) 0 0
\(923\) −17576.0 −0.626783
\(924\) 0 0
\(925\) −26752.0 −0.950919
\(926\) 0 0
\(927\) 16488.0 0.584182
\(928\) 0 0
\(929\) −17066.0 −0.602710 −0.301355 − 0.953512i \(-0.597439\pi\)
−0.301355 + 0.953512i \(0.597439\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −18624.0 −0.653507
\(934\) 0 0
\(935\) −3528.00 −0.123399
\(936\) 0 0
\(937\) 30821.0 1.07458 0.537288 − 0.843399i \(-0.319449\pi\)
0.537288 + 0.843399i \(0.319449\pi\)
\(938\) 0 0
\(939\) 10011.0 0.347920
\(940\) 0 0
\(941\) −4179.00 −0.144773 −0.0723866 − 0.997377i \(-0.523062\pi\)
−0.0723866 + 0.997377i \(0.523062\pi\)
\(942\) 0 0
\(943\) 14208.0 0.490643
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9776.00 −0.335457 −0.167728 − 0.985833i \(-0.553643\pi\)
−0.167728 + 0.985833i \(0.553643\pi\)
\(948\) 0 0
\(949\) 28184.0 0.964058
\(950\) 0 0
\(951\) −25911.0 −0.883514
\(952\) 0 0
\(953\) 14778.0 0.502315 0.251158 − 0.967946i \(-0.419189\pi\)
0.251158 + 0.967946i \(0.419189\pi\)
\(954\) 0 0
\(955\) −11984.0 −0.406066
\(956\) 0 0
\(957\) −5103.00 −0.172368
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −20766.0 −0.697056
\(962\) 0 0
\(963\) 8991.00 0.300863
\(964\) 0 0
\(965\) 23765.0 0.792769
\(966\) 0 0
\(967\) −8759.00 −0.291283 −0.145641 − 0.989337i \(-0.546525\pi\)
−0.145641 + 0.989337i \(0.546525\pi\)
\(968\) 0 0
\(969\) 4320.00 0.143218
\(970\) 0 0
\(971\) −33579.0 −1.10979 −0.554893 − 0.831922i \(-0.687241\pi\)
−0.554893 + 0.831922i \(0.687241\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 11856.0 0.389432
\(976\) 0 0
\(977\) 42634.0 1.39609 0.698046 − 0.716053i \(-0.254053\pi\)
0.698046 + 0.716053i \(0.254053\pi\)
\(978\) 0 0
\(979\) −2982.00 −0.0973495
\(980\) 0 0
\(981\) −14994.0 −0.487993
\(982\) 0 0
\(983\) 29148.0 0.945755 0.472877 − 0.881128i \(-0.343215\pi\)
0.472877 + 0.881128i \(0.343215\pi\)
\(984\) 0 0
\(985\) −3570.00 −0.115482
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7584.00 −0.243839
\(990\) 0 0
\(991\) 44797.0 1.43595 0.717974 − 0.696070i \(-0.245070\pi\)
0.717974 + 0.696070i \(0.245070\pi\)
\(992\) 0 0
\(993\) 5820.00 0.185994
\(994\) 0 0
\(995\) −1932.00 −0.0615563
\(996\) 0 0
\(997\) −10654.0 −0.338431 −0.169215 − 0.985579i \(-0.554123\pi\)
−0.169215 + 0.985579i \(0.554123\pi\)
\(998\) 0 0
\(999\) −9504.00 −0.300994
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.c.1.1 1
4.3 odd 2 1176.4.a.i.1.1 1
7.3 odd 6 336.4.q.a.289.1 2
7.5 odd 6 336.4.q.a.193.1 2
7.6 odd 2 2352.4.a.bg.1.1 1
28.3 even 6 168.4.q.a.121.1 yes 2
28.19 even 6 168.4.q.a.25.1 ✓ 2
28.27 even 2 1176.4.a.f.1.1 1
84.47 odd 6 504.4.s.e.361.1 2
84.59 odd 6 504.4.s.e.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.a.25.1 ✓ 2 28.19 even 6
168.4.q.a.121.1 yes 2 28.3 even 6
336.4.q.a.193.1 2 7.5 odd 6
336.4.q.a.289.1 2 7.3 odd 6
504.4.s.e.289.1 2 84.59 odd 6
504.4.s.e.361.1 2 84.47 odd 6
1176.4.a.f.1.1 1 28.27 even 2
1176.4.a.i.1.1 1 4.3 odd 2
2352.4.a.c.1.1 1 1.1 even 1 trivial
2352.4.a.bg.1.1 1 7.6 odd 2