Properties

Label 450.4.a.j.1.1
Level $450$
Weight $4$
Character 450.1
Self dual yes
Analytic conductor $26.551$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 450.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +26.0000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +26.0000 q^{7} -8.00000 q^{8} +28.0000 q^{11} +12.0000 q^{13} -52.0000 q^{14} +16.0000 q^{16} +64.0000 q^{17} -60.0000 q^{19} -56.0000 q^{22} +58.0000 q^{23} -24.0000 q^{26} +104.000 q^{28} -90.0000 q^{29} -128.000 q^{31} -32.0000 q^{32} -128.000 q^{34} +236.000 q^{37} +120.000 q^{38} -242.000 q^{41} +362.000 q^{43} +112.000 q^{44} -116.000 q^{46} -226.000 q^{47} +333.000 q^{49} +48.0000 q^{52} +108.000 q^{53} -208.000 q^{56} +180.000 q^{58} +20.0000 q^{59} +542.000 q^{61} +256.000 q^{62} +64.0000 q^{64} -434.000 q^{67} +256.000 q^{68} +1128.00 q^{71} +632.000 q^{73} -472.000 q^{74} -240.000 q^{76} +728.000 q^{77} -720.000 q^{79} +484.000 q^{82} +478.000 q^{83} -724.000 q^{86} -224.000 q^{88} +490.000 q^{89} +312.000 q^{91} +232.000 q^{92} +452.000 q^{94} +1456.00 q^{97} -666.000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 26.0000 1.40387 0.701934 0.712242i \(-0.252320\pi\)
0.701934 + 0.712242i \(0.252320\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 28.0000 0.767483 0.383742 0.923440i \(-0.374635\pi\)
0.383742 + 0.923440i \(0.374635\pi\)
\(12\) 0 0
\(13\) 12.0000 0.256015 0.128008 0.991773i \(-0.459142\pi\)
0.128008 + 0.991773i \(0.459142\pi\)
\(14\) −52.0000 −0.992685
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 64.0000 0.913075 0.456538 0.889704i \(-0.349089\pi\)
0.456538 + 0.889704i \(0.349089\pi\)
\(18\) 0 0
\(19\) −60.0000 −0.724471 −0.362235 0.932087i \(-0.617986\pi\)
−0.362235 + 0.932087i \(0.617986\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −56.0000 −0.542693
\(23\) 58.0000 0.525819 0.262909 0.964821i \(-0.415318\pi\)
0.262909 + 0.964821i \(0.415318\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −24.0000 −0.181030
\(27\) 0 0
\(28\) 104.000 0.701934
\(29\) −90.0000 −0.576296 −0.288148 0.957586i \(-0.593039\pi\)
−0.288148 + 0.957586i \(0.593039\pi\)
\(30\) 0 0
\(31\) −128.000 −0.741596 −0.370798 0.928714i \(-0.620916\pi\)
−0.370798 + 0.928714i \(0.620916\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −128.000 −0.645642
\(35\) 0 0
\(36\) 0 0
\(37\) 236.000 1.04860 0.524299 0.851534i \(-0.324327\pi\)
0.524299 + 0.851534i \(0.324327\pi\)
\(38\) 120.000 0.512278
\(39\) 0 0
\(40\) 0 0
\(41\) −242.000 −0.921806 −0.460903 0.887450i \(-0.652474\pi\)
−0.460903 + 0.887450i \(0.652474\pi\)
\(42\) 0 0
\(43\) 362.000 1.28383 0.641913 0.766778i \(-0.278141\pi\)
0.641913 + 0.766778i \(0.278141\pi\)
\(44\) 112.000 0.383742
\(45\) 0 0
\(46\) −116.000 −0.371810
\(47\) −226.000 −0.701393 −0.350697 0.936489i \(-0.614055\pi\)
−0.350697 + 0.936489i \(0.614055\pi\)
\(48\) 0 0
\(49\) 333.000 0.970845
\(50\) 0 0
\(51\) 0 0
\(52\) 48.0000 0.128008
\(53\) 108.000 0.279905 0.139952 0.990158i \(-0.455305\pi\)
0.139952 + 0.990158i \(0.455305\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −208.000 −0.496342
\(57\) 0 0
\(58\) 180.000 0.407503
\(59\) 20.0000 0.0441318 0.0220659 0.999757i \(-0.492976\pi\)
0.0220659 + 0.999757i \(0.492976\pi\)
\(60\) 0 0
\(61\) 542.000 1.13764 0.568820 0.822462i \(-0.307400\pi\)
0.568820 + 0.822462i \(0.307400\pi\)
\(62\) 256.000 0.524388
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −434.000 −0.791366 −0.395683 0.918387i \(-0.629492\pi\)
−0.395683 + 0.918387i \(0.629492\pi\)
\(68\) 256.000 0.456538
\(69\) 0 0
\(70\) 0 0
\(71\) 1128.00 1.88548 0.942739 0.333531i \(-0.108240\pi\)
0.942739 + 0.333531i \(0.108240\pi\)
\(72\) 0 0
\(73\) 632.000 1.01329 0.506644 0.862155i \(-0.330886\pi\)
0.506644 + 0.862155i \(0.330886\pi\)
\(74\) −472.000 −0.741471
\(75\) 0 0
\(76\) −240.000 −0.362235
\(77\) 728.000 1.07745
\(78\) 0 0
\(79\) −720.000 −1.02540 −0.512698 0.858569i \(-0.671354\pi\)
−0.512698 + 0.858569i \(0.671354\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 484.000 0.651815
\(83\) 478.000 0.632136 0.316068 0.948736i \(-0.397637\pi\)
0.316068 + 0.948736i \(0.397637\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −724.000 −0.907801
\(87\) 0 0
\(88\) −224.000 −0.271346
\(89\) 490.000 0.583594 0.291797 0.956480i \(-0.405747\pi\)
0.291797 + 0.956480i \(0.405747\pi\)
\(90\) 0 0
\(91\) 312.000 0.359412
\(92\) 232.000 0.262909
\(93\) 0 0
\(94\) 452.000 0.495960
\(95\) 0 0
\(96\) 0 0
\(97\) 1456.00 1.52407 0.762033 0.647538i \(-0.224201\pi\)
0.762033 + 0.647538i \(0.224201\pi\)
\(98\) −666.000 −0.686491
\(99\) 0 0
\(100\) 0 0
\(101\) 578.000 0.569437 0.284719 0.958611i \(-0.408100\pi\)
0.284719 + 0.958611i \(0.408100\pi\)
\(102\) 0 0
\(103\) 1462.00 1.39859 0.699297 0.714831i \(-0.253497\pi\)
0.699297 + 0.714831i \(0.253497\pi\)
\(104\) −96.0000 −0.0905151
\(105\) 0 0
\(106\) −216.000 −0.197922
\(107\) −966.000 −0.872773 −0.436387 0.899759i \(-0.643742\pi\)
−0.436387 + 0.899759i \(0.643742\pi\)
\(108\) 0 0
\(109\) 370.000 0.325134 0.162567 0.986698i \(-0.448023\pi\)
0.162567 + 0.986698i \(0.448023\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 416.000 0.350967
\(113\) 528.000 0.439558 0.219779 0.975550i \(-0.429466\pi\)
0.219779 + 0.975550i \(0.429466\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −360.000 −0.288148
\(117\) 0 0
\(118\) −40.0000 −0.0312059
\(119\) 1664.00 1.28184
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) −1084.00 −0.804432
\(123\) 0 0
\(124\) −512.000 −0.370798
\(125\) 0 0
\(126\) 0 0
\(127\) −1534.00 −1.07181 −0.535907 0.844277i \(-0.680030\pi\)
−0.535907 + 0.844277i \(0.680030\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −0.00800340 −0.00400170 0.999992i \(-0.501274\pi\)
−0.00400170 + 0.999992i \(0.501274\pi\)
\(132\) 0 0
\(133\) −1560.00 −1.01706
\(134\) 868.000 0.559580
\(135\) 0 0
\(136\) −512.000 −0.322821
\(137\) 1224.00 0.763309 0.381655 0.924305i \(-0.375354\pi\)
0.381655 + 0.924305i \(0.375354\pi\)
\(138\) 0 0
\(139\) 3100.00 1.89164 0.945822 0.324685i \(-0.105258\pi\)
0.945822 + 0.324685i \(0.105258\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2256.00 −1.33323
\(143\) 336.000 0.196488
\(144\) 0 0
\(145\) 0 0
\(146\) −1264.00 −0.716503
\(147\) 0 0
\(148\) 944.000 0.524299
\(149\) −250.000 −0.137455 −0.0687275 0.997635i \(-0.521894\pi\)
−0.0687275 + 0.997635i \(0.521894\pi\)
\(150\) 0 0
\(151\) 2152.00 1.15978 0.579892 0.814694i \(-0.303095\pi\)
0.579892 + 0.814694i \(0.303095\pi\)
\(152\) 480.000 0.256139
\(153\) 0 0
\(154\) −1456.00 −0.761869
\(155\) 0 0
\(156\) 0 0
\(157\) −524.000 −0.266368 −0.133184 0.991091i \(-0.542520\pi\)
−0.133184 + 0.991091i \(0.542520\pi\)
\(158\) 1440.00 0.725065
\(159\) 0 0
\(160\) 0 0
\(161\) 1508.00 0.738180
\(162\) 0 0
\(163\) −3518.00 −1.69050 −0.845249 0.534373i \(-0.820548\pi\)
−0.845249 + 0.534373i \(0.820548\pi\)
\(164\) −968.000 −0.460903
\(165\) 0 0
\(166\) −956.000 −0.446988
\(167\) 534.000 0.247438 0.123719 0.992317i \(-0.460518\pi\)
0.123719 + 0.992317i \(0.460518\pi\)
\(168\) 0 0
\(169\) −2053.00 −0.934456
\(170\) 0 0
\(171\) 0 0
\(172\) 1448.00 0.641913
\(173\) −4252.00 −1.86863 −0.934317 0.356444i \(-0.883989\pi\)
−0.934317 + 0.356444i \(0.883989\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 448.000 0.191871
\(177\) 0 0
\(178\) −980.000 −0.412664
\(179\) −2500.00 −1.04390 −0.521952 0.852975i \(-0.674796\pi\)
−0.521952 + 0.852975i \(0.674796\pi\)
\(180\) 0 0
\(181\) −2578.00 −1.05868 −0.529340 0.848410i \(-0.677561\pi\)
−0.529340 + 0.848410i \(0.677561\pi\)
\(182\) −624.000 −0.254143
\(183\) 0 0
\(184\) −464.000 −0.185905
\(185\) 0 0
\(186\) 0 0
\(187\) 1792.00 0.700770
\(188\) −904.000 −0.350697
\(189\) 0 0
\(190\) 0 0
\(191\) 768.000 0.290945 0.145473 0.989362i \(-0.453530\pi\)
0.145473 + 0.989362i \(0.453530\pi\)
\(192\) 0 0
\(193\) −2608.00 −0.972684 −0.486342 0.873769i \(-0.661669\pi\)
−0.486342 + 0.873769i \(0.661669\pi\)
\(194\) −2912.00 −1.07768
\(195\) 0 0
\(196\) 1332.00 0.485423
\(197\) −5116.00 −1.85025 −0.925127 0.379659i \(-0.876041\pi\)
−0.925127 + 0.379659i \(0.876041\pi\)
\(198\) 0 0
\(199\) −3480.00 −1.23965 −0.619826 0.784739i \(-0.712797\pi\)
−0.619826 + 0.784739i \(0.712797\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1156.00 −0.402653
\(203\) −2340.00 −0.809043
\(204\) 0 0
\(205\) 0 0
\(206\) −2924.00 −0.988955
\(207\) 0 0
\(208\) 192.000 0.0640039
\(209\) −1680.00 −0.556019
\(210\) 0 0
\(211\) 3132.00 1.02188 0.510938 0.859618i \(-0.329298\pi\)
0.510938 + 0.859618i \(0.329298\pi\)
\(212\) 432.000 0.139952
\(213\) 0 0
\(214\) 1932.00 0.617144
\(215\) 0 0
\(216\) 0 0
\(217\) −3328.00 −1.04110
\(218\) −740.000 −0.229904
\(219\) 0 0
\(220\) 0 0
\(221\) 768.000 0.233761
\(222\) 0 0
\(223\) 62.0000 0.0186181 0.00930903 0.999957i \(-0.497037\pi\)
0.00930903 + 0.999957i \(0.497037\pi\)
\(224\) −832.000 −0.248171
\(225\) 0 0
\(226\) −1056.00 −0.310814
\(227\) 5314.00 1.55376 0.776878 0.629651i \(-0.216802\pi\)
0.776878 + 0.629651i \(0.216802\pi\)
\(228\) 0 0
\(229\) −190.000 −0.0548277 −0.0274139 0.999624i \(-0.508727\pi\)
−0.0274139 + 0.999624i \(0.508727\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 720.000 0.203751
\(233\) 2408.00 0.677053 0.338526 0.940957i \(-0.390072\pi\)
0.338526 + 0.940957i \(0.390072\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 80.0000 0.0220659
\(237\) 0 0
\(238\) −3328.00 −0.906396
\(239\) 5680.00 1.53727 0.768637 0.639685i \(-0.220935\pi\)
0.768637 + 0.639685i \(0.220935\pi\)
\(240\) 0 0
\(241\) −278.000 −0.0743052 −0.0371526 0.999310i \(-0.511829\pi\)
−0.0371526 + 0.999310i \(0.511829\pi\)
\(242\) 1094.00 0.290599
\(243\) 0 0
\(244\) 2168.00 0.568820
\(245\) 0 0
\(246\) 0 0
\(247\) −720.000 −0.185476
\(248\) 1024.00 0.262194
\(249\) 0 0
\(250\) 0 0
\(251\) −3252.00 −0.817787 −0.408893 0.912582i \(-0.634085\pi\)
−0.408893 + 0.912582i \(0.634085\pi\)
\(252\) 0 0
\(253\) 1624.00 0.403557
\(254\) 3068.00 0.757888
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −1536.00 −0.372813 −0.186407 0.982473i \(-0.559684\pi\)
−0.186407 + 0.982473i \(0.559684\pi\)
\(258\) 0 0
\(259\) 6136.00 1.47209
\(260\) 0 0
\(261\) 0 0
\(262\) 24.0000 0.00565926
\(263\) 4858.00 1.13900 0.569500 0.821991i \(-0.307137\pi\)
0.569500 + 0.821991i \(0.307137\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3120.00 0.719171
\(267\) 0 0
\(268\) −1736.00 −0.395683
\(269\) −2610.00 −0.591578 −0.295789 0.955253i \(-0.595583\pi\)
−0.295789 + 0.955253i \(0.595583\pi\)
\(270\) 0 0
\(271\) −5168.00 −1.15843 −0.579213 0.815176i \(-0.696640\pi\)
−0.579213 + 0.815176i \(0.696640\pi\)
\(272\) 1024.00 0.228269
\(273\) 0 0
\(274\) −2448.00 −0.539741
\(275\) 0 0
\(276\) 0 0
\(277\) −1924.00 −0.417336 −0.208668 0.977987i \(-0.566913\pi\)
−0.208668 + 0.977987i \(0.566913\pi\)
\(278\) −6200.00 −1.33759
\(279\) 0 0
\(280\) 0 0
\(281\) −3042.00 −0.645803 −0.322901 0.946433i \(-0.604658\pi\)
−0.322901 + 0.946433i \(0.604658\pi\)
\(282\) 0 0
\(283\) −1718.00 −0.360864 −0.180432 0.983587i \(-0.557750\pi\)
−0.180432 + 0.983587i \(0.557750\pi\)
\(284\) 4512.00 0.942739
\(285\) 0 0
\(286\) −672.000 −0.138938
\(287\) −6292.00 −1.29409
\(288\) 0 0
\(289\) −817.000 −0.166294
\(290\) 0 0
\(291\) 0 0
\(292\) 2528.00 0.506644
\(293\) −2292.00 −0.456997 −0.228498 0.973544i \(-0.573382\pi\)
−0.228498 + 0.973544i \(0.573382\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1888.00 −0.370736
\(297\) 0 0
\(298\) 500.000 0.0971954
\(299\) 696.000 0.134618
\(300\) 0 0
\(301\) 9412.00 1.80232
\(302\) −4304.00 −0.820091
\(303\) 0 0
\(304\) −960.000 −0.181118
\(305\) 0 0
\(306\) 0 0
\(307\) 5406.00 1.00501 0.502503 0.864576i \(-0.332413\pi\)
0.502503 + 0.864576i \(0.332413\pi\)
\(308\) 2912.00 0.538723
\(309\) 0 0
\(310\) 0 0
\(311\) 5688.00 1.03710 0.518548 0.855048i \(-0.326473\pi\)
0.518548 + 0.855048i \(0.326473\pi\)
\(312\) 0 0
\(313\) 7352.00 1.32767 0.663833 0.747881i \(-0.268928\pi\)
0.663833 + 0.747881i \(0.268928\pi\)
\(314\) 1048.00 0.188351
\(315\) 0 0
\(316\) −2880.00 −0.512698
\(317\) 3484.00 0.617290 0.308645 0.951177i \(-0.400124\pi\)
0.308645 + 0.951177i \(0.400124\pi\)
\(318\) 0 0
\(319\) −2520.00 −0.442298
\(320\) 0 0
\(321\) 0 0
\(322\) −3016.00 −0.521972
\(323\) −3840.00 −0.661496
\(324\) 0 0
\(325\) 0 0
\(326\) 7036.00 1.19536
\(327\) 0 0
\(328\) 1936.00 0.325908
\(329\) −5876.00 −0.984664
\(330\) 0 0
\(331\) −7868.00 −1.30654 −0.653269 0.757125i \(-0.726603\pi\)
−0.653269 + 0.757125i \(0.726603\pi\)
\(332\) 1912.00 0.316068
\(333\) 0 0
\(334\) −1068.00 −0.174965
\(335\) 0 0
\(336\) 0 0
\(337\) 656.000 0.106037 0.0530187 0.998594i \(-0.483116\pi\)
0.0530187 + 0.998594i \(0.483116\pi\)
\(338\) 4106.00 0.660760
\(339\) 0 0
\(340\) 0 0
\(341\) −3584.00 −0.569163
\(342\) 0 0
\(343\) −260.000 −0.0409291
\(344\) −2896.00 −0.453901
\(345\) 0 0
\(346\) 8504.00 1.32132
\(347\) 5754.00 0.890176 0.445088 0.895487i \(-0.353172\pi\)
0.445088 + 0.895487i \(0.353172\pi\)
\(348\) 0 0
\(349\) −3110.00 −0.477004 −0.238502 0.971142i \(-0.576656\pi\)
−0.238502 + 0.971142i \(0.576656\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −896.000 −0.135673
\(353\) 7808.00 1.17727 0.588637 0.808397i \(-0.299665\pi\)
0.588637 + 0.808397i \(0.299665\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1960.00 0.291797
\(357\) 0 0
\(358\) 5000.00 0.738151
\(359\) 9240.00 1.35841 0.679204 0.733949i \(-0.262325\pi\)
0.679204 + 0.733949i \(0.262325\pi\)
\(360\) 0 0
\(361\) −3259.00 −0.475142
\(362\) 5156.00 0.748600
\(363\) 0 0
\(364\) 1248.00 0.179706
\(365\) 0 0
\(366\) 0 0
\(367\) −3214.00 −0.457137 −0.228569 0.973528i \(-0.573405\pi\)
−0.228569 + 0.973528i \(0.573405\pi\)
\(368\) 928.000 0.131455
\(369\) 0 0
\(370\) 0 0
\(371\) 2808.00 0.392949
\(372\) 0 0
\(373\) −348.000 −0.0483077 −0.0241538 0.999708i \(-0.507689\pi\)
−0.0241538 + 0.999708i \(0.507689\pi\)
\(374\) −3584.00 −0.495519
\(375\) 0 0
\(376\) 1808.00 0.247980
\(377\) −1080.00 −0.147541
\(378\) 0 0
\(379\) 4940.00 0.669527 0.334764 0.942302i \(-0.391344\pi\)
0.334764 + 0.942302i \(0.391344\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1536.00 −0.205729
\(383\) −6142.00 −0.819430 −0.409715 0.912214i \(-0.634372\pi\)
−0.409715 + 0.912214i \(0.634372\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5216.00 0.687791
\(387\) 0 0
\(388\) 5824.00 0.762033
\(389\) −3050.00 −0.397535 −0.198768 0.980047i \(-0.563694\pi\)
−0.198768 + 0.980047i \(0.563694\pi\)
\(390\) 0 0
\(391\) 3712.00 0.480112
\(392\) −2664.00 −0.343246
\(393\) 0 0
\(394\) 10232.0 1.30833
\(395\) 0 0
\(396\) 0 0
\(397\) 5396.00 0.682160 0.341080 0.940034i \(-0.389207\pi\)
0.341080 + 0.940034i \(0.389207\pi\)
\(398\) 6960.00 0.876566
\(399\) 0 0
\(400\) 0 0
\(401\) −14482.0 −1.80348 −0.901741 0.432276i \(-0.857711\pi\)
−0.901741 + 0.432276i \(0.857711\pi\)
\(402\) 0 0
\(403\) −1536.00 −0.189860
\(404\) 2312.00 0.284719
\(405\) 0 0
\(406\) 4680.00 0.572080
\(407\) 6608.00 0.804782
\(408\) 0 0
\(409\) −1090.00 −0.131778 −0.0658888 0.997827i \(-0.520988\pi\)
−0.0658888 + 0.997827i \(0.520988\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 5848.00 0.699297
\(413\) 520.000 0.0619553
\(414\) 0 0
\(415\) 0 0
\(416\) −384.000 −0.0452576
\(417\) 0 0
\(418\) 3360.00 0.393165
\(419\) 7180.00 0.837150 0.418575 0.908182i \(-0.362530\pi\)
0.418575 + 0.908182i \(0.362530\pi\)
\(420\) 0 0
\(421\) −8138.00 −0.942095 −0.471047 0.882108i \(-0.656124\pi\)
−0.471047 + 0.882108i \(0.656124\pi\)
\(422\) −6264.00 −0.722575
\(423\) 0 0
\(424\) −864.000 −0.0989612
\(425\) 0 0
\(426\) 0 0
\(427\) 14092.0 1.59710
\(428\) −3864.00 −0.436387
\(429\) 0 0
\(430\) 0 0
\(431\) 208.000 0.0232460 0.0116230 0.999932i \(-0.496300\pi\)
0.0116230 + 0.999932i \(0.496300\pi\)
\(432\) 0 0
\(433\) 12992.0 1.44193 0.720965 0.692971i \(-0.243699\pi\)
0.720965 + 0.692971i \(0.243699\pi\)
\(434\) 6656.00 0.736171
\(435\) 0 0
\(436\) 1480.00 0.162567
\(437\) −3480.00 −0.380940
\(438\) 0 0
\(439\) 1080.00 0.117416 0.0587080 0.998275i \(-0.481302\pi\)
0.0587080 + 0.998275i \(0.481302\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1536.00 −0.165294
\(443\) 9078.00 0.973609 0.486805 0.873511i \(-0.338162\pi\)
0.486805 + 0.873511i \(0.338162\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −124.000 −0.0131650
\(447\) 0 0
\(448\) 1664.00 0.175484
\(449\) −14310.0 −1.50408 −0.752039 0.659119i \(-0.770929\pi\)
−0.752039 + 0.659119i \(0.770929\pi\)
\(450\) 0 0
\(451\) −6776.00 −0.707471
\(452\) 2112.00 0.219779
\(453\) 0 0
\(454\) −10628.0 −1.09867
\(455\) 0 0
\(456\) 0 0
\(457\) −2344.00 −0.239929 −0.119965 0.992778i \(-0.538278\pi\)
−0.119965 + 0.992778i \(0.538278\pi\)
\(458\) 380.000 0.0387691
\(459\) 0 0
\(460\) 0 0
\(461\) −11382.0 −1.14992 −0.574959 0.818182i \(-0.694982\pi\)
−0.574959 + 0.818182i \(0.694982\pi\)
\(462\) 0 0
\(463\) 16062.0 1.61223 0.806117 0.591756i \(-0.201565\pi\)
0.806117 + 0.591756i \(0.201565\pi\)
\(464\) −1440.00 −0.144074
\(465\) 0 0
\(466\) −4816.00 −0.478749
\(467\) −17166.0 −1.70096 −0.850479 0.526008i \(-0.823688\pi\)
−0.850479 + 0.526008i \(0.823688\pi\)
\(468\) 0 0
\(469\) −11284.0 −1.11097
\(470\) 0 0
\(471\) 0 0
\(472\) −160.000 −0.0156030
\(473\) 10136.0 0.985315
\(474\) 0 0
\(475\) 0 0
\(476\) 6656.00 0.640919
\(477\) 0 0
\(478\) −11360.0 −1.08702
\(479\) −7520.00 −0.717323 −0.358661 0.933468i \(-0.616767\pi\)
−0.358661 + 0.933468i \(0.616767\pi\)
\(480\) 0 0
\(481\) 2832.00 0.268458
\(482\) 556.000 0.0525417
\(483\) 0 0
\(484\) −2188.00 −0.205485
\(485\) 0 0
\(486\) 0 0
\(487\) −11814.0 −1.09927 −0.549634 0.835406i \(-0.685233\pi\)
−0.549634 + 0.835406i \(0.685233\pi\)
\(488\) −4336.00 −0.402216
\(489\) 0 0
\(490\) 0 0
\(491\) −14052.0 −1.29156 −0.645782 0.763522i \(-0.723468\pi\)
−0.645782 + 0.763522i \(0.723468\pi\)
\(492\) 0 0
\(493\) −5760.00 −0.526202
\(494\) 1440.00 0.131151
\(495\) 0 0
\(496\) −2048.00 −0.185399
\(497\) 29328.0 2.64696
\(498\) 0 0
\(499\) 7620.00 0.683603 0.341802 0.939772i \(-0.388963\pi\)
0.341802 + 0.939772i \(0.388963\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 6504.00 0.578262
\(503\) 1818.00 0.161154 0.0805772 0.996748i \(-0.474324\pi\)
0.0805772 + 0.996748i \(0.474324\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3248.00 −0.285358
\(507\) 0 0
\(508\) −6136.00 −0.535907
\(509\) −17850.0 −1.55440 −0.777198 0.629256i \(-0.783360\pi\)
−0.777198 + 0.629256i \(0.783360\pi\)
\(510\) 0 0
\(511\) 16432.0 1.42252
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 3072.00 0.263619
\(515\) 0 0
\(516\) 0 0
\(517\) −6328.00 −0.538308
\(518\) −12272.0 −1.04093
\(519\) 0 0
\(520\) 0 0
\(521\) 19238.0 1.61772 0.808860 0.588001i \(-0.200085\pi\)
0.808860 + 0.588001i \(0.200085\pi\)
\(522\) 0 0
\(523\) −6278.00 −0.524891 −0.262445 0.964947i \(-0.584529\pi\)
−0.262445 + 0.964947i \(0.584529\pi\)
\(524\) −48.0000 −0.00400170
\(525\) 0 0
\(526\) −9716.00 −0.805395
\(527\) −8192.00 −0.677133
\(528\) 0 0
\(529\) −8803.00 −0.723514
\(530\) 0 0
\(531\) 0 0
\(532\) −6240.00 −0.508531
\(533\) −2904.00 −0.235997
\(534\) 0 0
\(535\) 0 0
\(536\) 3472.00 0.279790
\(537\) 0 0
\(538\) 5220.00 0.418309
\(539\) 9324.00 0.745108
\(540\) 0 0
\(541\) −9818.00 −0.780238 −0.390119 0.920764i \(-0.627566\pi\)
−0.390119 + 0.920764i \(0.627566\pi\)
\(542\) 10336.0 0.819131
\(543\) 0 0
\(544\) −2048.00 −0.161410
\(545\) 0 0
\(546\) 0 0
\(547\) −12514.0 −0.978172 −0.489086 0.872236i \(-0.662670\pi\)
−0.489086 + 0.872236i \(0.662670\pi\)
\(548\) 4896.00 0.381655
\(549\) 0 0
\(550\) 0 0
\(551\) 5400.00 0.417509
\(552\) 0 0
\(553\) −18720.0 −1.43952
\(554\) 3848.00 0.295101
\(555\) 0 0
\(556\) 12400.0 0.945822
\(557\) −10596.0 −0.806045 −0.403022 0.915190i \(-0.632040\pi\)
−0.403022 + 0.915190i \(0.632040\pi\)
\(558\) 0 0
\(559\) 4344.00 0.328679
\(560\) 0 0
\(561\) 0 0
\(562\) 6084.00 0.456651
\(563\) −14002.0 −1.04816 −0.524080 0.851669i \(-0.675591\pi\)
−0.524080 + 0.851669i \(0.675591\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3436.00 0.255169
\(567\) 0 0
\(568\) −9024.00 −0.666617
\(569\) 7330.00 0.540052 0.270026 0.962853i \(-0.412968\pi\)
0.270026 + 0.962853i \(0.412968\pi\)
\(570\) 0 0
\(571\) 5812.00 0.425963 0.212981 0.977056i \(-0.431683\pi\)
0.212981 + 0.977056i \(0.431683\pi\)
\(572\) 1344.00 0.0982438
\(573\) 0 0
\(574\) 12584.0 0.915063
\(575\) 0 0
\(576\) 0 0
\(577\) 16736.0 1.20750 0.603751 0.797173i \(-0.293672\pi\)
0.603751 + 0.797173i \(0.293672\pi\)
\(578\) 1634.00 0.117587
\(579\) 0 0
\(580\) 0 0
\(581\) 12428.0 0.887436
\(582\) 0 0
\(583\) 3024.00 0.214822
\(584\) −5056.00 −0.358251
\(585\) 0 0
\(586\) 4584.00 0.323146
\(587\) 7434.00 0.522716 0.261358 0.965242i \(-0.415830\pi\)
0.261358 + 0.965242i \(0.415830\pi\)
\(588\) 0 0
\(589\) 7680.00 0.537265
\(590\) 0 0
\(591\) 0 0
\(592\) 3776.00 0.262150
\(593\) −25872.0 −1.79163 −0.895814 0.444429i \(-0.853407\pi\)
−0.895814 + 0.444429i \(0.853407\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1000.00 −0.0687275
\(597\) 0 0
\(598\) −1392.00 −0.0951892
\(599\) 3720.00 0.253748 0.126874 0.991919i \(-0.459506\pi\)
0.126874 + 0.991919i \(0.459506\pi\)
\(600\) 0 0
\(601\) −12958.0 −0.879481 −0.439740 0.898125i \(-0.644930\pi\)
−0.439740 + 0.898125i \(0.644930\pi\)
\(602\) −18824.0 −1.27443
\(603\) 0 0
\(604\) 8608.00 0.579892
\(605\) 0 0
\(606\) 0 0
\(607\) −7214.00 −0.482384 −0.241192 0.970477i \(-0.577538\pi\)
−0.241192 + 0.970477i \(0.577538\pi\)
\(608\) 1920.00 0.128070
\(609\) 0 0
\(610\) 0 0
\(611\) −2712.00 −0.179568
\(612\) 0 0
\(613\) −4828.00 −0.318109 −0.159055 0.987270i \(-0.550845\pi\)
−0.159055 + 0.987270i \(0.550845\pi\)
\(614\) −10812.0 −0.710646
\(615\) 0 0
\(616\) −5824.00 −0.380934
\(617\) −27656.0 −1.80452 −0.902260 0.431193i \(-0.858093\pi\)
−0.902260 + 0.431193i \(0.858093\pi\)
\(618\) 0 0
\(619\) −21220.0 −1.37787 −0.688937 0.724821i \(-0.741922\pi\)
−0.688937 + 0.724821i \(0.741922\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −11376.0 −0.733338
\(623\) 12740.0 0.819289
\(624\) 0 0
\(625\) 0 0
\(626\) −14704.0 −0.938802
\(627\) 0 0
\(628\) −2096.00 −0.133184
\(629\) 15104.0 0.957450
\(630\) 0 0
\(631\) 17672.0 1.11491 0.557457 0.830206i \(-0.311777\pi\)
0.557457 + 0.830206i \(0.311777\pi\)
\(632\) 5760.00 0.362532
\(633\) 0 0
\(634\) −6968.00 −0.436490
\(635\) 0 0
\(636\) 0 0
\(637\) 3996.00 0.248551
\(638\) 5040.00 0.312752
\(639\) 0 0
\(640\) 0 0
\(641\) −7322.00 −0.451173 −0.225586 0.974223i \(-0.572430\pi\)
−0.225586 + 0.974223i \(0.572430\pi\)
\(642\) 0 0
\(643\) −8238.00 −0.505249 −0.252624 0.967564i \(-0.581294\pi\)
−0.252624 + 0.967564i \(0.581294\pi\)
\(644\) 6032.00 0.369090
\(645\) 0 0
\(646\) 7680.00 0.467749
\(647\) −6426.00 −0.390467 −0.195233 0.980757i \(-0.562546\pi\)
−0.195233 + 0.980757i \(0.562546\pi\)
\(648\) 0 0
\(649\) 560.000 0.0338705
\(650\) 0 0
\(651\) 0 0
\(652\) −14072.0 −0.845249
\(653\) 5908.00 0.354055 0.177027 0.984206i \(-0.443352\pi\)
0.177027 + 0.984206i \(0.443352\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3872.00 −0.230452
\(657\) 0 0
\(658\) 11752.0 0.696262
\(659\) 26780.0 1.58301 0.791503 0.611166i \(-0.209299\pi\)
0.791503 + 0.611166i \(0.209299\pi\)
\(660\) 0 0
\(661\) −24538.0 −1.44390 −0.721950 0.691945i \(-0.756754\pi\)
−0.721950 + 0.691945i \(0.756754\pi\)
\(662\) 15736.0 0.923863
\(663\) 0 0
\(664\) −3824.00 −0.223494
\(665\) 0 0
\(666\) 0 0
\(667\) −5220.00 −0.303027
\(668\) 2136.00 0.123719
\(669\) 0 0
\(670\) 0 0
\(671\) 15176.0 0.873119
\(672\) 0 0
\(673\) −28848.0 −1.65232 −0.826158 0.563439i \(-0.809478\pi\)
−0.826158 + 0.563439i \(0.809478\pi\)
\(674\) −1312.00 −0.0749798
\(675\) 0 0
\(676\) −8212.00 −0.467228
\(677\) 26884.0 1.52620 0.763099 0.646282i \(-0.223677\pi\)
0.763099 + 0.646282i \(0.223677\pi\)
\(678\) 0 0
\(679\) 37856.0 2.13959
\(680\) 0 0
\(681\) 0 0
\(682\) 7168.00 0.402459
\(683\) −14282.0 −0.800125 −0.400063 0.916488i \(-0.631012\pi\)
−0.400063 + 0.916488i \(0.631012\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 520.000 0.0289412
\(687\) 0 0
\(688\) 5792.00 0.320956
\(689\) 1296.00 0.0716599
\(690\) 0 0
\(691\) −3428.00 −0.188723 −0.0943613 0.995538i \(-0.530081\pi\)
−0.0943613 + 0.995538i \(0.530081\pi\)
\(692\) −17008.0 −0.934317
\(693\) 0 0
\(694\) −11508.0 −0.629449
\(695\) 0 0
\(696\) 0 0
\(697\) −15488.0 −0.841678
\(698\) 6220.00 0.337293
\(699\) 0 0
\(700\) 0 0
\(701\) −26942.0 −1.45162 −0.725810 0.687895i \(-0.758535\pi\)
−0.725810 + 0.687895i \(0.758535\pi\)
\(702\) 0 0
\(703\) −14160.0 −0.759679
\(704\) 1792.00 0.0959354
\(705\) 0 0
\(706\) −15616.0 −0.832459
\(707\) 15028.0 0.799415
\(708\) 0 0
\(709\) −1950.00 −0.103292 −0.0516458 0.998665i \(-0.516447\pi\)
−0.0516458 + 0.998665i \(0.516447\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3920.00 −0.206332
\(713\) −7424.00 −0.389945
\(714\) 0 0
\(715\) 0 0
\(716\) −10000.0 −0.521952
\(717\) 0 0
\(718\) −18480.0 −0.960540
\(719\) −12080.0 −0.626576 −0.313288 0.949658i \(-0.601430\pi\)
−0.313288 + 0.949658i \(0.601430\pi\)
\(720\) 0 0
\(721\) 38012.0 1.96344
\(722\) 6518.00 0.335976
\(723\) 0 0
\(724\) −10312.0 −0.529340
\(725\) 0 0
\(726\) 0 0
\(727\) 17226.0 0.878785 0.439393 0.898295i \(-0.355194\pi\)
0.439393 + 0.898295i \(0.355194\pi\)
\(728\) −2496.00 −0.127071
\(729\) 0 0
\(730\) 0 0
\(731\) 23168.0 1.17223
\(732\) 0 0
\(733\) −788.000 −0.0397073 −0.0198536 0.999803i \(-0.506320\pi\)
−0.0198536 + 0.999803i \(0.506320\pi\)
\(734\) 6428.00 0.323245
\(735\) 0 0
\(736\) −1856.00 −0.0929525
\(737\) −12152.0 −0.607360
\(738\) 0 0
\(739\) −2060.00 −0.102542 −0.0512709 0.998685i \(-0.516327\pi\)
−0.0512709 + 0.998685i \(0.516327\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −5616.00 −0.277857
\(743\) 3258.00 0.160867 0.0804337 0.996760i \(-0.474369\pi\)
0.0804337 + 0.996760i \(0.474369\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 696.000 0.0341587
\(747\) 0 0
\(748\) 7168.00 0.350385
\(749\) −25116.0 −1.22526
\(750\) 0 0
\(751\) −4528.00 −0.220012 −0.110006 0.993931i \(-0.535087\pi\)
−0.110006 + 0.993931i \(0.535087\pi\)
\(752\) −3616.00 −0.175348
\(753\) 0 0
\(754\) 2160.00 0.104327
\(755\) 0 0
\(756\) 0 0
\(757\) 18236.0 0.875560 0.437780 0.899082i \(-0.355765\pi\)
0.437780 + 0.899082i \(0.355765\pi\)
\(758\) −9880.00 −0.473427
\(759\) 0 0
\(760\) 0 0
\(761\) 18678.0 0.889720 0.444860 0.895600i \(-0.353253\pi\)
0.444860 + 0.895600i \(0.353253\pi\)
\(762\) 0 0
\(763\) 9620.00 0.456445
\(764\) 3072.00 0.145473
\(765\) 0 0
\(766\) 12284.0 0.579424
\(767\) 240.000 0.0112984
\(768\) 0 0
\(769\) 27390.0 1.28441 0.642203 0.766534i \(-0.278020\pi\)
0.642203 + 0.766534i \(0.278020\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10432.0 −0.486342
\(773\) −9252.00 −0.430493 −0.215247 0.976560i \(-0.569056\pi\)
−0.215247 + 0.976560i \(0.569056\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −11648.0 −0.538839
\(777\) 0 0
\(778\) 6100.00 0.281100
\(779\) 14520.0 0.667822
\(780\) 0 0
\(781\) 31584.0 1.44707
\(782\) −7424.00 −0.339491
\(783\) 0 0
\(784\) 5328.00 0.242711
\(785\) 0 0
\(786\) 0 0
\(787\) 5726.00 0.259352 0.129676 0.991556i \(-0.458606\pi\)
0.129676 + 0.991556i \(0.458606\pi\)
\(788\) −20464.0 −0.925127
\(789\) 0 0
\(790\) 0 0
\(791\) 13728.0 0.617082
\(792\) 0 0
\(793\) 6504.00 0.291253
\(794\) −10792.0 −0.482360
\(795\) 0 0
\(796\) −13920.0 −0.619826
\(797\) −27236.0 −1.21048 −0.605238 0.796045i \(-0.706922\pi\)
−0.605238 + 0.796045i \(0.706922\pi\)
\(798\) 0 0
\(799\) −14464.0 −0.640425
\(800\) 0 0
\(801\) 0 0
\(802\) 28964.0 1.27525
\(803\) 17696.0 0.777682
\(804\) 0 0
\(805\) 0 0
\(806\) 3072.00 0.134251
\(807\) 0 0
\(808\) −4624.00 −0.201326
\(809\) −10950.0 −0.475873 −0.237937 0.971281i \(-0.576471\pi\)
−0.237937 + 0.971281i \(0.576471\pi\)
\(810\) 0 0
\(811\) −8828.00 −0.382236 −0.191118 0.981567i \(-0.561211\pi\)
−0.191118 + 0.981567i \(0.561211\pi\)
\(812\) −9360.00 −0.404522
\(813\) 0 0
\(814\) −13216.0 −0.569067
\(815\) 0 0
\(816\) 0 0
\(817\) −21720.0 −0.930094
\(818\) 2180.00 0.0931808
\(819\) 0 0
\(820\) 0 0
\(821\) 16058.0 0.682616 0.341308 0.939951i \(-0.389130\pi\)
0.341308 + 0.939951i \(0.389130\pi\)
\(822\) 0 0
\(823\) 41862.0 1.77305 0.886523 0.462684i \(-0.153113\pi\)
0.886523 + 0.462684i \(0.153113\pi\)
\(824\) −11696.0 −0.494478
\(825\) 0 0
\(826\) −1040.00 −0.0438090
\(827\) 12154.0 0.511047 0.255524 0.966803i \(-0.417752\pi\)
0.255524 + 0.966803i \(0.417752\pi\)
\(828\) 0 0
\(829\) −15390.0 −0.644773 −0.322386 0.946608i \(-0.604485\pi\)
−0.322386 + 0.946608i \(0.604485\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 768.000 0.0320019
\(833\) 21312.0 0.886455
\(834\) 0 0
\(835\) 0 0
\(836\) −6720.00 −0.278010
\(837\) 0 0
\(838\) −14360.0 −0.591955
\(839\) 4280.00 0.176117 0.0880584 0.996115i \(-0.471934\pi\)
0.0880584 + 0.996115i \(0.471934\pi\)
\(840\) 0 0
\(841\) −16289.0 −0.667883
\(842\) 16276.0 0.666162
\(843\) 0 0
\(844\) 12528.0 0.510938
\(845\) 0 0
\(846\) 0 0
\(847\) −14222.0 −0.576947
\(848\) 1728.00 0.0699761
\(849\) 0 0
\(850\) 0 0
\(851\) 13688.0 0.551373
\(852\) 0 0
\(853\) 14452.0 0.580102 0.290051 0.957011i \(-0.406328\pi\)
0.290051 + 0.957011i \(0.406328\pi\)
\(854\) −28184.0 −1.12932
\(855\) 0 0
\(856\) 7728.00 0.308572
\(857\) 22584.0 0.900181 0.450090 0.892983i \(-0.351392\pi\)
0.450090 + 0.892983i \(0.351392\pi\)
\(858\) 0 0
\(859\) −26740.0 −1.06212 −0.531058 0.847336i \(-0.678205\pi\)
−0.531058 + 0.847336i \(0.678205\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −416.000 −0.0164374
\(863\) 498.000 0.0196432 0.00982162 0.999952i \(-0.496874\pi\)
0.00982162 + 0.999952i \(0.496874\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −25984.0 −1.01960
\(867\) 0 0
\(868\) −13312.0 −0.520552
\(869\) −20160.0 −0.786975
\(870\) 0 0
\(871\) −5208.00 −0.202602
\(872\) −2960.00 −0.114952
\(873\) 0 0
\(874\) 6960.00 0.269366
\(875\) 0 0
\(876\) 0 0
\(877\) −13244.0 −0.509941 −0.254970 0.966949i \(-0.582066\pi\)
−0.254970 + 0.966949i \(0.582066\pi\)
\(878\) −2160.00 −0.0830256
\(879\) 0 0
\(880\) 0 0
\(881\) −40842.0 −1.56186 −0.780932 0.624616i \(-0.785255\pi\)
−0.780932 + 0.624616i \(0.785255\pi\)
\(882\) 0 0
\(883\) −12078.0 −0.460314 −0.230157 0.973154i \(-0.573924\pi\)
−0.230157 + 0.973154i \(0.573924\pi\)
\(884\) 3072.00 0.116881
\(885\) 0 0
\(886\) −18156.0 −0.688446
\(887\) 18294.0 0.692506 0.346253 0.938141i \(-0.387454\pi\)
0.346253 + 0.938141i \(0.387454\pi\)
\(888\) 0 0
\(889\) −39884.0 −1.50469
\(890\) 0 0
\(891\) 0 0
\(892\) 248.000 0.00930903
\(893\) 13560.0 0.508139
\(894\) 0 0
\(895\) 0 0
\(896\) −3328.00 −0.124086
\(897\) 0 0
\(898\) 28620.0 1.06354
\(899\) 11520.0 0.427379
\(900\) 0 0
\(901\) 6912.00 0.255574
\(902\) 13552.0 0.500257
\(903\) 0 0
\(904\) −4224.00 −0.155407
\(905\) 0 0
\(906\) 0 0
\(907\) 22566.0 0.826121 0.413060 0.910704i \(-0.364460\pi\)
0.413060 + 0.910704i \(0.364460\pi\)
\(908\) 21256.0 0.776878
\(909\) 0 0
\(910\) 0 0
\(911\) 6768.00 0.246140 0.123070 0.992398i \(-0.460726\pi\)
0.123070 + 0.992398i \(0.460726\pi\)
\(912\) 0 0
\(913\) 13384.0 0.485154
\(914\) 4688.00 0.169656
\(915\) 0 0
\(916\) −760.000 −0.0274139
\(917\) −312.000 −0.0112357
\(918\) 0 0
\(919\) 22200.0 0.796856 0.398428 0.917200i \(-0.369556\pi\)
0.398428 + 0.917200i \(0.369556\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 22764.0 0.813115
\(923\) 13536.0 0.482712
\(924\) 0 0
\(925\) 0 0
\(926\) −32124.0 −1.14002
\(927\) 0 0
\(928\) 2880.00 0.101876
\(929\) 6330.00 0.223553 0.111776 0.993733i \(-0.464346\pi\)
0.111776 + 0.993733i \(0.464346\pi\)
\(930\) 0 0
\(931\) −19980.0 −0.703349
\(932\) 9632.00 0.338526
\(933\) 0 0
\(934\) 34332.0 1.20276
\(935\) 0 0
\(936\) 0 0
\(937\) −19544.0 −0.681403 −0.340702 0.940172i \(-0.610665\pi\)
−0.340702 + 0.940172i \(0.610665\pi\)
\(938\) 22568.0 0.785577
\(939\) 0 0
\(940\) 0 0
\(941\) 9898.00 0.342896 0.171448 0.985193i \(-0.445155\pi\)
0.171448 + 0.985193i \(0.445155\pi\)
\(942\) 0 0
\(943\) −14036.0 −0.484703
\(944\) 320.000 0.0110330
\(945\) 0 0
\(946\) −20272.0 −0.696723
\(947\) −41406.0 −1.42082 −0.710409 0.703789i \(-0.751490\pi\)
−0.710409 + 0.703789i \(0.751490\pi\)
\(948\) 0 0
\(949\) 7584.00 0.259417
\(950\) 0 0
\(951\) 0 0
\(952\) −13312.0 −0.453198
\(953\) −25432.0 −0.864453 −0.432226 0.901765i \(-0.642272\pi\)
−0.432226 + 0.901765i \(0.642272\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 22720.0 0.768637
\(957\) 0 0
\(958\) 15040.0 0.507224
\(959\) 31824.0 1.07159
\(960\) 0 0
\(961\) −13407.0 −0.450035
\(962\) −5664.00 −0.189828
\(963\) 0 0
\(964\) −1112.00 −0.0371526
\(965\) 0 0
\(966\) 0 0
\(967\) 12106.0 0.402588 0.201294 0.979531i \(-0.435485\pi\)
0.201294 + 0.979531i \(0.435485\pi\)
\(968\) 4376.00 0.145300
\(969\) 0 0
\(970\) 0 0
\(971\) −7812.00 −0.258186 −0.129093 0.991632i \(-0.541207\pi\)
−0.129093 + 0.991632i \(0.541207\pi\)
\(972\) 0 0
\(973\) 80600.0 2.65562
\(974\) 23628.0 0.777300
\(975\) 0 0
\(976\) 8672.00 0.284410
\(977\) −12576.0 −0.411814 −0.205907 0.978572i \(-0.566014\pi\)
−0.205907 + 0.978572i \(0.566014\pi\)
\(978\) 0 0
\(979\) 13720.0 0.447899
\(980\) 0 0
\(981\) 0 0
\(982\) 28104.0 0.913274
\(983\) −4342.00 −0.140883 −0.0704417 0.997516i \(-0.522441\pi\)
−0.0704417 + 0.997516i \(0.522441\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 11520.0 0.372081
\(987\) 0 0
\(988\) −2880.00 −0.0927379
\(989\) 20996.0 0.675060
\(990\) 0 0
\(991\) 26272.0 0.842137 0.421068 0.907029i \(-0.361655\pi\)
0.421068 + 0.907029i \(0.361655\pi\)
\(992\) 4096.00 0.131097
\(993\) 0 0
\(994\) −58656.0 −1.87169
\(995\) 0 0
\(996\) 0 0
\(997\) 44796.0 1.42297 0.711486 0.702700i \(-0.248022\pi\)
0.711486 + 0.702700i \(0.248022\pi\)
\(998\) −15240.0 −0.483381
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 450.4.a.j.1.1 1
3.2 odd 2 50.4.a.d.1.1 1
5.2 odd 4 90.4.c.b.19.1 2
5.3 odd 4 90.4.c.b.19.2 2
5.4 even 2 450.4.a.k.1.1 1
12.11 even 2 400.4.a.h.1.1 1
15.2 even 4 10.4.b.a.9.2 yes 2
15.8 even 4 10.4.b.a.9.1 2
15.14 odd 2 50.4.a.b.1.1 1
20.3 even 4 720.4.f.f.289.1 2
20.7 even 4 720.4.f.f.289.2 2
21.20 even 2 2450.4.a.bb.1.1 1
24.5 odd 2 1600.4.a.u.1.1 1
24.11 even 2 1600.4.a.bg.1.1 1
60.23 odd 4 80.4.c.a.49.1 2
60.47 odd 4 80.4.c.a.49.2 2
60.59 even 2 400.4.a.n.1.1 1
105.62 odd 4 490.4.c.b.99.2 2
105.83 odd 4 490.4.c.b.99.1 2
105.104 even 2 2450.4.a.o.1.1 1
120.29 odd 2 1600.4.a.bh.1.1 1
120.53 even 4 320.4.c.d.129.1 2
120.59 even 2 1600.4.a.t.1.1 1
120.77 even 4 320.4.c.d.129.2 2
120.83 odd 4 320.4.c.c.129.2 2
120.107 odd 4 320.4.c.c.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.b.a.9.1 2 15.8 even 4
10.4.b.a.9.2 yes 2 15.2 even 4
50.4.a.b.1.1 1 15.14 odd 2
50.4.a.d.1.1 1 3.2 odd 2
80.4.c.a.49.1 2 60.23 odd 4
80.4.c.a.49.2 2 60.47 odd 4
90.4.c.b.19.1 2 5.2 odd 4
90.4.c.b.19.2 2 5.3 odd 4
320.4.c.c.129.1 2 120.107 odd 4
320.4.c.c.129.2 2 120.83 odd 4
320.4.c.d.129.1 2 120.53 even 4
320.4.c.d.129.2 2 120.77 even 4
400.4.a.h.1.1 1 12.11 even 2
400.4.a.n.1.1 1 60.59 even 2
450.4.a.j.1.1 1 1.1 even 1 trivial
450.4.a.k.1.1 1 5.4 even 2
490.4.c.b.99.1 2 105.83 odd 4
490.4.c.b.99.2 2 105.62 odd 4
720.4.f.f.289.1 2 20.3 even 4
720.4.f.f.289.2 2 20.7 even 4
1600.4.a.t.1.1 1 120.59 even 2
1600.4.a.u.1.1 1 24.5 odd 2
1600.4.a.bg.1.1 1 24.11 even 2
1600.4.a.bh.1.1 1 120.29 odd 2
2450.4.a.o.1.1 1 105.104 even 2
2450.4.a.bb.1.1 1 21.20 even 2