Properties

Label 784.4.f.g.783.2
Level $784$
Weight $4$
Character 784.783
Analytic conductor $46.257$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.12258833328.1
Defining polynomial: \(x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 783.2
Root \(-2.61524 - 4.52973i\) of defining polynomial
Character \(\chi\) \(=\) 784.783
Dual form 784.4.f.g.783.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.29424 q^{3} +19.8510i q^{5} +59.3829 q^{9} +O(q^{10})\) \(q-9.29424 q^{3} +19.8510i q^{5} +59.3829 q^{9} -11.8356i q^{11} +19.9862i q^{13} -184.500i q^{15} +1.81702i q^{17} +7.32284 q^{19} -106.472i q^{23} -269.062 q^{25} -300.975 q^{27} -191.679 q^{29} -125.150 q^{31} +110.003i q^{33} +316.828 q^{37} -185.756i q^{39} -321.806i q^{41} +74.3089i q^{43} +1178.81i q^{45} -154.562 q^{47} -16.8879i q^{51} -319.421 q^{53} +234.948 q^{55} -68.0602 q^{57} +61.1169 q^{59} +309.324i q^{61} -396.745 q^{65} -594.102i q^{67} +989.576i q^{69} +48.6774i q^{71} -770.269i q^{73} +2500.73 q^{75} -960.207i q^{79} +1193.99 q^{81} +1064.55 q^{83} -36.0697 q^{85} +1781.51 q^{87} +655.567i q^{89} +1163.18 q^{93} +145.366i q^{95} -704.225i q^{97} -702.830i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 14q^{3} + 156q^{9} + O(q^{10}) \) \( 6q - 14q^{3} + 156q^{9} + 286q^{19} - 612q^{25} - 362q^{27} - 348q^{29} + 410q^{31} + 498q^{37} + 150q^{47} + 1290q^{53} - 918q^{55} - 6q^{57} + 642q^{59} + 1224q^{65} + 8276q^{75} - 450q^{81} - 24q^{83} + 3786q^{85} + 7284q^{87} + 5982q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −9.29424 −1.78868 −0.894339 0.447390i \(-0.852353\pi\)
−0.894339 + 0.447390i \(0.852353\pi\)
\(4\) 0 0
\(5\) 19.8510i 1.77553i 0.460300 + 0.887763i \(0.347742\pi\)
−0.460300 + 0.887763i \(0.652258\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 59.3829 2.19937
\(10\) 0 0
\(11\) − 11.8356i − 0.324414i −0.986757 0.162207i \(-0.948139\pi\)
0.986757 0.162207i \(-0.0518613\pi\)
\(12\) 0 0
\(13\) 19.9862i 0.426398i 0.977009 + 0.213199i \(0.0683882\pi\)
−0.977009 + 0.213199i \(0.931612\pi\)
\(14\) 0 0
\(15\) − 184.500i − 3.17584i
\(16\) 0 0
\(17\) 1.81702i 0.0259231i 0.999916 + 0.0129616i \(0.00412591\pi\)
−0.999916 + 0.0129616i \(0.995874\pi\)
\(18\) 0 0
\(19\) 7.32284 0.0884197 0.0442098 0.999022i \(-0.485923\pi\)
0.0442098 + 0.999022i \(0.485923\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 106.472i − 0.965258i −0.875825 0.482629i \(-0.839682\pi\)
0.875825 0.482629i \(-0.160318\pi\)
\(24\) 0 0
\(25\) −269.062 −2.15249
\(26\) 0 0
\(27\) −300.975 −2.14528
\(28\) 0 0
\(29\) −191.679 −1.22737 −0.613687 0.789549i \(-0.710315\pi\)
−0.613687 + 0.789549i \(0.710315\pi\)
\(30\) 0 0
\(31\) −125.150 −0.725087 −0.362543 0.931967i \(-0.618092\pi\)
−0.362543 + 0.931967i \(0.618092\pi\)
\(32\) 0 0
\(33\) 110.003i 0.580273i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 316.828 1.40773 0.703867 0.710332i \(-0.251455\pi\)
0.703867 + 0.710332i \(0.251455\pi\)
\(38\) 0 0
\(39\) − 185.756i − 0.762688i
\(40\) 0 0
\(41\) − 321.806i − 1.22580i −0.790162 0.612898i \(-0.790003\pi\)
0.790162 0.612898i \(-0.209997\pi\)
\(42\) 0 0
\(43\) 74.3089i 0.263535i 0.991281 + 0.131767i \(0.0420652\pi\)
−0.991281 + 0.131767i \(0.957935\pi\)
\(44\) 0 0
\(45\) 1178.81i 3.90504i
\(46\) 0 0
\(47\) −154.562 −0.479685 −0.239842 0.970812i \(-0.577096\pi\)
−0.239842 + 0.970812i \(0.577096\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 16.8879i − 0.0463681i
\(52\) 0 0
\(53\) −319.421 −0.827847 −0.413923 0.910312i \(-0.635842\pi\)
−0.413923 + 0.910312i \(0.635842\pi\)
\(54\) 0 0
\(55\) 234.948 0.576006
\(56\) 0 0
\(57\) −68.0602 −0.158154
\(58\) 0 0
\(59\) 61.1169 0.134860 0.0674300 0.997724i \(-0.478520\pi\)
0.0674300 + 0.997724i \(0.478520\pi\)
\(60\) 0 0
\(61\) 309.324i 0.649260i 0.945841 + 0.324630i \(0.105240\pi\)
−0.945841 + 0.324630i \(0.894760\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −396.745 −0.757080
\(66\) 0 0
\(67\) − 594.102i − 1.08330i −0.840604 0.541650i \(-0.817800\pi\)
0.840604 0.541650i \(-0.182200\pi\)
\(68\) 0 0
\(69\) 989.576i 1.72654i
\(70\) 0 0
\(71\) 48.6774i 0.0813654i 0.999172 + 0.0406827i \(0.0129533\pi\)
−0.999172 + 0.0406827i \(0.987047\pi\)
\(72\) 0 0
\(73\) − 770.269i − 1.23497i −0.786581 0.617487i \(-0.788151\pi\)
0.786581 0.617487i \(-0.211849\pi\)
\(74\) 0 0
\(75\) 2500.73 3.85012
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 960.207i − 1.36749i −0.729721 0.683745i \(-0.760350\pi\)
0.729721 0.683745i \(-0.239650\pi\)
\(80\) 0 0
\(81\) 1193.99 1.63785
\(82\) 0 0
\(83\) 1064.55 1.40782 0.703912 0.710288i \(-0.251435\pi\)
0.703912 + 0.710288i \(0.251435\pi\)
\(84\) 0 0
\(85\) −36.0697 −0.0460272
\(86\) 0 0
\(87\) 1781.51 2.19538
\(88\) 0 0
\(89\) 655.567i 0.780786i 0.920648 + 0.390393i \(0.127661\pi\)
−0.920648 + 0.390393i \(0.872339\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1163.18 1.29695
\(94\) 0 0
\(95\) 145.366i 0.156991i
\(96\) 0 0
\(97\) − 704.225i − 0.737147i −0.929599 0.368573i \(-0.879846\pi\)
0.929599 0.368573i \(-0.120154\pi\)
\(98\) 0 0
\(99\) − 702.830i − 0.713506i
\(100\) 0 0
\(101\) − 690.058i − 0.679835i −0.940455 0.339918i \(-0.889601\pi\)
0.940455 0.339918i \(-0.110399\pi\)
\(102\) 0 0
\(103\) −1398.27 −1.33763 −0.668814 0.743429i \(-0.733198\pi\)
−0.668814 + 0.743429i \(0.733198\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 31.1751i 0.0281665i 0.999901 + 0.0140832i \(0.00448298\pi\)
−0.999901 + 0.0140832i \(0.995517\pi\)
\(108\) 0 0
\(109\) 478.157 0.420176 0.210088 0.977683i \(-0.432625\pi\)
0.210088 + 0.977683i \(0.432625\pi\)
\(110\) 0 0
\(111\) −2944.67 −2.51798
\(112\) 0 0
\(113\) 1861.07 1.54933 0.774666 0.632371i \(-0.217918\pi\)
0.774666 + 0.632371i \(0.217918\pi\)
\(114\) 0 0
\(115\) 2113.57 1.71384
\(116\) 0 0
\(117\) 1186.84i 0.937805i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1190.92 0.894755
\(122\) 0 0
\(123\) 2990.94i 2.19256i
\(124\) 0 0
\(125\) − 2859.77i − 2.04628i
\(126\) 0 0
\(127\) 1583.04i 1.10608i 0.833155 + 0.553040i \(0.186532\pi\)
−0.833155 + 0.553040i \(0.813468\pi\)
\(128\) 0 0
\(129\) − 690.645i − 0.471379i
\(130\) 0 0
\(131\) 2197.20 1.46542 0.732712 0.680539i \(-0.238254\pi\)
0.732712 + 0.680539i \(0.238254\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 5974.65i − 3.80900i
\(136\) 0 0
\(137\) 2499.40 1.55867 0.779336 0.626607i \(-0.215557\pi\)
0.779336 + 0.626607i \(0.215557\pi\)
\(138\) 0 0
\(139\) −2709.21 −1.65318 −0.826591 0.562804i \(-0.809723\pi\)
−0.826591 + 0.562804i \(0.809723\pi\)
\(140\) 0 0
\(141\) 1436.54 0.858001
\(142\) 0 0
\(143\) 236.548 0.138329
\(144\) 0 0
\(145\) − 3805.02i − 2.17924i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1687.58 −0.927867 −0.463933 0.885870i \(-0.653562\pi\)
−0.463933 + 0.885870i \(0.653562\pi\)
\(150\) 0 0
\(151\) − 2554.77i − 1.37685i −0.725309 0.688424i \(-0.758303\pi\)
0.725309 0.688424i \(-0.241697\pi\)
\(152\) 0 0
\(153\) 107.900i 0.0570145i
\(154\) 0 0
\(155\) − 2484.36i − 1.28741i
\(156\) 0 0
\(157\) − 288.483i − 0.146646i −0.997308 0.0733230i \(-0.976640\pi\)
0.997308 0.0733230i \(-0.0233604\pi\)
\(158\) 0 0
\(159\) 2968.78 1.48075
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 731.683i − 0.351594i −0.984426 0.175797i \(-0.943750\pi\)
0.984426 0.175797i \(-0.0562502\pi\)
\(164\) 0 0
\(165\) −2183.66 −1.03029
\(166\) 0 0
\(167\) 2121.63 0.983092 0.491546 0.870852i \(-0.336432\pi\)
0.491546 + 0.870852i \(0.336432\pi\)
\(168\) 0 0
\(169\) 1797.55 0.818185
\(170\) 0 0
\(171\) 434.851 0.194467
\(172\) 0 0
\(173\) 3179.74i 1.39741i 0.715412 + 0.698703i \(0.246239\pi\)
−0.715412 + 0.698703i \(0.753761\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −568.035 −0.241221
\(178\) 0 0
\(179\) 2198.12i 0.917849i 0.888475 + 0.458924i \(0.151765\pi\)
−0.888475 + 0.458924i \(0.848235\pi\)
\(180\) 0 0
\(181\) 1266.98i 0.520299i 0.965568 + 0.260150i \(0.0837719\pi\)
−0.965568 + 0.260150i \(0.916228\pi\)
\(182\) 0 0
\(183\) − 2874.93i − 1.16132i
\(184\) 0 0
\(185\) 6289.34i 2.49947i
\(186\) 0 0
\(187\) 21.5055 0.00840983
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1741.49i 0.659737i 0.944027 + 0.329869i \(0.107004\pi\)
−0.944027 + 0.329869i \(0.892996\pi\)
\(192\) 0 0
\(193\) −1278.61 −0.476874 −0.238437 0.971158i \(-0.576635\pi\)
−0.238437 + 0.971158i \(0.576635\pi\)
\(194\) 0 0
\(195\) 3687.45 1.35417
\(196\) 0 0
\(197\) 760.622 0.275087 0.137543 0.990496i \(-0.456079\pi\)
0.137543 + 0.990496i \(0.456079\pi\)
\(198\) 0 0
\(199\) −4438.89 −1.58123 −0.790615 0.612313i \(-0.790239\pi\)
−0.790615 + 0.612313i \(0.790239\pi\)
\(200\) 0 0
\(201\) 5521.73i 1.93767i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 6388.17 2.17643
\(206\) 0 0
\(207\) − 6322.62i − 2.12296i
\(208\) 0 0
\(209\) − 86.6699i − 0.0286846i
\(210\) 0 0
\(211\) 2084.90i 0.680239i 0.940382 + 0.340119i \(0.110467\pi\)
−0.940382 + 0.340119i \(0.889533\pi\)
\(212\) 0 0
\(213\) − 452.420i − 0.145537i
\(214\) 0 0
\(215\) −1475.11 −0.467913
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 7159.06i 2.20897i
\(220\) 0 0
\(221\) −36.3154 −0.0110536
\(222\) 0 0
\(223\) −399.456 −0.119953 −0.0599766 0.998200i \(-0.519103\pi\)
−0.0599766 + 0.998200i \(0.519103\pi\)
\(224\) 0 0
\(225\) −15977.7 −4.73413
\(226\) 0 0
\(227\) 2932.62 0.857466 0.428733 0.903431i \(-0.358960\pi\)
0.428733 + 0.903431i \(0.358960\pi\)
\(228\) 0 0
\(229\) − 1505.24i − 0.434363i −0.976131 0.217182i \(-0.930314\pi\)
0.976131 0.217182i \(-0.0696864\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 934.190 0.262664 0.131332 0.991338i \(-0.458075\pi\)
0.131332 + 0.991338i \(0.458075\pi\)
\(234\) 0 0
\(235\) − 3068.21i − 0.851693i
\(236\) 0 0
\(237\) 8924.39i 2.44600i
\(238\) 0 0
\(239\) 6859.78i 1.85658i 0.371860 + 0.928289i \(0.378720\pi\)
−0.371860 + 0.928289i \(0.621280\pi\)
\(240\) 0 0
\(241\) 1294.25i 0.345933i 0.984928 + 0.172966i \(0.0553352\pi\)
−0.984928 + 0.172966i \(0.944665\pi\)
\(242\) 0 0
\(243\) −2970.94 −0.784304
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 146.356i 0.0377019i
\(248\) 0 0
\(249\) −9894.17 −2.51814
\(250\) 0 0
\(251\) −178.426 −0.0448691 −0.0224345 0.999748i \(-0.507142\pi\)
−0.0224345 + 0.999748i \(0.507142\pi\)
\(252\) 0 0
\(253\) −1260.16 −0.313144
\(254\) 0 0
\(255\) 335.241 0.0823278
\(256\) 0 0
\(257\) − 5899.74i − 1.43197i −0.698117 0.715984i \(-0.745978\pi\)
0.698117 0.715984i \(-0.254022\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −11382.5 −2.69945
\(262\) 0 0
\(263\) 2019.42i 0.473470i 0.971574 + 0.236735i \(0.0760774\pi\)
−0.971574 + 0.236735i \(0.923923\pi\)
\(264\) 0 0
\(265\) − 6340.83i − 1.46986i
\(266\) 0 0
\(267\) − 6092.99i − 1.39657i
\(268\) 0 0
\(269\) − 1849.77i − 0.419265i −0.977780 0.209632i \(-0.932773\pi\)
0.977780 0.209632i \(-0.0672267\pi\)
\(270\) 0 0
\(271\) −7198.86 −1.61365 −0.806826 0.590790i \(-0.798816\pi\)
−0.806826 + 0.590790i \(0.798816\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3184.50i 0.698300i
\(276\) 0 0
\(277\) 5534.83 1.20056 0.600281 0.799789i \(-0.295056\pi\)
0.600281 + 0.799789i \(0.295056\pi\)
\(278\) 0 0
\(279\) −7431.80 −1.59473
\(280\) 0 0
\(281\) 752.919 0.159841 0.0799207 0.996801i \(-0.474533\pi\)
0.0799207 + 0.996801i \(0.474533\pi\)
\(282\) 0 0
\(283\) −5623.58 −1.18123 −0.590614 0.806954i \(-0.701114\pi\)
−0.590614 + 0.806954i \(0.701114\pi\)
\(284\) 0 0
\(285\) − 1351.06i − 0.280807i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4909.70 0.999328
\(290\) 0 0
\(291\) 6545.24i 1.31852i
\(292\) 0 0
\(293\) 507.111i 0.101112i 0.998721 + 0.0505559i \(0.0160993\pi\)
−0.998721 + 0.0505559i \(0.983901\pi\)
\(294\) 0 0
\(295\) 1213.23i 0.239447i
\(296\) 0 0
\(297\) 3562.21i 0.695960i
\(298\) 0 0
\(299\) 2127.97 0.411584
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6413.57i 1.21601i
\(304\) 0 0
\(305\) −6140.38 −1.15278
\(306\) 0 0
\(307\) 7402.96 1.37625 0.688126 0.725591i \(-0.258434\pi\)
0.688126 + 0.725591i \(0.258434\pi\)
\(308\) 0 0
\(309\) 12995.9 2.39259
\(310\) 0 0
\(311\) −4284.78 −0.781245 −0.390623 0.920551i \(-0.627740\pi\)
−0.390623 + 0.920551i \(0.627740\pi\)
\(312\) 0 0
\(313\) 1502.00i 0.271239i 0.990761 + 0.135620i \(0.0433025\pi\)
−0.990761 + 0.135620i \(0.956698\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7097.55 1.25753 0.628767 0.777594i \(-0.283560\pi\)
0.628767 + 0.777594i \(0.283560\pi\)
\(318\) 0 0
\(319\) 2268.63i 0.398178i
\(320\) 0 0
\(321\) − 289.749i − 0.0503808i
\(322\) 0 0
\(323\) 13.3058i 0.00229211i
\(324\) 0 0
\(325\) − 5377.52i − 0.917818i
\(326\) 0 0
\(327\) −4444.11 −0.751559
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 11497.2i − 1.90919i −0.297907 0.954595i \(-0.596288\pi\)
0.297907 0.954595i \(-0.403712\pi\)
\(332\) 0 0
\(333\) 18814.2 3.09612
\(334\) 0 0
\(335\) 11793.5 1.92343
\(336\) 0 0
\(337\) 6020.26 0.973128 0.486564 0.873645i \(-0.338250\pi\)
0.486564 + 0.873645i \(0.338250\pi\)
\(338\) 0 0
\(339\) −17297.2 −2.77126
\(340\) 0 0
\(341\) 1481.23i 0.235229i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −19644.1 −3.06551
\(346\) 0 0
\(347\) − 8890.39i − 1.37539i −0.725999 0.687696i \(-0.758622\pi\)
0.725999 0.687696i \(-0.241378\pi\)
\(348\) 0 0
\(349\) − 6396.47i − 0.981075i −0.871420 0.490538i \(-0.836800\pi\)
0.871420 0.490538i \(-0.163200\pi\)
\(350\) 0 0
\(351\) − 6015.33i − 0.914743i
\(352\) 0 0
\(353\) − 11472.5i − 1.72981i −0.501940 0.864903i \(-0.667380\pi\)
0.501940 0.864903i \(-0.332620\pi\)
\(354\) 0 0
\(355\) −966.295 −0.144466
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 195.405i 0.0287272i 0.999897 + 0.0143636i \(0.00457224\pi\)
−0.999897 + 0.0143636i \(0.995428\pi\)
\(360\) 0 0
\(361\) −6805.38 −0.992182
\(362\) 0 0
\(363\) −11068.7 −1.60043
\(364\) 0 0
\(365\) 15290.6 2.19273
\(366\) 0 0
\(367\) 10227.7 1.45472 0.727362 0.686254i \(-0.240746\pi\)
0.727362 + 0.686254i \(0.240746\pi\)
\(368\) 0 0
\(369\) − 19109.8i − 2.69598i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7734.15 −1.07362 −0.536808 0.843704i \(-0.680370\pi\)
−0.536808 + 0.843704i \(0.680370\pi\)
\(374\) 0 0
\(375\) 26579.4i 3.66014i
\(376\) 0 0
\(377\) − 3830.93i − 0.523350i
\(378\) 0 0
\(379\) 9739.99i 1.32008i 0.751231 + 0.660039i \(0.229460\pi\)
−0.751231 + 0.660039i \(0.770540\pi\)
\(380\) 0 0
\(381\) − 14713.2i − 1.97842i
\(382\) 0 0
\(383\) 2525.17 0.336894 0.168447 0.985711i \(-0.446125\pi\)
0.168447 + 0.985711i \(0.446125\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4412.68i 0.579610i
\(388\) 0 0
\(389\) −2665.26 −0.347388 −0.173694 0.984800i \(-0.555570\pi\)
−0.173694 + 0.984800i \(0.555570\pi\)
\(390\) 0 0
\(391\) 193.462 0.0250225
\(392\) 0 0
\(393\) −20421.3 −2.62117
\(394\) 0 0
\(395\) 19061.1 2.42801
\(396\) 0 0
\(397\) − 7966.78i − 1.00716i −0.863950 0.503578i \(-0.832017\pi\)
0.863950 0.503578i \(-0.167983\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2607.89 0.324768 0.162384 0.986728i \(-0.448082\pi\)
0.162384 + 0.986728i \(0.448082\pi\)
\(402\) 0 0
\(403\) − 2501.28i − 0.309175i
\(404\) 0 0
\(405\) 23701.9i 2.90805i
\(406\) 0 0
\(407\) − 3749.83i − 0.456689i
\(408\) 0 0
\(409\) − 8794.35i − 1.06321i −0.846993 0.531605i \(-0.821589\pi\)
0.846993 0.531605i \(-0.178411\pi\)
\(410\) 0 0
\(411\) −23230.0 −2.78796
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 21132.3i 2.49963i
\(416\) 0 0
\(417\) 25180.0 2.95701
\(418\) 0 0
\(419\) 333.409 0.0388737 0.0194368 0.999811i \(-0.493813\pi\)
0.0194368 + 0.999811i \(0.493813\pi\)
\(420\) 0 0
\(421\) −1186.07 −0.137305 −0.0686526 0.997641i \(-0.521870\pi\)
−0.0686526 + 0.997641i \(0.521870\pi\)
\(422\) 0 0
\(423\) −9178.34 −1.05500
\(424\) 0 0
\(425\) − 488.892i − 0.0557994i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2198.53 −0.247427
\(430\) 0 0
\(431\) − 12163.1i − 1.35934i −0.733519 0.679669i \(-0.762124\pi\)
0.733519 0.679669i \(-0.237876\pi\)
\(432\) 0 0
\(433\) − 14229.4i − 1.57926i −0.613582 0.789631i \(-0.710272\pi\)
0.613582 0.789631i \(-0.289728\pi\)
\(434\) 0 0
\(435\) 35364.7i 3.89795i
\(436\) 0 0
\(437\) − 779.677i − 0.0853478i
\(438\) 0 0
\(439\) 7955.09 0.864865 0.432432 0.901666i \(-0.357655\pi\)
0.432432 + 0.901666i \(0.357655\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6411.91i 0.687673i 0.939030 + 0.343837i \(0.111727\pi\)
−0.939030 + 0.343837i \(0.888273\pi\)
\(444\) 0 0
\(445\) −13013.6 −1.38631
\(446\) 0 0
\(447\) 15684.8 1.65965
\(448\) 0 0
\(449\) −5918.99 −0.622125 −0.311063 0.950389i \(-0.600685\pi\)
−0.311063 + 0.950389i \(0.600685\pi\)
\(450\) 0 0
\(451\) −3808.76 −0.397666
\(452\) 0 0
\(453\) 23744.6i 2.46274i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 868.613 0.0889103 0.0444551 0.999011i \(-0.485845\pi\)
0.0444551 + 0.999011i \(0.485845\pi\)
\(458\) 0 0
\(459\) − 546.878i − 0.0556124i
\(460\) 0 0
\(461\) 4645.89i 0.469372i 0.972071 + 0.234686i \(0.0754062\pi\)
−0.972071 + 0.234686i \(0.924594\pi\)
\(462\) 0 0
\(463\) 288.831i 0.0289916i 0.999895 + 0.0144958i \(0.00461433\pi\)
−0.999895 + 0.0144958i \(0.995386\pi\)
\(464\) 0 0
\(465\) 23090.2i 2.30276i
\(466\) 0 0
\(467\) −11317.4 −1.12143 −0.560716 0.828008i \(-0.689474\pi\)
−0.560716 + 0.828008i \(0.689474\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2681.23i 0.262302i
\(472\) 0 0
\(473\) 879.488 0.0854945
\(474\) 0 0
\(475\) −1970.30 −0.190323
\(476\) 0 0
\(477\) −18968.2 −1.82074
\(478\) 0 0
\(479\) 4324.22 0.412482 0.206241 0.978501i \(-0.433877\pi\)
0.206241 + 0.978501i \(0.433877\pi\)
\(480\) 0 0
\(481\) 6332.17i 0.600254i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13979.6 1.30882
\(486\) 0 0
\(487\) 11480.5i 1.06824i 0.845410 + 0.534119i \(0.179356\pi\)
−0.845410 + 0.534119i \(0.820644\pi\)
\(488\) 0 0
\(489\) 6800.44i 0.628888i
\(490\) 0 0
\(491\) − 10052.9i − 0.923994i −0.886881 0.461997i \(-0.847133\pi\)
0.886881 0.461997i \(-0.152867\pi\)
\(492\) 0 0
\(493\) − 348.285i − 0.0318174i
\(494\) 0 0
\(495\) 13951.9 1.26685
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 7204.13i 0.646295i 0.946349 + 0.323147i \(0.104741\pi\)
−0.946349 + 0.323147i \(0.895259\pi\)
\(500\) 0 0
\(501\) −19718.9 −1.75843
\(502\) 0 0
\(503\) 6434.36 0.570366 0.285183 0.958473i \(-0.407946\pi\)
0.285183 + 0.958473i \(0.407946\pi\)
\(504\) 0 0
\(505\) 13698.3 1.20707
\(506\) 0 0
\(507\) −16706.9 −1.46347
\(508\) 0 0
\(509\) − 13015.0i − 1.13336i −0.823939 0.566679i \(-0.808228\pi\)
0.823939 0.566679i \(-0.191772\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2203.99 −0.189685
\(514\) 0 0
\(515\) − 27757.1i − 2.37500i
\(516\) 0 0
\(517\) 1829.33i 0.155617i
\(518\) 0 0
\(519\) − 29553.3i − 2.49951i
\(520\) 0 0
\(521\) 5144.91i 0.432635i 0.976323 + 0.216317i \(0.0694046\pi\)
−0.976323 + 0.216317i \(0.930595\pi\)
\(522\) 0 0
\(523\) 11197.7 0.936213 0.468106 0.883672i \(-0.344936\pi\)
0.468106 + 0.883672i \(0.344936\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 227.402i − 0.0187965i
\(528\) 0 0
\(529\) 830.726 0.0682769
\(530\) 0 0
\(531\) 3629.30 0.296607
\(532\) 0 0
\(533\) 6431.67 0.522677
\(534\) 0 0
\(535\) −618.857 −0.0500104
\(536\) 0 0
\(537\) − 20429.8i − 1.64174i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −21827.4 −1.73463 −0.867313 0.497762i \(-0.834155\pi\)
−0.867313 + 0.497762i \(0.834155\pi\)
\(542\) 0 0
\(543\) − 11775.6i − 0.930647i
\(544\) 0 0
\(545\) 9491.89i 0.746033i
\(546\) 0 0
\(547\) − 20005.0i − 1.56371i −0.623459 0.781856i \(-0.714273\pi\)
0.623459 0.781856i \(-0.285727\pi\)
\(548\) 0 0
\(549\) 18368.5i 1.42796i
\(550\) 0 0
\(551\) −1403.63 −0.108524
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 58454.7i − 4.47074i
\(556\) 0 0
\(557\) −7400.91 −0.562992 −0.281496 0.959562i \(-0.590831\pi\)
−0.281496 + 0.959562i \(0.590831\pi\)
\(558\) 0 0
\(559\) −1485.15 −0.112371
\(560\) 0 0
\(561\) −199.877 −0.0150425
\(562\) 0 0
\(563\) 1080.79 0.0809059 0.0404529 0.999181i \(-0.487120\pi\)
0.0404529 + 0.999181i \(0.487120\pi\)
\(564\) 0 0
\(565\) 36944.0i 2.75088i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4208.13 0.310042 0.155021 0.987911i \(-0.450455\pi\)
0.155021 + 0.987911i \(0.450455\pi\)
\(570\) 0 0
\(571\) 12881.7i 0.944105i 0.881570 + 0.472052i \(0.156487\pi\)
−0.881570 + 0.472052i \(0.843513\pi\)
\(572\) 0 0
\(573\) − 16185.8i − 1.18006i
\(574\) 0 0
\(575\) 28647.5i 2.07771i
\(576\) 0 0
\(577\) 13271.6i 0.957546i 0.877939 + 0.478773i \(0.158918\pi\)
−0.877939 + 0.478773i \(0.841082\pi\)
\(578\) 0 0
\(579\) 11883.7 0.852974
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3780.53i 0.268565i
\(584\) 0 0
\(585\) −23559.9 −1.66510
\(586\) 0 0
\(587\) −16892.3 −1.18777 −0.593885 0.804550i \(-0.702407\pi\)
−0.593885 + 0.804550i \(0.702407\pi\)
\(588\) 0 0
\(589\) −916.456 −0.0641119
\(590\) 0 0
\(591\) −7069.40 −0.492041
\(592\) 0 0
\(593\) − 10957.3i − 0.758787i −0.925235 0.379394i \(-0.876133\pi\)
0.925235 0.379394i \(-0.123867\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 41256.1 2.82831
\(598\) 0 0
\(599\) − 11575.6i − 0.789596i −0.918768 0.394798i \(-0.870815\pi\)
0.918768 0.394798i \(-0.129185\pi\)
\(600\) 0 0
\(601\) − 17612.6i − 1.19540i −0.801721 0.597699i \(-0.796082\pi\)
0.801721 0.597699i \(-0.203918\pi\)
\(602\) 0 0
\(603\) − 35279.5i − 2.38257i
\(604\) 0 0
\(605\) 23640.9i 1.58866i
\(606\) 0 0
\(607\) 16363.0 1.09416 0.547080 0.837080i \(-0.315739\pi\)
0.547080 + 0.837080i \(0.315739\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 3089.10i − 0.204536i
\(612\) 0 0
\(613\) 1007.72 0.0663974 0.0331987 0.999449i \(-0.489431\pi\)
0.0331987 + 0.999449i \(0.489431\pi\)
\(614\) 0 0
\(615\) −59373.2 −3.89294
\(616\) 0 0
\(617\) −14986.9 −0.977876 −0.488938 0.872318i \(-0.662616\pi\)
−0.488938 + 0.872318i \(0.662616\pi\)
\(618\) 0 0
\(619\) −17668.2 −1.14725 −0.573624 0.819119i \(-0.694463\pi\)
−0.573624 + 0.819119i \(0.694463\pi\)
\(620\) 0 0
\(621\) 32045.4i 2.07075i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 23136.5 1.48074
\(626\) 0 0
\(627\) 805.531i 0.0513075i
\(628\) 0 0
\(629\) 575.684i 0.0364929i
\(630\) 0 0
\(631\) 3794.54i 0.239395i 0.992810 + 0.119698i \(0.0381925\pi\)
−0.992810 + 0.119698i \(0.961807\pi\)
\(632\) 0 0
\(633\) − 19377.6i − 1.21673i
\(634\) 0 0
\(635\) −31424.9 −1.96387
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2890.61i 0.178952i
\(640\) 0 0
\(641\) 14251.5 0.878157 0.439079 0.898449i \(-0.355305\pi\)
0.439079 + 0.898449i \(0.355305\pi\)
\(642\) 0 0
\(643\) 10643.5 0.652782 0.326391 0.945235i \(-0.394167\pi\)
0.326391 + 0.945235i \(0.394167\pi\)
\(644\) 0 0
\(645\) 13710.0 0.836946
\(646\) 0 0
\(647\) 18948.4 1.15137 0.575686 0.817671i \(-0.304735\pi\)
0.575686 + 0.817671i \(0.304735\pi\)
\(648\) 0 0
\(649\) − 723.353i − 0.0437505i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21346.3 1.27924 0.639620 0.768691i \(-0.279092\pi\)
0.639620 + 0.768691i \(0.279092\pi\)
\(654\) 0 0
\(655\) 43616.7i 2.60190i
\(656\) 0 0
\(657\) − 45740.8i − 2.71616i
\(658\) 0 0
\(659\) − 3245.68i − 0.191857i −0.995388 0.0959284i \(-0.969418\pi\)
0.995388 0.0959284i \(-0.0305820\pi\)
\(660\) 0 0
\(661\) 15989.9i 0.940898i 0.882427 + 0.470449i \(0.155908\pi\)
−0.882427 + 0.470449i \(0.844092\pi\)
\(662\) 0 0
\(663\) 337.524 0.0197713
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 20408.4i 1.18473i
\(668\) 0 0
\(669\) 3712.64 0.214558
\(670\) 0 0
\(671\) 3661.02 0.210629
\(672\) 0 0
\(673\) −3473.65 −0.198959 −0.0994796 0.995040i \(-0.531718\pi\)
−0.0994796 + 0.995040i \(0.531718\pi\)
\(674\) 0 0
\(675\) 80980.8 4.61771
\(676\) 0 0
\(677\) − 27077.2i − 1.53717i −0.639750 0.768583i \(-0.720962\pi\)
0.639750 0.768583i \(-0.279038\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −27256.5 −1.53373
\(682\) 0 0
\(683\) 6615.17i 0.370604i 0.982682 + 0.185302i \(0.0593263\pi\)
−0.982682 + 0.185302i \(0.940674\pi\)
\(684\) 0 0
\(685\) 49615.5i 2.76746i
\(686\) 0 0
\(687\) 13990.1i 0.776936i
\(688\) 0 0
\(689\) − 6384.01i − 0.352992i
\(690\) 0 0
\(691\) 15808.2 0.870294 0.435147 0.900359i \(-0.356696\pi\)
0.435147 + 0.900359i \(0.356696\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 53780.5i − 2.93527i
\(696\) 0 0
\(697\) 584.730 0.0317765
\(698\) 0 0
\(699\) −8682.59 −0.469822
\(700\) 0 0
\(701\) 9851.70 0.530804 0.265402 0.964138i \(-0.414495\pi\)
0.265402 + 0.964138i \(0.414495\pi\)
\(702\) 0 0
\(703\) 2320.08 0.124471
\(704\) 0 0
\(705\) 28516.7i 1.52340i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10767.2 −0.570342 −0.285171 0.958477i \(-0.592050\pi\)
−0.285171 + 0.958477i \(0.592050\pi\)
\(710\) 0 0
\(711\) − 57019.9i − 3.00761i
\(712\) 0 0
\(713\) 13325.0i 0.699896i
\(714\) 0 0
\(715\) 4695.71i 0.245608i
\(716\) 0 0
\(717\) − 63756.4i − 3.32082i
\(718\) 0 0
\(719\) 20738.0 1.07565 0.537827 0.843055i \(-0.319245\pi\)
0.537827 + 0.843055i \(0.319245\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 12029.0i − 0.618762i
\(724\) 0 0
\(725\) 51573.5 2.64192
\(726\) 0 0
\(727\) −2738.00 −0.139679 −0.0698396 0.997558i \(-0.522249\pi\)
−0.0698396 + 0.997558i \(0.522249\pi\)
\(728\) 0 0
\(729\) −4625.18 −0.234984
\(730\) 0 0
\(731\) −135.021 −0.00683165
\(732\) 0 0
\(733\) − 17216.1i − 0.867519i −0.901029 0.433760i \(-0.857187\pi\)
0.901029 0.433760i \(-0.142813\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7031.53 −0.351438
\(738\) 0 0
\(739\) − 5523.75i − 0.274959i −0.990505 0.137479i \(-0.956100\pi\)
0.990505 0.137479i \(-0.0439001\pi\)
\(740\) 0 0
\(741\) − 1360.26i − 0.0674366i
\(742\) 0 0
\(743\) − 20424.6i − 1.00849i −0.863561 0.504244i \(-0.831771\pi\)
0.863561 0.504244i \(-0.168229\pi\)
\(744\) 0 0
\(745\) − 33500.2i − 1.64745i
\(746\) 0 0
\(747\) 63216.0 3.09632
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 10763.8i − 0.523003i −0.965203 0.261502i \(-0.915782\pi\)
0.965203 0.261502i \(-0.0842177\pi\)
\(752\) 0 0
\(753\) 1658.33 0.0802563
\(754\) 0 0
\(755\) 50714.7 2.44463
\(756\) 0 0
\(757\) 8591.14 0.412484 0.206242 0.978501i \(-0.433877\pi\)
0.206242 + 0.978501i \(0.433877\pi\)
\(758\) 0 0
\(759\) 11712.2 0.560113
\(760\) 0 0
\(761\) 29877.8i 1.42322i 0.702575 + 0.711610i \(0.252034\pi\)
−0.702575 + 0.711610i \(0.747966\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2141.93 −0.101231
\(766\) 0 0
\(767\) 1221.49i 0.0575040i
\(768\) 0 0
\(769\) 3093.61i 0.145070i 0.997366 + 0.0725348i \(0.0231089\pi\)
−0.997366 + 0.0725348i \(0.976891\pi\)
\(770\) 0 0
\(771\) 54833.6i 2.56133i
\(772\) 0 0
\(773\) 24031.7i 1.11819i 0.829104 + 0.559095i \(0.188851\pi\)
−0.829104 + 0.559095i \(0.811149\pi\)
\(774\) 0 0
\(775\) 33673.2 1.56075
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 2356.53i − 0.108385i
\(780\) 0 0
\(781\) 576.125 0.0263961
\(782\) 0 0
\(783\) 57690.5 2.63307
\(784\) 0 0
\(785\) 5726.67 0.260374
\(786\) 0 0
\(787\) −11593.2 −0.525099 −0.262550 0.964918i \(-0.584563\pi\)
−0.262550 + 0.964918i \(0.584563\pi\)
\(788\) 0 0
\(789\) − 18769.0i − 0.846886i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6182.20 −0.276843
\(794\) 0 0
\(795\) 58933.2i 2.62911i
\(796\) 0 0
\(797\) − 11467.4i − 0.509658i −0.966986 0.254829i \(-0.917981\pi\)
0.966986 0.254829i \(-0.0820192\pi\)
\(798\) 0 0
\(799\) − 280.843i − 0.0124349i
\(800\) 0 0
\(801\) 38929.5i 1.71723i
\(802\) 0 0
\(803\) −9116.57 −0.400643
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 17192.2i 0.749930i
\(808\) 0 0
\(809\) 36381.7 1.58110 0.790551 0.612396i \(-0.209794\pi\)
0.790551 + 0.612396i \(0.209794\pi\)
\(810\) 0 0
\(811\) −35717.1 −1.54648 −0.773241 0.634113i \(-0.781366\pi\)
−0.773241 + 0.634113i \(0.781366\pi\)
\(812\) 0 0
\(813\) 66907.9 2.88630
\(814\) 0 0
\(815\) 14524.6 0.624264
\(816\) 0 0
\(817\) 544.152i 0.0233017i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −31319.1 −1.33136 −0.665678 0.746239i \(-0.731858\pi\)
−0.665678 + 0.746239i \(0.731858\pi\)
\(822\) 0 0
\(823\) − 4176.34i − 0.176887i −0.996081 0.0884435i \(-0.971811\pi\)
0.996081 0.0884435i \(-0.0281893\pi\)
\(824\) 0 0
\(825\) − 29597.5i − 1.24903i
\(826\) 0 0
\(827\) − 7930.70i − 0.333467i −0.986002 0.166734i \(-0.946678\pi\)
0.986002 0.166734i \(-0.0533220\pi\)
\(828\) 0 0
\(829\) 2847.17i 0.119284i 0.998220 + 0.0596419i \(0.0189959\pi\)
−0.998220 + 0.0596419i \(0.981004\pi\)
\(830\) 0 0
\(831\) −51442.0 −2.14742
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 42116.4i 1.74551i
\(836\) 0 0
\(837\) 37667.1 1.55552
\(838\) 0 0
\(839\) 25648.9 1.05542 0.527711 0.849424i \(-0.323051\pi\)
0.527711 + 0.849424i \(0.323051\pi\)
\(840\) 0 0
\(841\) 12351.8 0.506449
\(842\) 0 0
\(843\) −6997.82 −0.285905
\(844\) 0 0
\(845\) 35683.2i 1.45271i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 52266.9 2.11283
\(850\) 0 0
\(851\) − 33733.3i − 1.35883i
\(852\) 0 0
\(853\) 1004.85i 0.0403348i 0.999797 + 0.0201674i \(0.00641991\pi\)
−0.999797 + 0.0201674i \(0.993580\pi\)
\(854\) 0 0
\(855\) 8632.23i 0.345282i
\(856\) 0 0
\(857\) 35060.5i 1.39748i 0.715374 + 0.698742i \(0.246256\pi\)
−0.715374 + 0.698742i \(0.753744\pi\)
\(858\) 0 0
\(859\) −9006.42 −0.357736 −0.178868 0.983873i \(-0.557243\pi\)
−0.178868 + 0.983873i \(0.557243\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 43533.3i − 1.71714i −0.512696 0.858570i \(-0.671353\pi\)
0.512696 0.858570i \(-0.328647\pi\)
\(864\) 0 0
\(865\) −63121.0 −2.48113
\(866\) 0 0
\(867\) −45631.9 −1.78748
\(868\) 0 0
\(869\) −11364.6 −0.443633
\(870\) 0 0
\(871\) 11873.8 0.461916
\(872\) 0 0
\(873\) − 41818.9i − 1.62126i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −23537.6 −0.906280 −0.453140 0.891439i \(-0.649696\pi\)
−0.453140 + 0.891439i \(0.649696\pi\)
\(878\) 0 0
\(879\) − 4713.21i − 0.180856i
\(880\) 0 0
\(881\) 41234.7i 1.57688i 0.615111 + 0.788440i \(0.289111\pi\)
−0.615111 + 0.788440i \(0.710889\pi\)
\(882\) 0 0
\(883\) − 2981.04i − 0.113613i −0.998385 0.0568063i \(-0.981908\pi\)
0.998385 0.0568063i \(-0.0180917\pi\)
\(884\) 0 0
\(885\) − 11276.1i − 0.428294i
\(886\) 0 0
\(887\) −23841.2 −0.902490 −0.451245 0.892400i \(-0.649020\pi\)
−0.451245 + 0.892400i \(0.649020\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 14131.6i − 0.531342i
\(892\) 0 0
\(893\) −1131.83 −0.0424136
\(894\) 0 0
\(895\) −43634.8 −1.62967
\(896\) 0 0
\(897\) −19777.8 −0.736191
\(898\) 0 0
\(899\) 23988.7 0.889953
\(900\) 0 0
\(901\) − 580.396i − 0.0214604i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −25150.9 −0.923805
\(906\) 0 0
\(907\) 33942.4i 1.24260i 0.783572 + 0.621301i \(0.213395\pi\)
−0.783572 + 0.621301i \(0.786605\pi\)
\(908\) 0 0
\(909\) − 40977.7i − 1.49521i
\(910\) 0 0
\(911\) 18217.3i 0.662531i 0.943538 + 0.331265i \(0.107476\pi\)
−0.943538 + 0.331265i \(0.892524\pi\)
\(912\) 0 0
\(913\) − 12599.5i − 0.456718i
\(914\) 0 0
\(915\) 57070.2 2.06195
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 40860.7i − 1.46667i −0.679867 0.733335i \(-0.737963\pi\)
0.679867 0.733335i \(-0.262037\pi\)
\(920\) 0 0
\(921\) −68804.9 −2.46167
\(922\) 0 0
\(923\) −972.875 −0.0346940
\(924\) 0 0
\(925\) −85246.2 −3.03014
\(926\) 0 0
\(927\) −83033.4 −2.94194
\(928\) 0 0
\(929\) − 30596.3i − 1.08055i −0.841488 0.540276i \(-0.818320\pi\)
0.841488 0.540276i \(-0.181680\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 39823.7 1.39740
\(934\) 0 0
\(935\) 426.906i 0.0149319i
\(936\) 0 0
\(937\) − 28125.2i − 0.980586i −0.871558 0.490293i \(-0.836890\pi\)
0.871558 0.490293i \(-0.163110\pi\)
\(938\) 0 0
\(939\) − 13959.9i − 0.485159i
\(940\) 0 0
\(941\) − 30098.1i − 1.04269i −0.853347 0.521344i \(-0.825431\pi\)
0.853347 0.521344i \(-0.174569\pi\)
\(942\) 0 0
\(943\) −34263.3 −1.18321
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41414.0i 1.42109i 0.703650 + 0.710547i \(0.251552\pi\)
−0.703650 + 0.710547i \(0.748448\pi\)
\(948\) 0 0
\(949\) 15394.7 0.526590
\(950\) 0 0
\(951\) −65966.3 −2.24932
\(952\) 0 0
\(953\) −33248.1 −1.13013 −0.565063 0.825047i \(-0.691148\pi\)
−0.565063 + 0.825047i \(0.691148\pi\)
\(954\) 0 0
\(955\) −34570.3 −1.17138
\(956\) 0 0
\(957\) − 21085.2i − 0.712212i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −14128.4 −0.474249
\(962\) 0 0
\(963\) 1851.27i 0.0619485i
\(964\) 0 0
\(965\) − 25381.8i − 0.846702i
\(966\) 0 0
\(967\) − 39769.8i − 1.32255i −0.750142 0.661277i \(-0.770015\pi\)
0.750142 0.661277i \(-0.229985\pi\)
\(968\) 0 0
\(969\) − 123.667i − 0.00409985i
\(970\) 0 0
\(971\) −26812.5 −0.886153 −0.443077 0.896484i \(-0.646113\pi\)
−0.443077 + 0.896484i \(0.646113\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 49979.9i 1.64168i
\(976\) 0 0
\(977\) 15822.0 0.518106 0.259053 0.965863i \(-0.416590\pi\)
0.259053 + 0.965863i \(0.416590\pi\)
\(978\) 0 0
\(979\) 7759.00 0.253298
\(980\) 0 0
\(981\) 28394.4 0.924120
\(982\) 0 0
\(983\) 35033.3 1.13671 0.568356 0.822783i \(-0.307580\pi\)
0.568356 + 0.822783i \(0.307580\pi\)
\(984\) 0 0
\(985\) 15099.1i 0.488424i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7911.82 0.254379
\(990\) 0 0
\(991\) − 25557.0i − 0.819217i −0.912261 0.409608i \(-0.865665\pi\)
0.912261 0.409608i \(-0.134335\pi\)
\(992\) 0 0
\(993\) 106858.i 3.41493i
\(994\) 0 0
\(995\) − 88116.4i − 2.80752i
\(996\) 0 0
\(997\) − 29598.9i − 0.940226i −0.882606 0.470113i \(-0.844213\pi\)
0.882606 0.470113i \(-0.155787\pi\)
\(998\) 0 0
\(999\) −95357.1 −3.01999
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 784.4.f.g.783.2 6
4.3 odd 2 784.4.f.h.783.6 6
7.4 even 3 112.4.p.g.47.3 yes 6
7.5 odd 6 112.4.p.f.31.1 6
7.6 odd 2 784.4.f.h.783.5 6
28.11 odd 6 112.4.p.f.47.1 yes 6
28.19 even 6 112.4.p.g.31.3 yes 6
28.27 even 2 inner 784.4.f.g.783.1 6
56.5 odd 6 448.4.p.g.255.3 6
56.11 odd 6 448.4.p.g.383.3 6
56.19 even 6 448.4.p.f.255.1 6
56.53 even 6 448.4.p.f.383.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.4.p.f.31.1 6 7.5 odd 6
112.4.p.f.47.1 yes 6 28.11 odd 6
112.4.p.g.31.3 yes 6 28.19 even 6
112.4.p.g.47.3 yes 6 7.4 even 3
448.4.p.f.255.1 6 56.19 even 6
448.4.p.f.383.1 6 56.53 even 6
448.4.p.g.255.3 6 56.5 odd 6
448.4.p.g.383.3 6 56.11 odd 6
784.4.f.g.783.1 6 28.27 even 2 inner
784.4.f.g.783.2 6 1.1 even 1 trivial
784.4.f.h.783.5 6 7.6 odd 2
784.4.f.h.783.6 6 4.3 odd 2