Properties

Label 2304.2.f.g.1151.4
Level $2304$
Weight $2$
Character 2304.1151
Analytic conductor $18.398$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1151
Dual form 2304.2.f.g.1151.3

$q$-expansion

\(f(q)\) \(=\) \(q+3.41421 q^{5} +4.82843i q^{7} +O(q^{10})\) \(q+3.41421 q^{5} +4.82843i q^{7} +2.82843i q^{11} +2.82843i q^{13} -5.41421i q^{17} +5.65685 q^{19} -1.17157 q^{23} +6.65685 q^{25} -0.585786 q^{29} +3.17157i q^{31} +16.4853i q^{35} -3.65685i q^{37} +2.58579i q^{41} -9.65685 q^{43} +12.4853 q^{47} -16.3137 q^{49} -5.07107 q^{53} +9.65685i q^{55} -2.34315i q^{59} +7.65685i q^{61} +9.65685i q^{65} -12.0000 q^{67} +4.48528 q^{71} -4.00000 q^{73} -13.6569 q^{77} -6.48528i q^{79} -5.17157i q^{83} -18.4853i q^{85} +12.2426i q^{89} -13.6569 q^{91} +19.3137 q^{95} +13.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{5} + O(q^{10}) \) \( 4q + 8q^{5} - 16q^{23} + 4q^{25} - 8q^{29} - 16q^{43} + 16q^{47} - 20q^{49} + 8q^{53} - 48q^{67} - 16q^{71} - 16q^{73} - 32q^{77} - 32q^{91} + 32q^{95} + 32q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.41421 1.52688 0.763441 0.645877i \(-0.223508\pi\)
0.763441 + 0.645877i \(0.223508\pi\)
\(6\) 0 0
\(7\) 4.82843i 1.82497i 0.409106 + 0.912487i \(0.365841\pi\)
−0.409106 + 0.912487i \(0.634159\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i 0.919866 + 0.392232i \(0.128297\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.41421i − 1.31314i −0.754265 0.656570i \(-0.772007\pi\)
0.754265 0.656570i \(-0.227993\pi\)
\(18\) 0 0
\(19\) 5.65685 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.17157 −0.244290 −0.122145 0.992512i \(-0.538977\pi\)
−0.122145 + 0.992512i \(0.538977\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.585786 −0.108778 −0.0543889 0.998520i \(-0.517321\pi\)
−0.0543889 + 0.998520i \(0.517321\pi\)
\(30\) 0 0
\(31\) 3.17157i 0.569631i 0.958582 + 0.284816i \(0.0919324\pi\)
−0.958582 + 0.284816i \(0.908068\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 16.4853i 2.78652i
\(36\) 0 0
\(37\) − 3.65685i − 0.601183i −0.953753 0.300592i \(-0.902816\pi\)
0.953753 0.300592i \(-0.0971841\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.58579i 0.403832i 0.979403 + 0.201916i \(0.0647168\pi\)
−0.979403 + 0.201916i \(0.935283\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.4853 1.82117 0.910583 0.413327i \(-0.135633\pi\)
0.910583 + 0.413327i \(0.135633\pi\)
\(48\) 0 0
\(49\) −16.3137 −2.33053
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.07107 −0.696565 −0.348282 0.937390i \(-0.613235\pi\)
−0.348282 + 0.937390i \(0.613235\pi\)
\(54\) 0 0
\(55\) 9.65685i 1.30213i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2.34315i − 0.305052i −0.988299 0.152526i \(-0.951259\pi\)
0.988299 0.152526i \(-0.0487407\pi\)
\(60\) 0 0
\(61\) 7.65685i 0.980360i 0.871621 + 0.490180i \(0.163069\pi\)
−0.871621 + 0.490180i \(0.836931\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.65685i 1.19779i
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.48528 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −13.6569 −1.55634
\(78\) 0 0
\(79\) − 6.48528i − 0.729651i −0.931076 0.364826i \(-0.881129\pi\)
0.931076 0.364826i \(-0.118871\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 5.17157i − 0.567654i −0.958876 0.283827i \(-0.908396\pi\)
0.958876 0.283827i \(-0.0916041\pi\)
\(84\) 0 0
\(85\) − 18.4853i − 2.00501i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.2426i 1.29772i 0.760909 + 0.648859i \(0.224753\pi\)
−0.760909 + 0.648859i \(0.775247\pi\)
\(90\) 0 0
\(91\) −13.6569 −1.43163
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 19.3137 1.98154
\(96\) 0 0
\(97\) 13.6569 1.38664 0.693322 0.720628i \(-0.256146\pi\)
0.693322 + 0.720628i \(0.256146\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.7279 1.06747 0.533734 0.845652i \(-0.320788\pi\)
0.533734 + 0.845652i \(0.320788\pi\)
\(102\) 0 0
\(103\) 3.17157i 0.312504i 0.987717 + 0.156252i \(0.0499413\pi\)
−0.987717 + 0.156252i \(0.950059\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3137i 1.09374i 0.837218 + 0.546869i \(0.184180\pi\)
−0.837218 + 0.546869i \(0.815820\pi\)
\(108\) 0 0
\(109\) − 10.8284i − 1.03718i −0.855024 0.518588i \(-0.826458\pi\)
0.855024 0.518588i \(-0.173542\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 12.2426i − 1.15169i −0.817559 0.575845i \(-0.804673\pi\)
0.817559 0.575845i \(-0.195327\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 26.1421 2.39645
\(120\) 0 0
\(121\) 3.00000 0.272727
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) − 4.82843i − 0.428454i −0.976784 0.214227i \(-0.931277\pi\)
0.976784 0.214227i \(-0.0687232\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685i 0.494242i 0.968985 + 0.247121i \(0.0794845\pi\)
−0.968985 + 0.247121i \(0.920516\pi\)
\(132\) 0 0
\(133\) 27.3137i 2.36840i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 10.5858i − 0.904405i −0.891915 0.452202i \(-0.850638\pi\)
0.891915 0.452202i \(-0.149362\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.72792 −0.551173 −0.275586 0.961276i \(-0.588872\pi\)
−0.275586 + 0.961276i \(0.588872\pi\)
\(150\) 0 0
\(151\) − 6.48528i − 0.527765i −0.964555 0.263882i \(-0.914997\pi\)
0.964555 0.263882i \(-0.0850031\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10.8284i 0.869760i
\(156\) 0 0
\(157\) 0.343146i 0.0273860i 0.999906 + 0.0136930i \(0.00435876\pi\)
−0.999906 + 0.0136930i \(0.995641\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 5.65685i − 0.445823i
\(162\) 0 0
\(163\) −1.65685 −0.129775 −0.0648874 0.997893i \(-0.520669\pi\)
−0.0648874 + 0.997893i \(0.520669\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.65685 0.437741 0.218870 0.975754i \(-0.429763\pi\)
0.218870 + 0.975754i \(0.429763\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.5858 1.56511 0.782554 0.622582i \(-0.213916\pi\)
0.782554 + 0.622582i \(0.213916\pi\)
\(174\) 0 0
\(175\) 32.1421i 2.42972i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 19.3137i 1.44357i 0.692115 + 0.721787i \(0.256679\pi\)
−0.692115 + 0.721787i \(0.743321\pi\)
\(180\) 0 0
\(181\) 6.14214i 0.456541i 0.973598 + 0.228271i \(0.0733071\pi\)
−0.973598 + 0.228271i \(0.926693\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 12.4853i − 0.917936i
\(186\) 0 0
\(187\) 15.3137 1.11985
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.3431 −0.748404 −0.374202 0.927347i \(-0.622083\pi\)
−0.374202 + 0.927347i \(0.622083\pi\)
\(192\) 0 0
\(193\) 5.31371 0.382489 0.191245 0.981542i \(-0.438748\pi\)
0.191245 + 0.981542i \(0.438748\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.3848 1.16737 0.583683 0.811981i \(-0.301611\pi\)
0.583683 + 0.811981i \(0.301611\pi\)
\(198\) 0 0
\(199\) − 4.82843i − 0.342278i −0.985247 0.171139i \(-0.945255\pi\)
0.985247 0.171139i \(-0.0547447\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.82843i − 0.198517i
\(204\) 0 0
\(205\) 8.82843i 0.616604i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000i 1.10674i
\(210\) 0 0
\(211\) −23.3137 −1.60498 −0.802491 0.596664i \(-0.796492\pi\)
−0.802491 + 0.596664i \(0.796492\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −32.9706 −2.24857
\(216\) 0 0
\(217\) −15.3137 −1.03956
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.3137 1.03011
\(222\) 0 0
\(223\) 17.7990i 1.19191i 0.803018 + 0.595954i \(0.203226\pi\)
−0.803018 + 0.595954i \(0.796774\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 26.8284i − 1.78067i −0.455311 0.890333i \(-0.650472\pi\)
0.455311 0.890333i \(-0.349528\pi\)
\(228\) 0 0
\(229\) 5.17157i 0.341747i 0.985293 + 0.170874i \(0.0546590\pi\)
−0.985293 + 0.170874i \(0.945341\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 17.8995i − 1.17263i −0.810081 0.586317i \(-0.800577\pi\)
0.810081 0.586317i \(-0.199423\pi\)
\(234\) 0 0
\(235\) 42.6274 2.78071
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 20.9706 1.35083 0.675416 0.737437i \(-0.263964\pi\)
0.675416 + 0.737437i \(0.263964\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −55.6985 −3.55845
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.4853i 1.54550i 0.634712 + 0.772749i \(0.281119\pi\)
−0.634712 + 0.772749i \(0.718881\pi\)
\(252\) 0 0
\(253\) − 3.31371i − 0.208331i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.89949i 0.617514i 0.951141 + 0.308757i \(0.0999129\pi\)
−0.951141 + 0.308757i \(0.900087\pi\)
\(258\) 0 0
\(259\) 17.6569 1.09714
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.3431 1.13109 0.565543 0.824719i \(-0.308666\pi\)
0.565543 + 0.824719i \(0.308666\pi\)
\(264\) 0 0
\(265\) −17.3137 −1.06357
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −32.3848 −1.97453 −0.987267 0.159070i \(-0.949150\pi\)
−0.987267 + 0.159070i \(0.949150\pi\)
\(270\) 0 0
\(271\) − 1.51472i − 0.0920126i −0.998941 0.0460063i \(-0.985351\pi\)
0.998941 0.0460063i \(-0.0146494\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 18.8284i 1.13540i
\(276\) 0 0
\(277\) − 21.1716i − 1.27208i −0.771658 0.636038i \(-0.780572\pi\)
0.771658 0.636038i \(-0.219428\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 6.10051i − 0.363926i −0.983305 0.181963i \(-0.941755\pi\)
0.983305 0.181963i \(-0.0582450\pi\)
\(282\) 0 0
\(283\) 3.31371 0.196980 0.0984898 0.995138i \(-0.468599\pi\)
0.0984898 + 0.995138i \(0.468599\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.4853 −0.736983
\(288\) 0 0
\(289\) −12.3137 −0.724336
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.8701 1.68661 0.843303 0.537438i \(-0.180608\pi\)
0.843303 + 0.537438i \(0.180608\pi\)
\(294\) 0 0
\(295\) − 8.00000i − 0.465778i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 3.31371i − 0.191637i
\(300\) 0 0
\(301\) − 46.6274i − 2.68756i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 26.1421i 1.49689i
\(306\) 0 0
\(307\) 0.686292 0.0391687 0.0195844 0.999808i \(-0.493766\pi\)
0.0195844 + 0.999808i \(0.493766\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.6569 0.774409 0.387205 0.921994i \(-0.373441\pi\)
0.387205 + 0.921994i \(0.373441\pi\)
\(312\) 0 0
\(313\) −9.31371 −0.526442 −0.263221 0.964736i \(-0.584785\pi\)
−0.263221 + 0.964736i \(0.584785\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.2426 0.799946 0.399973 0.916527i \(-0.369019\pi\)
0.399973 + 0.916527i \(0.369019\pi\)
\(318\) 0 0
\(319\) − 1.65685i − 0.0927660i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 30.6274i − 1.70416i
\(324\) 0 0
\(325\) 18.8284i 1.04441i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 60.2843i 3.32358i
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −40.9706 −2.23846
\(336\) 0 0
\(337\) 23.3137 1.26998 0.634989 0.772521i \(-0.281004\pi\)
0.634989 + 0.772521i \(0.281004\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.97056 −0.485783
\(342\) 0 0
\(343\) − 44.9706i − 2.42818i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 16.4853i − 0.884976i −0.896774 0.442488i \(-0.854096\pi\)
0.896774 0.442488i \(-0.145904\pi\)
\(348\) 0 0
\(349\) − 11.6569i − 0.623977i −0.950086 0.311989i \(-0.899005\pi\)
0.950086 0.311989i \(-0.100995\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 18.8701i − 1.00435i −0.864765 0.502176i \(-0.832533\pi\)
0.864765 0.502176i \(-0.167467\pi\)
\(354\) 0 0
\(355\) 15.3137 0.812767
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.4853 −1.08117 −0.540586 0.841289i \(-0.681797\pi\)
−0.540586 + 0.841289i \(0.681797\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −13.6569 −0.714832
\(366\) 0 0
\(367\) − 20.8284i − 1.08724i −0.839333 0.543618i \(-0.817054\pi\)
0.839333 0.543618i \(-0.182946\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 24.4853i − 1.27121i
\(372\) 0 0
\(373\) 14.9706i 0.775146i 0.921839 + 0.387573i \(0.126687\pi\)
−0.921839 + 0.387573i \(0.873313\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.65685i − 0.0853323i
\(378\) 0 0
\(379\) −20.2843 −1.04193 −0.520967 0.853577i \(-0.674428\pi\)
−0.520967 + 0.853577i \(0.674428\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.6274 −0.747426 −0.373713 0.927544i \(-0.621916\pi\)
−0.373713 + 0.927544i \(0.621916\pi\)
\(384\) 0 0
\(385\) −46.6274 −2.37635
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.44365 −0.123898 −0.0619490 0.998079i \(-0.519732\pi\)
−0.0619490 + 0.998079i \(0.519732\pi\)
\(390\) 0 0
\(391\) 6.34315i 0.320787i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 22.1421i − 1.11409i
\(396\) 0 0
\(397\) 18.9706i 0.952105i 0.879417 + 0.476053i \(0.157933\pi\)
−0.879417 + 0.476053i \(0.842067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 13.4142i − 0.669874i −0.942241 0.334937i \(-0.891285\pi\)
0.942241 0.334937i \(-0.108715\pi\)
\(402\) 0 0
\(403\) −8.97056 −0.446856
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.3431 0.512691
\(408\) 0 0
\(409\) −10.3431 −0.511436 −0.255718 0.966751i \(-0.582312\pi\)
−0.255718 + 0.966751i \(0.582312\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.3137 0.556711
\(414\) 0 0
\(415\) − 17.6569i − 0.866741i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 11.7990i − 0.576418i −0.957567 0.288209i \(-0.906940\pi\)
0.957567 0.288209i \(-0.0930599\pi\)
\(420\) 0 0
\(421\) 7.51472i 0.366245i 0.983090 + 0.183122i \(0.0586205\pi\)
−0.983090 + 0.183122i \(0.941380\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 36.0416i − 1.74828i
\(426\) 0 0
\(427\) −36.9706 −1.78913
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.4853 1.37209 0.686044 0.727560i \(-0.259346\pi\)
0.686044 + 0.727560i \(0.259346\pi\)
\(432\) 0 0
\(433\) 28.6274 1.37575 0.687873 0.725831i \(-0.258545\pi\)
0.687873 + 0.725831i \(0.258545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.62742 −0.317032
\(438\) 0 0
\(439\) − 7.85786i − 0.375035i −0.982261 0.187518i \(-0.939956\pi\)
0.982261 0.187518i \(-0.0600442\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 17.4558i − 0.829352i −0.909969 0.414676i \(-0.863895\pi\)
0.909969 0.414676i \(-0.136105\pi\)
\(444\) 0 0
\(445\) 41.7990i 1.98146i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 15.2721i − 0.720734i −0.932811 0.360367i \(-0.882651\pi\)
0.932811 0.360367i \(-0.117349\pi\)
\(450\) 0 0
\(451\) −7.31371 −0.344389
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −46.6274 −2.18593
\(456\) 0 0
\(457\) 8.97056 0.419625 0.209813 0.977742i \(-0.432715\pi\)
0.209813 + 0.977742i \(0.432715\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.10051 0.377278 0.188639 0.982046i \(-0.439592\pi\)
0.188639 + 0.982046i \(0.439592\pi\)
\(462\) 0 0
\(463\) − 33.7990i − 1.57077i −0.619006 0.785386i \(-0.712464\pi\)
0.619006 0.785386i \(-0.287536\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 6.14214i − 0.284224i −0.989851 0.142112i \(-0.954611\pi\)
0.989851 0.142112i \(-0.0453893\pi\)
\(468\) 0 0
\(469\) − 57.9411i − 2.67547i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 27.3137i − 1.25589i
\(474\) 0 0
\(475\) 37.6569 1.72781
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.17157 0.419060 0.209530 0.977802i \(-0.432807\pi\)
0.209530 + 0.977802i \(0.432807\pi\)
\(480\) 0 0
\(481\) 10.3431 0.471607
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 46.6274 2.11724
\(486\) 0 0
\(487\) − 28.8284i − 1.30634i −0.757211 0.653170i \(-0.773439\pi\)
0.757211 0.653170i \(-0.226561\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 35.3137i − 1.59369i −0.604187 0.796843i \(-0.706502\pi\)
0.604187 0.796843i \(-0.293498\pi\)
\(492\) 0 0
\(493\) 3.17157i 0.142840i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 21.6569i 0.971443i
\(498\) 0 0
\(499\) 6.62742 0.296684 0.148342 0.988936i \(-0.452606\pi\)
0.148342 + 0.988936i \(0.452606\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15.7990 −0.704442 −0.352221 0.935917i \(-0.614573\pi\)
−0.352221 + 0.935917i \(0.614573\pi\)
\(504\) 0 0
\(505\) 36.6274 1.62990
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −13.2721 −0.588275 −0.294137 0.955763i \(-0.595032\pi\)
−0.294137 + 0.955763i \(0.595032\pi\)
\(510\) 0 0
\(511\) − 19.3137i − 0.854388i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10.8284i 0.477158i
\(516\) 0 0
\(517\) 35.3137i 1.55310i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.07107i 0.134546i 0.997735 + 0.0672730i \(0.0214298\pi\)
−0.997735 + 0.0672730i \(0.978570\pi\)
\(522\) 0 0
\(523\) 45.6569 1.99643 0.998217 0.0596823i \(-0.0190088\pi\)
0.998217 + 0.0596823i \(0.0190088\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.1716 0.748005
\(528\) 0 0
\(529\) −21.6274 −0.940322
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.31371 −0.316792
\(534\) 0 0
\(535\) 38.6274i 1.67001i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 46.1421i − 1.98748i
\(540\) 0 0
\(541\) 35.7990i 1.53912i 0.638575 + 0.769559i \(0.279524\pi\)
−0.638575 + 0.769559i \(0.720476\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 36.9706i − 1.58364i
\(546\) 0 0
\(547\) 9.65685 0.412897 0.206449 0.978457i \(-0.433809\pi\)
0.206449 + 0.978457i \(0.433809\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.31371 −0.141169
\(552\) 0 0
\(553\) 31.3137 1.33159
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.07107 0.384353 0.192177 0.981360i \(-0.438445\pi\)
0.192177 + 0.981360i \(0.438445\pi\)
\(558\) 0 0
\(559\) − 27.3137i − 1.15525i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 25.4558i − 1.07284i −0.843952 0.536418i \(-0.819777\pi\)
0.843952 0.536418i \(-0.180223\pi\)
\(564\) 0 0
\(565\) − 41.7990i − 1.75850i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.10051i 0.0880578i 0.999030 + 0.0440289i \(0.0140194\pi\)
−0.999030 + 0.0440289i \(0.985981\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.79899 −0.325240
\(576\) 0 0
\(577\) −29.3137 −1.22035 −0.610173 0.792268i \(-0.708900\pi\)
−0.610173 + 0.792268i \(0.708900\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 24.9706 1.03595
\(582\) 0 0
\(583\) − 14.3431i − 0.594032i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.6569i 1.22407i 0.790831 + 0.612035i \(0.209649\pi\)
−0.790831 + 0.612035i \(0.790351\pi\)
\(588\) 0 0
\(589\) 17.9411i 0.739251i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 15.0711i − 0.618895i −0.950917 0.309447i \(-0.899856\pi\)
0.950917 0.309447i \(-0.100144\pi\)
\(594\) 0 0
\(595\) 89.2548 3.65909
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −44.4853 −1.81762 −0.908810 0.417211i \(-0.863008\pi\)
−0.908810 + 0.417211i \(0.863008\pi\)
\(600\) 0 0
\(601\) −21.3137 −0.869404 −0.434702 0.900574i \(-0.643146\pi\)
−0.434702 + 0.900574i \(0.643146\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.2426 0.416423
\(606\) 0 0
\(607\) 19.1716i 0.778150i 0.921206 + 0.389075i \(0.127205\pi\)
−0.921206 + 0.389075i \(0.872795\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35.3137i 1.42864i
\(612\) 0 0
\(613\) 11.6569i 0.470816i 0.971897 + 0.235408i \(0.0756426\pi\)
−0.971897 + 0.235408i \(0.924357\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.5858i 1.23134i 0.788005 + 0.615669i \(0.211114\pi\)
−0.788005 + 0.615669i \(0.788886\pi\)
\(618\) 0 0
\(619\) −33.9411 −1.36421 −0.682105 0.731255i \(-0.738935\pi\)
−0.682105 + 0.731255i \(0.738935\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −59.1127 −2.36830
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −19.7990 −0.789437
\(630\) 0 0
\(631\) − 1.51472i − 0.0603000i −0.999545 0.0301500i \(-0.990402\pi\)
0.999545 0.0301500i \(-0.00959850\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 16.4853i − 0.654198i
\(636\) 0 0
\(637\) − 46.1421i − 1.82822i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 38.8701i 1.53527i 0.640884 + 0.767637i \(0.278568\pi\)
−0.640884 + 0.767637i \(0.721432\pi\)
\(642\) 0 0
\(643\) −1.65685 −0.0653400 −0.0326700 0.999466i \(-0.510401\pi\)
−0.0326700 + 0.999466i \(0.510401\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 23.7990 0.935635 0.467817 0.883825i \(-0.345040\pi\)
0.467817 + 0.883825i \(0.345040\pi\)
\(648\) 0 0
\(649\) 6.62742 0.260149
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.7279 −1.20248 −0.601238 0.799070i \(-0.705326\pi\)
−0.601238 + 0.799070i \(0.705326\pi\)
\(654\) 0 0
\(655\) 19.3137i 0.754649i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 24.9706i − 0.972715i −0.873760 0.486358i \(-0.838325\pi\)
0.873760 0.486358i \(-0.161675\pi\)
\(660\) 0 0
\(661\) 7.65685i 0.297817i 0.988851 + 0.148909i \(0.0475760\pi\)
−0.988851 + 0.148909i \(0.952424\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 93.2548i 3.61627i
\(666\) 0 0
\(667\) 0.686292 0.0265733
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −21.6569 −0.836054
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.72792 0.258575 0.129288 0.991607i \(-0.458731\pi\)
0.129288 + 0.991607i \(0.458731\pi\)
\(678\) 0 0
\(679\) 65.9411i 2.53059i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 6.14214i − 0.235022i −0.993072 0.117511i \(-0.962508\pi\)
0.993072 0.117511i \(-0.0374916\pi\)
\(684\) 0 0
\(685\) − 36.1421i − 1.38092i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 14.3431i − 0.546430i
\(690\) 0 0
\(691\) 36.9706 1.40643 0.703213 0.710979i \(-0.251748\pi\)
0.703213 + 0.710979i \(0.251748\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −40.9706 −1.55410
\(696\) 0 0
\(697\) 14.0000 0.530288
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.2426 −0.840093 −0.420046 0.907503i \(-0.637986\pi\)
−0.420046 + 0.907503i \(0.637986\pi\)
\(702\) 0 0
\(703\) − 20.6863i − 0.780198i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 51.7990i 1.94810i
\(708\) 0 0
\(709\) − 48.0833i − 1.80580i −0.429846 0.902902i \(-0.641432\pi\)
0.429846 0.902902i \(-0.358568\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 3.71573i − 0.139155i
\(714\) 0 0
\(715\) −27.3137 −1.02147
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 11.5147 0.429427 0.214713 0.976677i \(-0.431118\pi\)
0.214713 + 0.976677i \(0.431118\pi\)
\(720\) 0 0
\(721\) −15.3137 −0.570312
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.89949 −0.144824
\(726\) 0 0
\(727\) 3.17157i 0.117627i 0.998269 + 0.0588136i \(0.0187317\pi\)
−0.998269 + 0.0588136i \(0.981268\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 52.2843i 1.93380i
\(732\) 0 0
\(733\) 7.51472i 0.277562i 0.990323 + 0.138781i \(0.0443185\pi\)
−0.990323 + 0.138781i \(0.955682\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 33.9411i − 1.25024i
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.9706 −0.622590 −0.311295 0.950313i \(-0.600763\pi\)
−0.311295 + 0.950313i \(0.600763\pi\)
\(744\) 0 0
\(745\) −22.9706 −0.841576
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −54.6274 −1.99604
\(750\) 0 0
\(751\) 35.1716i 1.28343i 0.766944 + 0.641714i \(0.221777\pi\)
−0.766944 + 0.641714i \(0.778223\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 22.1421i − 0.805835i
\(756\) 0 0
\(757\) 16.4853i 0.599168i 0.954070 + 0.299584i \(0.0968478\pi\)
−0.954070 + 0.299584i \(0.903152\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 33.6985i − 1.22157i −0.791797 0.610785i \(-0.790854\pi\)
0.791797 0.610785i \(-0.209146\pi\)
\(762\) 0 0
\(763\) 52.2843 1.89282
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.62742 0.239302
\(768\) 0 0
\(769\) 1.31371 0.0473735 0.0236868 0.999719i \(-0.492460\pi\)
0.0236868 + 0.999719i \(0.492460\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −15.4142 −0.554411 −0.277205 0.960811i \(-0.589408\pi\)
−0.277205 + 0.960811i \(0.589408\pi\)
\(774\) 0 0
\(775\) 21.1127i 0.758391i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14.6274i 0.524082i
\(780\) 0 0
\(781\) 12.6863i 0.453951i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.17157i 0.0418152i
\(786\) 0 0
\(787\) −37.6569 −1.34232 −0.671161 0.741312i \(-0.734204\pi\)
−0.671161 + 0.741312i \(0.734204\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 59.1127 2.10181
\(792\) 0 0
\(793\) −21.6569 −0.769057
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.07107 −0.0379392 −0.0189696 0.999820i \(-0.506039\pi\)
−0.0189696 + 0.999820i \(0.506039\pi\)
\(798\) 0 0
\(799\) − 67.5980i − 2.39144i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 11.3137i − 0.399252i
\(804\) 0 0
\(805\) − 19.3137i − 0.680719i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 50.6690i − 1.78143i −0.454563 0.890714i \(-0.650205\pi\)
0.454563 0.890714i \(-0.349795\pi\)
\(810\) 0 0
\(811\) 47.5980 1.67139 0.835696 0.549193i \(-0.185065\pi\)
0.835696 + 0.549193i \(0.185065\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.65685 −0.198151
\(816\) 0 0
\(817\) −54.6274 −1.91117
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.58579 −0.299646 −0.149823 0.988713i \(-0.547870\pi\)
−0.149823 + 0.988713i \(0.547870\pi\)
\(822\) 0 0
\(823\) − 32.4264i − 1.13031i −0.824984 0.565157i \(-0.808816\pi\)
0.824984 0.565157i \(-0.191184\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 48.9706i − 1.70287i −0.524457 0.851437i \(-0.675732\pi\)
0.524457 0.851437i \(-0.324268\pi\)
\(828\) 0 0
\(829\) 43.7990i 1.52120i 0.649220 + 0.760601i \(0.275096\pi\)
−0.649220 + 0.760601i \(0.724904\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 88.3259i 3.06031i
\(834\) 0 0
\(835\) 19.3137 0.668378
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −20.4853 −0.707230 −0.353615 0.935391i \(-0.615048\pi\)
−0.353615 + 0.935391i \(0.615048\pi\)
\(840\) 0 0
\(841\) −28.6569 −0.988167
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 17.0711 0.587263
\(846\) 0 0
\(847\) 14.4853i 0.497720i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.28427i 0.146863i
\(852\) 0 0
\(853\) 14.2843i 0.489084i 0.969639 + 0.244542i \(0.0786376\pi\)
−0.969639 + 0.244542i \(0.921362\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 2.10051i − 0.0717519i −0.999356 0.0358759i \(-0.988578\pi\)
0.999356 0.0358759i \(-0.0114221\pi\)
\(858\) 0 0
\(859\) −9.65685 −0.329488 −0.164744 0.986336i \(-0.552680\pi\)
−0.164744 + 0.986336i \(0.552680\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 32.9706 1.12233 0.561166 0.827704i \(-0.310353\pi\)
0.561166 + 0.827704i \(0.310353\pi\)
\(864\) 0 0
\(865\) 70.2843 2.38974
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 18.3431 0.622249
\(870\) 0 0
\(871\) − 33.9411i − 1.15005i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 27.3137i 0.923372i
\(876\) 0 0
\(877\) 26.9706i 0.910731i 0.890305 + 0.455366i \(0.150491\pi\)
−0.890305 + 0.455366i \(0.849509\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 12.7279i − 0.428815i −0.976744 0.214407i \(-0.931218\pi\)
0.976744 0.214407i \(-0.0687820\pi\)
\(882\) 0 0
\(883\) −16.2843 −0.548009 −0.274005 0.961728i \(-0.588348\pi\)
−0.274005 + 0.961728i \(0.588348\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.6569 −0.458552 −0.229276 0.973361i \(-0.573636\pi\)
−0.229276 + 0.973361i \(0.573636\pi\)
\(888\) 0 0
\(889\) 23.3137 0.781917
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 70.6274 2.36346
\(894\) 0 0
\(895\) 65.9411i 2.20417i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1.85786i − 0.0619632i
\(900\) 0 0
\(901\) 27.4558i 0.914687i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.9706i 0.697085i
\(906\) 0 0
\(907\) −25.6569 −0.851922 −0.425961 0.904742i \(-0.640064\pi\)
−0.425961 + 0.904742i \(0.640064\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 55.5980 1.84204 0.921022 0.389511i \(-0.127356\pi\)
0.921022 + 0.389511i \(0.127356\pi\)
\(912\) 0 0
\(913\) 14.6274 0.484097
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −27.3137 −0.901978
\(918\) 0 0
\(919\) 27.4558i 0.905685i 0.891591 + 0.452842i \(0.149590\pi\)
−0.891591 + 0.452842i \(0.850410\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.6863i 0.417574i
\(924\) 0 0
\(925\) − 24.3431i − 0.800398i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 8.72792i − 0.286354i −0.989697 0.143177i \(-0.954268\pi\)
0.989697 0.143177i \(-0.0457318\pi\)
\(930\) 0 0
\(931\) −92.2843 −3.02449
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 52.2843 1.70988
\(936\) 0 0
\(937\) 11.3726 0.371526 0.185763 0.982595i \(-0.440524\pi\)
0.185763 + 0.982595i \(0.440524\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 46.7279 1.52329 0.761643 0.647996i \(-0.224393\pi\)
0.761643 + 0.647996i \(0.224393\pi\)
\(942\) 0 0
\(943\) − 3.02944i − 0.0986521i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47.5980i 1.54673i 0.633963 + 0.773363i \(0.281427\pi\)
−0.633963 + 0.773363i \(0.718573\pi\)
\(948\) 0 0
\(949\) − 11.3137i − 0.367259i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 35.5563i − 1.15178i −0.817526 0.575892i \(-0.804655\pi\)
0.817526 0.575892i \(-0.195345\pi\)
\(954\) 0 0
\(955\) −35.3137 −1.14272
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 51.1127 1.65052
\(960\) 0 0
\(961\) 20.9411 0.675520
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 18.1421 0.584016
\(966\) 0 0
\(967\) − 50.0833i − 1.61057i −0.592889 0.805285i \(-0.702013\pi\)
0.592889 0.805285i \(-0.297987\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 26.8284i − 0.860965i −0.902599 0.430483i \(-0.858343\pi\)
0.902599 0.430483i \(-0.141657\pi\)
\(972\) 0 0
\(973\) − 57.9411i − 1.85751i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 52.5269i − 1.68048i −0.542211 0.840242i \(-0.682413\pi\)
0.542211 0.840242i \(-0.317587\pi\)
\(978\) 0 0
\(979\) −34.6274 −1.10670
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 45.6569 1.45623 0.728114 0.685456i \(-0.240397\pi\)
0.728114 + 0.685456i \(0.240397\pi\)
\(984\) 0 0
\(985\) 55.9411 1.78243
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.3137 0.359755
\(990\) 0 0
\(991\) 25.7990i 0.819532i 0.912191 + 0.409766i \(0.134390\pi\)
−0.912191 + 0.409766i \(0.865610\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 16.4853i − 0.522619i
\(996\) 0 0
\(997\) 61.5980i 1.95083i 0.220381 + 0.975414i \(0.429270\pi\)
−0.220381 + 0.975414i \(0.570730\pi\)
\(998\) 0 0
\(999\) 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 2304.2.f.g.1151.4 4
3.2 odd 2 2304.2.f.b.1151.2 4
4.3 odd 2 2304.2.f.h.1151.3 4
8.3 odd 2 2304.2.f.b.1151.1 4
8.5 even 2 2304.2.f.a.1151.2 4
12.11 even 2 2304.2.f.a.1151.1 4
16.3 odd 4 1152.2.c.b.1151.1 yes 4
16.5 even 4 1152.2.c.d.1151.4 yes 4
16.11 odd 4 1152.2.c.a.1151.4 yes 4
16.13 even 4 1152.2.c.c.1151.1 yes 4
24.5 odd 2 2304.2.f.h.1151.4 4
24.11 even 2 inner 2304.2.f.g.1151.3 4
48.5 odd 4 1152.2.c.a.1151.1 4
48.11 even 4 1152.2.c.d.1151.1 yes 4
48.29 odd 4 1152.2.c.b.1151.4 yes 4
48.35 even 4 1152.2.c.c.1151.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.c.a.1151.1 4 48.5 odd 4
1152.2.c.a.1151.4 yes 4 16.11 odd 4
1152.2.c.b.1151.1 yes 4 16.3 odd 4
1152.2.c.b.1151.4 yes 4 48.29 odd 4
1152.2.c.c.1151.1 yes 4 16.13 even 4
1152.2.c.c.1151.4 yes 4 48.35 even 4
1152.2.c.d.1151.1 yes 4 48.11 even 4
1152.2.c.d.1151.4 yes 4 16.5 even 4
2304.2.f.a.1151.1 4 12.11 even 2
2304.2.f.a.1151.2 4 8.5 even 2
2304.2.f.b.1151.1 4 8.3 odd 2
2304.2.f.b.1151.2 4 3.2 odd 2
2304.2.f.g.1151.3 4 24.11 even 2 inner
2304.2.f.g.1151.4 4 1.1 even 1 trivial
2304.2.f.h.1151.3 4 4.3 odd 2
2304.2.f.h.1151.4 4 24.5 odd 2