Properties

Label 1152.2.c.a.1151.1
Level $1152$
Weight $2$
Character 1152.1151
Analytic conductor $9.199$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1152.1151
Dual form 1152.2.c.a.1151.4

$q$-expansion

\(f(q)\) \(=\) \(q-3.41421i q^{5} -4.82843i q^{7} +O(q^{10})\) \(q-3.41421i q^{5} -4.82843i q^{7} +2.82843 q^{11} +2.82843 q^{13} +5.41421i q^{17} -5.65685i q^{19} -1.17157 q^{23} -6.65685 q^{25} -0.585786i q^{29} +3.17157i q^{31} -16.4853 q^{35} +3.65685 q^{37} +2.58579i q^{41} -9.65685i q^{43} -12.4853 q^{47} -16.3137 q^{49} +5.07107i q^{53} -9.65685i q^{55} -2.34315 q^{59} +7.65685 q^{61} -9.65685i q^{65} +12.0000i q^{67} +4.48528 q^{71} +4.00000 q^{73} -13.6569i q^{77} -6.48528i q^{79} +5.17157 q^{83} +18.4853 q^{85} +12.2426i q^{89} -13.6569i q^{91} -19.3137 q^{95} +13.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 16q^{23} - 4q^{25} - 32q^{35} - 8q^{37} - 16q^{47} - 20q^{49} - 32q^{59} + 8q^{61} - 16q^{71} + 16q^{73} + 32q^{83} + 40q^{85} - 32q^{95} + 32q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(5\) − 3.41421i − 1.52688i −0.645877 0.763441i \(-0.723508\pi\)
0.645877 0.763441i \(-0.276492\pi\)
\(6\) 0 0
\(7\) − 4.82843i − 1.82497i −0.409106 0.912487i \(-0.634159\pi\)
0.409106 0.912487i \(-0.365841\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.41421i 1.31314i 0.754265 + 0.656570i \(0.227993\pi\)
−0.754265 + 0.656570i \(0.772007\pi\)
\(18\) 0 0
\(19\) − 5.65685i − 1.29777i −0.760886 0.648886i \(-0.775235\pi\)
0.760886 0.648886i \(-0.224765\pi\)
\(20\) 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
\(29\) − 0.585786i − 0.108778i −0.998520 0.0543889i \(-0.982679\pi\)
0.998520 0.0543889i \(-0.0173211\pi\)
\(30\) 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
\(35\) −16.4853 −2.78652
\(36\) 0 0
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 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
\(43\) − 9.65685i − 1.47266i −0.676625 0.736328i \(-0.736558\pi\)
0.676625 0.736328i \(-0.263442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.4853 −1.82117 −0.910583 0.413327i \(-0.864367\pi\)
−0.910583 + 0.413327i \(0.864367\pi\)
\(48\) 0 0
\(49\) −16.3137 −2.33053
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.07107i 0.696565i 0.937390 + 0.348282i \(0.113235\pi\)
−0.937390 + 0.348282i \(0.886765\pi\)
\(54\) 0 0
\(55\) − 9.65685i − 1.30213i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.34315 −0.305052 −0.152526 0.988299i \(-0.548741\pi\)
−0.152526 + 0.988299i \(0.548741\pi\)
\(60\) 0 0
\(61\) 7.65685 0.980360 0.490180 0.871621i \(-0.336931\pi\)
0.490180 + 0.871621i \(0.336931\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 9.65685i − 1.19779i
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 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 \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 13.6569i − 1.55634i
\(78\) 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
\(83\) 5.17157 0.567654 0.283827 0.958876i \(-0.408396\pi\)
0.283827 + 0.958876i \(0.408396\pi\)
\(84\) 0 0
\(85\) 18.4853 2.00501
\(86\) 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
\(91\) − 13.6569i − 1.43163i
\(92\) 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
\(101\) − 10.7279i − 1.06747i −0.845652 0.533734i \(-0.820788\pi\)
0.845652 0.533734i \(-0.179212\pi\)
\(102\) 0 0
\(103\) − 3.17157i − 0.312504i −0.987717 0.156252i \(-0.950059\pi\)
0.987717 0.156252i \(-0.0499413\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 0 0
\(109\) −10.8284 −1.03718 −0.518588 0.855024i \(-0.673542\pi\)
−0.518588 + 0.855024i \(0.673542\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.2426i 1.15169i 0.817559 + 0.575845i \(0.195327\pi\)
−0.817559 + 0.575845i \(0.804673\pi\)
\(114\) 0 0
\(115\) 4.00000i 0.373002i
\(116\) 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
\(125\) 5.65685i 0.505964i
\(126\) 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
\(131\) −5.65685 −0.494242 −0.247121 0.968985i \(-0.579484\pi\)
−0.247121 + 0.968985i \(0.579484\pi\)
\(132\) 0 0
\(133\) −27.3137 −2.36840
\(134\) 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
\(139\) − 12.0000i − 1.01783i −0.860818 0.508913i \(-0.830047\pi\)
0.860818 0.508913i \(-0.169953\pi\)
\(140\) 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
\(149\) 6.72792i 0.551173i 0.961276 + 0.275586i \(0.0888720\pi\)
−0.961276 + 0.275586i \(0.911128\pi\)
\(150\) 0 0
\(151\) 6.48528i 0.527765i 0.964555 + 0.263882i \(0.0850031\pi\)
−0.964555 + 0.263882i \(0.914997\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10.8284 0.869760
\(156\) 0 0
\(157\) 0.343146 0.0273860 0.0136930 0.999906i \(-0.495641\pi\)
0.0136930 + 0.999906i \(0.495641\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65685i 0.445823i
\(162\) 0 0
\(163\) 1.65685i 0.129775i 0.997893 + 0.0648874i \(0.0206688\pi\)
−0.997893 + 0.0648874i \(0.979331\pi\)
\(164\) 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
\(173\) 20.5858i 1.56511i 0.622582 + 0.782554i \(0.286084\pi\)
−0.622582 + 0.782554i \(0.713916\pi\)
\(174\) 0 0
\(175\) 32.1421i 2.42972i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −19.3137 −1.44357 −0.721787 0.692115i \(-0.756679\pi\)
−0.721787 + 0.692115i \(0.756679\pi\)
\(180\) 0 0
\(181\) −6.14214 −0.456541 −0.228271 0.973598i \(-0.573307\pi\)
−0.228271 + 0.973598i \(0.573307\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 12.4853i − 0.917936i
\(186\) 0 0
\(187\) 15.3137i 1.11985i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.3431 0.748404 0.374202 0.927347i \(-0.377917\pi\)
0.374202 + 0.927347i \(0.377917\pi\)
\(192\) 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
\(197\) − 16.3848i − 1.16737i −0.811981 0.583683i \(-0.801611\pi\)
0.811981 0.583683i \(-0.198389\pi\)
\(198\) 0 0
\(199\) 4.82843i 0.342278i 0.985247 + 0.171139i \(0.0547447\pi\)
−0.985247 + 0.171139i \(0.945255\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.82843 −0.198517
\(204\) 0 0
\(205\) 8.82843 0.616604
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 16.0000i − 1.10674i
\(210\) 0 0
\(211\) 23.3137i 1.60498i 0.596664 + 0.802491i \(0.296492\pi\)
−0.596664 + 0.802491i \(0.703508\pi\)
\(212\) 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
\(221\) 15.3137i 1.03011i
\(222\) 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
\(227\) 26.8284 1.78067 0.890333 0.455311i \(-0.150472\pi\)
0.890333 + 0.455311i \(0.150472\pi\)
\(228\) 0 0
\(229\) −5.17157 −0.341747 −0.170874 0.985293i \(-0.554659\pi\)
−0.170874 + 0.985293i \(0.554659\pi\)
\(230\) 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
\(235\) 42.6274i 2.78071i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 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
\(245\) 55.6985i 3.55845i
\(246\) 0 0
\(247\) − 16.0000i − 1.01806i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.4853 1.54550 0.772749 0.634712i \(-0.218881\pi\)
0.772749 + 0.634712i \(0.218881\pi\)
\(252\) 0 0
\(253\) −3.31371 −0.208331
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 9.89949i − 0.617514i −0.951141 0.308757i \(-0.900087\pi\)
0.951141 0.308757i \(-0.0999129\pi\)
\(258\) 0 0
\(259\) − 17.6569i − 1.09714i
\(260\) 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
\(269\) − 32.3848i − 1.97453i −0.159070 0.987267i \(-0.550850\pi\)
0.159070 0.987267i \(-0.449150\pi\)
\(270\) 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
\(275\) −18.8284 −1.13540
\(276\) 0 0
\(277\) 21.1716 1.27208 0.636038 0.771658i \(-0.280572\pi\)
0.636038 + 0.771658i \(0.280572\pi\)
\(278\) 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
\(283\) 3.31371i 0.196980i 0.995138 + 0.0984898i \(0.0314012\pi\)
−0.995138 + 0.0984898i \(0.968599\pi\)
\(284\) 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
\(293\) − 28.8701i − 1.68661i −0.537438 0.843303i \(-0.680608\pi\)
0.537438 0.843303i \(-0.319392\pi\)
\(294\) 0 0
\(295\) 8.00000i 0.465778i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.31371 −0.191637
\(300\) 0 0
\(301\) −46.6274 −2.68756
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 26.1421i − 1.49689i
\(306\) 0 0
\(307\) − 0.686292i − 0.0391687i −0.999808 0.0195844i \(-0.993766\pi\)
0.999808 0.0195844i \(-0.00623429\pi\)
\(308\) 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 \(-0.415215\pi\)
0.263221 + 0.964736i \(0.415215\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.2426i 0.799946i 0.916527 + 0.399973i \(0.130981\pi\)
−0.916527 + 0.399973i \(0.869019\pi\)
\(318\) 0 0
\(319\) − 1.65685i − 0.0927660i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 30.6274 1.70416
\(324\) 0 0
\(325\) −18.8284 −1.04441
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 60.2843i 3.32358i
\(330\) 0 0
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) 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
\(341\) 8.97056i 0.485783i
\(342\) 0 0
\(343\) 44.9706i 2.42818i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.4853 −0.884976 −0.442488 0.896774i \(-0.645904\pi\)
−0.442488 + 0.896774i \(0.645904\pi\)
\(348\) 0 0
\(349\) −11.6569 −0.623977 −0.311989 0.950086i \(-0.600995\pi\)
−0.311989 + 0.950086i \(0.600995\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.8701i 1.00435i 0.864765 + 0.502176i \(0.167467\pi\)
−0.864765 + 0.502176i \(0.832533\pi\)
\(354\) 0 0
\(355\) − 15.3137i − 0.812767i
\(356\) 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
\(365\) − 13.6569i − 0.714832i
\(366\) 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
\(371\) 24.4853 1.27121
\(372\) 0 0
\(373\) −14.9706 −0.775146 −0.387573 0.921839i \(-0.626687\pi\)
−0.387573 + 0.921839i \(0.626687\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.65685i − 0.0853323i
\(378\) 0 0
\(379\) − 20.2843i − 1.04193i −0.853577 0.520967i \(-0.825572\pi\)
0.853577 0.520967i \(-0.174428\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 14.6274 0.747426 0.373713 0.927544i \(-0.378084\pi\)
0.373713 + 0.927544i \(0.378084\pi\)
\(384\) 0 0
\(385\) −46.6274 −2.37635
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.44365i 0.123898i 0.998079 + 0.0619490i \(0.0197316\pi\)
−0.998079 + 0.0619490i \(0.980268\pi\)
\(390\) 0 0
\(391\) − 6.34315i − 0.320787i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −22.1421 −1.11409
\(396\) 0 0
\(397\) 18.9706 0.952105 0.476053 0.879417i \(-0.342067\pi\)
0.476053 + 0.879417i \(0.342067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.4142i 0.669874i 0.942241 + 0.334937i \(0.108715\pi\)
−0.942241 + 0.334937i \(0.891285\pi\)
\(402\) 0 0
\(403\) 8.97056i 0.446856i
\(404\) 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 \(-0.417688\pi\)
0.255718 + 0.966751i \(0.417688\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.3137i 0.556711i
\(414\) 0 0
\(415\) − 17.6569i − 0.866741i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.7990 0.576418 0.288209 0.957567i \(-0.406940\pi\)
0.288209 + 0.957567i \(0.406940\pi\)
\(420\) 0 0
\(421\) −7.51472 −0.366245 −0.183122 0.983090i \(-0.558620\pi\)
−0.183122 + 0.983090i \(0.558620\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 36.0416i − 1.74828i
\(426\) 0 0
\(427\) − 36.9706i − 1.78913i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −28.4853 −1.37209 −0.686044 0.727560i \(-0.740654\pi\)
−0.686044 + 0.727560i \(0.740654\pi\)
\(432\) 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
\(437\) 6.62742i 0.317032i
\(438\) 0 0
\(439\) 7.85786i 0.375035i 0.982261 + 0.187518i \(0.0600442\pi\)
−0.982261 + 0.187518i \(0.939956\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −17.4558 −0.829352 −0.414676 0.909969i \(-0.636105\pi\)
−0.414676 + 0.909969i \(0.636105\pi\)
\(444\) 0 0
\(445\) 41.7990 1.98146
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.2721i 0.720734i 0.932811 + 0.360367i \(0.117349\pi\)
−0.932811 + 0.360367i \(0.882651\pi\)
\(450\) 0 0
\(451\) 7.31371i 0.344389i
\(452\) 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 \(-0.567285\pi\)
−0.209813 + 0.977742i \(0.567285\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.10051i 0.377278i 0.982046 + 0.188639i \(0.0604076\pi\)
−0.982046 + 0.188639i \(0.939592\pi\)
\(462\) 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
\(467\) 6.14214 0.284224 0.142112 0.989851i \(-0.454611\pi\)
0.142112 + 0.989851i \(0.454611\pi\)
\(468\) 0 0
\(469\) 57.9411 2.67547
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 27.3137i − 1.25589i
\(474\) 0 0
\(475\) 37.6569i 1.72781i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.17157 −0.419060 −0.209530 0.977802i \(-0.567193\pi\)
−0.209530 + 0.977802i \(0.567193\pi\)
\(480\) 0 0
\(481\) 10.3431 0.471607
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 46.6274i − 2.11724i
\(486\) 0 0
\(487\) 28.8284i 1.30634i 0.757211 + 0.653170i \(0.226561\pi\)
−0.757211 + 0.653170i \(0.773439\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −35.3137 −1.59369 −0.796843 0.604187i \(-0.793498\pi\)
−0.796843 + 0.604187i \(0.793498\pi\)
\(492\) 0 0
\(493\) 3.17157 0.142840
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 21.6569i − 0.971443i
\(498\) 0 0
\(499\) − 6.62742i − 0.296684i −0.988936 0.148342i \(-0.952606\pi\)
0.988936 0.148342i \(-0.0473936\pi\)
\(500\) 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
\(509\) − 13.2721i − 0.588275i −0.955763 0.294137i \(-0.904968\pi\)
0.955763 0.294137i \(-0.0950323\pi\)
\(510\) 0 0
\(511\) − 19.3137i − 0.854388i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.8284 −0.477158
\(516\) 0 0
\(517\) −35.3137 −1.55310
\(518\) 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
\(523\) 45.6569i 1.99643i 0.0596823 + 0.998217i \(0.480991\pi\)
−0.0596823 + 0.998217i \(0.519009\pi\)
\(524\) 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
\(533\) 7.31371i 0.316792i
\(534\) 0 0
\(535\) − 38.6274i − 1.67001i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −46.1421 −1.98748
\(540\) 0 0
\(541\) 35.7990 1.53912 0.769559 0.638575i \(-0.220476\pi\)
0.769559 + 0.638575i \(0.220476\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 36.9706i 1.58364i
\(546\) 0 0
\(547\) − 9.65685i − 0.412897i −0.978457 0.206449i \(-0.933809\pi\)
0.978457 0.206449i \(-0.0661906\pi\)
\(548\) 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
\(557\) 9.07107i 0.384353i 0.981360 + 0.192177i \(0.0615547\pi\)
−0.981360 + 0.192177i \(0.938445\pi\)
\(558\) 0 0
\(559\) − 27.3137i − 1.15525i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.4558 1.07284 0.536418 0.843952i \(-0.319777\pi\)
0.536418 + 0.843952i \(0.319777\pi\)
\(564\) 0 0
\(565\) 41.7990 1.75850
\(566\) 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
\(571\) − 24.0000i − 1.00437i −0.864761 0.502184i \(-0.832530\pi\)
0.864761 0.502184i \(-0.167470\pi\)
\(572\) 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
\(581\) − 24.9706i − 1.03595i
\(582\) 0 0
\(583\) 14.3431i 0.594032i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.6569 1.22407 0.612035 0.790831i \(-0.290351\pi\)
0.612035 + 0.790831i \(0.290351\pi\)
\(588\) 0 0
\(589\) 17.9411 0.739251
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.0711i 0.618895i 0.950917 + 0.309447i \(0.100144\pi\)
−0.950917 + 0.309447i \(0.899856\pi\)
\(594\) 0 0
\(595\) − 89.2548i − 3.65909i
\(596\) 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 \(-0.356854\pi\)
0.434702 + 0.900574i \(0.356854\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.2426i 0.416423i
\(606\) 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
\(611\) −35.3137 −1.42864
\(612\) 0 0
\(613\) −11.6569 −0.470816 −0.235408 0.971897i \(-0.575643\pi\)
−0.235408 + 0.971897i \(0.575643\pi\)
\(614\) 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
\(619\) − 33.9411i − 1.36421i −0.731255 0.682105i \(-0.761065\pi\)
0.731255 0.682105i \(-0.238935\pi\)
\(620\) 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
\(629\) 19.7990i 0.789437i
\(630\) 0 0
\(631\) 1.51472i 0.0603000i 0.999545 + 0.0301500i \(0.00959850\pi\)
−0.999545 + 0.0301500i \(0.990402\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.4853 −0.654198
\(636\) 0 0
\(637\) −46.1421 −1.82822
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 38.8701i − 1.53527i −0.640884 0.767637i \(-0.721432\pi\)
0.640884 0.767637i \(-0.278568\pi\)
\(642\) 0 0
\(643\) 1.65685i 0.0653400i 0.999466 + 0.0326700i \(0.0104010\pi\)
−0.999466 + 0.0326700i \(0.989599\pi\)
\(644\) 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
\(653\) − 30.7279i − 1.20248i −0.799070 0.601238i \(-0.794674\pi\)
0.799070 0.601238i \(-0.205326\pi\)
\(654\) 0 0
\(655\) 19.3137i 0.754649i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.9706 0.972715 0.486358 0.873760i \(-0.338325\pi\)
0.486358 + 0.873760i \(0.338325\pi\)
\(660\) 0 0
\(661\) −7.65685 −0.297817 −0.148909 0.988851i \(-0.547576\pi\)
−0.148909 + 0.988851i \(0.547576\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 93.2548i 3.61627i
\(666\) 0 0
\(667\) 0.686292i 0.0265733i
\(668\) 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
\(677\) − 6.72792i − 0.258575i −0.991607 0.129288i \(-0.958731\pi\)
0.991607 0.129288i \(-0.0412690\pi\)
\(678\) 0 0
\(679\) − 65.9411i − 2.53059i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.14214 −0.235022 −0.117511 0.993072i \(-0.537492\pi\)
−0.117511 + 0.993072i \(0.537492\pi\)
\(684\) 0 0
\(685\) −36.1421 −1.38092
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14.3431i 0.546430i
\(690\) 0 0
\(691\) − 36.9706i − 1.40643i −0.710979 0.703213i \(-0.751748\pi\)
0.710979 0.703213i \(-0.248252\pi\)
\(692\) 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
\(701\) − 22.2426i − 0.840093i −0.907503 0.420046i \(-0.862014\pi\)
0.907503 0.420046i \(-0.137986\pi\)
\(702\) 0 0
\(703\) − 20.6863i − 0.780198i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −51.7990 −1.94810
\(708\) 0 0
\(709\) 48.0833 1.80580 0.902902 0.429846i \(-0.141432\pi\)
0.902902 + 0.429846i \(0.141432\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 3.71573i − 0.139155i
\(714\) 0 0
\(715\) − 27.3137i − 1.02147i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −11.5147 −0.429427 −0.214713 0.976677i \(-0.568882\pi\)
−0.214713 + 0.976677i \(0.568882\pi\)
\(720\) 0 0
\(721\) −15.3137 −0.570312
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.89949i 0.144824i
\(726\) 0 0
\(727\) − 3.17157i − 0.117627i −0.998269 0.0588136i \(-0.981268\pi\)
0.998269 0.0588136i \(-0.0187317\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 52.2843 1.93380
\(732\) 0 0
\(733\) 7.51472 0.277562 0.138781 0.990323i \(-0.455682\pi\)
0.138781 + 0.990323i \(0.455682\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.9411i 1.25024i
\(738\) 0 0
\(739\) 8.00000i 0.294285i 0.989115 + 0.147142i \(0.0470076\pi\)
−0.989115 + 0.147142i \(0.952992\pi\)
\(740\) 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
\(749\) − 54.6274i − 1.99604i
\(750\) 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
\(755\) 22.1421 0.805835
\(756\) 0 0
\(757\) −16.4853 −0.599168 −0.299584 0.954070i \(-0.596848\pi\)
−0.299584 + 0.954070i \(0.596848\pi\)
\(758\) 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
\(763\) 52.2843i 1.89282i
\(764\) 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
\(773\) 15.4142i 0.554411i 0.960811 + 0.277205i \(0.0894082\pi\)
−0.960811 + 0.277205i \(0.910592\pi\)
\(774\) 0 0
\(775\) − 21.1127i − 0.758391i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14.6274 0.524082
\(780\) 0 0
\(781\) 12.6863 0.453951
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1.17157i − 0.0418152i
\(786\) 0 0
\(787\) 37.6569i 1.34232i 0.741312 + 0.671161i \(0.234204\pi\)
−0.741312 + 0.671161i \(0.765796\pi\)
\(788\) 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
\(797\) − 1.07107i − 0.0379392i −0.999820 0.0189696i \(-0.993961\pi\)
0.999820 0.0189696i \(-0.00603857\pi\)
\(798\) 0 0
\(799\) − 67.5980i − 2.39144i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.3137 0.399252
\(804\) 0 0
\(805\) 19.3137 0.680719
\(806\) 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
\(811\) 47.5980i 1.67139i 0.549193 + 0.835696i \(0.314935\pi\)
−0.549193 + 0.835696i \(0.685065\pi\)
\(812\) 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
\(821\) 8.58579i 0.299646i 0.988713 + 0.149823i \(0.0478704\pi\)
−0.988713 + 0.149823i \(0.952130\pi\)
\(822\) 0 0
\(823\) 32.4264i 1.13031i 0.824984 + 0.565157i \(0.191184\pi\)
−0.824984 + 0.565157i \(0.808816\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −48.9706 −1.70287 −0.851437 0.524457i \(-0.824268\pi\)
−0.851437 + 0.524457i \(0.824268\pi\)
\(828\) 0 0
\(829\) 43.7990 1.52120 0.760601 0.649220i \(-0.224904\pi\)
0.760601 + 0.649220i \(0.224904\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 88.3259i − 3.06031i
\(834\) 0 0
\(835\) − 19.3137i − 0.668378i
\(836\) 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
\(845\) 17.0711i 0.587263i
\(846\) 0 0
\(847\) 14.4853i 0.497720i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.28427 −0.146863
\(852\) 0 0
\(853\) −14.2843 −0.489084 −0.244542 0.969639i \(-0.578638\pi\)
−0.244542 + 0.969639i \(0.578638\pi\)
\(854\) 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
\(859\) − 9.65685i − 0.329488i −0.986336 0.164744i \(-0.947320\pi\)
0.986336 0.164744i \(-0.0526797\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.9706 −1.12233 −0.561166 0.827704i \(-0.689647\pi\)
−0.561166 + 0.827704i \(0.689647\pi\)
\(864\) 0 0
\(865\) 70.2843 2.38974
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 18.3431i − 0.622249i
\(870\) 0 0
\(871\) 33.9411i 1.15005i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 27.3137 0.923372
\(876\) 0 0
\(877\) 26.9706 0.910731 0.455366 0.890305i \(-0.349509\pi\)
0.455366 + 0.890305i \(0.349509\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 12.7279i 0.428815i 0.976744 + 0.214407i \(0.0687820\pi\)
−0.976744 + 0.214407i \(0.931218\pi\)
\(882\) 0 0
\(883\) 16.2843i 0.548009i 0.961728 + 0.274005i \(0.0883484\pi\)
−0.961728 + 0.274005i \(0.911652\pi\)
\(884\) 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
\(893\) 70.6274i 2.36346i
\(894\) 0 0
\(895\) 65.9411i 2.20417i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.85786 0.0619632
\(900\) 0 0
\(901\) −27.4558 −0.914687
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.9706i 0.697085i
\(906\) 0 0
\(907\) − 25.6569i − 0.851922i −0.904742 0.425961i \(-0.859936\pi\)
0.904742 0.425961i \(-0.140064\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −55.5980 −1.84204 −0.921022 0.389511i \(-0.872644\pi\)
−0.921022 + 0.389511i \(0.872644\pi\)
\(912\) 0 0
\(913\) 14.6274 0.484097
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 27.3137i 0.901978i
\(918\) 0 0
\(919\) − 27.4558i − 0.905685i −0.891591 0.452842i \(-0.850410\pi\)
0.891591 0.452842i \(-0.149590\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.6863 0.417574
\(924\) 0 0
\(925\) −24.3431 −0.800398
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.72792i 0.286354i 0.989697 + 0.143177i \(0.0457318\pi\)
−0.989697 + 0.143177i \(0.954268\pi\)
\(930\) 0 0
\(931\) 92.2843i 3.02449i
\(932\) 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 \(-0.559476\pi\)
−0.185763 + 0.982595i \(0.559476\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 46.7279i 1.52329i 0.647996 + 0.761643i \(0.275607\pi\)
−0.647996 + 0.761643i \(0.724393\pi\)
\(942\) 0 0
\(943\) − 3.02944i − 0.0986521i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −47.5980 −1.54673 −0.773363 0.633963i \(-0.781427\pi\)
−0.773363 + 0.633963i \(0.781427\pi\)
\(948\) 0 0
\(949\) 11.3137 0.367259
\(950\) 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
\(955\) − 35.3137i − 1.14272i
\(956\) 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
\(965\) − 18.1421i − 0.584016i
\(966\) 0 0
\(967\) 50.0833i 1.61057i 0.592889 + 0.805285i \(0.297987\pi\)
−0.592889 + 0.805285i \(0.702013\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −26.8284 −0.860965 −0.430483 0.902599i \(-0.641657\pi\)
−0.430483 + 0.902599i \(0.641657\pi\)
\(972\) 0 0
\(973\) −57.9411 −1.85751
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 52.5269i 1.68048i 0.542211 + 0.840242i \(0.317587\pi\)
−0.542211 + 0.840242i \(0.682413\pi\)
\(978\) 0 0
\(979\) 34.6274i 1.10670i
\(980\) 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
\(989\) 11.3137i 0.359755i
\(990\) 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
\(995\) 16.4853 0.522619
\(996\) 0 0
\(997\) −61.5980 −1.95083 −0.975414 0.220381i \(-0.929270\pi\)
−0.975414 + 0.220381i \(0.929270\pi\)
\(998\) 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 1152.2.c.a.1151.1 4
3.2 odd 2 1152.2.c.d.1151.4 yes 4
4.3 odd 2 1152.2.c.d.1151.1 yes 4
8.3 odd 2 1152.2.c.c.1151.4 yes 4
8.5 even 2 1152.2.c.b.1151.4 yes 4
12.11 even 2 inner 1152.2.c.a.1151.4 yes 4
16.3 odd 4 2304.2.f.a.1151.1 4
16.5 even 4 2304.2.f.h.1151.4 4
16.11 odd 4 2304.2.f.g.1151.3 4
16.13 even 4 2304.2.f.b.1151.2 4
24.5 odd 2 1152.2.c.c.1151.1 yes 4
24.11 even 2 1152.2.c.b.1151.1 yes 4
48.5 odd 4 2304.2.f.a.1151.2 4
48.11 even 4 2304.2.f.b.1151.1 4
48.29 odd 4 2304.2.f.g.1151.4 4
48.35 even 4 2304.2.f.h.1151.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.c.a.1151.1 4 1.1 even 1 trivial
1152.2.c.a.1151.4 yes 4 12.11 even 2 inner
1152.2.c.b.1151.1 yes 4 24.11 even 2
1152.2.c.b.1151.4 yes 4 8.5 even 2
1152.2.c.c.1151.1 yes 4 24.5 odd 2
1152.2.c.c.1151.4 yes 4 8.3 odd 2
1152.2.c.d.1151.1 yes 4 4.3 odd 2
1152.2.c.d.1151.4 yes 4 3.2 odd 2
2304.2.f.a.1151.1 4 16.3 odd 4
2304.2.f.a.1151.2 4 48.5 odd 4
2304.2.f.b.1151.1 4 48.11 even 4
2304.2.f.b.1151.2 4 16.13 even 4
2304.2.f.g.1151.3 4 16.11 odd 4
2304.2.f.g.1151.4 4 48.29 odd 4
2304.2.f.h.1151.3 4 48.35 even 4
2304.2.f.h.1151.4 4 16.5 even 4