Properties

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

$q$-expansion

\(f(q)\) \(=\) \(q-0.585786 q^{5} -0.828427i q^{7} +O(q^{10})\) \(q-0.585786 q^{5} -0.828427i q^{7} +2.82843i q^{11} +2.82843i q^{13} -2.58579i q^{17} +5.65685 q^{19} -6.82843 q^{23} -4.65685 q^{25} +3.41421 q^{29} +8.82843i q^{31} +0.485281i q^{35} -7.65685i q^{37} +5.41421i q^{41} -1.65685 q^{43} -4.48528 q^{47} +6.31371 q^{49} -9.07107 q^{53} -1.65685i q^{55} +13.6569i q^{59} +3.65685i q^{61} -1.65685i q^{65} +12.0000 q^{67} -12.4853 q^{71} -4.00000 q^{73} +2.34315 q^{77} +10.4853i q^{79} +10.8284i q^{83} +1.51472i q^{85} +3.75736i q^{89} +2.34315 q^{91} -3.31371 q^{95} +2.34315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{5} + O(q^{10}) \) \( 4q - 8q^{5} - 16q^{23} + 4q^{25} + 8q^{29} + 16q^{43} + 16q^{47} - 20q^{49} - 8q^{53} + 48q^{67} - 16q^{71} - 16q^{73} + 32q^{77} + 32q^{91} + 32q^{95} + 32q^{97} + O(q^{100}) \)

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
\(5\) −0.585786 −0.261972 −0.130986 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 0 0
\(7\) − 0.828427i − 0.313116i −0.987669 0.156558i \(-0.949960\pi\)
0.987669 0.156558i \(-0.0500398\pi\)
\(8\) 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
\(17\) − 2.58579i − 0.627145i −0.949564 0.313573i \(-0.898474\pi\)
0.949564 0.313573i \(-0.101526\pi\)
\(18\) 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
\(23\) −6.82843 −1.42383 −0.711913 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.41421 0.634004 0.317002 0.948425i \(-0.397324\pi\)
0.317002 + 0.948425i \(0.397324\pi\)
\(30\) 0 0
\(31\) 8.82843i 1.58563i 0.609461 + 0.792816i \(0.291386\pi\)
−0.609461 + 0.792816i \(0.708614\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.485281i 0.0820275i
\(36\) 0 0
\(37\) − 7.65685i − 1.25878i −0.777090 0.629390i \(-0.783305\pi\)
0.777090 0.629390i \(-0.216695\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.41421i 0.845558i 0.906233 + 0.422779i \(0.138945\pi\)
−0.906233 + 0.422779i \(0.861055\pi\)
\(42\) 0 0
\(43\) −1.65685 −0.252668 −0.126334 0.991988i \(-0.540321\pi\)
−0.126334 + 0.991988i \(0.540321\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.48528 −0.654246 −0.327123 0.944982i \(-0.606079\pi\)
−0.327123 + 0.944982i \(0.606079\pi\)
\(48\) 0 0
\(49\) 6.31371 0.901958
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.07107 −1.24601 −0.623003 0.782219i \(-0.714088\pi\)
−0.623003 + 0.782219i \(0.714088\pi\)
\(54\) 0 0
\(55\) − 1.65685i − 0.223410i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.6569i 1.77797i 0.457935 + 0.888985i \(0.348589\pi\)
−0.457935 + 0.888985i \(0.651411\pi\)
\(60\) 0 0
\(61\) 3.65685i 0.468212i 0.972211 + 0.234106i \(0.0752163\pi\)
−0.972211 + 0.234106i \(0.924784\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1.65685i − 0.205507i
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.4853 −1.48173 −0.740865 0.671654i \(-0.765584\pi\)
−0.740865 + 0.671654i \(0.765584\pi\)
\(72\) 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
\(77\) 2.34315 0.267026
\(78\) 0 0
\(79\) 10.4853i 1.17969i 0.807518 + 0.589843i \(0.200810\pi\)
−0.807518 + 0.589843i \(0.799190\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.8284i 1.18857i 0.804253 + 0.594287i \(0.202566\pi\)
−0.804253 + 0.594287i \(0.797434\pi\)
\(84\) 0 0
\(85\) 1.51472i 0.164294i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.75736i 0.398279i 0.979971 + 0.199140i \(0.0638147\pi\)
−0.979971 + 0.199140i \(0.936185\pi\)
\(90\) 0 0
\(91\) 2.34315 0.245628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.31371 −0.339979
\(96\) 0 0
\(97\) 2.34315 0.237910 0.118955 0.992900i \(-0.462046\pi\)
0.118955 + 0.992900i \(0.462046\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.7279 1.46548 0.732742 0.680507i \(-0.238240\pi\)
0.732742 + 0.680507i \(0.238240\pi\)
\(102\) 0 0
\(103\) 8.82843i 0.869891i 0.900457 + 0.434945i \(0.143232\pi\)
−0.900457 + 0.434945i \(0.856768\pi\)
\(104\) 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
\(109\) 5.17157i 0.495347i 0.968844 + 0.247673i \(0.0796660\pi\)
−0.968844 + 0.247673i \(0.920334\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 3.75736i − 0.353463i −0.984259 0.176731i \(-0.943448\pi\)
0.984259 0.176731i \(-0.0565524\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.14214 −0.196369
\(120\) 0 0
\(121\) 3.00000 0.272727
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 0.828427i 0.0735110i 0.999324 + 0.0367555i \(0.0117023\pi\)
−0.999324 + 0.0367555i \(0.988298\pi\)
\(128\) 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
\(133\) − 4.68629i − 0.406353i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 13.4142i − 1.14605i −0.819537 0.573027i \(-0.805769\pi\)
0.819537 0.573027i \(-0.194231\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 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
\(149\) −18.7279 −1.53425 −0.767126 0.641497i \(-0.778314\pi\)
−0.767126 + 0.641497i \(0.778314\pi\)
\(150\) 0 0
\(151\) 10.4853i 0.853280i 0.904422 + 0.426640i \(0.140303\pi\)
−0.904422 + 0.426640i \(0.859697\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 5.17157i − 0.415391i
\(156\) 0 0
\(157\) − 11.6569i − 0.930318i −0.885227 0.465159i \(-0.845997\pi\)
0.885227 0.465159i \(-0.154003\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65685i 0.445823i
\(162\) 0 0
\(163\) −9.65685 −0.756383 −0.378192 0.925727i \(-0.623454\pi\)
−0.378192 + 0.925727i \(0.623454\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.65685 −0.437741 −0.218870 0.975754i \(-0.570237\pi\)
−0.218870 + 0.975754i \(0.570237\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −23.4142 −1.78015 −0.890075 0.455814i \(-0.849348\pi\)
−0.890075 + 0.455814i \(0.849348\pi\)
\(174\) 0 0
\(175\) 3.85786i 0.291627i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.31371i 0.247678i 0.992302 + 0.123839i \(0.0395207\pi\)
−0.992302 + 0.123839i \(0.960479\pi\)
\(180\) 0 0
\(181\) 22.1421i 1.64581i 0.568178 + 0.822906i \(0.307649\pi\)
−0.568178 + 0.822906i \(0.692351\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.48528i 0.329764i
\(186\) 0 0
\(187\) 7.31371 0.534831
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −21.6569 −1.56703 −0.783517 0.621370i \(-0.786577\pi\)
−0.783517 + 0.621370i \(0.786577\pi\)
\(192\) 0 0
\(193\) −17.3137 −1.24627 −0.623134 0.782115i \(-0.714141\pi\)
−0.623134 + 0.782115i \(0.714141\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.3848 1.45236 0.726178 0.687507i \(-0.241295\pi\)
0.726178 + 0.687507i \(0.241295\pi\)
\(198\) 0 0
\(199\) 0.828427i 0.0587256i 0.999569 + 0.0293628i \(0.00934782\pi\)
−0.999569 + 0.0293628i \(0.990652\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 2.82843i − 0.198517i
\(204\) 0 0
\(205\) − 3.17157i − 0.221512i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000i 1.10674i
\(210\) 0 0
\(211\) 0.686292 0.0472463 0.0236231 0.999721i \(-0.492480\pi\)
0.0236231 + 0.999721i \(0.492480\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.970563 0.0661918
\(216\) 0 0
\(217\) 7.31371 0.496487
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.31371 0.491973
\(222\) 0 0
\(223\) − 21.7990i − 1.45977i −0.683571 0.729884i \(-0.739574\pi\)
0.683571 0.729884i \(-0.260426\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.1716i 1.40521i 0.711582 + 0.702603i \(0.247979\pi\)
−0.711582 + 0.702603i \(0.752021\pi\)
\(228\) 0 0
\(229\) − 10.8284i − 0.715563i −0.933805 0.357781i \(-0.883533\pi\)
0.933805 0.357781i \(-0.116467\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.89949i 0.124440i 0.998062 + 0.0622200i \(0.0198181\pi\)
−0.998062 + 0.0622200i \(0.980182\pi\)
\(234\) 0 0
\(235\) 2.62742 0.171394
\(236\) 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
\(241\) −12.9706 −0.835507 −0.417754 0.908560i \(-0.637183\pi\)
−0.417754 + 0.908560i \(0.637183\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.69848 −0.236288
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 7.51472i − 0.474325i −0.971470 0.237162i \(-0.923783\pi\)
0.971470 0.237162i \(-0.0762174\pi\)
\(252\) 0 0
\(253\) − 19.3137i − 1.21424i
\(254\) 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
\(259\) −6.34315 −0.394144
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 29.6569 1.82872 0.914360 0.404902i \(-0.132694\pi\)
0.914360 + 0.404902i \(0.132694\pi\)
\(264\) 0 0
\(265\) 5.31371 0.326419
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.38478 −0.267345 −0.133672 0.991026i \(-0.542677\pi\)
−0.133672 + 0.991026i \(0.542677\pi\)
\(270\) 0 0
\(271\) − 18.4853i − 1.12290i −0.827510 0.561450i \(-0.810244\pi\)
0.827510 0.561450i \(-0.189756\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 13.1716i − 0.794276i
\(276\) 0 0
\(277\) 26.8284i 1.61196i 0.591940 + 0.805982i \(0.298362\pi\)
−0.591940 + 0.805982i \(0.701638\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 25.8995i − 1.54503i −0.634994 0.772517i \(-0.718997\pi\)
0.634994 0.772517i \(-0.281003\pi\)
\(282\) 0 0
\(283\) 19.3137 1.14808 0.574040 0.818827i \(-0.305375\pi\)
0.574040 + 0.818827i \(0.305375\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.48528 0.264758
\(288\) 0 0
\(289\) 10.3137 0.606689
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.8701 1.45292 0.726462 0.687206i \(-0.241163\pi\)
0.726462 + 0.687206i \(0.241163\pi\)
\(294\) 0 0
\(295\) − 8.00000i − 0.465778i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 19.3137i − 1.11694i
\(300\) 0 0
\(301\) 1.37258i 0.0791144i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 2.14214i − 0.122658i
\(306\) 0 0
\(307\) −23.3137 −1.33058 −0.665292 0.746583i \(-0.731693\pi\)
−0.665292 + 0.746583i \(0.731693\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.34315 0.132868 0.0664338 0.997791i \(-0.478838\pi\)
0.0664338 + 0.997791i \(0.478838\pi\)
\(312\) 0 0
\(313\) 13.3137 0.752535 0.376268 0.926511i \(-0.377207\pi\)
0.376268 + 0.926511i \(0.377207\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.75736 −0.323366 −0.161683 0.986843i \(-0.551692\pi\)
−0.161683 + 0.986843i \(0.551692\pi\)
\(318\) 0 0
\(319\) 9.65685i 0.540680i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 14.6274i − 0.813891i
\(324\) 0 0
\(325\) − 13.1716i − 0.730627i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.71573i 0.204855i
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.02944 −0.384059
\(336\) 0 0
\(337\) 0.686292 0.0373847 0.0186923 0.999825i \(-0.494050\pi\)
0.0186923 + 0.999825i \(0.494050\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −24.9706 −1.35223
\(342\) 0 0
\(343\) − 11.0294i − 0.595534i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 0.485281i − 0.0260513i −0.999915 0.0130256i \(-0.995854\pi\)
0.999915 0.0130256i \(-0.00414631\pi\)
\(348\) 0 0
\(349\) 0.343146i 0.0183682i 0.999958 + 0.00918409i \(0.00292343\pi\)
−0.999958 + 0.00918409i \(0.997077\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.8701i 1.85595i 0.372648 + 0.927973i \(0.378450\pi\)
−0.372648 + 0.927973i \(0.621550\pi\)
\(354\) 0 0
\(355\) 7.31371 0.388171
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.51472 −0.185500 −0.0927499 0.995689i \(-0.529566\pi\)
−0.0927499 + 0.995689i \(0.529566\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.34315 0.122646
\(366\) 0 0
\(367\) − 15.1716i − 0.791950i −0.918261 0.395975i \(-0.870407\pi\)
0.918261 0.395975i \(-0.129593\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.51472i 0.390145i
\(372\) 0 0
\(373\) 18.9706i 0.982259i 0.871087 + 0.491129i \(0.163416\pi\)
−0.871087 + 0.491129i \(0.836584\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.65685i 0.497353i
\(378\) 0 0
\(379\) −36.2843 −1.86380 −0.931899 0.362718i \(-0.881849\pi\)
−0.931899 + 0.362718i \(0.881849\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 30.6274 1.56499 0.782494 0.622658i \(-0.213947\pi\)
0.782494 + 0.622658i \(0.213947\pi\)
\(384\) 0 0
\(385\) −1.37258 −0.0699533
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 33.5563 1.70137 0.850687 0.525672i \(-0.176186\pi\)
0.850687 + 0.525672i \(0.176186\pi\)
\(390\) 0 0
\(391\) 17.6569i 0.892946i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 6.14214i − 0.309044i
\(396\) 0 0
\(397\) 14.9706i 0.751351i 0.926751 + 0.375676i \(0.122589\pi\)
−0.926751 + 0.375676i \(0.877411\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 10.5858i − 0.528629i −0.964437 0.264314i \(-0.914854\pi\)
0.964437 0.264314i \(-0.0851457\pi\)
\(402\) 0 0
\(403\) −24.9706 −1.24387
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.6569 1.07349
\(408\) 0 0
\(409\) −21.6569 −1.07086 −0.535431 0.844579i \(-0.679851\pi\)
−0.535431 + 0.844579i \(0.679851\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.3137 0.556711
\(414\) 0 0
\(415\) − 6.34315i − 0.311373i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 27.7990i − 1.35807i −0.734106 0.679035i \(-0.762399\pi\)
0.734106 0.679035i \(-0.237601\pi\)
\(420\) 0 0
\(421\) − 24.4853i − 1.19334i −0.802487 0.596670i \(-0.796490\pi\)
0.802487 0.596670i \(-0.203510\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.0416i 0.584105i
\(426\) 0 0
\(427\) 3.02944 0.146605
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.5147 0.554644 0.277322 0.960777i \(-0.410553\pi\)
0.277322 + 0.960777i \(0.410553\pi\)
\(432\) 0 0
\(433\) −16.6274 −0.799063 −0.399531 0.916720i \(-0.630827\pi\)
−0.399531 + 0.916720i \(0.630827\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −38.6274 −1.84780
\(438\) 0 0
\(439\) − 36.1421i − 1.72497i −0.506083 0.862485i \(-0.668907\pi\)
0.506083 0.862485i \(-0.331093\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 33.4558i − 1.58954i −0.606914 0.794768i \(-0.707593\pi\)
0.606914 0.794768i \(-0.292407\pi\)
\(444\) 0 0
\(445\) − 2.20101i − 0.104338i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 40.7279i − 1.92207i −0.276428 0.961035i \(-0.589151\pi\)
0.276428 0.961035i \(-0.410849\pi\)
\(450\) 0 0
\(451\) −15.3137 −0.721094
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.37258 −0.0643477
\(456\) 0 0
\(457\) −24.9706 −1.16807 −0.584037 0.811727i \(-0.698528\pi\)
−0.584037 + 0.811727i \(0.698528\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −27.8995 −1.29941 −0.649705 0.760187i \(-0.725107\pi\)
−0.649705 + 0.760187i \(0.725107\pi\)
\(462\) 0 0
\(463\) 5.79899i 0.269502i 0.990879 + 0.134751i \(0.0430234\pi\)
−0.990879 + 0.134751i \(0.956977\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 22.1421i − 1.02462i −0.858802 0.512308i \(-0.828791\pi\)
0.858802 0.512308i \(-0.171209\pi\)
\(468\) 0 0
\(469\) − 9.94113i − 0.459039i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 4.68629i − 0.215476i
\(474\) 0 0
\(475\) −26.3431 −1.20871
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.8284 0.677528 0.338764 0.940871i \(-0.389991\pi\)
0.338764 + 0.940871i \(0.389991\pi\)
\(480\) 0 0
\(481\) 21.6569 0.987468
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.37258 −0.0623258
\(486\) 0 0
\(487\) − 23.1716i − 1.05000i −0.851101 0.525002i \(-0.824065\pi\)
0.851101 0.525002i \(-0.175935\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.6863i 0.572524i 0.958151 + 0.286262i \(0.0924128\pi\)
−0.958151 + 0.286262i \(0.907587\pi\)
\(492\) 0 0
\(493\) − 8.82843i − 0.397612i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.3431i 0.463953i
\(498\) 0 0
\(499\) 38.6274 1.72920 0.864600 0.502460i \(-0.167572\pi\)
0.864600 + 0.502460i \(0.167572\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 23.7990 1.06114 0.530572 0.847640i \(-0.321977\pi\)
0.530572 + 0.847640i \(0.321977\pi\)
\(504\) 0 0
\(505\) −8.62742 −0.383915
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 38.7279 1.71658 0.858292 0.513161i \(-0.171526\pi\)
0.858292 + 0.513161i \(0.171526\pi\)
\(510\) 0 0
\(511\) 3.31371i 0.146590i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 5.17157i − 0.227887i
\(516\) 0 0
\(517\) − 12.6863i − 0.557942i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 11.0711i − 0.485032i −0.970147 0.242516i \(-0.922027\pi\)
0.970147 0.242516i \(-0.0779727\pi\)
\(522\) 0 0
\(523\) −34.3431 −1.50172 −0.750860 0.660461i \(-0.770361\pi\)
−0.750860 + 0.660461i \(0.770361\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.8284 0.994422
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15.3137 −0.663310
\(534\) 0 0
\(535\) − 6.62742i − 0.286528i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17.8579i 0.769193i
\(540\) 0 0
\(541\) 3.79899i 0.163331i 0.996660 + 0.0816657i \(0.0260240\pi\)
−0.996660 + 0.0816657i \(0.973976\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 3.02944i − 0.129767i
\(546\) 0 0
\(547\) 1.65685 0.0708420 0.0354210 0.999372i \(-0.488723\pi\)
0.0354210 + 0.999372i \(0.488723\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19.3137 0.822792
\(552\) 0 0
\(553\) 8.68629 0.369379
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.07107 0.214868 0.107434 0.994212i \(-0.465737\pi\)
0.107434 + 0.994212i \(0.465737\pi\)
\(558\) 0 0
\(559\) − 4.68629i − 0.198209i
\(560\) 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
\(565\) 2.20101i 0.0925972i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 21.8995i 0.918075i 0.888417 + 0.459037i \(0.151806\pi\)
−0.888417 + 0.459037i \(0.848194\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 31.7990 1.32611
\(576\) 0 0
\(577\) −6.68629 −0.278354 −0.139177 0.990268i \(-0.544446\pi\)
−0.139177 + 0.990268i \(0.544446\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.97056 0.372162
\(582\) 0 0
\(583\) − 25.6569i − 1.06260i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 18.3431i − 0.757103i −0.925580 0.378551i \(-0.876422\pi\)
0.925580 0.378551i \(-0.123578\pi\)
\(588\) 0 0
\(589\) 49.9411i 2.05779i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 0.928932i − 0.0381467i −0.999818 0.0190733i \(-0.993928\pi\)
0.999818 0.0190733i \(-0.00607160\pi\)
\(594\) 0 0
\(595\) 1.25483 0.0514432
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −27.5147 −1.12422 −0.562110 0.827062i \(-0.690010\pi\)
−0.562110 + 0.827062i \(0.690010\pi\)
\(600\) 0 0
\(601\) 1.31371 0.0535873 0.0267936 0.999641i \(-0.491470\pi\)
0.0267936 + 0.999641i \(0.491470\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.75736 −0.0714468
\(606\) 0 0
\(607\) 24.8284i 1.00775i 0.863775 + 0.503877i \(0.168094\pi\)
−0.863775 + 0.503877i \(0.831906\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 12.6863i − 0.513232i
\(612\) 0 0
\(613\) − 0.343146i − 0.0138595i −0.999976 0.00692976i \(-0.997794\pi\)
0.999976 0.00692976i \(-0.00220583\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.4142i 1.34521i 0.740004 + 0.672603i \(0.234824\pi\)
−0.740004 + 0.672603i \(0.765176\pi\)
\(618\) 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
\(623\) 3.11270 0.124708
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −19.7990 −0.789437
\(630\) 0 0
\(631\) − 18.4853i − 0.735887i −0.929848 0.367944i \(-0.880062\pi\)
0.929848 0.367944i \(-0.119938\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 0.485281i − 0.0192578i
\(636\) 0 0
\(637\) 17.8579i 0.707554i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 14.8701i − 0.587332i −0.955908 0.293666i \(-0.905125\pi\)
0.955908 0.293666i \(-0.0948753\pi\)
\(642\) 0 0
\(643\) −9.65685 −0.380829 −0.190415 0.981704i \(-0.560983\pi\)
−0.190415 + 0.981704i \(0.560983\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.7990 −0.621122 −0.310561 0.950553i \(-0.600517\pi\)
−0.310561 + 0.950553i \(0.600517\pi\)
\(648\) 0 0
\(649\) −38.6274 −1.51626
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.27208 0.206312 0.103156 0.994665i \(-0.467106\pi\)
0.103156 + 0.994665i \(0.467106\pi\)
\(654\) 0 0
\(655\) − 3.31371i − 0.129477i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 8.97056i − 0.349444i −0.984618 0.174722i \(-0.944097\pi\)
0.984618 0.174722i \(-0.0559026\pi\)
\(660\) 0 0
\(661\) 3.65685i 0.142235i 0.997468 + 0.0711176i \(0.0226566\pi\)
−0.997468 + 0.0711176i \(0.977343\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.74517i 0.106453i
\(666\) 0 0
\(667\) −23.3137 −0.902710
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.3431 −0.399293
\(672\) 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
\(677\) 18.7279 0.719773 0.359886 0.932996i \(-0.382815\pi\)
0.359886 + 0.932996i \(0.382815\pi\)
\(678\) 0 0
\(679\) − 1.94113i − 0.0744936i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 22.1421i − 0.847245i −0.905839 0.423623i \(-0.860758\pi\)
0.905839 0.423623i \(-0.139242\pi\)
\(684\) 0 0
\(685\) 7.85786i 0.300234i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 25.6569i − 0.977448i
\(690\) 0 0
\(691\) −3.02944 −0.115245 −0.0576226 0.998338i \(-0.518352\pi\)
−0.0576226 + 0.998338i \(0.518352\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.02944 −0.266642
\(696\) 0 0
\(697\) 14.0000 0.530288
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.7574 0.519608 0.259804 0.965661i \(-0.416342\pi\)
0.259804 + 0.965661i \(0.416342\pi\)
\(702\) 0 0
\(703\) − 43.3137i − 1.63361i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 12.2010i − 0.458866i
\(708\) 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
\(713\) − 60.2843i − 2.25766i
\(714\) 0 0
\(715\) 4.68629 0.175257
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 28.4853 1.06232 0.531161 0.847271i \(-0.321756\pi\)
0.531161 + 0.847271i \(0.321756\pi\)
\(720\) 0 0
\(721\) 7.31371 0.272377
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −15.8995 −0.590492
\(726\) 0 0
\(727\) 8.82843i 0.327428i 0.986508 + 0.163714i \(0.0523475\pi\)
−0.986508 + 0.163714i \(0.947653\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.28427i 0.158459i
\(732\) 0 0
\(733\) − 24.4853i − 0.904385i −0.891920 0.452192i \(-0.850642\pi\)
0.891920 0.452192i \(-0.149358\pi\)
\(734\) 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 \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.9706 0.622590 0.311295 0.950313i \(-0.399237\pi\)
0.311295 + 0.950313i \(0.399237\pi\)
\(744\) 0 0
\(745\) 10.9706 0.401930
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.37258 0.342467
\(750\) 0 0
\(751\) 40.8284i 1.48985i 0.667148 + 0.744925i \(0.267515\pi\)
−0.667148 + 0.744925i \(0.732485\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 6.14214i − 0.223535i
\(756\) 0 0
\(757\) 0.485281i 0.0176379i 0.999961 + 0.00881893i \(0.00280719\pi\)
−0.999961 + 0.00881893i \(0.997193\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 25.6985i 0.931569i 0.884898 + 0.465785i \(0.154228\pi\)
−0.884898 + 0.465785i \(0.845772\pi\)
\(762\) 0 0
\(763\) 4.28427 0.155101
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −38.6274 −1.39476
\(768\) 0 0
\(769\) −21.3137 −0.768592 −0.384296 0.923210i \(-0.625556\pi\)
−0.384296 + 0.923210i \(0.625556\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.5858 0.452679 0.226340 0.974048i \(-0.427324\pi\)
0.226340 + 0.974048i \(0.427324\pi\)
\(774\) 0 0
\(775\) − 41.1127i − 1.47681i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 30.6274i 1.09734i
\(780\) 0 0
\(781\) − 35.3137i − 1.26362i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.82843i 0.243717i
\(786\) 0 0
\(787\) 26.3431 0.939032 0.469516 0.882924i \(-0.344428\pi\)
0.469516 + 0.882924i \(0.344428\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.11270 −0.110675
\(792\) 0 0
\(793\) −10.3431 −0.367296
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −13.0711 −0.463001 −0.231500 0.972835i \(-0.574363\pi\)
−0.231500 + 0.972835i \(0.574363\pi\)
\(798\) 0 0
\(799\) 11.5980i 0.410307i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 11.3137i − 0.399252i
\(804\) 0 0
\(805\) − 3.31371i − 0.116793i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 42.6690i 1.50016i 0.661345 + 0.750082i \(0.269986\pi\)
−0.661345 + 0.750082i \(0.730014\pi\)
\(810\) 0 0
\(811\) 31.5980 1.10956 0.554778 0.831999i \(-0.312803\pi\)
0.554778 + 0.831999i \(0.312803\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.65685 0.198151
\(816\) 0 0
\(817\) −9.37258 −0.327905
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11.4142 0.398359 0.199179 0.979963i \(-0.436172\pi\)
0.199179 + 0.979963i \(0.436172\pi\)
\(822\) 0 0
\(823\) 52.4264i 1.82747i 0.406311 + 0.913735i \(0.366815\pi\)
−0.406311 + 0.913735i \(0.633185\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.0294i 0.522625i 0.965254 + 0.261312i \(0.0841553\pi\)
−0.965254 + 0.261312i \(0.915845\pi\)
\(828\) 0 0
\(829\) − 4.20101i − 0.145907i −0.997335 0.0729536i \(-0.976758\pi\)
0.997335 0.0729536i \(-0.0232425\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 16.3259i − 0.565659i
\(834\) 0 0
\(835\) 3.31371 0.114676
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.51472 −0.121342 −0.0606708 0.998158i \(-0.519324\pi\)
−0.0606708 + 0.998158i \(0.519324\pi\)
\(840\) 0 0
\(841\) −17.3431 −0.598040
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.92893 −0.100758
\(846\) 0 0
\(847\) − 2.48528i − 0.0853953i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 52.2843i 1.79228i
\(852\) 0 0
\(853\) 42.2843i 1.44779i 0.689912 + 0.723893i \(0.257649\pi\)
−0.689912 + 0.723893i \(0.742351\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 21.8995i − 0.748072i −0.927414 0.374036i \(-0.877974\pi\)
0.927414 0.374036i \(-0.122026\pi\)
\(858\) 0 0
\(859\) −1.65685 −0.0565311 −0.0282656 0.999600i \(-0.508998\pi\)
−0.0282656 + 0.999600i \(0.508998\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.970563 −0.0330383 −0.0165192 0.999864i \(-0.505258\pi\)
−0.0165192 + 0.999864i \(0.505258\pi\)
\(864\) 0 0
\(865\) 13.7157 0.466349
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −29.6569 −1.00604
\(870\) 0 0
\(871\) 33.9411i 1.15005i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 4.68629i − 0.158426i
\(876\) 0 0
\(877\) 6.97056i 0.235379i 0.993050 + 0.117690i \(0.0375488\pi\)
−0.993050 + 0.117690i \(0.962451\pi\)
\(878\) 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
\(883\) −40.2843 −1.35567 −0.677837 0.735212i \(-0.737082\pi\)
−0.677837 + 0.735212i \(0.737082\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.34315 −0.0786751 −0.0393376 0.999226i \(-0.512525\pi\)
−0.0393376 + 0.999226i \(0.512525\pi\)
\(888\) 0 0
\(889\) 0.686292 0.0230175
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −25.3726 −0.849061
\(894\) 0 0
\(895\) − 1.94113i − 0.0648847i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 30.1421i 1.00530i
\(900\) 0 0
\(901\) 23.4558i 0.781427i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 12.9706i − 0.431156i
\(906\) 0 0
\(907\) 14.3431 0.476256 0.238128 0.971234i \(-0.423466\pi\)
0.238128 + 0.971234i \(0.423466\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −23.5980 −0.781836 −0.390918 0.920426i \(-0.627842\pi\)
−0.390918 + 0.920426i \(0.627842\pi\)
\(912\) 0 0
\(913\) −30.6274 −1.01362
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.68629 0.154755
\(918\) 0 0
\(919\) − 23.4558i − 0.773737i −0.922135 0.386868i \(-0.873557\pi\)
0.922135 0.386868i \(-0.126443\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 35.3137i − 1.16236i
\(924\) 0 0
\(925\) 35.6569i 1.17239i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 16.7279i 0.548825i 0.961612 + 0.274413i \(0.0884834\pi\)
−0.961612 + 0.274413i \(0.911517\pi\)
\(930\) 0 0
\(931\) 35.7157 1.17054
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.28427 −0.140111
\(936\) 0 0
\(937\) 56.6274 1.84994 0.924969 0.380044i \(-0.124091\pi\)
0.924969 + 0.380044i \(0.124091\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −21.2721 −0.693450 −0.346725 0.937967i \(-0.612706\pi\)
−0.346725 + 0.937967i \(0.612706\pi\)
\(942\) 0 0
\(943\) − 36.9706i − 1.20393i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31.5980i 1.02680i 0.858151 + 0.513398i \(0.171614\pi\)
−0.858151 + 0.513398i \(0.828386\pi\)
\(948\) 0 0
\(949\) − 11.3137i − 0.367259i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 4.44365i − 0.143944i −0.997407 0.0719720i \(-0.977071\pi\)
0.997407 0.0719720i \(-0.0229292\pi\)
\(954\) 0 0
\(955\) 12.6863 0.410519
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.1127 −0.358848
\(960\) 0 0
\(961\) −46.9411 −1.51423
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 10.1421 0.326487
\(966\) 0 0
\(967\) 46.0833i 1.48194i 0.671539 + 0.740969i \(0.265633\pi\)
−0.671539 + 0.740969i \(0.734367\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 21.1716i 0.679428i 0.940529 + 0.339714i \(0.110330\pi\)
−0.940529 + 0.339714i \(0.889670\pi\)
\(972\) 0 0
\(973\) − 9.94113i − 0.318698i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.5269i 0.400771i 0.979717 + 0.200386i \(0.0642195\pi\)
−0.979717 + 0.200386i \(0.935780\pi\)
\(978\) 0 0
\(979\) −10.6274 −0.339654
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 34.3431 1.09538 0.547688 0.836683i \(-0.315508\pi\)
0.547688 + 0.836683i \(0.315508\pi\)
\(984\) 0 0
\(985\) −11.9411 −0.380476
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.3137 0.359755
\(990\) 0 0
\(991\) − 13.7990i − 0.438339i −0.975687 0.219170i \(-0.929665\pi\)
0.975687 0.219170i \(-0.0703348\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 0.485281i − 0.0153845i
\(996\) 0 0
\(997\) 17.5980i 0.557334i 0.960388 + 0.278667i \(0.0898925\pi\)
−0.960388 + 0.278667i \(0.910107\pi\)
\(998\) 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.a.1151.3 4
3.2 odd 2 2304.2.f.h.1151.1 4
4.3 odd 2 2304.2.f.b.1151.4 4
8.3 odd 2 2304.2.f.h.1151.2 4
8.5 even 2 2304.2.f.g.1151.1 4
12.11 even 2 2304.2.f.g.1151.2 4
16.3 odd 4 1152.2.c.a.1151.3 yes 4
16.5 even 4 1152.2.c.c.1151.2 yes 4
16.11 odd 4 1152.2.c.b.1151.2 yes 4
16.13 even 4 1152.2.c.d.1151.3 yes 4
24.5 odd 2 2304.2.f.b.1151.3 4
24.11 even 2 inner 2304.2.f.a.1151.4 4
48.5 odd 4 1152.2.c.b.1151.3 yes 4
48.11 even 4 1152.2.c.c.1151.3 yes 4
48.29 odd 4 1152.2.c.a.1151.2 4
48.35 even 4 1152.2.c.d.1151.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.c.a.1151.2 4 48.29 odd 4
1152.2.c.a.1151.3 yes 4 16.3 odd 4
1152.2.c.b.1151.2 yes 4 16.11 odd 4
1152.2.c.b.1151.3 yes 4 48.5 odd 4
1152.2.c.c.1151.2 yes 4 16.5 even 4
1152.2.c.c.1151.3 yes 4 48.11 even 4
1152.2.c.d.1151.2 yes 4 48.35 even 4
1152.2.c.d.1151.3 yes 4 16.13 even 4
2304.2.f.a.1151.3 4 1.1 even 1 trivial
2304.2.f.a.1151.4 4 24.11 even 2 inner
2304.2.f.b.1151.3 4 24.5 odd 2
2304.2.f.b.1151.4 4 4.3 odd 2
2304.2.f.g.1151.1 4 8.5 even 2
2304.2.f.g.1151.2 4 12.11 even 2
2304.2.f.h.1151.1 4 3.2 odd 2
2304.2.f.h.1151.2 4 8.3 odd 2