Properties

Label 2304.2.c.h.2303.1
Level $2304$
Weight $2$
Character 2304.2303
Analytic conductor $18.398$
Analytic rank $0$
Dimension $2$
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2303.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2303
Dual form 2304.2.c.h.2303.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{5} -2.82843i q^{7} +O(q^{10})\) \(q-1.41421i q^{5} -2.82843i q^{7} +4.00000 q^{11} +2.00000 q^{13} +1.41421i q^{17} +5.65685i q^{19} +4.00000 q^{23} +3.00000 q^{25} -7.07107i q^{29} +8.48528i q^{31} -4.00000 q^{35} +8.00000 q^{37} +4.24264i q^{41} -11.3137i q^{43} -12.0000 q^{47} -1.00000 q^{49} -12.7279i q^{53} -5.65685i q^{55} +8.00000 q^{61} -2.82843i q^{65} +5.65685i q^{67} +4.00000 q^{71} -8.00000 q^{73} -11.3137i q^{77} +2.82843i q^{79} -12.0000 q^{83} +2.00000 q^{85} -15.5563i q^{89} -5.65685i q^{91} +8.00000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 8q^{11} + 4q^{13} + 8q^{23} + 6q^{25} - 8q^{35} + 16q^{37} - 24q^{47} - 2q^{49} + 16q^{61} + 8q^{71} - 16q^{73} - 24q^{83} + 4q^{85} + 16q^{95} + O(q^{100}) \)

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
\(5\) − 1.41421i − 0.632456i −0.948683 0.316228i \(-0.897584\pi\)
0.948683 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) 5.65685i 1.29777i 0.760886 + 0.648886i \(0.224765\pi\)
−0.760886 + 0.648886i \(0.775235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 7.07107i − 1.31306i −0.754298 0.656532i \(-0.772023\pi\)
0.754298 0.656532i \(-0.227977\pi\)
\(30\) 0 0
\(31\) 8.48528i 1.52400i 0.647576 + 0.762001i \(0.275783\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.24264i 0.662589i 0.943527 + 0.331295i \(0.107485\pi\)
−0.943527 + 0.331295i \(0.892515\pi\)
\(42\) 0 0
\(43\) − 11.3137i − 1.72532i −0.505781 0.862662i \(-0.668795\pi\)
0.505781 0.862662i \(-0.331205\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 12.7279i − 1.74831i −0.485643 0.874157i \(-0.661414\pi\)
0.485643 0.874157i \(-0.338586\pi\)
\(54\) 0 0
\(55\) − 5.65685i − 0.762770i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 2.82843i − 0.350823i
\(66\) 0 0
\(67\) 5.65685i 0.691095i 0.938401 + 0.345547i \(0.112307\pi\)
−0.938401 + 0.345547i \(0.887693\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 11.3137i − 1.28932i
\(78\) 0 0
\(79\) 2.82843i 0.318223i 0.987261 + 0.159111i \(0.0508629\pi\)
−0.987261 + 0.159111i \(0.949137\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 15.5563i − 1.64897i −0.565884 0.824485i \(-0.691465\pi\)
0.565884 0.824485i \(-0.308535\pi\)
\(90\) 0 0
\(91\) − 5.65685i − 0.592999i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 9.89949i − 0.985037i −0.870302 0.492518i \(-0.836076\pi\)
0.870302 0.492518i \(-0.163924\pi\)
\(102\) 0 0
\(103\) − 8.48528i − 0.836080i −0.908429 0.418040i \(-0.862717\pi\)
0.908429 0.418040i \(-0.137283\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.41421i 0.133038i 0.997785 + 0.0665190i \(0.0211893\pi\)
−0.997785 + 0.0665190i \(0.978811\pi\)
\(114\) 0 0
\(115\) − 5.65685i − 0.527504i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.3137i − 1.01193i
\(126\) 0 0
\(127\) − 2.82843i − 0.250982i −0.992095 0.125491i \(-0.959949\pi\)
0.992095 0.125491i \(-0.0400507\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.3848i 1.57072i 0.619041 + 0.785359i \(0.287521\pi\)
−0.619041 + 0.785359i \(0.712479\pi\)
\(138\) 0 0
\(139\) 16.9706i 1.43942i 0.694273 + 0.719712i \(0.255726\pi\)
−0.694273 + 0.719712i \(0.744274\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 9.89949i − 0.810998i −0.914095 0.405499i \(-0.867098\pi\)
0.914095 0.405499i \(-0.132902\pi\)
\(150\) 0 0
\(151\) 8.48528i 0.690522i 0.938507 + 0.345261i \(0.112210\pi\)
−0.938507 + 0.345261i \(0.887790\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 11.3137i − 0.891645i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 7.07107i − 0.537603i −0.963196 0.268802i \(-0.913372\pi\)
0.963196 0.268802i \(-0.0866276\pi\)
\(174\) 0 0
\(175\) − 8.48528i − 0.641427i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 11.3137i − 0.831800i
\(186\) 0 0
\(187\) 5.65685i 0.413670i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.41421i 0.100759i 0.998730 + 0.0503793i \(0.0160430\pi\)
−0.998730 + 0.0503793i \(0.983957\pi\)
\(198\) 0 0
\(199\) 25.4558i 1.80452i 0.431196 + 0.902258i \(0.358092\pi\)
−0.431196 + 0.902258i \(0.641908\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −20.0000 −1.40372
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.6274i 1.56517i
\(210\) 0 0
\(211\) − 5.65685i − 0.389434i −0.980859 0.194717i \(-0.937621\pi\)
0.980859 0.194717i \(-0.0623788\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.82843i 0.190261i
\(222\) 0 0
\(223\) − 14.1421i − 0.947027i −0.880786 0.473514i \(-0.842985\pi\)
0.880786 0.473514i \(-0.157015\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.5563i 1.01913i 0.860432 + 0.509565i \(0.170194\pi\)
−0.860432 + 0.509565i \(0.829806\pi\)
\(234\) 0 0
\(235\) 16.9706i 1.10704i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.41421i 0.0903508i
\(246\) 0 0
\(247\) 11.3137i 0.719874i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1.41421i − 0.0882162i −0.999027 0.0441081i \(-0.985955\pi\)
0.999027 0.0441081i \(-0.0140446\pi\)
\(258\) 0 0
\(259\) − 22.6274i − 1.40600i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −18.0000 −1.10573
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.07107i 0.431131i 0.976489 + 0.215565i \(0.0691594\pi\)
−0.976489 + 0.215565i \(0.930841\pi\)
\(270\) 0 0
\(271\) − 2.82843i − 0.171815i −0.996303 0.0859074i \(-0.972621\pi\)
0.996303 0.0859074i \(-0.0273789\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 26.8701i − 1.60293i −0.598040 0.801467i \(-0.704053\pi\)
0.598040 0.801467i \(-0.295947\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 1.41421i − 0.0826192i −0.999146 0.0413096i \(-0.986847\pi\)
0.999146 0.0413096i \(-0.0131530\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 11.3137i − 0.647821i
\(306\) 0 0
\(307\) − 16.9706i − 0.968561i −0.874913 0.484281i \(-0.839081\pi\)
0.874913 0.484281i \(-0.160919\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.3848i 1.03259i 0.856410 + 0.516296i \(0.172690\pi\)
−0.856410 + 0.516296i \(0.827310\pi\)
\(318\) 0 0
\(319\) − 28.2843i − 1.58362i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 33.9411i 1.87123i
\(330\) 0 0
\(331\) − 5.65685i − 0.310929i −0.987841 0.155464i \(-0.950313\pi\)
0.987841 0.155464i \(-0.0496874\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −24.0000 −1.30736 −0.653682 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.9411i 1.83801i
\(342\) 0 0
\(343\) − 16.9706i − 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 9.89949i − 0.526897i −0.964673 0.263448i \(-0.915140\pi\)
0.964673 0.263448i \(-0.0848599\pi\)
\(354\) 0 0
\(355\) − 5.65685i − 0.300235i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.3137i 0.592187i
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) 0 0
\(373\) −24.0000 −1.24267 −0.621336 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 14.1421i − 0.728357i
\(378\) 0 0
\(379\) 16.9706i 0.871719i 0.900015 + 0.435860i \(0.143556\pi\)
−0.900015 + 0.435860i \(0.856444\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.7279i 0.645331i 0.946513 + 0.322666i \(0.104579\pi\)
−0.946513 + 0.322666i \(0.895421\pi\)
\(390\) 0 0
\(391\) 5.65685i 0.286079i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0416i 1.20058i 0.799782 + 0.600291i \(0.204949\pi\)
−0.799782 + 0.600291i \(0.795051\pi\)
\(402\) 0 0
\(403\) 16.9706i 0.845364i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 32.0000 1.58618
\(408\) 0 0
\(409\) 16.0000 0.791149 0.395575 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 16.9706i 0.833052i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.24264i 0.205798i
\(426\) 0 0
\(427\) − 22.6274i − 1.09502i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.6274i 1.08242i
\(438\) 0 0
\(439\) 8.48528i 0.404980i 0.979284 + 0.202490i \(0.0649034\pi\)
−0.979284 + 0.202490i \(0.935097\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) −22.0000 −1.04290
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 12.7279i − 0.600668i −0.953834 0.300334i \(-0.902902\pi\)
0.953834 0.300334i \(-0.0970981\pi\)
\(450\) 0 0
\(451\) 16.9706i 0.799113i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 29.6985i 1.38320i 0.722282 + 0.691598i \(0.243093\pi\)
−0.722282 + 0.691598i \(0.756907\pi\)
\(462\) 0 0
\(463\) 25.4558i 1.18303i 0.806293 + 0.591517i \(0.201471\pi\)
−0.806293 + 0.591517i \(0.798529\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 45.2548i − 2.08082i
\(474\) 0 0
\(475\) 16.9706i 0.778663i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 8.48528i − 0.384505i −0.981346 0.192252i \(-0.938421\pi\)
0.981346 0.192252i \(-0.0615792\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 10.0000 0.450377
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 11.3137i − 0.507489i
\(498\) 0 0
\(499\) 22.6274i 1.01294i 0.862257 + 0.506471i \(0.169050\pi\)
−0.862257 + 0.506471i \(0.830950\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 38.1838i 1.69247i 0.532813 + 0.846233i \(0.321135\pi\)
−0.532813 + 0.846233i \(0.678865\pi\)
\(510\) 0 0
\(511\) 22.6274i 1.00098i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −48.0000 −2.11104
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.6985i 1.30111i 0.759457 + 0.650557i \(0.225465\pi\)
−0.759457 + 0.650557i \(0.774535\pi\)
\(522\) 0 0
\(523\) − 39.5980i − 1.73150i −0.500478 0.865749i \(-0.666842\pi\)
0.500478 0.865749i \(-0.333158\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.48528i 0.367538i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 19.7990i − 0.848096i
\(546\) 0 0
\(547\) 33.9411i 1.45122i 0.688107 + 0.725609i \(0.258442\pi\)
−0.688107 + 0.725609i \(0.741558\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 40.0000 1.70406
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.3848i 0.778988i 0.921029 + 0.389494i \(0.127350\pi\)
−0.921029 + 0.389494i \(0.872650\pi\)
\(558\) 0 0
\(559\) − 22.6274i − 0.957038i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.07107i 0.296435i 0.988955 + 0.148217i \(0.0473535\pi\)
−0.988955 + 0.148217i \(0.952646\pi\)
\(570\) 0 0
\(571\) − 33.9411i − 1.42039i −0.704004 0.710196i \(-0.748606\pi\)
0.704004 0.710196i \(-0.251394\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 33.9411i 1.40812i
\(582\) 0 0
\(583\) − 50.9117i − 2.10855i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 32.0000 1.32078 0.660391 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 0 0
\(589\) −48.0000 −1.97781
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 43.8406i 1.80032i 0.435561 + 0.900159i \(0.356550\pi\)
−0.435561 + 0.900159i \(0.643450\pi\)
\(594\) 0 0
\(595\) − 5.65685i − 0.231908i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 46.0000 1.87638 0.938190 0.346122i \(-0.112502\pi\)
0.938190 + 0.346122i \(0.112502\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 7.07107i − 0.287480i
\(606\) 0 0
\(607\) 19.7990i 0.803616i 0.915724 + 0.401808i \(0.131618\pi\)
−0.915724 + 0.401808i \(0.868382\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.07107i 0.284670i 0.989819 + 0.142335i \(0.0454611\pi\)
−0.989819 + 0.142335i \(0.954539\pi\)
\(618\) 0 0
\(619\) 33.9411i 1.36421i 0.731255 + 0.682105i \(0.238935\pi\)
−0.731255 + 0.682105i \(0.761065\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −44.0000 −1.76282
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.3137i 0.451107i
\(630\) 0 0
\(631\) − 42.4264i − 1.68897i −0.535580 0.844484i \(-0.679907\pi\)
0.535580 0.844484i \(-0.320093\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.41421i 0.0558581i 0.999610 + 0.0279290i \(0.00889125\pi\)
−0.999610 + 0.0279290i \(0.991109\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.00000 −0.157256 −0.0786281 0.996904i \(-0.525054\pi\)
−0.0786281 + 0.996904i \(0.525054\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 29.6985i − 1.16219i −0.813835 0.581096i \(-0.802624\pi\)
0.813835 0.581096i \(-0.197376\pi\)
\(654\) 0 0
\(655\) − 11.3137i − 0.442063i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 22.6274i − 0.877454i
\(666\) 0 0
\(667\) − 28.2843i − 1.09517i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 32.0000 1.23535
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.2132i 0.815290i 0.913141 + 0.407645i \(0.133650\pi\)
−0.913141 + 0.407645i \(0.866350\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 26.0000 0.993409
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 25.4558i − 0.969790i
\(690\) 0 0
\(691\) 22.6274i 0.860788i 0.902641 + 0.430394i \(0.141625\pi\)
−0.902641 + 0.430394i \(0.858375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.8701i 1.01487i 0.861691 + 0.507434i \(0.169406\pi\)
−0.861691 + 0.507434i \(0.830594\pi\)
\(702\) 0 0
\(703\) 45.2548i 1.70682i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −28.0000 −1.05305
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 33.9411i 1.27111i
\(714\) 0 0
\(715\) − 11.3137i − 0.423109i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 21.2132i − 0.787839i
\(726\) 0 0
\(727\) 14.1421i 0.524503i 0.965000 + 0.262251i \(0.0844650\pi\)
−0.965000 + 0.262251i \(0.915535\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22.6274i 0.833492i
\(738\) 0 0
\(739\) − 33.9411i − 1.24854i −0.781207 0.624272i \(-0.785396\pi\)
0.781207 0.624272i \(-0.214604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 19.7990i 0.722475i 0.932474 + 0.361238i \(0.117646\pi\)
−0.932474 + 0.361238i \(0.882354\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26.8701i 0.974039i 0.873391 + 0.487019i \(0.161916\pi\)
−0.873391 + 0.487019i \(0.838084\pi\)
\(762\) 0 0
\(763\) − 39.5980i − 1.43354i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 35.3553i − 1.27164i −0.771836 0.635822i \(-0.780661\pi\)
0.771836 0.635822i \(-0.219339\pi\)
\(774\) 0 0
\(775\) 25.4558i 0.914401i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.3137i 0.403804i
\(786\) 0 0
\(787\) − 28.2843i − 1.00823i −0.863638 0.504113i \(-0.831820\pi\)
0.863638 0.504113i \(-0.168180\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.5563i 0.551034i 0.961296 + 0.275517i \(0.0888491\pi\)
−0.961296 + 0.275517i \(0.911151\pi\)
\(798\) 0 0
\(799\) − 16.9706i − 0.600375i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −32.0000 −1.12926
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 18.3848i − 0.646374i −0.946335 0.323187i \(-0.895246\pi\)
0.946335 0.323187i \(-0.104754\pi\)
\(810\) 0 0
\(811\) 16.9706i 0.595917i 0.954579 + 0.297959i \(0.0963057\pi\)
−0.954579 + 0.297959i \(0.903694\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 64.0000 2.23908
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.41421i 0.0493564i 0.999695 + 0.0246782i \(0.00785611\pi\)
−0.999695 + 0.0246782i \(0.992144\pi\)
\(822\) 0 0
\(823\) 8.48528i 0.295778i 0.989004 + 0.147889i \(0.0472479\pi\)
−0.989004 + 0.147889i \(0.952752\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1.41421i − 0.0489996i
\(834\) 0 0
\(835\) 22.6274i 0.783054i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −52.0000 −1.79524 −0.897620 0.440771i \(-0.854705\pi\)
−0.897620 + 0.440771i \(0.854705\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.7279i 0.437854i
\(846\) 0 0
\(847\) − 14.1421i − 0.485930i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) 24.0000 0.821744 0.410872 0.911693i \(-0.365224\pi\)
0.410872 + 0.911693i \(0.365224\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 52.3259i − 1.78742i −0.448646 0.893709i \(-0.648094\pi\)
0.448646 0.893709i \(-0.351906\pi\)
\(858\) 0 0
\(859\) 33.9411i 1.15806i 0.815308 + 0.579028i \(0.196568\pi\)
−0.815308 + 0.579028i \(0.803432\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.3137i 0.383791i
\(870\) 0 0
\(871\) 11.3137i 0.383350i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −32.0000 −1.08180
\(876\) 0 0
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) 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
\(883\) 33.9411i 1.14221i 0.820877 + 0.571105i \(0.193485\pi\)
−0.820877 + 0.571105i \(0.806515\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 67.8823i − 2.27159i
\(894\) 0 0
\(895\) − 33.9411i − 1.13453i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 8.48528i − 0.282060i
\(906\) 0 0
\(907\) 33.9411i 1.12700i 0.826117 + 0.563498i \(0.190545\pi\)
−0.826117 + 0.563498i \(0.809455\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 22.6274i − 0.747223i
\(918\) 0 0
\(919\) 31.1127i 1.02631i 0.858295 + 0.513157i \(0.171524\pi\)
−0.858295 + 0.513157i \(0.828476\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 24.0000 0.789115
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 32.5269i − 1.06717i −0.845745 0.533587i \(-0.820844\pi\)
0.845745 0.533587i \(-0.179156\pi\)
\(930\) 0 0
\(931\) − 5.65685i − 0.185396i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 15.5563i − 0.507122i −0.967319 0.253561i \(-0.918398\pi\)
0.967319 0.253561i \(-0.0816019\pi\)
\(942\) 0 0
\(943\) 16.9706i 0.552638i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.0000 0.519930 0.259965 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 41.0122i 1.32852i 0.747504 + 0.664258i \(0.231252\pi\)
−0.747504 + 0.664258i \(0.768748\pi\)
\(954\) 0 0
\(955\) 22.6274i 0.732206i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 52.0000 1.67917
\(960\) 0 0
\(961\) −41.0000 −1.32258
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 14.1421i − 0.455251i
\(966\) 0 0
\(967\) − 31.1127i − 1.00052i −0.865876 0.500258i \(-0.833238\pi\)
0.865876 0.500258i \(-0.166762\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 48.0000 1.53881
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21.2132i 0.678671i 0.940666 + 0.339335i \(0.110202\pi\)
−0.940666 + 0.339335i \(0.889798\pi\)
\(978\) 0 0
\(979\) − 62.2254i − 1.98873i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 45.2548i − 1.43902i
\(990\) 0 0
\(991\) − 48.0833i − 1.52742i −0.645562 0.763708i \(-0.723377\pi\)
0.645562 0.763708i \(-0.276623\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 36.0000 1.14128
\(996\) 0 0
\(997\) 40.0000 1.26681 0.633406 0.773819i \(-0.281656\pi\)
0.633406 + 0.773819i \(0.281656\pi\)
\(998\) 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.c.h.2303.1 2
3.2 odd 2 2304.2.c.b.2303.2 2
4.3 odd 2 2304.2.c.b.2303.1 2
8.3 odd 2 2304.2.c.g.2303.2 2
8.5 even 2 2304.2.c.a.2303.2 2
12.11 even 2 inner 2304.2.c.h.2303.2 2
16.3 odd 4 1152.2.f.d.575.1 yes 4
16.5 even 4 1152.2.f.a.575.4 yes 4
16.11 odd 4 1152.2.f.d.575.3 yes 4
16.13 even 4 1152.2.f.a.575.2 yes 4
24.5 odd 2 2304.2.c.g.2303.1 2
24.11 even 2 2304.2.c.a.2303.1 2
48.5 odd 4 1152.2.f.d.575.2 yes 4
48.11 even 4 1152.2.f.a.575.1 4
48.29 odd 4 1152.2.f.d.575.4 yes 4
48.35 even 4 1152.2.f.a.575.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.f.a.575.1 4 48.11 even 4
1152.2.f.a.575.2 yes 4 16.13 even 4
1152.2.f.a.575.3 yes 4 48.35 even 4
1152.2.f.a.575.4 yes 4 16.5 even 4
1152.2.f.d.575.1 yes 4 16.3 odd 4
1152.2.f.d.575.2 yes 4 48.5 odd 4
1152.2.f.d.575.3 yes 4 16.11 odd 4
1152.2.f.d.575.4 yes 4 48.29 odd 4
2304.2.c.a.2303.1 2 24.11 even 2
2304.2.c.a.2303.2 2 8.5 even 2
2304.2.c.b.2303.1 2 4.3 odd 2
2304.2.c.b.2303.2 2 3.2 odd 2
2304.2.c.g.2303.1 2 24.5 odd 2
2304.2.c.g.2303.2 2 8.3 odd 2
2304.2.c.h.2303.1 2 1.1 even 1 trivial
2304.2.c.h.2303.2 2 12.11 even 2 inner