Properties

Label 5400.2.f.e.649.1
Level $5400$
Weight $2$
Character 5400.649
Analytic conductor $43.119$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5400.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(43.1192170915\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5400.649
Dual form 5400.2.f.e.649.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{7} +O(q^{10})\) \(q-3.00000i q^{7} -4.00000 q^{11} -1.00000i q^{13} +4.00000i q^{17} +1.00000 q^{19} +4.00000i q^{23} -4.00000 q^{31} -9.00000i q^{37} +8.00000i q^{43} +12.0000i q^{47} -2.00000 q^{49} -8.00000i q^{53} +4.00000 q^{59} -5.00000 q^{61} +11.0000i q^{67} -8.00000 q^{71} -1.00000i q^{73} +12.0000i q^{77} +5.00000 q^{79} +8.00000i q^{83} +12.0000 q^{89} -3.00000 q^{91} +5.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 8q^{11} + 2q^{19} - 8q^{31} - 4q^{49} + 8q^{59} - 10q^{61} - 16q^{71} + 10q^{79} + 24q^{89} - 6q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\) \(2701\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 0
\(6\) 0 0
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.00000i − 1.47959i −0.672832 0.739795i \(-0.734922\pi\)
0.672832 0.739795i \(-0.265078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 8.00000i − 1.09888i −0.835532 0.549442i \(-0.814840\pi\)
0.835532 0.549442i \(-0.185160\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.0000i 1.34386i 0.740613 + 0.671932i \(0.234535\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) − 1.00000i − 0.117041i −0.998286 0.0585206i \(-0.981362\pi\)
0.998286 0.0585206i \(-0.0186383\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000i 1.36753i
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.00000i 0.507673i 0.967247 + 0.253837i \(0.0816925\pi\)
−0.967247 + 0.253837i \(0.918307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 1.00000i − 0.0985329i −0.998786 0.0492665i \(-0.984312\pi\)
0.998786 0.0492665i \(-0.0156884\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 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
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) − 3.00000i − 0.260133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 0 0
\(139\) −9.00000 −0.763370 −0.381685 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.00000 0.655386 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 15.0000i 1.17489i 0.809264 + 0.587445i \(0.199866\pi\)
−0.809264 + 0.587445i \(0.800134\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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
\(181\) 21.0000 1.56092 0.780459 0.625207i \(-0.214986\pi\)
0.780459 + 0.625207i \(0.214986\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 16.0000i − 1.17004i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) − 23.0000i − 1.65558i −0.561041 0.827788i \(-0.689599\pi\)
0.561041 0.827788i \(-0.310401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) −25.0000 −1.77220 −0.886102 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000i 0.814613i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(241\) −15.0000 −0.966235 −0.483117 0.875556i \(-0.660496\pi\)
−0.483117 + 0.875556i \(0.660496\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.00000i − 0.0636285i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) − 16.0000i − 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 8.00000i − 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) 0 0
\(259\) −27.0000 −1.67770
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) 16.0000i 0.951101i 0.879688 + 0.475551i \(0.157751\pi\)
−0.879688 + 0.475551i \(0.842249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.0000i 0.701047i 0.936554 + 0.350524i \(0.113996\pi\)
−0.936554 + 0.350524i \(0.886004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) − 3.00000i − 0.169570i −0.996399 0.0847850i \(-0.972980\pi\)
0.996399 0.0847850i \(-0.0270203\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.0000i 1.34797i 0.738743 + 0.673987i \(0.235420\pi\)
−0.738743 + 0.673987i \(0.764580\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000i 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 36.0000 1.98474
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 3.00000i − 0.163420i −0.996656 0.0817102i \(-0.973962\pi\)
0.996656 0.0817102i \(-0.0260382\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) − 15.0000i − 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) 0 0
\(349\) 33.0000 1.76645 0.883225 0.468950i \(-0.155368\pi\)
0.883225 + 0.468950i \(0.155368\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.00000i 0.425797i 0.977074 + 0.212899i \(0.0682904\pi\)
−0.977074 + 0.212899i \(0.931710\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 23.0000i − 1.20059i −0.799779 0.600295i \(-0.795050\pi\)
0.799779 0.600295i \(-0.204950\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 1.00000i 0.0517780i 0.999665 + 0.0258890i \(0.00824165\pi\)
−0.999665 + 0.0258890i \(0.991758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11.0000 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 36.0000i 1.78445i
\(408\) 0 0
\(409\) 39.0000 1.92843 0.964213 0.265129i \(-0.0854146\pi\)
0.964213 + 0.265129i \(0.0854146\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 15.0000i 0.725901i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) 0 0
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.00000i 0.191346i
\(438\) 0 0
\(439\) −36.0000 −1.71819 −0.859093 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8.00000i 0.380091i 0.981775 + 0.190046i \(0.0608636\pi\)
−0.981775 + 0.190046i \(0.939136\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000i 1.77757i 0.458329 + 0.888783i \(0.348448\pi\)
−0.458329 + 0.888783i \(0.651552\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) 19.0000i 0.883005i 0.897260 + 0.441502i \(0.145554\pi\)
−0.897260 + 0.441502i \(0.854446\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) 33.0000 1.52380
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 32.0000i − 1.47136i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) −9.00000 −0.410365
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 11.0000i − 0.498458i −0.968445 0.249229i \(-0.919823\pi\)
0.968445 0.249229i \(-0.0801771\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000i 1.07655i
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 48.0000i − 2.11104i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 0 0
\(523\) 29.0000i 1.26808i 0.773300 + 0.634041i \(0.218605\pi\)
−0.773300 + 0.634041i \(0.781395\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 16.0000i − 0.696971i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8.00000 0.344584
\(540\) 0 0
\(541\) 9.00000 0.386940 0.193470 0.981106i \(-0.438026\pi\)
0.193470 + 0.981106i \(0.438026\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 7.00000i − 0.299298i −0.988739 0.149649i \(-0.952186\pi\)
0.988739 0.149649i \(-0.0478144\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 15.0000i − 0.637865i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 28.0000i − 1.18640i −0.805056 0.593199i \(-0.797865\pi\)
0.805056 0.593199i \(-0.202135\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 32.0000i − 1.34864i −0.738440 0.674320i \(-0.764437\pi\)
0.738440 0.674320i \(-0.235563\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.0000 1.17382 0.586911 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) −33.0000 −1.38101 −0.690504 0.723329i \(-0.742611\pi\)
−0.690504 + 0.723329i \(0.742611\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 13.0000i − 0.541197i −0.962692 0.270599i \(-0.912778\pi\)
0.962692 0.270599i \(-0.0872216\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 32.0000i 1.32530i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 40.0000i 1.64260i 0.570494 + 0.821302i \(0.306752\pi\)
−0.570494 + 0.821302i \(0.693248\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 5.00000i 0.202944i 0.994838 + 0.101472i \(0.0323552\pi\)
−0.994838 + 0.101472i \(0.967645\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) − 35.0000i − 1.41364i −0.707395 0.706818i \(-0.750130\pi\)
0.707395 0.706818i \(-0.249870\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 36.0000i − 1.44931i −0.689114 0.724653i \(-0.742000\pi\)
0.689114 0.724653i \(-0.258000\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 36.0000i − 1.44231i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 0 0
\(643\) 24.0000i 0.946468i 0.880937 + 0.473234i \(0.156913\pi\)
−0.880937 + 0.473234i \(0.843087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 32.0000i − 1.25805i −0.777385 0.629025i \(-0.783454\pi\)
0.777385 0.629025i \(-0.216546\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) − 19.0000i − 0.732396i −0.930537 0.366198i \(-0.880659\pi\)
0.930537 0.366198i \(-0.119341\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 44.0000i 1.69106i 0.533930 + 0.845529i \(0.320715\pi\)
−0.533930 + 0.845529i \(0.679285\pi\)
\(678\) 0 0
\(679\) 15.0000 0.575647
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.0000i 0.612223i 0.951996 + 0.306111i \(0.0990280\pi\)
−0.951996 + 0.306111i \(0.900972\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) − 9.00000i − 0.339441i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.00000 0.262891 0.131445 0.991323i \(-0.458038\pi\)
0.131445 + 0.991323i \(0.458038\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 16.0000i − 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12.0000i 0.445055i 0.974926 + 0.222528i \(0.0714308\pi\)
−0.974926 + 0.222528i \(0.928569\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) − 2.00000i − 0.0738717i −0.999318 0.0369358i \(-0.988240\pi\)
0.999318 0.0369358i \(-0.0117597\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 44.0000i − 1.62076i
\(738\) 0 0
\(739\) −32.0000 −1.17714 −0.588570 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.00000i 0.146746i 0.997305 + 0.0733729i \(0.0233763\pi\)
−0.997305 + 0.0733729i \(0.976624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) 3.00000 0.109472 0.0547358 0.998501i \(-0.482568\pi\)
0.0547358 + 0.998501i \(0.482568\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 31.0000i − 1.12671i −0.826214 0.563357i \(-0.809510\pi\)
0.826214 0.563357i \(-0.190490\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.00000 0.145000 0.0724999 0.997368i \(-0.476902\pi\)
0.0724999 + 0.997368i \(0.476902\pi\)
\(762\) 0 0
\(763\) − 42.0000i − 1.52050i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 4.00000i − 0.144432i
\(768\) 0 0
\(769\) −47.0000 −1.69486 −0.847432 0.530904i \(-0.821852\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 24.0000i − 0.863220i −0.902060 0.431610i \(-0.857946\pi\)
0.902060 0.431610i \(-0.142054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.00000i 0.249523i 0.992187 + 0.124762i \(0.0398166\pi\)
−0.992187 + 0.124762i \(0.960183\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) 0 0
\(793\) 5.00000i 0.177555i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 16.0000i − 0.566749i −0.959009 0.283375i \(-0.908546\pi\)
0.959009 0.283375i \(-0.0914540\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.00000i 0.141157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −40.0000 −1.40633 −0.703163 0.711029i \(-0.748229\pi\)
−0.703163 + 0.711029i \(0.748229\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.00000i 0.279885i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) 25.0000i 0.871445i 0.900081 + 0.435723i \(0.143507\pi\)
−0.900081 + 0.435723i \(0.856493\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 20.0000i − 0.695468i −0.937593 0.347734i \(-0.886951\pi\)
0.937593 0.347734i \(-0.113049\pi\)
\(828\) 0 0
\(829\) 35.0000 1.21560 0.607800 0.794090i \(-0.292052\pi\)
0.607800 + 0.794090i \(0.292052\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 8.00000i − 0.277184i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 15.0000i − 0.515406i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) 0 0
\(853\) 1.00000i 0.0342393i 0.999853 + 0.0171197i \(0.00544963\pi\)
−0.999853 + 0.0171197i \(0.994550\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 24.0000i − 0.819824i −0.912125 0.409912i \(-0.865559\pi\)
0.912125 0.409912i \(-0.134441\pi\)
\(858\) 0 0
\(859\) 27.0000 0.921228 0.460614 0.887601i \(-0.347629\pi\)
0.460614 + 0.887601i \(0.347629\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 44.0000i − 1.49778i −0.662696 0.748889i \(-0.730588\pi\)
0.662696 0.748889i \(-0.269412\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 11.0000 0.372721
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 33.0000i − 1.11433i −0.830402 0.557165i \(-0.811889\pi\)
0.830402 0.557165i \(-0.188111\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 28.0000 0.943344 0.471672 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) 0 0
\(883\) 7.00000i 0.235569i 0.993039 + 0.117784i \(0.0375792\pi\)
−0.993039 + 0.117784i \(0.962421\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 56.0000i − 1.88030i −0.340766 0.940148i \(-0.610687\pi\)
0.340766 0.940148i \(-0.389313\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 32.0000 1.06607
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 5.00000i − 0.166022i −0.996549 0.0830111i \(-0.973546\pi\)
0.996549 0.0830111i \(-0.0264537\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) − 32.0000i − 1.05905i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 48.0000i − 1.58510i
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.00000i 0.263323i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −40.0000 −1.31236 −0.656179 0.754606i \(-0.727828\pi\)
−0.656179 + 0.754606i \(0.727828\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 51.0000i 1.66610i 0.553200 + 0.833049i \(0.313407\pi\)
−0.553200 + 0.833049i \(0.686593\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 0 0
\(949\) −1.00000 −0.0324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 36.0000i − 1.16615i −0.812417 0.583077i \(-0.801849\pi\)
0.812417 0.583077i \(-0.198151\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 59.0000i 1.89731i 0.316310 + 0.948656i \(0.397556\pi\)
−0.316310 + 0.948656i \(0.602444\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 27.0000i 0.865580i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.0000i 0.767828i 0.923369 + 0.383914i \(0.125424\pi\)
−0.923369 + 0.383914i \(0.874576\pi\)
\(978\) 0 0
\(979\) −48.0000 −1.53409
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 25.0000 0.794151 0.397076 0.917786i \(-0.370025\pi\)
0.397076 + 0.917786i \(0.370025\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 26.0000i − 0.823428i −0.911313 0.411714i \(-0.864930\pi\)
0.911313 0.411714i \(-0.135070\pi\)
\(998\) 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 5400.2.f.e.649.1 2
3.2 odd 2 5400.2.f.v.649.1 2
5.2 odd 4 5400.2.a.bn.1.1 1
5.3 odd 4 216.2.a.a.1.1 1
5.4 even 2 inner 5400.2.f.e.649.2 2
15.2 even 4 5400.2.a.bp.1.1 1
15.8 even 4 216.2.a.d.1.1 yes 1
15.14 odd 2 5400.2.f.v.649.2 2
20.3 even 4 432.2.a.a.1.1 1
40.3 even 4 1728.2.a.bb.1.1 1
40.13 odd 4 1728.2.a.ba.1.1 1
45.13 odd 12 648.2.i.h.217.1 2
45.23 even 12 648.2.i.a.217.1 2
45.38 even 12 648.2.i.a.433.1 2
45.43 odd 12 648.2.i.h.433.1 2
60.23 odd 4 432.2.a.h.1.1 1
120.53 even 4 1728.2.a.a.1.1 1
120.83 odd 4 1728.2.a.b.1.1 1
180.23 odd 12 1296.2.i.a.865.1 2
180.43 even 12 1296.2.i.q.433.1 2
180.83 odd 12 1296.2.i.a.433.1 2
180.103 even 12 1296.2.i.q.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.a.a.1.1 1 5.3 odd 4
216.2.a.d.1.1 yes 1 15.8 even 4
432.2.a.a.1.1 1 20.3 even 4
432.2.a.h.1.1 1 60.23 odd 4
648.2.i.a.217.1 2 45.23 even 12
648.2.i.a.433.1 2 45.38 even 12
648.2.i.h.217.1 2 45.13 odd 12
648.2.i.h.433.1 2 45.43 odd 12
1296.2.i.a.433.1 2 180.83 odd 12
1296.2.i.a.865.1 2 180.23 odd 12
1296.2.i.q.433.1 2 180.43 even 12
1296.2.i.q.865.1 2 180.103 even 12
1728.2.a.a.1.1 1 120.53 even 4
1728.2.a.b.1.1 1 120.83 odd 4
1728.2.a.ba.1.1 1 40.13 odd 4
1728.2.a.bb.1.1 1 40.3 even 4
5400.2.a.bn.1.1 1 5.2 odd 4
5400.2.a.bp.1.1 1 15.2 even 4
5400.2.f.e.649.1 2 1.1 even 1 trivial
5400.2.f.e.649.2 2 5.4 even 2 inner
5400.2.f.v.649.1 2 3.2 odd 2
5400.2.f.v.649.2 2 15.14 odd 2