Properties

Label 4800.2.a.cr.1.1
Level $4800$
Weight $2$
Character 4800.1
Self dual yes
Analytic conductor $38.328$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.3281929702\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4800.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +4.00000 q^{7} +1.00000 q^{9} +4.00000 q^{13} +8.00000 q^{19} +4.00000 q^{21} -4.00000 q^{23} +1.00000 q^{27} +6.00000 q^{29} -8.00000 q^{31} -4.00000 q^{37} +4.00000 q^{39} +6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{47} +9.00000 q^{49} -12.0000 q^{53} +8.00000 q^{57} +6.00000 q^{61} +4.00000 q^{63} +12.0000 q^{67} -4.00000 q^{69} -16.0000 q^{71} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +6.00000 q^{87} -10.0000 q^{89} +16.0000 q^{91} -8.00000 q^{93} -8.00000 q^{97} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 32.0000 2.77475
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000 0.742307
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 0 0
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −24.0000 −1.72756 −0.863779 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) −16.0000 −1.09630
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −32.0000 −2.17230
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 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
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 32.0000 2.03611
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 16.0000 0.968364
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) −20.0000 −1.16841 −0.584206 0.811605i \(-0.698594\pi\)
−0.584206 + 0.811605i \(0.698594\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −48.0000 −2.49204
\(372\) 0 0
\(373\) 20.0000 1.03556 0.517780 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 0 0
\(399\) 32.0000 1.60200
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 24.0000 1.16144
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −40.0000 −1.92228 −0.961139 0.276066i \(-0.910969\pi\)
−0.961139 + 0.276066i \(0.910969\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) 48.0000 2.21643
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) −16.0000 −0.728025
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −64.0000 −2.87079
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 4.00000 0.178707
\(502\) 0 0
\(503\) 28.0000 1.24846 0.624229 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 48.0000 2.04487
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.00000 −0.169485 −0.0847427 0.996403i \(-0.527007\pi\)
−0.0847427 + 0.996403i \(0.527007\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 0 0
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) −64.0000 −2.63707
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) 8.00000 0.328521 0.164260 0.986417i \(-0.447476\pi\)
0.164260 + 0.986417i \(0.447476\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.0000 0.811775 0.405887 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 0 0
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000 0.322068 0.161034 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −40.0000 −1.60257
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 36.0000 1.42637
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 0 0
\(653\) −20.0000 −0.782660 −0.391330 0.920250i \(-0.627985\pi\)
−0.391330 + 0.920250i \(0.627985\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) 0 0
\(689\) −48.0000 −1.82865
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −56.0000 −2.10610
\(708\) 0 0
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.0000 −0.597531
\(718\) 0 0
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) 14.0000 0.520666
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) 32.0000 1.17555
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 16.0000 0.583072
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 40.0000 1.44810
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 0 0
\(773\) −28.0000 −1.00709 −0.503545 0.863969i \(-0.667971\pi\)
−0.503545 + 0.863969i \(0.667971\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −16.0000 −0.573997
\(778\) 0 0
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 32.0000 1.13779
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 22.0000 0.773479 0.386739 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) 0 0
\(819\) 16.0000 0.559085
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −4.00000 −0.138758
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −26.0000 −0.895488
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −44.0000 −1.51186
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 12.0000 0.410872 0.205436 0.978671i \(-0.434139\pi\)
0.205436 + 0.978671i \(0.434139\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.0000 −1.63965 −0.819824 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) 0 0
\(859\) −56.0000 −1.91070 −0.955348 0.295484i \(-0.904519\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −28.0000 −0.945493 −0.472746 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) 0 0
\(879\) −20.0000 −0.674583
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 0 0
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 64.0000 2.11347
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) −64.0000 −2.10659
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) 72.0000 2.35970
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 0 0
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) −48.0000 −1.55487 −0.777436 0.628962i \(-0.783480\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 16.0000 0.513464 0.256732 0.966483i \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(972\) 0 0
\(973\) −32.0000 −1.02587
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 0 0
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 4800.2.a.cr.1.1 1
4.3 odd 2 4800.2.a.e.1.1 1
5.2 odd 4 960.2.f.e.769.1 2
5.3 odd 4 960.2.f.e.769.2 2
5.4 even 2 4800.2.a.c.1.1 1
8.3 odd 2 2400.2.a.s.1.1 1
8.5 even 2 2400.2.a.o.1.1 1
15.2 even 4 2880.2.f.m.1729.1 2
15.8 even 4 2880.2.f.m.1729.2 2
20.3 even 4 960.2.f.d.769.1 2
20.7 even 4 960.2.f.d.769.2 2
20.19 odd 2 4800.2.a.cp.1.1 1
24.5 odd 2 7200.2.a.by.1.1 1
24.11 even 2 7200.2.a.a.1.1 1
40.3 even 4 480.2.f.c.289.2 yes 2
40.13 odd 4 480.2.f.d.289.1 yes 2
40.19 odd 2 2400.2.a.p.1.1 1
40.27 even 4 480.2.f.c.289.1 2
40.29 even 2 2400.2.a.t.1.1 1
40.37 odd 4 480.2.f.d.289.2 yes 2
60.23 odd 4 2880.2.f.o.1729.2 2
60.47 odd 4 2880.2.f.o.1729.1 2
80.3 even 4 3840.2.d.q.2689.2 2
80.13 odd 4 3840.2.d.a.2689.2 2
80.27 even 4 3840.2.d.q.2689.1 2
80.37 odd 4 3840.2.d.a.2689.1 2
80.43 even 4 3840.2.d.p.2689.1 2
80.53 odd 4 3840.2.d.bf.2689.1 2
80.67 even 4 3840.2.d.p.2689.2 2
80.77 odd 4 3840.2.d.bf.2689.2 2
120.29 odd 2 7200.2.a.c.1.1 1
120.53 even 4 1440.2.f.d.289.1 2
120.59 even 2 7200.2.a.ca.1.1 1
120.77 even 4 1440.2.f.d.289.2 2
120.83 odd 4 1440.2.f.b.289.1 2
120.107 odd 4 1440.2.f.b.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.2.f.c.289.1 2 40.27 even 4
480.2.f.c.289.2 yes 2 40.3 even 4
480.2.f.d.289.1 yes 2 40.13 odd 4
480.2.f.d.289.2 yes 2 40.37 odd 4
960.2.f.d.769.1 2 20.3 even 4
960.2.f.d.769.2 2 20.7 even 4
960.2.f.e.769.1 2 5.2 odd 4
960.2.f.e.769.2 2 5.3 odd 4
1440.2.f.b.289.1 2 120.83 odd 4
1440.2.f.b.289.2 2 120.107 odd 4
1440.2.f.d.289.1 2 120.53 even 4
1440.2.f.d.289.2 2 120.77 even 4
2400.2.a.o.1.1 1 8.5 even 2
2400.2.a.p.1.1 1 40.19 odd 2
2400.2.a.s.1.1 1 8.3 odd 2
2400.2.a.t.1.1 1 40.29 even 2
2880.2.f.m.1729.1 2 15.2 even 4
2880.2.f.m.1729.2 2 15.8 even 4
2880.2.f.o.1729.1 2 60.47 odd 4
2880.2.f.o.1729.2 2 60.23 odd 4
3840.2.d.a.2689.1 2 80.37 odd 4
3840.2.d.a.2689.2 2 80.13 odd 4
3840.2.d.p.2689.1 2 80.43 even 4
3840.2.d.p.2689.2 2 80.67 even 4
3840.2.d.q.2689.1 2 80.27 even 4
3840.2.d.q.2689.2 2 80.3 even 4
3840.2.d.bf.2689.1 2 80.53 odd 4
3840.2.d.bf.2689.2 2 80.77 odd 4
4800.2.a.c.1.1 1 5.4 even 2
4800.2.a.e.1.1 1 4.3 odd 2
4800.2.a.cp.1.1 1 20.19 odd 2
4800.2.a.cr.1.1 1 1.1 even 1 trivial
7200.2.a.a.1.1 1 24.11 even 2
7200.2.a.c.1.1 1 120.29 odd 2
7200.2.a.by.1.1 1 24.5 odd 2
7200.2.a.ca.1.1 1 120.59 even 2