Properties

Label 7728.2.a.k.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{13} -4.00000 q^{15} +4.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} +11.0000 q^{25} -1.00000 q^{27} +10.0000 q^{29} +2.00000 q^{31} +3.00000 q^{33} -4.00000 q^{35} -10.0000 q^{37} +2.00000 q^{39} +7.00000 q^{41} +4.00000 q^{43} +4.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} -9.00000 q^{53} -12.0000 q^{55} -7.00000 q^{57} -9.00000 q^{59} -1.00000 q^{61} -1.00000 q^{63} -8.00000 q^{65} -1.00000 q^{69} -8.00000 q^{71} +4.00000 q^{73} -11.0000 q^{75} +3.00000 q^{77} -14.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +16.0000 q^{85} -10.0000 q^{87} +6.00000 q^{89} +2.00000 q^{91} -2.00000 q^{93} +28.0000 q^{95} -2.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) −11.0000 −1.27017
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 16.0000 1.73544
\(86\) 0 0
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 28.0000 2.87274
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) 20.0000 1.93347 0.966736 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −7.00000 −0.631169
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 40.0000 3.32182
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 17.0000 1.33154 0.665771 0.746156i \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) 0 0
\(165\) 12.0000 0.934199
\(166\) 0 0
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 7.00000 0.535303
\(172\) 0 0
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) −11.0000 −0.831522
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 1.00000 0.0739221
\(184\) 0 0
\(185\) −40.0000 −2.94086
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −5.00000 −0.361787 −0.180894 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) 0 0
\(193\) −19.0000 −1.36765 −0.683825 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 0 0
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.0000 −0.701862
\(204\) 0 0
\(205\) 28.0000 1.95560
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) 23.0000 1.51988 0.759941 0.649992i \(-0.225228\pi\)
0.759941 + 0.649992i \(0.225228\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 14.0000 0.909398
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) −14.0000 −0.890799
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) −16.0000 −1.00196
\(256\) 0 0
\(257\) −5.00000 −0.311891 −0.155946 0.987766i \(-0.549842\pi\)
−0.155946 + 0.987766i \(0.549842\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) 29.0000 1.78822 0.894108 0.447851i \(-0.147810\pi\)
0.894108 + 0.447851i \(0.147810\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) −33.0000 −1.98997
\(276\) 0 0
\(277\) 3.00000 0.180253 0.0901263 0.995930i \(-0.471273\pi\)
0.0901263 + 0.995930i \(0.471273\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) −28.0000 −1.65858
\(286\) 0 0
\(287\) −7.00000 −0.413197
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) 16.0000 0.934730 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 0 0
\(295\) −36.0000 −2.09600
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) 11.0000 0.623753 0.311876 0.950123i \(-0.399043\pi\)
0.311876 + 0.950123i \(0.399043\pi\)
\(312\) 0 0
\(313\) −17.0000 −0.960897 −0.480448 0.877023i \(-0.659526\pi\)
−0.480448 + 0.877023i \(0.659526\pi\)
\(314\) 0 0
\(315\) −4.00000 −0.225374
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 28.0000 1.55796
\(324\) 0 0
\(325\) −22.0000 −1.22034
\(326\) 0 0
\(327\) 8.00000 0.442401
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) −23.0000 −1.26419 −0.632097 0.774889i \(-0.717806\pi\)
−0.632097 + 0.774889i \(0.717806\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 22.0000 1.17094 0.585471 0.810693i \(-0.300910\pi\)
0.585471 + 0.810693i \(0.300910\pi\)
\(354\) 0 0
\(355\) −32.0000 −1.69838
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 0 0
\(369\) 7.00000 0.364405
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 0 0
\(375\) −24.0000 −1.23935
\(376\) 0 0
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 0 0
\(381\) −17.0000 −0.870936
\(382\) 0 0
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 3.00000 0.151330
\(394\) 0 0
\(395\) −56.0000 −2.81767
\(396\) 0 0
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 0 0
\(399\) 7.00000 0.350438
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 0 0
\(405\) 4.00000 0.198762
\(406\) 0 0
\(407\) 30.0000 1.48704
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 0 0
\(413\) 9.00000 0.442861
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) −3.00000 −0.145865
\(424\) 0 0
\(425\) 44.0000 2.13431
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −7.00000 −0.337178 −0.168589 0.985686i \(-0.553921\pi\)
−0.168589 + 0.985686i \(0.553921\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) −40.0000 −1.91785
\(436\) 0 0
\(437\) 7.00000 0.334855
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) 0 0
\(447\) 11.0000 0.520282
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −21.0000 −0.988851
\(452\) 0 0
\(453\) 19.0000 0.892698
\(454\) 0 0
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 21.0000 0.975953 0.487976 0.872857i \(-0.337735\pi\)
0.487976 + 0.872857i \(0.337735\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13.0000 0.599008
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 77.0000 3.53300
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 0 0
\(489\) −17.0000 −0.768767
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) 40.0000 1.80151
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −9.00000 −0.402090
\(502\) 0 0
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) −29.0000 −1.28540 −0.642701 0.766117i \(-0.722186\pi\)
−0.642701 + 0.766117i \(0.722186\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 0 0
\(513\) −7.00000 −0.309058
\(514\) 0 0
\(515\) 52.0000 2.29139
\(516\) 0 0
\(517\) 9.00000 0.395820
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −1.00000 −0.0437269 −0.0218635 0.999761i \(-0.506960\pi\)
−0.0218635 + 0.999761i \(0.506960\pi\)
\(524\) 0 0
\(525\) 11.0000 0.480079
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) 0 0
\(533\) −14.0000 −0.606407
\(534\) 0 0
\(535\) 80.0000 3.45870
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −9.00000 −0.386940 −0.193470 0.981106i \(-0.561974\pi\)
−0.193470 + 0.981106i \(0.561974\pi\)
\(542\) 0 0
\(543\) 6.00000 0.257485
\(544\) 0 0
\(545\) −32.0000 −1.37073
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 70.0000 2.98210
\(552\) 0 0
\(553\) 14.0000 0.595341
\(554\) 0 0
\(555\) 40.0000 1.69791
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 40.0000 1.68281
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −35.0000 −1.46728 −0.733638 0.679540i \(-0.762179\pi\)
−0.733638 + 0.679540i \(0.762179\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 5.00000 0.208878
\(574\) 0 0
\(575\) 11.0000 0.458732
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 19.0000 0.789613
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) 0 0
\(585\) −8.00000 −0.330759
\(586\) 0 0
\(587\) 9.00000 0.371470 0.185735 0.982600i \(-0.440533\pi\)
0.185735 + 0.982600i \(0.440533\pi\)
\(588\) 0 0
\(589\) 14.0000 0.576860
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 0 0
\(597\) −11.0000 −0.450200
\(598\) 0 0
\(599\) −10.0000 −0.408589 −0.204294 0.978909i \(-0.565490\pi\)
−0.204294 + 0.978909i \(0.565490\pi\)
\(600\) 0 0
\(601\) 40.0000 1.63163 0.815817 0.578310i \(-0.196288\pi\)
0.815817 + 0.578310i \(0.196288\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.00000 −0.325246
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 0 0
\(615\) −28.0000 −1.12907
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 21.0000 0.838659
\(628\) 0 0
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 46.0000 1.83123 0.915616 0.402055i \(-0.131704\pi\)
0.915616 + 0.402055i \(0.131704\pi\)
\(632\) 0 0
\(633\) 15.0000 0.596196
\(634\) 0 0
\(635\) 68.0000 2.69850
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 13.0000 0.513469 0.256735 0.966482i \(-0.417353\pi\)
0.256735 + 0.966482i \(0.417353\pi\)
\(642\) 0 0
\(643\) −35.0000 −1.38027 −0.690133 0.723683i \(-0.742448\pi\)
−0.690133 + 0.723683i \(0.742448\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) 27.0000 1.05984
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) −1.00000 −0.0388955 −0.0194477 0.999811i \(-0.506191\pi\)
−0.0194477 + 0.999811i \(0.506191\pi\)
\(662\) 0 0
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) −28.0000 −1.08579
\(666\) 0 0
\(667\) 10.0000 0.387202
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −15.0000 −0.578208 −0.289104 0.957298i \(-0.593357\pi\)
−0.289104 + 0.957298i \(0.593357\pi\)
\(674\) 0 0
\(675\) −11.0000 −0.423390
\(676\) 0 0
\(677\) −28.0000 −1.07613 −0.538064 0.842904i \(-0.680844\pi\)
−0.538064 + 0.842904i \(0.680844\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 8.00000 0.306111 0.153056 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −23.0000 −0.877505
\(688\) 0 0
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) 64.0000 2.42766
\(696\) 0 0
\(697\) 28.0000 1.06058
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −33.0000 −1.24639 −0.623196 0.782065i \(-0.714166\pi\)
−0.623196 + 0.782065i \(0.714166\pi\)
\(702\) 0 0
\(703\) −70.0000 −2.64010
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) −3.00000 −0.112827
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) 0 0
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) 0 0
\(723\) −25.0000 −0.929760
\(724\) 0 0
\(725\) 110.000 4.08530
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 14.0000 0.514303
\(742\) 0 0
\(743\) 45.0000 1.65089 0.825445 0.564483i \(-0.190924\pi\)
0.825445 + 0.564483i \(0.190924\pi\)
\(744\) 0 0
\(745\) −44.0000 −1.61204
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) −30.0000 −1.09326
\(754\) 0 0
\(755\) −76.0000 −2.76592
\(756\) 0 0
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) 0 0
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 16.0000 0.578481
\(766\) 0 0
\(767\) 18.0000 0.649942
\(768\) 0 0
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) 5.00000 0.180071
\(772\) 0 0
\(773\) −44.0000 −1.58257 −0.791285 0.611448i \(-0.790588\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) 0 0
\(775\) 22.0000 0.790263
\(776\) 0 0
\(777\) −10.0000 −0.358748
\(778\) 0 0
\(779\) 49.0000 1.75561
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) −10.0000 −0.357371
\(784\) 0 0
\(785\) −52.0000 −1.85596
\(786\) 0 0
\(787\) 13.0000 0.463400 0.231700 0.972787i \(-0.425571\pi\)
0.231700 + 0.972787i \(0.425571\pi\)
\(788\) 0 0
\(789\) −29.0000 −1.03243
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) 36.0000 1.27679
\(796\) 0 0
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 68.0000 2.38194
\(816\) 0 0
\(817\) 28.0000 0.979596
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 0 0
\(823\) 19.0000 0.662298 0.331149 0.943578i \(-0.392564\pi\)
0.331149 + 0.943578i \(0.392564\pi\)
\(824\) 0 0
\(825\) 33.0000 1.14891
\(826\) 0 0
\(827\) −19.0000 −0.660695 −0.330347 0.943859i \(-0.607166\pi\)
−0.330347 + 0.943859i \(0.607166\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −3.00000 −0.104069
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 2.00000 0.0688837
\(844\) 0 0
\(845\) −36.0000 −1.23844
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) −10.0000 −0.342796
\(852\) 0 0
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 0 0
\(855\) 28.0000 0.957580
\(856\) 0 0
\(857\) −13.0000 −0.444072 −0.222036 0.975039i \(-0.571270\pi\)
−0.222036 + 0.975039i \(0.571270\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) 7.00000 0.238559
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) 88.0000 2.99209
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 42.0000 1.42475
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −21.0000 −0.709120 −0.354560 0.935033i \(-0.615369\pi\)
−0.354560 + 0.935033i \(0.615369\pi\)
\(878\) 0 0
\(879\) −16.0000 −0.539667
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) 40.0000 1.34611 0.673054 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) 0 0
\(885\) 36.0000 1.21013
\(886\) 0 0
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) −17.0000 −0.570162
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) −21.0000 −0.702738
\(894\) 0 0
\(895\) −48.0000 −1.60446
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) 0 0
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −24.0000 −0.797787
\(906\) 0 0
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) 0 0
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.595713
\(914\) 0 0
\(915\) 4.00000 0.132236
\(916\) 0 0
\(917\) 3.00000 0.0990687
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −110.000 −3.61678
\(926\) 0 0
\(927\) 13.0000 0.426976
\(928\) 0 0
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 0 0
\(933\) −11.0000 −0.360124
\(934\) 0 0
\(935\) −48.0000 −1.56977
\(936\) 0 0
\(937\) −59.0000 −1.92745 −0.963723 0.266904i \(-0.913999\pi\)
−0.963723 + 0.266904i \(0.913999\pi\)
\(938\) 0 0
\(939\) 17.0000 0.554774
\(940\) 0 0
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 0 0
\(943\) 7.00000 0.227951
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 35.0000 1.13376 0.566881 0.823800i \(-0.308150\pi\)
0.566881 + 0.823800i \(0.308150\pi\)
\(954\) 0 0
\(955\) −20.0000 −0.647185
\(956\) 0 0
\(957\) 30.0000 0.969762
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) 0 0
\(965\) −76.0000 −2.44653
\(966\) 0 0
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) 0 0
\(969\) −28.0000 −0.899490
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) 0 0
\(975\) 22.0000 0.704564
\(976\) 0 0
\(977\) 47.0000 1.50366 0.751832 0.659355i \(-0.229171\pi\)
0.751832 + 0.659355i \(0.229171\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −8.00000 −0.255420
\(982\) 0 0
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) 72.0000 2.29411
\(986\) 0 0
\(987\) −3.00000 −0.0954911
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −19.0000 −0.603555 −0.301777 0.953378i \(-0.597580\pi\)
−0.301777 + 0.953378i \(0.597580\pi\)
\(992\) 0 0
\(993\) 23.0000 0.729883
\(994\) 0 0
\(995\) 44.0000 1.39489
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 7728.2.a.k.1.1 1
4.3 odd 2 3864.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.f.1.1 1 4.3 odd 2
7728.2.a.k.1.1 1 1.1 even 1 trivial