Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -4.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -4.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -1.30278 q^{13} +4.30278 q^{15} +1.60555 q^{17} -5.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} +13.5139 q^{25} -1.00000 q^{27} -8.21110 q^{29} -3.00000 q^{31} -5.00000 q^{33} -4.30278 q^{35} -9.00000 q^{37} +1.30278 q^{39} +2.21110 q^{41} +12.5139 q^{43} -4.30278 q^{45} +1.39445 q^{47} +1.00000 q^{49} -1.60555 q^{51} +5.51388 q^{53} -21.5139 q^{55} +5.60555 q^{57} +6.90833 q^{59} -11.9083 q^{61} +1.00000 q^{63} +5.60555 q^{65} +1.09167 q^{67} +1.00000 q^{69} +9.90833 q^{71} +12.2111 q^{73} -13.5139 q^{75} +5.00000 q^{77} +1.00000 q^{79} +1.00000 q^{81} +1.60555 q^{83} -6.90833 q^{85} +8.21110 q^{87} +0.0916731 q^{89} -1.30278 q^{91} +3.00000 q^{93} +24.1194 q^{95} -10.3944 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 5q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 5q^{5} + 2q^{7} + 2q^{9} + 10q^{11} + q^{13} + 5q^{15} - 4q^{17} - 4q^{19} - 2q^{21} - 2q^{23} + 9q^{25} - 2q^{27} - 2q^{29} - 6q^{31} - 10q^{33} - 5q^{35} - 18q^{37} - q^{39} - 10q^{41} + 7q^{43} - 5q^{45} + 10q^{47} + 2q^{49} + 4q^{51} - 7q^{53} - 25q^{55} + 4q^{57} + 3q^{59} - 13q^{61} + 2q^{63} + 4q^{65} + 13q^{67} + 2q^{69} + 9q^{71} + 10q^{73} - 9q^{75} + 10q^{77} + 2q^{79} + 2q^{81} - 4q^{83} - 3q^{85} + 2q^{87} + 11q^{89} + q^{91} + 6q^{93} + 23q^{95} - 28q^{97} + 10q^{99} + O(q^{100}) \)

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
\(5\) −4.30278 −1.92426 −0.962130 0.272591i \(-0.912119\pi\)
−0.962130 + 0.272591i \(0.912119\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −1.30278 −0.361325 −0.180662 0.983545i \(-0.557824\pi\)
−0.180662 + 0.983545i \(0.557824\pi\)
\(14\) 0 0
\(15\) 4.30278 1.11097
\(16\) 0 0
\(17\) 1.60555 0.389403 0.194702 0.980863i \(-0.437626\pi\)
0.194702 + 0.980863i \(0.437626\pi\)
\(18\) 0 0
\(19\) −5.60555 −1.28600 −0.643001 0.765865i \(-0.722311\pi\)
−0.643001 + 0.765865i \(0.722311\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 13.5139 2.70278
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.21110 −1.52476 −0.762382 0.647128i \(-0.775970\pi\)
−0.762382 + 0.647128i \(0.775970\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) −4.30278 −0.727302
\(36\) 0 0
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) 1.30278 0.208611
\(40\) 0 0
\(41\) 2.21110 0.345316 0.172658 0.984982i \(-0.444764\pi\)
0.172658 + 0.984982i \(0.444764\pi\)
\(42\) 0 0
\(43\) 12.5139 1.90835 0.954174 0.299252i \(-0.0967370\pi\)
0.954174 + 0.299252i \(0.0967370\pi\)
\(44\) 0 0
\(45\) −4.30278 −0.641420
\(46\) 0 0
\(47\) 1.39445 0.203401 0.101701 0.994815i \(-0.467572\pi\)
0.101701 + 0.994815i \(0.467572\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.60555 −0.224822
\(52\) 0 0
\(53\) 5.51388 0.757389 0.378695 0.925522i \(-0.376373\pi\)
0.378695 + 0.925522i \(0.376373\pi\)
\(54\) 0 0
\(55\) −21.5139 −2.90093
\(56\) 0 0
\(57\) 5.60555 0.742473
\(58\) 0 0
\(59\) 6.90833 0.899388 0.449694 0.893183i \(-0.351533\pi\)
0.449694 + 0.893183i \(0.351533\pi\)
\(60\) 0 0
\(61\) −11.9083 −1.52471 −0.762353 0.647162i \(-0.775956\pi\)
−0.762353 + 0.647162i \(0.775956\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 5.60555 0.695283
\(66\) 0 0
\(67\) 1.09167 0.133369 0.0666845 0.997774i \(-0.478758\pi\)
0.0666845 + 0.997774i \(0.478758\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 9.90833 1.17590 0.587951 0.808897i \(-0.299935\pi\)
0.587951 + 0.808897i \(0.299935\pi\)
\(72\) 0 0
\(73\) 12.2111 1.42920 0.714601 0.699533i \(-0.246608\pi\)
0.714601 + 0.699533i \(0.246608\pi\)
\(74\) 0 0
\(75\) −13.5139 −1.56045
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.60555 0.176232 0.0881161 0.996110i \(-0.471915\pi\)
0.0881161 + 0.996110i \(0.471915\pi\)
\(84\) 0 0
\(85\) −6.90833 −0.749313
\(86\) 0 0
\(87\) 8.21110 0.880323
\(88\) 0 0
\(89\) 0.0916731 0.00971733 0.00485866 0.999988i \(-0.498453\pi\)
0.00485866 + 0.999988i \(0.498453\pi\)
\(90\) 0 0
\(91\) −1.30278 −0.136568
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) 24.1194 2.47460
\(96\) 0 0
\(97\) −10.3944 −1.05540 −0.527698 0.849432i \(-0.676945\pi\)
−0.527698 + 0.849432i \(0.676945\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 5.69722 0.566895 0.283448 0.958988i \(-0.408522\pi\)
0.283448 + 0.958988i \(0.408522\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 4.30278 0.419908
\(106\) 0 0
\(107\) 10.5139 1.01641 0.508207 0.861235i \(-0.330308\pi\)
0.508207 + 0.861235i \(0.330308\pi\)
\(108\) 0 0
\(109\) 6.90833 0.661698 0.330849 0.943684i \(-0.392665\pi\)
0.330849 + 0.943684i \(0.392665\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) 0 0
\(113\) 1.69722 0.159661 0.0798307 0.996808i \(-0.474562\pi\)
0.0798307 + 0.996808i \(0.474562\pi\)
\(114\) 0 0
\(115\) 4.30278 0.401236
\(116\) 0 0
\(117\) −1.30278 −0.120442
\(118\) 0 0
\(119\) 1.60555 0.147181
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −2.21110 −0.199368
\(124\) 0 0
\(125\) −36.6333 −3.27658
\(126\) 0 0
\(127\) −5.30278 −0.470545 −0.235273 0.971929i \(-0.575598\pi\)
−0.235273 + 0.971929i \(0.575598\pi\)
\(128\) 0 0
\(129\) −12.5139 −1.10179
\(130\) 0 0
\(131\) −17.6056 −1.53820 −0.769102 0.639126i \(-0.779296\pi\)
−0.769102 + 0.639126i \(0.779296\pi\)
\(132\) 0 0
\(133\) −5.60555 −0.486063
\(134\) 0 0
\(135\) 4.30278 0.370324
\(136\) 0 0
\(137\) −4.81665 −0.411515 −0.205757 0.978603i \(-0.565966\pi\)
−0.205757 + 0.978603i \(0.565966\pi\)
\(138\) 0 0
\(139\) −5.09167 −0.431870 −0.215935 0.976408i \(-0.569280\pi\)
−0.215935 + 0.976408i \(0.569280\pi\)
\(140\) 0 0
\(141\) −1.39445 −0.117434
\(142\) 0 0
\(143\) −6.51388 −0.544718
\(144\) 0 0
\(145\) 35.3305 2.93404
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 1.39445 0.114238 0.0571188 0.998367i \(-0.481809\pi\)
0.0571188 + 0.998367i \(0.481809\pi\)
\(150\) 0 0
\(151\) −9.39445 −0.764509 −0.382255 0.924057i \(-0.624852\pi\)
−0.382255 + 0.924057i \(0.624852\pi\)
\(152\) 0 0
\(153\) 1.60555 0.129801
\(154\) 0 0
\(155\) 12.9083 1.03682
\(156\) 0 0
\(157\) 17.8167 1.42192 0.710962 0.703231i \(-0.248260\pi\)
0.710962 + 0.703231i \(0.248260\pi\)
\(158\) 0 0
\(159\) −5.51388 −0.437279
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −18.7250 −1.46665 −0.733327 0.679876i \(-0.762033\pi\)
−0.733327 + 0.679876i \(0.762033\pi\)
\(164\) 0 0
\(165\) 21.5139 1.67485
\(166\) 0 0
\(167\) 18.8167 1.45608 0.728038 0.685537i \(-0.240432\pi\)
0.728038 + 0.685537i \(0.240432\pi\)
\(168\) 0 0
\(169\) −11.3028 −0.869444
\(170\) 0 0
\(171\) −5.60555 −0.428667
\(172\) 0 0
\(173\) −3.78890 −0.288065 −0.144032 0.989573i \(-0.546007\pi\)
−0.144032 + 0.989573i \(0.546007\pi\)
\(174\) 0 0
\(175\) 13.5139 1.02155
\(176\) 0 0
\(177\) −6.90833 −0.519262
\(178\) 0 0
\(179\) 6.30278 0.471092 0.235546 0.971863i \(-0.424312\pi\)
0.235546 + 0.971863i \(0.424312\pi\)
\(180\) 0 0
\(181\) 14.8167 1.10131 0.550657 0.834732i \(-0.314377\pi\)
0.550657 + 0.834732i \(0.314377\pi\)
\(182\) 0 0
\(183\) 11.9083 0.880289
\(184\) 0 0
\(185\) 38.7250 2.84712
\(186\) 0 0
\(187\) 8.02776 0.587048
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −2.60555 −0.188531 −0.0942655 0.995547i \(-0.530050\pi\)
−0.0942655 + 0.995547i \(0.530050\pi\)
\(192\) 0 0
\(193\) −27.0278 −1.94550 −0.972750 0.231856i \(-0.925520\pi\)
−0.972750 + 0.231856i \(0.925520\pi\)
\(194\) 0 0
\(195\) −5.60555 −0.401422
\(196\) 0 0
\(197\) −17.9083 −1.27592 −0.637958 0.770071i \(-0.720221\pi\)
−0.637958 + 0.770071i \(0.720221\pi\)
\(198\) 0 0
\(199\) 5.51388 0.390868 0.195434 0.980717i \(-0.437388\pi\)
0.195434 + 0.980717i \(0.437388\pi\)
\(200\) 0 0
\(201\) −1.09167 −0.0770007
\(202\) 0 0
\(203\) −8.21110 −0.576306
\(204\) 0 0
\(205\) −9.51388 −0.664478
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −28.0278 −1.93872
\(210\) 0 0
\(211\) −1.42221 −0.0979086 −0.0489543 0.998801i \(-0.515589\pi\)
−0.0489543 + 0.998801i \(0.515589\pi\)
\(212\) 0 0
\(213\) −9.90833 −0.678907
\(214\) 0 0
\(215\) −53.8444 −3.67216
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) −12.2111 −0.825150
\(220\) 0 0
\(221\) −2.09167 −0.140701
\(222\) 0 0
\(223\) −9.09167 −0.608823 −0.304412 0.952541i \(-0.598460\pi\)
−0.304412 + 0.952541i \(0.598460\pi\)
\(224\) 0 0
\(225\) 13.5139 0.900925
\(226\) 0 0
\(227\) −19.3305 −1.28301 −0.641506 0.767118i \(-0.721690\pi\)
−0.641506 + 0.767118i \(0.721690\pi\)
\(228\) 0 0
\(229\) −15.5139 −1.02519 −0.512593 0.858632i \(-0.671315\pi\)
−0.512593 + 0.858632i \(0.671315\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) 25.3305 1.65946 0.829729 0.558166i \(-0.188495\pi\)
0.829729 + 0.558166i \(0.188495\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) 24.9083 1.61119 0.805593 0.592470i \(-0.201847\pi\)
0.805593 + 0.592470i \(0.201847\pi\)
\(240\) 0 0
\(241\) −24.0278 −1.54776 −0.773882 0.633330i \(-0.781688\pi\)
−0.773882 + 0.633330i \(0.781688\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −4.30278 −0.274894
\(246\) 0 0
\(247\) 7.30278 0.464664
\(248\) 0 0
\(249\) −1.60555 −0.101748
\(250\) 0 0
\(251\) −2.18335 −0.137812 −0.0689058 0.997623i \(-0.521951\pi\)
−0.0689058 + 0.997623i \(0.521951\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 0 0
\(255\) 6.90833 0.432616
\(256\) 0 0
\(257\) −9.02776 −0.563136 −0.281568 0.959541i \(-0.590854\pi\)
−0.281568 + 0.959541i \(0.590854\pi\)
\(258\) 0 0
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) −8.21110 −0.508254
\(262\) 0 0
\(263\) −17.6056 −1.08560 −0.542802 0.839860i \(-0.682637\pi\)
−0.542802 + 0.839860i \(0.682637\pi\)
\(264\) 0 0
\(265\) −23.7250 −1.45741
\(266\) 0 0
\(267\) −0.0916731 −0.00561030
\(268\) 0 0
\(269\) −9.90833 −0.604121 −0.302061 0.953289i \(-0.597675\pi\)
−0.302061 + 0.953289i \(0.597675\pi\)
\(270\) 0 0
\(271\) 3.60555 0.219022 0.109511 0.993986i \(-0.465072\pi\)
0.109511 + 0.993986i \(0.465072\pi\)
\(272\) 0 0
\(273\) 1.30278 0.0788476
\(274\) 0 0
\(275\) 67.5694 4.07459
\(276\) 0 0
\(277\) −8.69722 −0.522566 −0.261283 0.965262i \(-0.584146\pi\)
−0.261283 + 0.965262i \(0.584146\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 6.18335 0.368868 0.184434 0.982845i \(-0.440955\pi\)
0.184434 + 0.982845i \(0.440955\pi\)
\(282\) 0 0
\(283\) −13.6972 −0.814215 −0.407108 0.913380i \(-0.633463\pi\)
−0.407108 + 0.913380i \(0.633463\pi\)
\(284\) 0 0
\(285\) −24.1194 −1.42871
\(286\) 0 0
\(287\) 2.21110 0.130517
\(288\) 0 0
\(289\) −14.4222 −0.848365
\(290\) 0 0
\(291\) 10.3944 0.609333
\(292\) 0 0
\(293\) 32.8444 1.91879 0.959395 0.282064i \(-0.0910192\pi\)
0.959395 + 0.282064i \(0.0910192\pi\)
\(294\) 0 0
\(295\) −29.7250 −1.73066
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) 1.30278 0.0753415
\(300\) 0 0
\(301\) 12.5139 0.721288
\(302\) 0 0
\(303\) −5.69722 −0.327297
\(304\) 0 0
\(305\) 51.2389 2.93393
\(306\) 0 0
\(307\) −3.78890 −0.216244 −0.108122 0.994138i \(-0.534484\pi\)
−0.108122 + 0.994138i \(0.534484\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −8.51388 −0.482778 −0.241389 0.970428i \(-0.577603\pi\)
−0.241389 + 0.970428i \(0.577603\pi\)
\(312\) 0 0
\(313\) −13.2111 −0.746736 −0.373368 0.927683i \(-0.621797\pi\)
−0.373368 + 0.927683i \(0.621797\pi\)
\(314\) 0 0
\(315\) −4.30278 −0.242434
\(316\) 0 0
\(317\) −12.4861 −0.701290 −0.350645 0.936508i \(-0.614038\pi\)
−0.350645 + 0.936508i \(0.614038\pi\)
\(318\) 0 0
\(319\) −41.0555 −2.29867
\(320\) 0 0
\(321\) −10.5139 −0.586827
\(322\) 0 0
\(323\) −9.00000 −0.500773
\(324\) 0 0
\(325\) −17.6056 −0.976580
\(326\) 0 0
\(327\) −6.90833 −0.382031
\(328\) 0 0
\(329\) 1.39445 0.0768784
\(330\) 0 0
\(331\) 25.6333 1.40893 0.704467 0.709737i \(-0.251186\pi\)
0.704467 + 0.709737i \(0.251186\pi\)
\(332\) 0 0
\(333\) −9.00000 −0.493197
\(334\) 0 0
\(335\) −4.69722 −0.256637
\(336\) 0 0
\(337\) −19.3028 −1.05149 −0.525745 0.850642i \(-0.676213\pi\)
−0.525745 + 0.850642i \(0.676213\pi\)
\(338\) 0 0
\(339\) −1.69722 −0.0921806
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −4.30278 −0.231654
\(346\) 0 0
\(347\) 28.6333 1.53712 0.768558 0.639780i \(-0.220974\pi\)
0.768558 + 0.639780i \(0.220974\pi\)
\(348\) 0 0
\(349\) −20.9083 −1.11920 −0.559599 0.828764i \(-0.689045\pi\)
−0.559599 + 0.828764i \(0.689045\pi\)
\(350\) 0 0
\(351\) 1.30278 0.0695370
\(352\) 0 0
\(353\) −11.0000 −0.585471 −0.292735 0.956193i \(-0.594566\pi\)
−0.292735 + 0.956193i \(0.594566\pi\)
\(354\) 0 0
\(355\) −42.6333 −2.26274
\(356\) 0 0
\(357\) −1.60555 −0.0849748
\(358\) 0 0
\(359\) −18.5416 −0.978590 −0.489295 0.872118i \(-0.662746\pi\)
−0.489295 + 0.872118i \(0.662746\pi\)
\(360\) 0 0
\(361\) 12.4222 0.653800
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) −52.5416 −2.75015
\(366\) 0 0
\(367\) −3.48612 −0.181974 −0.0909870 0.995852i \(-0.529002\pi\)
−0.0909870 + 0.995852i \(0.529002\pi\)
\(368\) 0 0
\(369\) 2.21110 0.115105
\(370\) 0 0
\(371\) 5.51388 0.286266
\(372\) 0 0
\(373\) 1.78890 0.0926256 0.0463128 0.998927i \(-0.485253\pi\)
0.0463128 + 0.998927i \(0.485253\pi\)
\(374\) 0 0
\(375\) 36.6333 1.89174
\(376\) 0 0
\(377\) 10.6972 0.550935
\(378\) 0 0
\(379\) −6.42221 −0.329887 −0.164943 0.986303i \(-0.552744\pi\)
−0.164943 + 0.986303i \(0.552744\pi\)
\(380\) 0 0
\(381\) 5.30278 0.271669
\(382\) 0 0
\(383\) 16.8167 0.859291 0.429645 0.902998i \(-0.358639\pi\)
0.429645 + 0.902998i \(0.358639\pi\)
\(384\) 0 0
\(385\) −21.5139 −1.09645
\(386\) 0 0
\(387\) 12.5139 0.636116
\(388\) 0 0
\(389\) 6.63331 0.336322 0.168161 0.985760i \(-0.446217\pi\)
0.168161 + 0.985760i \(0.446217\pi\)
\(390\) 0 0
\(391\) −1.60555 −0.0811962
\(392\) 0 0
\(393\) 17.6056 0.888083
\(394\) 0 0
\(395\) −4.30278 −0.216496
\(396\) 0 0
\(397\) −34.6056 −1.73680 −0.868401 0.495862i \(-0.834852\pi\)
−0.868401 + 0.495862i \(0.834852\pi\)
\(398\) 0 0
\(399\) 5.60555 0.280629
\(400\) 0 0
\(401\) 15.4222 0.770148 0.385074 0.922886i \(-0.374176\pi\)
0.385074 + 0.922886i \(0.374176\pi\)
\(402\) 0 0
\(403\) 3.90833 0.194688
\(404\) 0 0
\(405\) −4.30278 −0.213807
\(406\) 0 0
\(407\) −45.0000 −2.23057
\(408\) 0 0
\(409\) −12.0278 −0.594734 −0.297367 0.954763i \(-0.596109\pi\)
−0.297367 + 0.954763i \(0.596109\pi\)
\(410\) 0 0
\(411\) 4.81665 0.237588
\(412\) 0 0
\(413\) 6.90833 0.339937
\(414\) 0 0
\(415\) −6.90833 −0.339116
\(416\) 0 0
\(417\) 5.09167 0.249340
\(418\) 0 0
\(419\) 10.1194 0.494366 0.247183 0.968969i \(-0.420495\pi\)
0.247183 + 0.968969i \(0.420495\pi\)
\(420\) 0 0
\(421\) 13.3028 0.648338 0.324169 0.945999i \(-0.394915\pi\)
0.324169 + 0.945999i \(0.394915\pi\)
\(422\) 0 0
\(423\) 1.39445 0.0678004
\(424\) 0 0
\(425\) 21.6972 1.05247
\(426\) 0 0
\(427\) −11.9083 −0.576284
\(428\) 0 0
\(429\) 6.51388 0.314493
\(430\) 0 0
\(431\) −23.7250 −1.14279 −0.571396 0.820674i \(-0.693598\pi\)
−0.571396 + 0.820674i \(0.693598\pi\)
\(432\) 0 0
\(433\) 15.0278 0.722188 0.361094 0.932529i \(-0.382403\pi\)
0.361094 + 0.932529i \(0.382403\pi\)
\(434\) 0 0
\(435\) −35.3305 −1.69397
\(436\) 0 0
\(437\) 5.60555 0.268150
\(438\) 0 0
\(439\) −5.78890 −0.276289 −0.138145 0.990412i \(-0.544114\pi\)
−0.138145 + 0.990412i \(0.544114\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.8444 −1.18039 −0.590197 0.807259i \(-0.700950\pi\)
−0.590197 + 0.807259i \(0.700950\pi\)
\(444\) 0 0
\(445\) −0.394449 −0.0186987
\(446\) 0 0
\(447\) −1.39445 −0.0659552
\(448\) 0 0
\(449\) −3.27502 −0.154558 −0.0772789 0.997010i \(-0.524623\pi\)
−0.0772789 + 0.997010i \(0.524623\pi\)
\(450\) 0 0
\(451\) 11.0555 0.520584
\(452\) 0 0
\(453\) 9.39445 0.441390
\(454\) 0 0
\(455\) 5.60555 0.262792
\(456\) 0 0
\(457\) −36.1194 −1.68960 −0.844798 0.535086i \(-0.820279\pi\)
−0.844798 + 0.535086i \(0.820279\pi\)
\(458\) 0 0
\(459\) −1.60555 −0.0749407
\(460\) 0 0
\(461\) −24.7250 −1.15156 −0.575779 0.817606i \(-0.695301\pi\)
−0.575779 + 0.817606i \(0.695301\pi\)
\(462\) 0 0
\(463\) −2.81665 −0.130901 −0.0654505 0.997856i \(-0.520848\pi\)
−0.0654505 + 0.997856i \(0.520848\pi\)
\(464\) 0 0
\(465\) −12.9083 −0.598609
\(466\) 0 0
\(467\) −16.3944 −0.758645 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(468\) 0 0
\(469\) 1.09167 0.0504088
\(470\) 0 0
\(471\) −17.8167 −0.820948
\(472\) 0 0
\(473\) 62.5694 2.87694
\(474\) 0 0
\(475\) −75.7527 −3.47577
\(476\) 0 0
\(477\) 5.51388 0.252463
\(478\) 0 0
\(479\) 3.78890 0.173119 0.0865596 0.996247i \(-0.472413\pi\)
0.0865596 + 0.996247i \(0.472413\pi\)
\(480\) 0 0
\(481\) 11.7250 0.534613
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 44.7250 2.03086
\(486\) 0 0
\(487\) −20.8167 −0.943293 −0.471646 0.881788i \(-0.656340\pi\)
−0.471646 + 0.881788i \(0.656340\pi\)
\(488\) 0 0
\(489\) 18.7250 0.846773
\(490\) 0 0
\(491\) −28.9361 −1.30587 −0.652934 0.757415i \(-0.726462\pi\)
−0.652934 + 0.757415i \(0.726462\pi\)
\(492\) 0 0
\(493\) −13.1833 −0.593748
\(494\) 0 0
\(495\) −21.5139 −0.966977
\(496\) 0 0
\(497\) 9.90833 0.444449
\(498\) 0 0
\(499\) −21.0917 −0.944193 −0.472096 0.881547i \(-0.656503\pi\)
−0.472096 + 0.881547i \(0.656503\pi\)
\(500\) 0 0
\(501\) −18.8167 −0.840666
\(502\) 0 0
\(503\) 14.7250 0.656554 0.328277 0.944581i \(-0.393532\pi\)
0.328277 + 0.944581i \(0.393532\pi\)
\(504\) 0 0
\(505\) −24.5139 −1.09085
\(506\) 0 0
\(507\) 11.3028 0.501974
\(508\) 0 0
\(509\) 31.4500 1.39400 0.696998 0.717074i \(-0.254519\pi\)
0.696998 + 0.717074i \(0.254519\pi\)
\(510\) 0 0
\(511\) 12.2111 0.540187
\(512\) 0 0
\(513\) 5.60555 0.247491
\(514\) 0 0
\(515\) −17.2111 −0.758412
\(516\) 0 0
\(517\) 6.97224 0.306639
\(518\) 0 0
\(519\) 3.78890 0.166314
\(520\) 0 0
\(521\) −21.6333 −0.947772 −0.473886 0.880586i \(-0.657149\pi\)
−0.473886 + 0.880586i \(0.657149\pi\)
\(522\) 0 0
\(523\) −28.4222 −1.24282 −0.621408 0.783487i \(-0.713439\pi\)
−0.621408 + 0.783487i \(0.713439\pi\)
\(524\) 0 0
\(525\) −13.5139 −0.589794
\(526\) 0 0
\(527\) −4.81665 −0.209817
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.90833 0.299796
\(532\) 0 0
\(533\) −2.88057 −0.124771
\(534\) 0 0
\(535\) −45.2389 −1.95585
\(536\) 0 0
\(537\) −6.30278 −0.271985
\(538\) 0 0
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) 15.8167 0.680011 0.340006 0.940423i \(-0.389571\pi\)
0.340006 + 0.940423i \(0.389571\pi\)
\(542\) 0 0
\(543\) −14.8167 −0.635843
\(544\) 0 0
\(545\) −29.7250 −1.27328
\(546\) 0 0
\(547\) −24.7250 −1.05716 −0.528582 0.848882i \(-0.677276\pi\)
−0.528582 + 0.848882i \(0.677276\pi\)
\(548\) 0 0
\(549\) −11.9083 −0.508235
\(550\) 0 0
\(551\) 46.0278 1.96085
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) 0 0
\(555\) −38.7250 −1.64378
\(556\) 0 0
\(557\) −31.0278 −1.31469 −0.657344 0.753591i \(-0.728320\pi\)
−0.657344 + 0.753591i \(0.728320\pi\)
\(558\) 0 0
\(559\) −16.3028 −0.689534
\(560\) 0 0
\(561\) −8.02776 −0.338932
\(562\) 0 0
\(563\) 3.66947 0.154650 0.0773248 0.997006i \(-0.475362\pi\)
0.0773248 + 0.997006i \(0.475362\pi\)
\(564\) 0 0
\(565\) −7.30278 −0.307230
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 25.2111 1.05690 0.528452 0.848963i \(-0.322773\pi\)
0.528452 + 0.848963i \(0.322773\pi\)
\(570\) 0 0
\(571\) 4.21110 0.176229 0.0881146 0.996110i \(-0.471916\pi\)
0.0881146 + 0.996110i \(0.471916\pi\)
\(572\) 0 0
\(573\) 2.60555 0.108848
\(574\) 0 0
\(575\) −13.5139 −0.563568
\(576\) 0 0
\(577\) −7.57779 −0.315468 −0.157734 0.987482i \(-0.550419\pi\)
−0.157734 + 0.987482i \(0.550419\pi\)
\(578\) 0 0
\(579\) 27.0278 1.12324
\(580\) 0 0
\(581\) 1.60555 0.0666095
\(582\) 0 0
\(583\) 27.5694 1.14181
\(584\) 0 0
\(585\) 5.60555 0.231761
\(586\) 0 0
\(587\) −22.1472 −0.914112 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(588\) 0 0
\(589\) 16.8167 0.692918
\(590\) 0 0
\(591\) 17.9083 0.736650
\(592\) 0 0
\(593\) 0.972244 0.0399253 0.0199626 0.999801i \(-0.493645\pi\)
0.0199626 + 0.999801i \(0.493645\pi\)
\(594\) 0 0
\(595\) −6.90833 −0.283214
\(596\) 0 0
\(597\) −5.51388 −0.225668
\(598\) 0 0
\(599\) 21.4861 0.877899 0.438950 0.898512i \(-0.355351\pi\)
0.438950 + 0.898512i \(0.355351\pi\)
\(600\) 0 0
\(601\) −5.93608 −0.242138 −0.121069 0.992644i \(-0.538632\pi\)
−0.121069 + 0.992644i \(0.538632\pi\)
\(602\) 0 0
\(603\) 1.09167 0.0444564
\(604\) 0 0
\(605\) −60.2389 −2.44906
\(606\) 0 0
\(607\) 42.5139 1.72559 0.862793 0.505558i \(-0.168713\pi\)
0.862793 + 0.505558i \(0.168713\pi\)
\(608\) 0 0
\(609\) 8.21110 0.332731
\(610\) 0 0
\(611\) −1.81665 −0.0734939
\(612\) 0 0
\(613\) −18.8167 −0.759997 −0.379999 0.924987i \(-0.624076\pi\)
−0.379999 + 0.924987i \(0.624076\pi\)
\(614\) 0 0
\(615\) 9.51388 0.383637
\(616\) 0 0
\(617\) −17.7250 −0.713581 −0.356790 0.934184i \(-0.616129\pi\)
−0.356790 + 0.934184i \(0.616129\pi\)
\(618\) 0 0
\(619\) −35.1194 −1.41157 −0.705785 0.708427i \(-0.749405\pi\)
−0.705785 + 0.708427i \(0.749405\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 0.0916731 0.00367280
\(624\) 0 0
\(625\) 90.0555 3.60222
\(626\) 0 0
\(627\) 28.0278 1.11932
\(628\) 0 0
\(629\) −14.4500 −0.576158
\(630\) 0 0
\(631\) 48.0278 1.91195 0.955977 0.293440i \(-0.0948002\pi\)
0.955977 + 0.293440i \(0.0948002\pi\)
\(632\) 0 0
\(633\) 1.42221 0.0565276
\(634\) 0 0
\(635\) 22.8167 0.905451
\(636\) 0 0
\(637\) −1.30278 −0.0516179
\(638\) 0 0
\(639\) 9.90833 0.391967
\(640\) 0 0
\(641\) 43.5416 1.71979 0.859896 0.510470i \(-0.170529\pi\)
0.859896 + 0.510470i \(0.170529\pi\)
\(642\) 0 0
\(643\) 46.5694 1.83652 0.918259 0.395981i \(-0.129595\pi\)
0.918259 + 0.395981i \(0.129595\pi\)
\(644\) 0 0
\(645\) 53.8444 2.12012
\(646\) 0 0
\(647\) 23.3305 0.917218 0.458609 0.888638i \(-0.348348\pi\)
0.458609 + 0.888638i \(0.348348\pi\)
\(648\) 0 0
\(649\) 34.5416 1.35588
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) 0 0
\(653\) 12.4861 0.488620 0.244310 0.969697i \(-0.421439\pi\)
0.244310 + 0.969697i \(0.421439\pi\)
\(654\) 0 0
\(655\) 75.7527 2.95990
\(656\) 0 0
\(657\) 12.2111 0.476400
\(658\) 0 0
\(659\) 0.633308 0.0246702 0.0123351 0.999924i \(-0.496074\pi\)
0.0123351 + 0.999924i \(0.496074\pi\)
\(660\) 0 0
\(661\) −0.816654 −0.0317642 −0.0158821 0.999874i \(-0.505056\pi\)
−0.0158821 + 0.999874i \(0.505056\pi\)
\(662\) 0 0
\(663\) 2.09167 0.0812339
\(664\) 0 0
\(665\) 24.1194 0.935311
\(666\) 0 0
\(667\) 8.21110 0.317935
\(668\) 0 0
\(669\) 9.09167 0.351504
\(670\) 0 0
\(671\) −59.5416 −2.29858
\(672\) 0 0
\(673\) 16.6333 0.641167 0.320583 0.947220i \(-0.396121\pi\)
0.320583 + 0.947220i \(0.396121\pi\)
\(674\) 0 0
\(675\) −13.5139 −0.520149
\(676\) 0 0
\(677\) −24.1472 −0.928052 −0.464026 0.885822i \(-0.653596\pi\)
−0.464026 + 0.885822i \(0.653596\pi\)
\(678\) 0 0
\(679\) −10.3944 −0.398902
\(680\) 0 0
\(681\) 19.3305 0.740748
\(682\) 0 0
\(683\) −27.4222 −1.04928 −0.524641 0.851324i \(-0.675800\pi\)
−0.524641 + 0.851324i \(0.675800\pi\)
\(684\) 0 0
\(685\) 20.7250 0.791861
\(686\) 0 0
\(687\) 15.5139 0.591891
\(688\) 0 0
\(689\) −7.18335 −0.273664
\(690\) 0 0
\(691\) 13.4861 0.513036 0.256518 0.966539i \(-0.417425\pi\)
0.256518 + 0.966539i \(0.417425\pi\)
\(692\) 0 0
\(693\) 5.00000 0.189934
\(694\) 0 0
\(695\) 21.9083 0.831030
\(696\) 0 0
\(697\) 3.55004 0.134467
\(698\) 0 0
\(699\) −25.3305 −0.958089
\(700\) 0 0
\(701\) 28.5416 1.07800 0.539001 0.842305i \(-0.318802\pi\)
0.539001 + 0.842305i \(0.318802\pi\)
\(702\) 0 0
\(703\) 50.4500 1.90276
\(704\) 0 0
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) 5.69722 0.214266
\(708\) 0 0
\(709\) −7.66947 −0.288033 −0.144016 0.989575i \(-0.546002\pi\)
−0.144016 + 0.989575i \(0.546002\pi\)
\(710\) 0 0
\(711\) 1.00000 0.0375029
\(712\) 0 0
\(713\) 3.00000 0.112351
\(714\) 0 0
\(715\) 28.0278 1.04818
\(716\) 0 0
\(717\) −24.9083 −0.930219
\(718\) 0 0
\(719\) 0.211103 0.00787280 0.00393640 0.999992i \(-0.498747\pi\)
0.00393640 + 0.999992i \(0.498747\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 24.0278 0.893602
\(724\) 0 0
\(725\) −110.964 −4.12109
\(726\) 0 0
\(727\) −46.8722 −1.73839 −0.869196 0.494467i \(-0.835363\pi\)
−0.869196 + 0.494467i \(0.835363\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 20.0917 0.743117
\(732\) 0 0
\(733\) −19.3944 −0.716350 −0.358175 0.933654i \(-0.616601\pi\)
−0.358175 + 0.933654i \(0.616601\pi\)
\(734\) 0 0
\(735\) 4.30278 0.158710
\(736\) 0 0
\(737\) 5.45837 0.201061
\(738\) 0 0
\(739\) 14.5778 0.536253 0.268126 0.963384i \(-0.413596\pi\)
0.268126 + 0.963384i \(0.413596\pi\)
\(740\) 0 0
\(741\) −7.30278 −0.268274
\(742\) 0 0
\(743\) −12.4861 −0.458071 −0.229036 0.973418i \(-0.573557\pi\)
−0.229036 + 0.973418i \(0.573557\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 1.60555 0.0587440
\(748\) 0 0
\(749\) 10.5139 0.384169
\(750\) 0 0
\(751\) −12.3028 −0.448935 −0.224467 0.974482i \(-0.572064\pi\)
−0.224467 + 0.974482i \(0.572064\pi\)
\(752\) 0 0
\(753\) 2.18335 0.0795656
\(754\) 0 0
\(755\) 40.4222 1.47111
\(756\) 0 0
\(757\) −46.6333 −1.69492 −0.847458 0.530862i \(-0.821868\pi\)
−0.847458 + 0.530862i \(0.821868\pi\)
\(758\) 0 0
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 6.90833 0.250098
\(764\) 0 0
\(765\) −6.90833 −0.249771
\(766\) 0 0
\(767\) −9.00000 −0.324971
\(768\) 0 0
\(769\) 37.4500 1.35048 0.675240 0.737598i \(-0.264040\pi\)
0.675240 + 0.737598i \(0.264040\pi\)
\(770\) 0 0
\(771\) 9.02776 0.325127
\(772\) 0 0
\(773\) −25.8444 −0.929559 −0.464779 0.885427i \(-0.653866\pi\)
−0.464779 + 0.885427i \(0.653866\pi\)
\(774\) 0 0
\(775\) −40.5416 −1.45630
\(776\) 0 0
\(777\) 9.00000 0.322873
\(778\) 0 0
\(779\) −12.3944 −0.444077
\(780\) 0 0
\(781\) 49.5416 1.77274
\(782\) 0 0
\(783\) 8.21110 0.293441
\(784\) 0 0
\(785\) −76.6611 −2.73615
\(786\) 0 0
\(787\) 29.1472 1.03898 0.519492 0.854475i \(-0.326121\pi\)
0.519492 + 0.854475i \(0.326121\pi\)
\(788\) 0 0
\(789\) 17.6056 0.626774
\(790\) 0 0
\(791\) 1.69722 0.0603464
\(792\) 0 0
\(793\) 15.5139 0.550914
\(794\) 0 0
\(795\) 23.7250 0.841438
\(796\) 0 0
\(797\) −2.97224 −0.105282 −0.0526411 0.998613i \(-0.516764\pi\)
−0.0526411 + 0.998613i \(0.516764\pi\)
\(798\) 0 0
\(799\) 2.23886 0.0792051
\(800\) 0 0
\(801\) 0.0916731 0.00323911
\(802\) 0 0
\(803\) 61.0555 2.15460
\(804\) 0 0
\(805\) 4.30278 0.151653
\(806\) 0 0
\(807\) 9.90833 0.348790
\(808\) 0 0
\(809\) −20.9361 −0.736073 −0.368037 0.929811i \(-0.619970\pi\)
−0.368037 + 0.929811i \(0.619970\pi\)
\(810\) 0 0
\(811\) 17.6056 0.618215 0.309107 0.951027i \(-0.399970\pi\)
0.309107 + 0.951027i \(0.399970\pi\)
\(812\) 0 0
\(813\) −3.60555 −0.126452
\(814\) 0 0
\(815\) 80.5694 2.82222
\(816\) 0 0
\(817\) −70.1472 −2.45414
\(818\) 0 0
\(819\) −1.30278 −0.0455227
\(820\) 0 0
\(821\) −26.6056 −0.928540 −0.464270 0.885694i \(-0.653683\pi\)
−0.464270 + 0.885694i \(0.653683\pi\)
\(822\) 0 0
\(823\) 11.5139 0.401349 0.200674 0.979658i \(-0.435687\pi\)
0.200674 + 0.979658i \(0.435687\pi\)
\(824\) 0 0
\(825\) −67.5694 −2.35246
\(826\) 0 0
\(827\) −0.908327 −0.0315856 −0.0157928 0.999875i \(-0.505027\pi\)
−0.0157928 + 0.999875i \(0.505027\pi\)
\(828\) 0 0
\(829\) 34.4500 1.19650 0.598248 0.801311i \(-0.295864\pi\)
0.598248 + 0.801311i \(0.295864\pi\)
\(830\) 0 0
\(831\) 8.69722 0.301703
\(832\) 0 0
\(833\) 1.60555 0.0556291
\(834\) 0 0
\(835\) −80.9638 −2.80187
\(836\) 0 0
\(837\) 3.00000 0.103695
\(838\) 0 0
\(839\) −34.6972 −1.19788 −0.598941 0.800793i \(-0.704411\pi\)
−0.598941 + 0.800793i \(0.704411\pi\)
\(840\) 0 0
\(841\) 38.4222 1.32490
\(842\) 0 0
\(843\) −6.18335 −0.212966
\(844\) 0 0
\(845\) 48.6333 1.67304
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 0 0
\(849\) 13.6972 0.470088
\(850\) 0 0
\(851\) 9.00000 0.308516
\(852\) 0 0
\(853\) −7.76114 −0.265736 −0.132868 0.991134i \(-0.542419\pi\)
−0.132868 + 0.991134i \(0.542419\pi\)
\(854\) 0 0
\(855\) 24.1194 0.824867
\(856\) 0 0
\(857\) 0.238859 0.00815927 0.00407963 0.999992i \(-0.498701\pi\)
0.00407963 + 0.999992i \(0.498701\pi\)
\(858\) 0 0
\(859\) −28.7889 −0.982265 −0.491132 0.871085i \(-0.663417\pi\)
−0.491132 + 0.871085i \(0.663417\pi\)
\(860\) 0 0
\(861\) −2.21110 −0.0753542
\(862\) 0 0
\(863\) 56.2389 1.91439 0.957197 0.289439i \(-0.0934687\pi\)
0.957197 + 0.289439i \(0.0934687\pi\)
\(864\) 0 0
\(865\) 16.3028 0.554311
\(866\) 0 0
\(867\) 14.4222 0.489804
\(868\) 0 0
\(869\) 5.00000 0.169613
\(870\) 0 0
\(871\) −1.42221 −0.0481896
\(872\) 0 0
\(873\) −10.3944 −0.351799
\(874\) 0 0
\(875\) −36.6333 −1.23843
\(876\) 0 0
\(877\) −29.2111 −0.986389 −0.493194 0.869919i \(-0.664171\pi\)
−0.493194 + 0.869919i \(0.664171\pi\)
\(878\) 0 0
\(879\) −32.8444 −1.10781
\(880\) 0 0
\(881\) −22.1833 −0.747376 −0.373688 0.927554i \(-0.621907\pi\)
−0.373688 + 0.927554i \(0.621907\pi\)
\(882\) 0 0
\(883\) 36.5139 1.22879 0.614395 0.788999i \(-0.289400\pi\)
0.614395 + 0.788999i \(0.289400\pi\)
\(884\) 0 0
\(885\) 29.7250 0.999194
\(886\) 0 0
\(887\) −34.9361 −1.17304 −0.586519 0.809935i \(-0.699502\pi\)
−0.586519 + 0.809935i \(0.699502\pi\)
\(888\) 0 0
\(889\) −5.30278 −0.177849
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) −7.81665 −0.261574
\(894\) 0 0
\(895\) −27.1194 −0.906503
\(896\) 0 0
\(897\) −1.30278 −0.0434984
\(898\) 0 0
\(899\) 24.6333 0.821567
\(900\) 0 0
\(901\) 8.85281 0.294930
\(902\) 0 0
\(903\) −12.5139 −0.416436
\(904\) 0 0
\(905\) −63.7527 −2.11921
\(906\) 0 0
\(907\) −53.1472 −1.76472 −0.882362 0.470572i \(-0.844048\pi\)
−0.882362 + 0.470572i \(0.844048\pi\)
\(908\) 0 0
\(909\) 5.69722 0.188965
\(910\) 0 0
\(911\) −26.2389 −0.869332 −0.434666 0.900592i \(-0.643134\pi\)
−0.434666 + 0.900592i \(0.643134\pi\)
\(912\) 0 0
\(913\) 8.02776 0.265680
\(914\) 0 0
\(915\) −51.2389 −1.69390
\(916\) 0 0
\(917\) −17.6056 −0.581387
\(918\) 0 0
\(919\) 17.6056 0.580754 0.290377 0.956912i \(-0.406219\pi\)
0.290377 + 0.956912i \(0.406219\pi\)
\(920\) 0 0
\(921\) 3.78890 0.124848
\(922\) 0 0
\(923\) −12.9083 −0.424883
\(924\) 0 0
\(925\) −121.625 −3.99900
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) 0.669468 0.0219645 0.0109823 0.999940i \(-0.496504\pi\)
0.0109823 + 0.999940i \(0.496504\pi\)
\(930\) 0 0
\(931\) −5.60555 −0.183715
\(932\) 0 0
\(933\) 8.51388 0.278732
\(934\) 0 0
\(935\) −34.5416 −1.12963
\(936\) 0 0
\(937\) −33.4222 −1.09186 −0.545928 0.837832i \(-0.683823\pi\)
−0.545928 + 0.837832i \(0.683823\pi\)
\(938\) 0 0
\(939\) 13.2111 0.431128
\(940\) 0 0
\(941\) 56.8444 1.85307 0.926537 0.376203i \(-0.122770\pi\)
0.926537 + 0.376203i \(0.122770\pi\)
\(942\) 0 0
\(943\) −2.21110 −0.0720034
\(944\) 0 0
\(945\) 4.30278 0.139969
\(946\) 0 0
\(947\) −29.6333 −0.962953 −0.481477 0.876459i \(-0.659899\pi\)
−0.481477 + 0.876459i \(0.659899\pi\)
\(948\) 0 0
\(949\) −15.9083 −0.516406
\(950\) 0 0
\(951\) 12.4861 0.404890
\(952\) 0 0
\(953\) −11.6695 −0.378011 −0.189006 0.981976i \(-0.560526\pi\)
−0.189006 + 0.981976i \(0.560526\pi\)
\(954\) 0 0
\(955\) 11.2111 0.362783
\(956\) 0 0
\(957\) 41.0555 1.32714
\(958\) 0 0
\(959\) −4.81665 −0.155538
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 10.5139 0.338805
\(964\) 0 0
\(965\) 116.294 3.74365
\(966\) 0 0
\(967\) 45.2666 1.45568 0.727838 0.685749i \(-0.240525\pi\)
0.727838 + 0.685749i \(0.240525\pi\)
\(968\) 0 0
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 45.9083 1.47327 0.736634 0.676291i \(-0.236414\pi\)
0.736634 + 0.676291i \(0.236414\pi\)
\(972\) 0 0
\(973\) −5.09167 −0.163232
\(974\) 0 0
\(975\) 17.6056 0.563829
\(976\) 0 0
\(977\) −30.1472 −0.964494 −0.482247 0.876035i \(-0.660179\pi\)
−0.482247 + 0.876035i \(0.660179\pi\)
\(978\) 0 0
\(979\) 0.458365 0.0146494
\(980\) 0 0
\(981\) 6.90833 0.220566
\(982\) 0 0
\(983\) 1.18335 0.0377429 0.0188714 0.999822i \(-0.493993\pi\)
0.0188714 + 0.999822i \(0.493993\pi\)
\(984\) 0 0
\(985\) 77.0555 2.45519
\(986\) 0 0
\(987\) −1.39445 −0.0443858
\(988\) 0 0
\(989\) −12.5139 −0.397918
\(990\) 0 0
\(991\) −17.4861 −0.555465 −0.277732 0.960658i \(-0.589583\pi\)
−0.277732 + 0.960658i \(0.589583\pi\)
\(992\) 0 0
\(993\) −25.6333 −0.813448
\(994\) 0 0
\(995\) −23.7250 −0.752132
\(996\) 0 0
\(997\) 36.0278 1.14101 0.570505 0.821294i \(-0.306747\pi\)
0.570505 + 0.821294i \(0.306747\pi\)
\(998\) 0 0
\(999\) 9.00000 0.284747
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.x.1.1 2
4.3 odd 2 483.2.a.f.1.2 2
12.11 even 2 1449.2.a.j.1.1 2
28.27 even 2 3381.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.f.1.2 2 4.3 odd 2
1449.2.a.j.1.1 2 12.11 even 2
3381.2.a.p.1.2 2 28.27 even 2
7728.2.a.x.1.1 2 1.1 even 1 trivial