Properties

Label 2352.4.a.cm.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 152 x^{2} - 177 x + 2922\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.92665\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -15.8130 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -15.8130 q^{5} +9.00000 q^{9} +30.2479 q^{11} -61.6298 q^{13} +47.4389 q^{15} +56.1448 q^{17} +139.400 q^{19} +8.14480 q^{23} +125.049 q^{25} -27.0000 q^{27} -0.217857 q^{29} -176.581 q^{31} -90.7438 q^{33} +211.080 q^{37} +184.889 q^{39} -293.305 q^{41} -434.591 q^{43} -142.317 q^{45} -483.396 q^{47} -168.434 q^{51} +20.5073 q^{53} -478.309 q^{55} -418.201 q^{57} +231.348 q^{59} -838.701 q^{61} +974.549 q^{65} +624.040 q^{67} -24.4344 q^{69} -227.106 q^{71} +43.0494 q^{73} -375.148 q^{75} +309.060 q^{79} +81.0000 q^{81} -1233.99 q^{83} -887.815 q^{85} +0.653571 q^{87} +1144.36 q^{89} +529.742 q^{93} -2204.33 q^{95} +1688.12 q^{97} +272.231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{3} + 4q^{5} + 36q^{9} + O(q^{10}) \) \( 4q - 12q^{3} + 4q^{5} + 36q^{9} - 14q^{11} + 22q^{13} - 12q^{15} + 96q^{17} + 26q^{19} - 96q^{23} + 110q^{25} - 108q^{27} - 76q^{29} - 238q^{31} + 42q^{33} + 562q^{37} - 66q^{39} + 428q^{41} + 258q^{43} + 36q^{45} + 80q^{47} - 288q^{51} - 1476q^{55} - 78q^{57} - 262q^{59} - 276q^{61} + 2196q^{65} - 150q^{67} + 288q^{69} + 848q^{71} - 218q^{73} - 330q^{75} - 1762q^{79} + 324q^{81} - 3450q^{83} + 1452q^{85} + 228q^{87} - 344q^{89} + 714q^{93} - 2004q^{95} - 622q^{97} - 126q^{99} + O(q^{100}) \)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −15.8130 −1.41435 −0.707177 0.707037i \(-0.750031\pi\)
−0.707177 + 0.707037i \(0.750031\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 30.2479 0.829099 0.414550 0.910027i \(-0.363939\pi\)
0.414550 + 0.910027i \(0.363939\pi\)
\(12\) 0 0
\(13\) −61.6298 −1.31485 −0.657424 0.753521i \(-0.728354\pi\)
−0.657424 + 0.753521i \(0.728354\pi\)
\(14\) 0 0
\(15\) 47.4389 0.816577
\(16\) 0 0
\(17\) 56.1448 0.801007 0.400503 0.916295i \(-0.368835\pi\)
0.400503 + 0.916295i \(0.368835\pi\)
\(18\) 0 0
\(19\) 139.400 1.68319 0.841596 0.540107i \(-0.181616\pi\)
0.841596 + 0.540107i \(0.181616\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.14480 0.0738395 0.0369197 0.999318i \(-0.488245\pi\)
0.0369197 + 0.999318i \(0.488245\pi\)
\(24\) 0 0
\(25\) 125.049 1.00040
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −0.217857 −0.00139500 −0.000697500 1.00000i \(-0.500222\pi\)
−0.000697500 1.00000i \(0.500222\pi\)
\(30\) 0 0
\(31\) −176.581 −1.02306 −0.511529 0.859266i \(-0.670921\pi\)
−0.511529 + 0.859266i \(0.670921\pi\)
\(32\) 0 0
\(33\) −90.7438 −0.478681
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 211.080 0.937874 0.468937 0.883232i \(-0.344637\pi\)
0.468937 + 0.883232i \(0.344637\pi\)
\(38\) 0 0
\(39\) 184.889 0.759128
\(40\) 0 0
\(41\) −293.305 −1.11723 −0.558616 0.829427i \(-0.688667\pi\)
−0.558616 + 0.829427i \(0.688667\pi\)
\(42\) 0 0
\(43\) −434.591 −1.54127 −0.770634 0.637278i \(-0.780060\pi\)
−0.770634 + 0.637278i \(0.780060\pi\)
\(44\) 0 0
\(45\) −142.317 −0.471451
\(46\) 0 0
\(47\) −483.396 −1.50022 −0.750112 0.661311i \(-0.770000\pi\)
−0.750112 + 0.661311i \(0.770000\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −168.434 −0.462461
\(52\) 0 0
\(53\) 20.5073 0.0531489 0.0265745 0.999647i \(-0.491540\pi\)
0.0265745 + 0.999647i \(0.491540\pi\)
\(54\) 0 0
\(55\) −478.309 −1.17264
\(56\) 0 0
\(57\) −418.201 −0.971792
\(58\) 0 0
\(59\) 231.348 0.510490 0.255245 0.966876i \(-0.417844\pi\)
0.255245 + 0.966876i \(0.417844\pi\)
\(60\) 0 0
\(61\) −838.701 −1.76041 −0.880203 0.474598i \(-0.842593\pi\)
−0.880203 + 0.474598i \(0.842593\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 974.549 1.85966
\(66\) 0 0
\(67\) 624.040 1.13789 0.568945 0.822375i \(-0.307352\pi\)
0.568945 + 0.822375i \(0.307352\pi\)
\(68\) 0 0
\(69\) −24.4344 −0.0426312
\(70\) 0 0
\(71\) −227.106 −0.379613 −0.189806 0.981822i \(-0.560786\pi\)
−0.189806 + 0.981822i \(0.560786\pi\)
\(72\) 0 0
\(73\) 43.0494 0.0690213 0.0345107 0.999404i \(-0.489013\pi\)
0.0345107 + 0.999404i \(0.489013\pi\)
\(74\) 0 0
\(75\) −375.148 −0.577579
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 309.060 0.440152 0.220076 0.975483i \(-0.429370\pi\)
0.220076 + 0.975483i \(0.429370\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1233.99 −1.63190 −0.815951 0.578122i \(-0.803786\pi\)
−0.815951 + 0.578122i \(0.803786\pi\)
\(84\) 0 0
\(85\) −887.815 −1.13291
\(86\) 0 0
\(87\) 0.653571 0.000805404 0
\(88\) 0 0
\(89\) 1144.36 1.36294 0.681470 0.731846i \(-0.261341\pi\)
0.681470 + 0.731846i \(0.261341\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 529.742 0.590663
\(94\) 0 0
\(95\) −2204.33 −2.38063
\(96\) 0 0
\(97\) 1688.12 1.76704 0.883520 0.468394i \(-0.155167\pi\)
0.883520 + 0.468394i \(0.155167\pi\)
\(98\) 0 0
\(99\) 272.231 0.276366
\(100\) 0 0
\(101\) −1487.86 −1.46582 −0.732909 0.680327i \(-0.761838\pi\)
−0.732909 + 0.680327i \(0.761838\pi\)
\(102\) 0 0
\(103\) 778.408 0.744648 0.372324 0.928103i \(-0.378561\pi\)
0.372324 + 0.928103i \(0.378561\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1302.08 −1.17642 −0.588209 0.808709i \(-0.700167\pi\)
−0.588209 + 0.808709i \(0.700167\pi\)
\(108\) 0 0
\(109\) 469.082 0.412201 0.206100 0.978531i \(-0.433923\pi\)
0.206100 + 0.978531i \(0.433923\pi\)
\(110\) 0 0
\(111\) −633.240 −0.541482
\(112\) 0 0
\(113\) 1653.86 1.37684 0.688418 0.725314i \(-0.258306\pi\)
0.688418 + 0.725314i \(0.258306\pi\)
\(114\) 0 0
\(115\) −128.793 −0.104435
\(116\) 0 0
\(117\) −554.668 −0.438283
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −416.063 −0.312594
\(122\) 0 0
\(123\) 879.914 0.645034
\(124\) 0 0
\(125\) −0.781685 −0.000559328 0
\(126\) 0 0
\(127\) −163.760 −0.114420 −0.0572100 0.998362i \(-0.518220\pi\)
−0.0572100 + 0.998362i \(0.518220\pi\)
\(128\) 0 0
\(129\) 1303.77 0.889851
\(130\) 0 0
\(131\) 521.894 0.348077 0.174038 0.984739i \(-0.444318\pi\)
0.174038 + 0.984739i \(0.444318\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 426.950 0.272192
\(136\) 0 0
\(137\) 2676.89 1.66936 0.834679 0.550736i \(-0.185653\pi\)
0.834679 + 0.550736i \(0.185653\pi\)
\(138\) 0 0
\(139\) −853.692 −0.520930 −0.260465 0.965483i \(-0.583876\pi\)
−0.260465 + 0.965483i \(0.583876\pi\)
\(140\) 0 0
\(141\) 1450.19 0.866154
\(142\) 0 0
\(143\) −1864.17 −1.09014
\(144\) 0 0
\(145\) 3.44496 0.00197302
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 557.686 0.306627 0.153313 0.988178i \(-0.451006\pi\)
0.153313 + 0.988178i \(0.451006\pi\)
\(150\) 0 0
\(151\) −1769.35 −0.953560 −0.476780 0.879023i \(-0.658196\pi\)
−0.476780 + 0.879023i \(0.658196\pi\)
\(152\) 0 0
\(153\) 505.303 0.267002
\(154\) 0 0
\(155\) 2792.26 1.44697
\(156\) 0 0
\(157\) −2404.89 −1.22249 −0.611247 0.791440i \(-0.709332\pi\)
−0.611247 + 0.791440i \(0.709332\pi\)
\(158\) 0 0
\(159\) −61.5219 −0.0306855
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3378.79 −1.62360 −0.811802 0.583932i \(-0.801513\pi\)
−0.811802 + 0.583932i \(0.801513\pi\)
\(164\) 0 0
\(165\) 1434.93 0.677024
\(166\) 0 0
\(167\) 805.900 0.373428 0.186714 0.982414i \(-0.440216\pi\)
0.186714 + 0.982414i \(0.440216\pi\)
\(168\) 0 0
\(169\) 1601.23 0.728826
\(170\) 0 0
\(171\) 1254.60 0.561064
\(172\) 0 0
\(173\) −840.504 −0.369378 −0.184689 0.982797i \(-0.559128\pi\)
−0.184689 + 0.982797i \(0.559128\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −694.043 −0.294731
\(178\) 0 0
\(179\) −1604.61 −0.670024 −0.335012 0.942214i \(-0.608740\pi\)
−0.335012 + 0.942214i \(0.608740\pi\)
\(180\) 0 0
\(181\) −3779.43 −1.55206 −0.776029 0.630697i \(-0.782769\pi\)
−0.776029 + 0.630697i \(0.782769\pi\)
\(182\) 0 0
\(183\) 2516.10 1.01637
\(184\) 0 0
\(185\) −3337.80 −1.32649
\(186\) 0 0
\(187\) 1698.26 0.664114
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1849.60 0.700691 0.350346 0.936621i \(-0.386064\pi\)
0.350346 + 0.936621i \(0.386064\pi\)
\(192\) 0 0
\(193\) −353.003 −0.131656 −0.0658282 0.997831i \(-0.520969\pi\)
−0.0658282 + 0.997831i \(0.520969\pi\)
\(194\) 0 0
\(195\) −2923.65 −1.07368
\(196\) 0 0
\(197\) 4448.22 1.60874 0.804372 0.594125i \(-0.202502\pi\)
0.804372 + 0.594125i \(0.202502\pi\)
\(198\) 0 0
\(199\) 2240.15 0.797991 0.398995 0.916953i \(-0.369359\pi\)
0.398995 + 0.916953i \(0.369359\pi\)
\(200\) 0 0
\(201\) −1872.12 −0.656961
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4638.01 1.58016
\(206\) 0 0
\(207\) 73.3032 0.0246132
\(208\) 0 0
\(209\) 4216.58 1.39553
\(210\) 0 0
\(211\) −1224.34 −0.399463 −0.199732 0.979851i \(-0.564007\pi\)
−0.199732 + 0.979851i \(0.564007\pi\)
\(212\) 0 0
\(213\) 681.318 0.219170
\(214\) 0 0
\(215\) 6872.17 2.17990
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −129.148 −0.0398495
\(220\) 0 0
\(221\) −3460.19 −1.05320
\(222\) 0 0
\(223\) 3457.37 1.03822 0.519108 0.854708i \(-0.326264\pi\)
0.519108 + 0.854708i \(0.326264\pi\)
\(224\) 0 0
\(225\) 1125.44 0.333465
\(226\) 0 0
\(227\) 5178.22 1.51405 0.757027 0.653383i \(-0.226651\pi\)
0.757027 + 0.653383i \(0.226651\pi\)
\(228\) 0 0
\(229\) 2311.88 0.667134 0.333567 0.942726i \(-0.391748\pi\)
0.333567 + 0.942726i \(0.391748\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4742.97 1.33357 0.666787 0.745249i \(-0.267669\pi\)
0.666787 + 0.745249i \(0.267669\pi\)
\(234\) 0 0
\(235\) 7643.91 2.12185
\(236\) 0 0
\(237\) −927.181 −0.254122
\(238\) 0 0
\(239\) 1412.59 0.382313 0.191157 0.981560i \(-0.438776\pi\)
0.191157 + 0.981560i \(0.438776\pi\)
\(240\) 0 0
\(241\) −1881.15 −0.502804 −0.251402 0.967883i \(-0.580892\pi\)
−0.251402 + 0.967883i \(0.580892\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8591.22 −2.21314
\(248\) 0 0
\(249\) 3701.96 0.942179
\(250\) 0 0
\(251\) −4519.64 −1.13656 −0.568281 0.822834i \(-0.692391\pi\)
−0.568281 + 0.822834i \(0.692391\pi\)
\(252\) 0 0
\(253\) 246.363 0.0612202
\(254\) 0 0
\(255\) 2663.44 0.654084
\(256\) 0 0
\(257\) 7912.65 1.92054 0.960268 0.279079i \(-0.0900291\pi\)
0.960268 + 0.279079i \(0.0900291\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.96071 −0.000465000 0
\(262\) 0 0
\(263\) 3431.16 0.804466 0.402233 0.915537i \(-0.368234\pi\)
0.402233 + 0.915537i \(0.368234\pi\)
\(264\) 0 0
\(265\) −324.281 −0.0751714
\(266\) 0 0
\(267\) −3433.07 −0.786894
\(268\) 0 0
\(269\) 8299.19 1.88108 0.940540 0.339682i \(-0.110319\pi\)
0.940540 + 0.339682i \(0.110319\pi\)
\(270\) 0 0
\(271\) 5924.49 1.32800 0.663998 0.747734i \(-0.268858\pi\)
0.663998 + 0.747734i \(0.268858\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3782.49 0.829427
\(276\) 0 0
\(277\) 1520.60 0.329834 0.164917 0.986307i \(-0.447264\pi\)
0.164917 + 0.986307i \(0.447264\pi\)
\(278\) 0 0
\(279\) −1589.22 −0.341019
\(280\) 0 0
\(281\) −352.333 −0.0747986 −0.0373993 0.999300i \(-0.511907\pi\)
−0.0373993 + 0.999300i \(0.511907\pi\)
\(282\) 0 0
\(283\) −4288.49 −0.900792 −0.450396 0.892829i \(-0.648717\pi\)
−0.450396 + 0.892829i \(0.648717\pi\)
\(284\) 0 0
\(285\) 6613.00 1.37446
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1760.76 −0.358388
\(290\) 0 0
\(291\) −5064.37 −1.02020
\(292\) 0 0
\(293\) 2661.99 0.530767 0.265384 0.964143i \(-0.414501\pi\)
0.265384 + 0.964143i \(0.414501\pi\)
\(294\) 0 0
\(295\) −3658.29 −0.722013
\(296\) 0 0
\(297\) −816.694 −0.159560
\(298\) 0 0
\(299\) −501.962 −0.0970877
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4463.58 0.846290
\(304\) 0 0
\(305\) 13262.3 2.48983
\(306\) 0 0
\(307\) −145.970 −0.0271366 −0.0135683 0.999908i \(-0.504319\pi\)
−0.0135683 + 0.999908i \(0.504319\pi\)
\(308\) 0 0
\(309\) −2335.22 −0.429923
\(310\) 0 0
\(311\) −3416.51 −0.622935 −0.311467 0.950257i \(-0.600820\pi\)
−0.311467 + 0.950257i \(0.600820\pi\)
\(312\) 0 0
\(313\) 3682.35 0.664980 0.332490 0.943107i \(-0.392111\pi\)
0.332490 + 0.943107i \(0.392111\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1376.78 −0.243936 −0.121968 0.992534i \(-0.538920\pi\)
−0.121968 + 0.992534i \(0.538920\pi\)
\(318\) 0 0
\(319\) −6.58972 −0.00115659
\(320\) 0 0
\(321\) 3906.24 0.679205
\(322\) 0 0
\(323\) 7826.61 1.34825
\(324\) 0 0
\(325\) −7706.77 −1.31537
\(326\) 0 0
\(327\) −1407.24 −0.237984
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2174.93 −0.361163 −0.180581 0.983560i \(-0.557798\pi\)
−0.180581 + 0.983560i \(0.557798\pi\)
\(332\) 0 0
\(333\) 1899.72 0.312625
\(334\) 0 0
\(335\) −9867.92 −1.60938
\(336\) 0 0
\(337\) −6321.14 −1.02176 −0.510882 0.859651i \(-0.670681\pi\)
−0.510882 + 0.859651i \(0.670681\pi\)
\(338\) 0 0
\(339\) −4961.59 −0.794916
\(340\) 0 0
\(341\) −5341.20 −0.848217
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 386.380 0.0602956
\(346\) 0 0
\(347\) 11624.4 1.79836 0.899181 0.437577i \(-0.144163\pi\)
0.899181 + 0.437577i \(0.144163\pi\)
\(348\) 0 0
\(349\) −9841.79 −1.50951 −0.754755 0.656007i \(-0.772244\pi\)
−0.754755 + 0.656007i \(0.772244\pi\)
\(350\) 0 0
\(351\) 1664.00 0.253043
\(352\) 0 0
\(353\) 4531.88 0.683308 0.341654 0.939826i \(-0.389013\pi\)
0.341654 + 0.939826i \(0.389013\pi\)
\(354\) 0 0
\(355\) 3591.22 0.536907
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2280.10 0.335207 0.167603 0.985855i \(-0.446397\pi\)
0.167603 + 0.985855i \(0.446397\pi\)
\(360\) 0 0
\(361\) 12573.5 1.83314
\(362\) 0 0
\(363\) 1248.19 0.180476
\(364\) 0 0
\(365\) −680.739 −0.0976205
\(366\) 0 0
\(367\) −6337.51 −0.901404 −0.450702 0.892674i \(-0.648826\pi\)
−0.450702 + 0.892674i \(0.648826\pi\)
\(368\) 0 0
\(369\) −2639.74 −0.372410
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3624.96 −0.503199 −0.251600 0.967831i \(-0.580957\pi\)
−0.251600 + 0.967831i \(0.580957\pi\)
\(374\) 0 0
\(375\) 2.34506 0.000322928 0
\(376\) 0 0
\(377\) 13.4265 0.00183421
\(378\) 0 0
\(379\) −268.622 −0.0364068 −0.0182034 0.999834i \(-0.505795\pi\)
−0.0182034 + 0.999834i \(0.505795\pi\)
\(380\) 0 0
\(381\) 491.280 0.0660604
\(382\) 0 0
\(383\) 2137.39 0.285158 0.142579 0.989783i \(-0.454460\pi\)
0.142579 + 0.989783i \(0.454460\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3911.32 −0.513756
\(388\) 0 0
\(389\) 1463.23 0.190717 0.0953583 0.995443i \(-0.469600\pi\)
0.0953583 + 0.995443i \(0.469600\pi\)
\(390\) 0 0
\(391\) 457.288 0.0591459
\(392\) 0 0
\(393\) −1565.68 −0.200962
\(394\) 0 0
\(395\) −4887.15 −0.622530
\(396\) 0 0
\(397\) −1237.44 −0.156436 −0.0782180 0.996936i \(-0.524923\pi\)
−0.0782180 + 0.996936i \(0.524923\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6179.89 −0.769598 −0.384799 0.923000i \(-0.625729\pi\)
−0.384799 + 0.923000i \(0.625729\pi\)
\(402\) 0 0
\(403\) 10882.6 1.34517
\(404\) 0 0
\(405\) −1280.85 −0.157150
\(406\) 0 0
\(407\) 6384.73 0.777591
\(408\) 0 0
\(409\) −2254.98 −0.272620 −0.136310 0.990666i \(-0.543524\pi\)
−0.136310 + 0.990666i \(0.543524\pi\)
\(410\) 0 0
\(411\) −8030.67 −0.963805
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 19513.0 2.30808
\(416\) 0 0
\(417\) 2561.08 0.300759
\(418\) 0 0
\(419\) 1404.53 0.163761 0.0818806 0.996642i \(-0.473907\pi\)
0.0818806 + 0.996642i \(0.473907\pi\)
\(420\) 0 0
\(421\) 13068.1 1.51283 0.756414 0.654094i \(-0.226950\pi\)
0.756414 + 0.654094i \(0.226950\pi\)
\(422\) 0 0
\(423\) −4350.56 −0.500074
\(424\) 0 0
\(425\) 7020.88 0.801323
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5592.52 0.629392
\(430\) 0 0
\(431\) 5362.79 0.599342 0.299671 0.954043i \(-0.403123\pi\)
0.299671 + 0.954043i \(0.403123\pi\)
\(432\) 0 0
\(433\) −2495.82 −0.277001 −0.138501 0.990362i \(-0.544228\pi\)
−0.138501 + 0.990362i \(0.544228\pi\)
\(434\) 0 0
\(435\) −10.3349 −0.00113913
\(436\) 0 0
\(437\) 1135.39 0.124286
\(438\) 0 0
\(439\) −3762.12 −0.409012 −0.204506 0.978865i \(-0.565559\pi\)
−0.204506 + 0.978865i \(0.565559\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11445.3 −1.22750 −0.613748 0.789502i \(-0.710339\pi\)
−0.613748 + 0.789502i \(0.710339\pi\)
\(444\) 0 0
\(445\) −18095.7 −1.92768
\(446\) 0 0
\(447\) −1673.06 −0.177031
\(448\) 0 0
\(449\) −9420.02 −0.990107 −0.495054 0.868862i \(-0.664852\pi\)
−0.495054 + 0.868862i \(0.664852\pi\)
\(450\) 0 0
\(451\) −8871.85 −0.926296
\(452\) 0 0
\(453\) 5308.05 0.550538
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11464.0 1.17344 0.586719 0.809790i \(-0.300419\pi\)
0.586719 + 0.809790i \(0.300419\pi\)
\(458\) 0 0
\(459\) −1515.91 −0.154154
\(460\) 0 0
\(461\) −9751.10 −0.985149 −0.492575 0.870270i \(-0.663944\pi\)
−0.492575 + 0.870270i \(0.663944\pi\)
\(462\) 0 0
\(463\) −2182.03 −0.219023 −0.109512 0.993986i \(-0.534929\pi\)
−0.109512 + 0.993986i \(0.534929\pi\)
\(464\) 0 0
\(465\) −8376.78 −0.835406
\(466\) 0 0
\(467\) −3196.34 −0.316722 −0.158361 0.987381i \(-0.550621\pi\)
−0.158361 + 0.987381i \(0.550621\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 7214.68 0.705807
\(472\) 0 0
\(473\) −13145.5 −1.27786
\(474\) 0 0
\(475\) 17432.0 1.68386
\(476\) 0 0
\(477\) 184.566 0.0177163
\(478\) 0 0
\(479\) 12143.6 1.15836 0.579180 0.815200i \(-0.303373\pi\)
0.579180 + 0.815200i \(0.303373\pi\)
\(480\) 0 0
\(481\) −13008.8 −1.23316
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −26694.2 −2.49922
\(486\) 0 0
\(487\) 20302.5 1.88910 0.944551 0.328364i \(-0.106497\pi\)
0.944551 + 0.328364i \(0.106497\pi\)
\(488\) 0 0
\(489\) 10136.4 0.937389
\(490\) 0 0
\(491\) −2562.41 −0.235519 −0.117760 0.993042i \(-0.537571\pi\)
−0.117760 + 0.993042i \(0.537571\pi\)
\(492\) 0 0
\(493\) −12.2315 −0.00111740
\(494\) 0 0
\(495\) −4304.78 −0.390880
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 11656.1 1.04569 0.522844 0.852428i \(-0.324871\pi\)
0.522844 + 0.852428i \(0.324871\pi\)
\(500\) 0 0
\(501\) −2417.70 −0.215598
\(502\) 0 0
\(503\) 18532.8 1.64281 0.821407 0.570342i \(-0.193189\pi\)
0.821407 + 0.570342i \(0.193189\pi\)
\(504\) 0 0
\(505\) 23527.5 2.07318
\(506\) 0 0
\(507\) −4803.69 −0.420788
\(508\) 0 0
\(509\) 18732.1 1.63121 0.815605 0.578609i \(-0.196404\pi\)
0.815605 + 0.578609i \(0.196404\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3763.81 −0.323931
\(514\) 0 0
\(515\) −12308.9 −1.05320
\(516\) 0 0
\(517\) −14621.7 −1.24383
\(518\) 0 0
\(519\) 2521.51 0.213260
\(520\) 0 0
\(521\) −3323.25 −0.279452 −0.139726 0.990190i \(-0.544622\pi\)
−0.139726 + 0.990190i \(0.544622\pi\)
\(522\) 0 0
\(523\) 23297.4 1.94785 0.973925 0.226869i \(-0.0728488\pi\)
0.973925 + 0.226869i \(0.0728488\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9914.08 −0.819476
\(528\) 0 0
\(529\) −12100.7 −0.994548
\(530\) 0 0
\(531\) 2082.13 0.170163
\(532\) 0 0
\(533\) 18076.3 1.46899
\(534\) 0 0
\(535\) 20589.7 1.66387
\(536\) 0 0
\(537\) 4813.83 0.386838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9050.01 0.719206 0.359603 0.933105i \(-0.382912\pi\)
0.359603 + 0.933105i \(0.382912\pi\)
\(542\) 0 0
\(543\) 11338.3 0.896081
\(544\) 0 0
\(545\) −7417.56 −0.582997
\(546\) 0 0
\(547\) 12316.7 0.962749 0.481374 0.876515i \(-0.340138\pi\)
0.481374 + 0.876515i \(0.340138\pi\)
\(548\) 0 0
\(549\) −7548.31 −0.586802
\(550\) 0 0
\(551\) −30.3694 −0.00234805
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10013.4 0.765847
\(556\) 0 0
\(557\) 24904.1 1.89447 0.947236 0.320538i \(-0.103864\pi\)
0.947236 + 0.320538i \(0.103864\pi\)
\(558\) 0 0
\(559\) 26783.8 2.02653
\(560\) 0 0
\(561\) −5094.79 −0.383426
\(562\) 0 0
\(563\) 4146.59 0.310405 0.155203 0.987883i \(-0.450397\pi\)
0.155203 + 0.987883i \(0.450397\pi\)
\(564\) 0 0
\(565\) −26152.5 −1.94733
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7297.48 0.537656 0.268828 0.963188i \(-0.413364\pi\)
0.268828 + 0.963188i \(0.413364\pi\)
\(570\) 0 0
\(571\) 11564.9 0.847593 0.423797 0.905757i \(-0.360697\pi\)
0.423797 + 0.905757i \(0.360697\pi\)
\(572\) 0 0
\(573\) −5548.79 −0.404544
\(574\) 0 0
\(575\) 1018.50 0.0738687
\(576\) 0 0
\(577\) 9473.45 0.683509 0.341755 0.939789i \(-0.388979\pi\)
0.341755 + 0.939789i \(0.388979\pi\)
\(578\) 0 0
\(579\) 1059.01 0.0760119
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 620.303 0.0440657
\(584\) 0 0
\(585\) 8770.94 0.619887
\(586\) 0 0
\(587\) −7336.02 −0.515826 −0.257913 0.966168i \(-0.583035\pi\)
−0.257913 + 0.966168i \(0.583035\pi\)
\(588\) 0 0
\(589\) −24615.4 −1.72200
\(590\) 0 0
\(591\) −13344.7 −0.928809
\(592\) 0 0
\(593\) 18081.5 1.25214 0.626068 0.779768i \(-0.284663\pi\)
0.626068 + 0.779768i \(0.284663\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6720.45 −0.460720
\(598\) 0 0
\(599\) 192.433 0.0131262 0.00656310 0.999978i \(-0.497911\pi\)
0.00656310 + 0.999978i \(0.497911\pi\)
\(600\) 0 0
\(601\) −5055.76 −0.343143 −0.171571 0.985172i \(-0.554884\pi\)
−0.171571 + 0.985172i \(0.554884\pi\)
\(602\) 0 0
\(603\) 5616.36 0.379297
\(604\) 0 0
\(605\) 6579.19 0.442119
\(606\) 0 0
\(607\) −20235.7 −1.35312 −0.676560 0.736388i \(-0.736530\pi\)
−0.676560 + 0.736388i \(0.736530\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 29791.6 1.97257
\(612\) 0 0
\(613\) −3840.51 −0.253045 −0.126523 0.991964i \(-0.540382\pi\)
−0.126523 + 0.991964i \(0.540382\pi\)
\(614\) 0 0
\(615\) −13914.0 −0.912306
\(616\) 0 0
\(617\) −5720.89 −0.373281 −0.186641 0.982428i \(-0.559760\pi\)
−0.186641 + 0.982428i \(0.559760\pi\)
\(618\) 0 0
\(619\) −22940.3 −1.48957 −0.744787 0.667302i \(-0.767449\pi\)
−0.744787 + 0.667302i \(0.767449\pi\)
\(620\) 0 0
\(621\) −219.910 −0.0142104
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −15618.8 −0.999604
\(626\) 0 0
\(627\) −12649.7 −0.805712
\(628\) 0 0
\(629\) 11851.0 0.751244
\(630\) 0 0
\(631\) −7634.81 −0.481675 −0.240837 0.970565i \(-0.577422\pi\)
−0.240837 + 0.970565i \(0.577422\pi\)
\(632\) 0 0
\(633\) 3673.01 0.230630
\(634\) 0 0
\(635\) 2589.53 0.161830
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2043.95 −0.126538
\(640\) 0 0
\(641\) −2779.60 −0.171275 −0.0856376 0.996326i \(-0.527293\pi\)
−0.0856376 + 0.996326i \(0.527293\pi\)
\(642\) 0 0
\(643\) −18304.5 −1.12264 −0.561322 0.827598i \(-0.689707\pi\)
−0.561322 + 0.827598i \(0.689707\pi\)
\(644\) 0 0
\(645\) −20616.5 −1.25856
\(646\) 0 0
\(647\) −6552.53 −0.398156 −0.199078 0.979984i \(-0.563795\pi\)
−0.199078 + 0.979984i \(0.563795\pi\)
\(648\) 0 0
\(649\) 6997.78 0.423247
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31986.1 −1.91686 −0.958432 0.285322i \(-0.907900\pi\)
−0.958432 + 0.285322i \(0.907900\pi\)
\(654\) 0 0
\(655\) −8252.68 −0.492304
\(656\) 0 0
\(657\) 387.445 0.0230071
\(658\) 0 0
\(659\) −517.327 −0.0305799 −0.0152900 0.999883i \(-0.504867\pi\)
−0.0152900 + 0.999883i \(0.504867\pi\)
\(660\) 0 0
\(661\) 19656.8 1.15667 0.578336 0.815798i \(-0.303702\pi\)
0.578336 + 0.815798i \(0.303702\pi\)
\(662\) 0 0
\(663\) 10380.6 0.608067
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.77440 −0.000103006 0
\(668\) 0 0
\(669\) −10372.1 −0.599415
\(670\) 0 0
\(671\) −25369.0 −1.45955
\(672\) 0 0
\(673\) 25836.1 1.47981 0.739904 0.672713i \(-0.234871\pi\)
0.739904 + 0.672713i \(0.234871\pi\)
\(674\) 0 0
\(675\) −3376.33 −0.192526
\(676\) 0 0
\(677\) 26526.2 1.50589 0.752944 0.658085i \(-0.228633\pi\)
0.752944 + 0.658085i \(0.228633\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −15534.7 −0.874140
\(682\) 0 0
\(683\) −33878.8 −1.89800 −0.949002 0.315269i \(-0.897905\pi\)
−0.949002 + 0.315269i \(0.897905\pi\)
\(684\) 0 0
\(685\) −42329.5 −2.36106
\(686\) 0 0
\(687\) −6935.65 −0.385170
\(688\) 0 0
\(689\) −1263.86 −0.0698828
\(690\) 0 0
\(691\) 1001.26 0.0551227 0.0275614 0.999620i \(-0.491226\pi\)
0.0275614 + 0.999620i \(0.491226\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13499.4 0.736779
\(696\) 0 0
\(697\) −16467.5 −0.894910
\(698\) 0 0
\(699\) −14228.9 −0.769939
\(700\) 0 0
\(701\) 7713.38 0.415592 0.207796 0.978172i \(-0.433371\pi\)
0.207796 + 0.978172i \(0.433371\pi\)
\(702\) 0 0
\(703\) 29424.7 1.57862
\(704\) 0 0
\(705\) −22931.7 −1.22505
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10035.1 0.531561 0.265781 0.964034i \(-0.414370\pi\)
0.265781 + 0.964034i \(0.414370\pi\)
\(710\) 0 0
\(711\) 2781.54 0.146717
\(712\) 0 0
\(713\) −1438.21 −0.0755421
\(714\) 0 0
\(715\) 29478.1 1.54184
\(716\) 0 0
\(717\) −4237.77 −0.220729
\(718\) 0 0
\(719\) 20119.4 1.04357 0.521785 0.853077i \(-0.325266\pi\)
0.521785 + 0.853077i \(0.325266\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 5643.46 0.290294
\(724\) 0 0
\(725\) −27.2429 −0.00139555
\(726\) 0 0
\(727\) 7567.09 0.386035 0.193018 0.981195i \(-0.438173\pi\)
0.193018 + 0.981195i \(0.438173\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −24400.0 −1.23457
\(732\) 0 0
\(733\) 30800.0 1.55201 0.776006 0.630726i \(-0.217243\pi\)
0.776006 + 0.630726i \(0.217243\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18875.9 0.943424
\(738\) 0 0
\(739\) 13319.3 0.663003 0.331501 0.943455i \(-0.392445\pi\)
0.331501 + 0.943455i \(0.392445\pi\)
\(740\) 0 0
\(741\) 25773.7 1.27776
\(742\) 0 0
\(743\) 10747.7 0.530682 0.265341 0.964155i \(-0.414516\pi\)
0.265341 + 0.964155i \(0.414516\pi\)
\(744\) 0 0
\(745\) −8818.66 −0.433679
\(746\) 0 0
\(747\) −11105.9 −0.543967
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −12213.4 −0.593439 −0.296720 0.954965i \(-0.595893\pi\)
−0.296720 + 0.954965i \(0.595893\pi\)
\(752\) 0 0
\(753\) 13558.9 0.656195
\(754\) 0 0
\(755\) 27978.6 1.34867
\(756\) 0 0
\(757\) −30173.2 −1.44870 −0.724349 0.689434i \(-0.757860\pi\)
−0.724349 + 0.689434i \(0.757860\pi\)
\(758\) 0 0
\(759\) −739.090 −0.0353455
\(760\) 0 0
\(761\) 15207.3 0.724396 0.362198 0.932101i \(-0.382026\pi\)
0.362198 + 0.932101i \(0.382026\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −7990.33 −0.377636
\(766\) 0 0
\(767\) −14257.9 −0.671216
\(768\) 0 0
\(769\) −9368.65 −0.439327 −0.219663 0.975576i \(-0.570496\pi\)
−0.219663 + 0.975576i \(0.570496\pi\)
\(770\) 0 0
\(771\) −23738.0 −1.10882
\(772\) 0 0
\(773\) 32516.3 1.51298 0.756488 0.654008i \(-0.226914\pi\)
0.756488 + 0.654008i \(0.226914\pi\)
\(774\) 0 0
\(775\) −22081.3 −1.02346
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −40886.8 −1.88052
\(780\) 0 0
\(781\) −6869.48 −0.314737
\(782\) 0 0
\(783\) 5.88214 0.000268468 0
\(784\) 0 0
\(785\) 38028.5 1.72904
\(786\) 0 0
\(787\) 40250.3 1.82308 0.911542 0.411207i \(-0.134893\pi\)
0.911542 + 0.411207i \(0.134893\pi\)
\(788\) 0 0
\(789\) −10293.5 −0.464458
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 51689.0 2.31467
\(794\) 0 0
\(795\) 972.842 0.0434002
\(796\) 0 0
\(797\) 21387.5 0.950544 0.475272 0.879839i \(-0.342350\pi\)
0.475272 + 0.879839i \(0.342350\pi\)
\(798\) 0 0
\(799\) −27140.1 −1.20169
\(800\) 0 0
\(801\) 10299.2 0.454313
\(802\) 0 0
\(803\) 1302.16 0.0572255
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24897.6 −1.08604
\(808\) 0 0
\(809\) −19634.1 −0.853274 −0.426637 0.904423i \(-0.640302\pi\)
−0.426637 + 0.904423i \(0.640302\pi\)
\(810\) 0 0
\(811\) −35238.1 −1.52574 −0.762872 0.646549i \(-0.776211\pi\)
−0.762872 + 0.646549i \(0.776211\pi\)
\(812\) 0 0
\(813\) −17773.5 −0.766719
\(814\) 0 0
\(815\) 53428.7 2.29635
\(816\) 0 0
\(817\) −60582.2 −2.59425
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7054.80 −0.299895 −0.149948 0.988694i \(-0.547911\pi\)
−0.149948 + 0.988694i \(0.547911\pi\)
\(822\) 0 0
\(823\) −6188.32 −0.262104 −0.131052 0.991376i \(-0.541835\pi\)
−0.131052 + 0.991376i \(0.541835\pi\)
\(824\) 0 0
\(825\) −11347.5 −0.478870
\(826\) 0 0
\(827\) 24000.0 1.00915 0.504573 0.863369i \(-0.331650\pi\)
0.504573 + 0.863369i \(0.331650\pi\)
\(828\) 0 0
\(829\) −8708.86 −0.364863 −0.182431 0.983219i \(-0.558397\pi\)
−0.182431 + 0.983219i \(0.558397\pi\)
\(830\) 0 0
\(831\) −4561.80 −0.190430
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −12743.7 −0.528158
\(836\) 0 0
\(837\) 4767.67 0.196888
\(838\) 0 0
\(839\) −7820.46 −0.321802 −0.160901 0.986971i \(-0.551440\pi\)
−0.160901 + 0.986971i \(0.551440\pi\)
\(840\) 0 0
\(841\) −24389.0 −0.999998
\(842\) 0 0
\(843\) 1057.00 0.0431850
\(844\) 0 0
\(845\) −25320.2 −1.03082
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 12865.5 0.520073
\(850\) 0 0
\(851\) 1719.20 0.0692521
\(852\) 0 0
\(853\) 31176.8 1.25143 0.625717 0.780050i \(-0.284807\pi\)
0.625717 + 0.780050i \(0.284807\pi\)
\(854\) 0 0
\(855\) −19839.0 −0.793543
\(856\) 0 0
\(857\) 895.728 0.0357030 0.0178515 0.999841i \(-0.494317\pi\)
0.0178515 + 0.999841i \(0.494317\pi\)
\(858\) 0 0
\(859\) −9115.02 −0.362049 −0.181025 0.983479i \(-0.557941\pi\)
−0.181025 + 0.983479i \(0.557941\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 21126.6 0.833325 0.416662 0.909061i \(-0.363200\pi\)
0.416662 + 0.909061i \(0.363200\pi\)
\(864\) 0 0
\(865\) 13290.8 0.522430
\(866\) 0 0
\(867\) 5282.29 0.206916
\(868\) 0 0
\(869\) 9348.43 0.364930
\(870\) 0 0
\(871\) −38459.5 −1.49615
\(872\) 0 0
\(873\) 15193.1 0.589013
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −18749.4 −0.721920 −0.360960 0.932581i \(-0.617551\pi\)
−0.360960 + 0.932581i \(0.617551\pi\)
\(878\) 0 0
\(879\) −7985.96 −0.306439
\(880\) 0 0
\(881\) 37000.3 1.41495 0.707475 0.706739i \(-0.249834\pi\)
0.707475 + 0.706739i \(0.249834\pi\)
\(882\) 0 0
\(883\) −21618.3 −0.823913 −0.411956 0.911204i \(-0.635154\pi\)
−0.411956 + 0.911204i \(0.635154\pi\)
\(884\) 0 0
\(885\) 10974.9 0.416854
\(886\) 0 0
\(887\) −8303.70 −0.314330 −0.157165 0.987572i \(-0.550235\pi\)
−0.157165 + 0.987572i \(0.550235\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2450.08 0.0921221
\(892\) 0 0
\(893\) −67385.6 −2.52516
\(894\) 0 0
\(895\) 25373.6 0.947650
\(896\) 0 0
\(897\) 1505.89 0.0560536
\(898\) 0 0
\(899\) 38.4693 0.00142717
\(900\) 0 0
\(901\) 1151.38 0.0425726
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 59763.9 2.19516
\(906\) 0 0
\(907\) 34544.5 1.26464 0.632321 0.774706i \(-0.282102\pi\)
0.632321 + 0.774706i \(0.282102\pi\)
\(908\) 0 0
\(909\) −13390.7 −0.488606
\(910\) 0 0
\(911\) 10505.3 0.382059 0.191030 0.981584i \(-0.438817\pi\)
0.191030 + 0.981584i \(0.438817\pi\)
\(912\) 0 0
\(913\) −37325.6 −1.35301
\(914\) 0 0
\(915\) −39787.0 −1.43751
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −32201.6 −1.15586 −0.577929 0.816087i \(-0.696139\pi\)
−0.577929 + 0.816087i \(0.696139\pi\)
\(920\) 0 0
\(921\) 437.909 0.0156673
\(922\) 0 0
\(923\) 13996.5 0.499133
\(924\) 0 0
\(925\) 26395.4 0.938245
\(926\) 0 0
\(927\) 7005.67 0.248216
\(928\) 0 0
\(929\) −37034.8 −1.30794 −0.653968 0.756522i \(-0.726897\pi\)
−0.653968 + 0.756522i \(0.726897\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 10249.5 0.359651
\(934\) 0 0
\(935\) −26854.6 −0.939292
\(936\) 0 0
\(937\) −7207.18 −0.251279 −0.125639 0.992076i \(-0.540098\pi\)
−0.125639 + 0.992076i \(0.540098\pi\)
\(938\) 0 0
\(939\) −11047.1 −0.383927
\(940\) 0 0
\(941\) −21718.9 −0.752407 −0.376203 0.926537i \(-0.622771\pi\)
−0.376203 + 0.926537i \(0.622771\pi\)
\(942\) 0 0
\(943\) −2388.91 −0.0824958
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 131.343 0.00450693 0.00225347 0.999997i \(-0.499283\pi\)
0.00225347 + 0.999997i \(0.499283\pi\)
\(948\) 0 0
\(949\) −2653.13 −0.0907525
\(950\) 0 0
\(951\) 4130.33 0.140836
\(952\) 0 0
\(953\) −39810.0 −1.35317 −0.676586 0.736363i \(-0.736541\pi\)
−0.676586 + 0.736363i \(0.736541\pi\)
\(954\) 0 0
\(955\) −29247.6 −0.991025
\(956\) 0 0
\(957\) 19.7692 0.000667760 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1389.69 0.0466480
\(962\) 0 0
\(963\) −11718.7 −0.392139
\(964\) 0 0
\(965\) 5582.01 0.186209
\(966\) 0 0
\(967\) −18858.5 −0.627145 −0.313573 0.949564i \(-0.601526\pi\)
−0.313573 + 0.949564i \(0.601526\pi\)
\(968\) 0 0
\(969\) −23479.8 −0.778412
\(970\) 0 0
\(971\) 48355.1 1.59813 0.799067 0.601242i \(-0.205327\pi\)
0.799067 + 0.601242i \(0.205327\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 23120.3 0.759428
\(976\) 0 0
\(977\) −5085.72 −0.166537 −0.0832685 0.996527i \(-0.526536\pi\)
−0.0832685 + 0.996527i \(0.526536\pi\)
\(978\) 0 0
\(979\) 34614.4 1.13001
\(980\) 0 0
\(981\) 4221.73 0.137400
\(982\) 0 0
\(983\) −48617.7 −1.57748 −0.788741 0.614726i \(-0.789267\pi\)
−0.788741 + 0.614726i \(0.789267\pi\)
\(984\) 0 0
\(985\) −70339.5 −2.27533
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3539.66 −0.113806
\(990\) 0 0
\(991\) −35000.1 −1.12191 −0.560955 0.827846i \(-0.689566\pi\)
−0.560955 + 0.827846i \(0.689566\pi\)
\(992\) 0 0
\(993\) 6524.79 0.208517
\(994\) 0 0
\(995\) −35423.4 −1.12864
\(996\) 0 0
\(997\) 32807.9 1.04216 0.521081 0.853507i \(-0.325529\pi\)
0.521081 + 0.853507i \(0.325529\pi\)
\(998\) 0 0
\(999\) −5699.16 −0.180494
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 2352.4.a.cm.1.1 4
4.3 odd 2 1176.4.a.bd.1.1 4
7.2 even 3 336.4.q.m.193.4 8
7.4 even 3 336.4.q.m.289.4 8
7.6 odd 2 2352.4.a.cp.1.4 4
28.11 odd 6 168.4.q.f.121.4 yes 8
28.23 odd 6 168.4.q.f.25.4 8
28.27 even 2 1176.4.a.ba.1.4 4
84.11 even 6 504.4.s.j.289.1 8
84.23 even 6 504.4.s.j.361.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.f.25.4 8 28.23 odd 6
168.4.q.f.121.4 yes 8 28.11 odd 6
336.4.q.m.193.4 8 7.2 even 3
336.4.q.m.289.4 8 7.4 even 3
504.4.s.j.289.1 8 84.11 even 6
504.4.s.j.361.1 8 84.23 even 6
1176.4.a.ba.1.4 4 28.27 even 2
1176.4.a.bd.1.1 4 4.3 odd 2
2352.4.a.cm.1.1 4 1.1 even 1 trivial
2352.4.a.cp.1.4 4 7.6 odd 2