Properties

Label 117.4.a.e.1.2
Level $117$
Weight $4$
Character 117.1
Self dual yes
Analytic conductor $6.903$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.90322347067\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 117.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.64575 q^{2} -1.00000 q^{4} -10.5830 q^{5} -22.0000 q^{7} -23.8118 q^{8} +O(q^{10})\) \(q+2.64575 q^{2} -1.00000 q^{4} -10.5830 q^{5} -22.0000 q^{7} -23.8118 q^{8} -28.0000 q^{10} -5.29150 q^{11} +13.0000 q^{13} -58.2065 q^{14} -55.0000 q^{16} +116.413 q^{17} -126.000 q^{19} +10.5830 q^{20} -14.0000 q^{22} +31.7490 q^{23} -13.0000 q^{25} +34.3948 q^{26} +22.0000 q^{28} +52.9150 q^{29} -182.000 q^{31} +44.9778 q^{32} +308.000 q^{34} +232.826 q^{35} -86.0000 q^{37} -333.365 q^{38} +252.000 q^{40} -444.486 q^{41} +96.0000 q^{43} +5.29150 q^{44} +84.0000 q^{46} +365.114 q^{47} +141.000 q^{49} -34.3948 q^{50} -13.0000 q^{52} -190.494 q^{53} +56.0000 q^{55} +523.859 q^{56} +140.000 q^{58} -587.357 q^{59} +574.000 q^{61} -481.527 q^{62} +559.000 q^{64} -137.579 q^{65} -530.000 q^{67} -116.413 q^{68} +616.000 q^{70} +809.600 q^{71} -154.000 q^{73} -227.535 q^{74} +126.000 q^{76} +116.413 q^{77} -460.000 q^{79} +582.065 q^{80} -1176.00 q^{82} -322.782 q^{83} -1232.00 q^{85} +253.992 q^{86} +126.000 q^{88} +1439.29 q^{89} -286.000 q^{91} -31.7490 q^{92} +966.000 q^{94} +1333.46 q^{95} +70.0000 q^{97} +373.051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 44 q^{7} - 56 q^{10} + 26 q^{13} - 110 q^{16} - 252 q^{19} - 28 q^{22} - 26 q^{25} + 44 q^{28} - 364 q^{31} + 616 q^{34} - 172 q^{37} + 504 q^{40} + 192 q^{43} + 168 q^{46} + 282 q^{49} - 26 q^{52} + 112 q^{55} + 280 q^{58} + 1148 q^{61} + 1118 q^{64} - 1060 q^{67} + 1232 q^{70} - 308 q^{73} + 252 q^{76} - 920 q^{79} - 2352 q^{82} - 2464 q^{85} + 252 q^{88} - 572 q^{91} + 1932 q^{94} + 140 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.64575 0.935414 0.467707 0.883883i \(-0.345080\pi\)
0.467707 + 0.883883i \(0.345080\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.125000
\(5\) −10.5830 −0.946573 −0.473286 0.880909i \(-0.656932\pi\)
−0.473286 + 0.880909i \(0.656932\pi\)
\(6\) 0 0
\(7\) −22.0000 −1.18789 −0.593944 0.804506i \(-0.702430\pi\)
−0.593944 + 0.804506i \(0.702430\pi\)
\(8\) −23.8118 −1.05234
\(9\) 0 0
\(10\) −28.0000 −0.885438
\(11\) −5.29150 −0.145041 −0.0725204 0.997367i \(-0.523104\pi\)
−0.0725204 + 0.997367i \(0.523104\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) −58.2065 −1.11117
\(15\) 0 0
\(16\) −55.0000 −0.859375
\(17\) 116.413 1.66084 0.830421 0.557136i \(-0.188100\pi\)
0.830421 + 0.557136i \(0.188100\pi\)
\(18\) 0 0
\(19\) −126.000 −1.52139 −0.760694 0.649110i \(-0.775141\pi\)
−0.760694 + 0.649110i \(0.775141\pi\)
\(20\) 10.5830 0.118322
\(21\) 0 0
\(22\) −14.0000 −0.135673
\(23\) 31.7490 0.287832 0.143916 0.989590i \(-0.454031\pi\)
0.143916 + 0.989590i \(0.454031\pi\)
\(24\) 0 0
\(25\) −13.0000 −0.104000
\(26\) 34.3948 0.259437
\(27\) 0 0
\(28\) 22.0000 0.148486
\(29\) 52.9150 0.338830 0.169415 0.985545i \(-0.445812\pi\)
0.169415 + 0.985545i \(0.445812\pi\)
\(30\) 0 0
\(31\) −182.000 −1.05446 −0.527228 0.849724i \(-0.676769\pi\)
−0.527228 + 0.849724i \(0.676769\pi\)
\(32\) 44.9778 0.248469
\(33\) 0 0
\(34\) 308.000 1.55358
\(35\) 232.826 1.12442
\(36\) 0 0
\(37\) −86.0000 −0.382117 −0.191058 0.981579i \(-0.561192\pi\)
−0.191058 + 0.981579i \(0.561192\pi\)
\(38\) −333.365 −1.42313
\(39\) 0 0
\(40\) 252.000 0.996117
\(41\) −444.486 −1.69310 −0.846550 0.532310i \(-0.821324\pi\)
−0.846550 + 0.532310i \(0.821324\pi\)
\(42\) 0 0
\(43\) 96.0000 0.340462 0.170231 0.985404i \(-0.445549\pi\)
0.170231 + 0.985404i \(0.445549\pi\)
\(44\) 5.29150 0.0181301
\(45\) 0 0
\(46\) 84.0000 0.269242
\(47\) 365.114 1.13313 0.566567 0.824016i \(-0.308271\pi\)
0.566567 + 0.824016i \(0.308271\pi\)
\(48\) 0 0
\(49\) 141.000 0.411079
\(50\) −34.3948 −0.0972831
\(51\) 0 0
\(52\) −13.0000 −0.0346688
\(53\) −190.494 −0.493705 −0.246853 0.969053i \(-0.579396\pi\)
−0.246853 + 0.969053i \(0.579396\pi\)
\(54\) 0 0
\(55\) 56.0000 0.137292
\(56\) 523.859 1.25006
\(57\) 0 0
\(58\) 140.000 0.316947
\(59\) −587.357 −1.29606 −0.648028 0.761616i \(-0.724406\pi\)
−0.648028 + 0.761616i \(0.724406\pi\)
\(60\) 0 0
\(61\) 574.000 1.20481 0.602403 0.798192i \(-0.294210\pi\)
0.602403 + 0.798192i \(0.294210\pi\)
\(62\) −481.527 −0.986354
\(63\) 0 0
\(64\) 559.000 1.09180
\(65\) −137.579 −0.262532
\(66\) 0 0
\(67\) −530.000 −0.966415 −0.483208 0.875506i \(-0.660528\pi\)
−0.483208 + 0.875506i \(0.660528\pi\)
\(68\) −116.413 −0.207605
\(69\) 0 0
\(70\) 616.000 1.05180
\(71\) 809.600 1.35327 0.676633 0.736321i \(-0.263439\pi\)
0.676633 + 0.736321i \(0.263439\pi\)
\(72\) 0 0
\(73\) −154.000 −0.246909 −0.123454 0.992350i \(-0.539397\pi\)
−0.123454 + 0.992350i \(0.539397\pi\)
\(74\) −227.535 −0.357437
\(75\) 0 0
\(76\) 126.000 0.190174
\(77\) 116.413 0.172292
\(78\) 0 0
\(79\) −460.000 −0.655114 −0.327557 0.944831i \(-0.606225\pi\)
−0.327557 + 0.944831i \(0.606225\pi\)
\(80\) 582.065 0.813461
\(81\) 0 0
\(82\) −1176.00 −1.58375
\(83\) −322.782 −0.426866 −0.213433 0.976958i \(-0.568465\pi\)
−0.213433 + 0.976958i \(0.568465\pi\)
\(84\) 0 0
\(85\) −1232.00 −1.57211
\(86\) 253.992 0.318473
\(87\) 0 0
\(88\) 126.000 0.152632
\(89\) 1439.29 1.71421 0.857103 0.515145i \(-0.172262\pi\)
0.857103 + 0.515145i \(0.172262\pi\)
\(90\) 0 0
\(91\) −286.000 −0.329461
\(92\) −31.7490 −0.0359790
\(93\) 0 0
\(94\) 966.000 1.05995
\(95\) 1333.46 1.44010
\(96\) 0 0
\(97\) 70.0000 0.0732724 0.0366362 0.999329i \(-0.488336\pi\)
0.0366362 + 0.999329i \(0.488336\pi\)
\(98\) 373.051 0.384529
\(99\) 0 0
\(100\) 13.0000 0.0130000
\(101\) −1460.45 −1.43882 −0.719409 0.694586i \(-0.755587\pi\)
−0.719409 + 0.694586i \(0.755587\pi\)
\(102\) 0 0
\(103\) −1428.00 −1.36607 −0.683034 0.730387i \(-0.739340\pi\)
−0.683034 + 0.730387i \(0.739340\pi\)
\(104\) −309.553 −0.291867
\(105\) 0 0
\(106\) −504.000 −0.461819
\(107\) −1619.20 −1.46293 −0.731467 0.681877i \(-0.761164\pi\)
−0.731467 + 0.681877i \(0.761164\pi\)
\(108\) 0 0
\(109\) −338.000 −0.297014 −0.148507 0.988911i \(-0.547447\pi\)
−0.148507 + 0.988911i \(0.547447\pi\)
\(110\) 148.162 0.128425
\(111\) 0 0
\(112\) 1210.00 1.02084
\(113\) −1682.70 −1.40084 −0.700420 0.713731i \(-0.747004\pi\)
−0.700420 + 0.713731i \(0.747004\pi\)
\(114\) 0 0
\(115\) −336.000 −0.272454
\(116\) −52.9150 −0.0423538
\(117\) 0 0
\(118\) −1554.00 −1.21235
\(119\) −2561.09 −1.97289
\(120\) 0 0
\(121\) −1303.00 −0.978963
\(122\) 1518.66 1.12699
\(123\) 0 0
\(124\) 182.000 0.131807
\(125\) 1460.45 1.04502
\(126\) 0 0
\(127\) −376.000 −0.262713 −0.131357 0.991335i \(-0.541933\pi\)
−0.131357 + 0.991335i \(0.541933\pi\)
\(128\) 1119.15 0.772813
\(129\) 0 0
\(130\) −364.000 −0.245576
\(131\) −687.895 −0.458792 −0.229396 0.973333i \(-0.573675\pi\)
−0.229396 + 0.973333i \(0.573675\pi\)
\(132\) 0 0
\(133\) 2772.00 1.80724
\(134\) −1402.25 −0.903998
\(135\) 0 0
\(136\) −2772.00 −1.74777
\(137\) 1396.96 0.871168 0.435584 0.900148i \(-0.356542\pi\)
0.435584 + 0.900148i \(0.356542\pi\)
\(138\) 0 0
\(139\) 2100.00 1.28144 0.640718 0.767776i \(-0.278637\pi\)
0.640718 + 0.767776i \(0.278637\pi\)
\(140\) −232.826 −0.140553
\(141\) 0 0
\(142\) 2142.00 1.26586
\(143\) −68.7895 −0.0402271
\(144\) 0 0
\(145\) −560.000 −0.320727
\(146\) −407.446 −0.230962
\(147\) 0 0
\(148\) 86.0000 0.0477646
\(149\) −2000.19 −1.09974 −0.549872 0.835249i \(-0.685323\pi\)
−0.549872 + 0.835249i \(0.685323\pi\)
\(150\) 0 0
\(151\) 3526.00 1.90028 0.950138 0.311828i \(-0.100941\pi\)
0.950138 + 0.311828i \(0.100941\pi\)
\(152\) 3000.28 1.60102
\(153\) 0 0
\(154\) 308.000 0.161165
\(155\) 1926.11 0.998120
\(156\) 0 0
\(157\) 3066.00 1.55856 0.779278 0.626678i \(-0.215586\pi\)
0.779278 + 0.626678i \(0.215586\pi\)
\(158\) −1217.05 −0.612803
\(159\) 0 0
\(160\) −476.000 −0.235194
\(161\) −698.478 −0.341912
\(162\) 0 0
\(163\) −3442.00 −1.65398 −0.826988 0.562219i \(-0.809948\pi\)
−0.826988 + 0.562219i \(0.809948\pi\)
\(164\) 444.486 0.211637
\(165\) 0 0
\(166\) −854.000 −0.399297
\(167\) 2693.37 1.24802 0.624011 0.781416i \(-0.285502\pi\)
0.624011 + 0.781416i \(0.285502\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) −3259.57 −1.47057
\(171\) 0 0
\(172\) −96.0000 −0.0425577
\(173\) 3492.39 1.53481 0.767404 0.641164i \(-0.221548\pi\)
0.767404 + 0.641164i \(0.221548\pi\)
\(174\) 0 0
\(175\) 286.000 0.123540
\(176\) 291.033 0.124644
\(177\) 0 0
\(178\) 3808.00 1.60349
\(179\) 169.328 0.0707049 0.0353524 0.999375i \(-0.488745\pi\)
0.0353524 + 0.999375i \(0.488745\pi\)
\(180\) 0 0
\(181\) 3374.00 1.38557 0.692783 0.721146i \(-0.256384\pi\)
0.692783 + 0.721146i \(0.256384\pi\)
\(182\) −756.685 −0.308182
\(183\) 0 0
\(184\) −756.000 −0.302897
\(185\) 910.138 0.361701
\(186\) 0 0
\(187\) −616.000 −0.240890
\(188\) −365.114 −0.141642
\(189\) 0 0
\(190\) 3528.00 1.34709
\(191\) −1185.30 −0.449032 −0.224516 0.974470i \(-0.572080\pi\)
−0.224516 + 0.974470i \(0.572080\pi\)
\(192\) 0 0
\(193\) −1542.00 −0.575107 −0.287553 0.957765i \(-0.592842\pi\)
−0.287553 + 0.957765i \(0.592842\pi\)
\(194\) 185.203 0.0685401
\(195\) 0 0
\(196\) −141.000 −0.0513848
\(197\) 2127.18 0.769318 0.384659 0.923059i \(-0.374319\pi\)
0.384659 + 0.923059i \(0.374319\pi\)
\(198\) 0 0
\(199\) −952.000 −0.339123 −0.169562 0.985520i \(-0.554235\pi\)
−0.169562 + 0.985520i \(0.554235\pi\)
\(200\) 309.553 0.109443
\(201\) 0 0
\(202\) −3864.00 −1.34589
\(203\) −1164.13 −0.402492
\(204\) 0 0
\(205\) 4704.00 1.60264
\(206\) −3778.13 −1.27784
\(207\) 0 0
\(208\) −715.000 −0.238348
\(209\) 666.729 0.220663
\(210\) 0 0
\(211\) −1640.00 −0.535082 −0.267541 0.963547i \(-0.586211\pi\)
−0.267541 + 0.963547i \(0.586211\pi\)
\(212\) 190.494 0.0617132
\(213\) 0 0
\(214\) −4284.00 −1.36845
\(215\) −1015.97 −0.322272
\(216\) 0 0
\(217\) 4004.00 1.25258
\(218\) −894.264 −0.277831
\(219\) 0 0
\(220\) −56.0000 −0.0171615
\(221\) 1513.37 0.460635
\(222\) 0 0
\(223\) −4886.00 −1.46722 −0.733612 0.679569i \(-0.762167\pi\)
−0.733612 + 0.679569i \(0.762167\pi\)
\(224\) −989.511 −0.295154
\(225\) 0 0
\(226\) −4452.00 −1.31037
\(227\) 1867.90 0.546154 0.273077 0.961992i \(-0.411959\pi\)
0.273077 + 0.961992i \(0.411959\pi\)
\(228\) 0 0
\(229\) 5558.00 1.60386 0.801928 0.597421i \(-0.203808\pi\)
0.801928 + 0.597421i \(0.203808\pi\)
\(230\) −888.972 −0.254857
\(231\) 0 0
\(232\) −1260.00 −0.356565
\(233\) −3577.06 −1.00575 −0.502877 0.864358i \(-0.667725\pi\)
−0.502877 + 0.864358i \(0.667725\pi\)
\(234\) 0 0
\(235\) −3864.00 −1.07259
\(236\) 587.357 0.162007
\(237\) 0 0
\(238\) −6776.00 −1.84547
\(239\) −2936.78 −0.794832 −0.397416 0.917639i \(-0.630093\pi\)
−0.397416 + 0.917639i \(0.630093\pi\)
\(240\) 0 0
\(241\) −602.000 −0.160906 −0.0804528 0.996758i \(-0.525637\pi\)
−0.0804528 + 0.996758i \(0.525637\pi\)
\(242\) −3447.41 −0.915736
\(243\) 0 0
\(244\) −574.000 −0.150601
\(245\) −1492.20 −0.389116
\(246\) 0 0
\(247\) −1638.00 −0.421957
\(248\) 4333.74 1.10965
\(249\) 0 0
\(250\) 3864.00 0.977523
\(251\) −3524.14 −0.886222 −0.443111 0.896467i \(-0.646125\pi\)
−0.443111 + 0.896467i \(0.646125\pi\)
\(252\) 0 0
\(253\) −168.000 −0.0417473
\(254\) −994.802 −0.245746
\(255\) 0 0
\(256\) −1511.00 −0.368896
\(257\) 2942.08 0.714092 0.357046 0.934087i \(-0.383784\pi\)
0.357046 + 0.934087i \(0.383784\pi\)
\(258\) 0 0
\(259\) 1892.00 0.453912
\(260\) 137.579 0.0328165
\(261\) 0 0
\(262\) −1820.00 −0.429160
\(263\) 857.223 0.200984 0.100492 0.994938i \(-0.467958\pi\)
0.100492 + 0.994938i \(0.467958\pi\)
\(264\) 0 0
\(265\) 2016.00 0.467328
\(266\) 7334.02 1.69052
\(267\) 0 0
\(268\) 530.000 0.120802
\(269\) 328.073 0.0743605 0.0371802 0.999309i \(-0.488162\pi\)
0.0371802 + 0.999309i \(0.488162\pi\)
\(270\) 0 0
\(271\) −2814.00 −0.630769 −0.315384 0.948964i \(-0.602133\pi\)
−0.315384 + 0.948964i \(0.602133\pi\)
\(272\) −6402.72 −1.42729
\(273\) 0 0
\(274\) 3696.00 0.814903
\(275\) 68.7895 0.0150842
\(276\) 0 0
\(277\) −3190.00 −0.691944 −0.345972 0.938245i \(-0.612451\pi\)
−0.345972 + 0.938245i \(0.612451\pi\)
\(278\) 5556.08 1.19867
\(279\) 0 0
\(280\) −5544.00 −1.18328
\(281\) 6116.98 1.29861 0.649303 0.760530i \(-0.275061\pi\)
0.649303 + 0.760530i \(0.275061\pi\)
\(282\) 0 0
\(283\) −4788.00 −1.00571 −0.502857 0.864370i \(-0.667718\pi\)
−0.502857 + 0.864370i \(0.667718\pi\)
\(284\) −809.600 −0.169158
\(285\) 0 0
\(286\) −182.000 −0.0376290
\(287\) 9778.70 2.01121
\(288\) 0 0
\(289\) 8639.00 1.75840
\(290\) −1481.62 −0.300013
\(291\) 0 0
\(292\) 154.000 0.0308636
\(293\) −6699.04 −1.33571 −0.667854 0.744293i \(-0.732787\pi\)
−0.667854 + 0.744293i \(0.732787\pi\)
\(294\) 0 0
\(295\) 6216.00 1.22681
\(296\) 2047.81 0.402117
\(297\) 0 0
\(298\) −5292.00 −1.02872
\(299\) 412.737 0.0798301
\(300\) 0 0
\(301\) −2112.00 −0.404431
\(302\) 9328.92 1.77755
\(303\) 0 0
\(304\) 6930.00 1.30744
\(305\) −6074.65 −1.14044
\(306\) 0 0
\(307\) −406.000 −0.0754777 −0.0377388 0.999288i \(-0.512015\pi\)
−0.0377388 + 0.999288i \(0.512015\pi\)
\(308\) −116.413 −0.0215365
\(309\) 0 0
\(310\) 5096.00 0.933656
\(311\) 8286.49 1.51088 0.755440 0.655217i \(-0.227423\pi\)
0.755440 + 0.655217i \(0.227423\pi\)
\(312\) 0 0
\(313\) −5586.00 −1.00875 −0.504376 0.863484i \(-0.668277\pi\)
−0.504376 + 0.863484i \(0.668277\pi\)
\(314\) 8111.87 1.45790
\(315\) 0 0
\(316\) 460.000 0.0818893
\(317\) −8392.32 −1.48694 −0.743470 0.668770i \(-0.766821\pi\)
−0.743470 + 0.668770i \(0.766821\pi\)
\(318\) 0 0
\(319\) −280.000 −0.0491442
\(320\) −5915.90 −1.03347
\(321\) 0 0
\(322\) −1848.00 −0.319829
\(323\) −14668.0 −2.52679
\(324\) 0 0
\(325\) −169.000 −0.0288444
\(326\) −9106.68 −1.54715
\(327\) 0 0
\(328\) 10584.0 1.78172
\(329\) −8032.50 −1.34604
\(330\) 0 0
\(331\) −4426.00 −0.734970 −0.367485 0.930030i \(-0.619781\pi\)
−0.367485 + 0.930030i \(0.619781\pi\)
\(332\) 322.782 0.0533583
\(333\) 0 0
\(334\) 7126.00 1.16742
\(335\) 5608.99 0.914782
\(336\) 0 0
\(337\) 8370.00 1.35295 0.676473 0.736467i \(-0.263507\pi\)
0.676473 + 0.736467i \(0.263507\pi\)
\(338\) 447.132 0.0719549
\(339\) 0 0
\(340\) 1232.00 0.196513
\(341\) 963.053 0.152939
\(342\) 0 0
\(343\) 4444.00 0.699573
\(344\) −2285.93 −0.358282
\(345\) 0 0
\(346\) 9240.00 1.43568
\(347\) 6095.81 0.943056 0.471528 0.881851i \(-0.343703\pi\)
0.471528 + 0.881851i \(0.343703\pi\)
\(348\) 0 0
\(349\) 4354.00 0.667806 0.333903 0.942607i \(-0.391634\pi\)
0.333903 + 0.942607i \(0.391634\pi\)
\(350\) 756.685 0.115561
\(351\) 0 0
\(352\) −238.000 −0.0360382
\(353\) 3407.73 0.513810 0.256905 0.966437i \(-0.417297\pi\)
0.256905 + 0.966437i \(0.417297\pi\)
\(354\) 0 0
\(355\) −8568.00 −1.28096
\(356\) −1439.29 −0.214276
\(357\) 0 0
\(358\) 448.000 0.0661384
\(359\) −7762.63 −1.14121 −0.570607 0.821223i \(-0.693292\pi\)
−0.570607 + 0.821223i \(0.693292\pi\)
\(360\) 0 0
\(361\) 9017.00 1.31462
\(362\) 8926.76 1.29608
\(363\) 0 0
\(364\) 286.000 0.0411826
\(365\) 1629.78 0.233717
\(366\) 0 0
\(367\) −7784.00 −1.10714 −0.553572 0.832802i \(-0.686735\pi\)
−0.553572 + 0.832802i \(0.686735\pi\)
\(368\) −1746.20 −0.247355
\(369\) 0 0
\(370\) 2408.00 0.338340
\(371\) 4190.87 0.586467
\(372\) 0 0
\(373\) −8510.00 −1.18132 −0.590658 0.806922i \(-0.701132\pi\)
−0.590658 + 0.806922i \(0.701132\pi\)
\(374\) −1629.78 −0.225332
\(375\) 0 0
\(376\) −8694.00 −1.19244
\(377\) 687.895 0.0939746
\(378\) 0 0
\(379\) 1650.00 0.223627 0.111814 0.993729i \(-0.464334\pi\)
0.111814 + 0.993729i \(0.464334\pi\)
\(380\) −1333.46 −0.180013
\(381\) 0 0
\(382\) −3136.00 −0.420031
\(383\) −8662.19 −1.15566 −0.577829 0.816158i \(-0.696100\pi\)
−0.577829 + 0.816158i \(0.696100\pi\)
\(384\) 0 0
\(385\) −1232.00 −0.163087
\(386\) −4079.75 −0.537963
\(387\) 0 0
\(388\) −70.0000 −0.00915905
\(389\) 2423.51 0.315879 0.157939 0.987449i \(-0.449515\pi\)
0.157939 + 0.987449i \(0.449515\pi\)
\(390\) 0 0
\(391\) 3696.00 0.478043
\(392\) −3357.46 −0.432595
\(393\) 0 0
\(394\) 5628.00 0.719631
\(395\) 4868.18 0.620114
\(396\) 0 0
\(397\) −1414.00 −0.178757 −0.0893786 0.995998i \(-0.528488\pi\)
−0.0893786 + 0.995998i \(0.528488\pi\)
\(398\) −2518.76 −0.317221
\(399\) 0 0
\(400\) 715.000 0.0893750
\(401\) −5228.00 −0.651058 −0.325529 0.945532i \(-0.605542\pi\)
−0.325529 + 0.945532i \(0.605542\pi\)
\(402\) 0 0
\(403\) −2366.00 −0.292454
\(404\) 1460.45 0.179852
\(405\) 0 0
\(406\) −3080.00 −0.376497
\(407\) 455.069 0.0554225
\(408\) 0 0
\(409\) 5782.00 0.699026 0.349513 0.936932i \(-0.386347\pi\)
0.349513 + 0.936932i \(0.386347\pi\)
\(410\) 12445.6 1.49913
\(411\) 0 0
\(412\) 1428.00 0.170759
\(413\) 12921.8 1.53957
\(414\) 0 0
\(415\) 3416.00 0.404060
\(416\) 584.711 0.0689130
\(417\) 0 0
\(418\) 1764.00 0.206412
\(419\) −11482.6 −1.33881 −0.669403 0.742899i \(-0.733450\pi\)
−0.669403 + 0.742899i \(0.733450\pi\)
\(420\) 0 0
\(421\) −14194.0 −1.64317 −0.821583 0.570088i \(-0.806909\pi\)
−0.821583 + 0.570088i \(0.806909\pi\)
\(422\) −4339.03 −0.500523
\(423\) 0 0
\(424\) 4536.00 0.519546
\(425\) −1513.37 −0.172728
\(426\) 0 0
\(427\) −12628.0 −1.43118
\(428\) 1619.20 0.182867
\(429\) 0 0
\(430\) −2688.00 −0.301458
\(431\) 5222.71 0.583687 0.291844 0.956466i \(-0.405731\pi\)
0.291844 + 0.956466i \(0.405731\pi\)
\(432\) 0 0
\(433\) −686.000 −0.0761364 −0.0380682 0.999275i \(-0.512120\pi\)
−0.0380682 + 0.999275i \(0.512120\pi\)
\(434\) 10593.6 1.17168
\(435\) 0 0
\(436\) 338.000 0.0371268
\(437\) −4000.38 −0.437904
\(438\) 0 0
\(439\) −1372.00 −0.149162 −0.0745809 0.997215i \(-0.523762\pi\)
−0.0745809 + 0.997215i \(0.523762\pi\)
\(440\) −1333.46 −0.144478
\(441\) 0 0
\(442\) 4004.00 0.430884
\(443\) 2338.84 0.250839 0.125420 0.992104i \(-0.459972\pi\)
0.125420 + 0.992104i \(0.459972\pi\)
\(444\) 0 0
\(445\) −15232.0 −1.62262
\(446\) −12927.1 −1.37246
\(447\) 0 0
\(448\) −12298.0 −1.29693
\(449\) 17250.3 1.81312 0.906561 0.422074i \(-0.138698\pi\)
0.906561 + 0.422074i \(0.138698\pi\)
\(450\) 0 0
\(451\) 2352.00 0.245568
\(452\) 1682.70 0.175105
\(453\) 0 0
\(454\) 4942.00 0.510880
\(455\) 3026.74 0.311859
\(456\) 0 0
\(457\) −17866.0 −1.82874 −0.914372 0.404875i \(-0.867315\pi\)
−0.914372 + 0.404875i \(0.867315\pi\)
\(458\) 14705.1 1.50027
\(459\) 0 0
\(460\) 336.000 0.0340567
\(461\) 2106.02 0.212770 0.106385 0.994325i \(-0.466072\pi\)
0.106385 + 0.994325i \(0.466072\pi\)
\(462\) 0 0
\(463\) −13718.0 −1.37695 −0.688477 0.725258i \(-0.741720\pi\)
−0.688477 + 0.725258i \(0.741720\pi\)
\(464\) −2910.33 −0.291182
\(465\) 0 0
\(466\) −9464.00 −0.940797
\(467\) 4095.62 0.405830 0.202915 0.979196i \(-0.434958\pi\)
0.202915 + 0.979196i \(0.434958\pi\)
\(468\) 0 0
\(469\) 11660.0 1.14799
\(470\) −10223.2 −1.00332
\(471\) 0 0
\(472\) 13986.0 1.36389
\(473\) −507.984 −0.0493808
\(474\) 0 0
\(475\) 1638.00 0.158224
\(476\) 2561.09 0.246612
\(477\) 0 0
\(478\) −7770.00 −0.743497
\(479\) 8715.10 0.831322 0.415661 0.909520i \(-0.363550\pi\)
0.415661 + 0.909520i \(0.363550\pi\)
\(480\) 0 0
\(481\) −1118.00 −0.105980
\(482\) −1592.74 −0.150513
\(483\) 0 0
\(484\) 1303.00 0.122370
\(485\) −740.810 −0.0693577
\(486\) 0 0
\(487\) 4506.00 0.419274 0.209637 0.977779i \(-0.432772\pi\)
0.209637 + 0.977779i \(0.432772\pi\)
\(488\) −13668.0 −1.26787
\(489\) 0 0
\(490\) −3948.00 −0.363985
\(491\) −21621.1 −1.98726 −0.993631 0.112683i \(-0.964056\pi\)
−0.993631 + 0.112683i \(0.964056\pi\)
\(492\) 0 0
\(493\) 6160.00 0.562743
\(494\) −4333.74 −0.394705
\(495\) 0 0
\(496\) 10010.0 0.906174
\(497\) −17811.2 −1.60753
\(498\) 0 0
\(499\) −786.000 −0.0705134 −0.0352567 0.999378i \(-0.511225\pi\)
−0.0352567 + 0.999378i \(0.511225\pi\)
\(500\) −1460.45 −0.130627
\(501\) 0 0
\(502\) −9324.00 −0.828985
\(503\) 2106.02 0.186685 0.0933426 0.995634i \(-0.470245\pi\)
0.0933426 + 0.995634i \(0.470245\pi\)
\(504\) 0 0
\(505\) 15456.0 1.36195
\(506\) −444.486 −0.0390510
\(507\) 0 0
\(508\) 376.000 0.0328392
\(509\) 8392.32 0.730812 0.365406 0.930848i \(-0.380930\pi\)
0.365406 + 0.930848i \(0.380930\pi\)
\(510\) 0 0
\(511\) 3388.00 0.293300
\(512\) −12951.0 −1.11788
\(513\) 0 0
\(514\) 7784.00 0.667972
\(515\) 15112.5 1.29308
\(516\) 0 0
\(517\) −1932.00 −0.164351
\(518\) 5005.76 0.424596
\(519\) 0 0
\(520\) 3276.00 0.276273
\(521\) 3905.13 0.328382 0.164191 0.986429i \(-0.447499\pi\)
0.164191 + 0.986429i \(0.447499\pi\)
\(522\) 0 0
\(523\) −17668.0 −1.47718 −0.738592 0.674152i \(-0.764509\pi\)
−0.738592 + 0.674152i \(0.764509\pi\)
\(524\) 687.895 0.0573489
\(525\) 0 0
\(526\) 2268.00 0.188003
\(527\) −21187.2 −1.75129
\(528\) 0 0
\(529\) −11159.0 −0.917153
\(530\) 5333.83 0.437145
\(531\) 0 0
\(532\) −2772.00 −0.225905
\(533\) −5778.32 −0.469581
\(534\) 0 0
\(535\) 17136.0 1.38477
\(536\) 12620.2 1.01700
\(537\) 0 0
\(538\) 868.000 0.0695579
\(539\) −746.102 −0.0596232
\(540\) 0 0
\(541\) −1650.00 −0.131126 −0.0655629 0.997848i \(-0.520884\pi\)
−0.0655629 + 0.997848i \(0.520884\pi\)
\(542\) −7445.14 −0.590030
\(543\) 0 0
\(544\) 5236.00 0.412668
\(545\) 3577.06 0.281145
\(546\) 0 0
\(547\) 3796.00 0.296719 0.148359 0.988934i \(-0.452601\pi\)
0.148359 + 0.988934i \(0.452601\pi\)
\(548\) −1396.96 −0.108896
\(549\) 0 0
\(550\) 182.000 0.0141100
\(551\) −6667.29 −0.515492
\(552\) 0 0
\(553\) 10120.0 0.778203
\(554\) −8439.95 −0.647254
\(555\) 0 0
\(556\) −2100.00 −0.160180
\(557\) −2063.69 −0.156986 −0.0784930 0.996915i \(-0.525011\pi\)
−0.0784930 + 0.996915i \(0.525011\pi\)
\(558\) 0 0
\(559\) 1248.00 0.0944271
\(560\) −12805.4 −0.966301
\(561\) 0 0
\(562\) 16184.0 1.21473
\(563\) 26034.2 1.94886 0.974432 0.224683i \(-0.0721346\pi\)
0.974432 + 0.224683i \(0.0721346\pi\)
\(564\) 0 0
\(565\) 17808.0 1.32600
\(566\) −12667.9 −0.940759
\(567\) 0 0
\(568\) −19278.0 −1.42410
\(569\) −3640.55 −0.268225 −0.134112 0.990966i \(-0.542818\pi\)
−0.134112 + 0.990966i \(0.542818\pi\)
\(570\) 0 0
\(571\) −19612.0 −1.43737 −0.718684 0.695337i \(-0.755255\pi\)
−0.718684 + 0.695337i \(0.755255\pi\)
\(572\) 68.7895 0.00502838
\(573\) 0 0
\(574\) 25872.0 1.88132
\(575\) −412.737 −0.0299345
\(576\) 0 0
\(577\) 15722.0 1.13434 0.567171 0.823600i \(-0.308038\pi\)
0.567171 + 0.823600i \(0.308038\pi\)
\(578\) 22856.6 1.64483
\(579\) 0 0
\(580\) 560.000 0.0400909
\(581\) 7101.20 0.507069
\(582\) 0 0
\(583\) 1008.00 0.0716074
\(584\) 3667.01 0.259832
\(585\) 0 0
\(586\) −17724.0 −1.24944
\(587\) 2725.12 0.191615 0.0958074 0.995400i \(-0.469457\pi\)
0.0958074 + 0.995400i \(0.469457\pi\)
\(588\) 0 0
\(589\) 22932.0 1.60424
\(590\) 16446.0 1.14758
\(591\) 0 0
\(592\) 4730.00 0.328381
\(593\) 18012.3 1.24734 0.623672 0.781686i \(-0.285640\pi\)
0.623672 + 0.781686i \(0.285640\pi\)
\(594\) 0 0
\(595\) 27104.0 1.86749
\(596\) 2000.19 0.137468
\(597\) 0 0
\(598\) 1092.00 0.0746742
\(599\) 12181.0 0.830891 0.415446 0.909618i \(-0.363626\pi\)
0.415446 + 0.909618i \(0.363626\pi\)
\(600\) 0 0
\(601\) 5950.00 0.403836 0.201918 0.979402i \(-0.435283\pi\)
0.201918 + 0.979402i \(0.435283\pi\)
\(602\) −5587.83 −0.378310
\(603\) 0 0
\(604\) −3526.00 −0.237535
\(605\) 13789.7 0.926660
\(606\) 0 0
\(607\) 14168.0 0.947383 0.473691 0.880691i \(-0.342921\pi\)
0.473691 + 0.880691i \(0.342921\pi\)
\(608\) −5667.20 −0.378019
\(609\) 0 0
\(610\) −16072.0 −1.06678
\(611\) 4746.48 0.314275
\(612\) 0 0
\(613\) −6326.00 −0.416810 −0.208405 0.978043i \(-0.566827\pi\)
−0.208405 + 0.978043i \(0.566827\pi\)
\(614\) −1074.18 −0.0706029
\(615\) 0 0
\(616\) −2772.00 −0.181310
\(617\) 8805.06 0.574519 0.287260 0.957853i \(-0.407256\pi\)
0.287260 + 0.957853i \(0.407256\pi\)
\(618\) 0 0
\(619\) −24486.0 −1.58994 −0.794972 0.606646i \(-0.792515\pi\)
−0.794972 + 0.606646i \(0.792515\pi\)
\(620\) −1926.11 −0.124765
\(621\) 0 0
\(622\) 21924.0 1.41330
\(623\) −31664.4 −2.03628
\(624\) 0 0
\(625\) −13831.0 −0.885184
\(626\) −14779.2 −0.943601
\(627\) 0 0
\(628\) −3066.00 −0.194820
\(629\) −10011.5 −0.634635
\(630\) 0 0
\(631\) 22430.0 1.41509 0.707547 0.706666i \(-0.249802\pi\)
0.707547 + 0.706666i \(0.249802\pi\)
\(632\) 10953.4 0.689404
\(633\) 0 0
\(634\) −22204.0 −1.39090
\(635\) 3979.21 0.248677
\(636\) 0 0
\(637\) 1833.00 0.114013
\(638\) −740.810 −0.0459702
\(639\) 0 0
\(640\) −11844.0 −0.731524
\(641\) 27484.1 1.69353 0.846767 0.531964i \(-0.178546\pi\)
0.846767 + 0.531964i \(0.178546\pi\)
\(642\) 0 0
\(643\) 16478.0 1.01062 0.505310 0.862938i \(-0.331378\pi\)
0.505310 + 0.862938i \(0.331378\pi\)
\(644\) 698.478 0.0427390
\(645\) 0 0
\(646\) −38808.0 −2.36359
\(647\) −26563.3 −1.61408 −0.807042 0.590494i \(-0.798933\pi\)
−0.807042 + 0.590494i \(0.798933\pi\)
\(648\) 0 0
\(649\) 3108.00 0.187981
\(650\) −447.132 −0.0269815
\(651\) 0 0
\(652\) 3442.00 0.206747
\(653\) −20287.6 −1.21580 −0.607899 0.794014i \(-0.707987\pi\)
−0.607899 + 0.794014i \(0.707987\pi\)
\(654\) 0 0
\(655\) 7280.00 0.434280
\(656\) 24446.7 1.45501
\(657\) 0 0
\(658\) −21252.0 −1.25910
\(659\) −656.146 −0.0387858 −0.0193929 0.999812i \(-0.506173\pi\)
−0.0193929 + 0.999812i \(0.506173\pi\)
\(660\) 0 0
\(661\) −14238.0 −0.837812 −0.418906 0.908030i \(-0.637586\pi\)
−0.418906 + 0.908030i \(0.637586\pi\)
\(662\) −11710.1 −0.687501
\(663\) 0 0
\(664\) 7686.00 0.449209
\(665\) −29336.1 −1.71068
\(666\) 0 0
\(667\) 1680.00 0.0975260
\(668\) −2693.37 −0.156003
\(669\) 0 0
\(670\) 14840.0 0.855700
\(671\) −3037.32 −0.174746
\(672\) 0 0
\(673\) −4874.00 −0.279166 −0.139583 0.990210i \(-0.544576\pi\)
−0.139583 + 0.990210i \(0.544576\pi\)
\(674\) 22144.9 1.26557
\(675\) 0 0
\(676\) −169.000 −0.00961538
\(677\) −21801.0 −1.23764 −0.618818 0.785534i \(-0.712388\pi\)
−0.618818 + 0.785534i \(0.712388\pi\)
\(678\) 0 0
\(679\) −1540.00 −0.0870394
\(680\) 29336.1 1.65439
\(681\) 0 0
\(682\) 2548.00 0.143062
\(683\) −8746.85 −0.490028 −0.245014 0.969520i \(-0.578793\pi\)
−0.245014 + 0.969520i \(0.578793\pi\)
\(684\) 0 0
\(685\) −14784.0 −0.824624
\(686\) 11757.7 0.654390
\(687\) 0 0
\(688\) −5280.00 −0.292584
\(689\) −2476.42 −0.136929
\(690\) 0 0
\(691\) 294.000 0.0161857 0.00809283 0.999967i \(-0.497424\pi\)
0.00809283 + 0.999967i \(0.497424\pi\)
\(692\) −3492.39 −0.191851
\(693\) 0 0
\(694\) 16128.0 0.882148
\(695\) −22224.3 −1.21297
\(696\) 0 0
\(697\) −51744.0 −2.81197
\(698\) 11519.6 0.624675
\(699\) 0 0
\(700\) −286.000 −0.0154425
\(701\) −15758.1 −0.849037 −0.424519 0.905419i \(-0.639557\pi\)
−0.424519 + 0.905419i \(0.639557\pi\)
\(702\) 0 0
\(703\) 10836.0 0.581348
\(704\) −2957.95 −0.158355
\(705\) 0 0
\(706\) 9016.00 0.480626
\(707\) 32130.0 1.70916
\(708\) 0 0
\(709\) −6722.00 −0.356065 −0.178032 0.984025i \(-0.556973\pi\)
−0.178032 + 0.984025i \(0.556973\pi\)
\(710\) −22668.8 −1.19823
\(711\) 0 0
\(712\) −34272.0 −1.80393
\(713\) −5778.32 −0.303506
\(714\) 0 0
\(715\) 728.000 0.0380778
\(716\) −169.328 −0.00883811
\(717\) 0 0
\(718\) −20538.0 −1.06751
\(719\) 31.7490 0.00164679 0.000823393 1.00000i \(-0.499738\pi\)
0.000823393 1.00000i \(0.499738\pi\)
\(720\) 0 0
\(721\) 31416.0 1.62274
\(722\) 23856.7 1.22972
\(723\) 0 0
\(724\) −3374.00 −0.173196
\(725\) −687.895 −0.0352383
\(726\) 0 0
\(727\) 12824.0 0.654217 0.327109 0.944987i \(-0.393926\pi\)
0.327109 + 0.944987i \(0.393926\pi\)
\(728\) 6810.16 0.346705
\(729\) 0 0
\(730\) 4312.00 0.218622
\(731\) 11175.7 0.565453
\(732\) 0 0
\(733\) −29610.0 −1.49205 −0.746023 0.665920i \(-0.768039\pi\)
−0.746023 + 0.665920i \(0.768039\pi\)
\(734\) −20594.5 −1.03564
\(735\) 0 0
\(736\) 1428.00 0.0715174
\(737\) 2804.50 0.140170
\(738\) 0 0
\(739\) −15622.0 −0.777625 −0.388812 0.921317i \(-0.627115\pi\)
−0.388812 + 0.921317i \(0.627115\pi\)
\(740\) −910.138 −0.0452126
\(741\) 0 0
\(742\) 11088.0 0.548589
\(743\) −8588.11 −0.424047 −0.212024 0.977265i \(-0.568005\pi\)
−0.212024 + 0.977265i \(0.568005\pi\)
\(744\) 0 0
\(745\) 21168.0 1.04099
\(746\) −22515.3 −1.10502
\(747\) 0 0
\(748\) 616.000 0.0301112
\(749\) 35622.4 1.73780
\(750\) 0 0
\(751\) 29468.0 1.43183 0.715914 0.698189i \(-0.246010\pi\)
0.715914 + 0.698189i \(0.246010\pi\)
\(752\) −20081.3 −0.973787
\(753\) 0 0
\(754\) 1820.00 0.0879052
\(755\) −37315.7 −1.79875
\(756\) 0 0
\(757\) −35030.0 −1.68189 −0.840943 0.541124i \(-0.817999\pi\)
−0.840943 + 0.541124i \(0.817999\pi\)
\(758\) 4365.49 0.209184
\(759\) 0 0
\(760\) −31752.0 −1.51548
\(761\) −22330.1 −1.06369 −0.531844 0.846842i \(-0.678501\pi\)
−0.531844 + 0.846842i \(0.678501\pi\)
\(762\) 0 0
\(763\) 7436.00 0.352819
\(764\) 1185.30 0.0561290
\(765\) 0 0
\(766\) −22918.0 −1.08102
\(767\) −7635.64 −0.359461
\(768\) 0 0
\(769\) −27342.0 −1.28216 −0.641078 0.767476i \(-0.721512\pi\)
−0.641078 + 0.767476i \(0.721512\pi\)
\(770\) −3259.57 −0.152554
\(771\) 0 0
\(772\) 1542.00 0.0718883
\(773\) 11884.7 0.552993 0.276496 0.961015i \(-0.410827\pi\)
0.276496 + 0.961015i \(0.410827\pi\)
\(774\) 0 0
\(775\) 2366.00 0.109664
\(776\) −1666.82 −0.0771076
\(777\) 0 0
\(778\) 6412.00 0.295477
\(779\) 56005.3 2.57586
\(780\) 0 0
\(781\) −4284.00 −0.196279
\(782\) 9778.70 0.447168
\(783\) 0 0
\(784\) −7755.00 −0.353271
\(785\) −32447.5 −1.47529
\(786\) 0 0
\(787\) 22666.0 1.02663 0.513314 0.858201i \(-0.328418\pi\)
0.513314 + 0.858201i \(0.328418\pi\)
\(788\) −2127.18 −0.0961647
\(789\) 0 0
\(790\) 12880.0 0.580063
\(791\) 37019.4 1.66404
\(792\) 0 0
\(793\) 7462.00 0.334153
\(794\) −3741.09 −0.167212
\(795\) 0 0
\(796\) 952.000 0.0423904
\(797\) 582.065 0.0258693 0.0129346 0.999916i \(-0.495883\pi\)
0.0129346 + 0.999916i \(0.495883\pi\)
\(798\) 0 0
\(799\) 42504.0 1.88196
\(800\) −584.711 −0.0258408
\(801\) 0 0
\(802\) −13832.0 −0.609009
\(803\) 814.891 0.0358118
\(804\) 0 0
\(805\) 7392.00 0.323644
\(806\) −6259.85 −0.273565
\(807\) 0 0
\(808\) 34776.0 1.51413
\(809\) 793.725 0.0344943 0.0172472 0.999851i \(-0.494510\pi\)
0.0172472 + 0.999851i \(0.494510\pi\)
\(810\) 0 0
\(811\) −9478.00 −0.410379 −0.205190 0.978722i \(-0.565781\pi\)
−0.205190 + 0.978722i \(0.565781\pi\)
\(812\) 1164.13 0.0503115
\(813\) 0 0
\(814\) 1204.00 0.0518430
\(815\) 36426.7 1.56561
\(816\) 0 0
\(817\) −12096.0 −0.517975
\(818\) 15297.7 0.653879
\(819\) 0 0
\(820\) −4704.00 −0.200330
\(821\) 3227.82 0.137213 0.0686063 0.997644i \(-0.478145\pi\)
0.0686063 + 0.997644i \(0.478145\pi\)
\(822\) 0 0
\(823\) 40476.0 1.71434 0.857172 0.515031i \(-0.172219\pi\)
0.857172 + 0.515031i \(0.172219\pi\)
\(824\) 34003.2 1.43757
\(825\) 0 0
\(826\) 34188.0 1.44014
\(827\) 7169.99 0.301481 0.150741 0.988573i \(-0.451834\pi\)
0.150741 + 0.988573i \(0.451834\pi\)
\(828\) 0 0
\(829\) 27482.0 1.15137 0.575687 0.817670i \(-0.304735\pi\)
0.575687 + 0.817670i \(0.304735\pi\)
\(830\) 9037.89 0.377963
\(831\) 0 0
\(832\) 7267.00 0.302810
\(833\) 16414.2 0.682737
\(834\) 0 0
\(835\) −28504.0 −1.18134
\(836\) −666.729 −0.0275829
\(837\) 0 0
\(838\) −30380.0 −1.25234
\(839\) 19128.8 0.787126 0.393563 0.919298i \(-0.371242\pi\)
0.393563 + 0.919298i \(0.371242\pi\)
\(840\) 0 0
\(841\) −21589.0 −0.885194
\(842\) −37553.8 −1.53704
\(843\) 0 0
\(844\) 1640.00 0.0668852
\(845\) −1788.53 −0.0728133
\(846\) 0 0
\(847\) 28666.0 1.16290
\(848\) 10477.2 0.424278
\(849\) 0 0
\(850\) −4004.00 −0.161572
\(851\) −2730.42 −0.109985
\(852\) 0 0
\(853\) 31962.0 1.28295 0.641476 0.767143i \(-0.278322\pi\)
0.641476 + 0.767143i \(0.278322\pi\)
\(854\) −33410.5 −1.33874
\(855\) 0 0
\(856\) 38556.0 1.53951
\(857\) −4931.68 −0.196573 −0.0982865 0.995158i \(-0.531336\pi\)
−0.0982865 + 0.995158i \(0.531336\pi\)
\(858\) 0 0
\(859\) 11704.0 0.464884 0.232442 0.972610i \(-0.425328\pi\)
0.232442 + 0.972610i \(0.425328\pi\)
\(860\) 1015.97 0.0402840
\(861\) 0 0
\(862\) 13818.0 0.545989
\(863\) 2280.64 0.0899581 0.0449790 0.998988i \(-0.485678\pi\)
0.0449790 + 0.998988i \(0.485678\pi\)
\(864\) 0 0
\(865\) −36960.0 −1.45281
\(866\) −1814.99 −0.0712191
\(867\) 0 0
\(868\) −4004.00 −0.156572
\(869\) 2434.09 0.0950183
\(870\) 0 0
\(871\) −6890.00 −0.268035
\(872\) 8048.38 0.312560
\(873\) 0 0
\(874\) −10584.0 −0.409621
\(875\) −32130.0 −1.24136
\(876\) 0 0
\(877\) −1006.00 −0.0387346 −0.0193673 0.999812i \(-0.506165\pi\)
−0.0193673 + 0.999812i \(0.506165\pi\)
\(878\) −3629.97 −0.139528
\(879\) 0 0
\(880\) −3080.00 −0.117985
\(881\) 40681.1 1.55571 0.777855 0.628444i \(-0.216308\pi\)
0.777855 + 0.628444i \(0.216308\pi\)
\(882\) 0 0
\(883\) 124.000 0.00472586 0.00236293 0.999997i \(-0.499248\pi\)
0.00236293 + 0.999997i \(0.499248\pi\)
\(884\) −1513.37 −0.0575793
\(885\) 0 0
\(886\) 6188.00 0.234639
\(887\) −16573.0 −0.627358 −0.313679 0.949529i \(-0.601562\pi\)
−0.313679 + 0.949529i \(0.601562\pi\)
\(888\) 0 0
\(889\) 8272.00 0.312074
\(890\) −40300.1 −1.51782
\(891\) 0 0
\(892\) 4886.00 0.183403
\(893\) −46004.3 −1.72394
\(894\) 0 0
\(895\) −1792.00 −0.0669273
\(896\) −24621.4 −0.918016
\(897\) 0 0
\(898\) 45640.0 1.69602
\(899\) −9630.53 −0.357282
\(900\) 0 0
\(901\) −22176.0 −0.819966
\(902\) 6222.81 0.229708
\(903\) 0 0
\(904\) 40068.0 1.47416
\(905\) −35707.1 −1.31154
\(906\) 0 0
\(907\) −42996.0 −1.57404 −0.787022 0.616924i \(-0.788378\pi\)
−0.787022 + 0.616924i \(0.788378\pi\)
\(908\) −1867.90 −0.0682692
\(909\) 0 0
\(910\) 8008.00 0.291717
\(911\) 53169.0 1.93366 0.966832 0.255413i \(-0.0822113\pi\)
0.966832 + 0.255413i \(0.0822113\pi\)
\(912\) 0 0
\(913\) 1708.00 0.0619130
\(914\) −47269.0 −1.71063
\(915\) 0 0
\(916\) −5558.00 −0.200482
\(917\) 15133.7 0.544993
\(918\) 0 0
\(919\) 8244.00 0.295913 0.147957 0.988994i \(-0.452730\pi\)
0.147957 + 0.988994i \(0.452730\pi\)
\(920\) 8000.75 0.286714
\(921\) 0 0
\(922\) 5572.00 0.199028
\(923\) 10524.8 0.375328
\(924\) 0 0
\(925\) 1118.00 0.0397401
\(926\) −36294.4 −1.28802
\(927\) 0 0
\(928\) 2380.00 0.0841889
\(929\) 16192.0 0.571843 0.285922 0.958253i \(-0.407700\pi\)
0.285922 + 0.958253i \(0.407700\pi\)
\(930\) 0 0
\(931\) −17766.0 −0.625410
\(932\) 3577.06 0.125719
\(933\) 0 0
\(934\) 10836.0 0.379620
\(935\) 6519.13 0.228020
\(936\) 0 0
\(937\) −18214.0 −0.635032 −0.317516 0.948253i \(-0.602849\pi\)
−0.317516 + 0.948253i \(0.602849\pi\)
\(938\) 30849.5 1.07385
\(939\) 0 0
\(940\) 3864.00 0.134074
\(941\) 13916.7 0.482115 0.241057 0.970511i \(-0.422506\pi\)
0.241057 + 0.970511i \(0.422506\pi\)
\(942\) 0 0
\(943\) −14112.0 −0.487328
\(944\) 32304.6 1.11380
\(945\) 0 0
\(946\) −1344.00 −0.0461916
\(947\) −40400.6 −1.38632 −0.693159 0.720784i \(-0.743782\pi\)
−0.693159 + 0.720784i \(0.743782\pi\)
\(948\) 0 0
\(949\) −2002.00 −0.0684802
\(950\) 4333.74 0.148005
\(951\) 0 0
\(952\) 60984.0 2.07616
\(953\) −21271.8 −0.723046 −0.361523 0.932363i \(-0.617743\pi\)
−0.361523 + 0.932363i \(0.617743\pi\)
\(954\) 0 0
\(955\) 12544.0 0.425041
\(956\) 2936.78 0.0993540
\(957\) 0 0
\(958\) 23058.0 0.777631
\(959\) −30733.0 −1.03485
\(960\) 0 0
\(961\) 3333.00 0.111879
\(962\) −2957.95 −0.0991353
\(963\) 0 0
\(964\) 602.000 0.0201132
\(965\) 16319.0 0.544380
\(966\) 0 0
\(967\) 9578.00 0.318519 0.159259 0.987237i \(-0.449089\pi\)
0.159259 + 0.987237i \(0.449089\pi\)
\(968\) 31026.7 1.03020
\(969\) 0 0
\(970\) −1960.00 −0.0648782
\(971\) −44956.6 −1.48581 −0.742907 0.669394i \(-0.766554\pi\)
−0.742907 + 0.669394i \(0.766554\pi\)
\(972\) 0 0
\(973\) −46200.0 −1.52220
\(974\) 11921.8 0.392195
\(975\) 0 0
\(976\) −31570.0 −1.03538
\(977\) 32765.0 1.07292 0.536461 0.843925i \(-0.319761\pi\)
0.536461 + 0.843925i \(0.319761\pi\)
\(978\) 0 0
\(979\) −7616.00 −0.248630
\(980\) 1492.20 0.0486395
\(981\) 0 0
\(982\) −57204.0 −1.85891
\(983\) −47438.3 −1.53921 −0.769607 0.638518i \(-0.779548\pi\)
−0.769607 + 0.638518i \(0.779548\pi\)
\(984\) 0 0
\(985\) −22512.0 −0.728215
\(986\) 16297.8 0.526398
\(987\) 0 0
\(988\) 1638.00 0.0527447
\(989\) 3047.91 0.0979957
\(990\) 0 0
\(991\) −6848.00 −0.219509 −0.109755 0.993959i \(-0.535007\pi\)
−0.109755 + 0.993959i \(0.535007\pi\)
\(992\) −8185.95 −0.262000
\(993\) 0 0
\(994\) −47124.0 −1.50370
\(995\) 10075.0 0.321005
\(996\) 0 0
\(997\) −5810.00 −0.184558 −0.0922791 0.995733i \(-0.529415\pi\)
−0.0922791 + 0.995733i \(0.529415\pi\)
\(998\) −2079.56 −0.0659593
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 117.4.a.e.1.2 yes 2
3.2 odd 2 inner 117.4.a.e.1.1 2
4.3 odd 2 1872.4.a.ba.1.1 2
12.11 even 2 1872.4.a.ba.1.2 2
13.12 even 2 1521.4.a.p.1.1 2
39.38 odd 2 1521.4.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.a.e.1.1 2 3.2 odd 2 inner
117.4.a.e.1.2 yes 2 1.1 even 1 trivial
1521.4.a.p.1.1 2 13.12 even 2
1521.4.a.p.1.2 2 39.38 odd 2
1872.4.a.ba.1.1 2 4.3 odd 2
1872.4.a.ba.1.2 2 12.11 even 2