Properties

Label 2352.4.a.bp.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -9.82475 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -9.82475 q^{5} +9.00000 q^{9} +14.1752 q^{11} +26.1238 q^{13} +29.4743 q^{15} +78.5980 q^{17} +73.1752 q^{19} +96.0000 q^{23} -28.4743 q^{25} -27.0000 q^{27} +173.021 q^{29} +67.2475 q^{31} -42.5257 q^{33} -301.670 q^{37} -78.3713 q^{39} -472.042 q^{41} +463.670 q^{43} -88.4228 q^{45} -91.1960 q^{47} -235.794 q^{51} -163.268 q^{53} -139.268 q^{55} -219.526 q^{57} -600.526 q^{59} -571.691 q^{61} -256.659 q^{65} +539.423 q^{67} -288.000 q^{69} +1064.39 q^{71} +442.680 q^{73} +85.4228 q^{75} +45.7010 q^{79} +81.0000 q^{81} -686.464 q^{83} -772.206 q^{85} -519.062 q^{87} -660.145 q^{89} -201.743 q^{93} -718.929 q^{95} -658.320 q^{97} +127.577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 3q^{5} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 3q^{5} + 18q^{9} + 51q^{11} - 61q^{13} - 9q^{15} - 24q^{17} + 169q^{19} + 192q^{23} + 11q^{25} - 54q^{27} - 39q^{29} - 92q^{31} - 153q^{33} - 173q^{37} + 183q^{39} - 174q^{41} + 497q^{43} + 27q^{45} + 180q^{47} + 72q^{51} + 285q^{53} + 333q^{55} - 507q^{57} - 1269q^{59} - 328q^{61} - 1374q^{65} + 875q^{67} - 576q^{69} + 1404q^{71} + 1361q^{73} - 33q^{75} + 182q^{79} + 162q^{81} - 399q^{83} - 2088q^{85} + 117q^{87} - 822q^{89} + 276q^{93} + 510q^{95} - 841q^{97} + 459q^{99} + O(q^{100}) \)

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
\(5\) −9.82475 −0.878753 −0.439376 0.898303i \(-0.644801\pi\)
−0.439376 + 0.898303i \(0.644801\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 14.1752 0.388545 0.194273 0.980948i \(-0.437765\pi\)
0.194273 + 0.980948i \(0.437765\pi\)
\(12\) 0 0
\(13\) 26.1238 0.557341 0.278670 0.960387i \(-0.410106\pi\)
0.278670 + 0.960387i \(0.410106\pi\)
\(14\) 0 0
\(15\) 29.4743 0.507348
\(16\) 0 0
\(17\) 78.5980 1.12134 0.560671 0.828039i \(-0.310543\pi\)
0.560671 + 0.828039i \(0.310543\pi\)
\(18\) 0 0
\(19\) 73.1752 0.883555 0.441778 0.897125i \(-0.354348\pi\)
0.441778 + 0.897125i \(0.354348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 96.0000 0.870321 0.435161 0.900353i \(-0.356692\pi\)
0.435161 + 0.900353i \(0.356692\pi\)
\(24\) 0 0
\(25\) −28.4743 −0.227794
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 173.021 1.10790 0.553951 0.832549i \(-0.313120\pi\)
0.553951 + 0.832549i \(0.313120\pi\)
\(30\) 0 0
\(31\) 67.2475 0.389613 0.194807 0.980842i \(-0.437592\pi\)
0.194807 + 0.980842i \(0.437592\pi\)
\(32\) 0 0
\(33\) −42.5257 −0.224327
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −301.670 −1.34039 −0.670193 0.742187i \(-0.733789\pi\)
−0.670193 + 0.742187i \(0.733789\pi\)
\(38\) 0 0
\(39\) −78.3713 −0.321781
\(40\) 0 0
\(41\) −472.042 −1.79806 −0.899031 0.437886i \(-0.855727\pi\)
−0.899031 + 0.437886i \(0.855727\pi\)
\(42\) 0 0
\(43\) 463.670 1.64440 0.822198 0.569201i \(-0.192747\pi\)
0.822198 + 0.569201i \(0.192747\pi\)
\(44\) 0 0
\(45\) −88.4228 −0.292918
\(46\) 0 0
\(47\) −91.1960 −0.283028 −0.141514 0.989936i \(-0.545197\pi\)
−0.141514 + 0.989936i \(0.545197\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −235.794 −0.647407
\(52\) 0 0
\(53\) −163.268 −0.423144 −0.211572 0.977362i \(-0.567858\pi\)
−0.211572 + 0.977362i \(0.567858\pi\)
\(54\) 0 0
\(55\) −139.268 −0.341435
\(56\) 0 0
\(57\) −219.526 −0.510121
\(58\) 0 0
\(59\) −600.526 −1.32512 −0.662558 0.749011i \(-0.730529\pi\)
−0.662558 + 0.749011i \(0.730529\pi\)
\(60\) 0 0
\(61\) −571.691 −1.19996 −0.599980 0.800015i \(-0.704825\pi\)
−0.599980 + 0.800015i \(0.704825\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −256.659 −0.489764
\(66\) 0 0
\(67\) 539.423 0.983597 0.491798 0.870709i \(-0.336340\pi\)
0.491798 + 0.870709i \(0.336340\pi\)
\(68\) 0 0
\(69\) −288.000 −0.502480
\(70\) 0 0
\(71\) 1064.39 1.77916 0.889578 0.456783i \(-0.150998\pi\)
0.889578 + 0.456783i \(0.150998\pi\)
\(72\) 0 0
\(73\) 442.680 0.709751 0.354875 0.934914i \(-0.384523\pi\)
0.354875 + 0.934914i \(0.384523\pi\)
\(74\) 0 0
\(75\) 85.4228 0.131517
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 45.7010 0.0650856 0.0325428 0.999470i \(-0.489639\pi\)
0.0325428 + 0.999470i \(0.489639\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −686.464 −0.907822 −0.453911 0.891047i \(-0.649972\pi\)
−0.453911 + 0.891047i \(0.649972\pi\)
\(84\) 0 0
\(85\) −772.206 −0.985382
\(86\) 0 0
\(87\) −519.062 −0.639647
\(88\) 0 0
\(89\) −660.145 −0.786238 −0.393119 0.919488i \(-0.628604\pi\)
−0.393119 + 0.919488i \(0.628604\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −201.743 −0.224943
\(94\) 0 0
\(95\) −718.929 −0.776427
\(96\) 0 0
\(97\) −658.320 −0.689095 −0.344548 0.938769i \(-0.611968\pi\)
−0.344548 + 0.938769i \(0.611968\pi\)
\(98\) 0 0
\(99\) 127.577 0.129515
\(100\) 0 0
\(101\) 1803.90 1.77717 0.888586 0.458709i \(-0.151688\pi\)
0.888586 + 0.458709i \(0.151688\pi\)
\(102\) 0 0
\(103\) −1074.91 −1.02829 −0.514145 0.857703i \(-0.671891\pi\)
−0.514145 + 0.857703i \(0.671891\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1659.35 1.49921 0.749605 0.661885i \(-0.230243\pi\)
0.749605 + 0.661885i \(0.230243\pi\)
\(108\) 0 0
\(109\) 598.928 0.526302 0.263151 0.964755i \(-0.415238\pi\)
0.263151 + 0.964755i \(0.415238\pi\)
\(110\) 0 0
\(111\) 905.011 0.773872
\(112\) 0 0
\(113\) −781.650 −0.650720 −0.325360 0.945590i \(-0.605486\pi\)
−0.325360 + 0.945590i \(0.605486\pi\)
\(114\) 0 0
\(115\) −943.176 −0.764797
\(116\) 0 0
\(117\) 235.114 0.185780
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1130.06 −0.849033
\(122\) 0 0
\(123\) 1416.12 1.03811
\(124\) 0 0
\(125\) 1507.85 1.07893
\(126\) 0 0
\(127\) −25.3722 −0.0177277 −0.00886385 0.999961i \(-0.502821\pi\)
−0.00886385 + 0.999961i \(0.502821\pi\)
\(128\) 0 0
\(129\) −1391.01 −0.949393
\(130\) 0 0
\(131\) −475.577 −0.317186 −0.158593 0.987344i \(-0.550696\pi\)
−0.158593 + 0.987344i \(0.550696\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 265.268 0.169116
\(136\) 0 0
\(137\) 1058.54 0.660123 0.330062 0.943959i \(-0.392930\pi\)
0.330062 + 0.943959i \(0.392930\pi\)
\(138\) 0 0
\(139\) 2580.99 1.57494 0.787470 0.616352i \(-0.211390\pi\)
0.787470 + 0.616352i \(0.211390\pi\)
\(140\) 0 0
\(141\) 273.588 0.163406
\(142\) 0 0
\(143\) 370.311 0.216552
\(144\) 0 0
\(145\) −1699.89 −0.973571
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −101.875 −0.0560131 −0.0280066 0.999608i \(-0.508916\pi\)
−0.0280066 + 0.999608i \(0.508916\pi\)
\(150\) 0 0
\(151\) 1778.42 0.958450 0.479225 0.877692i \(-0.340918\pi\)
0.479225 + 0.877692i \(0.340918\pi\)
\(152\) 0 0
\(153\) 707.382 0.373781
\(154\) 0 0
\(155\) −660.690 −0.342374
\(156\) 0 0
\(157\) −1399.26 −0.711292 −0.355646 0.934621i \(-0.615739\pi\)
−0.355646 + 0.934621i \(0.615739\pi\)
\(158\) 0 0
\(159\) 489.805 0.244302
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3615.43 −1.73731 −0.868656 0.495415i \(-0.835016\pi\)
−0.868656 + 0.495415i \(0.835016\pi\)
\(164\) 0 0
\(165\) 417.805 0.197128
\(166\) 0 0
\(167\) −1076.72 −0.498917 −0.249459 0.968385i \(-0.580253\pi\)
−0.249459 + 0.968385i \(0.580253\pi\)
\(168\) 0 0
\(169\) −1514.55 −0.689372
\(170\) 0 0
\(171\) 658.577 0.294518
\(172\) 0 0
\(173\) −1873.63 −0.823407 −0.411704 0.911318i \(-0.635066\pi\)
−0.411704 + 0.911318i \(0.635066\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1801.58 0.765056
\(178\) 0 0
\(179\) 2692.29 1.12420 0.562099 0.827070i \(-0.309994\pi\)
0.562099 + 0.827070i \(0.309994\pi\)
\(180\) 0 0
\(181\) −461.670 −0.189589 −0.0947947 0.995497i \(-0.530219\pi\)
−0.0947947 + 0.995497i \(0.530219\pi\)
\(182\) 0 0
\(183\) 1715.07 0.692797
\(184\) 0 0
\(185\) 2963.84 1.17787
\(186\) 0 0
\(187\) 1114.15 0.435692
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3764.83 1.42625 0.713124 0.701038i \(-0.247280\pi\)
0.713124 + 0.701038i \(0.247280\pi\)
\(192\) 0 0
\(193\) 2926.20 1.09136 0.545680 0.837994i \(-0.316272\pi\)
0.545680 + 0.837994i \(0.316272\pi\)
\(194\) 0 0
\(195\) 769.978 0.282766
\(196\) 0 0
\(197\) 5045.55 1.82477 0.912387 0.409330i \(-0.134237\pi\)
0.912387 + 0.409330i \(0.134237\pi\)
\(198\) 0 0
\(199\) 4292.25 1.52899 0.764496 0.644628i \(-0.222988\pi\)
0.764496 + 0.644628i \(0.222988\pi\)
\(200\) 0 0
\(201\) −1618.27 −0.567880
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4637.69 1.58005
\(206\) 0 0
\(207\) 864.000 0.290107
\(208\) 0 0
\(209\) 1037.28 0.343301
\(210\) 0 0
\(211\) −3625.76 −1.18297 −0.591487 0.806315i \(-0.701459\pi\)
−0.591487 + 0.806315i \(0.701459\pi\)
\(212\) 0 0
\(213\) −3193.18 −1.02720
\(214\) 0 0
\(215\) −4555.45 −1.44502
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1328.04 −0.409775
\(220\) 0 0
\(221\) 2053.28 0.624969
\(222\) 0 0
\(223\) −3145.00 −0.944417 −0.472208 0.881487i \(-0.656543\pi\)
−0.472208 + 0.881487i \(0.656543\pi\)
\(224\) 0 0
\(225\) −256.268 −0.0759313
\(226\) 0 0
\(227\) −1545.75 −0.451959 −0.225980 0.974132i \(-0.572558\pi\)
−0.225980 + 0.974132i \(0.572558\pi\)
\(228\) 0 0
\(229\) 4072.15 1.17509 0.587544 0.809192i \(-0.300095\pi\)
0.587544 + 0.809192i \(0.300095\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3015.71 −0.847922 −0.423961 0.905680i \(-0.639361\pi\)
−0.423961 + 0.905680i \(0.639361\pi\)
\(234\) 0 0
\(235\) 895.978 0.248711
\(236\) 0 0
\(237\) −137.103 −0.0375772
\(238\) 0 0
\(239\) −4647.38 −1.25780 −0.628900 0.777486i \(-0.716495\pi\)
−0.628900 + 0.777486i \(0.716495\pi\)
\(240\) 0 0
\(241\) −1118.28 −0.298898 −0.149449 0.988769i \(-0.547750\pi\)
−0.149449 + 0.988769i \(0.547750\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1911.61 0.492441
\(248\) 0 0
\(249\) 2059.39 0.524131
\(250\) 0 0
\(251\) 883.518 0.222180 0.111090 0.993810i \(-0.464566\pi\)
0.111090 + 0.993810i \(0.464566\pi\)
\(252\) 0 0
\(253\) 1360.82 0.338159
\(254\) 0 0
\(255\) 2316.62 0.568911
\(256\) 0 0
\(257\) 5868.89 1.42448 0.712240 0.701937i \(-0.247681\pi\)
0.712240 + 0.701937i \(0.247681\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1557.19 0.369301
\(262\) 0 0
\(263\) −4875.61 −1.14313 −0.571565 0.820557i \(-0.693664\pi\)
−0.571565 + 0.820557i \(0.693664\pi\)
\(264\) 0 0
\(265\) 1604.07 0.371839
\(266\) 0 0
\(267\) 1980.43 0.453935
\(268\) 0 0
\(269\) 5527.10 1.25276 0.626382 0.779516i \(-0.284535\pi\)
0.626382 + 0.779516i \(0.284535\pi\)
\(270\) 0 0
\(271\) −3224.73 −0.722836 −0.361418 0.932404i \(-0.617707\pi\)
−0.361418 + 0.932404i \(0.617707\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −403.630 −0.0885083
\(276\) 0 0
\(277\) −1958.04 −0.424720 −0.212360 0.977191i \(-0.568115\pi\)
−0.212360 + 0.977191i \(0.568115\pi\)
\(278\) 0 0
\(279\) 605.228 0.129871
\(280\) 0 0
\(281\) 8531.16 1.81113 0.905563 0.424212i \(-0.139449\pi\)
0.905563 + 0.424212i \(0.139449\pi\)
\(282\) 0 0
\(283\) 7909.86 1.66146 0.830728 0.556678i \(-0.187924\pi\)
0.830728 + 0.556678i \(0.187924\pi\)
\(284\) 0 0
\(285\) 2156.79 0.448270
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1264.65 0.257408
\(290\) 0 0
\(291\) 1974.96 0.397849
\(292\) 0 0
\(293\) 8321.41 1.65919 0.829594 0.558367i \(-0.188572\pi\)
0.829594 + 0.558367i \(0.188572\pi\)
\(294\) 0 0
\(295\) 5900.02 1.16445
\(296\) 0 0
\(297\) −382.732 −0.0747756
\(298\) 0 0
\(299\) 2507.88 0.485065
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −5411.69 −1.02605
\(304\) 0 0
\(305\) 5616.72 1.05447
\(306\) 0 0
\(307\) 2541.79 0.472533 0.236267 0.971688i \(-0.424076\pi\)
0.236267 + 0.971688i \(0.424076\pi\)
\(308\) 0 0
\(309\) 3224.72 0.593683
\(310\) 0 0
\(311\) −607.709 −0.110804 −0.0554020 0.998464i \(-0.517644\pi\)
−0.0554020 + 0.998464i \(0.517644\pi\)
\(312\) 0 0
\(313\) −11027.3 −1.99137 −0.995684 0.0928043i \(-0.970417\pi\)
−0.995684 + 0.0928043i \(0.970417\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7261.60 1.28660 0.643300 0.765614i \(-0.277565\pi\)
0.643300 + 0.765614i \(0.277565\pi\)
\(318\) 0 0
\(319\) 2452.61 0.430470
\(320\) 0 0
\(321\) −4978.05 −0.865570
\(322\) 0 0
\(323\) 5751.43 0.990768
\(324\) 0 0
\(325\) −743.855 −0.126959
\(326\) 0 0
\(327\) −1796.78 −0.303860
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6632.73 −1.10141 −0.550706 0.834699i \(-0.685642\pi\)
−0.550706 + 0.834699i \(0.685642\pi\)
\(332\) 0 0
\(333\) −2715.03 −0.446795
\(334\) 0 0
\(335\) −5299.69 −0.864338
\(336\) 0 0
\(337\) 8104.59 1.31005 0.655023 0.755609i \(-0.272659\pi\)
0.655023 + 0.755609i \(0.272659\pi\)
\(338\) 0 0
\(339\) 2344.95 0.375694
\(340\) 0 0
\(341\) 953.250 0.151382
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2829.53 0.441556
\(346\) 0 0
\(347\) 5937.28 0.918530 0.459265 0.888299i \(-0.348113\pi\)
0.459265 + 0.888299i \(0.348113\pi\)
\(348\) 0 0
\(349\) −268.472 −0.0411775 −0.0205888 0.999788i \(-0.506554\pi\)
−0.0205888 + 0.999788i \(0.506554\pi\)
\(350\) 0 0
\(351\) −705.341 −0.107260
\(352\) 0 0
\(353\) −2662.81 −0.401493 −0.200747 0.979643i \(-0.564337\pi\)
−0.200747 + 0.979643i \(0.564337\pi\)
\(354\) 0 0
\(355\) −10457.4 −1.56344
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1563.27 0.229823 0.114911 0.993376i \(-0.463342\pi\)
0.114911 + 0.993376i \(0.463342\pi\)
\(360\) 0 0
\(361\) −1504.38 −0.219330
\(362\) 0 0
\(363\) 3390.19 0.490189
\(364\) 0 0
\(365\) −4349.22 −0.623695
\(366\) 0 0
\(367\) 1210.82 0.172218 0.0861091 0.996286i \(-0.472557\pi\)
0.0861091 + 0.996286i \(0.472557\pi\)
\(368\) 0 0
\(369\) −4248.37 −0.599354
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4771.86 −0.662406 −0.331203 0.943559i \(-0.607455\pi\)
−0.331203 + 0.943559i \(0.607455\pi\)
\(374\) 0 0
\(375\) −4523.54 −0.622919
\(376\) 0 0
\(377\) 4519.95 0.617479
\(378\) 0 0
\(379\) 4118.66 0.558209 0.279104 0.960261i \(-0.409962\pi\)
0.279104 + 0.960261i \(0.409962\pi\)
\(380\) 0 0
\(381\) 76.1166 0.0102351
\(382\) 0 0
\(383\) 3535.98 0.471749 0.235875 0.971783i \(-0.424204\pi\)
0.235875 + 0.971783i \(0.424204\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4173.03 0.548132
\(388\) 0 0
\(389\) 9606.46 1.25210 0.626050 0.779783i \(-0.284671\pi\)
0.626050 + 0.779783i \(0.284671\pi\)
\(390\) 0 0
\(391\) 7545.41 0.975928
\(392\) 0 0
\(393\) 1426.73 0.183127
\(394\) 0 0
\(395\) −449.001 −0.0571941
\(396\) 0 0
\(397\) −5367.07 −0.678503 −0.339252 0.940696i \(-0.610174\pi\)
−0.339252 + 0.940696i \(0.610174\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14065.8 −1.75166 −0.875828 0.482624i \(-0.839684\pi\)
−0.875828 + 0.482624i \(0.839684\pi\)
\(402\) 0 0
\(403\) 1756.76 0.217147
\(404\) 0 0
\(405\) −795.805 −0.0976392
\(406\) 0 0
\(407\) −4276.25 −0.520801
\(408\) 0 0
\(409\) −3272.44 −0.395628 −0.197814 0.980240i \(-0.563384\pi\)
−0.197814 + 0.980240i \(0.563384\pi\)
\(410\) 0 0
\(411\) −3175.61 −0.381122
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6744.34 0.797751
\(416\) 0 0
\(417\) −7742.97 −0.909293
\(418\) 0 0
\(419\) 8077.07 0.941744 0.470872 0.882202i \(-0.343939\pi\)
0.470872 + 0.882202i \(0.343939\pi\)
\(420\) 0 0
\(421\) 8051.68 0.932101 0.466051 0.884758i \(-0.345676\pi\)
0.466051 + 0.884758i \(0.345676\pi\)
\(422\) 0 0
\(423\) −820.764 −0.0943426
\(424\) 0 0
\(425\) −2238.02 −0.255435
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1110.93 −0.125026
\(430\) 0 0
\(431\) 9900.22 1.10644 0.553221 0.833034i \(-0.313398\pi\)
0.553221 + 0.833034i \(0.313398\pi\)
\(432\) 0 0
\(433\) 511.795 0.0568021 0.0284010 0.999597i \(-0.490958\pi\)
0.0284010 + 0.999597i \(0.490958\pi\)
\(434\) 0 0
\(435\) 5099.66 0.562092
\(436\) 0 0
\(437\) 7024.82 0.768977
\(438\) 0 0
\(439\) −1302.26 −0.141580 −0.0707898 0.997491i \(-0.522552\pi\)
−0.0707898 + 0.997491i \(0.522552\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8200.32 0.879479 0.439739 0.898125i \(-0.355071\pi\)
0.439739 + 0.898125i \(0.355071\pi\)
\(444\) 0 0
\(445\) 6485.76 0.690909
\(446\) 0 0
\(447\) 305.626 0.0323392
\(448\) 0 0
\(449\) −5788.12 −0.608370 −0.304185 0.952613i \(-0.598384\pi\)
−0.304185 + 0.952613i \(0.598384\pi\)
\(450\) 0 0
\(451\) −6691.31 −0.698628
\(452\) 0 0
\(453\) −5335.27 −0.553362
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7259.18 0.743041 0.371521 0.928425i \(-0.378836\pi\)
0.371521 + 0.928425i \(0.378836\pi\)
\(458\) 0 0
\(459\) −2122.15 −0.215802
\(460\) 0 0
\(461\) 5966.92 0.602835 0.301418 0.953492i \(-0.402540\pi\)
0.301418 + 0.953492i \(0.402540\pi\)
\(462\) 0 0
\(463\) 8884.02 0.891740 0.445870 0.895098i \(-0.352894\pi\)
0.445870 + 0.895098i \(0.352894\pi\)
\(464\) 0 0
\(465\) 1982.07 0.197669
\(466\) 0 0
\(467\) −4029.36 −0.399265 −0.199632 0.979871i \(-0.563975\pi\)
−0.199632 + 0.979871i \(0.563975\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4197.77 0.410665
\(472\) 0 0
\(473\) 6572.64 0.638923
\(474\) 0 0
\(475\) −2083.61 −0.201269
\(476\) 0 0
\(477\) −1469.41 −0.141048
\(478\) 0 0
\(479\) −17440.7 −1.66364 −0.831820 0.555045i \(-0.812701\pi\)
−0.831820 + 0.555045i \(0.812701\pi\)
\(480\) 0 0
\(481\) −7880.76 −0.747052
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6467.83 0.605544
\(486\) 0 0
\(487\) 4272.28 0.397526 0.198763 0.980048i \(-0.436308\pi\)
0.198763 + 0.980048i \(0.436308\pi\)
\(488\) 0 0
\(489\) 10846.3 1.00304
\(490\) 0 0
\(491\) −2013.29 −0.185048 −0.0925240 0.995710i \(-0.529493\pi\)
−0.0925240 + 0.995710i \(0.529493\pi\)
\(492\) 0 0
\(493\) 13599.1 1.24234
\(494\) 0 0
\(495\) −1253.41 −0.113812
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4880.46 0.437834 0.218917 0.975743i \(-0.429748\pi\)
0.218917 + 0.975743i \(0.429748\pi\)
\(500\) 0 0
\(501\) 3230.16 0.288050
\(502\) 0 0
\(503\) 12961.9 1.14899 0.574496 0.818508i \(-0.305198\pi\)
0.574496 + 0.818508i \(0.305198\pi\)
\(504\) 0 0
\(505\) −17722.8 −1.56170
\(506\) 0 0
\(507\) 4543.65 0.398009
\(508\) 0 0
\(509\) −8155.38 −0.710178 −0.355089 0.934832i \(-0.615550\pi\)
−0.355089 + 0.934832i \(0.615550\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1975.73 −0.170040
\(514\) 0 0
\(515\) 10560.7 0.903612
\(516\) 0 0
\(517\) −1292.73 −0.109969
\(518\) 0 0
\(519\) 5620.89 0.475394
\(520\) 0 0
\(521\) 332.018 0.0279193 0.0139597 0.999903i \(-0.495556\pi\)
0.0139597 + 0.999903i \(0.495556\pi\)
\(522\) 0 0
\(523\) 7254.83 0.606561 0.303281 0.952901i \(-0.401918\pi\)
0.303281 + 0.952901i \(0.401918\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5285.52 0.436890
\(528\) 0 0
\(529\) −2951.00 −0.242541
\(530\) 0 0
\(531\) −5404.73 −0.441705
\(532\) 0 0
\(533\) −12331.5 −1.00213
\(534\) 0 0
\(535\) −16302.7 −1.31744
\(536\) 0 0
\(537\) −8076.87 −0.649055
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5410.19 0.429949 0.214974 0.976620i \(-0.431033\pi\)
0.214974 + 0.976620i \(0.431033\pi\)
\(542\) 0 0
\(543\) 1385.01 0.109459
\(544\) 0 0
\(545\) −5884.32 −0.462489
\(546\) 0 0
\(547\) 18673.9 1.45967 0.729834 0.683625i \(-0.239597\pi\)
0.729834 + 0.683625i \(0.239597\pi\)
\(548\) 0 0
\(549\) −5145.22 −0.399987
\(550\) 0 0
\(551\) 12660.8 0.978893
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8891.51 −0.680042
\(556\) 0 0
\(557\) −8209.48 −0.624500 −0.312250 0.950000i \(-0.601083\pi\)
−0.312250 + 0.950000i \(0.601083\pi\)
\(558\) 0 0
\(559\) 12112.8 0.916489
\(560\) 0 0
\(561\) −3342.44 −0.251547
\(562\) 0 0
\(563\) 17522.2 1.31167 0.655837 0.754902i \(-0.272316\pi\)
0.655837 + 0.754902i \(0.272316\pi\)
\(564\) 0 0
\(565\) 7679.51 0.571822
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9056.83 −0.667279 −0.333640 0.942701i \(-0.608277\pi\)
−0.333640 + 0.942701i \(0.608277\pi\)
\(570\) 0 0
\(571\) −10716.5 −0.785417 −0.392709 0.919663i \(-0.628462\pi\)
−0.392709 + 0.919663i \(0.628462\pi\)
\(572\) 0 0
\(573\) −11294.5 −0.823444
\(574\) 0 0
\(575\) −2733.53 −0.198254
\(576\) 0 0
\(577\) −3217.23 −0.232123 −0.116062 0.993242i \(-0.537027\pi\)
−0.116062 + 0.993242i \(0.537027\pi\)
\(578\) 0 0
\(579\) −8778.59 −0.630097
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2314.37 −0.164411
\(584\) 0 0
\(585\) −2309.93 −0.163255
\(586\) 0 0
\(587\) 3248.45 0.228412 0.114206 0.993457i \(-0.463568\pi\)
0.114206 + 0.993457i \(0.463568\pi\)
\(588\) 0 0
\(589\) 4920.85 0.344245
\(590\) 0 0
\(591\) −15136.6 −1.05353
\(592\) 0 0
\(593\) 21110.4 1.46189 0.730945 0.682437i \(-0.239080\pi\)
0.730945 + 0.682437i \(0.239080\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12876.7 −0.882764
\(598\) 0 0
\(599\) 15738.0 1.07352 0.536758 0.843736i \(-0.319649\pi\)
0.536758 + 0.843736i \(0.319649\pi\)
\(600\) 0 0
\(601\) 4856.96 0.329650 0.164825 0.986323i \(-0.447294\pi\)
0.164825 + 0.986323i \(0.447294\pi\)
\(602\) 0 0
\(603\) 4854.80 0.327866
\(604\) 0 0
\(605\) 11102.6 0.746089
\(606\) 0 0
\(607\) −17258.6 −1.15404 −0.577022 0.816729i \(-0.695785\pi\)
−0.577022 + 0.816729i \(0.695785\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2382.38 −0.157743
\(612\) 0 0
\(613\) 16117.1 1.06193 0.530964 0.847394i \(-0.321830\pi\)
0.530964 + 0.847394i \(0.321830\pi\)
\(614\) 0 0
\(615\) −13913.1 −0.912243
\(616\) 0 0
\(617\) −17507.3 −1.14233 −0.571164 0.820836i \(-0.693508\pi\)
−0.571164 + 0.820836i \(0.693508\pi\)
\(618\) 0 0
\(619\) 9718.49 0.631048 0.315524 0.948918i \(-0.397820\pi\)
0.315524 + 0.948918i \(0.397820\pi\)
\(620\) 0 0
\(621\) −2592.00 −0.167493
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11254.9 −0.720316
\(626\) 0 0
\(627\) −3111.83 −0.198205
\(628\) 0 0
\(629\) −23710.7 −1.50303
\(630\) 0 0
\(631\) −5144.78 −0.324580 −0.162290 0.986743i \(-0.551888\pi\)
−0.162290 + 0.986743i \(0.551888\pi\)
\(632\) 0 0
\(633\) 10877.3 0.682990
\(634\) 0 0
\(635\) 249.275 0.0155783
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 9579.53 0.593052
\(640\) 0 0
\(641\) 21686.1 1.33627 0.668135 0.744040i \(-0.267093\pi\)
0.668135 + 0.744040i \(0.267093\pi\)
\(642\) 0 0
\(643\) −14171.1 −0.869132 −0.434566 0.900640i \(-0.643098\pi\)
−0.434566 + 0.900640i \(0.643098\pi\)
\(644\) 0 0
\(645\) 13666.3 0.834281
\(646\) 0 0
\(647\) 20053.3 1.21851 0.609254 0.792975i \(-0.291469\pi\)
0.609254 + 0.792975i \(0.291469\pi\)
\(648\) 0 0
\(649\) −8512.60 −0.514867
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23641.2 1.41677 0.708385 0.705827i \(-0.249424\pi\)
0.708385 + 0.705827i \(0.249424\pi\)
\(654\) 0 0
\(655\) 4672.43 0.278728
\(656\) 0 0
\(657\) 3984.12 0.236584
\(658\) 0 0
\(659\) −2367.34 −0.139937 −0.0699685 0.997549i \(-0.522290\pi\)
−0.0699685 + 0.997549i \(0.522290\pi\)
\(660\) 0 0
\(661\) 389.981 0.0229478 0.0114739 0.999934i \(-0.496348\pi\)
0.0114739 + 0.999934i \(0.496348\pi\)
\(662\) 0 0
\(663\) −6159.83 −0.360826
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16610.0 0.964230
\(668\) 0 0
\(669\) 9435.01 0.545259
\(670\) 0 0
\(671\) −8103.86 −0.466239
\(672\) 0 0
\(673\) 28764.5 1.64753 0.823766 0.566930i \(-0.191869\pi\)
0.823766 + 0.566930i \(0.191869\pi\)
\(674\) 0 0
\(675\) 768.805 0.0438390
\(676\) 0 0
\(677\) 1762.12 0.100035 0.0500176 0.998748i \(-0.484072\pi\)
0.0500176 + 0.998748i \(0.484072\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 4637.24 0.260939
\(682\) 0 0
\(683\) 2228.87 0.124869 0.0624344 0.998049i \(-0.480114\pi\)
0.0624344 + 0.998049i \(0.480114\pi\)
\(684\) 0 0
\(685\) −10399.9 −0.580085
\(686\) 0 0
\(687\) −12216.4 −0.678437
\(688\) 0 0
\(689\) −4265.18 −0.235835
\(690\) 0 0
\(691\) 17344.3 0.954859 0.477429 0.878670i \(-0.341569\pi\)
0.477429 + 0.878670i \(0.341569\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −25357.6 −1.38398
\(696\) 0 0
\(697\) −37101.5 −2.01624
\(698\) 0 0
\(699\) 9047.14 0.489548
\(700\) 0 0
\(701\) −23697.7 −1.27682 −0.638409 0.769697i \(-0.720407\pi\)
−0.638409 + 0.769697i \(0.720407\pi\)
\(702\) 0 0
\(703\) −22074.8 −1.18431
\(704\) 0 0
\(705\) −2687.93 −0.143594
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −22791.0 −1.20724 −0.603620 0.797272i \(-0.706275\pi\)
−0.603620 + 0.797272i \(0.706275\pi\)
\(710\) 0 0
\(711\) 411.309 0.0216952
\(712\) 0 0
\(713\) 6455.76 0.339089
\(714\) 0 0
\(715\) −3638.21 −0.190296
\(716\) 0 0
\(717\) 13942.2 0.726191
\(718\) 0 0
\(719\) 31844.8 1.65175 0.825876 0.563852i \(-0.190681\pi\)
0.825876 + 0.563852i \(0.190681\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3354.83 0.172569
\(724\) 0 0
\(725\) −4926.64 −0.252373
\(726\) 0 0
\(727\) −22500.0 −1.14784 −0.573920 0.818912i \(-0.694578\pi\)
−0.573920 + 0.818912i \(0.694578\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 36443.6 1.84393
\(732\) 0 0
\(733\) 3222.75 0.162394 0.0811972 0.996698i \(-0.474126\pi\)
0.0811972 + 0.996698i \(0.474126\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7646.45 0.382172
\(738\) 0 0
\(739\) −18359.5 −0.913893 −0.456946 0.889494i \(-0.651057\pi\)
−0.456946 + 0.889494i \(0.651057\pi\)
\(740\) 0 0
\(741\) −5734.84 −0.284311
\(742\) 0 0
\(743\) −12890.4 −0.636478 −0.318239 0.948010i \(-0.603091\pi\)
−0.318239 + 0.948010i \(0.603091\pi\)
\(744\) 0 0
\(745\) 1000.90 0.0492217
\(746\) 0 0
\(747\) −6178.18 −0.302607
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −15390.2 −0.747798 −0.373899 0.927469i \(-0.621979\pi\)
−0.373899 + 0.927469i \(0.621979\pi\)
\(752\) 0 0
\(753\) −2650.55 −0.128276
\(754\) 0 0
\(755\) −17472.6 −0.842241
\(756\) 0 0
\(757\) 13974.6 0.670957 0.335479 0.942048i \(-0.391102\pi\)
0.335479 + 0.942048i \(0.391102\pi\)
\(758\) 0 0
\(759\) −4082.47 −0.195236
\(760\) 0 0
\(761\) 3375.14 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6949.85 −0.328461
\(766\) 0 0
\(767\) −15688.0 −0.738540
\(768\) 0 0
\(769\) 31253.9 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(770\) 0 0
\(771\) −17606.7 −0.822423
\(772\) 0 0
\(773\) −23163.5 −1.07779 −0.538897 0.842372i \(-0.681159\pi\)
−0.538897 + 0.842372i \(0.681159\pi\)
\(774\) 0 0
\(775\) −1914.82 −0.0887516
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −34541.8 −1.58869
\(780\) 0 0
\(781\) 15088.0 0.691283
\(782\) 0 0
\(783\) −4671.56 −0.213216
\(784\) 0 0
\(785\) 13747.4 0.625050
\(786\) 0 0
\(787\) −9902.95 −0.448541 −0.224271 0.974527i \(-0.572000\pi\)
−0.224271 + 0.974527i \(0.572000\pi\)
\(788\) 0 0
\(789\) 14626.8 0.659986
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −14934.7 −0.668786
\(794\) 0 0
\(795\) −4812.21 −0.214681
\(796\) 0 0
\(797\) −16844.5 −0.748638 −0.374319 0.927300i \(-0.622123\pi\)
−0.374319 + 0.927300i \(0.622123\pi\)
\(798\) 0 0
\(799\) −7167.83 −0.317371
\(800\) 0 0
\(801\) −5941.30 −0.262079
\(802\) 0 0
\(803\) 6275.10 0.275770
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −16581.3 −0.723284
\(808\) 0 0
\(809\) −34430.7 −1.49631 −0.748157 0.663522i \(-0.769061\pi\)
−0.748157 + 0.663522i \(0.769061\pi\)
\(810\) 0 0
\(811\) −62.4498 −0.00270396 −0.00135198 0.999999i \(-0.500430\pi\)
−0.00135198 + 0.999999i \(0.500430\pi\)
\(812\) 0 0
\(813\) 9674.20 0.417329
\(814\) 0 0
\(815\) 35520.7 1.52667
\(816\) 0 0
\(817\) 33929.2 1.45292
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 38951.7 1.65581 0.827906 0.560866i \(-0.189532\pi\)
0.827906 + 0.560866i \(0.189532\pi\)
\(822\) 0 0
\(823\) 35482.3 1.50284 0.751420 0.659825i \(-0.229369\pi\)
0.751420 + 0.659825i \(0.229369\pi\)
\(824\) 0 0
\(825\) 1210.89 0.0511003
\(826\) 0 0
\(827\) −7945.87 −0.334105 −0.167053 0.985948i \(-0.553425\pi\)
−0.167053 + 0.985948i \(0.553425\pi\)
\(828\) 0 0
\(829\) −40636.4 −1.70249 −0.851243 0.524772i \(-0.824151\pi\)
−0.851243 + 0.524772i \(0.824151\pi\)
\(830\) 0 0
\(831\) 5874.13 0.245212
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 10578.5 0.438425
\(836\) 0 0
\(837\) −1815.68 −0.0749811
\(838\) 0 0
\(839\) 27723.5 1.14079 0.570394 0.821371i \(-0.306791\pi\)
0.570394 + 0.821371i \(0.306791\pi\)
\(840\) 0 0
\(841\) 5547.19 0.227446
\(842\) 0 0
\(843\) −25593.5 −1.04565
\(844\) 0 0
\(845\) 14880.1 0.605787
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −23729.6 −0.959243
\(850\) 0 0
\(851\) −28960.3 −1.16657
\(852\) 0 0
\(853\) −37599.1 −1.50922 −0.754612 0.656171i \(-0.772175\pi\)
−0.754612 + 0.656171i \(0.772175\pi\)
\(854\) 0 0
\(855\) −6470.36 −0.258809
\(856\) 0 0
\(857\) −25610.0 −1.02079 −0.510397 0.859939i \(-0.670502\pi\)
−0.510397 + 0.859939i \(0.670502\pi\)
\(858\) 0 0
\(859\) −3979.39 −0.158062 −0.0790309 0.996872i \(-0.525183\pi\)
−0.0790309 + 0.996872i \(0.525183\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24820.3 0.979020 0.489510 0.871998i \(-0.337176\pi\)
0.489510 + 0.871998i \(0.337176\pi\)
\(864\) 0 0
\(865\) 18407.9 0.723571
\(866\) 0 0
\(867\) −3793.94 −0.148615
\(868\) 0 0
\(869\) 647.823 0.0252887
\(870\) 0 0
\(871\) 14091.7 0.548198
\(872\) 0 0
\(873\) −5924.88 −0.229698
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 50019.4 1.92593 0.962963 0.269635i \(-0.0869030\pi\)
0.962963 + 0.269635i \(0.0869030\pi\)
\(878\) 0 0
\(879\) −24964.2 −0.957933
\(880\) 0 0
\(881\) 10386.4 0.397193 0.198596 0.980081i \(-0.436362\pi\)
0.198596 + 0.980081i \(0.436362\pi\)
\(882\) 0 0
\(883\) −36366.4 −1.38599 −0.692994 0.720944i \(-0.743709\pi\)
−0.692994 + 0.720944i \(0.743709\pi\)
\(884\) 0 0
\(885\) −17700.0 −0.672295
\(886\) 0 0
\(887\) −7607.31 −0.287969 −0.143984 0.989580i \(-0.545992\pi\)
−0.143984 + 0.989580i \(0.545992\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1148.20 0.0431717
\(892\) 0 0
\(893\) −6673.29 −0.250071
\(894\) 0 0
\(895\) −26451.1 −0.987891
\(896\) 0 0
\(897\) −7523.64 −0.280053
\(898\) 0 0
\(899\) 11635.2 0.431653
\(900\) 0 0
\(901\) −12832.6 −0.474489
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4535.80 0.166602
\(906\) 0 0
\(907\) −25803.1 −0.944628 −0.472314 0.881430i \(-0.656581\pi\)
−0.472314 + 0.881430i \(0.656581\pi\)
\(908\) 0 0
\(909\) 16235.1 0.592391
\(910\) 0 0
\(911\) −8165.21 −0.296954 −0.148477 0.988916i \(-0.547437\pi\)
−0.148477 + 0.988916i \(0.547437\pi\)
\(912\) 0 0
\(913\) −9730.80 −0.352730
\(914\) 0 0
\(915\) −16850.2 −0.608797
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −593.607 −0.0213072 −0.0106536 0.999943i \(-0.503391\pi\)
−0.0106536 + 0.999943i \(0.503391\pi\)
\(920\) 0 0
\(921\) −7625.37 −0.272817
\(922\) 0 0
\(923\) 27805.9 0.991596
\(924\) 0 0
\(925\) 8589.84 0.305332
\(926\) 0 0
\(927\) −9674.17 −0.342763
\(928\) 0 0
\(929\) 33147.4 1.17065 0.585323 0.810800i \(-0.300968\pi\)
0.585323 + 0.810800i \(0.300968\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1823.13 0.0639727
\(934\) 0 0
\(935\) −10946.2 −0.382866
\(936\) 0 0
\(937\) 24883.8 0.867575 0.433788 0.901015i \(-0.357177\pi\)
0.433788 + 0.901015i \(0.357177\pi\)
\(938\) 0 0
\(939\) 33081.8 1.14972
\(940\) 0 0
\(941\) 26130.3 0.905232 0.452616 0.891706i \(-0.350491\pi\)
0.452616 + 0.891706i \(0.350491\pi\)
\(942\) 0 0
\(943\) −45316.0 −1.56489
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7968.69 −0.273440 −0.136720 0.990610i \(-0.543656\pi\)
−0.136720 + 0.990610i \(0.543656\pi\)
\(948\) 0 0
\(949\) 11564.5 0.395573
\(950\) 0 0
\(951\) −21784.8 −0.742819
\(952\) 0 0
\(953\) −6742.36 −0.229178 −0.114589 0.993413i \(-0.536555\pi\)
−0.114589 + 0.993413i \(0.536555\pi\)
\(954\) 0 0
\(955\) −36988.5 −1.25332
\(956\) 0 0
\(957\) −7357.84 −0.248532
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −25268.8 −0.848202
\(962\) 0 0
\(963\) 14934.2 0.499737
\(964\) 0 0
\(965\) −28749.2 −0.959035
\(966\) 0 0
\(967\) 7202.50 0.239521 0.119760 0.992803i \(-0.461787\pi\)
0.119760 + 0.992803i \(0.461787\pi\)
\(968\) 0 0
\(969\) −17254.3 −0.572020
\(970\) 0 0
\(971\) 13705.7 0.452973 0.226487 0.974014i \(-0.427276\pi\)
0.226487 + 0.974014i \(0.427276\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2231.56 0.0732997
\(976\) 0 0
\(977\) −56548.0 −1.85172 −0.925860 0.377868i \(-0.876657\pi\)
−0.925860 + 0.377868i \(0.876657\pi\)
\(978\) 0 0
\(979\) −9357.71 −0.305489
\(980\) 0 0
\(981\) 5390.35 0.175434
\(982\) 0 0
\(983\) 17363.7 0.563395 0.281698 0.959503i \(-0.409103\pi\)
0.281698 + 0.959503i \(0.409103\pi\)
\(984\) 0 0
\(985\) −49571.2 −1.60352
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 44512.3 1.43115
\(990\) 0 0
\(991\) −49039.4 −1.57194 −0.785968 0.618267i \(-0.787835\pi\)
−0.785968 + 0.618267i \(0.787835\pi\)
\(992\) 0 0
\(993\) 19898.2 0.635901
\(994\) 0 0
\(995\) −42170.3 −1.34361
\(996\) 0 0
\(997\) 9116.60 0.289595 0.144797 0.989461i \(-0.453747\pi\)
0.144797 + 0.989461i \(0.453747\pi\)
\(998\) 0 0
\(999\) 8145.10 0.257957
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.bp.1.1 2
4.3 odd 2 588.4.a.h.1.1 2
7.3 odd 6 336.4.q.h.289.1 4
7.5 odd 6 336.4.q.h.193.1 4
7.6 odd 2 2352.4.a.cb.1.2 2
12.11 even 2 1764.4.a.p.1.2 2
28.3 even 6 84.4.i.b.37.1 yes 4
28.11 odd 6 588.4.i.i.373.2 4
28.19 even 6 84.4.i.b.25.1 4
28.23 odd 6 588.4.i.i.361.2 4
28.27 even 2 588.4.a.g.1.2 2
84.11 even 6 1764.4.k.z.1549.1 4
84.23 even 6 1764.4.k.z.361.1 4
84.47 odd 6 252.4.k.d.109.2 4
84.59 odd 6 252.4.k.d.37.2 4
84.83 odd 2 1764.4.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.b.25.1 4 28.19 even 6
84.4.i.b.37.1 yes 4 28.3 even 6
252.4.k.d.37.2 4 84.59 odd 6
252.4.k.d.109.2 4 84.47 odd 6
336.4.q.h.193.1 4 7.5 odd 6
336.4.q.h.289.1 4 7.3 odd 6
588.4.a.g.1.2 2 28.27 even 2
588.4.a.h.1.1 2 4.3 odd 2
588.4.i.i.361.2 4 28.23 odd 6
588.4.i.i.373.2 4 28.11 odd 6
1764.4.a.p.1.2 2 12.11 even 2
1764.4.a.x.1.1 2 84.83 odd 2
1764.4.k.z.361.1 4 84.23 even 6
1764.4.k.z.1549.1 4 84.11 even 6
2352.4.a.bp.1.1 2 1.1 even 1 trivial
2352.4.a.cb.1.2 2 7.6 odd 2