Properties

Label 20.4.a.a.1.1
Level $20$
Weight $4$
Character 20.1
Self dual yes
Analytic conductor $1.180$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 20.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.18003820011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 20.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} -11.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} -11.0000 q^{9} -60.0000 q^{11} +86.0000 q^{13} +20.0000 q^{15} +18.0000 q^{17} +44.0000 q^{19} -64.0000 q^{21} +48.0000 q^{23} +25.0000 q^{25} -152.000 q^{27} -186.000 q^{29} +176.000 q^{31} -240.000 q^{33} -80.0000 q^{35} +254.000 q^{37} +344.000 q^{39} +186.000 q^{41} -100.000 q^{43} -55.0000 q^{45} +168.000 q^{47} -87.0000 q^{49} +72.0000 q^{51} -498.000 q^{53} -300.000 q^{55} +176.000 q^{57} -252.000 q^{59} -58.0000 q^{61} +176.000 q^{63} +430.000 q^{65} -1036.00 q^{67} +192.000 q^{69} +168.000 q^{71} +506.000 q^{73} +100.000 q^{75} +960.000 q^{77} +272.000 q^{79} -311.000 q^{81} +948.000 q^{83} +90.0000 q^{85} -744.000 q^{87} -1014.00 q^{89} -1376.00 q^{91} +704.000 q^{93} +220.000 q^{95} -766.000 q^{97} +660.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 4.00000 0.769800 0.384900 0.922958i \(-0.374236\pi\)
0.384900 + 0.922958i \(0.374236\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −16.0000 −0.863919 −0.431959 0.901893i \(-0.642178\pi\)
−0.431959 + 0.901893i \(0.642178\pi\)
\(8\) 0 0
\(9\) −11.0000 −0.407407
\(10\) 0 0
\(11\) −60.0000 −1.64461 −0.822304 0.569049i \(-0.807311\pi\)
−0.822304 + 0.569049i \(0.807311\pi\)
\(12\) 0 0
\(13\) 86.0000 1.83478 0.917389 0.397992i \(-0.130293\pi\)
0.917389 + 0.397992i \(0.130293\pi\)
\(14\) 0 0
\(15\) 20.0000 0.344265
\(16\) 0 0
\(17\) 18.0000 0.256802 0.128401 0.991722i \(-0.459015\pi\)
0.128401 + 0.991722i \(0.459015\pi\)
\(18\) 0 0
\(19\) 44.0000 0.531279 0.265639 0.964072i \(-0.414417\pi\)
0.265639 + 0.964072i \(0.414417\pi\)
\(20\) 0 0
\(21\) −64.0000 −0.665045
\(22\) 0 0
\(23\) 48.0000 0.435161 0.217580 0.976042i \(-0.430184\pi\)
0.217580 + 0.976042i \(0.430184\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −152.000 −1.08342
\(28\) 0 0
\(29\) −186.000 −1.19101 −0.595506 0.803351i \(-0.703048\pi\)
−0.595506 + 0.803351i \(0.703048\pi\)
\(30\) 0 0
\(31\) 176.000 1.01969 0.509847 0.860265i \(-0.329702\pi\)
0.509847 + 0.860265i \(0.329702\pi\)
\(32\) 0 0
\(33\) −240.000 −1.26602
\(34\) 0 0
\(35\) −80.0000 −0.386356
\(36\) 0 0
\(37\) 254.000 1.12858 0.564288 0.825578i \(-0.309151\pi\)
0.564288 + 0.825578i \(0.309151\pi\)
\(38\) 0 0
\(39\) 344.000 1.41241
\(40\) 0 0
\(41\) 186.000 0.708496 0.354248 0.935152i \(-0.384737\pi\)
0.354248 + 0.935152i \(0.384737\pi\)
\(42\) 0 0
\(43\) −100.000 −0.354648 −0.177324 0.984153i \(-0.556744\pi\)
−0.177324 + 0.984153i \(0.556744\pi\)
\(44\) 0 0
\(45\) −55.0000 −0.182198
\(46\) 0 0
\(47\) 168.000 0.521390 0.260695 0.965421i \(-0.416048\pi\)
0.260695 + 0.965421i \(0.416048\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) 72.0000 0.197687
\(52\) 0 0
\(53\) −498.000 −1.29067 −0.645335 0.763899i \(-0.723282\pi\)
−0.645335 + 0.763899i \(0.723282\pi\)
\(54\) 0 0
\(55\) −300.000 −0.735491
\(56\) 0 0
\(57\) 176.000 0.408978
\(58\) 0 0
\(59\) −252.000 −0.556061 −0.278031 0.960572i \(-0.589682\pi\)
−0.278031 + 0.960572i \(0.589682\pi\)
\(60\) 0 0
\(61\) −58.0000 −0.121740 −0.0608700 0.998146i \(-0.519388\pi\)
−0.0608700 + 0.998146i \(0.519388\pi\)
\(62\) 0 0
\(63\) 176.000 0.351967
\(64\) 0 0
\(65\) 430.000 0.820537
\(66\) 0 0
\(67\) −1036.00 −1.88907 −0.944534 0.328414i \(-0.893486\pi\)
−0.944534 + 0.328414i \(0.893486\pi\)
\(68\) 0 0
\(69\) 192.000 0.334987
\(70\) 0 0
\(71\) 168.000 0.280816 0.140408 0.990094i \(-0.455159\pi\)
0.140408 + 0.990094i \(0.455159\pi\)
\(72\) 0 0
\(73\) 506.000 0.811272 0.405636 0.914035i \(-0.367050\pi\)
0.405636 + 0.914035i \(0.367050\pi\)
\(74\) 0 0
\(75\) 100.000 0.153960
\(76\) 0 0
\(77\) 960.000 1.42081
\(78\) 0 0
\(79\) 272.000 0.387372 0.193686 0.981064i \(-0.437956\pi\)
0.193686 + 0.981064i \(0.437956\pi\)
\(80\) 0 0
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) 948.000 1.25369 0.626846 0.779143i \(-0.284345\pi\)
0.626846 + 0.779143i \(0.284345\pi\)
\(84\) 0 0
\(85\) 90.0000 0.114846
\(86\) 0 0
\(87\) −744.000 −0.916841
\(88\) 0 0
\(89\) −1014.00 −1.20768 −0.603841 0.797104i \(-0.706364\pi\)
−0.603841 + 0.797104i \(0.706364\pi\)
\(90\) 0 0
\(91\) −1376.00 −1.58510
\(92\) 0 0
\(93\) 704.000 0.784961
\(94\) 0 0
\(95\) 220.000 0.237595
\(96\) 0 0
\(97\) −766.000 −0.801809 −0.400905 0.916120i \(-0.631304\pi\)
−0.400905 + 0.916120i \(0.631304\pi\)
\(98\) 0 0
\(99\) 660.000 0.670025
\(100\) 0 0
\(101\) −1314.00 −1.29453 −0.647267 0.762264i \(-0.724088\pi\)
−0.647267 + 0.762264i \(0.724088\pi\)
\(102\) 0 0
\(103\) −448.000 −0.428570 −0.214285 0.976771i \(-0.568742\pi\)
−0.214285 + 0.976771i \(0.568742\pi\)
\(104\) 0 0
\(105\) −320.000 −0.297417
\(106\) 0 0
\(107\) 1548.00 1.39861 0.699303 0.714826i \(-0.253494\pi\)
0.699303 + 0.714826i \(0.253494\pi\)
\(108\) 0 0
\(109\) 278.000 0.244290 0.122145 0.992512i \(-0.461023\pi\)
0.122145 + 0.992512i \(0.461023\pi\)
\(110\) 0 0
\(111\) 1016.00 0.868779
\(112\) 0 0
\(113\) −558.000 −0.464533 −0.232266 0.972652i \(-0.574614\pi\)
−0.232266 + 0.972652i \(0.574614\pi\)
\(114\) 0 0
\(115\) 240.000 0.194610
\(116\) 0 0
\(117\) −946.000 −0.747502
\(118\) 0 0
\(119\) −288.000 −0.221856
\(120\) 0 0
\(121\) 2269.00 1.70473
\(122\) 0 0
\(123\) 744.000 0.545400
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 344.000 0.240355 0.120177 0.992752i \(-0.461654\pi\)
0.120177 + 0.992752i \(0.461654\pi\)
\(128\) 0 0
\(129\) −400.000 −0.273008
\(130\) 0 0
\(131\) 780.000 0.520221 0.260110 0.965579i \(-0.416241\pi\)
0.260110 + 0.965579i \(0.416241\pi\)
\(132\) 0 0
\(133\) −704.000 −0.458982
\(134\) 0 0
\(135\) −760.000 −0.484521
\(136\) 0 0
\(137\) 666.000 0.415330 0.207665 0.978200i \(-0.433414\pi\)
0.207665 + 0.978200i \(0.433414\pi\)
\(138\) 0 0
\(139\) 884.000 0.539424 0.269712 0.962941i \(-0.413072\pi\)
0.269712 + 0.962941i \(0.413072\pi\)
\(140\) 0 0
\(141\) 672.000 0.401366
\(142\) 0 0
\(143\) −5160.00 −3.01749
\(144\) 0 0
\(145\) −930.000 −0.532637
\(146\) 0 0
\(147\) −348.000 −0.195255
\(148\) 0 0
\(149\) −114.000 −0.0626795 −0.0313397 0.999509i \(-0.509977\pi\)
−0.0313397 + 0.999509i \(0.509977\pi\)
\(150\) 0 0
\(151\) −40.0000 −0.0215573 −0.0107787 0.999942i \(-0.503431\pi\)
−0.0107787 + 0.999942i \(0.503431\pi\)
\(152\) 0 0
\(153\) −198.000 −0.104623
\(154\) 0 0
\(155\) 880.000 0.456021
\(156\) 0 0
\(157\) −154.000 −0.0782837 −0.0391418 0.999234i \(-0.512462\pi\)
−0.0391418 + 0.999234i \(0.512462\pi\)
\(158\) 0 0
\(159\) −1992.00 −0.993559
\(160\) 0 0
\(161\) −768.000 −0.375943
\(162\) 0 0
\(163\) 2180.00 1.04755 0.523775 0.851856i \(-0.324523\pi\)
0.523775 + 0.851856i \(0.324523\pi\)
\(164\) 0 0
\(165\) −1200.00 −0.566181
\(166\) 0 0
\(167\) 3696.00 1.71261 0.856303 0.516474i \(-0.172756\pi\)
0.856303 + 0.516474i \(0.172756\pi\)
\(168\) 0 0
\(169\) 5199.00 2.36641
\(170\) 0 0
\(171\) −484.000 −0.216447
\(172\) 0 0
\(173\) 1302.00 0.572192 0.286096 0.958201i \(-0.407642\pi\)
0.286096 + 0.958201i \(0.407642\pi\)
\(174\) 0 0
\(175\) −400.000 −0.172784
\(176\) 0 0
\(177\) −1008.00 −0.428056
\(178\) 0 0
\(179\) −4308.00 −1.79885 −0.899427 0.437070i \(-0.856016\pi\)
−0.899427 + 0.437070i \(0.856016\pi\)
\(180\) 0 0
\(181\) 1550.00 0.636523 0.318261 0.948003i \(-0.396901\pi\)
0.318261 + 0.948003i \(0.396901\pi\)
\(182\) 0 0
\(183\) −232.000 −0.0937155
\(184\) 0 0
\(185\) 1270.00 0.504715
\(186\) 0 0
\(187\) −1080.00 −0.422339
\(188\) 0 0
\(189\) 2432.00 0.935989
\(190\) 0 0
\(191\) 48.0000 0.0181841 0.00909204 0.999959i \(-0.497106\pi\)
0.00909204 + 0.999959i \(0.497106\pi\)
\(192\) 0 0
\(193\) 1058.00 0.394593 0.197297 0.980344i \(-0.436784\pi\)
0.197297 + 0.980344i \(0.436784\pi\)
\(194\) 0 0
\(195\) 1720.00 0.631650
\(196\) 0 0
\(197\) −3714.00 −1.34321 −0.671603 0.740911i \(-0.734394\pi\)
−0.671603 + 0.740911i \(0.734394\pi\)
\(198\) 0 0
\(199\) −1768.00 −0.629800 −0.314900 0.949125i \(-0.601971\pi\)
−0.314900 + 0.949125i \(0.601971\pi\)
\(200\) 0 0
\(201\) −4144.00 −1.45421
\(202\) 0 0
\(203\) 2976.00 1.02894
\(204\) 0 0
\(205\) 930.000 0.316849
\(206\) 0 0
\(207\) −528.000 −0.177288
\(208\) 0 0
\(209\) −2640.00 −0.873745
\(210\) 0 0
\(211\) −4036.00 −1.31682 −0.658412 0.752658i \(-0.728771\pi\)
−0.658412 + 0.752658i \(0.728771\pi\)
\(212\) 0 0
\(213\) 672.000 0.216172
\(214\) 0 0
\(215\) −500.000 −0.158603
\(216\) 0 0
\(217\) −2816.00 −0.880933
\(218\) 0 0
\(219\) 2024.00 0.624517
\(220\) 0 0
\(221\) 1548.00 0.471175
\(222\) 0 0
\(223\) 680.000 0.204198 0.102099 0.994774i \(-0.467444\pi\)
0.102099 + 0.994774i \(0.467444\pi\)
\(224\) 0 0
\(225\) −275.000 −0.0814815
\(226\) 0 0
\(227\) 2388.00 0.698225 0.349113 0.937081i \(-0.386483\pi\)
0.349113 + 0.937081i \(0.386483\pi\)
\(228\) 0 0
\(229\) −3874.00 −1.11791 −0.558954 0.829198i \(-0.688797\pi\)
−0.558954 + 0.829198i \(0.688797\pi\)
\(230\) 0 0
\(231\) 3840.00 1.09374
\(232\) 0 0
\(233\) 3162.00 0.889054 0.444527 0.895766i \(-0.353372\pi\)
0.444527 + 0.895766i \(0.353372\pi\)
\(234\) 0 0
\(235\) 840.000 0.233173
\(236\) 0 0
\(237\) 1088.00 0.298199
\(238\) 0 0
\(239\) 5424.00 1.46799 0.733995 0.679155i \(-0.237654\pi\)
0.733995 + 0.679155i \(0.237654\pi\)
\(240\) 0 0
\(241\) −3886.00 −1.03867 −0.519335 0.854571i \(-0.673820\pi\)
−0.519335 + 0.854571i \(0.673820\pi\)
\(242\) 0 0
\(243\) 2860.00 0.755017
\(244\) 0 0
\(245\) −435.000 −0.113433
\(246\) 0 0
\(247\) 3784.00 0.974778
\(248\) 0 0
\(249\) 3792.00 0.965093
\(250\) 0 0
\(251\) −5100.00 −1.28251 −0.641253 0.767329i \(-0.721585\pi\)
−0.641253 + 0.767329i \(0.721585\pi\)
\(252\) 0 0
\(253\) −2880.00 −0.715668
\(254\) 0 0
\(255\) 360.000 0.0884081
\(256\) 0 0
\(257\) 2178.00 0.528638 0.264319 0.964435i \(-0.414853\pi\)
0.264319 + 0.964435i \(0.414853\pi\)
\(258\) 0 0
\(259\) −4064.00 −0.974999
\(260\) 0 0
\(261\) 2046.00 0.485227
\(262\) 0 0
\(263\) −6144.00 −1.44051 −0.720257 0.693707i \(-0.755976\pi\)
−0.720257 + 0.693707i \(0.755976\pi\)
\(264\) 0 0
\(265\) −2490.00 −0.577206
\(266\) 0 0
\(267\) −4056.00 −0.929675
\(268\) 0 0
\(269\) 822.000 0.186313 0.0931566 0.995651i \(-0.470304\pi\)
0.0931566 + 0.995651i \(0.470304\pi\)
\(270\) 0 0
\(271\) 8480.00 1.90082 0.950412 0.310994i \(-0.100662\pi\)
0.950412 + 0.310994i \(0.100662\pi\)
\(272\) 0 0
\(273\) −5504.00 −1.22021
\(274\) 0 0
\(275\) −1500.00 −0.328921
\(276\) 0 0
\(277\) −1138.00 −0.246844 −0.123422 0.992354i \(-0.539387\pi\)
−0.123422 + 0.992354i \(0.539387\pi\)
\(278\) 0 0
\(279\) −1936.00 −0.415431
\(280\) 0 0
\(281\) 5706.00 1.21136 0.605679 0.795709i \(-0.292902\pi\)
0.605679 + 0.795709i \(0.292902\pi\)
\(282\) 0 0
\(283\) −3028.00 −0.636028 −0.318014 0.948086i \(-0.603016\pi\)
−0.318014 + 0.948086i \(0.603016\pi\)
\(284\) 0 0
\(285\) 880.000 0.182901
\(286\) 0 0
\(287\) −2976.00 −0.612083
\(288\) 0 0
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) −3064.00 −0.617233
\(292\) 0 0
\(293\) 3390.00 0.675925 0.337962 0.941160i \(-0.390262\pi\)
0.337962 + 0.941160i \(0.390262\pi\)
\(294\) 0 0
\(295\) −1260.00 −0.248678
\(296\) 0 0
\(297\) 9120.00 1.78180
\(298\) 0 0
\(299\) 4128.00 0.798423
\(300\) 0 0
\(301\) 1600.00 0.306387
\(302\) 0 0
\(303\) −5256.00 −0.996532
\(304\) 0 0
\(305\) −290.000 −0.0544438
\(306\) 0 0
\(307\) −4156.00 −0.772624 −0.386312 0.922368i \(-0.626251\pi\)
−0.386312 + 0.922368i \(0.626251\pi\)
\(308\) 0 0
\(309\) −1792.00 −0.329914
\(310\) 0 0
\(311\) 6552.00 1.19463 0.597315 0.802007i \(-0.296234\pi\)
0.597315 + 0.802007i \(0.296234\pi\)
\(312\) 0 0
\(313\) −1366.00 −0.246680 −0.123340 0.992364i \(-0.539361\pi\)
−0.123340 + 0.992364i \(0.539361\pi\)
\(314\) 0 0
\(315\) 880.000 0.157404
\(316\) 0 0
\(317\) 2598.00 0.460310 0.230155 0.973154i \(-0.426077\pi\)
0.230155 + 0.973154i \(0.426077\pi\)
\(318\) 0 0
\(319\) 11160.0 1.95875
\(320\) 0 0
\(321\) 6192.00 1.07665
\(322\) 0 0
\(323\) 792.000 0.136434
\(324\) 0 0
\(325\) 2150.00 0.366956
\(326\) 0 0
\(327\) 1112.00 0.188054
\(328\) 0 0
\(329\) −2688.00 −0.450438
\(330\) 0 0
\(331\) −3292.00 −0.546661 −0.273330 0.961920i \(-0.588125\pi\)
−0.273330 + 0.961920i \(0.588125\pi\)
\(332\) 0 0
\(333\) −2794.00 −0.459791
\(334\) 0 0
\(335\) −5180.00 −0.844817
\(336\) 0 0
\(337\) 6194.00 1.00121 0.500606 0.865675i \(-0.333110\pi\)
0.500606 + 0.865675i \(0.333110\pi\)
\(338\) 0 0
\(339\) −2232.00 −0.357598
\(340\) 0 0
\(341\) −10560.0 −1.67700
\(342\) 0 0
\(343\) 6880.00 1.08305
\(344\) 0 0
\(345\) 960.000 0.149811
\(346\) 0 0
\(347\) −10020.0 −1.55015 −0.775075 0.631870i \(-0.782288\pi\)
−0.775075 + 0.631870i \(0.782288\pi\)
\(348\) 0 0
\(349\) −3130.00 −0.480072 −0.240036 0.970764i \(-0.577159\pi\)
−0.240036 + 0.970764i \(0.577159\pi\)
\(350\) 0 0
\(351\) −13072.0 −1.98784
\(352\) 0 0
\(353\) 4194.00 0.632363 0.316181 0.948699i \(-0.397599\pi\)
0.316181 + 0.948699i \(0.397599\pi\)
\(354\) 0 0
\(355\) 840.000 0.125585
\(356\) 0 0
\(357\) −1152.00 −0.170785
\(358\) 0 0
\(359\) −4104.00 −0.603345 −0.301672 0.953412i \(-0.597545\pi\)
−0.301672 + 0.953412i \(0.597545\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) 0 0
\(363\) 9076.00 1.31230
\(364\) 0 0
\(365\) 2530.00 0.362812
\(366\) 0 0
\(367\) 7496.00 1.06618 0.533090 0.846059i \(-0.321031\pi\)
0.533090 + 0.846059i \(0.321031\pi\)
\(368\) 0 0
\(369\) −2046.00 −0.288646
\(370\) 0 0
\(371\) 7968.00 1.11503
\(372\) 0 0
\(373\) −5842.00 −0.810958 −0.405479 0.914104i \(-0.632895\pi\)
−0.405479 + 0.914104i \(0.632895\pi\)
\(374\) 0 0
\(375\) 500.000 0.0688530
\(376\) 0 0
\(377\) −15996.0 −2.18524
\(378\) 0 0
\(379\) −412.000 −0.0558391 −0.0279195 0.999610i \(-0.508888\pi\)
−0.0279195 + 0.999610i \(0.508888\pi\)
\(380\) 0 0
\(381\) 1376.00 0.185025
\(382\) 0 0
\(383\) 2568.00 0.342607 0.171304 0.985218i \(-0.445202\pi\)
0.171304 + 0.985218i \(0.445202\pi\)
\(384\) 0 0
\(385\) 4800.00 0.635404
\(386\) 0 0
\(387\) 1100.00 0.144486
\(388\) 0 0
\(389\) 13086.0 1.70562 0.852810 0.522221i \(-0.174896\pi\)
0.852810 + 0.522221i \(0.174896\pi\)
\(390\) 0 0
\(391\) 864.000 0.111750
\(392\) 0 0
\(393\) 3120.00 0.400466
\(394\) 0 0
\(395\) 1360.00 0.173238
\(396\) 0 0
\(397\) 10454.0 1.32159 0.660795 0.750566i \(-0.270219\pi\)
0.660795 + 0.750566i \(0.270219\pi\)
\(398\) 0 0
\(399\) −2816.00 −0.353324
\(400\) 0 0
\(401\) −10830.0 −1.34869 −0.674345 0.738417i \(-0.735574\pi\)
−0.674345 + 0.738417i \(0.735574\pi\)
\(402\) 0 0
\(403\) 15136.0 1.87091
\(404\) 0 0
\(405\) −1555.00 −0.190787
\(406\) 0 0
\(407\) −15240.0 −1.85607
\(408\) 0 0
\(409\) −8566.00 −1.03560 −0.517801 0.855501i \(-0.673249\pi\)
−0.517801 + 0.855501i \(0.673249\pi\)
\(410\) 0 0
\(411\) 2664.00 0.319721
\(412\) 0 0
\(413\) 4032.00 0.480392
\(414\) 0 0
\(415\) 4740.00 0.560669
\(416\) 0 0
\(417\) 3536.00 0.415249
\(418\) 0 0
\(419\) 13884.0 1.61880 0.809401 0.587257i \(-0.199792\pi\)
0.809401 + 0.587257i \(0.199792\pi\)
\(420\) 0 0
\(421\) 4286.00 0.496168 0.248084 0.968738i \(-0.420199\pi\)
0.248084 + 0.968738i \(0.420199\pi\)
\(422\) 0 0
\(423\) −1848.00 −0.212418
\(424\) 0 0
\(425\) 450.000 0.0513605
\(426\) 0 0
\(427\) 928.000 0.105173
\(428\) 0 0
\(429\) −20640.0 −2.32286
\(430\) 0 0
\(431\) 6336.00 0.708108 0.354054 0.935225i \(-0.384803\pi\)
0.354054 + 0.935225i \(0.384803\pi\)
\(432\) 0 0
\(433\) −8974.00 −0.995988 −0.497994 0.867180i \(-0.665930\pi\)
−0.497994 + 0.867180i \(0.665930\pi\)
\(434\) 0 0
\(435\) −3720.00 −0.410024
\(436\) 0 0
\(437\) 2112.00 0.231191
\(438\) 0 0
\(439\) −2968.00 −0.322676 −0.161338 0.986899i \(-0.551581\pi\)
−0.161338 + 0.986899i \(0.551581\pi\)
\(440\) 0 0
\(441\) 957.000 0.103337
\(442\) 0 0
\(443\) −12372.0 −1.32689 −0.663444 0.748226i \(-0.730906\pi\)
−0.663444 + 0.748226i \(0.730906\pi\)
\(444\) 0 0
\(445\) −5070.00 −0.540092
\(446\) 0 0
\(447\) −456.000 −0.0482507
\(448\) 0 0
\(449\) 11394.0 1.19759 0.598793 0.800904i \(-0.295647\pi\)
0.598793 + 0.800904i \(0.295647\pi\)
\(450\) 0 0
\(451\) −11160.0 −1.16520
\(452\) 0 0
\(453\) −160.000 −0.0165948
\(454\) 0 0
\(455\) −6880.00 −0.708878
\(456\) 0 0
\(457\) −358.000 −0.0366445 −0.0183222 0.999832i \(-0.505832\pi\)
−0.0183222 + 0.999832i \(0.505832\pi\)
\(458\) 0 0
\(459\) −2736.00 −0.278226
\(460\) 0 0
\(461\) −7530.00 −0.760753 −0.380376 0.924832i \(-0.624206\pi\)
−0.380376 + 0.924832i \(0.624206\pi\)
\(462\) 0 0
\(463\) −13768.0 −1.38197 −0.690986 0.722868i \(-0.742823\pi\)
−0.690986 + 0.722868i \(0.742823\pi\)
\(464\) 0 0
\(465\) 3520.00 0.351045
\(466\) 0 0
\(467\) 13380.0 1.32581 0.662904 0.748704i \(-0.269324\pi\)
0.662904 + 0.748704i \(0.269324\pi\)
\(468\) 0 0
\(469\) 16576.0 1.63200
\(470\) 0 0
\(471\) −616.000 −0.0602628
\(472\) 0 0
\(473\) 6000.00 0.583256
\(474\) 0 0
\(475\) 1100.00 0.106256
\(476\) 0 0
\(477\) 5478.00 0.525829
\(478\) 0 0
\(479\) −6336.00 −0.604383 −0.302191 0.953247i \(-0.597718\pi\)
−0.302191 + 0.953247i \(0.597718\pi\)
\(480\) 0 0
\(481\) 21844.0 2.07069
\(482\) 0 0
\(483\) −3072.00 −0.289401
\(484\) 0 0
\(485\) −3830.00 −0.358580
\(486\) 0 0
\(487\) −5008.00 −0.465984 −0.232992 0.972479i \(-0.574852\pi\)
−0.232992 + 0.972479i \(0.574852\pi\)
\(488\) 0 0
\(489\) 8720.00 0.806405
\(490\) 0 0
\(491\) 12900.0 1.18568 0.592840 0.805320i \(-0.298007\pi\)
0.592840 + 0.805320i \(0.298007\pi\)
\(492\) 0 0
\(493\) −3348.00 −0.305855
\(494\) 0 0
\(495\) 3300.00 0.299644
\(496\) 0 0
\(497\) −2688.00 −0.242602
\(498\) 0 0
\(499\) −8116.00 −0.728100 −0.364050 0.931379i \(-0.618606\pi\)
−0.364050 + 0.931379i \(0.618606\pi\)
\(500\) 0 0
\(501\) 14784.0 1.31836
\(502\) 0 0
\(503\) −4944.00 −0.438255 −0.219127 0.975696i \(-0.570321\pi\)
−0.219127 + 0.975696i \(0.570321\pi\)
\(504\) 0 0
\(505\) −6570.00 −0.578933
\(506\) 0 0
\(507\) 20796.0 1.82166
\(508\) 0 0
\(509\) −5466.00 −0.475985 −0.237992 0.971267i \(-0.576489\pi\)
−0.237992 + 0.971267i \(0.576489\pi\)
\(510\) 0 0
\(511\) −8096.00 −0.700873
\(512\) 0 0
\(513\) −6688.00 −0.575599
\(514\) 0 0
\(515\) −2240.00 −0.191663
\(516\) 0 0
\(517\) −10080.0 −0.857481
\(518\) 0 0
\(519\) 5208.00 0.440474
\(520\) 0 0
\(521\) 10074.0 0.847121 0.423560 0.905868i \(-0.360780\pi\)
0.423560 + 0.905868i \(0.360780\pi\)
\(522\) 0 0
\(523\) −13828.0 −1.15613 −0.578065 0.815991i \(-0.696192\pi\)
−0.578065 + 0.815991i \(0.696192\pi\)
\(524\) 0 0
\(525\) −1600.00 −0.133009
\(526\) 0 0
\(527\) 3168.00 0.261860
\(528\) 0 0
\(529\) −9863.00 −0.810635
\(530\) 0 0
\(531\) 2772.00 0.226543
\(532\) 0 0
\(533\) 15996.0 1.29993
\(534\) 0 0
\(535\) 7740.00 0.625475
\(536\) 0 0
\(537\) −17232.0 −1.38476
\(538\) 0 0
\(539\) 5220.00 0.417145
\(540\) 0 0
\(541\) −15226.0 −1.21001 −0.605006 0.796221i \(-0.706829\pi\)
−0.605006 + 0.796221i \(0.706829\pi\)
\(542\) 0 0
\(543\) 6200.00 0.489995
\(544\) 0 0
\(545\) 1390.00 0.109250
\(546\) 0 0
\(547\) −13228.0 −1.03398 −0.516991 0.855991i \(-0.672948\pi\)
−0.516991 + 0.855991i \(0.672948\pi\)
\(548\) 0 0
\(549\) 638.000 0.0495978
\(550\) 0 0
\(551\) −8184.00 −0.632759
\(552\) 0 0
\(553\) −4352.00 −0.334658
\(554\) 0 0
\(555\) 5080.00 0.388530
\(556\) 0 0
\(557\) −8490.00 −0.645840 −0.322920 0.946426i \(-0.604664\pi\)
−0.322920 + 0.946426i \(0.604664\pi\)
\(558\) 0 0
\(559\) −8600.00 −0.650700
\(560\) 0 0
\(561\) −4320.00 −0.325117
\(562\) 0 0
\(563\) −10284.0 −0.769838 −0.384919 0.922950i \(-0.625771\pi\)
−0.384919 + 0.922950i \(0.625771\pi\)
\(564\) 0 0
\(565\) −2790.00 −0.207745
\(566\) 0 0
\(567\) 4976.00 0.368558
\(568\) 0 0
\(569\) 1770.00 0.130408 0.0652041 0.997872i \(-0.479230\pi\)
0.0652041 + 0.997872i \(0.479230\pi\)
\(570\) 0 0
\(571\) 6068.00 0.444725 0.222362 0.974964i \(-0.428623\pi\)
0.222362 + 0.974964i \(0.428623\pi\)
\(572\) 0 0
\(573\) 192.000 0.0139981
\(574\) 0 0
\(575\) 1200.00 0.0870321
\(576\) 0 0
\(577\) 21506.0 1.55166 0.775829 0.630943i \(-0.217332\pi\)
0.775829 + 0.630943i \(0.217332\pi\)
\(578\) 0 0
\(579\) 4232.00 0.303758
\(580\) 0 0
\(581\) −15168.0 −1.08309
\(582\) 0 0
\(583\) 29880.0 2.12265
\(584\) 0 0
\(585\) −4730.00 −0.334293
\(586\) 0 0
\(587\) 12108.0 0.851364 0.425682 0.904873i \(-0.360034\pi\)
0.425682 + 0.904873i \(0.360034\pi\)
\(588\) 0 0
\(589\) 7744.00 0.541742
\(590\) 0 0
\(591\) −14856.0 −1.03400
\(592\) 0 0
\(593\) 15474.0 1.07157 0.535785 0.844354i \(-0.320016\pi\)
0.535785 + 0.844354i \(0.320016\pi\)
\(594\) 0 0
\(595\) −1440.00 −0.0992172
\(596\) 0 0
\(597\) −7072.00 −0.484820
\(598\) 0 0
\(599\) −2520.00 −0.171894 −0.0859469 0.996300i \(-0.527392\pi\)
−0.0859469 + 0.996300i \(0.527392\pi\)
\(600\) 0 0
\(601\) −12790.0 −0.868078 −0.434039 0.900894i \(-0.642912\pi\)
−0.434039 + 0.900894i \(0.642912\pi\)
\(602\) 0 0
\(603\) 11396.0 0.769620
\(604\) 0 0
\(605\) 11345.0 0.762380
\(606\) 0 0
\(607\) 11576.0 0.774062 0.387031 0.922067i \(-0.373501\pi\)
0.387031 + 0.922067i \(0.373501\pi\)
\(608\) 0 0
\(609\) 11904.0 0.792076
\(610\) 0 0
\(611\) 14448.0 0.956634
\(612\) 0 0
\(613\) 20126.0 1.32607 0.663035 0.748588i \(-0.269268\pi\)
0.663035 + 0.748588i \(0.269268\pi\)
\(614\) 0 0
\(615\) 3720.00 0.243910
\(616\) 0 0
\(617\) −27942.0 −1.82318 −0.911590 0.411100i \(-0.865145\pi\)
−0.911590 + 0.411100i \(0.865145\pi\)
\(618\) 0 0
\(619\) −22540.0 −1.46358 −0.731792 0.681528i \(-0.761316\pi\)
−0.731792 + 0.681528i \(0.761316\pi\)
\(620\) 0 0
\(621\) −7296.00 −0.471463
\(622\) 0 0
\(623\) 16224.0 1.04334
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −10560.0 −0.672609
\(628\) 0 0
\(629\) 4572.00 0.289821
\(630\) 0 0
\(631\) −5128.00 −0.323522 −0.161761 0.986830i \(-0.551717\pi\)
−0.161761 + 0.986830i \(0.551717\pi\)
\(632\) 0 0
\(633\) −16144.0 −1.01369
\(634\) 0 0
\(635\) 1720.00 0.107490
\(636\) 0 0
\(637\) −7482.00 −0.465381
\(638\) 0 0
\(639\) −1848.00 −0.114406
\(640\) 0 0
\(641\) −12798.0 −0.788597 −0.394298 0.918982i \(-0.629012\pi\)
−0.394298 + 0.918982i \(0.629012\pi\)
\(642\) 0 0
\(643\) −21148.0 −1.29704 −0.648519 0.761198i \(-0.724611\pi\)
−0.648519 + 0.761198i \(0.724611\pi\)
\(644\) 0 0
\(645\) −2000.00 −0.122093
\(646\) 0 0
\(647\) 16464.0 1.00041 0.500206 0.865906i \(-0.333258\pi\)
0.500206 + 0.865906i \(0.333258\pi\)
\(648\) 0 0
\(649\) 15120.0 0.914502
\(650\) 0 0
\(651\) −11264.0 −0.678143
\(652\) 0 0
\(653\) −24234.0 −1.45230 −0.726148 0.687538i \(-0.758691\pi\)
−0.726148 + 0.687538i \(0.758691\pi\)
\(654\) 0 0
\(655\) 3900.00 0.232650
\(656\) 0 0
\(657\) −5566.00 −0.330518
\(658\) 0 0
\(659\) −22836.0 −1.34987 −0.674935 0.737877i \(-0.735828\pi\)
−0.674935 + 0.737877i \(0.735828\pi\)
\(660\) 0 0
\(661\) 26318.0 1.54864 0.774320 0.632794i \(-0.218092\pi\)
0.774320 + 0.632794i \(0.218092\pi\)
\(662\) 0 0
\(663\) 6192.00 0.362711
\(664\) 0 0
\(665\) −3520.00 −0.205263
\(666\) 0 0
\(667\) −8928.00 −0.518281
\(668\) 0 0
\(669\) 2720.00 0.157192
\(670\) 0 0
\(671\) 3480.00 0.200214
\(672\) 0 0
\(673\) 28802.0 1.64968 0.824841 0.565365i \(-0.191265\pi\)
0.824841 + 0.565365i \(0.191265\pi\)
\(674\) 0 0
\(675\) −3800.00 −0.216685
\(676\) 0 0
\(677\) 2526.00 0.143400 0.0717002 0.997426i \(-0.477158\pi\)
0.0717002 + 0.997426i \(0.477158\pi\)
\(678\) 0 0
\(679\) 12256.0 0.692698
\(680\) 0 0
\(681\) 9552.00 0.537494
\(682\) 0 0
\(683\) −23076.0 −1.29279 −0.646397 0.763001i \(-0.723725\pi\)
−0.646397 + 0.763001i \(0.723725\pi\)
\(684\) 0 0
\(685\) 3330.00 0.185741
\(686\) 0 0
\(687\) −15496.0 −0.860567
\(688\) 0 0
\(689\) −42828.0 −2.36809
\(690\) 0 0
\(691\) 7868.00 0.433159 0.216579 0.976265i \(-0.430510\pi\)
0.216579 + 0.976265i \(0.430510\pi\)
\(692\) 0 0
\(693\) −10560.0 −0.578847
\(694\) 0 0
\(695\) 4420.00 0.241238
\(696\) 0 0
\(697\) 3348.00 0.181943
\(698\) 0 0
\(699\) 12648.0 0.684394
\(700\) 0 0
\(701\) 21510.0 1.15895 0.579473 0.814991i \(-0.303258\pi\)
0.579473 + 0.814991i \(0.303258\pi\)
\(702\) 0 0
\(703\) 11176.0 0.599589
\(704\) 0 0
\(705\) 3360.00 0.179496
\(706\) 0 0
\(707\) 21024.0 1.11837
\(708\) 0 0
\(709\) 30014.0 1.58984 0.794922 0.606712i \(-0.207512\pi\)
0.794922 + 0.606712i \(0.207512\pi\)
\(710\) 0 0
\(711\) −2992.00 −0.157818
\(712\) 0 0
\(713\) 8448.00 0.443731
\(714\) 0 0
\(715\) −25800.0 −1.34946
\(716\) 0 0
\(717\) 21696.0 1.13006
\(718\) 0 0
\(719\) −816.000 −0.0423250 −0.0211625 0.999776i \(-0.506737\pi\)
−0.0211625 + 0.999776i \(0.506737\pi\)
\(720\) 0 0
\(721\) 7168.00 0.370250
\(722\) 0 0
\(723\) −15544.0 −0.799568
\(724\) 0 0
\(725\) −4650.00 −0.238202
\(726\) 0 0
\(727\) −9952.00 −0.507702 −0.253851 0.967243i \(-0.581697\pi\)
−0.253851 + 0.967243i \(0.581697\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 0 0
\(731\) −1800.00 −0.0910744
\(732\) 0 0
\(733\) −33946.0 −1.71054 −0.855269 0.518185i \(-0.826608\pi\)
−0.855269 + 0.518185i \(0.826608\pi\)
\(734\) 0 0
\(735\) −1740.00 −0.0873209
\(736\) 0 0
\(737\) 62160.0 3.10677
\(738\) 0 0
\(739\) 23420.0 1.16579 0.582895 0.812548i \(-0.301920\pi\)
0.582895 + 0.812548i \(0.301920\pi\)
\(740\) 0 0
\(741\) 15136.0 0.750384
\(742\) 0 0
\(743\) −14592.0 −0.720496 −0.360248 0.932857i \(-0.617308\pi\)
−0.360248 + 0.932857i \(0.617308\pi\)
\(744\) 0 0
\(745\) −570.000 −0.0280311
\(746\) 0 0
\(747\) −10428.0 −0.510764
\(748\) 0 0
\(749\) −24768.0 −1.20828
\(750\) 0 0
\(751\) 9056.00 0.440024 0.220012 0.975497i \(-0.429390\pi\)
0.220012 + 0.975497i \(0.429390\pi\)
\(752\) 0 0
\(753\) −20400.0 −0.987274
\(754\) 0 0
\(755\) −200.000 −0.00964072
\(756\) 0 0
\(757\) −17554.0 −0.842815 −0.421408 0.906871i \(-0.638464\pi\)
−0.421408 + 0.906871i \(0.638464\pi\)
\(758\) 0 0
\(759\) −11520.0 −0.550922
\(760\) 0 0
\(761\) −36438.0 −1.73571 −0.867856 0.496816i \(-0.834502\pi\)
−0.867856 + 0.496816i \(0.834502\pi\)
\(762\) 0 0
\(763\) −4448.00 −0.211046
\(764\) 0 0
\(765\) −990.000 −0.0467889
\(766\) 0 0
\(767\) −21672.0 −1.02025
\(768\) 0 0
\(769\) −9022.00 −0.423071 −0.211536 0.977370i \(-0.567846\pi\)
−0.211536 + 0.977370i \(0.567846\pi\)
\(770\) 0 0
\(771\) 8712.00 0.406946
\(772\) 0 0
\(773\) 1470.00 0.0683987 0.0341994 0.999415i \(-0.489112\pi\)
0.0341994 + 0.999415i \(0.489112\pi\)
\(774\) 0 0
\(775\) 4400.00 0.203939
\(776\) 0 0
\(777\) −16256.0 −0.750554
\(778\) 0 0
\(779\) 8184.00 0.376409
\(780\) 0 0
\(781\) −10080.0 −0.461832
\(782\) 0 0
\(783\) 28272.0 1.29037
\(784\) 0 0
\(785\) −770.000 −0.0350095
\(786\) 0 0
\(787\) 5252.00 0.237883 0.118941 0.992901i \(-0.462050\pi\)
0.118941 + 0.992901i \(0.462050\pi\)
\(788\) 0 0
\(789\) −24576.0 −1.10891
\(790\) 0 0
\(791\) 8928.00 0.401319
\(792\) 0 0
\(793\) −4988.00 −0.223366
\(794\) 0 0
\(795\) −9960.00 −0.444333
\(796\) 0 0
\(797\) 12294.0 0.546394 0.273197 0.961958i \(-0.411919\pi\)
0.273197 + 0.961958i \(0.411919\pi\)
\(798\) 0 0
\(799\) 3024.00 0.133894
\(800\) 0 0
\(801\) 11154.0 0.492019
\(802\) 0 0
\(803\) −30360.0 −1.33422
\(804\) 0 0
\(805\) −3840.00 −0.168127
\(806\) 0 0
\(807\) 3288.00 0.143424
\(808\) 0 0
\(809\) 15546.0 0.675610 0.337805 0.941216i \(-0.390316\pi\)
0.337805 + 0.941216i \(0.390316\pi\)
\(810\) 0 0
\(811\) 19364.0 0.838424 0.419212 0.907888i \(-0.362306\pi\)
0.419212 + 0.907888i \(0.362306\pi\)
\(812\) 0 0
\(813\) 33920.0 1.46326
\(814\) 0 0
\(815\) 10900.0 0.468479
\(816\) 0 0
\(817\) −4400.00 −0.188417
\(818\) 0 0
\(819\) 15136.0 0.645781
\(820\) 0 0
\(821\) −7314.00 −0.310914 −0.155457 0.987843i \(-0.549685\pi\)
−0.155457 + 0.987843i \(0.549685\pi\)
\(822\) 0 0
\(823\) 11984.0 0.507577 0.253789 0.967260i \(-0.418323\pi\)
0.253789 + 0.967260i \(0.418323\pi\)
\(824\) 0 0
\(825\) −6000.00 −0.253204
\(826\) 0 0
\(827\) 13500.0 0.567643 0.283822 0.958877i \(-0.408398\pi\)
0.283822 + 0.958877i \(0.408398\pi\)
\(828\) 0 0
\(829\) −44602.0 −1.86863 −0.934313 0.356453i \(-0.883986\pi\)
−0.934313 + 0.356453i \(0.883986\pi\)
\(830\) 0 0
\(831\) −4552.00 −0.190021
\(832\) 0 0
\(833\) −1566.00 −0.0651365
\(834\) 0 0
\(835\) 18480.0 0.765900
\(836\) 0 0
\(837\) −26752.0 −1.10476
\(838\) 0 0
\(839\) 35448.0 1.45864 0.729321 0.684172i \(-0.239836\pi\)
0.729321 + 0.684172i \(0.239836\pi\)
\(840\) 0 0
\(841\) 10207.0 0.418508
\(842\) 0 0
\(843\) 22824.0 0.932503
\(844\) 0 0
\(845\) 25995.0 1.05829
\(846\) 0 0
\(847\) −36304.0 −1.47275
\(848\) 0 0
\(849\) −12112.0 −0.489615
\(850\) 0 0
\(851\) 12192.0 0.491112
\(852\) 0 0
\(853\) 12590.0 0.505362 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(854\) 0 0
\(855\) −2420.00 −0.0967980
\(856\) 0 0
\(857\) 24906.0 0.992734 0.496367 0.868113i \(-0.334667\pi\)
0.496367 + 0.868113i \(0.334667\pi\)
\(858\) 0 0
\(859\) 23204.0 0.921665 0.460833 0.887487i \(-0.347551\pi\)
0.460833 + 0.887487i \(0.347551\pi\)
\(860\) 0 0
\(861\) −11904.0 −0.471181
\(862\) 0 0
\(863\) 19848.0 0.782890 0.391445 0.920202i \(-0.371975\pi\)
0.391445 + 0.920202i \(0.371975\pi\)
\(864\) 0 0
\(865\) 6510.00 0.255892
\(866\) 0 0
\(867\) −18356.0 −0.719034
\(868\) 0 0
\(869\) −16320.0 −0.637075
\(870\) 0 0
\(871\) −89096.0 −3.46602
\(872\) 0 0
\(873\) 8426.00 0.326663
\(874\) 0 0
\(875\) −2000.00 −0.0772712
\(876\) 0 0
\(877\) 27542.0 1.06046 0.530232 0.847852i \(-0.322105\pi\)
0.530232 + 0.847852i \(0.322105\pi\)
\(878\) 0 0
\(879\) 13560.0 0.520327
\(880\) 0 0
\(881\) −20718.0 −0.792290 −0.396145 0.918188i \(-0.629652\pi\)
−0.396145 + 0.918188i \(0.629652\pi\)
\(882\) 0 0
\(883\) 25172.0 0.959349 0.479675 0.877446i \(-0.340755\pi\)
0.479675 + 0.877446i \(0.340755\pi\)
\(884\) 0 0
\(885\) −5040.00 −0.191432
\(886\) 0 0
\(887\) −12864.0 −0.486957 −0.243478 0.969906i \(-0.578289\pi\)
−0.243478 + 0.969906i \(0.578289\pi\)
\(888\) 0 0
\(889\) −5504.00 −0.207647
\(890\) 0 0
\(891\) 18660.0 0.701609
\(892\) 0 0
\(893\) 7392.00 0.277003
\(894\) 0 0
\(895\) −21540.0 −0.804472
\(896\) 0 0
\(897\) 16512.0 0.614626
\(898\) 0 0
\(899\) −32736.0 −1.21447
\(900\) 0 0
\(901\) −8964.00 −0.331447
\(902\) 0 0
\(903\) 6400.00 0.235857
\(904\) 0 0
\(905\) 7750.00 0.284662
\(906\) 0 0
\(907\) −23092.0 −0.845377 −0.422689 0.906275i \(-0.638914\pi\)
−0.422689 + 0.906275i \(0.638914\pi\)
\(908\) 0 0
\(909\) 14454.0 0.527403
\(910\) 0 0
\(911\) −14208.0 −0.516720 −0.258360 0.966049i \(-0.583182\pi\)
−0.258360 + 0.966049i \(0.583182\pi\)
\(912\) 0 0
\(913\) −56880.0 −2.06183
\(914\) 0 0
\(915\) −1160.00 −0.0419108
\(916\) 0 0
\(917\) −12480.0 −0.449428
\(918\) 0 0
\(919\) −26584.0 −0.954217 −0.477108 0.878844i \(-0.658315\pi\)
−0.477108 + 0.878844i \(0.658315\pi\)
\(920\) 0 0
\(921\) −16624.0 −0.594766
\(922\) 0 0
\(923\) 14448.0 0.515235
\(924\) 0 0
\(925\) 6350.00 0.225715
\(926\) 0 0
\(927\) 4928.00 0.174603
\(928\) 0 0
\(929\) 162.000 0.00572126 0.00286063 0.999996i \(-0.499089\pi\)
0.00286063 + 0.999996i \(0.499089\pi\)
\(930\) 0 0
\(931\) −3828.00 −0.134756
\(932\) 0 0
\(933\) 26208.0 0.919626
\(934\) 0 0
\(935\) −5400.00 −0.188876
\(936\) 0 0
\(937\) −29734.0 −1.03668 −0.518339 0.855175i \(-0.673449\pi\)
−0.518339 + 0.855175i \(0.673449\pi\)
\(938\) 0 0
\(939\) −5464.00 −0.189894
\(940\) 0 0
\(941\) 17142.0 0.593850 0.296925 0.954901i \(-0.404039\pi\)
0.296925 + 0.954901i \(0.404039\pi\)
\(942\) 0 0
\(943\) 8928.00 0.308309
\(944\) 0 0
\(945\) 12160.0 0.418587
\(946\) 0 0
\(947\) 26436.0 0.907133 0.453566 0.891223i \(-0.350152\pi\)
0.453566 + 0.891223i \(0.350152\pi\)
\(948\) 0 0
\(949\) 43516.0 1.48850
\(950\) 0 0
\(951\) 10392.0 0.354347
\(952\) 0 0
\(953\) 27882.0 0.947730 0.473865 0.880598i \(-0.342858\pi\)
0.473865 + 0.880598i \(0.342858\pi\)
\(954\) 0 0
\(955\) 240.000 0.00813217
\(956\) 0 0
\(957\) 44640.0 1.50784
\(958\) 0 0
\(959\) −10656.0 −0.358811
\(960\) 0 0
\(961\) 1185.00 0.0397771
\(962\) 0 0
\(963\) −17028.0 −0.569802
\(964\) 0 0
\(965\) 5290.00 0.176467
\(966\) 0 0
\(967\) 12656.0 0.420879 0.210439 0.977607i \(-0.432511\pi\)
0.210439 + 0.977607i \(0.432511\pi\)
\(968\) 0 0
\(969\) 3168.00 0.105027
\(970\) 0 0
\(971\) 2916.00 0.0963737 0.0481869 0.998838i \(-0.484656\pi\)
0.0481869 + 0.998838i \(0.484656\pi\)
\(972\) 0 0
\(973\) −14144.0 −0.466018
\(974\) 0 0
\(975\) 8600.00 0.282482
\(976\) 0 0
\(977\) −6894.00 −0.225751 −0.112875 0.993609i \(-0.536006\pi\)
−0.112875 + 0.993609i \(0.536006\pi\)
\(978\) 0 0
\(979\) 60840.0 1.98616
\(980\) 0 0
\(981\) −3058.00 −0.0995254
\(982\) 0 0
\(983\) −45264.0 −1.46866 −0.734332 0.678790i \(-0.762505\pi\)
−0.734332 + 0.678790i \(0.762505\pi\)
\(984\) 0 0
\(985\) −18570.0 −0.600700
\(986\) 0 0
\(987\) −10752.0 −0.346748
\(988\) 0 0
\(989\) −4800.00 −0.154329
\(990\) 0 0
\(991\) 52016.0 1.66735 0.833674 0.552256i \(-0.186233\pi\)
0.833674 + 0.552256i \(0.186233\pi\)
\(992\) 0 0
\(993\) −13168.0 −0.420820
\(994\) 0 0
\(995\) −8840.00 −0.281655
\(996\) 0 0
\(997\) −13858.0 −0.440208 −0.220104 0.975476i \(-0.570640\pi\)
−0.220104 + 0.975476i \(0.570640\pi\)
\(998\) 0 0
\(999\) −38608.0 −1.22273
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 20.4.a.a.1.1 1
3.2 odd 2 180.4.a.a.1.1 1
4.3 odd 2 80.4.a.c.1.1 1
5.2 odd 4 100.4.c.a.49.1 2
5.3 odd 4 100.4.c.a.49.2 2
5.4 even 2 100.4.a.a.1.1 1
7.2 even 3 980.4.i.e.361.1 2
7.3 odd 6 980.4.i.n.961.1 2
7.4 even 3 980.4.i.e.961.1 2
7.5 odd 6 980.4.i.n.361.1 2
7.6 odd 2 980.4.a.c.1.1 1
8.3 odd 2 320.4.a.k.1.1 1
8.5 even 2 320.4.a.d.1.1 1
9.2 odd 6 1620.4.i.j.1081.1 2
9.4 even 3 1620.4.i.d.541.1 2
9.5 odd 6 1620.4.i.j.541.1 2
9.7 even 3 1620.4.i.d.1081.1 2
11.10 odd 2 2420.4.a.d.1.1 1
12.11 even 2 720.4.a.k.1.1 1
15.2 even 4 900.4.d.k.649.1 2
15.8 even 4 900.4.d.k.649.2 2
15.14 odd 2 900.4.a.m.1.1 1
16.3 odd 4 1280.4.d.c.641.1 2
16.5 even 4 1280.4.d.n.641.1 2
16.11 odd 4 1280.4.d.c.641.2 2
16.13 even 4 1280.4.d.n.641.2 2
20.3 even 4 400.4.c.j.49.1 2
20.7 even 4 400.4.c.j.49.2 2
20.19 odd 2 400.4.a.o.1.1 1
40.19 odd 2 1600.4.a.p.1.1 1
40.29 even 2 1600.4.a.bl.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.4.a.a.1.1 1 1.1 even 1 trivial
80.4.a.c.1.1 1 4.3 odd 2
100.4.a.a.1.1 1 5.4 even 2
100.4.c.a.49.1 2 5.2 odd 4
100.4.c.a.49.2 2 5.3 odd 4
180.4.a.a.1.1 1 3.2 odd 2
320.4.a.d.1.1 1 8.5 even 2
320.4.a.k.1.1 1 8.3 odd 2
400.4.a.o.1.1 1 20.19 odd 2
400.4.c.j.49.1 2 20.3 even 4
400.4.c.j.49.2 2 20.7 even 4
720.4.a.k.1.1 1 12.11 even 2
900.4.a.m.1.1 1 15.14 odd 2
900.4.d.k.649.1 2 15.2 even 4
900.4.d.k.649.2 2 15.8 even 4
980.4.a.c.1.1 1 7.6 odd 2
980.4.i.e.361.1 2 7.2 even 3
980.4.i.e.961.1 2 7.4 even 3
980.4.i.n.361.1 2 7.5 odd 6
980.4.i.n.961.1 2 7.3 odd 6
1280.4.d.c.641.1 2 16.3 odd 4
1280.4.d.c.641.2 2 16.11 odd 4
1280.4.d.n.641.1 2 16.5 even 4
1280.4.d.n.641.2 2 16.13 even 4
1600.4.a.p.1.1 1 40.19 odd 2
1600.4.a.bl.1.1 1 40.29 even 2
1620.4.i.d.541.1 2 9.4 even 3
1620.4.i.d.1081.1 2 9.7 even 3
1620.4.i.j.541.1 2 9.5 odd 6
1620.4.i.j.1081.1 2 9.2 odd 6
2420.4.a.d.1.1 1 11.10 odd 2