Properties

Label 1575.4.a.w.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.53113 q^{2} +4.46887 q^{4} +7.00000 q^{7} +12.4689 q^{8} +O(q^{10})\) \(q-3.53113 q^{2} +4.46887 q^{4} +7.00000 q^{7} +12.4689 q^{8} +2.93774 q^{11} +19.0623 q^{13} -24.7179 q^{14} -79.7802 q^{16} +122.498 q^{17} +107.436 q^{19} -10.3735 q^{22} +210.623 q^{23} -67.3113 q^{26} +31.2821 q^{28} -95.4942 q^{29} -94.3074 q^{31} +181.963 q^{32} -432.556 q^{34} -97.1206 q^{37} -379.370 q^{38} +491.113 q^{41} +43.0039 q^{43} +13.1284 q^{44} -743.735 q^{46} +473.494 q^{47} +49.0000 q^{49} +85.1868 q^{52} -183.677 q^{53} +87.2821 q^{56} +337.202 q^{58} +760.615 q^{59} -198.747 q^{61} +333.012 q^{62} -4.29373 q^{64} +309.992 q^{67} +547.428 q^{68} -665.693 q^{71} -621.288 q^{73} +342.945 q^{74} +480.117 q^{76} +20.5642 q^{77} -24.7626 q^{79} -1734.18 q^{82} -406.724 q^{83} -151.852 q^{86} +36.6303 q^{88} -261.751 q^{89} +133.436 q^{91} +941.245 q^{92} -1671.97 q^{94} +1004.77 q^{97} -173.025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8} + 22 q^{11} + 22 q^{13} + 7 q^{14} - 87 q^{16} + 116 q^{17} + 102 q^{19} + 76 q^{22} + 260 q^{23} - 54 q^{26} + 119 q^{28} + 196 q^{29} + 150 q^{31} - 15 q^{32} - 462 q^{34} + 96 q^{37} - 404 q^{38} + 176 q^{41} + 344 q^{43} + 252 q^{44} - 520 q^{46} + 560 q^{47} + 98 q^{49} + 122 q^{52} + 326 q^{53} + 231 q^{56} + 1658 q^{58} + 844 q^{59} - 204 q^{61} + 1440 q^{62} - 839 q^{64} + 104 q^{67} + 466 q^{68} - 1670 q^{71} + 386 q^{73} + 1218 q^{74} + 412 q^{76} + 154 q^{77} - 888 q^{79} - 3162 q^{82} + 928 q^{83} + 1212 q^{86} + 428 q^{88} - 588 q^{89} + 154 q^{91} + 1560 q^{92} - 1280 q^{94} - 522 q^{97} + 49 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.53113 −1.24844 −0.624221 0.781248i \(-0.714584\pi\)
−0.624221 + 0.781248i \(0.714584\pi\)
\(3\) 0 0
\(4\) 4.46887 0.558609
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 12.4689 0.551051
\(9\) 0 0
\(10\) 0 0
\(11\) 2.93774 0.0805239 0.0402619 0.999189i \(-0.487181\pi\)
0.0402619 + 0.999189i \(0.487181\pi\)
\(12\) 0 0
\(13\) 19.0623 0.406686 0.203343 0.979108i \(-0.434819\pi\)
0.203343 + 0.979108i \(0.434819\pi\)
\(14\) −24.7179 −0.471867
\(15\) 0 0
\(16\) −79.7802 −1.24656
\(17\) 122.498 1.74766 0.873828 0.486236i \(-0.161630\pi\)
0.873828 + 0.486236i \(0.161630\pi\)
\(18\) 0 0
\(19\) 107.436 1.29723 0.648617 0.761115i \(-0.275348\pi\)
0.648617 + 0.761115i \(0.275348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −10.3735 −0.100529
\(23\) 210.623 1.90947 0.954736 0.297455i \(-0.0961379\pi\)
0.954736 + 0.297455i \(0.0961379\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −67.3113 −0.507724
\(27\) 0 0
\(28\) 31.2821 0.211134
\(29\) −95.4942 −0.611477 −0.305738 0.952116i \(-0.598903\pi\)
−0.305738 + 0.952116i \(0.598903\pi\)
\(30\) 0 0
\(31\) −94.3074 −0.546391 −0.273195 0.961959i \(-0.588081\pi\)
−0.273195 + 0.961959i \(0.588081\pi\)
\(32\) 181.963 1.00521
\(33\) 0 0
\(34\) −432.556 −2.18185
\(35\) 0 0
\(36\) 0 0
\(37\) −97.1206 −0.431528 −0.215764 0.976446i \(-0.569224\pi\)
−0.215764 + 0.976446i \(0.569224\pi\)
\(38\) −379.370 −1.61952
\(39\) 0 0
\(40\) 0 0
\(41\) 491.113 1.87071 0.935353 0.353716i \(-0.115082\pi\)
0.935353 + 0.353716i \(0.115082\pi\)
\(42\) 0 0
\(43\) 43.0039 0.152512 0.0762562 0.997088i \(-0.475703\pi\)
0.0762562 + 0.997088i \(0.475703\pi\)
\(44\) 13.1284 0.0449814
\(45\) 0 0
\(46\) −743.735 −2.38387
\(47\) 473.494 1.46949 0.734747 0.678341i \(-0.237301\pi\)
0.734747 + 0.678341i \(0.237301\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 85.1868 0.227178
\(53\) −183.677 −0.476038 −0.238019 0.971261i \(-0.576498\pi\)
−0.238019 + 0.971261i \(0.576498\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 87.2821 0.208278
\(57\) 0 0
\(58\) 337.202 0.763394
\(59\) 760.615 1.67837 0.839183 0.543849i \(-0.183034\pi\)
0.839183 + 0.543849i \(0.183034\pi\)
\(60\) 0 0
\(61\) −198.747 −0.417163 −0.208582 0.978005i \(-0.566885\pi\)
−0.208582 + 0.978005i \(0.566885\pi\)
\(62\) 333.012 0.682137
\(63\) 0 0
\(64\) −4.29373 −0.00838618
\(65\) 0 0
\(66\) 0 0
\(67\) 309.992 0.565247 0.282624 0.959231i \(-0.408795\pi\)
0.282624 + 0.959231i \(0.408795\pi\)
\(68\) 547.428 0.976256
\(69\) 0 0
\(70\) 0 0
\(71\) −665.693 −1.11272 −0.556360 0.830941i \(-0.687803\pi\)
−0.556360 + 0.830941i \(0.687803\pi\)
\(72\) 0 0
\(73\) −621.288 −0.996113 −0.498057 0.867145i \(-0.665953\pi\)
−0.498057 + 0.867145i \(0.665953\pi\)
\(74\) 342.945 0.538738
\(75\) 0 0
\(76\) 480.117 0.724647
\(77\) 20.5642 0.0304352
\(78\) 0 0
\(79\) −24.7626 −0.0352659 −0.0176330 0.999845i \(-0.505613\pi\)
−0.0176330 + 0.999845i \(0.505613\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1734.18 −2.33547
\(83\) −406.724 −0.537876 −0.268938 0.963157i \(-0.586673\pi\)
−0.268938 + 0.963157i \(0.586673\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −151.852 −0.190403
\(87\) 0 0
\(88\) 36.6303 0.0443728
\(89\) −261.751 −0.311748 −0.155874 0.987777i \(-0.549819\pi\)
−0.155874 + 0.987777i \(0.549819\pi\)
\(90\) 0 0
\(91\) 133.436 0.153713
\(92\) 941.245 1.06665
\(93\) 0 0
\(94\) −1671.97 −1.83458
\(95\) 0 0
\(96\) 0 0
\(97\) 1004.77 1.05175 0.525873 0.850563i \(-0.323739\pi\)
0.525873 + 0.850563i \(0.323739\pi\)
\(98\) −173.025 −0.178349
\(99\) 0 0
\(100\) 0 0
\(101\) 128.872 0.126962 0.0634812 0.997983i \(-0.479780\pi\)
0.0634812 + 0.997983i \(0.479780\pi\)
\(102\) 0 0
\(103\) −806.008 −0.771051 −0.385526 0.922697i \(-0.625980\pi\)
−0.385526 + 0.922697i \(0.625980\pi\)
\(104\) 237.685 0.224105
\(105\) 0 0
\(106\) 648.587 0.594305
\(107\) −769.712 −0.695429 −0.347714 0.937600i \(-0.613042\pi\)
−0.347714 + 0.937600i \(0.613042\pi\)
\(108\) 0 0
\(109\) −780.856 −0.686169 −0.343085 0.939304i \(-0.611472\pi\)
−0.343085 + 0.939304i \(0.611472\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −558.461 −0.471157
\(113\) −1115.65 −0.928771 −0.464386 0.885633i \(-0.653725\pi\)
−0.464386 + 0.885633i \(0.653725\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −426.751 −0.341576
\(117\) 0 0
\(118\) −2685.83 −2.09534
\(119\) 857.486 0.660552
\(120\) 0 0
\(121\) −1322.37 −0.993516
\(122\) 701.802 0.520804
\(123\) 0 0
\(124\) −421.448 −0.305219
\(125\) 0 0
\(126\) 0 0
\(127\) 1875.98 1.31076 0.655381 0.755299i \(-0.272508\pi\)
0.655381 + 0.755299i \(0.272508\pi\)
\(128\) −1440.54 −0.994744
\(129\) 0 0
\(130\) 0 0
\(131\) −364.203 −0.242905 −0.121452 0.992597i \(-0.538755\pi\)
−0.121452 + 0.992597i \(0.538755\pi\)
\(132\) 0 0
\(133\) 752.051 0.490309
\(134\) −1094.62 −0.705679
\(135\) 0 0
\(136\) 1527.41 0.963048
\(137\) 1603.13 0.999743 0.499872 0.866099i \(-0.333380\pi\)
0.499872 + 0.866099i \(0.333380\pi\)
\(138\) 0 0
\(139\) 2431.12 1.48349 0.741746 0.670681i \(-0.233998\pi\)
0.741746 + 0.670681i \(0.233998\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2350.65 1.38917
\(143\) 56.0000 0.0327479
\(144\) 0 0
\(145\) 0 0
\(146\) 2193.85 1.24359
\(147\) 0 0
\(148\) −434.020 −0.241055
\(149\) −2341.57 −1.28744 −0.643722 0.765260i \(-0.722611\pi\)
−0.643722 + 0.765260i \(0.722611\pi\)
\(150\) 0 0
\(151\) −2104.07 −1.13395 −0.566976 0.823734i \(-0.691887\pi\)
−0.566976 + 0.823734i \(0.691887\pi\)
\(152\) 1339.60 0.714843
\(153\) 0 0
\(154\) −72.6148 −0.0379966
\(155\) 0 0
\(156\) 0 0
\(157\) 593.467 0.301680 0.150840 0.988558i \(-0.451802\pi\)
0.150840 + 0.988558i \(0.451802\pi\)
\(158\) 87.4399 0.0440275
\(159\) 0 0
\(160\) 0 0
\(161\) 1474.36 0.721712
\(162\) 0 0
\(163\) −2178.71 −1.04693 −0.523465 0.852047i \(-0.675361\pi\)
−0.523465 + 0.852047i \(0.675361\pi\)
\(164\) 2194.72 1.04499
\(165\) 0 0
\(166\) 1436.19 0.671508
\(167\) 799.502 0.370463 0.185231 0.982695i \(-0.440697\pi\)
0.185231 + 0.982695i \(0.440697\pi\)
\(168\) 0 0
\(169\) −1833.63 −0.834606
\(170\) 0 0
\(171\) 0 0
\(172\) 192.179 0.0851947
\(173\) −1444.36 −0.634754 −0.317377 0.948299i \(-0.602802\pi\)
−0.317377 + 0.948299i \(0.602802\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −234.374 −0.100378
\(177\) 0 0
\(178\) 924.276 0.389199
\(179\) −3343.49 −1.39611 −0.698056 0.716043i \(-0.745952\pi\)
−0.698056 + 0.716043i \(0.745952\pi\)
\(180\) 0 0
\(181\) 2251.81 0.924729 0.462365 0.886690i \(-0.347001\pi\)
0.462365 + 0.886690i \(0.347001\pi\)
\(182\) −471.179 −0.191902
\(183\) 0 0
\(184\) 2626.23 1.05222
\(185\) 0 0
\(186\) 0 0
\(187\) 359.868 0.140728
\(188\) 2115.98 0.820873
\(189\) 0 0
\(190\) 0 0
\(191\) 1001.93 0.379565 0.189782 0.981826i \(-0.439222\pi\)
0.189782 + 0.981826i \(0.439222\pi\)
\(192\) 0 0
\(193\) 4054.97 1.51235 0.756173 0.654372i \(-0.227067\pi\)
0.756173 + 0.654372i \(0.227067\pi\)
\(194\) −3547.99 −1.31304
\(195\) 0 0
\(196\) 218.975 0.0798013
\(197\) −5140.23 −1.85902 −0.929508 0.368802i \(-0.879768\pi\)
−0.929508 + 0.368802i \(0.879768\pi\)
\(198\) 0 0
\(199\) 585.631 0.208614 0.104307 0.994545i \(-0.466737\pi\)
0.104307 + 0.994545i \(0.466737\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −455.062 −0.158505
\(203\) −668.459 −0.231116
\(204\) 0 0
\(205\) 0 0
\(206\) 2846.12 0.962614
\(207\) 0 0
\(208\) −1520.79 −0.506961
\(209\) 315.619 0.104458
\(210\) 0 0
\(211\) −1055.16 −0.344266 −0.172133 0.985074i \(-0.555066\pi\)
−0.172133 + 0.985074i \(0.555066\pi\)
\(212\) −820.829 −0.265919
\(213\) 0 0
\(214\) 2717.95 0.868203
\(215\) 0 0
\(216\) 0 0
\(217\) −660.152 −0.206516
\(218\) 2757.30 0.856643
\(219\) 0 0
\(220\) 0 0
\(221\) 2335.09 0.710747
\(222\) 0 0
\(223\) −4675.85 −1.40412 −0.702059 0.712119i \(-0.747736\pi\)
−0.702059 + 0.712119i \(0.747736\pi\)
\(224\) 1273.74 0.379935
\(225\) 0 0
\(226\) 3939.49 1.15952
\(227\) 5443.11 1.59151 0.795754 0.605621i \(-0.207075\pi\)
0.795754 + 0.605621i \(0.207075\pi\)
\(228\) 0 0
\(229\) −536.303 −0.154759 −0.0773797 0.997002i \(-0.524655\pi\)
−0.0773797 + 0.997002i \(0.524655\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1190.70 −0.336955
\(233\) −183.490 −0.0515916 −0.0257958 0.999667i \(-0.508212\pi\)
−0.0257958 + 0.999667i \(0.508212\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3399.09 0.937550
\(237\) 0 0
\(238\) −3027.90 −0.824661
\(239\) −643.218 −0.174085 −0.0870425 0.996205i \(-0.527742\pi\)
−0.0870425 + 0.996205i \(0.527742\pi\)
\(240\) 0 0
\(241\) −5755.61 −1.53839 −0.769194 0.639015i \(-0.779342\pi\)
−0.769194 + 0.639015i \(0.779342\pi\)
\(242\) 4669.46 1.24035
\(243\) 0 0
\(244\) −888.175 −0.233031
\(245\) 0 0
\(246\) 0 0
\(247\) 2047.97 0.527567
\(248\) −1175.91 −0.301089
\(249\) 0 0
\(250\) 0 0
\(251\) 5132.27 1.29062 0.645311 0.763920i \(-0.276728\pi\)
0.645311 + 0.763920i \(0.276728\pi\)
\(252\) 0 0
\(253\) 618.755 0.153758
\(254\) −6624.34 −1.63641
\(255\) 0 0
\(256\) 5121.09 1.25027
\(257\) 5041.74 1.22372 0.611859 0.790967i \(-0.290422\pi\)
0.611859 + 0.790967i \(0.290422\pi\)
\(258\) 0 0
\(259\) −679.844 −0.163102
\(260\) 0 0
\(261\) 0 0
\(262\) 1286.05 0.303253
\(263\) 7577.00 1.77649 0.888246 0.459367i \(-0.151924\pi\)
0.888246 + 0.459367i \(0.151924\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2655.59 −0.612122
\(267\) 0 0
\(268\) 1385.32 0.315752
\(269\) −1023.10 −0.231893 −0.115947 0.993255i \(-0.536990\pi\)
−0.115947 + 0.993255i \(0.536990\pi\)
\(270\) 0 0
\(271\) −2251.98 −0.504790 −0.252395 0.967624i \(-0.581218\pi\)
−0.252395 + 0.967624i \(0.581218\pi\)
\(272\) −9772.91 −2.17857
\(273\) 0 0
\(274\) −5660.87 −1.24812
\(275\) 0 0
\(276\) 0 0
\(277\) 8630.72 1.87209 0.936047 0.351875i \(-0.114456\pi\)
0.936047 + 0.351875i \(0.114456\pi\)
\(278\) −8584.61 −1.85205
\(279\) 0 0
\(280\) 0 0
\(281\) 7521.62 1.59680 0.798402 0.602124i \(-0.205679\pi\)
0.798402 + 0.602124i \(0.205679\pi\)
\(282\) 0 0
\(283\) −14.8169 −0.00311226 −0.00155613 0.999999i \(-0.500495\pi\)
−0.00155613 + 0.999999i \(0.500495\pi\)
\(284\) −2974.89 −0.621576
\(285\) 0 0
\(286\) −197.743 −0.0408839
\(287\) 3437.79 0.707060
\(288\) 0 0
\(289\) 10092.8 2.05430
\(290\) 0 0
\(291\) 0 0
\(292\) −2776.46 −0.556438
\(293\) 6913.39 1.37844 0.689222 0.724550i \(-0.257952\pi\)
0.689222 + 0.724550i \(0.257952\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1210.98 −0.237794
\(297\) 0 0
\(298\) 8268.39 1.60730
\(299\) 4014.94 0.776555
\(300\) 0 0
\(301\) 301.027 0.0576442
\(302\) 7429.74 1.41567
\(303\) 0 0
\(304\) −8571.25 −1.61709
\(305\) 0 0
\(306\) 0 0
\(307\) 7644.12 1.42108 0.710542 0.703655i \(-0.248450\pi\)
0.710542 + 0.703655i \(0.248450\pi\)
\(308\) 91.8987 0.0170014
\(309\) 0 0
\(310\) 0 0
\(311\) −7593.99 −1.38462 −0.692308 0.721602i \(-0.743406\pi\)
−0.692308 + 0.721602i \(0.743406\pi\)
\(312\) 0 0
\(313\) 9127.84 1.64836 0.824179 0.566329i \(-0.191637\pi\)
0.824179 + 0.566329i \(0.191637\pi\)
\(314\) −2095.61 −0.376631
\(315\) 0 0
\(316\) −110.661 −0.0196999
\(317\) −4929.81 −0.873456 −0.436728 0.899593i \(-0.643863\pi\)
−0.436728 + 0.899593i \(0.643863\pi\)
\(318\) 0 0
\(319\) −280.537 −0.0492385
\(320\) 0 0
\(321\) 0 0
\(322\) −5206.15 −0.901016
\(323\) 13160.7 2.26712
\(324\) 0 0
\(325\) 0 0
\(326\) 7693.30 1.30703
\(327\) 0 0
\(328\) 6123.62 1.03086
\(329\) 3314.46 0.555417
\(330\) 0 0
\(331\) 1221.67 0.202867 0.101433 0.994842i \(-0.467657\pi\)
0.101433 + 0.994842i \(0.467657\pi\)
\(332\) −1817.60 −0.300463
\(333\) 0 0
\(334\) −2823.14 −0.462502
\(335\) 0 0
\(336\) 0 0
\(337\) 8744.83 1.41354 0.706768 0.707446i \(-0.250153\pi\)
0.706768 + 0.707446i \(0.250153\pi\)
\(338\) 6474.79 1.04196
\(339\) 0 0
\(340\) 0 0
\(341\) −277.051 −0.0439975
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 536.210 0.0840421
\(345\) 0 0
\(346\) 5100.21 0.792454
\(347\) 4589.56 0.710031 0.355015 0.934860i \(-0.384476\pi\)
0.355015 + 0.934860i \(0.384476\pi\)
\(348\) 0 0
\(349\) −3989.89 −0.611960 −0.305980 0.952038i \(-0.598984\pi\)
−0.305980 + 0.952038i \(0.598984\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 534.561 0.0809437
\(353\) 2416.35 0.364333 0.182166 0.983268i \(-0.441689\pi\)
0.182166 + 0.983268i \(0.441689\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1169.73 −0.174145
\(357\) 0 0
\(358\) 11806.3 1.74297
\(359\) 2756.24 0.405206 0.202603 0.979261i \(-0.435060\pi\)
0.202603 + 0.979261i \(0.435060\pi\)
\(360\) 0 0
\(361\) 4683.45 0.682818
\(362\) −7951.44 −1.15447
\(363\) 0 0
\(364\) 596.307 0.0858654
\(365\) 0 0
\(366\) 0 0
\(367\) −11112.8 −1.58061 −0.790307 0.612711i \(-0.790079\pi\)
−0.790307 + 0.612711i \(0.790079\pi\)
\(368\) −16803.5 −2.38028
\(369\) 0 0
\(370\) 0 0
\(371\) −1285.74 −0.179925
\(372\) 0 0
\(373\) −6091.09 −0.845535 −0.422768 0.906238i \(-0.638941\pi\)
−0.422768 + 0.906238i \(0.638941\pi\)
\(374\) −1270.74 −0.175691
\(375\) 0 0
\(376\) 5903.94 0.809767
\(377\) −1820.33 −0.248679
\(378\) 0 0
\(379\) 3984.29 0.539998 0.269999 0.962861i \(-0.412977\pi\)
0.269999 + 0.962861i \(0.412977\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3537.93 −0.473865
\(383\) 318.475 0.0424890 0.0212445 0.999774i \(-0.493237\pi\)
0.0212445 + 0.999774i \(0.493237\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14318.6 −1.88808
\(387\) 0 0
\(388\) 4490.21 0.587515
\(389\) −3885.46 −0.506429 −0.253214 0.967410i \(-0.581488\pi\)
−0.253214 + 0.967410i \(0.581488\pi\)
\(390\) 0 0
\(391\) 25800.9 3.33710
\(392\) 610.975 0.0787216
\(393\) 0 0
\(394\) 18150.8 2.32088
\(395\) 0 0
\(396\) 0 0
\(397\) −4806.04 −0.607578 −0.303789 0.952739i \(-0.598252\pi\)
−0.303789 + 0.952739i \(0.598252\pi\)
\(398\) −2067.94 −0.260443
\(399\) 0 0
\(400\) 0 0
\(401\) −3618.59 −0.450633 −0.225316 0.974286i \(-0.572342\pi\)
−0.225316 + 0.974286i \(0.572342\pi\)
\(402\) 0 0
\(403\) −1797.71 −0.222209
\(404\) 575.911 0.0709223
\(405\) 0 0
\(406\) 2360.42 0.288536
\(407\) −285.315 −0.0347483
\(408\) 0 0
\(409\) −2109.05 −0.254978 −0.127489 0.991840i \(-0.540692\pi\)
−0.127489 + 0.991840i \(0.540692\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3601.94 −0.430716
\(413\) 5324.30 0.634363
\(414\) 0 0
\(415\) 0 0
\(416\) 3468.63 0.408806
\(417\) 0 0
\(418\) −1114.49 −0.130410
\(419\) 6905.91 0.805193 0.402597 0.915377i \(-0.368108\pi\)
0.402597 + 0.915377i \(0.368108\pi\)
\(420\) 0 0
\(421\) −9647.54 −1.11685 −0.558423 0.829556i \(-0.688593\pi\)
−0.558423 + 0.829556i \(0.688593\pi\)
\(422\) 3725.91 0.429797
\(423\) 0 0
\(424\) −2290.25 −0.262321
\(425\) 0 0
\(426\) 0 0
\(427\) −1391.23 −0.157673
\(428\) −3439.74 −0.388473
\(429\) 0 0
\(430\) 0 0
\(431\) 13002.7 1.45318 0.726589 0.687073i \(-0.241105\pi\)
0.726589 + 0.687073i \(0.241105\pi\)
\(432\) 0 0
\(433\) −7356.07 −0.816420 −0.408210 0.912888i \(-0.633847\pi\)
−0.408210 + 0.912888i \(0.633847\pi\)
\(434\) 2331.08 0.257824
\(435\) 0 0
\(436\) −3489.55 −0.383300
\(437\) 22628.4 2.47703
\(438\) 0 0
\(439\) 6909.21 0.751159 0.375579 0.926790i \(-0.377444\pi\)
0.375579 + 0.926790i \(0.377444\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −8245.50 −0.887327
\(443\) −14812.6 −1.58864 −0.794318 0.607502i \(-0.792172\pi\)
−0.794318 + 0.607502i \(0.792172\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 16511.0 1.75296
\(447\) 0 0
\(448\) −30.0561 −0.00316968
\(449\) 10654.5 1.11986 0.559932 0.828538i \(-0.310827\pi\)
0.559932 + 0.828538i \(0.310827\pi\)
\(450\) 0 0
\(451\) 1442.76 0.150636
\(452\) −4985.68 −0.518820
\(453\) 0 0
\(454\) −19220.3 −1.98691
\(455\) 0 0
\(456\) 0 0
\(457\) 5855.16 0.599328 0.299664 0.954045i \(-0.403125\pi\)
0.299664 + 0.954045i \(0.403125\pi\)
\(458\) 1893.76 0.193208
\(459\) 0 0
\(460\) 0 0
\(461\) −3204.74 −0.323774 −0.161887 0.986809i \(-0.551758\pi\)
−0.161887 + 0.986809i \(0.551758\pi\)
\(462\) 0 0
\(463\) −371.658 −0.0373054 −0.0186527 0.999826i \(-0.505938\pi\)
−0.0186527 + 0.999826i \(0.505938\pi\)
\(464\) 7618.54 0.762245
\(465\) 0 0
\(466\) 647.927 0.0644091
\(467\) −19752.3 −1.95723 −0.978614 0.205703i \(-0.934052\pi\)
−0.978614 + 0.205703i \(0.934052\pi\)
\(468\) 0 0
\(469\) 2169.95 0.213643
\(470\) 0 0
\(471\) 0 0
\(472\) 9484.01 0.924866
\(473\) 126.334 0.0122809
\(474\) 0 0
\(475\) 0 0
\(476\) 3832.00 0.368990
\(477\) 0 0
\(478\) 2271.28 0.217335
\(479\) 20762.0 1.98046 0.990232 0.139433i \(-0.0445279\pi\)
0.990232 + 0.139433i \(0.0445279\pi\)
\(480\) 0 0
\(481\) −1851.34 −0.175496
\(482\) 20323.8 1.92059
\(483\) 0 0
\(484\) −5909.50 −0.554987
\(485\) 0 0
\(486\) 0 0
\(487\) −17647.6 −1.64207 −0.821035 0.570878i \(-0.806603\pi\)
−0.821035 + 0.570878i \(0.806603\pi\)
\(488\) −2478.15 −0.229878
\(489\) 0 0
\(490\) 0 0
\(491\) 5637.46 0.518157 0.259078 0.965856i \(-0.416581\pi\)
0.259078 + 0.965856i \(0.416581\pi\)
\(492\) 0 0
\(493\) −11697.9 −1.06865
\(494\) −7231.64 −0.658638
\(495\) 0 0
\(496\) 7523.86 0.681112
\(497\) −4659.85 −0.420569
\(498\) 0 0
\(499\) −17474.1 −1.56764 −0.783818 0.620991i \(-0.786730\pi\)
−0.783818 + 0.620991i \(0.786730\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −18122.7 −1.61127
\(503\) 7444.81 0.659936 0.329968 0.943992i \(-0.392962\pi\)
0.329968 + 0.943992i \(0.392962\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2184.90 −0.191958
\(507\) 0 0
\(508\) 8383.53 0.732203
\(509\) 3384.48 0.294724 0.147362 0.989083i \(-0.452922\pi\)
0.147362 + 0.989083i \(0.452922\pi\)
\(510\) 0 0
\(511\) −4349.02 −0.376495
\(512\) −6558.89 −0.566142
\(513\) 0 0
\(514\) −17803.0 −1.52774
\(515\) 0 0
\(516\) 0 0
\(517\) 1391.00 0.118329
\(518\) 2400.62 0.203624
\(519\) 0 0
\(520\) 0 0
\(521\) −2973.12 −0.250009 −0.125005 0.992156i \(-0.539895\pi\)
−0.125005 + 0.992156i \(0.539895\pi\)
\(522\) 0 0
\(523\) −2689.02 −0.224823 −0.112412 0.993662i \(-0.535858\pi\)
−0.112412 + 0.993662i \(0.535858\pi\)
\(524\) −1627.57 −0.135689
\(525\) 0 0
\(526\) −26755.3 −2.21785
\(527\) −11552.5 −0.954903
\(528\) 0 0
\(529\) 32194.9 2.64608
\(530\) 0 0
\(531\) 0 0
\(532\) 3360.82 0.273891
\(533\) 9361.72 0.760790
\(534\) 0 0
\(535\) 0 0
\(536\) 3865.25 0.311480
\(537\) 0 0
\(538\) 3612.69 0.289506
\(539\) 143.949 0.0115034
\(540\) 0 0
\(541\) −14429.5 −1.14671 −0.573356 0.819306i \(-0.694359\pi\)
−0.573356 + 0.819306i \(0.694359\pi\)
\(542\) 7952.03 0.630201
\(543\) 0 0
\(544\) 22290.1 1.75677
\(545\) 0 0
\(546\) 0 0
\(547\) −13811.2 −1.07957 −0.539784 0.841804i \(-0.681494\pi\)
−0.539784 + 0.841804i \(0.681494\pi\)
\(548\) 7164.19 0.558466
\(549\) 0 0
\(550\) 0 0
\(551\) −10259.5 −0.793229
\(552\) 0 0
\(553\) −173.338 −0.0133293
\(554\) −30476.2 −2.33720
\(555\) 0 0
\(556\) 10864.4 0.828692
\(557\) −6033.26 −0.458954 −0.229477 0.973314i \(-0.573702\pi\)
−0.229477 + 0.973314i \(0.573702\pi\)
\(558\) 0 0
\(559\) 819.751 0.0620246
\(560\) 0 0
\(561\) 0 0
\(562\) −26559.8 −1.99352
\(563\) −6958.47 −0.520896 −0.260448 0.965488i \(-0.583870\pi\)
−0.260448 + 0.965488i \(0.583870\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 52.3202 0.00388548
\(567\) 0 0
\(568\) −8300.44 −0.613166
\(569\) 13396.4 0.987009 0.493505 0.869743i \(-0.335716\pi\)
0.493505 + 0.869743i \(0.335716\pi\)
\(570\) 0 0
\(571\) −8055.84 −0.590414 −0.295207 0.955433i \(-0.595389\pi\)
−0.295207 + 0.955433i \(0.595389\pi\)
\(572\) 250.257 0.0182933
\(573\) 0 0
\(574\) −12139.3 −0.882724
\(575\) 0 0
\(576\) 0 0
\(577\) 21456.9 1.54812 0.774059 0.633114i \(-0.218223\pi\)
0.774059 + 0.633114i \(0.218223\pi\)
\(578\) −35638.9 −2.56468
\(579\) 0 0
\(580\) 0 0
\(581\) −2847.07 −0.203298
\(582\) 0 0
\(583\) −539.596 −0.0383324
\(584\) −7746.76 −0.548910
\(585\) 0 0
\(586\) −24412.1 −1.72091
\(587\) 20156.3 1.41728 0.708638 0.705572i \(-0.249310\pi\)
0.708638 + 0.705572i \(0.249310\pi\)
\(588\) 0 0
\(589\) −10132.0 −0.708797
\(590\) 0 0
\(591\) 0 0
\(592\) 7748.30 0.537928
\(593\) 599.307 0.0415018 0.0207509 0.999785i \(-0.493394\pi\)
0.0207509 + 0.999785i \(0.493394\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10464.2 −0.719177
\(597\) 0 0
\(598\) −14177.3 −0.969485
\(599\) 5493.05 0.374691 0.187346 0.982294i \(-0.440012\pi\)
0.187346 + 0.982294i \(0.440012\pi\)
\(600\) 0 0
\(601\) 24292.8 1.64879 0.824396 0.566014i \(-0.191515\pi\)
0.824396 + 0.566014i \(0.191515\pi\)
\(602\) −1062.97 −0.0719655
\(603\) 0 0
\(604\) −9402.82 −0.633436
\(605\) 0 0
\(606\) 0 0
\(607\) −3029.50 −0.202576 −0.101288 0.994857i \(-0.532296\pi\)
−0.101288 + 0.994857i \(0.532296\pi\)
\(608\) 19549.3 1.30400
\(609\) 0 0
\(610\) 0 0
\(611\) 9025.87 0.597623
\(612\) 0 0
\(613\) 19339.6 1.27426 0.637129 0.770757i \(-0.280122\pi\)
0.637129 + 0.770757i \(0.280122\pi\)
\(614\) −26992.4 −1.77414
\(615\) 0 0
\(616\) 256.412 0.0167713
\(617\) −5743.91 −0.374783 −0.187391 0.982285i \(-0.560003\pi\)
−0.187391 + 0.982285i \(0.560003\pi\)
\(618\) 0 0
\(619\) −8243.35 −0.535264 −0.267632 0.963521i \(-0.586241\pi\)
−0.267632 + 0.963521i \(0.586241\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 26815.4 1.72861
\(623\) −1832.26 −0.117830
\(624\) 0 0
\(625\) 0 0
\(626\) −32231.6 −2.05788
\(627\) 0 0
\(628\) 2652.13 0.168521
\(629\) −11897.1 −0.754162
\(630\) 0 0
\(631\) −4376.56 −0.276114 −0.138057 0.990424i \(-0.544086\pi\)
−0.138057 + 0.990424i \(0.544086\pi\)
\(632\) −308.762 −0.0194334
\(633\) 0 0
\(634\) 17407.8 1.09046
\(635\) 0 0
\(636\) 0 0
\(637\) 934.051 0.0580980
\(638\) 990.613 0.0614714
\(639\) 0 0
\(640\) 0 0
\(641\) −11836.6 −0.729357 −0.364678 0.931133i \(-0.618821\pi\)
−0.364678 + 0.931133i \(0.618821\pi\)
\(642\) 0 0
\(643\) −1448.21 −0.0888209 −0.0444104 0.999013i \(-0.514141\pi\)
−0.0444104 + 0.999013i \(0.514141\pi\)
\(644\) 6588.72 0.403155
\(645\) 0 0
\(646\) −46472.0 −2.83037
\(647\) −8732.95 −0.530646 −0.265323 0.964160i \(-0.585479\pi\)
−0.265323 + 0.964160i \(0.585479\pi\)
\(648\) 0 0
\(649\) 2234.49 0.135149
\(650\) 0 0
\(651\) 0 0
\(652\) −9736.37 −0.584824
\(653\) −21978.4 −1.31712 −0.658562 0.752527i \(-0.728835\pi\)
−0.658562 + 0.752527i \(0.728835\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −39181.1 −2.33196
\(657\) 0 0
\(658\) −11703.8 −0.693406
\(659\) 27761.7 1.64103 0.820516 0.571623i \(-0.193686\pi\)
0.820516 + 0.571623i \(0.193686\pi\)
\(660\) 0 0
\(661\) −8573.72 −0.504507 −0.252254 0.967661i \(-0.581172\pi\)
−0.252254 + 0.967661i \(0.581172\pi\)
\(662\) −4313.86 −0.253267
\(663\) 0 0
\(664\) −5071.39 −0.296398
\(665\) 0 0
\(666\) 0 0
\(667\) −20113.2 −1.16760
\(668\) 3572.87 0.206944
\(669\) 0 0
\(670\) 0 0
\(671\) −583.868 −0.0335916
\(672\) 0 0
\(673\) −27159.2 −1.55559 −0.777795 0.628518i \(-0.783662\pi\)
−0.777795 + 0.628518i \(0.783662\pi\)
\(674\) −30879.1 −1.76472
\(675\) 0 0
\(676\) −8194.26 −0.466219
\(677\) −1392.30 −0.0790404 −0.0395202 0.999219i \(-0.512583\pi\)
−0.0395202 + 0.999219i \(0.512583\pi\)
\(678\) 0 0
\(679\) 7033.42 0.397523
\(680\) 0 0
\(681\) 0 0
\(682\) 978.302 0.0549283
\(683\) −8675.09 −0.486007 −0.243004 0.970025i \(-0.578133\pi\)
−0.243004 + 0.970025i \(0.578133\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1211.18 −0.0674096
\(687\) 0 0
\(688\) −3430.86 −0.190117
\(689\) −3501.30 −0.193598
\(690\) 0 0
\(691\) −21426.0 −1.17957 −0.589785 0.807561i \(-0.700787\pi\)
−0.589785 + 0.807561i \(0.700787\pi\)
\(692\) −6454.65 −0.354579
\(693\) 0 0
\(694\) −16206.3 −0.886433
\(695\) 0 0
\(696\) 0 0
\(697\) 60160.4 3.26935
\(698\) 14088.8 0.763997
\(699\) 0 0
\(700\) 0 0
\(701\) −24840.5 −1.33839 −0.669197 0.743085i \(-0.733362\pi\)
−0.669197 + 0.743085i \(0.733362\pi\)
\(702\) 0 0
\(703\) −10434.2 −0.559793
\(704\) −12.6139 −0.000675288 0
\(705\) 0 0
\(706\) −8532.45 −0.454848
\(707\) 902.101 0.0479873
\(708\) 0 0
\(709\) 12525.0 0.663450 0.331725 0.943376i \(-0.392369\pi\)
0.331725 + 0.943376i \(0.392369\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3263.74 −0.171789
\(713\) −19863.3 −1.04332
\(714\) 0 0
\(715\) 0 0
\(716\) −14941.6 −0.779881
\(717\) 0 0
\(718\) −9732.66 −0.505877
\(719\) −28085.0 −1.45674 −0.728369 0.685185i \(-0.759721\pi\)
−0.728369 + 0.685185i \(0.759721\pi\)
\(720\) 0 0
\(721\) −5642.05 −0.291430
\(722\) −16537.9 −0.852460
\(723\) 0 0
\(724\) 10063.1 0.516562
\(725\) 0 0
\(726\) 0 0
\(727\) 14326.2 0.730851 0.365426 0.930841i \(-0.380923\pi\)
0.365426 + 0.930841i \(0.380923\pi\)
\(728\) 1663.79 0.0847037
\(729\) 0 0
\(730\) 0 0
\(731\) 5267.89 0.266539
\(732\) 0 0
\(733\) −6727.85 −0.339016 −0.169508 0.985529i \(-0.554218\pi\)
−0.169508 + 0.985529i \(0.554218\pi\)
\(734\) 39240.8 1.97331
\(735\) 0 0
\(736\) 38325.5 1.91943
\(737\) 910.677 0.0455159
\(738\) 0 0
\(739\) −3418.51 −0.170165 −0.0850826 0.996374i \(-0.527115\pi\)
−0.0850826 + 0.996374i \(0.527115\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4540.11 0.224626
\(743\) 8095.50 0.399724 0.199862 0.979824i \(-0.435951\pi\)
0.199862 + 0.979824i \(0.435951\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 21508.4 1.05560
\(747\) 0 0
\(748\) 1608.20 0.0786119
\(749\) −5387.99 −0.262847
\(750\) 0 0
\(751\) 13446.8 0.653371 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(752\) −37775.4 −1.83182
\(753\) 0 0
\(754\) 6427.84 0.310462
\(755\) 0 0
\(756\) 0 0
\(757\) 2593.24 0.124508 0.0622541 0.998060i \(-0.480171\pi\)
0.0622541 + 0.998060i \(0.480171\pi\)
\(758\) −14069.0 −0.674156
\(759\) 0 0
\(760\) 0 0
\(761\) −27079.4 −1.28992 −0.644959 0.764217i \(-0.723125\pi\)
−0.644959 + 0.764217i \(0.723125\pi\)
\(762\) 0 0
\(763\) −5465.99 −0.259348
\(764\) 4477.48 0.212028
\(765\) 0 0
\(766\) −1124.58 −0.0530451
\(767\) 14499.0 0.682568
\(768\) 0 0
\(769\) 2138.72 0.100292 0.0501458 0.998742i \(-0.484031\pi\)
0.0501458 + 0.998742i \(0.484031\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18121.1 0.844810
\(773\) 25864.0 1.20345 0.601724 0.798704i \(-0.294481\pi\)
0.601724 + 0.798704i \(0.294481\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12528.4 0.579566
\(777\) 0 0
\(778\) 13720.1 0.632247
\(779\) 52763.1 2.42675
\(780\) 0 0
\(781\) −1955.63 −0.0896006
\(782\) −91106.2 −4.16618
\(783\) 0 0
\(784\) −3909.23 −0.178081
\(785\) 0 0
\(786\) 0 0
\(787\) 32371.3 1.46621 0.733107 0.680113i \(-0.238069\pi\)
0.733107 + 0.680113i \(0.238069\pi\)
\(788\) −22971.0 −1.03846
\(789\) 0 0
\(790\) 0 0
\(791\) −7809.52 −0.351043
\(792\) 0 0
\(793\) −3788.57 −0.169654
\(794\) 16970.8 0.758526
\(795\) 0 0
\(796\) 2617.11 0.116534
\(797\) 2024.33 0.0899691 0.0449845 0.998988i \(-0.485676\pi\)
0.0449845 + 0.998988i \(0.485676\pi\)
\(798\) 0 0
\(799\) 58002.1 2.56817
\(800\) 0 0
\(801\) 0 0
\(802\) 12777.7 0.562589
\(803\) −1825.18 −0.0802109
\(804\) 0 0
\(805\) 0 0
\(806\) 6347.95 0.277416
\(807\) 0 0
\(808\) 1606.88 0.0699628
\(809\) −12391.7 −0.538526 −0.269263 0.963067i \(-0.586780\pi\)
−0.269263 + 0.963067i \(0.586780\pi\)
\(810\) 0 0
\(811\) 14654.5 0.634511 0.317256 0.948340i \(-0.397239\pi\)
0.317256 + 0.948340i \(0.397239\pi\)
\(812\) −2987.26 −0.129104
\(813\) 0 0
\(814\) 1007.49 0.0433813
\(815\) 0 0
\(816\) 0 0
\(817\) 4620.16 0.197844
\(818\) 7447.33 0.318325
\(819\) 0 0
\(820\) 0 0
\(821\) −23887.9 −1.01546 −0.507731 0.861516i \(-0.669515\pi\)
−0.507731 + 0.861516i \(0.669515\pi\)
\(822\) 0 0
\(823\) 4008.41 0.169774 0.0848871 0.996391i \(-0.472947\pi\)
0.0848871 + 0.996391i \(0.472947\pi\)
\(824\) −10050.0 −0.424889
\(825\) 0 0
\(826\) −18800.8 −0.791966
\(827\) −45110.4 −1.89679 −0.948394 0.317096i \(-0.897292\pi\)
−0.948394 + 0.317096i \(0.897292\pi\)
\(828\) 0 0
\(829\) 16165.4 0.677260 0.338630 0.940920i \(-0.390036\pi\)
0.338630 + 0.940920i \(0.390036\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −81.8481 −0.00341054
\(833\) 6002.41 0.249665
\(834\) 0 0
\(835\) 0 0
\(836\) 1410.46 0.0583514
\(837\) 0 0
\(838\) −24385.7 −1.00524
\(839\) 25244.4 1.03878 0.519388 0.854538i \(-0.326160\pi\)
0.519388 + 0.854538i \(0.326160\pi\)
\(840\) 0 0
\(841\) −15269.9 −0.626096
\(842\) 34066.7 1.39432
\(843\) 0 0
\(844\) −4715.37 −0.192310
\(845\) 0 0
\(846\) 0 0
\(847\) −9256.59 −0.375514
\(848\) 14653.8 0.593412
\(849\) 0 0
\(850\) 0 0
\(851\) −20455.8 −0.823990
\(852\) 0 0
\(853\) 30168.1 1.21094 0.605472 0.795867i \(-0.292984\pi\)
0.605472 + 0.795867i \(0.292984\pi\)
\(854\) 4912.61 0.196846
\(855\) 0 0
\(856\) −9597.44 −0.383217
\(857\) −13393.6 −0.533857 −0.266929 0.963716i \(-0.586009\pi\)
−0.266929 + 0.963716i \(0.586009\pi\)
\(858\) 0 0
\(859\) 19060.4 0.757081 0.378541 0.925585i \(-0.376426\pi\)
0.378541 + 0.925585i \(0.376426\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −45914.3 −1.81421
\(863\) −9466.86 −0.373413 −0.186707 0.982416i \(-0.559781\pi\)
−0.186707 + 0.982416i \(0.559781\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 25975.2 1.01925
\(867\) 0 0
\(868\) −2950.13 −0.115362
\(869\) −72.7461 −0.00283975
\(870\) 0 0
\(871\) 5909.15 0.229878
\(872\) −9736.39 −0.378115
\(873\) 0 0
\(874\) −79903.8 −3.09243
\(875\) 0 0
\(876\) 0 0
\(877\) −37740.6 −1.45315 −0.726573 0.687090i \(-0.758888\pi\)
−0.726573 + 0.687090i \(0.758888\pi\)
\(878\) −24397.3 −0.937778
\(879\) 0 0
\(880\) 0 0
\(881\) −25991.5 −0.993957 −0.496979 0.867763i \(-0.665557\pi\)
−0.496979 + 0.867763i \(0.665557\pi\)
\(882\) 0 0
\(883\) −39420.3 −1.50238 −0.751189 0.660087i \(-0.770519\pi\)
−0.751189 + 0.660087i \(0.770519\pi\)
\(884\) 10435.2 0.397030
\(885\) 0 0
\(886\) 52305.0 1.98332
\(887\) 46005.2 1.74149 0.870745 0.491735i \(-0.163637\pi\)
0.870745 + 0.491735i \(0.163637\pi\)
\(888\) 0 0
\(889\) 13131.9 0.495421
\(890\) 0 0
\(891\) 0 0
\(892\) −20895.8 −0.784353
\(893\) 50870.2 1.90628
\(894\) 0 0
\(895\) 0 0
\(896\) −10083.8 −0.375978
\(897\) 0 0
\(898\) −37622.6 −1.39809
\(899\) 9005.81 0.334105
\(900\) 0 0
\(901\) −22500.1 −0.831950
\(902\) −5094.58 −0.188061
\(903\) 0 0
\(904\) −13910.8 −0.511801
\(905\) 0 0
\(906\) 0 0
\(907\) 2838.97 0.103932 0.0519661 0.998649i \(-0.483451\pi\)
0.0519661 + 0.998649i \(0.483451\pi\)
\(908\) 24324.6 0.889030
\(909\) 0 0
\(910\) 0 0
\(911\) −39890.9 −1.45076 −0.725382 0.688347i \(-0.758337\pi\)
−0.725382 + 0.688347i \(0.758337\pi\)
\(912\) 0 0
\(913\) −1194.85 −0.0433119
\(914\) −20675.3 −0.748226
\(915\) 0 0
\(916\) −2396.67 −0.0864500
\(917\) −2549.42 −0.0918094
\(918\) 0 0
\(919\) −646.475 −0.0232048 −0.0116024 0.999933i \(-0.503693\pi\)
−0.0116024 + 0.999933i \(0.503693\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 11316.3 0.404213
\(923\) −12689.6 −0.452528
\(924\) 0 0
\(925\) 0 0
\(926\) 1312.37 0.0465737
\(927\) 0 0
\(928\) −17376.4 −0.614665
\(929\) −51188.2 −1.80778 −0.903892 0.427760i \(-0.859303\pi\)
−0.903892 + 0.427760i \(0.859303\pi\)
\(930\) 0 0
\(931\) 5264.35 0.185319
\(932\) −819.993 −0.0288195
\(933\) 0 0
\(934\) 69747.8 2.44349
\(935\) 0 0
\(936\) 0 0
\(937\) −29786.1 −1.03849 −0.519247 0.854624i \(-0.673788\pi\)
−0.519247 + 0.854624i \(0.673788\pi\)
\(938\) −7662.36 −0.266722
\(939\) 0 0
\(940\) 0 0
\(941\) 44817.4 1.55261 0.776304 0.630358i \(-0.217092\pi\)
0.776304 + 0.630358i \(0.217092\pi\)
\(942\) 0 0
\(943\) 103439. 3.57206
\(944\) −60682.0 −2.09219
\(945\) 0 0
\(946\) −446.103 −0.0153320
\(947\) 54697.1 1.87689 0.938446 0.345425i \(-0.112265\pi\)
0.938446 + 0.345425i \(0.112265\pi\)
\(948\) 0 0
\(949\) −11843.2 −0.405105
\(950\) 0 0
\(951\) 0 0
\(952\) 10691.9 0.363998
\(953\) −7577.51 −0.257565 −0.128783 0.991673i \(-0.541107\pi\)
−0.128783 + 0.991673i \(0.541107\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2874.46 −0.0972454
\(957\) 0 0
\(958\) −73313.4 −2.47249
\(959\) 11221.9 0.377868
\(960\) 0 0
\(961\) −20897.1 −0.701457
\(962\) 6537.32 0.219097
\(963\) 0 0
\(964\) −25721.1 −0.859357
\(965\) 0 0
\(966\) 0 0
\(967\) −50779.0 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(968\) −16488.5 −0.547478
\(969\) 0 0
\(970\) 0 0
\(971\) 15313.2 0.506102 0.253051 0.967453i \(-0.418566\pi\)
0.253051 + 0.967453i \(0.418566\pi\)
\(972\) 0 0
\(973\) 17017.9 0.560707
\(974\) 62315.9 2.05003
\(975\) 0 0
\(976\) 15856.1 0.520021
\(977\) 46620.4 1.52663 0.763316 0.646025i \(-0.223570\pi\)
0.763316 + 0.646025i \(0.223570\pi\)
\(978\) 0 0
\(979\) −768.957 −0.0251031
\(980\) 0 0
\(981\) 0 0
\(982\) −19906.6 −0.646889
\(983\) −2824.37 −0.0916414 −0.0458207 0.998950i \(-0.514590\pi\)
−0.0458207 + 0.998950i \(0.514590\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 41306.6 1.33415
\(987\) 0 0
\(988\) 9152.11 0.294704
\(989\) 9057.59 0.291218
\(990\) 0 0
\(991\) 16951.4 0.543370 0.271685 0.962386i \(-0.412419\pi\)
0.271685 + 0.962386i \(0.412419\pi\)
\(992\) −17160.5 −0.549239
\(993\) 0 0
\(994\) 16454.5 0.525056
\(995\) 0 0
\(996\) 0 0
\(997\) −23847.8 −0.757540 −0.378770 0.925491i \(-0.623653\pi\)
−0.378770 + 0.925491i \(0.623653\pi\)
\(998\) 61703.4 1.95710
\(999\) 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 1575.4.a.w.1.1 2
3.2 odd 2 525.4.a.k.1.2 2
5.4 even 2 315.4.a.i.1.2 2
15.2 even 4 525.4.d.h.274.3 4
15.8 even 4 525.4.d.h.274.2 4
15.14 odd 2 105.4.a.f.1.1 2
35.34 odd 2 2205.4.a.z.1.2 2
60.59 even 2 1680.4.a.bg.1.1 2
105.104 even 2 735.4.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.1 2 15.14 odd 2
315.4.a.i.1.2 2 5.4 even 2
525.4.a.k.1.2 2 3.2 odd 2
525.4.d.h.274.2 4 15.8 even 4
525.4.d.h.274.3 4 15.2 even 4
735.4.a.p.1.1 2 105.104 even 2
1575.4.a.w.1.1 2 1.1 even 1 trivial
1680.4.a.bg.1.1 2 60.59 even 2
2205.4.a.z.1.2 2 35.34 odd 2