Properties

Label 1680.4.a.bg.1.1
Level $1680$
Weight $4$
Character 1680.1
Self dual yes
Analytic conductor $99.123$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,4,Mod(1,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1680.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(99.1232088096\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.53113\) of defining polynomial
Character \(\chi\) \(=\) 1680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +2.93774 q^{11} -19.0623 q^{13} -15.0000 q^{15} +122.498 q^{17} -107.436 q^{19} -21.0000 q^{21} -210.623 q^{23} +25.0000 q^{25} -27.0000 q^{27} +95.4942 q^{29} +94.3074 q^{31} -8.81323 q^{33} +35.0000 q^{35} +97.1206 q^{37} +57.1868 q^{39} -491.113 q^{41} +43.0039 q^{43} +45.0000 q^{45} -473.494 q^{47} +49.0000 q^{49} -367.494 q^{51} -183.677 q^{53} +14.6887 q^{55} +322.307 q^{57} +760.615 q^{59} -198.747 q^{61} +63.0000 q^{63} -95.3113 q^{65} +309.992 q^{67} +631.868 q^{69} -665.693 q^{71} +621.288 q^{73} -75.0000 q^{75} +20.5642 q^{77} +24.7626 q^{79} +81.0000 q^{81} +406.724 q^{83} +612.490 q^{85} -286.483 q^{87} +261.751 q^{89} -133.436 q^{91} -282.922 q^{93} -537.179 q^{95} -1004.77 q^{97} +26.4397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35} - 96 q^{37} + 66 q^{39} - 176 q^{41} + 344 q^{43} + 90 q^{45} - 560 q^{47} + 98 q^{49} - 348 q^{51} + 326 q^{53} + 110 q^{55} + 306 q^{57} + 844 q^{59} - 204 q^{61} + 126 q^{63} - 110 q^{65} + 104 q^{67} + 780 q^{69} - 1670 q^{71} - 386 q^{73} - 150 q^{75} + 154 q^{77} + 888 q^{79} + 162 q^{81} - 928 q^{83} + 580 q^{85} + 588 q^{87} + 588 q^{89} - 154 q^{91} + 450 q^{93} - 510 q^{95} + 522 q^{97} + 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 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 \(-0.565181\pi\)
−0.203343 + 0.979108i \(0.565181\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 0 0
\(17\) 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 \(-0.724652\pi\)
−0.648617 + 0.761115i \(0.724652\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) −210.623 −1.90947 −0.954736 0.297455i \(-0.903862\pi\)
−0.954736 + 0.297455i \(0.903862\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 95.4942 0.611477 0.305738 0.952116i \(-0.401097\pi\)
0.305738 + 0.952116i \(0.401097\pi\)
\(30\) 0 0
\(31\) 94.3074 0.546391 0.273195 0.961959i \(-0.411919\pi\)
0.273195 + 0.961959i \(0.411919\pi\)
\(32\) 0 0
\(33\) −8.81323 −0.0464905
\(34\) 0 0
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) 97.1206 0.431528 0.215764 0.976446i \(-0.430776\pi\)
0.215764 + 0.976446i \(0.430776\pi\)
\(38\) 0 0
\(39\) 57.1868 0.234800
\(40\) 0 0
\(41\) −491.113 −1.87071 −0.935353 0.353716i \(-0.884918\pi\)
−0.935353 + 0.353716i \(0.884918\pi\)
\(42\) 0 0
\(43\) 43.0039 0.152512 0.0762562 0.997088i \(-0.475703\pi\)
0.0762562 + 0.997088i \(0.475703\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) −473.494 −1.46949 −0.734747 0.678341i \(-0.762699\pi\)
−0.734747 + 0.678341i \(0.762699\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −367.494 −1.00901
\(52\) 0 0
\(53\) −183.677 −0.476038 −0.238019 0.971261i \(-0.576498\pi\)
−0.238019 + 0.971261i \(0.576498\pi\)
\(54\) 0 0
\(55\) 14.6887 0.0360114
\(56\) 0 0
\(57\) 322.307 0.748959
\(58\) 0 0
\(59\) 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 \(0.566885\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) −95.3113 −0.181876
\(66\) 0 0
\(67\) 309.992 0.565247 0.282624 0.959231i \(-0.408795\pi\)
0.282624 + 0.959231i \(0.408795\pi\)
\(68\) 0 0
\(69\) 631.868 1.10243
\(70\) 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 \(-0.334047\pi\)
0.498057 + 0.867145i \(0.334047\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) 20.5642 0.0304352
\(78\) 0 0
\(79\) 24.7626 0.0352659 0.0176330 0.999845i \(-0.494387\pi\)
0.0176330 + 0.999845i \(0.494387\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 406.724 0.537876 0.268938 0.963157i \(-0.413327\pi\)
0.268938 + 0.963157i \(0.413327\pi\)
\(84\) 0 0
\(85\) 612.490 0.781575
\(86\) 0 0
\(87\) −286.483 −0.353036
\(88\) 0 0
\(89\) 261.751 0.311748 0.155874 0.987777i \(-0.450181\pi\)
0.155874 + 0.987777i \(0.450181\pi\)
\(90\) 0 0
\(91\) −133.436 −0.153713
\(92\) 0 0
\(93\) −282.922 −0.315459
\(94\) 0 0
\(95\) −537.179 −0.580141
\(96\) 0 0
\(97\) −1004.77 −1.05175 −0.525873 0.850563i \(-0.676261\pi\)
−0.525873 + 0.850563i \(0.676261\pi\)
\(98\) 0 0
\(99\) 26.4397 0.0268413
\(100\) 0 0
\(101\) −128.872 −0.126962 −0.0634812 0.997983i \(-0.520220\pi\)
−0.0634812 + 0.997983i \(0.520220\pi\)
\(102\) 0 0
\(103\) −806.008 −0.771051 −0.385526 0.922697i \(-0.625980\pi\)
−0.385526 + 0.922697i \(0.625980\pi\)
\(104\) 0 0
\(105\) −105.000 −0.0975900
\(106\) 0 0
\(107\) 769.712 0.695429 0.347714 0.937600i \(-0.386958\pi\)
0.347714 + 0.937600i \(0.386958\pi\)
\(108\) 0 0
\(109\) −780.856 −0.686169 −0.343085 0.939304i \(-0.611472\pi\)
−0.343085 + 0.939304i \(0.611472\pi\)
\(110\) 0 0
\(111\) −291.362 −0.249143
\(112\) 0 0
\(113\) −1115.65 −0.928771 −0.464386 0.885633i \(-0.653725\pi\)
−0.464386 + 0.885633i \(0.653725\pi\)
\(114\) 0 0
\(115\) −1053.11 −0.853942
\(116\) 0 0
\(117\) −171.560 −0.135562
\(118\) 0 0
\(119\) 857.486 0.660552
\(120\) 0 0
\(121\) −1322.37 −0.993516
\(122\) 0 0
\(123\) 1473.34 1.08005
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1875.98 1.31076 0.655381 0.755299i \(-0.272508\pi\)
0.655381 + 0.755299i \(0.272508\pi\)
\(128\) 0 0
\(129\) −129.012 −0.0880530
\(130\) 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
\(134\) 0 0
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) 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 \(-0.766002\pi\)
−0.741746 + 0.670681i \(0.766002\pi\)
\(140\) 0 0
\(141\) 1420.48 0.848413
\(142\) 0 0
\(143\) −56.0000 −0.0327479
\(144\) 0 0
\(145\) 477.471 0.273461
\(146\) 0 0
\(147\) −147.000 −0.0824786
\(148\) 0 0
\(149\) 2341.57 1.28744 0.643722 0.765260i \(-0.277389\pi\)
0.643722 + 0.765260i \(0.277389\pi\)
\(150\) 0 0
\(151\) 2104.07 1.13395 0.566976 0.823734i \(-0.308113\pi\)
0.566976 + 0.823734i \(0.308113\pi\)
\(152\) 0 0
\(153\) 1102.48 0.582552
\(154\) 0 0
\(155\) 471.537 0.244353
\(156\) 0 0
\(157\) −593.467 −0.301680 −0.150840 0.988558i \(-0.548198\pi\)
−0.150840 + 0.988558i \(0.548198\pi\)
\(158\) 0 0
\(159\) 551.031 0.274840
\(160\) 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 \(0.675361\pi\)
\(164\) 0 0
\(165\) −44.0661 −0.0207912
\(166\) 0 0
\(167\) −799.502 −0.370463 −0.185231 0.982695i \(-0.559303\pi\)
−0.185231 + 0.982695i \(0.559303\pi\)
\(168\) 0 0
\(169\) −1833.63 −0.834606
\(170\) 0 0
\(171\) −966.922 −0.432412
\(172\) 0 0
\(173\) −1444.36 −0.634754 −0.317377 0.948299i \(-0.602802\pi\)
−0.317377 + 0.948299i \(0.602802\pi\)
\(174\) 0 0
\(175\) 175.000 0.0755929
\(176\) 0 0
\(177\) −2281.84 −0.969005
\(178\) 0 0
\(179\) −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 \(0.347001\pi\)
\(182\) 0 0
\(183\) 596.241 0.240849
\(184\) 0 0
\(185\) 485.603 0.192985
\(186\) 0 0
\(187\) 359.868 0.140728
\(188\) 0 0
\(189\) −189.000 −0.0727393
\(190\) 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 \(-0.772933\pi\)
−0.756173 + 0.654372i \(0.772933\pi\)
\(194\) 0 0
\(195\) 285.934 0.105006
\(196\) 0 0
\(197\) −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 \(-0.533263\pi\)
−0.104307 + 0.994545i \(0.533263\pi\)
\(200\) 0 0
\(201\) −929.977 −0.326346
\(202\) 0 0
\(203\) 668.459 0.231116
\(204\) 0 0
\(205\) −2455.56 −0.836605
\(206\) 0 0
\(207\) −1895.60 −0.636490
\(208\) 0 0
\(209\) −315.619 −0.104458
\(210\) 0 0
\(211\) 1055.16 0.344266 0.172133 0.985074i \(-0.444934\pi\)
0.172133 + 0.985074i \(0.444934\pi\)
\(212\) 0 0
\(213\) 1997.08 0.642430
\(214\) 0 0
\(215\) 215.019 0.0682056
\(216\) 0 0
\(217\) 660.152 0.206516
\(218\) 0 0
\(219\) −1863.86 −0.575106
\(220\) 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 \(0.747736\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −5443.11 −1.59151 −0.795754 0.605621i \(-0.792925\pi\)
−0.795754 + 0.605621i \(0.792925\pi\)
\(228\) 0 0
\(229\) −536.303 −0.154759 −0.0773797 0.997002i \(-0.524655\pi\)
−0.0773797 + 0.997002i \(0.524655\pi\)
\(230\) 0 0
\(231\) −61.6926 −0.0175717
\(232\) 0 0
\(233\) −183.490 −0.0515916 −0.0257958 0.999667i \(-0.508212\pi\)
−0.0257958 + 0.999667i \(0.508212\pi\)
\(234\) 0 0
\(235\) −2367.47 −0.657178
\(236\) 0 0
\(237\) −74.2878 −0.0203608
\(238\) 0 0
\(239\) −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 \(0.779342\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 245.000 0.0638877
\(246\) 0 0
\(247\) 2047.97 0.527567
\(248\) 0 0
\(249\) −1220.17 −0.310543
\(250\) 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
\(254\) 0 0
\(255\) −1837.47 −0.451243
\(256\) 0 0
\(257\) 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
\(261\) 859.448 0.203826
\(262\) 0 0
\(263\) −7577.00 −1.77649 −0.888246 0.459367i \(-0.848076\pi\)
−0.888246 + 0.459367i \(0.848076\pi\)
\(264\) 0 0
\(265\) −918.385 −0.212890
\(266\) 0 0
\(267\) −785.253 −0.179988
\(268\) 0 0
\(269\) 1023.10 0.231893 0.115947 0.993255i \(-0.463010\pi\)
0.115947 + 0.993255i \(0.463010\pi\)
\(270\) 0 0
\(271\) 2251.98 0.504790 0.252395 0.967624i \(-0.418782\pi\)
0.252395 + 0.967624i \(0.418782\pi\)
\(272\) 0 0
\(273\) 400.307 0.0887462
\(274\) 0 0
\(275\) 73.4436 0.0161048
\(276\) 0 0
\(277\) −8630.72 −1.87209 −0.936047 0.351875i \(-0.885544\pi\)
−0.936047 + 0.351875i \(0.885544\pi\)
\(278\) 0 0
\(279\) 848.767 0.182130
\(280\) 0 0
\(281\) −7521.62 −1.59680 −0.798402 0.602124i \(-0.794321\pi\)
−0.798402 + 0.602124i \(0.794321\pi\)
\(282\) 0 0
\(283\) −14.8169 −0.00311226 −0.00155613 0.999999i \(-0.500495\pi\)
−0.00155613 + 0.999999i \(0.500495\pi\)
\(284\) 0 0
\(285\) 1611.54 0.334945
\(286\) 0 0
\(287\) −3437.79 −0.707060
\(288\) 0 0
\(289\) 10092.8 2.05430
\(290\) 0 0
\(291\) 3014.32 0.607226
\(292\) 0 0
\(293\) 6913.39 1.37844 0.689222 0.724550i \(-0.257952\pi\)
0.689222 + 0.724550i \(0.257952\pi\)
\(294\) 0 0
\(295\) 3803.07 0.750588
\(296\) 0 0
\(297\) −79.3190 −0.0154968
\(298\) 0 0
\(299\) 4014.94 0.776555
\(300\) 0 0
\(301\) 301.027 0.0576442
\(302\) 0 0
\(303\) 386.615 0.0733018
\(304\) 0 0
\(305\) −993.735 −0.186561
\(306\) 0 0
\(307\) 7644.12 1.42108 0.710542 0.703655i \(-0.248450\pi\)
0.710542 + 0.703655i \(0.248450\pi\)
\(308\) 0 0
\(309\) 2418.02 0.445167
\(310\) 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 \(-0.808363\pi\)
−0.824179 + 0.566329i \(0.808363\pi\)
\(314\) 0 0
\(315\) 315.000 0.0563436
\(316\) 0 0
\(317\) −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
\(321\) −2309.14 −0.401506
\(322\) 0 0
\(323\) −13160.7 −2.26712
\(324\) 0 0
\(325\) −476.556 −0.0813372
\(326\) 0 0
\(327\) 2342.57 0.396160
\(328\) 0 0
\(329\) −3314.46 −0.555417
\(330\) 0 0
\(331\) −1221.67 −0.202867 −0.101433 0.994842i \(-0.532343\pi\)
−0.101433 + 0.994842i \(0.532343\pi\)
\(332\) 0 0
\(333\) 874.086 0.143843
\(334\) 0 0
\(335\) 1549.96 0.252786
\(336\) 0 0
\(337\) −8744.83 −1.41354 −0.706768 0.707446i \(-0.749847\pi\)
−0.706768 + 0.707446i \(0.749847\pi\)
\(338\) 0 0
\(339\) 3346.94 0.536226
\(340\) 0 0
\(341\) 277.051 0.0439975
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 3159.34 0.493023
\(346\) 0 0
\(347\) −4589.56 −0.710031 −0.355015 0.934860i \(-0.615524\pi\)
−0.355015 + 0.934860i \(0.615524\pi\)
\(348\) 0 0
\(349\) −3989.89 −0.611960 −0.305980 0.952038i \(-0.598984\pi\)
−0.305980 + 0.952038i \(0.598984\pi\)
\(350\) 0 0
\(351\) 514.681 0.0782668
\(352\) 0 0
\(353\) 2416.35 0.364333 0.182166 0.983268i \(-0.441689\pi\)
0.182166 + 0.983268i \(0.441689\pi\)
\(354\) 0 0
\(355\) −3328.46 −0.497624
\(356\) 0 0
\(357\) −2572.46 −0.381370
\(358\) 0 0
\(359\) 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
\(362\) 0 0
\(363\) 3967.11 0.573607
\(364\) 0 0
\(365\) 3106.44 0.445475
\(366\) 0 0
\(367\) −11112.8 −1.58061 −0.790307 0.612711i \(-0.790079\pi\)
−0.790307 + 0.612711i \(0.790079\pi\)
\(368\) 0 0
\(369\) −4420.02 −0.623569
\(370\) 0 0
\(371\) −1285.74 −0.179925
\(372\) 0 0
\(373\) 6091.09 0.845535 0.422768 0.906238i \(-0.361059\pi\)
0.422768 + 0.906238i \(0.361059\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 0 0
\(377\) −1820.33 −0.248679
\(378\) 0 0
\(379\) −3984.29 −0.539998 −0.269999 0.962861i \(-0.587023\pi\)
−0.269999 + 0.962861i \(0.587023\pi\)
\(380\) 0 0
\(381\) −5627.95 −0.756768
\(382\) 0 0
\(383\) −318.475 −0.0424890 −0.0212445 0.999774i \(-0.506763\pi\)
−0.0212445 + 0.999774i \(0.506763\pi\)
\(384\) 0 0
\(385\) 102.821 0.0136110
\(386\) 0 0
\(387\) 387.035 0.0508374
\(388\) 0 0
\(389\) 3885.46 0.506429 0.253214 0.967410i \(-0.418512\pi\)
0.253214 + 0.967410i \(0.418512\pi\)
\(390\) 0 0
\(391\) −25800.9 −3.33710
\(392\) 0 0
\(393\) 1092.61 0.140241
\(394\) 0 0
\(395\) 123.813 0.0157714
\(396\) 0 0
\(397\) 4806.04 0.607578 0.303789 0.952739i \(-0.401748\pi\)
0.303789 + 0.952739i \(0.401748\pi\)
\(398\) 0 0
\(399\) 2256.15 0.283080
\(400\) 0 0
\(401\) 3618.59 0.450633 0.225316 0.974286i \(-0.427658\pi\)
0.225316 + 0.974286i \(0.427658\pi\)
\(402\) 0 0
\(403\) −1797.71 −0.222209
\(404\) 0 0
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) 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
\(411\) −4809.40 −0.577202
\(412\) 0 0
\(413\) 5324.30 0.634363
\(414\) 0 0
\(415\) 2033.62 0.240546
\(416\) 0 0
\(417\) 7293.37 0.856494
\(418\) 0 0
\(419\) 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 \(0.688593\pi\)
\(422\) 0 0
\(423\) −4261.45 −0.489831
\(424\) 0 0
\(425\) 3062.45 0.349531
\(426\) 0 0
\(427\) −1391.23 −0.157673
\(428\) 0 0
\(429\) 168.000 0.0189070
\(430\) 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 \(-0.366153\pi\)
0.408210 + 0.912888i \(0.366153\pi\)
\(434\) 0 0
\(435\) −1432.41 −0.157883
\(436\) 0 0
\(437\) 22628.4 2.47703
\(438\) 0 0
\(439\) −6909.21 −0.751159 −0.375579 0.926790i \(-0.622556\pi\)
−0.375579 + 0.926790i \(0.622556\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 14812.6 1.58864 0.794318 0.607502i \(-0.207828\pi\)
0.794318 + 0.607502i \(0.207828\pi\)
\(444\) 0 0
\(445\) 1308.75 0.139418
\(446\) 0 0
\(447\) −7024.72 −0.743306
\(448\) 0 0
\(449\) −10654.5 −1.11986 −0.559932 0.828538i \(-0.689173\pi\)
−0.559932 + 0.828538i \(0.689173\pi\)
\(450\) 0 0
\(451\) −1442.76 −0.150636
\(452\) 0 0
\(453\) −6312.21 −0.654688
\(454\) 0 0
\(455\) −667.179 −0.0687425
\(456\) 0 0
\(457\) −5855.16 −0.599328 −0.299664 0.954045i \(-0.596875\pi\)
−0.299664 + 0.954045i \(0.596875\pi\)
\(458\) 0 0
\(459\) −3307.45 −0.336336
\(460\) 0 0
\(461\) 3204.74 0.323774 0.161887 0.986809i \(-0.448242\pi\)
0.161887 + 0.986809i \(0.448242\pi\)
\(462\) 0 0
\(463\) −371.658 −0.0373054 −0.0186527 0.999826i \(-0.505938\pi\)
−0.0186527 + 0.999826i \(0.505938\pi\)
\(464\) 0 0
\(465\) −1414.61 −0.141077
\(466\) 0 0
\(467\) 19752.3 1.95723 0.978614 0.205703i \(-0.0659482\pi\)
0.978614 + 0.205703i \(0.0659482\pi\)
\(468\) 0 0
\(469\) 2169.95 0.213643
\(470\) 0 0
\(471\) 1780.40 0.174175
\(472\) 0 0
\(473\) 126.334 0.0122809
\(474\) 0 0
\(475\) −2685.90 −0.259447
\(476\) 0 0
\(477\) −1653.09 −0.158679
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) 4423.07 0.416681
\(484\) 0 0
\(485\) −5023.87 −0.470355
\(486\) 0 0
\(487\) −17647.6 −1.64207 −0.821035 0.570878i \(-0.806603\pi\)
−0.821035 + 0.570878i \(0.806603\pi\)
\(488\) 0 0
\(489\) 6536.12 0.604445
\(490\) 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
\(494\) 0 0
\(495\) 132.198 0.0120038
\(496\) 0 0
\(497\) −4659.85 −0.420569
\(498\) 0 0
\(499\) 17474.1 1.56764 0.783818 0.620991i \(-0.213270\pi\)
0.783818 + 0.620991i \(0.213270\pi\)
\(500\) 0 0
\(501\) 2398.51 0.213887
\(502\) 0 0
\(503\) −7444.81 −0.659936 −0.329968 0.943992i \(-0.607038\pi\)
−0.329968 + 0.943992i \(0.607038\pi\)
\(504\) 0 0
\(505\) −644.358 −0.0567793
\(506\) 0 0
\(507\) 5500.89 0.481860
\(508\) 0 0
\(509\) −3384.48 −0.294724 −0.147362 0.989083i \(-0.547078\pi\)
−0.147362 + 0.989083i \(0.547078\pi\)
\(510\) 0 0
\(511\) 4349.02 0.376495
\(512\) 0 0
\(513\) 2900.77 0.249653
\(514\) 0 0
\(515\) −4030.04 −0.344825
\(516\) 0 0
\(517\) −1391.00 −0.118329
\(518\) 0 0
\(519\) 4333.07 0.366476
\(520\) 0 0
\(521\) 2973.12 0.250009 0.125005 0.992156i \(-0.460105\pi\)
0.125005 + 0.992156i \(0.460105\pi\)
\(522\) 0 0
\(523\) −2689.02 −0.224823 −0.112412 0.993662i \(-0.535858\pi\)
−0.112412 + 0.993662i \(0.535858\pi\)
\(524\) 0 0
\(525\) −525.000 −0.0436436
\(526\) 0 0
\(527\) 11552.5 0.954903
\(528\) 0 0
\(529\) 32194.9 2.64608
\(530\) 0 0
\(531\) 6845.53 0.559455
\(532\) 0 0
\(533\) 9361.72 0.760790
\(534\) 0 0
\(535\) 3848.56 0.311005
\(536\) 0 0
\(537\) 10030.5 0.806046
\(538\) 0 0
\(539\) 143.949 0.0115034
\(540\) 0 0
\(541\) −14429.5 −1.14671 −0.573356 0.819306i \(-0.694359\pi\)
−0.573356 + 0.819306i \(0.694359\pi\)
\(542\) 0 0
\(543\) −6755.44 −0.533893
\(544\) 0 0
\(545\) −3904.28 −0.306864
\(546\) 0 0
\(547\) −13811.2 −1.07957 −0.539784 0.841804i \(-0.681494\pi\)
−0.539784 + 0.841804i \(0.681494\pi\)
\(548\) 0 0
\(549\) −1788.72 −0.139054
\(550\) 0 0
\(551\) −10259.5 −0.793229
\(552\) 0 0
\(553\) 173.338 0.0133293
\(554\) 0 0
\(555\) −1456.81 −0.111420
\(556\) 0 0
\(557\) −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
\(561\) −1079.60 −0.0812493
\(562\) 0 0
\(563\) 6958.47 0.520896 0.260448 0.965488i \(-0.416130\pi\)
0.260448 + 0.965488i \(0.416130\pi\)
\(564\) 0 0
\(565\) −5578.23 −0.415359
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) −13396.4 −0.987009 −0.493505 0.869743i \(-0.664284\pi\)
−0.493505 + 0.869743i \(0.664284\pi\)
\(570\) 0 0
\(571\) 8055.84 0.590414 0.295207 0.955433i \(-0.404611\pi\)
0.295207 + 0.955433i \(0.404611\pi\)
\(572\) 0 0
\(573\) −3005.78 −0.219142
\(574\) 0 0
\(575\) −5265.56 −0.381894
\(576\) 0 0
\(577\) −21456.9 −1.54812 −0.774059 0.633114i \(-0.781777\pi\)
−0.774059 + 0.633114i \(0.781777\pi\)
\(578\) 0 0
\(579\) 12164.9 0.873153
\(580\) 0 0
\(581\) 2847.07 0.203298
\(582\) 0 0
\(583\) −539.596 −0.0383324
\(584\) 0 0
\(585\) −857.802 −0.0606252
\(586\) 0 0
\(587\) −20156.3 −1.41728 −0.708638 0.705572i \(-0.750690\pi\)
−0.708638 + 0.705572i \(0.750690\pi\)
\(588\) 0 0
\(589\) −10132.0 −0.708797
\(590\) 0 0
\(591\) 15420.7 1.07330
\(592\) 0 0
\(593\) 599.307 0.0415018 0.0207509 0.999785i \(-0.493394\pi\)
0.0207509 + 0.999785i \(0.493394\pi\)
\(594\) 0 0
\(595\) 4287.43 0.295408
\(596\) 0 0
\(597\) 1756.89 0.120444
\(598\) 0 0
\(599\) 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 \(0.191515\pi\)
\(602\) 0 0
\(603\) 2789.93 0.188416
\(604\) 0 0
\(605\) −6611.85 −0.444314
\(606\) 0 0
\(607\) −3029.50 −0.202576 −0.101288 0.994857i \(-0.532296\pi\)
−0.101288 + 0.994857i \(0.532296\pi\)
\(608\) 0 0
\(609\) −2005.38 −0.133435
\(610\) 0 0
\(611\) 9025.87 0.597623
\(612\) 0 0
\(613\) −19339.6 −1.27426 −0.637129 0.770757i \(-0.719878\pi\)
−0.637129 + 0.770757i \(0.719878\pi\)
\(614\) 0 0
\(615\) 7366.69 0.483014
\(616\) 0 0
\(617\) −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 \(-0.413759\pi\)
0.267632 + 0.963521i \(0.413759\pi\)
\(620\) 0 0
\(621\) 5686.81 0.367478
\(622\) 0 0
\(623\) 1832.26 0.117830
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 946.856 0.0603091
\(628\) 0 0
\(629\) 11897.1 0.754162
\(630\) 0 0
\(631\) 4376.56 0.276114 0.138057 0.990424i \(-0.455914\pi\)
0.138057 + 0.990424i \(0.455914\pi\)
\(632\) 0 0
\(633\) −3165.48 −0.198762
\(634\) 0 0
\(635\) 9379.92 0.586190
\(636\) 0 0
\(637\) −934.051 −0.0580980
\(638\) 0 0
\(639\) −5991.23 −0.370907
\(640\) 0 0
\(641\) 11836.6 0.729357 0.364678 0.931133i \(-0.381179\pi\)
0.364678 + 0.931133i \(0.381179\pi\)
\(642\) 0 0
\(643\) −1448.21 −0.0888209 −0.0444104 0.999013i \(-0.514141\pi\)
−0.0444104 + 0.999013i \(0.514141\pi\)
\(644\) 0 0
\(645\) −645.058 −0.0393785
\(646\) 0 0
\(647\) 8732.95 0.530646 0.265323 0.964160i \(-0.414521\pi\)
0.265323 + 0.964160i \(0.414521\pi\)
\(648\) 0 0
\(649\) 2234.49 0.135149
\(650\) 0 0
\(651\) −1980.46 −0.119232
\(652\) 0 0
\(653\) −21978.4 −1.31712 −0.658562 0.752527i \(-0.728835\pi\)
−0.658562 + 0.752527i \(0.728835\pi\)
\(654\) 0 0
\(655\) −1821.01 −0.108630
\(656\) 0 0
\(657\) 5591.59 0.332038
\(658\) 0 0
\(659\) 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 \(0.581172\pi\)
\(662\) 0 0
\(663\) 7005.27 0.410350
\(664\) 0 0
\(665\) −3760.25 −0.219273
\(666\) 0 0
\(667\) −20113.2 −1.16760
\(668\) 0 0
\(669\) 14027.6 0.810668
\(670\) 0 0
\(671\) −583.868 −0.0335916
\(672\) 0 0
\(673\) 27159.2 1.55559 0.777795 0.628518i \(-0.216338\pi\)
0.777795 + 0.628518i \(0.216338\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) −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
\(681\) 16329.3 0.918857
\(682\) 0 0
\(683\) 8675.09 0.486007 0.243004 0.970025i \(-0.421867\pi\)
0.243004 + 0.970025i \(0.421867\pi\)
\(684\) 0 0
\(685\) 8015.66 0.447099
\(686\) 0 0
\(687\) 1608.91 0.0893504
\(688\) 0 0
\(689\) 3501.30 0.193598
\(690\) 0 0
\(691\) 21426.0 1.17957 0.589785 0.807561i \(-0.299213\pi\)
0.589785 + 0.807561i \(0.299213\pi\)
\(692\) 0 0
\(693\) 185.078 0.0101451
\(694\) 0 0
\(695\) −12155.6 −0.663438
\(696\) 0 0
\(697\) −60160.4 −3.26935
\(698\) 0 0
\(699\) 550.470 0.0297864
\(700\) 0 0
\(701\) 24840.5 1.33839 0.669197 0.743085i \(-0.266638\pi\)
0.669197 + 0.743085i \(0.266638\pi\)
\(702\) 0 0
\(703\) −10434.2 −0.559793
\(704\) 0 0
\(705\) 7102.41 0.379422
\(706\) 0 0
\(707\) −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
\(711\) 222.863 0.0117553
\(712\) 0 0
\(713\) −19863.3 −1.04332
\(714\) 0 0
\(715\) −280.000 −0.0146453
\(716\) 0 0
\(717\) 1929.65 0.100508
\(718\) 0 0
\(719\) −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
\(722\) 0 0
\(723\) 17266.8 0.888189
\(724\) 0 0
\(725\) 2387.35 0.122295
\(726\) 0 0
\(727\) 14326.2 0.730851 0.365426 0.930841i \(-0.380923\pi\)
0.365426 + 0.930841i \(0.380923\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 5267.89 0.266539
\(732\) 0 0
\(733\) 6727.85 0.339016 0.169508 0.985529i \(-0.445782\pi\)
0.169508 + 0.985529i \(0.445782\pi\)
\(734\) 0 0
\(735\) −735.000 −0.0368856
\(736\) 0 0
\(737\) 910.677 0.0455159
\(738\) 0 0
\(739\) 3418.51 0.170165 0.0850826 0.996374i \(-0.472885\pi\)
0.0850826 + 0.996374i \(0.472885\pi\)
\(740\) 0 0
\(741\) −6143.91 −0.304591
\(742\) 0 0
\(743\) −8095.50 −0.399724 −0.199862 0.979824i \(-0.564049\pi\)
−0.199862 + 0.979824i \(0.564049\pi\)
\(744\) 0 0
\(745\) 11707.9 0.575762
\(746\) 0 0
\(747\) 3660.51 0.179292
\(748\) 0 0
\(749\) 5387.99 0.262847
\(750\) 0 0
\(751\) −13446.8 −0.653371 −0.326686 0.945133i \(-0.605932\pi\)
−0.326686 + 0.945133i \(0.605932\pi\)
\(752\) 0 0
\(753\) −15396.8 −0.745141
\(754\) 0 0
\(755\) 10520.4 0.507119
\(756\) 0 0
\(757\) −2593.24 −0.124508 −0.0622541 0.998060i \(-0.519829\pi\)
−0.0622541 + 0.998060i \(0.519829\pi\)
\(758\) 0 0
\(759\) 1856.26 0.0887722
\(760\) 0 0
\(761\) 27079.4 1.28992 0.644959 0.764217i \(-0.276875\pi\)
0.644959 + 0.764217i \(0.276875\pi\)
\(762\) 0 0
\(763\) −5465.99 −0.259348
\(764\) 0 0
\(765\) 5512.41 0.260525
\(766\) 0 0
\(767\) −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
\(771\) −15125.2 −0.706513
\(772\) 0 0
\(773\) 25864.0 1.20345 0.601724 0.798704i \(-0.294481\pi\)
0.601724 + 0.798704i \(0.294481\pi\)
\(774\) 0 0
\(775\) 2357.69 0.109278
\(776\) 0 0
\(777\) −2039.53 −0.0941671
\(778\) 0 0
\(779\) 52763.1 2.42675
\(780\) 0 0
\(781\) −1955.63 −0.0896006
\(782\) 0 0
\(783\) −2578.34 −0.117679
\(784\) 0 0
\(785\) −2967.33 −0.134916
\(786\) 0 0
\(787\) 32371.3 1.46621 0.733107 0.680113i \(-0.238069\pi\)
0.733107 + 0.680113i \(0.238069\pi\)
\(788\) 0 0
\(789\) 22731.0 1.02566
\(790\) 0 0
\(791\) −7809.52 −0.351043
\(792\) 0 0
\(793\) 3788.57 0.169654
\(794\) 0 0
\(795\) 2755.16 0.122912
\(796\) 0 0
\(797\) 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
\(801\) 2355.76 0.103916
\(802\) 0 0
\(803\) 1825.18 0.0802109
\(804\) 0 0
\(805\) −7371.79 −0.322760
\(806\) 0 0
\(807\) −3069.29 −0.133884
\(808\) 0 0
\(809\) 12391.7 0.538526 0.269263 0.963067i \(-0.413220\pi\)
0.269263 + 0.963067i \(0.413220\pi\)
\(810\) 0 0
\(811\) −14654.5 −0.634511 −0.317256 0.948340i \(-0.602761\pi\)
−0.317256 + 0.948340i \(0.602761\pi\)
\(812\) 0 0
\(813\) −6755.94 −0.291441
\(814\) 0 0
\(815\) −10893.5 −0.468201
\(816\) 0 0
\(817\) −4620.16 −0.197844
\(818\) 0 0
\(819\) −1200.92 −0.0512376
\(820\) 0 0
\(821\) 23887.9 1.01546 0.507731 0.861516i \(-0.330485\pi\)
0.507731 + 0.861516i \(0.330485\pi\)
\(822\) 0 0
\(823\) 4008.41 0.169774 0.0848871 0.996391i \(-0.472947\pi\)
0.0848871 + 0.996391i \(0.472947\pi\)
\(824\) 0 0
\(825\) −220.331 −0.00929810
\(826\) 0 0
\(827\) 45110.4 1.89679 0.948394 0.317096i \(-0.102708\pi\)
0.948394 + 0.317096i \(0.102708\pi\)
\(828\) 0 0
\(829\) 16165.4 0.677260 0.338630 0.940920i \(-0.390036\pi\)
0.338630 + 0.940920i \(0.390036\pi\)
\(830\) 0 0
\(831\) 25892.2 1.08085
\(832\) 0 0
\(833\) 6002.41 0.249665
\(834\) 0 0
\(835\) −3997.51 −0.165676
\(836\) 0 0
\(837\) −2546.30 −0.105153
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) 22564.9 0.921916
\(844\) 0 0
\(845\) −9168.15 −0.373247
\(846\) 0 0
\(847\) −9256.59 −0.375514
\(848\) 0 0
\(849\) 44.4506 0.00179687
\(850\) 0 0
\(851\) −20455.8 −0.823990
\(852\) 0 0
\(853\) −30168.1 −1.21094 −0.605472 0.795867i \(-0.707016\pi\)
−0.605472 + 0.795867i \(0.707016\pi\)
\(854\) 0 0
\(855\) −4834.61 −0.193380
\(856\) 0 0
\(857\) −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 \(-0.623574\pi\)
−0.378541 + 0.925585i \(0.623574\pi\)
\(860\) 0 0
\(861\) 10313.4 0.408222
\(862\) 0 0
\(863\) 9466.86 0.373413 0.186707 0.982416i \(-0.440219\pi\)
0.186707 + 0.982416i \(0.440219\pi\)
\(864\) 0 0
\(865\) −7221.79 −0.283871
\(866\) 0 0
\(867\) −30278.3 −1.18605
\(868\) 0 0
\(869\) 72.7461 0.00283975
\(870\) 0 0
\(871\) −5909.15 −0.229878
\(872\) 0 0
\(873\) −9042.97 −0.350582
\(874\) 0 0
\(875\) 875.000 0.0338062
\(876\) 0 0
\(877\) 37740.6 1.45315 0.726573 0.687090i \(-0.241112\pi\)
0.726573 + 0.687090i \(0.241112\pi\)
\(878\) 0 0
\(879\) −20740.2 −0.795845
\(880\) 0 0
\(881\) 25991.5 0.993957 0.496979 0.867763i \(-0.334443\pi\)
0.496979 + 0.867763i \(0.334443\pi\)
\(882\) 0 0
\(883\) −39420.3 −1.50238 −0.751189 0.660087i \(-0.770519\pi\)
−0.751189 + 0.660087i \(0.770519\pi\)
\(884\) 0 0
\(885\) −11409.2 −0.433352
\(886\) 0 0
\(887\) −46005.2 −1.74149 −0.870745 0.491735i \(-0.836363\pi\)
−0.870745 + 0.491735i \(0.836363\pi\)
\(888\) 0 0
\(889\) 13131.9 0.495421
\(890\) 0 0
\(891\) 237.957 0.00894710
\(892\) 0 0
\(893\) 50870.2 1.90628
\(894\) 0 0
\(895\) −16717.5 −0.624361
\(896\) 0 0
\(897\) −12044.8 −0.448345
\(898\) 0 0
\(899\) 9005.81 0.334105
\(900\) 0 0
\(901\) −22500.1 −0.831950
\(902\) 0 0
\(903\) −903.081 −0.0332809
\(904\) 0 0
\(905\) 11259.1 0.413551
\(906\) 0 0
\(907\) 2838.97 0.103932 0.0519661 0.998649i \(-0.483451\pi\)
0.0519661 + 0.998649i \(0.483451\pi\)
\(908\) 0 0
\(909\) −1159.84 −0.0423208
\(910\) 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
\(914\) 0 0
\(915\) 2981.21 0.107711
\(916\) 0 0
\(917\) −2549.42 −0.0918094
\(918\) 0 0
\(919\) 646.475 0.0232048 0.0116024 0.999933i \(-0.496307\pi\)
0.0116024 + 0.999933i \(0.496307\pi\)
\(920\) 0 0
\(921\) −22932.4 −0.820463
\(922\) 0 0
\(923\) 12689.6 0.452528
\(924\) 0 0
\(925\) 2428.02 0.0863056
\(926\) 0 0
\(927\) −7254.07 −0.257017
\(928\) 0 0
\(929\) 51188.2 1.80778 0.903892 0.427760i \(-0.140697\pi\)
0.903892 + 0.427760i \(0.140697\pi\)
\(930\) 0 0
\(931\) −5264.35 −0.185319
\(932\) 0 0
\(933\) 22782.0 0.799409
\(934\) 0 0
\(935\) 1799.34 0.0629355
\(936\) 0 0
\(937\) 29786.1 1.03849 0.519247 0.854624i \(-0.326212\pi\)
0.519247 + 0.854624i \(0.326212\pi\)
\(938\) 0 0
\(939\) 27383.5 0.951680
\(940\) 0 0
\(941\) −44817.4 −1.55261 −0.776304 0.630358i \(-0.782908\pi\)
−0.776304 + 0.630358i \(0.782908\pi\)
\(942\) 0 0
\(943\) 103439. 3.57206
\(944\) 0 0
\(945\) −945.000 −0.0325300
\(946\) 0 0
\(947\) −54697.1 −1.87689 −0.938446 0.345425i \(-0.887735\pi\)
−0.938446 + 0.345425i \(0.887735\pi\)
\(948\) 0 0
\(949\) −11843.2 −0.405105
\(950\) 0 0
\(951\) 14789.4 0.504290
\(952\) 0 0
\(953\) −7577.51 −0.257565 −0.128783 0.991673i \(-0.541107\pi\)
−0.128783 + 0.991673i \(0.541107\pi\)
\(954\) 0 0
\(955\) 5009.63 0.169746
\(956\) 0 0
\(957\) −841.612 −0.0284278
\(958\) 0 0
\(959\) 11221.9 0.377868
\(960\) 0 0
\(961\) −20897.1 −0.701457
\(962\) 0 0
\(963\) 6927.41 0.231810
\(964\) 0 0
\(965\) −20274.8 −0.676342
\(966\) 0 0
\(967\) −50779.0 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(968\) 0 0
\(969\) 39482.0 1.30892
\(970\) 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
\(974\) 0 0
\(975\) 1429.67 0.0469601
\(976\) 0 0
\(977\) 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
\(981\) −7027.70 −0.228723
\(982\) 0 0
\(983\) 2824.37 0.0916414 0.0458207 0.998950i \(-0.485410\pi\)
0.0458207 + 0.998950i \(0.485410\pi\)
\(984\) 0 0
\(985\) −25701.1 −0.831377
\(986\) 0 0
\(987\) 9943.38 0.320670
\(988\) 0 0
\(989\) −9057.59 −0.291218
\(990\) 0 0
\(991\) −16951.4 −0.543370 −0.271685 0.962386i \(-0.587581\pi\)
−0.271685 + 0.962386i \(0.587581\pi\)
\(992\) 0 0
\(993\) 3665.00 0.117125
\(994\) 0 0
\(995\) −2928.15 −0.0932952
\(996\) 0 0
\(997\) 23847.8 0.757540 0.378770 0.925491i \(-0.376347\pi\)
0.378770 + 0.925491i \(0.376347\pi\)
\(998\) 0 0
\(999\) −2622.26 −0.0830476
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 1680.4.a.bg.1.1 2
4.3 odd 2 105.4.a.f.1.1 2
12.11 even 2 315.4.a.i.1.2 2
20.3 even 4 525.4.d.h.274.3 4
20.7 even 4 525.4.d.h.274.2 4
20.19 odd 2 525.4.a.k.1.2 2
28.27 even 2 735.4.a.p.1.1 2
60.59 even 2 1575.4.a.w.1.1 2
84.83 odd 2 2205.4.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.1 2 4.3 odd 2
315.4.a.i.1.2 2 12.11 even 2
525.4.a.k.1.2 2 20.19 odd 2
525.4.d.h.274.2 4 20.7 even 4
525.4.d.h.274.3 4 20.3 even 4
735.4.a.p.1.1 2 28.27 even 2
1575.4.a.w.1.1 2 60.59 even 2
1680.4.a.bg.1.1 2 1.1 even 1 trivial
2205.4.a.z.1.2 2 84.83 odd 2