Properties

Label 464.4.a.n.1.6
Level $464$
Weight $4$
Character 464.1
Self dual yes
Analytic conductor $27.377$
Analytic rank $0$
Dimension $6$
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] = [464,4,Mod(1,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("464.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 464.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: \(27.3768862427\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 40x^{4} - 88x^{3} - 8x^{2} + 48x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(7.22553\) of defining polynomial
Character \(\chi\) \(=\) 464.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.88017 q^{3} +2.64112 q^{5} -3.30580 q^{7} +70.6177 q^{9} +O(q^{10})\) \(q+9.88017 q^{3} +2.64112 q^{5} -3.30580 q^{7} +70.6177 q^{9} +28.3440 q^{11} +85.7400 q^{13} +26.0947 q^{15} -111.092 q^{17} -99.8826 q^{19} -32.6619 q^{21} +110.105 q^{23} -118.024 q^{25} +430.950 q^{27} -29.0000 q^{29} +60.0577 q^{31} +280.044 q^{33} -8.73102 q^{35} +119.286 q^{37} +847.126 q^{39} -285.212 q^{41} +249.574 q^{43} +186.510 q^{45} +226.276 q^{47} -332.072 q^{49} -1097.60 q^{51} +387.165 q^{53} +74.8600 q^{55} -986.857 q^{57} -164.811 q^{59} +188.155 q^{61} -233.448 q^{63} +226.450 q^{65} +903.648 q^{67} +1087.86 q^{69} -820.374 q^{71} -394.228 q^{73} -1166.10 q^{75} -93.6997 q^{77} -605.439 q^{79} +2351.18 q^{81} +504.986 q^{83} -293.406 q^{85} -286.525 q^{87} -930.620 q^{89} -283.439 q^{91} +593.380 q^{93} -263.802 q^{95} +12.2577 q^{97} +2001.59 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 5 q^{3} - 5 q^{5} + 38 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 5 q^{3} - 5 q^{5} + 38 q^{7} + 47 q^{9} + 19 q^{11} + 13 q^{13} + 191 q^{15} - 218 q^{17} + 290 q^{19} - 266 q^{21} + 196 q^{23} - 13 q^{25} + 437 q^{27} - 174 q^{29} + 675 q^{31} + 291 q^{33} + 466 q^{35} - 238 q^{37} + 1297 q^{39} - 464 q^{41} + 579 q^{43} - 148 q^{45} + 975 q^{47} + 914 q^{49} + 576 q^{51} + 515 q^{53} + 1605 q^{55} - 340 q^{57} + 108 q^{59} + 1158 q^{61} + 1136 q^{63} + 1239 q^{65} + 80 q^{67} + 2568 q^{69} - 438 q^{71} + 262 q^{73} - 1766 q^{75} + 194 q^{77} + 237 q^{79} + 2554 q^{81} + 1288 q^{83} + 3112 q^{85} - 145 q^{87} - 252 q^{89} - 2450 q^{91} + 2131 q^{93} - 180 q^{95} + 380 q^{97} - 2264 q^{99}+O(q^{100}) \) 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
\(3\) 9.88017 1.90144 0.950719 0.310053i \(-0.100347\pi\)
0.950719 + 0.310053i \(0.100347\pi\)
\(4\) 0 0
\(5\) 2.64112 0.236229 0.118114 0.993000i \(-0.462315\pi\)
0.118114 + 0.993000i \(0.462315\pi\)
\(6\) 0 0
\(7\) −3.30580 −0.178496 −0.0892482 0.996009i \(-0.528446\pi\)
−0.0892482 + 0.996009i \(0.528446\pi\)
\(8\) 0 0
\(9\) 70.6177 2.61547
\(10\) 0 0
\(11\) 28.3440 0.776913 0.388457 0.921467i \(-0.373008\pi\)
0.388457 + 0.921467i \(0.373008\pi\)
\(12\) 0 0
\(13\) 85.7400 1.82923 0.914615 0.404325i \(-0.132494\pi\)
0.914615 + 0.404325i \(0.132494\pi\)
\(14\) 0 0
\(15\) 26.0947 0.449175
\(16\) 0 0
\(17\) −111.092 −1.58492 −0.792462 0.609922i \(-0.791201\pi\)
−0.792462 + 0.609922i \(0.791201\pi\)
\(18\) 0 0
\(19\) −99.8826 −1.20603 −0.603017 0.797729i \(-0.706035\pi\)
−0.603017 + 0.797729i \(0.706035\pi\)
\(20\) 0 0
\(21\) −32.6619 −0.339400
\(22\) 0 0
\(23\) 110.105 0.998196 0.499098 0.866546i \(-0.333665\pi\)
0.499098 + 0.866546i \(0.333665\pi\)
\(24\) 0 0
\(25\) −118.024 −0.944196
\(26\) 0 0
\(27\) 430.950 3.07172
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 60.0577 0.347957 0.173979 0.984749i \(-0.444338\pi\)
0.173979 + 0.984749i \(0.444338\pi\)
\(32\) 0 0
\(33\) 280.044 1.47725
\(34\) 0 0
\(35\) −8.73102 −0.0421660
\(36\) 0 0
\(37\) 119.286 0.530014 0.265007 0.964247i \(-0.414626\pi\)
0.265007 + 0.964247i \(0.414626\pi\)
\(38\) 0 0
\(39\) 847.126 3.47817
\(40\) 0 0
\(41\) −285.212 −1.08641 −0.543204 0.839601i \(-0.682789\pi\)
−0.543204 + 0.839601i \(0.682789\pi\)
\(42\) 0 0
\(43\) 249.574 0.885110 0.442555 0.896741i \(-0.354072\pi\)
0.442555 + 0.896741i \(0.354072\pi\)
\(44\) 0 0
\(45\) 186.510 0.617850
\(46\) 0 0
\(47\) 226.276 0.702251 0.351125 0.936328i \(-0.385799\pi\)
0.351125 + 0.936328i \(0.385799\pi\)
\(48\) 0 0
\(49\) −332.072 −0.968139
\(50\) 0 0
\(51\) −1097.60 −3.01363
\(52\) 0 0
\(53\) 387.165 1.00342 0.501709 0.865036i \(-0.332705\pi\)
0.501709 + 0.865036i \(0.332705\pi\)
\(54\) 0 0
\(55\) 74.8600 0.183529
\(56\) 0 0
\(57\) −986.857 −2.29320
\(58\) 0 0
\(59\) −164.811 −0.363670 −0.181835 0.983329i \(-0.558204\pi\)
−0.181835 + 0.983329i \(0.558204\pi\)
\(60\) 0 0
\(61\) 188.155 0.394932 0.197466 0.980310i \(-0.436729\pi\)
0.197466 + 0.980310i \(0.436729\pi\)
\(62\) 0 0
\(63\) −233.448 −0.466852
\(64\) 0 0
\(65\) 226.450 0.432117
\(66\) 0 0
\(67\) 903.648 1.64773 0.823867 0.566783i \(-0.191812\pi\)
0.823867 + 0.566783i \(0.191812\pi\)
\(68\) 0 0
\(69\) 1087.86 1.89801
\(70\) 0 0
\(71\) −820.374 −1.37127 −0.685637 0.727943i \(-0.740476\pi\)
−0.685637 + 0.727943i \(0.740476\pi\)
\(72\) 0 0
\(73\) −394.228 −0.632066 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(74\) 0 0
\(75\) −1166.10 −1.79533
\(76\) 0 0
\(77\) −93.6997 −0.138676
\(78\) 0 0
\(79\) −605.439 −0.862243 −0.431122 0.902294i \(-0.641882\pi\)
−0.431122 + 0.902294i \(0.641882\pi\)
\(80\) 0 0
\(81\) 2351.18 3.22521
\(82\) 0 0
\(83\) 504.986 0.667824 0.333912 0.942604i \(-0.391631\pi\)
0.333912 + 0.942604i \(0.391631\pi\)
\(84\) 0 0
\(85\) −293.406 −0.374405
\(86\) 0 0
\(87\) −286.525 −0.353088
\(88\) 0 0
\(89\) −930.620 −1.10838 −0.554189 0.832391i \(-0.686971\pi\)
−0.554189 + 0.832391i \(0.686971\pi\)
\(90\) 0 0
\(91\) −283.439 −0.326511
\(92\) 0 0
\(93\) 593.380 0.661620
\(94\) 0 0
\(95\) −263.802 −0.284900
\(96\) 0 0
\(97\) 12.2577 0.0128307 0.00641535 0.999979i \(-0.497958\pi\)
0.00641535 + 0.999979i \(0.497958\pi\)
\(98\) 0 0
\(99\) 2001.59 2.03199
\(100\) 0 0
\(101\) −1165.44 −1.14818 −0.574089 0.818793i \(-0.694644\pi\)
−0.574089 + 0.818793i \(0.694644\pi\)
\(102\) 0 0
\(103\) 216.871 0.207465 0.103733 0.994605i \(-0.466921\pi\)
0.103733 + 0.994605i \(0.466921\pi\)
\(104\) 0 0
\(105\) −86.2639 −0.0801761
\(106\) 0 0
\(107\) 1868.17 1.68787 0.843937 0.536443i \(-0.180232\pi\)
0.843937 + 0.536443i \(0.180232\pi\)
\(108\) 0 0
\(109\) −624.737 −0.548981 −0.274490 0.961590i \(-0.588509\pi\)
−0.274490 + 0.961590i \(0.588509\pi\)
\(110\) 0 0
\(111\) 1178.57 1.00779
\(112\) 0 0
\(113\) −1181.43 −0.983537 −0.491769 0.870726i \(-0.663649\pi\)
−0.491769 + 0.870726i \(0.663649\pi\)
\(114\) 0 0
\(115\) 290.801 0.235803
\(116\) 0 0
\(117\) 6054.76 4.78430
\(118\) 0 0
\(119\) 367.247 0.282903
\(120\) 0 0
\(121\) −527.616 −0.396406
\(122\) 0 0
\(123\) −2817.95 −2.06574
\(124\) 0 0
\(125\) −641.857 −0.459275
\(126\) 0 0
\(127\) −1693.76 −1.18344 −0.591720 0.806144i \(-0.701551\pi\)
−0.591720 + 0.806144i \(0.701551\pi\)
\(128\) 0 0
\(129\) 2465.84 1.68298
\(130\) 0 0
\(131\) 195.314 0.130265 0.0651323 0.997877i \(-0.479253\pi\)
0.0651323 + 0.997877i \(0.479253\pi\)
\(132\) 0 0
\(133\) 330.192 0.215273
\(134\) 0 0
\(135\) 1138.19 0.725628
\(136\) 0 0
\(137\) −319.479 −0.199233 −0.0996164 0.995026i \(-0.531762\pi\)
−0.0996164 + 0.995026i \(0.531762\pi\)
\(138\) 0 0
\(139\) 1096.70 0.669213 0.334607 0.942358i \(-0.391397\pi\)
0.334607 + 0.942358i \(0.391397\pi\)
\(140\) 0 0
\(141\) 2235.65 1.33529
\(142\) 0 0
\(143\) 2430.22 1.42115
\(144\) 0 0
\(145\) −76.5925 −0.0438666
\(146\) 0 0
\(147\) −3280.92 −1.84086
\(148\) 0 0
\(149\) 1194.25 0.656625 0.328312 0.944569i \(-0.393520\pi\)
0.328312 + 0.944569i \(0.393520\pi\)
\(150\) 0 0
\(151\) −2259.35 −1.21764 −0.608819 0.793309i \(-0.708357\pi\)
−0.608819 + 0.793309i \(0.708357\pi\)
\(152\) 0 0
\(153\) −7845.04 −4.14532
\(154\) 0 0
\(155\) 158.620 0.0821976
\(156\) 0 0
\(157\) −2592.33 −1.31777 −0.658887 0.752242i \(-0.728972\pi\)
−0.658887 + 0.752242i \(0.728972\pi\)
\(158\) 0 0
\(159\) 3825.25 1.90794
\(160\) 0 0
\(161\) −363.986 −0.178174
\(162\) 0 0
\(163\) −498.162 −0.239381 −0.119690 0.992811i \(-0.538190\pi\)
−0.119690 + 0.992811i \(0.538190\pi\)
\(164\) 0 0
\(165\) 739.629 0.348970
\(166\) 0 0
\(167\) −551.442 −0.255520 −0.127760 0.991805i \(-0.540779\pi\)
−0.127760 + 0.991805i \(0.540779\pi\)
\(168\) 0 0
\(169\) 5154.35 2.34609
\(170\) 0 0
\(171\) −7053.48 −3.15434
\(172\) 0 0
\(173\) 2539.86 1.11620 0.558099 0.829774i \(-0.311531\pi\)
0.558099 + 0.829774i \(0.311531\pi\)
\(174\) 0 0
\(175\) 390.165 0.168536
\(176\) 0 0
\(177\) −1628.36 −0.691496
\(178\) 0 0
\(179\) −739.618 −0.308836 −0.154418 0.988006i \(-0.549350\pi\)
−0.154418 + 0.988006i \(0.549350\pi\)
\(180\) 0 0
\(181\) −4127.70 −1.69508 −0.847539 0.530733i \(-0.821917\pi\)
−0.847539 + 0.530733i \(0.821917\pi\)
\(182\) 0 0
\(183\) 1859.01 0.750938
\(184\) 0 0
\(185\) 315.049 0.125205
\(186\) 0 0
\(187\) −3148.79 −1.23135
\(188\) 0 0
\(189\) −1424.63 −0.548290
\(190\) 0 0
\(191\) 1630.27 0.617605 0.308802 0.951126i \(-0.400072\pi\)
0.308802 + 0.951126i \(0.400072\pi\)
\(192\) 0 0
\(193\) −210.636 −0.0785592 −0.0392796 0.999228i \(-0.512506\pi\)
−0.0392796 + 0.999228i \(0.512506\pi\)
\(194\) 0 0
\(195\) 2237.36 0.821645
\(196\) 0 0
\(197\) −4418.37 −1.59795 −0.798974 0.601366i \(-0.794623\pi\)
−0.798974 + 0.601366i \(0.794623\pi\)
\(198\) 0 0
\(199\) 1436.52 0.511721 0.255860 0.966714i \(-0.417641\pi\)
0.255860 + 0.966714i \(0.417641\pi\)
\(200\) 0 0
\(201\) 8928.19 3.13307
\(202\) 0 0
\(203\) 95.8682 0.0331460
\(204\) 0 0
\(205\) −753.280 −0.256641
\(206\) 0 0
\(207\) 7775.37 2.61075
\(208\) 0 0
\(209\) −2831.07 −0.936983
\(210\) 0 0
\(211\) −3571.25 −1.16519 −0.582595 0.812763i \(-0.697963\pi\)
−0.582595 + 0.812763i \(0.697963\pi\)
\(212\) 0 0
\(213\) −8105.43 −2.60739
\(214\) 0 0
\(215\) 659.156 0.209089
\(216\) 0 0
\(217\) −198.539 −0.0621092
\(218\) 0 0
\(219\) −3895.03 −1.20184
\(220\) 0 0
\(221\) −9525.00 −2.89919
\(222\) 0 0
\(223\) 6382.81 1.91670 0.958351 0.285594i \(-0.0921909\pi\)
0.958351 + 0.285594i \(0.0921909\pi\)
\(224\) 0 0
\(225\) −8334.61 −2.46952
\(226\) 0 0
\(227\) −4565.57 −1.33492 −0.667462 0.744644i \(-0.732619\pi\)
−0.667462 + 0.744644i \(0.732619\pi\)
\(228\) 0 0
\(229\) 5084.23 1.46714 0.733570 0.679614i \(-0.237853\pi\)
0.733570 + 0.679614i \(0.237853\pi\)
\(230\) 0 0
\(231\) −925.769 −0.263684
\(232\) 0 0
\(233\) −5743.69 −1.61494 −0.807471 0.589907i \(-0.799164\pi\)
−0.807471 + 0.589907i \(0.799164\pi\)
\(234\) 0 0
\(235\) 597.623 0.165892
\(236\) 0 0
\(237\) −5981.84 −1.63950
\(238\) 0 0
\(239\) −5554.14 −1.50321 −0.751605 0.659614i \(-0.770720\pi\)
−0.751605 + 0.659614i \(0.770720\pi\)
\(240\) 0 0
\(241\) −2222.62 −0.594073 −0.297036 0.954866i \(-0.595998\pi\)
−0.297036 + 0.954866i \(0.595998\pi\)
\(242\) 0 0
\(243\) 11594.4 3.06082
\(244\) 0 0
\(245\) −877.041 −0.228702
\(246\) 0 0
\(247\) −8563.93 −2.20611
\(248\) 0 0
\(249\) 4989.34 1.26983
\(250\) 0 0
\(251\) −4973.67 −1.25074 −0.625369 0.780329i \(-0.715052\pi\)
−0.625369 + 0.780329i \(0.715052\pi\)
\(252\) 0 0
\(253\) 3120.82 0.775512
\(254\) 0 0
\(255\) −2898.90 −0.711908
\(256\) 0 0
\(257\) 15.0797 0.00366011 0.00183005 0.999998i \(-0.499417\pi\)
0.00183005 + 0.999998i \(0.499417\pi\)
\(258\) 0 0
\(259\) −394.336 −0.0946056
\(260\) 0 0
\(261\) −2047.91 −0.485680
\(262\) 0 0
\(263\) −3238.30 −0.759248 −0.379624 0.925141i \(-0.623947\pi\)
−0.379624 + 0.925141i \(0.623947\pi\)
\(264\) 0 0
\(265\) 1022.55 0.237037
\(266\) 0 0
\(267\) −9194.68 −2.10751
\(268\) 0 0
\(269\) 4147.30 0.940020 0.470010 0.882661i \(-0.344250\pi\)
0.470010 + 0.882661i \(0.344250\pi\)
\(270\) 0 0
\(271\) −1309.45 −0.293519 −0.146759 0.989172i \(-0.546884\pi\)
−0.146759 + 0.989172i \(0.546884\pi\)
\(272\) 0 0
\(273\) −2800.43 −0.620841
\(274\) 0 0
\(275\) −3345.29 −0.733558
\(276\) 0 0
\(277\) 6746.76 1.46344 0.731721 0.681604i \(-0.238717\pi\)
0.731721 + 0.681604i \(0.238717\pi\)
\(278\) 0 0
\(279\) 4241.13 0.910072
\(280\) 0 0
\(281\) 2896.25 0.614861 0.307430 0.951571i \(-0.400531\pi\)
0.307430 + 0.951571i \(0.400531\pi\)
\(282\) 0 0
\(283\) 8577.44 1.80168 0.900841 0.434150i \(-0.142951\pi\)
0.900841 + 0.434150i \(0.142951\pi\)
\(284\) 0 0
\(285\) −2606.41 −0.541720
\(286\) 0 0
\(287\) 942.856 0.193920
\(288\) 0 0
\(289\) 7428.36 1.51198
\(290\) 0 0
\(291\) 121.108 0.0243968
\(292\) 0 0
\(293\) −8204.00 −1.63578 −0.817888 0.575377i \(-0.804855\pi\)
−0.817888 + 0.575377i \(0.804855\pi\)
\(294\) 0 0
\(295\) −435.285 −0.0859093
\(296\) 0 0
\(297\) 12214.9 2.38646
\(298\) 0 0
\(299\) 9440.41 1.82593
\(300\) 0 0
\(301\) −825.044 −0.157989
\(302\) 0 0
\(303\) −11514.8 −2.18319
\(304\) 0 0
\(305\) 496.941 0.0932943
\(306\) 0 0
\(307\) 2506.45 0.465963 0.232981 0.972481i \(-0.425152\pi\)
0.232981 + 0.972481i \(0.425152\pi\)
\(308\) 0 0
\(309\) 2142.72 0.394482
\(310\) 0 0
\(311\) 6501.50 1.18542 0.592711 0.805415i \(-0.298058\pi\)
0.592711 + 0.805415i \(0.298058\pi\)
\(312\) 0 0
\(313\) 7244.35 1.30823 0.654113 0.756397i \(-0.273042\pi\)
0.654113 + 0.756397i \(0.273042\pi\)
\(314\) 0 0
\(315\) −616.564 −0.110284
\(316\) 0 0
\(317\) 8069.20 1.42969 0.714845 0.699283i \(-0.246497\pi\)
0.714845 + 0.699283i \(0.246497\pi\)
\(318\) 0 0
\(319\) −821.977 −0.144269
\(320\) 0 0
\(321\) 18457.8 3.20939
\(322\) 0 0
\(323\) 11096.1 1.91147
\(324\) 0 0
\(325\) −10119.4 −1.72715
\(326\) 0 0
\(327\) −6172.50 −1.04385
\(328\) 0 0
\(329\) −748.024 −0.125349
\(330\) 0 0
\(331\) −6351.97 −1.05479 −0.527396 0.849620i \(-0.676832\pi\)
−0.527396 + 0.849620i \(0.676832\pi\)
\(332\) 0 0
\(333\) 8423.71 1.38624
\(334\) 0 0
\(335\) 2386.64 0.389242
\(336\) 0 0
\(337\) −1126.41 −0.182076 −0.0910381 0.995847i \(-0.529019\pi\)
−0.0910381 + 0.995847i \(0.529019\pi\)
\(338\) 0 0
\(339\) −11672.7 −1.87014
\(340\) 0 0
\(341\) 1702.28 0.270333
\(342\) 0 0
\(343\) 2231.65 0.351306
\(344\) 0 0
\(345\) 2873.16 0.448365
\(346\) 0 0
\(347\) −5208.09 −0.805720 −0.402860 0.915262i \(-0.631984\pi\)
−0.402860 + 0.915262i \(0.631984\pi\)
\(348\) 0 0
\(349\) 9644.73 1.47928 0.739642 0.673000i \(-0.234995\pi\)
0.739642 + 0.673000i \(0.234995\pi\)
\(350\) 0 0
\(351\) 36949.6 5.61888
\(352\) 0 0
\(353\) 3082.55 0.464780 0.232390 0.972623i \(-0.425345\pi\)
0.232390 + 0.972623i \(0.425345\pi\)
\(354\) 0 0
\(355\) −2166.71 −0.323935
\(356\) 0 0
\(357\) 3628.46 0.537923
\(358\) 0 0
\(359\) −5668.89 −0.833405 −0.416702 0.909043i \(-0.636814\pi\)
−0.416702 + 0.909043i \(0.636814\pi\)
\(360\) 0 0
\(361\) 3117.53 0.454517
\(362\) 0 0
\(363\) −5212.94 −0.753742
\(364\) 0 0
\(365\) −1041.20 −0.149312
\(366\) 0 0
\(367\) 12831.6 1.82507 0.912537 0.408994i \(-0.134120\pi\)
0.912537 + 0.408994i \(0.134120\pi\)
\(368\) 0 0
\(369\) −20141.0 −2.84147
\(370\) 0 0
\(371\) −1279.89 −0.179107
\(372\) 0 0
\(373\) 5541.65 0.769265 0.384632 0.923070i \(-0.374328\pi\)
0.384632 + 0.923070i \(0.374328\pi\)
\(374\) 0 0
\(375\) −6341.65 −0.873284
\(376\) 0 0
\(377\) −2486.46 −0.339680
\(378\) 0 0
\(379\) −11614.1 −1.57408 −0.787039 0.616903i \(-0.788387\pi\)
−0.787039 + 0.616903i \(0.788387\pi\)
\(380\) 0 0
\(381\) −16734.6 −2.25024
\(382\) 0 0
\(383\) −2733.89 −0.364739 −0.182370 0.983230i \(-0.558377\pi\)
−0.182370 + 0.983230i \(0.558377\pi\)
\(384\) 0 0
\(385\) −247.472 −0.0327593
\(386\) 0 0
\(387\) 17624.4 2.31498
\(388\) 0 0
\(389\) −8079.05 −1.05302 −0.526509 0.850169i \(-0.676499\pi\)
−0.526509 + 0.850169i \(0.676499\pi\)
\(390\) 0 0
\(391\) −12231.8 −1.58206
\(392\) 0 0
\(393\) 1929.73 0.247690
\(394\) 0 0
\(395\) −1599.04 −0.203687
\(396\) 0 0
\(397\) −4371.33 −0.552622 −0.276311 0.961068i \(-0.589112\pi\)
−0.276311 + 0.961068i \(0.589112\pi\)
\(398\) 0 0
\(399\) 3262.35 0.409328
\(400\) 0 0
\(401\) −15230.3 −1.89667 −0.948337 0.317265i \(-0.897236\pi\)
−0.948337 + 0.317265i \(0.897236\pi\)
\(402\) 0 0
\(403\) 5149.35 0.636494
\(404\) 0 0
\(405\) 6209.74 0.761888
\(406\) 0 0
\(407\) 3381.05 0.411775
\(408\) 0 0
\(409\) −382.195 −0.0462061 −0.0231031 0.999733i \(-0.507355\pi\)
−0.0231031 + 0.999733i \(0.507355\pi\)
\(410\) 0 0
\(411\) −3156.50 −0.378829
\(412\) 0 0
\(413\) 544.831 0.0649138
\(414\) 0 0
\(415\) 1333.73 0.157759
\(416\) 0 0
\(417\) 10835.6 1.27247
\(418\) 0 0
\(419\) 5109.88 0.595785 0.297893 0.954599i \(-0.403716\pi\)
0.297893 + 0.954599i \(0.403716\pi\)
\(420\) 0 0
\(421\) 9354.78 1.08295 0.541477 0.840715i \(-0.317865\pi\)
0.541477 + 0.840715i \(0.317865\pi\)
\(422\) 0 0
\(423\) 15979.1 1.83672
\(424\) 0 0
\(425\) 13111.5 1.49648
\(426\) 0 0
\(427\) −622.004 −0.0704939
\(428\) 0 0
\(429\) 24010.9 2.70224
\(430\) 0 0
\(431\) 3609.19 0.403361 0.201680 0.979451i \(-0.435360\pi\)
0.201680 + 0.979451i \(0.435360\pi\)
\(432\) 0 0
\(433\) −3781.22 −0.419662 −0.209831 0.977738i \(-0.567291\pi\)
−0.209831 + 0.977738i \(0.567291\pi\)
\(434\) 0 0
\(435\) −756.746 −0.0834097
\(436\) 0 0
\(437\) −10997.6 −1.20386
\(438\) 0 0
\(439\) 1990.19 0.216370 0.108185 0.994131i \(-0.465496\pi\)
0.108185 + 0.994131i \(0.465496\pi\)
\(440\) 0 0
\(441\) −23450.1 −2.53214
\(442\) 0 0
\(443\) 2859.89 0.306721 0.153361 0.988170i \(-0.450990\pi\)
0.153361 + 0.988170i \(0.450990\pi\)
\(444\) 0 0
\(445\) −2457.88 −0.261831
\(446\) 0 0
\(447\) 11799.4 1.24853
\(448\) 0 0
\(449\) −1177.30 −0.123742 −0.0618709 0.998084i \(-0.519707\pi\)
−0.0618709 + 0.998084i \(0.519707\pi\)
\(450\) 0 0
\(451\) −8084.07 −0.844044
\(452\) 0 0
\(453\) −22322.8 −2.31527
\(454\) 0 0
\(455\) −748.597 −0.0771314
\(456\) 0 0
\(457\) 427.683 0.0437772 0.0218886 0.999760i \(-0.493032\pi\)
0.0218886 + 0.999760i \(0.493032\pi\)
\(458\) 0 0
\(459\) −47874.9 −4.86843
\(460\) 0 0
\(461\) 12610.0 1.27399 0.636994 0.770869i \(-0.280178\pi\)
0.636994 + 0.770869i \(0.280178\pi\)
\(462\) 0 0
\(463\) 4584.31 0.460153 0.230076 0.973173i \(-0.426102\pi\)
0.230076 + 0.973173i \(0.426102\pi\)
\(464\) 0 0
\(465\) 1567.19 0.156294
\(466\) 0 0
\(467\) 315.535 0.0312660 0.0156330 0.999878i \(-0.495024\pi\)
0.0156330 + 0.999878i \(0.495024\pi\)
\(468\) 0 0
\(469\) −2987.28 −0.294115
\(470\) 0 0
\(471\) −25612.6 −2.50566
\(472\) 0 0
\(473\) 7073.95 0.687654
\(474\) 0 0
\(475\) 11788.6 1.13873
\(476\) 0 0
\(477\) 27340.7 2.62441
\(478\) 0 0
\(479\) 9507.29 0.906888 0.453444 0.891285i \(-0.350195\pi\)
0.453444 + 0.891285i \(0.350195\pi\)
\(480\) 0 0
\(481\) 10227.6 0.969518
\(482\) 0 0
\(483\) −3596.24 −0.338788
\(484\) 0 0
\(485\) 32.3740 0.00303098
\(486\) 0 0
\(487\) −2848.85 −0.265079 −0.132540 0.991178i \(-0.542313\pi\)
−0.132540 + 0.991178i \(0.542313\pi\)
\(488\) 0 0
\(489\) −4921.92 −0.455168
\(490\) 0 0
\(491\) −16581.9 −1.52409 −0.762046 0.647522i \(-0.775805\pi\)
−0.762046 + 0.647522i \(0.775805\pi\)
\(492\) 0 0
\(493\) 3221.66 0.294313
\(494\) 0 0
\(495\) 5286.44 0.480015
\(496\) 0 0
\(497\) 2711.99 0.244768
\(498\) 0 0
\(499\) 14103.9 1.26528 0.632642 0.774444i \(-0.281970\pi\)
0.632642 + 0.774444i \(0.281970\pi\)
\(500\) 0 0
\(501\) −5448.34 −0.485856
\(502\) 0 0
\(503\) 1773.92 0.157247 0.0786236 0.996904i \(-0.474947\pi\)
0.0786236 + 0.996904i \(0.474947\pi\)
\(504\) 0 0
\(505\) −3078.08 −0.271233
\(506\) 0 0
\(507\) 50925.8 4.46094
\(508\) 0 0
\(509\) −14432.9 −1.25683 −0.628414 0.777879i \(-0.716296\pi\)
−0.628414 + 0.777879i \(0.716296\pi\)
\(510\) 0 0
\(511\) 1303.24 0.112822
\(512\) 0 0
\(513\) −43044.4 −3.70459
\(514\) 0 0
\(515\) 572.782 0.0490093
\(516\) 0 0
\(517\) 6413.58 0.545588
\(518\) 0 0
\(519\) 25094.3 2.12238
\(520\) 0 0
\(521\) 1855.69 0.156044 0.0780222 0.996952i \(-0.475140\pi\)
0.0780222 + 0.996952i \(0.475140\pi\)
\(522\) 0 0
\(523\) −10294.0 −0.860660 −0.430330 0.902672i \(-0.641603\pi\)
−0.430330 + 0.902672i \(0.641603\pi\)
\(524\) 0 0
\(525\) 3854.90 0.320460
\(526\) 0 0
\(527\) −6671.91 −0.551486
\(528\) 0 0
\(529\) −43.8612 −0.00360493
\(530\) 0 0
\(531\) −11638.5 −0.951167
\(532\) 0 0
\(533\) −24454.1 −1.98729
\(534\) 0 0
\(535\) 4934.05 0.398725
\(536\) 0 0
\(537\) −7307.55 −0.587233
\(538\) 0 0
\(539\) −9412.25 −0.752160
\(540\) 0 0
\(541\) −4340.91 −0.344973 −0.172486 0.985012i \(-0.555180\pi\)
−0.172486 + 0.985012i \(0.555180\pi\)
\(542\) 0 0
\(543\) −40782.3 −3.22309
\(544\) 0 0
\(545\) −1650.00 −0.129685
\(546\) 0 0
\(547\) 18868.8 1.47490 0.737452 0.675399i \(-0.236029\pi\)
0.737452 + 0.675399i \(0.236029\pi\)
\(548\) 0 0
\(549\) 13287.1 1.03293
\(550\) 0 0
\(551\) 2896.60 0.223955
\(552\) 0 0
\(553\) 2001.46 0.153907
\(554\) 0 0
\(555\) 3112.74 0.238069
\(556\) 0 0
\(557\) 6986.62 0.531477 0.265738 0.964045i \(-0.414384\pi\)
0.265738 + 0.964045i \(0.414384\pi\)
\(558\) 0 0
\(559\) 21398.5 1.61907
\(560\) 0 0
\(561\) −31110.5 −2.34133
\(562\) 0 0
\(563\) 19006.3 1.42277 0.711386 0.702802i \(-0.248068\pi\)
0.711386 + 0.702802i \(0.248068\pi\)
\(564\) 0 0
\(565\) −3120.30 −0.232340
\(566\) 0 0
\(567\) −7772.53 −0.575689
\(568\) 0 0
\(569\) −5475.13 −0.403391 −0.201696 0.979448i \(-0.564645\pi\)
−0.201696 + 0.979448i \(0.564645\pi\)
\(570\) 0 0
\(571\) −1264.86 −0.0927016 −0.0463508 0.998925i \(-0.514759\pi\)
−0.0463508 + 0.998925i \(0.514759\pi\)
\(572\) 0 0
\(573\) 16107.4 1.17434
\(574\) 0 0
\(575\) −12995.1 −0.942492
\(576\) 0 0
\(577\) 27670.3 1.99642 0.998208 0.0598463i \(-0.0190611\pi\)
0.998208 + 0.0598463i \(0.0190611\pi\)
\(578\) 0 0
\(579\) −2081.12 −0.149375
\(580\) 0 0
\(581\) −1669.38 −0.119204
\(582\) 0 0
\(583\) 10973.8 0.779569
\(584\) 0 0
\(585\) 15991.3 1.13019
\(586\) 0 0
\(587\) −6872.78 −0.483254 −0.241627 0.970369i \(-0.577681\pi\)
−0.241627 + 0.970369i \(0.577681\pi\)
\(588\) 0 0
\(589\) −5998.72 −0.419648
\(590\) 0 0
\(591\) −43654.2 −3.03840
\(592\) 0 0
\(593\) −4896.40 −0.339074 −0.169537 0.985524i \(-0.554227\pi\)
−0.169537 + 0.985524i \(0.554227\pi\)
\(594\) 0 0
\(595\) 969.943 0.0668299
\(596\) 0 0
\(597\) 14193.1 0.973006
\(598\) 0 0
\(599\) 11189.9 0.763282 0.381641 0.924311i \(-0.375359\pi\)
0.381641 + 0.924311i \(0.375359\pi\)
\(600\) 0 0
\(601\) −6271.18 −0.425635 −0.212818 0.977092i \(-0.568264\pi\)
−0.212818 + 0.977092i \(0.568264\pi\)
\(602\) 0 0
\(603\) 63813.5 4.30960
\(604\) 0 0
\(605\) −1393.50 −0.0936426
\(606\) 0 0
\(607\) 22094.6 1.47742 0.738710 0.674024i \(-0.235436\pi\)
0.738710 + 0.674024i \(0.235436\pi\)
\(608\) 0 0
\(609\) 947.194 0.0630250
\(610\) 0 0
\(611\) 19400.9 1.28458
\(612\) 0 0
\(613\) 26196.1 1.72602 0.863009 0.505189i \(-0.168577\pi\)
0.863009 + 0.505189i \(0.168577\pi\)
\(614\) 0 0
\(615\) −7442.53 −0.487987
\(616\) 0 0
\(617\) 1836.57 0.119834 0.0599169 0.998203i \(-0.480916\pi\)
0.0599169 + 0.998203i \(0.480916\pi\)
\(618\) 0 0
\(619\) 4634.79 0.300950 0.150475 0.988614i \(-0.451920\pi\)
0.150475 + 0.988614i \(0.451920\pi\)
\(620\) 0 0
\(621\) 47449.8 3.06617
\(622\) 0 0
\(623\) 3076.45 0.197841
\(624\) 0 0
\(625\) 13057.8 0.835702
\(626\) 0 0
\(627\) −27971.5 −1.78162
\(628\) 0 0
\(629\) −13251.7 −0.840031
\(630\) 0 0
\(631\) 16556.8 1.04456 0.522278 0.852775i \(-0.325082\pi\)
0.522278 + 0.852775i \(0.325082\pi\)
\(632\) 0 0
\(633\) −35284.6 −2.21554
\(634\) 0 0
\(635\) −4473.42 −0.279563
\(636\) 0 0
\(637\) −28471.8 −1.77095
\(638\) 0 0
\(639\) −57932.9 −3.58653
\(640\) 0 0
\(641\) −1437.66 −0.0885866 −0.0442933 0.999019i \(-0.514104\pi\)
−0.0442933 + 0.999019i \(0.514104\pi\)
\(642\) 0 0
\(643\) 9284.85 0.569454 0.284727 0.958609i \(-0.408097\pi\)
0.284727 + 0.958609i \(0.408097\pi\)
\(644\) 0 0
\(645\) 6512.57 0.397569
\(646\) 0 0
\(647\) −14249.5 −0.865852 −0.432926 0.901430i \(-0.642519\pi\)
−0.432926 + 0.901430i \(0.642519\pi\)
\(648\) 0 0
\(649\) −4671.40 −0.282540
\(650\) 0 0
\(651\) −1961.60 −0.118097
\(652\) 0 0
\(653\) −11031.1 −0.661071 −0.330535 0.943794i \(-0.607229\pi\)
−0.330535 + 0.943794i \(0.607229\pi\)
\(654\) 0 0
\(655\) 515.847 0.0307723
\(656\) 0 0
\(657\) −27839.4 −1.65315
\(658\) 0 0
\(659\) 23429.9 1.38497 0.692487 0.721430i \(-0.256515\pi\)
0.692487 + 0.721430i \(0.256515\pi\)
\(660\) 0 0
\(661\) 20939.6 1.23216 0.616079 0.787684i \(-0.288720\pi\)
0.616079 + 0.787684i \(0.288720\pi\)
\(662\) 0 0
\(663\) −94108.6 −5.51263
\(664\) 0 0
\(665\) 872.077 0.0508537
\(666\) 0 0
\(667\) −3193.05 −0.185360
\(668\) 0 0
\(669\) 63063.2 3.64449
\(670\) 0 0
\(671\) 5333.08 0.306828
\(672\) 0 0
\(673\) 10460.4 0.599138 0.299569 0.954075i \(-0.403157\pi\)
0.299569 + 0.954075i \(0.403157\pi\)
\(674\) 0 0
\(675\) −50862.6 −2.90030
\(676\) 0 0
\(677\) −1898.28 −0.107765 −0.0538824 0.998547i \(-0.517160\pi\)
−0.0538824 + 0.998547i \(0.517160\pi\)
\(678\) 0 0
\(679\) −40.5214 −0.00229024
\(680\) 0 0
\(681\) −45108.6 −2.53827
\(682\) 0 0
\(683\) 6001.01 0.336197 0.168098 0.985770i \(-0.446237\pi\)
0.168098 + 0.985770i \(0.446237\pi\)
\(684\) 0 0
\(685\) −843.781 −0.0470646
\(686\) 0 0
\(687\) 50233.0 2.78968
\(688\) 0 0
\(689\) 33195.5 1.83548
\(690\) 0 0
\(691\) −20066.6 −1.10473 −0.552367 0.833601i \(-0.686275\pi\)
−0.552367 + 0.833601i \(0.686275\pi\)
\(692\) 0 0
\(693\) −6616.85 −0.362704
\(694\) 0 0
\(695\) 2896.51 0.158088
\(696\) 0 0
\(697\) 31684.7 1.72187
\(698\) 0 0
\(699\) −56748.6 −3.07071
\(700\) 0 0
\(701\) −7897.60 −0.425518 −0.212759 0.977105i \(-0.568245\pi\)
−0.212759 + 0.977105i \(0.568245\pi\)
\(702\) 0 0
\(703\) −11914.6 −0.639215
\(704\) 0 0
\(705\) 5904.61 0.315433
\(706\) 0 0
\(707\) 3852.72 0.204946
\(708\) 0 0
\(709\) −8805.89 −0.466449 −0.233224 0.972423i \(-0.574928\pi\)
−0.233224 + 0.972423i \(0.574928\pi\)
\(710\) 0 0
\(711\) −42754.7 −2.25517
\(712\) 0 0
\(713\) 6612.66 0.347330
\(714\) 0 0
\(715\) 6418.49 0.335718
\(716\) 0 0
\(717\) −54875.8 −2.85826
\(718\) 0 0
\(719\) 9315.54 0.483186 0.241593 0.970378i \(-0.422330\pi\)
0.241593 + 0.970378i \(0.422330\pi\)
\(720\) 0 0
\(721\) −716.932 −0.0370318
\(722\) 0 0
\(723\) −21959.9 −1.12959
\(724\) 0 0
\(725\) 3422.71 0.175333
\(726\) 0 0
\(727\) 12393.7 0.632264 0.316132 0.948715i \(-0.397616\pi\)
0.316132 + 0.948715i \(0.397616\pi\)
\(728\) 0 0
\(729\) 51072.7 2.59476
\(730\) 0 0
\(731\) −27725.6 −1.40283
\(732\) 0 0
\(733\) −25018.2 −1.26067 −0.630333 0.776325i \(-0.717082\pi\)
−0.630333 + 0.776325i \(0.717082\pi\)
\(734\) 0 0
\(735\) −8665.31 −0.434864
\(736\) 0 0
\(737\) 25613.0 1.28015
\(738\) 0 0
\(739\) −2887.04 −0.143710 −0.0718548 0.997415i \(-0.522892\pi\)
−0.0718548 + 0.997415i \(0.522892\pi\)
\(740\) 0 0
\(741\) −84613.1 −4.19479
\(742\) 0 0
\(743\) −34004.2 −1.67899 −0.839497 0.543365i \(-0.817150\pi\)
−0.839497 + 0.543365i \(0.817150\pi\)
\(744\) 0 0
\(745\) 3154.17 0.155114
\(746\) 0 0
\(747\) 35660.9 1.74667
\(748\) 0 0
\(749\) −6175.79 −0.301279
\(750\) 0 0
\(751\) 37460.2 1.82016 0.910081 0.414431i \(-0.136019\pi\)
0.910081 + 0.414431i \(0.136019\pi\)
\(752\) 0 0
\(753\) −49140.7 −2.37820
\(754\) 0 0
\(755\) −5967.22 −0.287641
\(756\) 0 0
\(757\) 30483.5 1.46359 0.731797 0.681523i \(-0.238682\pi\)
0.731797 + 0.681523i \(0.238682\pi\)
\(758\) 0 0
\(759\) 30834.2 1.47459
\(760\) 0 0
\(761\) 19511.6 0.929430 0.464715 0.885460i \(-0.346157\pi\)
0.464715 + 0.885460i \(0.346157\pi\)
\(762\) 0 0
\(763\) 2065.25 0.0979912
\(764\) 0 0
\(765\) −20719.7 −0.979244
\(766\) 0 0
\(767\) −14130.9 −0.665236
\(768\) 0 0
\(769\) −17422.4 −0.816995 −0.408497 0.912760i \(-0.633947\pi\)
−0.408497 + 0.912760i \(0.633947\pi\)
\(770\) 0 0
\(771\) 148.990 0.00695947
\(772\) 0 0
\(773\) −3685.78 −0.171498 −0.0857491 0.996317i \(-0.527328\pi\)
−0.0857491 + 0.996317i \(0.527328\pi\)
\(774\) 0 0
\(775\) −7088.28 −0.328540
\(776\) 0 0
\(777\) −3896.11 −0.179887
\(778\) 0 0
\(779\) 28487.8 1.31024
\(780\) 0 0
\(781\) −23252.7 −1.06536
\(782\) 0 0
\(783\) −12497.5 −0.570403
\(784\) 0 0
\(785\) −6846.65 −0.311296
\(786\) 0 0
\(787\) 7764.73 0.351694 0.175847 0.984418i \(-0.443734\pi\)
0.175847 + 0.984418i \(0.443734\pi\)
\(788\) 0 0
\(789\) −31994.9 −1.44366
\(790\) 0 0
\(791\) 3905.58 0.175558
\(792\) 0 0
\(793\) 16132.4 0.722421
\(794\) 0 0
\(795\) 10103.0 0.450710
\(796\) 0 0
\(797\) 22754.0 1.01128 0.505639 0.862745i \(-0.331257\pi\)
0.505639 + 0.862745i \(0.331257\pi\)
\(798\) 0 0
\(799\) −25137.4 −1.11301
\(800\) 0 0
\(801\) −65718.2 −2.89893
\(802\) 0 0
\(803\) −11174.0 −0.491061
\(804\) 0 0
\(805\) −961.330 −0.0420900
\(806\) 0 0
\(807\) 40976.0 1.78739
\(808\) 0 0
\(809\) 9866.29 0.428776 0.214388 0.976749i \(-0.431224\pi\)
0.214388 + 0.976749i \(0.431224\pi\)
\(810\) 0 0
\(811\) −12456.6 −0.539346 −0.269673 0.962952i \(-0.586916\pi\)
−0.269673 + 0.962952i \(0.586916\pi\)
\(812\) 0 0
\(813\) −12937.6 −0.558108
\(814\) 0 0
\(815\) −1315.71 −0.0565487
\(816\) 0 0
\(817\) −24928.1 −1.06747
\(818\) 0 0
\(819\) −20015.8 −0.853980
\(820\) 0 0
\(821\) 22347.4 0.949976 0.474988 0.879992i \(-0.342452\pi\)
0.474988 + 0.879992i \(0.342452\pi\)
\(822\) 0 0
\(823\) 37377.5 1.58311 0.791554 0.611100i \(-0.209273\pi\)
0.791554 + 0.611100i \(0.209273\pi\)
\(824\) 0 0
\(825\) −33052.0 −1.39482
\(826\) 0 0
\(827\) 21218.3 0.892182 0.446091 0.894988i \(-0.352816\pi\)
0.446091 + 0.894988i \(0.352816\pi\)
\(828\) 0 0
\(829\) 12922.1 0.541378 0.270689 0.962667i \(-0.412749\pi\)
0.270689 + 0.962667i \(0.412749\pi\)
\(830\) 0 0
\(831\) 66659.1 2.78265
\(832\) 0 0
\(833\) 36890.4 1.53443
\(834\) 0 0
\(835\) −1456.42 −0.0603613
\(836\) 0 0
\(837\) 25881.9 1.06883
\(838\) 0 0
\(839\) −25253.6 −1.03915 −0.519577 0.854423i \(-0.673911\pi\)
−0.519577 + 0.854423i \(0.673911\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 28615.4 1.16912
\(844\) 0 0
\(845\) 13613.3 0.554213
\(846\) 0 0
\(847\) 1744.19 0.0707571
\(848\) 0 0
\(849\) 84746.5 3.42579
\(850\) 0 0
\(851\) 13134.0 0.529058
\(852\) 0 0
\(853\) 23958.9 0.961707 0.480854 0.876801i \(-0.340327\pi\)
0.480854 + 0.876801i \(0.340327\pi\)
\(854\) 0 0
\(855\) −18629.1 −0.745147
\(856\) 0 0
\(857\) −1289.87 −0.0514133 −0.0257066 0.999670i \(-0.508184\pi\)
−0.0257066 + 0.999670i \(0.508184\pi\)
\(858\) 0 0
\(859\) −12961.8 −0.514843 −0.257422 0.966299i \(-0.582873\pi\)
−0.257422 + 0.966299i \(0.582873\pi\)
\(860\) 0 0
\(861\) 9315.57 0.368727
\(862\) 0 0
\(863\) −39133.0 −1.54357 −0.771785 0.635883i \(-0.780636\pi\)
−0.771785 + 0.635883i \(0.780636\pi\)
\(864\) 0 0
\(865\) 6708.09 0.263678
\(866\) 0 0
\(867\) 73393.5 2.87494
\(868\) 0 0
\(869\) −17160.6 −0.669888
\(870\) 0 0
\(871\) 77478.8 3.01409
\(872\) 0 0
\(873\) 865.609 0.0335583
\(874\) 0 0
\(875\) 2121.85 0.0819790
\(876\) 0 0
\(877\) −12957.8 −0.498921 −0.249460 0.968385i \(-0.580253\pi\)
−0.249460 + 0.968385i \(0.580253\pi\)
\(878\) 0 0
\(879\) −81056.8 −3.11033
\(880\) 0 0
\(881\) −33767.4 −1.29132 −0.645659 0.763626i \(-0.723417\pi\)
−0.645659 + 0.763626i \(0.723417\pi\)
\(882\) 0 0
\(883\) 34290.1 1.30686 0.653428 0.756989i \(-0.273330\pi\)
0.653428 + 0.756989i \(0.273330\pi\)
\(884\) 0 0
\(885\) −4300.68 −0.163351
\(886\) 0 0
\(887\) −10538.1 −0.398911 −0.199456 0.979907i \(-0.563917\pi\)
−0.199456 + 0.979907i \(0.563917\pi\)
\(888\) 0 0
\(889\) 5599.23 0.211240
\(890\) 0 0
\(891\) 66641.9 2.50571
\(892\) 0 0
\(893\) −22601.1 −0.846938
\(894\) 0 0
\(895\) −1953.42 −0.0729560
\(896\) 0 0
\(897\) 93272.9 3.47190
\(898\) 0 0
\(899\) −1741.67 −0.0646141
\(900\) 0 0
\(901\) −43010.8 −1.59034
\(902\) 0 0
\(903\) −8151.57 −0.300407
\(904\) 0 0
\(905\) −10901.7 −0.400427
\(906\) 0 0
\(907\) −25835.0 −0.945797 −0.472899 0.881117i \(-0.656792\pi\)
−0.472899 + 0.881117i \(0.656792\pi\)
\(908\) 0 0
\(909\) −82300.9 −3.00302
\(910\) 0 0
\(911\) −6732.30 −0.244842 −0.122421 0.992478i \(-0.539066\pi\)
−0.122421 + 0.992478i \(0.539066\pi\)
\(912\) 0 0
\(913\) 14313.3 0.518841
\(914\) 0 0
\(915\) 4909.86 0.177393
\(916\) 0 0
\(917\) −645.669 −0.0232518
\(918\) 0 0
\(919\) −16257.6 −0.583557 −0.291778 0.956486i \(-0.594247\pi\)
−0.291778 + 0.956486i \(0.594247\pi\)
\(920\) 0 0
\(921\) 24764.1 0.886000
\(922\) 0 0
\(923\) −70338.9 −2.50838
\(924\) 0 0
\(925\) −14078.7 −0.500437
\(926\) 0 0
\(927\) 15314.9 0.542619
\(928\) 0 0
\(929\) −27354.8 −0.966073 −0.483037 0.875600i \(-0.660466\pi\)
−0.483037 + 0.875600i \(0.660466\pi\)
\(930\) 0 0
\(931\) 33168.2 1.16761
\(932\) 0 0
\(933\) 64235.9 2.25401
\(934\) 0 0
\(935\) −8316.32 −0.290880
\(936\) 0 0
\(937\) 38897.4 1.35616 0.678080 0.734988i \(-0.262812\pi\)
0.678080 + 0.734988i \(0.262812\pi\)
\(938\) 0 0
\(939\) 71575.4 2.48751
\(940\) 0 0
\(941\) 36998.7 1.28175 0.640874 0.767646i \(-0.278572\pi\)
0.640874 + 0.767646i \(0.278572\pi\)
\(942\) 0 0
\(943\) −31403.4 −1.08445
\(944\) 0 0
\(945\) −3762.63 −0.129522
\(946\) 0 0
\(947\) −37594.0 −1.29001 −0.645006 0.764177i \(-0.723145\pi\)
−0.645006 + 0.764177i \(0.723145\pi\)
\(948\) 0 0
\(949\) −33801.1 −1.15620
\(950\) 0 0
\(951\) 79725.0 2.71847
\(952\) 0 0
\(953\) 49341.0 1.67714 0.838568 0.544797i \(-0.183393\pi\)
0.838568 + 0.544797i \(0.183393\pi\)
\(954\) 0 0
\(955\) 4305.75 0.145896
\(956\) 0 0
\(957\) −8121.27 −0.274319
\(958\) 0 0
\(959\) 1056.13 0.0355624
\(960\) 0 0
\(961\) −26184.1 −0.878926
\(962\) 0 0
\(963\) 131926. 4.41458
\(964\) 0 0
\(965\) −556.315 −0.0185579
\(966\) 0 0
\(967\) −44067.6 −1.46548 −0.732740 0.680509i \(-0.761759\pi\)
−0.732740 + 0.680509i \(0.761759\pi\)
\(968\) 0 0
\(969\) 109632. 3.63454
\(970\) 0 0
\(971\) 51385.3 1.69828 0.849141 0.528166i \(-0.177120\pi\)
0.849141 + 0.528166i \(0.177120\pi\)
\(972\) 0 0
\(973\) −3625.46 −0.119452
\(974\) 0 0
\(975\) −99981.6 −3.28407
\(976\) 0 0
\(977\) 27297.1 0.893871 0.446935 0.894566i \(-0.352515\pi\)
0.446935 + 0.894566i \(0.352515\pi\)
\(978\) 0 0
\(979\) −26377.5 −0.861113
\(980\) 0 0
\(981\) −44117.4 −1.43584
\(982\) 0 0
\(983\) 3876.96 0.125794 0.0628972 0.998020i \(-0.479966\pi\)
0.0628972 + 0.998020i \(0.479966\pi\)
\(984\) 0 0
\(985\) −11669.4 −0.377481
\(986\) 0 0
\(987\) −7390.60 −0.238344
\(988\) 0 0
\(989\) 27479.4 0.883514
\(990\) 0 0
\(991\) −11942.6 −0.382814 −0.191407 0.981511i \(-0.561305\pi\)
−0.191407 + 0.981511i \(0.561305\pi\)
\(992\) 0 0
\(993\) −62758.6 −2.00562
\(994\) 0 0
\(995\) 3794.03 0.120883
\(996\) 0 0
\(997\) 60556.0 1.92360 0.961800 0.273755i \(-0.0882656\pi\)
0.961800 + 0.273755i \(0.0882656\pi\)
\(998\) 0 0
\(999\) 51406.3 1.62805
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 464.4.a.n.1.6 6
4.3 odd 2 232.4.a.e.1.1 6
8.3 odd 2 1856.4.a.bd.1.6 6
8.5 even 2 1856.4.a.bc.1.1 6
12.11 even 2 2088.4.a.l.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.e.1.1 6 4.3 odd 2
464.4.a.n.1.6 6 1.1 even 1 trivial
1856.4.a.bc.1.1 6 8.5 even 2
1856.4.a.bd.1.6 6 8.3 odd 2
2088.4.a.l.1.3 6 12.11 even 2