Properties

Label 2112.4.a.bl.1.2
Level $2112$
Weight $4$
Character 2112.1
Self dual yes
Analytic conductor $124.612$
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] = [2112,4,Mod(1,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2112.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2112.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: \(124.612033932\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 264)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2112.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +7.12311 q^{5} +9.12311 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +7.12311 q^{5} +9.12311 q^{7} +9.00000 q^{9} -11.0000 q^{11} -69.0928 q^{13} +21.3693 q^{15} -126.216 q^{17} +121.693 q^{19} +27.3693 q^{21} +2.60037 q^{23} -74.2614 q^{25} +27.0000 q^{27} +88.7993 q^{29} +132.924 q^{31} -33.0000 q^{33} +64.9848 q^{35} +411.848 q^{37} -207.278 q^{39} -283.538 q^{41} -364.371 q^{43} +64.1080 q^{45} -342.540 q^{47} -259.769 q^{49} -378.648 q^{51} +343.555 q^{53} -78.3542 q^{55} +365.080 q^{57} -493.140 q^{59} -619.589 q^{61} +82.1080 q^{63} -492.155 q^{65} +452.742 q^{67} +7.80112 q^{69} -698.756 q^{71} -793.576 q^{73} -222.784 q^{75} -100.354 q^{77} -340.941 q^{79} +81.0000 q^{81} -1122.10 q^{83} -899.049 q^{85} +266.398 q^{87} -185.318 q^{89} -630.341 q^{91} +398.773 q^{93} +866.833 q^{95} +1140.68 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 6 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 6 q^{5} + 10 q^{7} + 18 q^{9} - 22 q^{11} + 2 q^{13} + 18 q^{15} - 104 q^{17} - 4 q^{19} + 30 q^{21} - 102 q^{23} - 198 q^{25} + 54 q^{27} + 392 q^{29} - 64 q^{31} - 66 q^{33} + 64 q^{35} + 164 q^{37} + 6 q^{39} - 732 q^{41} - 168 q^{43} + 54 q^{45} - 314 q^{47} - 602 q^{49} - 312 q^{51} + 382 q^{53} - 66 q^{55} - 12 q^{57} - 508 q^{59} + 6 q^{61} + 90 q^{63} - 572 q^{65} - 216 q^{67} - 306 q^{69} - 878 q^{71} + 260 q^{73} - 594 q^{75} - 110 q^{77} + 118 q^{79} + 162 q^{81} - 496 q^{83} - 924 q^{85} + 1176 q^{87} - 1756 q^{89} - 568 q^{91} - 192 q^{93} + 1008 q^{95} + 1968 q^{97} - 198 q^{99}+O(q^{100}) \) 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
\(5\) 7.12311 0.637110 0.318555 0.947904i \(-0.396802\pi\)
0.318555 + 0.947904i \(0.396802\pi\)
\(6\) 0 0
\(7\) 9.12311 0.492601 0.246301 0.969193i \(-0.420785\pi\)
0.246301 + 0.969193i \(0.420785\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −69.0928 −1.47407 −0.737034 0.675855i \(-0.763774\pi\)
−0.737034 + 0.675855i \(0.763774\pi\)
\(14\) 0 0
\(15\) 21.3693 0.367836
\(16\) 0 0
\(17\) −126.216 −1.80070 −0.900349 0.435169i \(-0.856688\pi\)
−0.900349 + 0.435169i \(0.856688\pi\)
\(18\) 0 0
\(19\) 121.693 1.46939 0.734693 0.678400i \(-0.237326\pi\)
0.734693 + 0.678400i \(0.237326\pi\)
\(20\) 0 0
\(21\) 27.3693 0.284404
\(22\) 0 0
\(23\) 2.60037 0.0235746 0.0117873 0.999931i \(-0.496248\pi\)
0.0117873 + 0.999931i \(0.496248\pi\)
\(24\) 0 0
\(25\) −74.2614 −0.594091
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 88.7993 0.568607 0.284304 0.958734i \(-0.408238\pi\)
0.284304 + 0.958734i \(0.408238\pi\)
\(30\) 0 0
\(31\) 132.924 0.770126 0.385063 0.922890i \(-0.374180\pi\)
0.385063 + 0.922890i \(0.374180\pi\)
\(32\) 0 0
\(33\) −33.0000 −0.174078
\(34\) 0 0
\(35\) 64.9848 0.313841
\(36\) 0 0
\(37\) 411.848 1.82993 0.914966 0.403531i \(-0.132217\pi\)
0.914966 + 0.403531i \(0.132217\pi\)
\(38\) 0 0
\(39\) −207.278 −0.851054
\(40\) 0 0
\(41\) −283.538 −1.08003 −0.540014 0.841656i \(-0.681581\pi\)
−0.540014 + 0.841656i \(0.681581\pi\)
\(42\) 0 0
\(43\) −364.371 −1.29223 −0.646117 0.763238i \(-0.723608\pi\)
−0.646117 + 0.763238i \(0.723608\pi\)
\(44\) 0 0
\(45\) 64.1080 0.212370
\(46\) 0 0
\(47\) −342.540 −1.06308 −0.531538 0.847035i \(-0.678386\pi\)
−0.531538 + 0.847035i \(0.678386\pi\)
\(48\) 0 0
\(49\) −259.769 −0.757344
\(50\) 0 0
\(51\) −378.648 −1.03963
\(52\) 0 0
\(53\) 343.555 0.890394 0.445197 0.895433i \(-0.353134\pi\)
0.445197 + 0.895433i \(0.353134\pi\)
\(54\) 0 0
\(55\) −78.3542 −0.192096
\(56\) 0 0
\(57\) 365.080 0.848350
\(58\) 0 0
\(59\) −493.140 −1.08816 −0.544079 0.839034i \(-0.683121\pi\)
−0.544079 + 0.839034i \(0.683121\pi\)
\(60\) 0 0
\(61\) −619.589 −1.30050 −0.650248 0.759722i \(-0.725335\pi\)
−0.650248 + 0.759722i \(0.725335\pi\)
\(62\) 0 0
\(63\) 82.1080 0.164200
\(64\) 0 0
\(65\) −492.155 −0.939144
\(66\) 0 0
\(67\) 452.742 0.825542 0.412771 0.910835i \(-0.364561\pi\)
0.412771 + 0.910835i \(0.364561\pi\)
\(68\) 0 0
\(69\) 7.80112 0.0136108
\(70\) 0 0
\(71\) −698.756 −1.16799 −0.583993 0.811759i \(-0.698511\pi\)
−0.583993 + 0.811759i \(0.698511\pi\)
\(72\) 0 0
\(73\) −793.576 −1.27234 −0.636171 0.771548i \(-0.719483\pi\)
−0.636171 + 0.771548i \(0.719483\pi\)
\(74\) 0 0
\(75\) −222.784 −0.342999
\(76\) 0 0
\(77\) −100.354 −0.148525
\(78\) 0 0
\(79\) −340.941 −0.485556 −0.242778 0.970082i \(-0.578059\pi\)
−0.242778 + 0.970082i \(0.578059\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1122.10 −1.48393 −0.741966 0.670438i \(-0.766106\pi\)
−0.741966 + 0.670438i \(0.766106\pi\)
\(84\) 0 0
\(85\) −899.049 −1.14724
\(86\) 0 0
\(87\) 266.398 0.328285
\(88\) 0 0
\(89\) −185.318 −0.220716 −0.110358 0.993892i \(-0.535200\pi\)
−0.110358 + 0.993892i \(0.535200\pi\)
\(90\) 0 0
\(91\) −630.341 −0.726128
\(92\) 0 0
\(93\) 398.773 0.444632
\(94\) 0 0
\(95\) 866.833 0.936160
\(96\) 0 0
\(97\) 1140.68 1.19400 0.597002 0.802240i \(-0.296359\pi\)
0.597002 + 0.802240i \(0.296359\pi\)
\(98\) 0 0
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 1593.85 1.57024 0.785120 0.619344i \(-0.212601\pi\)
0.785120 + 0.619344i \(0.212601\pi\)
\(102\) 0 0
\(103\) 1068.65 1.02230 0.511150 0.859491i \(-0.329220\pi\)
0.511150 + 0.859491i \(0.329220\pi\)
\(104\) 0 0
\(105\) 194.955 0.181196
\(106\) 0 0
\(107\) 884.068 0.798748 0.399374 0.916788i \(-0.369227\pi\)
0.399374 + 0.916788i \(0.369227\pi\)
\(108\) 0 0
\(109\) −1894.82 −1.66505 −0.832527 0.553985i \(-0.813107\pi\)
−0.832527 + 0.553985i \(0.813107\pi\)
\(110\) 0 0
\(111\) 1235.55 1.05651
\(112\) 0 0
\(113\) −1067.36 −0.888570 −0.444285 0.895886i \(-0.646542\pi\)
−0.444285 + 0.895886i \(0.646542\pi\)
\(114\) 0 0
\(115\) 18.5227 0.0150196
\(116\) 0 0
\(117\) −621.835 −0.491356
\(118\) 0 0
\(119\) −1151.48 −0.887026
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −850.614 −0.623555
\(124\) 0 0
\(125\) −1419.36 −1.01561
\(126\) 0 0
\(127\) 128.968 0.0901106 0.0450553 0.998984i \(-0.485654\pi\)
0.0450553 + 0.998984i \(0.485654\pi\)
\(128\) 0 0
\(129\) −1093.11 −0.746072
\(130\) 0 0
\(131\) 391.454 0.261080 0.130540 0.991443i \(-0.458329\pi\)
0.130540 + 0.991443i \(0.458329\pi\)
\(132\) 0 0
\(133\) 1110.22 0.723821
\(134\) 0 0
\(135\) 192.324 0.122612
\(136\) 0 0
\(137\) 436.746 0.272363 0.136182 0.990684i \(-0.456517\pi\)
0.136182 + 0.990684i \(0.456517\pi\)
\(138\) 0 0
\(139\) −2030.62 −1.23910 −0.619549 0.784958i \(-0.712685\pi\)
−0.619549 + 0.784958i \(0.712685\pi\)
\(140\) 0 0
\(141\) −1027.62 −0.613767
\(142\) 0 0
\(143\) 760.021 0.444448
\(144\) 0 0
\(145\) 632.526 0.362265
\(146\) 0 0
\(147\) −779.307 −0.437253
\(148\) 0 0
\(149\) 2949.15 1.62150 0.810751 0.585391i \(-0.199059\pi\)
0.810751 + 0.585391i \(0.199059\pi\)
\(150\) 0 0
\(151\) −1138.47 −0.613560 −0.306780 0.951781i \(-0.599252\pi\)
−0.306780 + 0.951781i \(0.599252\pi\)
\(152\) 0 0
\(153\) −1135.94 −0.600232
\(154\) 0 0
\(155\) 946.833 0.490655
\(156\) 0 0
\(157\) 513.549 0.261055 0.130528 0.991445i \(-0.458333\pi\)
0.130528 + 0.991445i \(0.458333\pi\)
\(158\) 0 0
\(159\) 1030.66 0.514069
\(160\) 0 0
\(161\) 23.7235 0.0116129
\(162\) 0 0
\(163\) −508.864 −0.244523 −0.122262 0.992498i \(-0.539015\pi\)
−0.122262 + 0.992498i \(0.539015\pi\)
\(164\) 0 0
\(165\) −235.062 −0.110907
\(166\) 0 0
\(167\) 1161.03 0.537985 0.268992 0.963142i \(-0.413309\pi\)
0.268992 + 0.963142i \(0.413309\pi\)
\(168\) 0 0
\(169\) 2576.81 1.17288
\(170\) 0 0
\(171\) 1095.24 0.489795
\(172\) 0 0
\(173\) 1358.71 0.597114 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(174\) 0 0
\(175\) −677.494 −0.292650
\(176\) 0 0
\(177\) −1479.42 −0.628249
\(178\) 0 0
\(179\) −2645.09 −1.10449 −0.552243 0.833683i \(-0.686228\pi\)
−0.552243 + 0.833683i \(0.686228\pi\)
\(180\) 0 0
\(181\) −394.148 −0.161861 −0.0809303 0.996720i \(-0.525789\pi\)
−0.0809303 + 0.996720i \(0.525789\pi\)
\(182\) 0 0
\(183\) −1858.77 −0.750842
\(184\) 0 0
\(185\) 2933.64 1.16587
\(186\) 0 0
\(187\) 1388.37 0.542931
\(188\) 0 0
\(189\) 246.324 0.0948012
\(190\) 0 0
\(191\) −4774.26 −1.80866 −0.904328 0.426838i \(-0.859627\pi\)
−0.904328 + 0.426838i \(0.859627\pi\)
\(192\) 0 0
\(193\) 1349.61 0.503352 0.251676 0.967811i \(-0.419018\pi\)
0.251676 + 0.967811i \(0.419018\pi\)
\(194\) 0 0
\(195\) −1476.47 −0.542215
\(196\) 0 0
\(197\) −3964.09 −1.43365 −0.716826 0.697252i \(-0.754406\pi\)
−0.716826 + 0.697252i \(0.754406\pi\)
\(198\) 0 0
\(199\) −1692.39 −0.602866 −0.301433 0.953487i \(-0.597465\pi\)
−0.301433 + 0.953487i \(0.597465\pi\)
\(200\) 0 0
\(201\) 1358.23 0.476627
\(202\) 0 0
\(203\) 810.125 0.280097
\(204\) 0 0
\(205\) −2019.67 −0.688097
\(206\) 0 0
\(207\) 23.4034 0.00785819
\(208\) 0 0
\(209\) −1338.62 −0.443036
\(210\) 0 0
\(211\) 4689.42 1.53001 0.765007 0.644023i \(-0.222736\pi\)
0.765007 + 0.644023i \(0.222736\pi\)
\(212\) 0 0
\(213\) −2096.27 −0.674337
\(214\) 0 0
\(215\) −2595.45 −0.823295
\(216\) 0 0
\(217\) 1212.68 0.379365
\(218\) 0 0
\(219\) −2380.73 −0.734587
\(220\) 0 0
\(221\) 8720.61 2.65435
\(222\) 0 0
\(223\) −3586.97 −1.07714 −0.538568 0.842582i \(-0.681034\pi\)
−0.538568 + 0.842582i \(0.681034\pi\)
\(224\) 0 0
\(225\) −668.352 −0.198030
\(226\) 0 0
\(227\) −2910.47 −0.850990 −0.425495 0.904961i \(-0.639900\pi\)
−0.425495 + 0.904961i \(0.639900\pi\)
\(228\) 0 0
\(229\) −1808.06 −0.521746 −0.260873 0.965373i \(-0.584010\pi\)
−0.260873 + 0.965373i \(0.584010\pi\)
\(230\) 0 0
\(231\) −301.062 −0.0857509
\(232\) 0 0
\(233\) −1863.18 −0.523867 −0.261934 0.965086i \(-0.584360\pi\)
−0.261934 + 0.965086i \(0.584360\pi\)
\(234\) 0 0
\(235\) −2439.95 −0.677296
\(236\) 0 0
\(237\) −1022.82 −0.280336
\(238\) 0 0
\(239\) −1975.95 −0.534785 −0.267393 0.963588i \(-0.586162\pi\)
−0.267393 + 0.963588i \(0.586162\pi\)
\(240\) 0 0
\(241\) −6267.54 −1.67522 −0.837610 0.546269i \(-0.816048\pi\)
−0.837610 + 0.546269i \(0.816048\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −1850.36 −0.482511
\(246\) 0 0
\(247\) −8408.12 −2.16598
\(248\) 0 0
\(249\) −3366.30 −0.856748
\(250\) 0 0
\(251\) 1614.87 0.406094 0.203047 0.979169i \(-0.434916\pi\)
0.203047 + 0.979169i \(0.434916\pi\)
\(252\) 0 0
\(253\) −28.6041 −0.00710800
\(254\) 0 0
\(255\) −2697.15 −0.662361
\(256\) 0 0
\(257\) 2171.47 0.527053 0.263526 0.964652i \(-0.415114\pi\)
0.263526 + 0.964652i \(0.415114\pi\)
\(258\) 0 0
\(259\) 3757.34 0.901427
\(260\) 0 0
\(261\) 799.193 0.189536
\(262\) 0 0
\(263\) 704.379 0.165148 0.0825739 0.996585i \(-0.473686\pi\)
0.0825739 + 0.996585i \(0.473686\pi\)
\(264\) 0 0
\(265\) 2447.18 0.567279
\(266\) 0 0
\(267\) −555.955 −0.127430
\(268\) 0 0
\(269\) −4704.82 −1.06639 −0.533194 0.845993i \(-0.679008\pi\)
−0.533194 + 0.845993i \(0.679008\pi\)
\(270\) 0 0
\(271\) −569.817 −0.127727 −0.0638633 0.997959i \(-0.520342\pi\)
−0.0638633 + 0.997959i \(0.520342\pi\)
\(272\) 0 0
\(273\) −1891.02 −0.419230
\(274\) 0 0
\(275\) 816.875 0.179125
\(276\) 0 0
\(277\) 1128.04 0.244684 0.122342 0.992488i \(-0.460960\pi\)
0.122342 + 0.992488i \(0.460960\pi\)
\(278\) 0 0
\(279\) 1196.32 0.256709
\(280\) 0 0
\(281\) 1459.40 0.309825 0.154912 0.987928i \(-0.450490\pi\)
0.154912 + 0.987928i \(0.450490\pi\)
\(282\) 0 0
\(283\) 4000.93 0.840391 0.420196 0.907434i \(-0.361961\pi\)
0.420196 + 0.907434i \(0.361961\pi\)
\(284\) 0 0
\(285\) 2600.50 0.540492
\(286\) 0 0
\(287\) −2586.75 −0.532024
\(288\) 0 0
\(289\) 11017.5 2.24251
\(290\) 0 0
\(291\) 3422.03 0.689358
\(292\) 0 0
\(293\) −5234.86 −1.04377 −0.521884 0.853016i \(-0.674771\pi\)
−0.521884 + 0.853016i \(0.674771\pi\)
\(294\) 0 0
\(295\) −3512.69 −0.693277
\(296\) 0 0
\(297\) −297.000 −0.0580259
\(298\) 0 0
\(299\) −179.667 −0.0347505
\(300\) 0 0
\(301\) −3324.20 −0.636557
\(302\) 0 0
\(303\) 4781.56 0.906578
\(304\) 0 0
\(305\) −4413.40 −0.828559
\(306\) 0 0
\(307\) 1090.64 0.202756 0.101378 0.994848i \(-0.467675\pi\)
0.101378 + 0.994848i \(0.467675\pi\)
\(308\) 0 0
\(309\) 3205.94 0.590226
\(310\) 0 0
\(311\) 7203.53 1.31342 0.656712 0.754142i \(-0.271947\pi\)
0.656712 + 0.754142i \(0.271947\pi\)
\(312\) 0 0
\(313\) 6365.80 1.14957 0.574787 0.818303i \(-0.305085\pi\)
0.574787 + 0.818303i \(0.305085\pi\)
\(314\) 0 0
\(315\) 584.864 0.104614
\(316\) 0 0
\(317\) 615.672 0.109084 0.0545419 0.998511i \(-0.482630\pi\)
0.0545419 + 0.998511i \(0.482630\pi\)
\(318\) 0 0
\(319\) −976.792 −0.171442
\(320\) 0 0
\(321\) 2652.20 0.461158
\(322\) 0 0
\(323\) −15359.6 −2.64592
\(324\) 0 0
\(325\) 5130.93 0.875731
\(326\) 0 0
\(327\) −5684.46 −0.961319
\(328\) 0 0
\(329\) −3125.03 −0.523673
\(330\) 0 0
\(331\) −6983.55 −1.15967 −0.579835 0.814734i \(-0.696883\pi\)
−0.579835 + 0.814734i \(0.696883\pi\)
\(332\) 0 0
\(333\) 3706.64 0.609977
\(334\) 0 0
\(335\) 3224.93 0.525961
\(336\) 0 0
\(337\) −457.523 −0.0739550 −0.0369775 0.999316i \(-0.511773\pi\)
−0.0369775 + 0.999316i \(0.511773\pi\)
\(338\) 0 0
\(339\) −3202.07 −0.513016
\(340\) 0 0
\(341\) −1462.17 −0.232202
\(342\) 0 0
\(343\) −5499.12 −0.865670
\(344\) 0 0
\(345\) 55.5682 0.00867157
\(346\) 0 0
\(347\) −12210.8 −1.88907 −0.944537 0.328405i \(-0.893489\pi\)
−0.944537 + 0.328405i \(0.893489\pi\)
\(348\) 0 0
\(349\) 11783.5 1.80733 0.903665 0.428240i \(-0.140866\pi\)
0.903665 + 0.428240i \(0.140866\pi\)
\(350\) 0 0
\(351\) −1865.51 −0.283685
\(352\) 0 0
\(353\) 4080.02 0.615177 0.307589 0.951519i \(-0.400478\pi\)
0.307589 + 0.951519i \(0.400478\pi\)
\(354\) 0 0
\(355\) −4977.31 −0.744136
\(356\) 0 0
\(357\) −3454.44 −0.512125
\(358\) 0 0
\(359\) 19.3224 0.00284067 0.00142033 0.999999i \(-0.499548\pi\)
0.00142033 + 0.999999i \(0.499548\pi\)
\(360\) 0 0
\(361\) 7950.23 1.15909
\(362\) 0 0
\(363\) 363.000 0.0524864
\(364\) 0 0
\(365\) −5652.72 −0.810622
\(366\) 0 0
\(367\) 5976.46 0.850051 0.425026 0.905181i \(-0.360265\pi\)
0.425026 + 0.905181i \(0.360265\pi\)
\(368\) 0 0
\(369\) −2551.84 −0.360010
\(370\) 0 0
\(371\) 3134.29 0.438609
\(372\) 0 0
\(373\) −4504.78 −0.625332 −0.312666 0.949863i \(-0.601222\pi\)
−0.312666 + 0.949863i \(0.601222\pi\)
\(374\) 0 0
\(375\) −4258.08 −0.586363
\(376\) 0 0
\(377\) −6135.39 −0.838166
\(378\) 0 0
\(379\) −2908.71 −0.394223 −0.197112 0.980381i \(-0.563156\pi\)
−0.197112 + 0.980381i \(0.563156\pi\)
\(380\) 0 0
\(381\) 386.903 0.0520254
\(382\) 0 0
\(383\) 5238.97 0.698953 0.349476 0.936945i \(-0.386359\pi\)
0.349476 + 0.936945i \(0.386359\pi\)
\(384\) 0 0
\(385\) −714.833 −0.0946267
\(386\) 0 0
\(387\) −3279.34 −0.430745
\(388\) 0 0
\(389\) −798.514 −0.104078 −0.0520389 0.998645i \(-0.516572\pi\)
−0.0520389 + 0.998645i \(0.516572\pi\)
\(390\) 0 0
\(391\) −328.208 −0.0424507
\(392\) 0 0
\(393\) 1174.36 0.150735
\(394\) 0 0
\(395\) −2428.56 −0.309352
\(396\) 0 0
\(397\) 2848.72 0.360134 0.180067 0.983654i \(-0.442368\pi\)
0.180067 + 0.983654i \(0.442368\pi\)
\(398\) 0 0
\(399\) 3330.66 0.417899
\(400\) 0 0
\(401\) 5835.84 0.726754 0.363377 0.931642i \(-0.381624\pi\)
0.363377 + 0.931642i \(0.381624\pi\)
\(402\) 0 0
\(403\) −9184.11 −1.13522
\(404\) 0 0
\(405\) 576.972 0.0707900
\(406\) 0 0
\(407\) −4530.33 −0.551745
\(408\) 0 0
\(409\) −6620.63 −0.800414 −0.400207 0.916425i \(-0.631062\pi\)
−0.400207 + 0.916425i \(0.631062\pi\)
\(410\) 0 0
\(411\) 1310.24 0.157249
\(412\) 0 0
\(413\) −4498.97 −0.536029
\(414\) 0 0
\(415\) −7992.83 −0.945427
\(416\) 0 0
\(417\) −6091.85 −0.715394
\(418\) 0 0
\(419\) 7460.09 0.869808 0.434904 0.900477i \(-0.356782\pi\)
0.434904 + 0.900477i \(0.356782\pi\)
\(420\) 0 0
\(421\) 7696.17 0.890946 0.445473 0.895295i \(-0.353036\pi\)
0.445473 + 0.895295i \(0.353036\pi\)
\(422\) 0 0
\(423\) −3082.86 −0.354359
\(424\) 0 0
\(425\) 9372.97 1.06978
\(426\) 0 0
\(427\) −5652.58 −0.640626
\(428\) 0 0
\(429\) 2280.06 0.256602
\(430\) 0 0
\(431\) −7818.87 −0.873832 −0.436916 0.899502i \(-0.643929\pi\)
−0.436916 + 0.899502i \(0.643929\pi\)
\(432\) 0 0
\(433\) 1615.80 0.179331 0.0896654 0.995972i \(-0.471420\pi\)
0.0896654 + 0.995972i \(0.471420\pi\)
\(434\) 0 0
\(435\) 1897.58 0.209154
\(436\) 0 0
\(437\) 316.448 0.0346401
\(438\) 0 0
\(439\) 5391.47 0.586153 0.293076 0.956089i \(-0.405321\pi\)
0.293076 + 0.956089i \(0.405321\pi\)
\(440\) 0 0
\(441\) −2337.92 −0.252448
\(442\) 0 0
\(443\) −16376.9 −1.75641 −0.878206 0.478282i \(-0.841260\pi\)
−0.878206 + 0.478282i \(0.841260\pi\)
\(444\) 0 0
\(445\) −1320.04 −0.140620
\(446\) 0 0
\(447\) 8847.45 0.936175
\(448\) 0 0
\(449\) 669.178 0.0703351 0.0351675 0.999381i \(-0.488804\pi\)
0.0351675 + 0.999381i \(0.488804\pi\)
\(450\) 0 0
\(451\) 3118.92 0.325641
\(452\) 0 0
\(453\) −3415.42 −0.354239
\(454\) 0 0
\(455\) −4489.98 −0.462624
\(456\) 0 0
\(457\) −738.065 −0.0755475 −0.0377738 0.999286i \(-0.512027\pi\)
−0.0377738 + 0.999286i \(0.512027\pi\)
\(458\) 0 0
\(459\) −3407.83 −0.346544
\(460\) 0 0
\(461\) 6899.51 0.697054 0.348527 0.937299i \(-0.386682\pi\)
0.348527 + 0.937299i \(0.386682\pi\)
\(462\) 0 0
\(463\) −12475.5 −1.25224 −0.626119 0.779728i \(-0.715358\pi\)
−0.626119 + 0.779728i \(0.715358\pi\)
\(464\) 0 0
\(465\) 2840.50 0.283280
\(466\) 0 0
\(467\) −1099.37 −0.108935 −0.0544677 0.998516i \(-0.517346\pi\)
−0.0544677 + 0.998516i \(0.517346\pi\)
\(468\) 0 0
\(469\) 4130.42 0.406663
\(470\) 0 0
\(471\) 1540.65 0.150720
\(472\) 0 0
\(473\) 4008.08 0.389623
\(474\) 0 0
\(475\) −9037.10 −0.872949
\(476\) 0 0
\(477\) 3091.99 0.296798
\(478\) 0 0
\(479\) 18437.8 1.75876 0.879379 0.476122i \(-0.157958\pi\)
0.879379 + 0.476122i \(0.157958\pi\)
\(480\) 0 0
\(481\) −28455.8 −2.69744
\(482\) 0 0
\(483\) 71.1704 0.00670469
\(484\) 0 0
\(485\) 8125.17 0.760711
\(486\) 0 0
\(487\) −505.412 −0.0470276 −0.0235138 0.999724i \(-0.507485\pi\)
−0.0235138 + 0.999724i \(0.507485\pi\)
\(488\) 0 0
\(489\) −1526.59 −0.141176
\(490\) 0 0
\(491\) −6189.24 −0.568873 −0.284436 0.958695i \(-0.591806\pi\)
−0.284436 + 0.958695i \(0.591806\pi\)
\(492\) 0 0
\(493\) −11207.9 −1.02389
\(494\) 0 0
\(495\) −705.187 −0.0640320
\(496\) 0 0
\(497\) −6374.82 −0.575352
\(498\) 0 0
\(499\) 20437.4 1.83348 0.916738 0.399489i \(-0.130813\pi\)
0.916738 + 0.399489i \(0.130813\pi\)
\(500\) 0 0
\(501\) 3483.10 0.310606
\(502\) 0 0
\(503\) −6786.22 −0.601556 −0.300778 0.953694i \(-0.597246\pi\)
−0.300778 + 0.953694i \(0.597246\pi\)
\(504\) 0 0
\(505\) 11353.2 1.00042
\(506\) 0 0
\(507\) 7730.44 0.677162
\(508\) 0 0
\(509\) −10153.5 −0.884178 −0.442089 0.896971i \(-0.645763\pi\)
−0.442089 + 0.896971i \(0.645763\pi\)
\(510\) 0 0
\(511\) −7239.87 −0.626758
\(512\) 0 0
\(513\) 3285.72 0.282783
\(514\) 0 0
\(515\) 7612.09 0.651318
\(516\) 0 0
\(517\) 3767.94 0.320529
\(518\) 0 0
\(519\) 4076.12 0.344744
\(520\) 0 0
\(521\) 11854.0 0.996803 0.498402 0.866946i \(-0.333920\pi\)
0.498402 + 0.866946i \(0.333920\pi\)
\(522\) 0 0
\(523\) −21263.7 −1.77781 −0.888906 0.458089i \(-0.848534\pi\)
−0.888906 + 0.458089i \(0.848534\pi\)
\(524\) 0 0
\(525\) −2032.48 −0.168962
\(526\) 0 0
\(527\) −16777.2 −1.38676
\(528\) 0 0
\(529\) −12160.2 −0.999444
\(530\) 0 0
\(531\) −4438.26 −0.362720
\(532\) 0 0
\(533\) 19590.4 1.59204
\(534\) 0 0
\(535\) 6297.31 0.508891
\(536\) 0 0
\(537\) −7935.26 −0.637676
\(538\) 0 0
\(539\) 2857.46 0.228348
\(540\) 0 0
\(541\) 15215.4 1.20917 0.604586 0.796540i \(-0.293338\pi\)
0.604586 + 0.796540i \(0.293338\pi\)
\(542\) 0 0
\(543\) −1182.44 −0.0934503
\(544\) 0 0
\(545\) −13497.0 −1.06082
\(546\) 0 0
\(547\) −21733.8 −1.69885 −0.849425 0.527709i \(-0.823051\pi\)
−0.849425 + 0.527709i \(0.823051\pi\)
\(548\) 0 0
\(549\) −5576.30 −0.433499
\(550\) 0 0
\(551\) 10806.3 0.835503
\(552\) 0 0
\(553\) −3110.44 −0.239185
\(554\) 0 0
\(555\) 8800.92 0.673114
\(556\) 0 0
\(557\) 3105.20 0.236214 0.118107 0.993001i \(-0.462317\pi\)
0.118107 + 0.993001i \(0.462317\pi\)
\(558\) 0 0
\(559\) 25175.4 1.90484
\(560\) 0 0
\(561\) 4165.12 0.313461
\(562\) 0 0
\(563\) 22463.6 1.68158 0.840788 0.541364i \(-0.182092\pi\)
0.840788 + 0.541364i \(0.182092\pi\)
\(564\) 0 0
\(565\) −7602.89 −0.566117
\(566\) 0 0
\(567\) 738.972 0.0547335
\(568\) 0 0
\(569\) −20903.7 −1.54012 −0.770060 0.637971i \(-0.779774\pi\)
−0.770060 + 0.637971i \(0.779774\pi\)
\(570\) 0 0
\(571\) 21072.8 1.54443 0.772216 0.635360i \(-0.219148\pi\)
0.772216 + 0.635360i \(0.219148\pi\)
\(572\) 0 0
\(573\) −14322.8 −1.04423
\(574\) 0 0
\(575\) −193.107 −0.0140054
\(576\) 0 0
\(577\) 23453.7 1.69219 0.846093 0.533035i \(-0.178949\pi\)
0.846093 + 0.533035i \(0.178949\pi\)
\(578\) 0 0
\(579\) 4048.83 0.290611
\(580\) 0 0
\(581\) −10237.0 −0.730987
\(582\) 0 0
\(583\) −3779.10 −0.268464
\(584\) 0 0
\(585\) −4429.40 −0.313048
\(586\) 0 0
\(587\) −7575.68 −0.532677 −0.266339 0.963879i \(-0.585814\pi\)
−0.266339 + 0.963879i \(0.585814\pi\)
\(588\) 0 0
\(589\) 16176.0 1.13161
\(590\) 0 0
\(591\) −11892.3 −0.827719
\(592\) 0 0
\(593\) −16147.0 −1.11818 −0.559089 0.829108i \(-0.688849\pi\)
−0.559089 + 0.829108i \(0.688849\pi\)
\(594\) 0 0
\(595\) −8202.12 −0.565133
\(596\) 0 0
\(597\) −5077.17 −0.348065
\(598\) 0 0
\(599\) −13524.3 −0.922521 −0.461260 0.887265i \(-0.652603\pi\)
−0.461260 + 0.887265i \(0.652603\pi\)
\(600\) 0 0
\(601\) 7485.97 0.508085 0.254042 0.967193i \(-0.418240\pi\)
0.254042 + 0.967193i \(0.418240\pi\)
\(602\) 0 0
\(603\) 4074.68 0.275181
\(604\) 0 0
\(605\) 861.896 0.0579191
\(606\) 0 0
\(607\) −20024.4 −1.33899 −0.669494 0.742818i \(-0.733489\pi\)
−0.669494 + 0.742818i \(0.733489\pi\)
\(608\) 0 0
\(609\) 2430.37 0.161714
\(610\) 0 0
\(611\) 23667.0 1.56705
\(612\) 0 0
\(613\) 20034.0 1.32001 0.660005 0.751262i \(-0.270554\pi\)
0.660005 + 0.751262i \(0.270554\pi\)
\(614\) 0 0
\(615\) −6059.01 −0.397273
\(616\) 0 0
\(617\) −22895.8 −1.49392 −0.746961 0.664868i \(-0.768488\pi\)
−0.746961 + 0.664868i \(0.768488\pi\)
\(618\) 0 0
\(619\) 25283.6 1.64174 0.820869 0.571117i \(-0.193490\pi\)
0.820869 + 0.571117i \(0.193490\pi\)
\(620\) 0 0
\(621\) 70.2101 0.00453693
\(622\) 0 0
\(623\) −1690.68 −0.108725
\(624\) 0 0
\(625\) −827.579 −0.0529650
\(626\) 0 0
\(627\) −4015.87 −0.255787
\(628\) 0 0
\(629\) −51981.8 −3.29515
\(630\) 0 0
\(631\) 22308.5 1.40743 0.703716 0.710482i \(-0.251523\pi\)
0.703716 + 0.710482i \(0.251523\pi\)
\(632\) 0 0
\(633\) 14068.2 0.883353
\(634\) 0 0
\(635\) 918.651 0.0574103
\(636\) 0 0
\(637\) 17948.2 1.11638
\(638\) 0 0
\(639\) −6288.80 −0.389329
\(640\) 0 0
\(641\) −1708.47 −0.105274 −0.0526370 0.998614i \(-0.516763\pi\)
−0.0526370 + 0.998614i \(0.516763\pi\)
\(642\) 0 0
\(643\) −28912.2 −1.77323 −0.886614 0.462511i \(-0.846949\pi\)
−0.886614 + 0.462511i \(0.846949\pi\)
\(644\) 0 0
\(645\) −7786.36 −0.475330
\(646\) 0 0
\(647\) 1462.57 0.0888710 0.0444355 0.999012i \(-0.485851\pi\)
0.0444355 + 0.999012i \(0.485851\pi\)
\(648\) 0 0
\(649\) 5424.54 0.328092
\(650\) 0 0
\(651\) 3638.05 0.219026
\(652\) 0 0
\(653\) −5203.71 −0.311849 −0.155924 0.987769i \(-0.549836\pi\)
−0.155924 + 0.987769i \(0.549836\pi\)
\(654\) 0 0
\(655\) 2788.37 0.166337
\(656\) 0 0
\(657\) −7142.18 −0.424114
\(658\) 0 0
\(659\) −10989.8 −0.649624 −0.324812 0.945779i \(-0.605301\pi\)
−0.324812 + 0.945779i \(0.605301\pi\)
\(660\) 0 0
\(661\) −6936.07 −0.408142 −0.204071 0.978956i \(-0.565417\pi\)
−0.204071 + 0.978956i \(0.565417\pi\)
\(662\) 0 0
\(663\) 26161.8 1.53249
\(664\) 0 0
\(665\) 7908.21 0.461154
\(666\) 0 0
\(667\) 230.911 0.0134047
\(668\) 0 0
\(669\) −10760.9 −0.621885
\(670\) 0 0
\(671\) 6815.48 0.392114
\(672\) 0 0
\(673\) −4903.59 −0.280861 −0.140430 0.990091i \(-0.544849\pi\)
−0.140430 + 0.990091i \(0.544849\pi\)
\(674\) 0 0
\(675\) −2005.06 −0.114333
\(676\) 0 0
\(677\) −9622.58 −0.546271 −0.273136 0.961976i \(-0.588061\pi\)
−0.273136 + 0.961976i \(0.588061\pi\)
\(678\) 0 0
\(679\) 10406.5 0.588168
\(680\) 0 0
\(681\) −8731.41 −0.491319
\(682\) 0 0
\(683\) 11375.1 0.637274 0.318637 0.947877i \(-0.396775\pi\)
0.318637 + 0.947877i \(0.396775\pi\)
\(684\) 0 0
\(685\) 3110.99 0.173525
\(686\) 0 0
\(687\) −5424.17 −0.301230
\(688\) 0 0
\(689\) −23737.2 −1.31250
\(690\) 0 0
\(691\) 29834.2 1.64247 0.821235 0.570589i \(-0.193285\pi\)
0.821235 + 0.570589i \(0.193285\pi\)
\(692\) 0 0
\(693\) −903.187 −0.0495083
\(694\) 0 0
\(695\) −14464.3 −0.789442
\(696\) 0 0
\(697\) 35787.0 1.94480
\(698\) 0 0
\(699\) −5589.54 −0.302455
\(700\) 0 0
\(701\) 25289.0 1.36255 0.681277 0.732025i \(-0.261425\pi\)
0.681277 + 0.732025i \(0.261425\pi\)
\(702\) 0 0
\(703\) 50119.1 2.68888
\(704\) 0 0
\(705\) −7319.84 −0.391037
\(706\) 0 0
\(707\) 14540.9 0.773502
\(708\) 0 0
\(709\) −19263.0 −1.02036 −0.510182 0.860066i \(-0.670422\pi\)
−0.510182 + 0.860066i \(0.670422\pi\)
\(710\) 0 0
\(711\) −3068.47 −0.161852
\(712\) 0 0
\(713\) 345.653 0.0181554
\(714\) 0 0
\(715\) 5413.71 0.283163
\(716\) 0 0
\(717\) −5927.85 −0.308758
\(718\) 0 0
\(719\) −25465.3 −1.32086 −0.660428 0.750889i \(-0.729625\pi\)
−0.660428 + 0.750889i \(0.729625\pi\)
\(720\) 0 0
\(721\) 9749.39 0.503587
\(722\) 0 0
\(723\) −18802.6 −0.967188
\(724\) 0 0
\(725\) −6594.35 −0.337804
\(726\) 0 0
\(727\) −6887.80 −0.351382 −0.175691 0.984445i \(-0.556216\pi\)
−0.175691 + 0.984445i \(0.556216\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 45989.4 2.32692
\(732\) 0 0
\(733\) 26293.8 1.32494 0.662471 0.749087i \(-0.269508\pi\)
0.662471 + 0.749087i \(0.269508\pi\)
\(734\) 0 0
\(735\) −5551.08 −0.278578
\(736\) 0 0
\(737\) −4980.17 −0.248910
\(738\) 0 0
\(739\) 13718.4 0.682869 0.341434 0.939906i \(-0.389087\pi\)
0.341434 + 0.939906i \(0.389087\pi\)
\(740\) 0 0
\(741\) −25224.4 −1.25053
\(742\) 0 0
\(743\) 8200.99 0.404933 0.202466 0.979289i \(-0.435104\pi\)
0.202466 + 0.979289i \(0.435104\pi\)
\(744\) 0 0
\(745\) 21007.1 1.03308
\(746\) 0 0
\(747\) −10098.9 −0.494644
\(748\) 0 0
\(749\) 8065.45 0.393465
\(750\) 0 0
\(751\) 6068.39 0.294859 0.147429 0.989073i \(-0.452900\pi\)
0.147429 + 0.989073i \(0.452900\pi\)
\(752\) 0 0
\(753\) 4844.60 0.234458
\(754\) 0 0
\(755\) −8109.45 −0.390905
\(756\) 0 0
\(757\) −8179.76 −0.392733 −0.196366 0.980531i \(-0.562914\pi\)
−0.196366 + 0.980531i \(0.562914\pi\)
\(758\) 0 0
\(759\) −85.8123 −0.00410381
\(760\) 0 0
\(761\) 9406.98 0.448098 0.224049 0.974578i \(-0.428072\pi\)
0.224049 + 0.974578i \(0.428072\pi\)
\(762\) 0 0
\(763\) −17286.6 −0.820208
\(764\) 0 0
\(765\) −8091.44 −0.382414
\(766\) 0 0
\(767\) 34072.4 1.60402
\(768\) 0 0
\(769\) 27573.0 1.29299 0.646493 0.762920i \(-0.276235\pi\)
0.646493 + 0.762920i \(0.276235\pi\)
\(770\) 0 0
\(771\) 6514.41 0.304294
\(772\) 0 0
\(773\) −28787.9 −1.33949 −0.669747 0.742589i \(-0.733598\pi\)
−0.669747 + 0.742589i \(0.733598\pi\)
\(774\) 0 0
\(775\) −9871.13 −0.457525
\(776\) 0 0
\(777\) 11272.0 0.520439
\(778\) 0 0
\(779\) −34504.6 −1.58698
\(780\) 0 0
\(781\) 7686.31 0.352161
\(782\) 0 0
\(783\) 2397.58 0.109428
\(784\) 0 0
\(785\) 3658.06 0.166321
\(786\) 0 0
\(787\) 23922.8 1.08355 0.541776 0.840523i \(-0.317752\pi\)
0.541776 + 0.840523i \(0.317752\pi\)
\(788\) 0 0
\(789\) 2113.14 0.0953481
\(790\) 0 0
\(791\) −9737.60 −0.437711
\(792\) 0 0
\(793\) 42809.1 1.91702
\(794\) 0 0
\(795\) 7341.53 0.327519
\(796\) 0 0
\(797\) −30362.4 −1.34942 −0.674712 0.738081i \(-0.735732\pi\)
−0.674712 + 0.738081i \(0.735732\pi\)
\(798\) 0 0
\(799\) 43234.0 1.91428
\(800\) 0 0
\(801\) −1667.86 −0.0735719
\(802\) 0 0
\(803\) 8729.33 0.383626
\(804\) 0 0
\(805\) 168.985 0.00739867
\(806\) 0 0
\(807\) −14114.5 −0.615679
\(808\) 0 0
\(809\) −32005.6 −1.39092 −0.695462 0.718563i \(-0.744800\pi\)
−0.695462 + 0.718563i \(0.744800\pi\)
\(810\) 0 0
\(811\) −7624.99 −0.330147 −0.165074 0.986281i \(-0.552786\pi\)
−0.165074 + 0.986281i \(0.552786\pi\)
\(812\) 0 0
\(813\) −1709.45 −0.0737429
\(814\) 0 0
\(815\) −3624.69 −0.155788
\(816\) 0 0
\(817\) −44341.5 −1.89879
\(818\) 0 0
\(819\) −5673.07 −0.242043
\(820\) 0 0
\(821\) −13109.1 −0.557262 −0.278631 0.960398i \(-0.589881\pi\)
−0.278631 + 0.960398i \(0.589881\pi\)
\(822\) 0 0
\(823\) 15410.0 0.652683 0.326342 0.945252i \(-0.394184\pi\)
0.326342 + 0.945252i \(0.394184\pi\)
\(824\) 0 0
\(825\) 2450.63 0.103418
\(826\) 0 0
\(827\) −3795.54 −0.159594 −0.0797968 0.996811i \(-0.525427\pi\)
−0.0797968 + 0.996811i \(0.525427\pi\)
\(828\) 0 0
\(829\) 41605.8 1.74310 0.871550 0.490306i \(-0.163115\pi\)
0.871550 + 0.490306i \(0.163115\pi\)
\(830\) 0 0
\(831\) 3384.12 0.141268
\(832\) 0 0
\(833\) 32787.0 1.36375
\(834\) 0 0
\(835\) 8270.17 0.342756
\(836\) 0 0
\(837\) 3588.95 0.148211
\(838\) 0 0
\(839\) 32795.3 1.34949 0.674744 0.738052i \(-0.264254\pi\)
0.674744 + 0.738052i \(0.264254\pi\)
\(840\) 0 0
\(841\) −16503.7 −0.676686
\(842\) 0 0
\(843\) 4378.21 0.178878
\(844\) 0 0
\(845\) 18354.9 0.747253
\(846\) 0 0
\(847\) 1103.90 0.0447819
\(848\) 0 0
\(849\) 12002.8 0.485200
\(850\) 0 0
\(851\) 1070.96 0.0431399
\(852\) 0 0
\(853\) 17928.0 0.719629 0.359814 0.933024i \(-0.382840\pi\)
0.359814 + 0.933024i \(0.382840\pi\)
\(854\) 0 0
\(855\) 7801.50 0.312053
\(856\) 0 0
\(857\) 41976.6 1.67315 0.836577 0.547849i \(-0.184553\pi\)
0.836577 + 0.547849i \(0.184553\pi\)
\(858\) 0 0
\(859\) −12120.4 −0.481424 −0.240712 0.970597i \(-0.577381\pi\)
−0.240712 + 0.970597i \(0.577381\pi\)
\(860\) 0 0
\(861\) −7760.24 −0.307164
\(862\) 0 0
\(863\) −20949.8 −0.826350 −0.413175 0.910652i \(-0.635580\pi\)
−0.413175 + 0.910652i \(0.635580\pi\)
\(864\) 0 0
\(865\) 9678.22 0.380427
\(866\) 0 0
\(867\) 33052.4 1.29471
\(868\) 0 0
\(869\) 3750.35 0.146401
\(870\) 0 0
\(871\) −31281.2 −1.21690
\(872\) 0 0
\(873\) 10266.1 0.398001
\(874\) 0 0
\(875\) −12949.0 −0.500291
\(876\) 0 0
\(877\) 32033.3 1.23340 0.616698 0.787200i \(-0.288470\pi\)
0.616698 + 0.787200i \(0.288470\pi\)
\(878\) 0 0
\(879\) −15704.6 −0.602620
\(880\) 0 0
\(881\) 6181.28 0.236382 0.118191 0.992991i \(-0.462290\pi\)
0.118191 + 0.992991i \(0.462290\pi\)
\(882\) 0 0
\(883\) −44940.1 −1.71275 −0.856373 0.516358i \(-0.827287\pi\)
−0.856373 + 0.516358i \(0.827287\pi\)
\(884\) 0 0
\(885\) −10538.1 −0.400264
\(886\) 0 0
\(887\) −34284.2 −1.29780 −0.648901 0.760873i \(-0.724771\pi\)
−0.648901 + 0.760873i \(0.724771\pi\)
\(888\) 0 0
\(889\) 1176.59 0.0443886
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 0 0
\(893\) −41684.7 −1.56207
\(894\) 0 0
\(895\) −18841.2 −0.703679
\(896\) 0 0
\(897\) −539.001 −0.0200632
\(898\) 0 0
\(899\) 11803.6 0.437899
\(900\) 0 0
\(901\) −43362.1 −1.60333
\(902\) 0 0
\(903\) −9972.59 −0.367516
\(904\) 0 0
\(905\) −2807.56 −0.103123
\(906\) 0 0
\(907\) 38002.6 1.39124 0.695620 0.718410i \(-0.255130\pi\)
0.695620 + 0.718410i \(0.255130\pi\)
\(908\) 0 0
\(909\) 14344.7 0.523413
\(910\) 0 0
\(911\) −51842.2 −1.88541 −0.942706 0.333626i \(-0.891728\pi\)
−0.942706 + 0.333626i \(0.891728\pi\)
\(912\) 0 0
\(913\) 12343.1 0.447422
\(914\) 0 0
\(915\) −13240.2 −0.478369
\(916\) 0 0
\(917\) 3571.28 0.128609
\(918\) 0 0
\(919\) −18972.6 −0.681008 −0.340504 0.940243i \(-0.610598\pi\)
−0.340504 + 0.940243i \(0.610598\pi\)
\(920\) 0 0
\(921\) 3271.92 0.117061
\(922\) 0 0
\(923\) 48279.0 1.72169
\(924\) 0 0
\(925\) −30584.4 −1.08715
\(926\) 0 0
\(927\) 9617.83 0.340767
\(928\) 0 0
\(929\) −29463.8 −1.04055 −0.520277 0.853997i \(-0.674171\pi\)
−0.520277 + 0.853997i \(0.674171\pi\)
\(930\) 0 0
\(931\) −31612.1 −1.11283
\(932\) 0 0
\(933\) 21610.6 0.758306
\(934\) 0 0
\(935\) 9889.54 0.345907
\(936\) 0 0
\(937\) −1999.36 −0.0697080 −0.0348540 0.999392i \(-0.511097\pi\)
−0.0348540 + 0.999392i \(0.511097\pi\)
\(938\) 0 0
\(939\) 19097.4 0.663707
\(940\) 0 0
\(941\) 31401.2 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(942\) 0 0
\(943\) −737.304 −0.0254612
\(944\) 0 0
\(945\) 1754.59 0.0603988
\(946\) 0 0
\(947\) 39890.0 1.36880 0.684398 0.729109i \(-0.260065\pi\)
0.684398 + 0.729109i \(0.260065\pi\)
\(948\) 0 0
\(949\) 54830.4 1.87552
\(950\) 0 0
\(951\) 1847.02 0.0629796
\(952\) 0 0
\(953\) −14905.2 −0.506639 −0.253319 0.967383i \(-0.581522\pi\)
−0.253319 + 0.967383i \(0.581522\pi\)
\(954\) 0 0
\(955\) −34007.6 −1.15231
\(956\) 0 0
\(957\) −2930.38 −0.0989818
\(958\) 0 0
\(959\) 3984.48 0.134166
\(960\) 0 0
\(961\) −12122.2 −0.406906
\(962\) 0 0
\(963\) 7956.61 0.266249
\(964\) 0 0
\(965\) 9613.41 0.320691
\(966\) 0 0
\(967\) 2049.74 0.0681645 0.0340822 0.999419i \(-0.489149\pi\)
0.0340822 + 0.999419i \(0.489149\pi\)
\(968\) 0 0
\(969\) −46078.8 −1.52762
\(970\) 0 0
\(971\) 10459.3 0.345681 0.172841 0.984950i \(-0.444705\pi\)
0.172841 + 0.984950i \(0.444705\pi\)
\(972\) 0 0
\(973\) −18525.5 −0.610382
\(974\) 0 0
\(975\) 15392.8 0.505603
\(976\) 0 0
\(977\) −6009.04 −0.196772 −0.0983860 0.995148i \(-0.531368\pi\)
−0.0983860 + 0.995148i \(0.531368\pi\)
\(978\) 0 0
\(979\) 2038.50 0.0665483
\(980\) 0 0
\(981\) −17053.4 −0.555018
\(982\) 0 0
\(983\) 36301.5 1.17786 0.588931 0.808183i \(-0.299549\pi\)
0.588931 + 0.808183i \(0.299549\pi\)
\(984\) 0 0
\(985\) −28236.6 −0.913394
\(986\) 0 0
\(987\) −9375.08 −0.302343
\(988\) 0 0
\(989\) −947.501 −0.0304639
\(990\) 0 0
\(991\) 46565.1 1.49262 0.746311 0.665597i \(-0.231823\pi\)
0.746311 + 0.665597i \(0.231823\pi\)
\(992\) 0 0
\(993\) −20950.7 −0.669536
\(994\) 0 0
\(995\) −12055.1 −0.384092
\(996\) 0 0
\(997\) 31997.2 1.01641 0.508205 0.861236i \(-0.330309\pi\)
0.508205 + 0.861236i \(0.330309\pi\)
\(998\) 0 0
\(999\) 11119.9 0.352170
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 2112.4.a.bl.1.2 2
4.3 odd 2 2112.4.a.be.1.2 2
8.3 odd 2 528.4.a.r.1.1 2
8.5 even 2 264.4.a.e.1.1 2
24.5 odd 2 792.4.a.h.1.2 2
24.11 even 2 1584.4.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
264.4.a.e.1.1 2 8.5 even 2
528.4.a.r.1.1 2 8.3 odd 2
792.4.a.h.1.2 2 24.5 odd 2
1584.4.a.bf.1.2 2 24.11 even 2
2112.4.a.be.1.2 2 4.3 odd 2
2112.4.a.bl.1.2 2 1.1 even 1 trivial