Properties

Label 1840.4.a.k.1.3
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $1$
Dimension $4$
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] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, 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("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 60x^{2} - 45x + 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.01192\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.98808 q^{3} +5.00000 q^{5} -2.98517 q^{7} -18.0714 q^{9} +O(q^{10})\) \(q-2.98808 q^{3} +5.00000 q^{5} -2.98517 q^{7} -18.0714 q^{9} -68.6262 q^{11} -12.0028 q^{13} -14.9404 q^{15} +106.208 q^{17} +48.4891 q^{19} +8.91992 q^{21} +23.0000 q^{23} +25.0000 q^{25} +134.677 q^{27} +135.226 q^{29} +230.782 q^{31} +205.060 q^{33} -14.9259 q^{35} -107.956 q^{37} +35.8651 q^{39} +394.637 q^{41} -136.063 q^{43} -90.3570 q^{45} +50.4512 q^{47} -334.089 q^{49} -317.357 q^{51} -414.707 q^{53} -343.131 q^{55} -144.889 q^{57} +183.586 q^{59} -98.4751 q^{61} +53.9463 q^{63} -60.0138 q^{65} -136.925 q^{67} -68.7257 q^{69} +708.800 q^{71} -689.605 q^{73} -74.7019 q^{75} +204.861 q^{77} -546.583 q^{79} +85.5038 q^{81} +20.2622 q^{83} +531.039 q^{85} -404.065 q^{87} -1087.31 q^{89} +35.8303 q^{91} -689.595 q^{93} +242.445 q^{95} -1115.90 q^{97} +1240.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9} - 21 q^{11} + 70 q^{13} - 70 q^{15} + 56 q^{17} - 173 q^{19} - 120 q^{21} + 92 q^{23} + 100 q^{25} - 389 q^{27} - 118 q^{29} - 17 q^{31} - 89 q^{33} - 40 q^{35} - 343 q^{37} + 221 q^{39} + 139 q^{41} + 50 q^{43} + 320 q^{45} - 367 q^{47} - 124 q^{49} - 439 q^{51} - 353 q^{53} - 105 q^{55} - 238 q^{57} + 453 q^{59} - 327 q^{61} + 1723 q^{63} + 350 q^{65} + 455 q^{67} - 322 q^{69} - 195 q^{71} - 633 q^{73} - 350 q^{75} - 2 q^{77} + 1140 q^{79} + 1456 q^{81} + 1199 q^{83} + 280 q^{85} + 1775 q^{87} - 2170 q^{89} - 557 q^{91} - 3241 q^{93} - 865 q^{95} - 703 q^{97} + 2745 q^{99}+O(q^{100}) \) 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
\(3\) −2.98808 −0.575055 −0.287528 0.957772i \(-0.592833\pi\)
−0.287528 + 0.957772i \(0.592833\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −2.98517 −0.161184 −0.0805921 0.996747i \(-0.525681\pi\)
−0.0805921 + 0.996747i \(0.525681\pi\)
\(8\) 0 0
\(9\) −18.0714 −0.669311
\(10\) 0 0
\(11\) −68.6262 −1.88105 −0.940527 0.339720i \(-0.889668\pi\)
−0.940527 + 0.339720i \(0.889668\pi\)
\(12\) 0 0
\(13\) −12.0028 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(14\) 0 0
\(15\) −14.9404 −0.257173
\(16\) 0 0
\(17\) 106.208 1.51525 0.757623 0.652693i \(-0.226361\pi\)
0.757623 + 0.652693i \(0.226361\pi\)
\(18\) 0 0
\(19\) 48.4891 0.585482 0.292741 0.956192i \(-0.405433\pi\)
0.292741 + 0.956192i \(0.405433\pi\)
\(20\) 0 0
\(21\) 8.91992 0.0926898
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 134.677 0.959946
\(28\) 0 0
\(29\) 135.226 0.865890 0.432945 0.901420i \(-0.357474\pi\)
0.432945 + 0.901420i \(0.357474\pi\)
\(30\) 0 0
\(31\) 230.782 1.33709 0.668544 0.743673i \(-0.266918\pi\)
0.668544 + 0.743673i \(0.266918\pi\)
\(32\) 0 0
\(33\) 205.060 1.08171
\(34\) 0 0
\(35\) −14.9259 −0.0720838
\(36\) 0 0
\(37\) −107.956 −0.479672 −0.239836 0.970813i \(-0.577094\pi\)
−0.239836 + 0.970813i \(0.577094\pi\)
\(38\) 0 0
\(39\) 35.8651 0.147257
\(40\) 0 0
\(41\) 394.637 1.50322 0.751610 0.659608i \(-0.229278\pi\)
0.751610 + 0.659608i \(0.229278\pi\)
\(42\) 0 0
\(43\) −136.063 −0.482543 −0.241272 0.970458i \(-0.577564\pi\)
−0.241272 + 0.970458i \(0.577564\pi\)
\(44\) 0 0
\(45\) −90.3570 −0.299325
\(46\) 0 0
\(47\) 50.4512 0.156576 0.0782880 0.996931i \(-0.475055\pi\)
0.0782880 + 0.996931i \(0.475055\pi\)
\(48\) 0 0
\(49\) −334.089 −0.974020
\(50\) 0 0
\(51\) −317.357 −0.871350
\(52\) 0 0
\(53\) −414.707 −1.07480 −0.537400 0.843327i \(-0.680594\pi\)
−0.537400 + 0.843327i \(0.680594\pi\)
\(54\) 0 0
\(55\) −343.131 −0.841232
\(56\) 0 0
\(57\) −144.889 −0.336685
\(58\) 0 0
\(59\) 183.586 0.405100 0.202550 0.979272i \(-0.435077\pi\)
0.202550 + 0.979272i \(0.435077\pi\)
\(60\) 0 0
\(61\) −98.4751 −0.206696 −0.103348 0.994645i \(-0.532956\pi\)
−0.103348 + 0.994645i \(0.532956\pi\)
\(62\) 0 0
\(63\) 53.9463 0.107882
\(64\) 0 0
\(65\) −60.0138 −0.114520
\(66\) 0 0
\(67\) −136.925 −0.249672 −0.124836 0.992177i \(-0.539840\pi\)
−0.124836 + 0.992177i \(0.539840\pi\)
\(68\) 0 0
\(69\) −68.7257 −0.119907
\(70\) 0 0
\(71\) 708.800 1.18478 0.592388 0.805653i \(-0.298185\pi\)
0.592388 + 0.805653i \(0.298185\pi\)
\(72\) 0 0
\(73\) −689.605 −1.10565 −0.552823 0.833299i \(-0.686449\pi\)
−0.552823 + 0.833299i \(0.686449\pi\)
\(74\) 0 0
\(75\) −74.7019 −0.115011
\(76\) 0 0
\(77\) 204.861 0.303196
\(78\) 0 0
\(79\) −546.583 −0.778423 −0.389212 0.921148i \(-0.627252\pi\)
−0.389212 + 0.921148i \(0.627252\pi\)
\(80\) 0 0
\(81\) 85.5038 0.117289
\(82\) 0 0
\(83\) 20.2622 0.0267959 0.0133980 0.999910i \(-0.495735\pi\)
0.0133980 + 0.999910i \(0.495735\pi\)
\(84\) 0 0
\(85\) 531.039 0.677638
\(86\) 0 0
\(87\) −404.065 −0.497935
\(88\) 0 0
\(89\) −1087.31 −1.29499 −0.647495 0.762069i \(-0.724183\pi\)
−0.647495 + 0.762069i \(0.724183\pi\)
\(90\) 0 0
\(91\) 35.8303 0.0412751
\(92\) 0 0
\(93\) −689.595 −0.768900
\(94\) 0 0
\(95\) 242.445 0.261836
\(96\) 0 0
\(97\) −1115.90 −1.16807 −0.584036 0.811728i \(-0.698527\pi\)
−0.584036 + 0.811728i \(0.698527\pi\)
\(98\) 0 0
\(99\) 1240.17 1.25901
\(100\) 0 0
\(101\) −134.565 −0.132572 −0.0662859 0.997801i \(-0.521115\pi\)
−0.0662859 + 0.997801i \(0.521115\pi\)
\(102\) 0 0
\(103\) 925.953 0.885795 0.442898 0.896572i \(-0.353951\pi\)
0.442898 + 0.896572i \(0.353951\pi\)
\(104\) 0 0
\(105\) 44.5996 0.0414522
\(106\) 0 0
\(107\) −1871.13 −1.69055 −0.845273 0.534334i \(-0.820562\pi\)
−0.845273 + 0.534334i \(0.820562\pi\)
\(108\) 0 0
\(109\) −978.242 −0.859620 −0.429810 0.902919i \(-0.641419\pi\)
−0.429810 + 0.902919i \(0.641419\pi\)
\(110\) 0 0
\(111\) 322.581 0.275838
\(112\) 0 0
\(113\) −1760.45 −1.46557 −0.732786 0.680459i \(-0.761780\pi\)
−0.732786 + 0.680459i \(0.761780\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) 0 0
\(117\) 216.907 0.171393
\(118\) 0 0
\(119\) −317.049 −0.244234
\(120\) 0 0
\(121\) 3378.56 2.53836
\(122\) 0 0
\(123\) −1179.21 −0.864434
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1260.87 0.880980 0.440490 0.897758i \(-0.354805\pi\)
0.440490 + 0.897758i \(0.354805\pi\)
\(128\) 0 0
\(129\) 406.566 0.277489
\(130\) 0 0
\(131\) −444.283 −0.296314 −0.148157 0.988964i \(-0.547334\pi\)
−0.148157 + 0.988964i \(0.547334\pi\)
\(132\) 0 0
\(133\) −144.748 −0.0943705
\(134\) 0 0
\(135\) 673.384 0.429301
\(136\) 0 0
\(137\) 429.072 0.267577 0.133789 0.991010i \(-0.457286\pi\)
0.133789 + 0.991010i \(0.457286\pi\)
\(138\) 0 0
\(139\) 249.755 0.152402 0.0762012 0.997092i \(-0.475721\pi\)
0.0762012 + 0.997092i \(0.475721\pi\)
\(140\) 0 0
\(141\) −150.752 −0.0900398
\(142\) 0 0
\(143\) 823.704 0.481689
\(144\) 0 0
\(145\) 676.129 0.387238
\(146\) 0 0
\(147\) 998.282 0.560115
\(148\) 0 0
\(149\) −1564.18 −0.860020 −0.430010 0.902824i \(-0.641490\pi\)
−0.430010 + 0.902824i \(0.641490\pi\)
\(150\) 0 0
\(151\) 1860.04 1.00244 0.501219 0.865321i \(-0.332885\pi\)
0.501219 + 0.865321i \(0.332885\pi\)
\(152\) 0 0
\(153\) −1919.32 −1.01417
\(154\) 0 0
\(155\) 1153.91 0.597964
\(156\) 0 0
\(157\) 2834.52 1.44089 0.720445 0.693512i \(-0.243938\pi\)
0.720445 + 0.693512i \(0.243938\pi\)
\(158\) 0 0
\(159\) 1239.18 0.618070
\(160\) 0 0
\(161\) −68.6590 −0.0336092
\(162\) 0 0
\(163\) −2114.51 −1.01608 −0.508041 0.861333i \(-0.669630\pi\)
−0.508041 + 0.861333i \(0.669630\pi\)
\(164\) 0 0
\(165\) 1025.30 0.483755
\(166\) 0 0
\(167\) −2487.41 −1.15258 −0.576291 0.817244i \(-0.695501\pi\)
−0.576291 + 0.817244i \(0.695501\pi\)
\(168\) 0 0
\(169\) −2052.93 −0.934426
\(170\) 0 0
\(171\) −876.266 −0.391870
\(172\) 0 0
\(173\) 3672.32 1.61388 0.806939 0.590634i \(-0.201122\pi\)
0.806939 + 0.590634i \(0.201122\pi\)
\(174\) 0 0
\(175\) −74.6293 −0.0322368
\(176\) 0 0
\(177\) −548.569 −0.232955
\(178\) 0 0
\(179\) 3224.63 1.34648 0.673240 0.739424i \(-0.264902\pi\)
0.673240 + 0.739424i \(0.264902\pi\)
\(180\) 0 0
\(181\) −2224.23 −0.913401 −0.456700 0.889621i \(-0.650969\pi\)
−0.456700 + 0.889621i \(0.650969\pi\)
\(182\) 0 0
\(183\) 294.251 0.118862
\(184\) 0 0
\(185\) −539.780 −0.214516
\(186\) 0 0
\(187\) −7288.64 −2.85026
\(188\) 0 0
\(189\) −402.033 −0.154728
\(190\) 0 0
\(191\) −3273.53 −1.24013 −0.620063 0.784552i \(-0.712893\pi\)
−0.620063 + 0.784552i \(0.712893\pi\)
\(192\) 0 0
\(193\) 5126.86 1.91212 0.956062 0.293166i \(-0.0947089\pi\)
0.956062 + 0.293166i \(0.0947089\pi\)
\(194\) 0 0
\(195\) 179.326 0.0658553
\(196\) 0 0
\(197\) 3262.57 1.17994 0.589971 0.807425i \(-0.299139\pi\)
0.589971 + 0.807425i \(0.299139\pi\)
\(198\) 0 0
\(199\) 168.282 0.0599457 0.0299728 0.999551i \(-0.490458\pi\)
0.0299728 + 0.999551i \(0.490458\pi\)
\(200\) 0 0
\(201\) 409.141 0.143575
\(202\) 0 0
\(203\) −403.673 −0.139568
\(204\) 0 0
\(205\) 1973.19 0.672260
\(206\) 0 0
\(207\) −415.642 −0.139561
\(208\) 0 0
\(209\) −3327.62 −1.10132
\(210\) 0 0
\(211\) −4197.01 −1.36936 −0.684678 0.728846i \(-0.740057\pi\)
−0.684678 + 0.728846i \(0.740057\pi\)
\(212\) 0 0
\(213\) −2117.95 −0.681312
\(214\) 0 0
\(215\) −680.313 −0.215800
\(216\) 0 0
\(217\) −688.925 −0.215518
\(218\) 0 0
\(219\) 2060.59 0.635807
\(220\) 0 0
\(221\) −1274.79 −0.388015
\(222\) 0 0
\(223\) −5192.66 −1.55931 −0.779656 0.626208i \(-0.784606\pi\)
−0.779656 + 0.626208i \(0.784606\pi\)
\(224\) 0 0
\(225\) −451.785 −0.133862
\(226\) 0 0
\(227\) −4701.24 −1.37459 −0.687295 0.726378i \(-0.741202\pi\)
−0.687295 + 0.726378i \(0.741202\pi\)
\(228\) 0 0
\(229\) −2125.17 −0.613255 −0.306628 0.951830i \(-0.599201\pi\)
−0.306628 + 0.951830i \(0.599201\pi\)
\(230\) 0 0
\(231\) −612.141 −0.174355
\(232\) 0 0
\(233\) −1646.67 −0.462992 −0.231496 0.972836i \(-0.574362\pi\)
−0.231496 + 0.972836i \(0.574362\pi\)
\(234\) 0 0
\(235\) 252.256 0.0700229
\(236\) 0 0
\(237\) 1633.23 0.447636
\(238\) 0 0
\(239\) −2877.88 −0.778888 −0.389444 0.921050i \(-0.627333\pi\)
−0.389444 + 0.921050i \(0.627333\pi\)
\(240\) 0 0
\(241\) −4244.54 −1.13450 −0.567251 0.823545i \(-0.691993\pi\)
−0.567251 + 0.823545i \(0.691993\pi\)
\(242\) 0 0
\(243\) −3891.76 −1.02739
\(244\) 0 0
\(245\) −1670.44 −0.435595
\(246\) 0 0
\(247\) −582.003 −0.149927
\(248\) 0 0
\(249\) −60.5449 −0.0154092
\(250\) 0 0
\(251\) −4401.99 −1.10698 −0.553488 0.832857i \(-0.686704\pi\)
−0.553488 + 0.832857i \(0.686704\pi\)
\(252\) 0 0
\(253\) −1578.40 −0.392227
\(254\) 0 0
\(255\) −1586.78 −0.389680
\(256\) 0 0
\(257\) 1694.78 0.411352 0.205676 0.978620i \(-0.434061\pi\)
0.205676 + 0.978620i \(0.434061\pi\)
\(258\) 0 0
\(259\) 322.268 0.0773156
\(260\) 0 0
\(261\) −2443.72 −0.579550
\(262\) 0 0
\(263\) 727.462 0.170560 0.0852799 0.996357i \(-0.472822\pi\)
0.0852799 + 0.996357i \(0.472822\pi\)
\(264\) 0 0
\(265\) −2073.54 −0.480666
\(266\) 0 0
\(267\) 3248.95 0.744691
\(268\) 0 0
\(269\) 94.9937 0.0215311 0.0107656 0.999942i \(-0.496573\pi\)
0.0107656 + 0.999942i \(0.496573\pi\)
\(270\) 0 0
\(271\) −5321.54 −1.19284 −0.596422 0.802671i \(-0.703411\pi\)
−0.596422 + 0.802671i \(0.703411\pi\)
\(272\) 0 0
\(273\) −107.064 −0.0237355
\(274\) 0 0
\(275\) −1715.66 −0.376211
\(276\) 0 0
\(277\) 2561.25 0.555562 0.277781 0.960644i \(-0.410401\pi\)
0.277781 + 0.960644i \(0.410401\pi\)
\(278\) 0 0
\(279\) −4170.56 −0.894928
\(280\) 0 0
\(281\) −8753.73 −1.85838 −0.929188 0.369606i \(-0.879493\pi\)
−0.929188 + 0.369606i \(0.879493\pi\)
\(282\) 0 0
\(283\) 2400.14 0.504147 0.252073 0.967708i \(-0.418888\pi\)
0.252073 + 0.967708i \(0.418888\pi\)
\(284\) 0 0
\(285\) −724.445 −0.150570
\(286\) 0 0
\(287\) −1178.06 −0.242295
\(288\) 0 0
\(289\) 6367.09 1.29597
\(290\) 0 0
\(291\) 3334.41 0.671706
\(292\) 0 0
\(293\) 1045.56 0.208472 0.104236 0.994553i \(-0.466760\pi\)
0.104236 + 0.994553i \(0.466760\pi\)
\(294\) 0 0
\(295\) 917.931 0.181166
\(296\) 0 0
\(297\) −9242.36 −1.80571
\(298\) 0 0
\(299\) −276.063 −0.0533952
\(300\) 0 0
\(301\) 406.171 0.0777784
\(302\) 0 0
\(303\) 402.091 0.0762361
\(304\) 0 0
\(305\) −492.376 −0.0924372
\(306\) 0 0
\(307\) 7905.84 1.46974 0.734870 0.678208i \(-0.237243\pi\)
0.734870 + 0.678208i \(0.237243\pi\)
\(308\) 0 0
\(309\) −2766.82 −0.509381
\(310\) 0 0
\(311\) 9227.63 1.68248 0.841240 0.540662i \(-0.181826\pi\)
0.841240 + 0.540662i \(0.181826\pi\)
\(312\) 0 0
\(313\) 2712.57 0.489852 0.244926 0.969542i \(-0.421236\pi\)
0.244926 + 0.969542i \(0.421236\pi\)
\(314\) 0 0
\(315\) 269.731 0.0482465
\(316\) 0 0
\(317\) −3440.29 −0.609546 −0.304773 0.952425i \(-0.598581\pi\)
−0.304773 + 0.952425i \(0.598581\pi\)
\(318\) 0 0
\(319\) −9280.04 −1.62879
\(320\) 0 0
\(321\) 5591.06 0.972158
\(322\) 0 0
\(323\) 5149.92 0.887149
\(324\) 0 0
\(325\) −300.069 −0.0512149
\(326\) 0 0
\(327\) 2923.06 0.494329
\(328\) 0 0
\(329\) −150.606 −0.0252376
\(330\) 0 0
\(331\) −3259.75 −0.541305 −0.270652 0.962677i \(-0.587239\pi\)
−0.270652 + 0.962677i \(0.587239\pi\)
\(332\) 0 0
\(333\) 1950.92 0.321050
\(334\) 0 0
\(335\) −684.623 −0.111657
\(336\) 0 0
\(337\) 7163.38 1.15790 0.578952 0.815361i \(-0.303462\pi\)
0.578952 + 0.815361i \(0.303462\pi\)
\(338\) 0 0
\(339\) 5260.37 0.842785
\(340\) 0 0
\(341\) −15837.7 −2.51513
\(342\) 0 0
\(343\) 2021.23 0.318181
\(344\) 0 0
\(345\) −343.629 −0.0536242
\(346\) 0 0
\(347\) 1833.45 0.283645 0.141822 0.989892i \(-0.454704\pi\)
0.141822 + 0.989892i \(0.454704\pi\)
\(348\) 0 0
\(349\) −10400.3 −1.59517 −0.797583 0.603209i \(-0.793888\pi\)
−0.797583 + 0.603209i \(0.793888\pi\)
\(350\) 0 0
\(351\) −1616.49 −0.245818
\(352\) 0 0
\(353\) −3156.48 −0.475928 −0.237964 0.971274i \(-0.576480\pi\)
−0.237964 + 0.971274i \(0.576480\pi\)
\(354\) 0 0
\(355\) 3544.00 0.529848
\(356\) 0 0
\(357\) 947.365 0.140448
\(358\) 0 0
\(359\) 9324.16 1.37078 0.685390 0.728176i \(-0.259632\pi\)
0.685390 + 0.728176i \(0.259632\pi\)
\(360\) 0 0
\(361\) −4507.81 −0.657211
\(362\) 0 0
\(363\) −10095.4 −1.45970
\(364\) 0 0
\(365\) −3448.02 −0.494460
\(366\) 0 0
\(367\) −3802.68 −0.540867 −0.270434 0.962739i \(-0.587167\pi\)
−0.270434 + 0.962739i \(0.587167\pi\)
\(368\) 0 0
\(369\) −7131.65 −1.00612
\(370\) 0 0
\(371\) 1237.97 0.173241
\(372\) 0 0
\(373\) −8928.03 −1.23935 −0.619673 0.784860i \(-0.712735\pi\)
−0.619673 + 0.784860i \(0.712735\pi\)
\(374\) 0 0
\(375\) −373.509 −0.0514345
\(376\) 0 0
\(377\) −1623.08 −0.221732
\(378\) 0 0
\(379\) −3798.13 −0.514768 −0.257384 0.966309i \(-0.582860\pi\)
−0.257384 + 0.966309i \(0.582860\pi\)
\(380\) 0 0
\(381\) −3767.58 −0.506612
\(382\) 0 0
\(383\) −1456.41 −0.194306 −0.0971530 0.995269i \(-0.530974\pi\)
−0.0971530 + 0.995269i \(0.530974\pi\)
\(384\) 0 0
\(385\) 1024.31 0.135593
\(386\) 0 0
\(387\) 2458.84 0.322972
\(388\) 0 0
\(389\) −8211.34 −1.07026 −0.535130 0.844769i \(-0.679738\pi\)
−0.535130 + 0.844769i \(0.679738\pi\)
\(390\) 0 0
\(391\) 2442.78 0.315950
\(392\) 0 0
\(393\) 1327.55 0.170397
\(394\) 0 0
\(395\) −2732.92 −0.348121
\(396\) 0 0
\(397\) 914.365 0.115594 0.0577968 0.998328i \(-0.481592\pi\)
0.0577968 + 0.998328i \(0.481592\pi\)
\(398\) 0 0
\(399\) 432.519 0.0542683
\(400\) 0 0
\(401\) 9414.97 1.17247 0.586236 0.810140i \(-0.300609\pi\)
0.586236 + 0.810140i \(0.300609\pi\)
\(402\) 0 0
\(403\) −2770.02 −0.342394
\(404\) 0 0
\(405\) 427.519 0.0524533
\(406\) 0 0
\(407\) 7408.62 0.902289
\(408\) 0 0
\(409\) 3299.41 0.398889 0.199444 0.979909i \(-0.436086\pi\)
0.199444 + 0.979909i \(0.436086\pi\)
\(410\) 0 0
\(411\) −1282.10 −0.153872
\(412\) 0 0
\(413\) −548.037 −0.0652957
\(414\) 0 0
\(415\) 101.311 0.0119835
\(416\) 0 0
\(417\) −746.286 −0.0876398
\(418\) 0 0
\(419\) 3554.64 0.414452 0.207226 0.978293i \(-0.433556\pi\)
0.207226 + 0.978293i \(0.433556\pi\)
\(420\) 0 0
\(421\) 11655.5 1.34930 0.674649 0.738139i \(-0.264295\pi\)
0.674649 + 0.738139i \(0.264295\pi\)
\(422\) 0 0
\(423\) −911.725 −0.104798
\(424\) 0 0
\(425\) 2655.19 0.303049
\(426\) 0 0
\(427\) 293.965 0.0333161
\(428\) 0 0
\(429\) −2461.29 −0.276998
\(430\) 0 0
\(431\) −4727.38 −0.528330 −0.264165 0.964478i \(-0.585096\pi\)
−0.264165 + 0.964478i \(0.585096\pi\)
\(432\) 0 0
\(433\) 3936.24 0.436868 0.218434 0.975852i \(-0.429905\pi\)
0.218434 + 0.975852i \(0.429905\pi\)
\(434\) 0 0
\(435\) −2020.33 −0.222683
\(436\) 0 0
\(437\) 1115.25 0.122081
\(438\) 0 0
\(439\) −6859.77 −0.745784 −0.372892 0.927875i \(-0.621634\pi\)
−0.372892 + 0.927875i \(0.621634\pi\)
\(440\) 0 0
\(441\) 6037.45 0.651922
\(442\) 0 0
\(443\) −16814.4 −1.80334 −0.901668 0.432430i \(-0.857656\pi\)
−0.901668 + 0.432430i \(0.857656\pi\)
\(444\) 0 0
\(445\) −5436.53 −0.579137
\(446\) 0 0
\(447\) 4673.90 0.494559
\(448\) 0 0
\(449\) 10673.8 1.12189 0.560943 0.827855i \(-0.310439\pi\)
0.560943 + 0.827855i \(0.310439\pi\)
\(450\) 0 0
\(451\) −27082.5 −2.82764
\(452\) 0 0
\(453\) −5557.94 −0.576457
\(454\) 0 0
\(455\) 179.152 0.0184588
\(456\) 0 0
\(457\) 15493.0 1.58585 0.792926 0.609318i \(-0.208557\pi\)
0.792926 + 0.609318i \(0.208557\pi\)
\(458\) 0 0
\(459\) 14303.7 1.45455
\(460\) 0 0
\(461\) −10793.0 −1.09041 −0.545207 0.838302i \(-0.683549\pi\)
−0.545207 + 0.838302i \(0.683549\pi\)
\(462\) 0 0
\(463\) 6165.87 0.618904 0.309452 0.950915i \(-0.399854\pi\)
0.309452 + 0.950915i \(0.399854\pi\)
\(464\) 0 0
\(465\) −3447.97 −0.343862
\(466\) 0 0
\(467\) 10736.4 1.06386 0.531929 0.846789i \(-0.321467\pi\)
0.531929 + 0.846789i \(0.321467\pi\)
\(468\) 0 0
\(469\) 408.744 0.0402431
\(470\) 0 0
\(471\) −8469.77 −0.828591
\(472\) 0 0
\(473\) 9337.47 0.907690
\(474\) 0 0
\(475\) 1212.23 0.117096
\(476\) 0 0
\(477\) 7494.35 0.719377
\(478\) 0 0
\(479\) −6333.30 −0.604125 −0.302062 0.953288i \(-0.597675\pi\)
−0.302062 + 0.953288i \(0.597675\pi\)
\(480\) 0 0
\(481\) 1295.77 0.122832
\(482\) 0 0
\(483\) 205.158 0.0193272
\(484\) 0 0
\(485\) −5579.52 −0.522377
\(486\) 0 0
\(487\) 16833.3 1.56630 0.783149 0.621834i \(-0.213612\pi\)
0.783149 + 0.621834i \(0.213612\pi\)
\(488\) 0 0
\(489\) 6318.33 0.584304
\(490\) 0 0
\(491\) 4.80428 0.000441577 0 0.000220788 1.00000i \(-0.499930\pi\)
0.000220788 1.00000i \(0.499930\pi\)
\(492\) 0 0
\(493\) 14362.0 1.31204
\(494\) 0 0
\(495\) 6200.86 0.563046
\(496\) 0 0
\(497\) −2115.89 −0.190967
\(498\) 0 0
\(499\) −21206.9 −1.90251 −0.951253 0.308412i \(-0.900203\pi\)
−0.951253 + 0.308412i \(0.900203\pi\)
\(500\) 0 0
\(501\) 7432.56 0.662799
\(502\) 0 0
\(503\) −14513.7 −1.28655 −0.643275 0.765635i \(-0.722425\pi\)
−0.643275 + 0.765635i \(0.722425\pi\)
\(504\) 0 0
\(505\) −672.827 −0.0592879
\(506\) 0 0
\(507\) 6134.32 0.537347
\(508\) 0 0
\(509\) −2545.77 −0.221689 −0.110844 0.993838i \(-0.535355\pi\)
−0.110844 + 0.993838i \(0.535355\pi\)
\(510\) 0 0
\(511\) 2058.59 0.178213
\(512\) 0 0
\(513\) 6530.35 0.562032
\(514\) 0 0
\(515\) 4629.77 0.396140
\(516\) 0 0
\(517\) −3462.28 −0.294528
\(518\) 0 0
\(519\) −10973.2 −0.928070
\(520\) 0 0
\(521\) −21343.1 −1.79473 −0.897367 0.441285i \(-0.854523\pi\)
−0.897367 + 0.441285i \(0.854523\pi\)
\(522\) 0 0
\(523\) 789.191 0.0659827 0.0329913 0.999456i \(-0.489497\pi\)
0.0329913 + 0.999456i \(0.489497\pi\)
\(524\) 0 0
\(525\) 222.998 0.0185380
\(526\) 0 0
\(527\) 24510.9 2.02602
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −3317.66 −0.271138
\(532\) 0 0
\(533\) −4736.74 −0.384936
\(534\) 0 0
\(535\) −9355.63 −0.756036
\(536\) 0 0
\(537\) −9635.42 −0.774300
\(538\) 0 0
\(539\) 22927.2 1.83218
\(540\) 0 0
\(541\) 13405.1 1.06531 0.532653 0.846334i \(-0.321195\pi\)
0.532653 + 0.846334i \(0.321195\pi\)
\(542\) 0 0
\(543\) 6646.16 0.525256
\(544\) 0 0
\(545\) −4891.21 −0.384434
\(546\) 0 0
\(547\) −10088.8 −0.788605 −0.394303 0.918981i \(-0.629014\pi\)
−0.394303 + 0.918981i \(0.629014\pi\)
\(548\) 0 0
\(549\) 1779.58 0.138344
\(550\) 0 0
\(551\) 6556.98 0.506963
\(552\) 0 0
\(553\) 1631.65 0.125470
\(554\) 0 0
\(555\) 1612.90 0.123359
\(556\) 0 0
\(557\) −11657.7 −0.886809 −0.443405 0.896322i \(-0.646230\pi\)
−0.443405 + 0.896322i \(0.646230\pi\)
\(558\) 0 0
\(559\) 1633.13 0.123567
\(560\) 0 0
\(561\) 21779.0 1.63906
\(562\) 0 0
\(563\) −4839.43 −0.362270 −0.181135 0.983458i \(-0.557977\pi\)
−0.181135 + 0.983458i \(0.557977\pi\)
\(564\) 0 0
\(565\) −8802.27 −0.655424
\(566\) 0 0
\(567\) −255.244 −0.0189052
\(568\) 0 0
\(569\) −646.680 −0.0476454 −0.0238227 0.999716i \(-0.507584\pi\)
−0.0238227 + 0.999716i \(0.507584\pi\)
\(570\) 0 0
\(571\) −3263.66 −0.239194 −0.119597 0.992822i \(-0.538160\pi\)
−0.119597 + 0.992822i \(0.538160\pi\)
\(572\) 0 0
\(573\) 9781.54 0.713141
\(574\) 0 0
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) −8980.46 −0.647940 −0.323970 0.946067i \(-0.605018\pi\)
−0.323970 + 0.946067i \(0.605018\pi\)
\(578\) 0 0
\(579\) −15319.5 −1.09958
\(580\) 0 0
\(581\) −60.4861 −0.00431908
\(582\) 0 0
\(583\) 28459.8 2.02176
\(584\) 0 0
\(585\) 1084.53 0.0766495
\(586\) 0 0
\(587\) 6538.05 0.459718 0.229859 0.973224i \(-0.426174\pi\)
0.229859 + 0.973224i \(0.426174\pi\)
\(588\) 0 0
\(589\) 11190.4 0.782841
\(590\) 0 0
\(591\) −9748.80 −0.678532
\(592\) 0 0
\(593\) 7803.41 0.540384 0.270192 0.962806i \(-0.412913\pi\)
0.270192 + 0.962806i \(0.412913\pi\)
\(594\) 0 0
\(595\) −1585.24 −0.109225
\(596\) 0 0
\(597\) −502.839 −0.0344721
\(598\) 0 0
\(599\) 27812.0 1.89711 0.948555 0.316613i \(-0.102546\pi\)
0.948555 + 0.316613i \(0.102546\pi\)
\(600\) 0 0
\(601\) 483.216 0.0327966 0.0163983 0.999866i \(-0.494780\pi\)
0.0163983 + 0.999866i \(0.494780\pi\)
\(602\) 0 0
\(603\) 2474.42 0.167108
\(604\) 0 0
\(605\) 16892.8 1.13519
\(606\) 0 0
\(607\) 18104.0 1.21058 0.605289 0.796006i \(-0.293058\pi\)
0.605289 + 0.796006i \(0.293058\pi\)
\(608\) 0 0
\(609\) 1206.20 0.0802592
\(610\) 0 0
\(611\) −605.554 −0.0400951
\(612\) 0 0
\(613\) 2191.19 0.144374 0.0721870 0.997391i \(-0.477002\pi\)
0.0721870 + 0.997391i \(0.477002\pi\)
\(614\) 0 0
\(615\) −5896.03 −0.386587
\(616\) 0 0
\(617\) −8243.06 −0.537849 −0.268925 0.963161i \(-0.586668\pi\)
−0.268925 + 0.963161i \(0.586668\pi\)
\(618\) 0 0
\(619\) −18298.0 −1.18814 −0.594069 0.804414i \(-0.702479\pi\)
−0.594069 + 0.804414i \(0.702479\pi\)
\(620\) 0 0
\(621\) 3097.57 0.200163
\(622\) 0 0
\(623\) 3245.80 0.208732
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 9943.19 0.633322
\(628\) 0 0
\(629\) −11465.8 −0.726821
\(630\) 0 0
\(631\) 26472.0 1.67010 0.835051 0.550172i \(-0.185438\pi\)
0.835051 + 0.550172i \(0.185438\pi\)
\(632\) 0 0
\(633\) 12541.0 0.787455
\(634\) 0 0
\(635\) 6304.37 0.393986
\(636\) 0 0
\(637\) 4009.99 0.249421
\(638\) 0 0
\(639\) −12809.0 −0.792984
\(640\) 0 0
\(641\) 7411.93 0.456714 0.228357 0.973577i \(-0.426665\pi\)
0.228357 + 0.973577i \(0.426665\pi\)
\(642\) 0 0
\(643\) −8405.96 −0.515550 −0.257775 0.966205i \(-0.582989\pi\)
−0.257775 + 0.966205i \(0.582989\pi\)
\(644\) 0 0
\(645\) 2032.83 0.124097
\(646\) 0 0
\(647\) −27878.3 −1.69398 −0.846992 0.531605i \(-0.821589\pi\)
−0.846992 + 0.531605i \(0.821589\pi\)
\(648\) 0 0
\(649\) −12598.8 −0.762014
\(650\) 0 0
\(651\) 2058.56 0.123934
\(652\) 0 0
\(653\) −18871.1 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(654\) 0 0
\(655\) −2221.42 −0.132516
\(656\) 0 0
\(657\) 12462.1 0.740021
\(658\) 0 0
\(659\) 12144.4 0.717872 0.358936 0.933362i \(-0.383140\pi\)
0.358936 + 0.933362i \(0.383140\pi\)
\(660\) 0 0
\(661\) −32504.2 −1.91266 −0.956328 0.292295i \(-0.905581\pi\)
−0.956328 + 0.292295i \(0.905581\pi\)
\(662\) 0 0
\(663\) 3809.16 0.223130
\(664\) 0 0
\(665\) −723.742 −0.0422038
\(666\) 0 0
\(667\) 3110.19 0.180551
\(668\) 0 0
\(669\) 15516.1 0.896690
\(670\) 0 0
\(671\) 6757.98 0.388806
\(672\) 0 0
\(673\) −21387.8 −1.22502 −0.612511 0.790462i \(-0.709840\pi\)
−0.612511 + 0.790462i \(0.709840\pi\)
\(674\) 0 0
\(675\) 3366.92 0.191989
\(676\) 0 0
\(677\) 10494.6 0.595776 0.297888 0.954601i \(-0.403718\pi\)
0.297888 + 0.954601i \(0.403718\pi\)
\(678\) 0 0
\(679\) 3331.17 0.188275
\(680\) 0 0
\(681\) 14047.6 0.790466
\(682\) 0 0
\(683\) −24014.2 −1.34535 −0.672677 0.739937i \(-0.734855\pi\)
−0.672677 + 0.739937i \(0.734855\pi\)
\(684\) 0 0
\(685\) 2145.36 0.119664
\(686\) 0 0
\(687\) 6350.18 0.352656
\(688\) 0 0
\(689\) 4977.63 0.275229
\(690\) 0 0
\(691\) 10825.8 0.595995 0.297997 0.954567i \(-0.403681\pi\)
0.297997 + 0.954567i \(0.403681\pi\)
\(692\) 0 0
\(693\) −3702.13 −0.202933
\(694\) 0 0
\(695\) 1248.77 0.0681564
\(696\) 0 0
\(697\) 41913.6 2.27775
\(698\) 0 0
\(699\) 4920.38 0.266246
\(700\) 0 0
\(701\) −171.021 −0.00921451 −0.00460726 0.999989i \(-0.501467\pi\)
−0.00460726 + 0.999989i \(0.501467\pi\)
\(702\) 0 0
\(703\) −5234.69 −0.280840
\(704\) 0 0
\(705\) −753.760 −0.0402670
\(706\) 0 0
\(707\) 401.701 0.0213685
\(708\) 0 0
\(709\) 28732.7 1.52197 0.760987 0.648767i \(-0.224715\pi\)
0.760987 + 0.648767i \(0.224715\pi\)
\(710\) 0 0
\(711\) 9877.53 0.521007
\(712\) 0 0
\(713\) 5307.99 0.278802
\(714\) 0 0
\(715\) 4118.52 0.215418
\(716\) 0 0
\(717\) 8599.31 0.447904
\(718\) 0 0
\(719\) −6976.22 −0.361849 −0.180924 0.983497i \(-0.557909\pi\)
−0.180924 + 0.983497i \(0.557909\pi\)
\(720\) 0 0
\(721\) −2764.13 −0.142776
\(722\) 0 0
\(723\) 12683.0 0.652401
\(724\) 0 0
\(725\) 3380.65 0.173178
\(726\) 0 0
\(727\) −13114.6 −0.669042 −0.334521 0.942388i \(-0.608574\pi\)
−0.334521 + 0.942388i \(0.608574\pi\)
\(728\) 0 0
\(729\) 9320.28 0.473519
\(730\) 0 0
\(731\) −14450.9 −0.731172
\(732\) 0 0
\(733\) −22681.7 −1.14293 −0.571466 0.820626i \(-0.693625\pi\)
−0.571466 + 0.820626i \(0.693625\pi\)
\(734\) 0 0
\(735\) 4991.41 0.250491
\(736\) 0 0
\(737\) 9396.62 0.469646
\(738\) 0 0
\(739\) −24237.8 −1.20650 −0.603249 0.797553i \(-0.706127\pi\)
−0.603249 + 0.797553i \(0.706127\pi\)
\(740\) 0 0
\(741\) 1739.07 0.0862163
\(742\) 0 0
\(743\) −27356.7 −1.35077 −0.675384 0.737466i \(-0.736022\pi\)
−0.675384 + 0.737466i \(0.736022\pi\)
\(744\) 0 0
\(745\) −7820.92 −0.384612
\(746\) 0 0
\(747\) −366.166 −0.0179348
\(748\) 0 0
\(749\) 5585.63 0.272489
\(750\) 0 0
\(751\) −22297.8 −1.08343 −0.541716 0.840561i \(-0.682225\pi\)
−0.541716 + 0.840561i \(0.682225\pi\)
\(752\) 0 0
\(753\) 13153.5 0.636573
\(754\) 0 0
\(755\) 9300.21 0.448304
\(756\) 0 0
\(757\) −16088.4 −0.772447 −0.386224 0.922405i \(-0.626221\pi\)
−0.386224 + 0.922405i \(0.626221\pi\)
\(758\) 0 0
\(759\) 4716.39 0.225552
\(760\) 0 0
\(761\) 22500.2 1.07179 0.535894 0.844285i \(-0.319975\pi\)
0.535894 + 0.844285i \(0.319975\pi\)
\(762\) 0 0
\(763\) 2920.22 0.138557
\(764\) 0 0
\(765\) −9596.62 −0.453551
\(766\) 0 0
\(767\) −2203.54 −0.103736
\(768\) 0 0
\(769\) −22677.7 −1.06343 −0.531716 0.846923i \(-0.678453\pi\)
−0.531716 + 0.846923i \(0.678453\pi\)
\(770\) 0 0
\(771\) −5064.13 −0.236550
\(772\) 0 0
\(773\) 20465.1 0.952235 0.476118 0.879382i \(-0.342044\pi\)
0.476118 + 0.879382i \(0.342044\pi\)
\(774\) 0 0
\(775\) 5769.56 0.267418
\(776\) 0 0
\(777\) −962.960 −0.0444607
\(778\) 0 0
\(779\) 19135.6 0.880108
\(780\) 0 0
\(781\) −48642.3 −2.22863
\(782\) 0 0
\(783\) 18211.8 0.831208
\(784\) 0 0
\(785\) 14172.6 0.644386
\(786\) 0 0
\(787\) 28850.8 1.30676 0.653379 0.757031i \(-0.273351\pi\)
0.653379 + 0.757031i \(0.273351\pi\)
\(788\) 0 0
\(789\) −2173.71 −0.0980813
\(790\) 0 0
\(791\) 5255.26 0.236227
\(792\) 0 0
\(793\) 1181.97 0.0529295
\(794\) 0 0
\(795\) 6195.88 0.276409
\(796\) 0 0
\(797\) −39332.9 −1.74811 −0.874054 0.485828i \(-0.838518\pi\)
−0.874054 + 0.485828i \(0.838518\pi\)
\(798\) 0 0
\(799\) 5358.31 0.237251
\(800\) 0 0
\(801\) 19649.1 0.866752
\(802\) 0 0
\(803\) 47325.0 2.07978
\(804\) 0 0
\(805\) −343.295 −0.0150305
\(806\) 0 0
\(807\) −283.848 −0.0123816
\(808\) 0 0
\(809\) 33946.5 1.47527 0.737637 0.675198i \(-0.235942\pi\)
0.737637 + 0.675198i \(0.235942\pi\)
\(810\) 0 0
\(811\) 20772.2 0.899398 0.449699 0.893180i \(-0.351531\pi\)
0.449699 + 0.893180i \(0.351531\pi\)
\(812\) 0 0
\(813\) 15901.2 0.685951
\(814\) 0 0
\(815\) −10572.6 −0.454406
\(816\) 0 0
\(817\) −6597.56 −0.282521
\(818\) 0 0
\(819\) −647.504 −0.0276259
\(820\) 0 0
\(821\) 6466.37 0.274882 0.137441 0.990510i \(-0.456112\pi\)
0.137441 + 0.990510i \(0.456112\pi\)
\(822\) 0 0
\(823\) 16503.7 0.699008 0.349504 0.936935i \(-0.386350\pi\)
0.349504 + 0.936935i \(0.386350\pi\)
\(824\) 0 0
\(825\) 5126.51 0.216342
\(826\) 0 0
\(827\) −36540.8 −1.53645 −0.768227 0.640177i \(-0.778861\pi\)
−0.768227 + 0.640177i \(0.778861\pi\)
\(828\) 0 0
\(829\) −24874.5 −1.04213 −0.521066 0.853516i \(-0.674465\pi\)
−0.521066 + 0.853516i \(0.674465\pi\)
\(830\) 0 0
\(831\) −7653.21 −0.319479
\(832\) 0 0
\(833\) −35482.8 −1.47588
\(834\) 0 0
\(835\) −12437.0 −0.515450
\(836\) 0 0
\(837\) 31081.0 1.28353
\(838\) 0 0
\(839\) −15814.3 −0.650741 −0.325370 0.945587i \(-0.605489\pi\)
−0.325370 + 0.945587i \(0.605489\pi\)
\(840\) 0 0
\(841\) −6102.97 −0.250234
\(842\) 0 0
\(843\) 26156.8 1.06867
\(844\) 0 0
\(845\) −10264.7 −0.417888
\(846\) 0 0
\(847\) −10085.6 −0.409144
\(848\) 0 0
\(849\) −7171.80 −0.289912
\(850\) 0 0
\(851\) −2482.99 −0.100019
\(852\) 0 0
\(853\) −31093.2 −1.24808 −0.624039 0.781394i \(-0.714509\pi\)
−0.624039 + 0.781394i \(0.714509\pi\)
\(854\) 0 0
\(855\) −4381.33 −0.175250
\(856\) 0 0
\(857\) 34212.4 1.36368 0.681840 0.731501i \(-0.261180\pi\)
0.681840 + 0.731501i \(0.261180\pi\)
\(858\) 0 0
\(859\) −4821.57 −0.191513 −0.0957567 0.995405i \(-0.530527\pi\)
−0.0957567 + 0.995405i \(0.530527\pi\)
\(860\) 0 0
\(861\) 3520.13 0.139333
\(862\) 0 0
\(863\) −19145.6 −0.755185 −0.377593 0.925972i \(-0.623248\pi\)
−0.377593 + 0.925972i \(0.623248\pi\)
\(864\) 0 0
\(865\) 18361.6 0.721748
\(866\) 0 0
\(867\) −19025.4 −0.745254
\(868\) 0 0
\(869\) 37509.9 1.46426
\(870\) 0 0
\(871\) 1643.47 0.0639345
\(872\) 0 0
\(873\) 20166.0 0.781804
\(874\) 0 0
\(875\) −373.147 −0.0144168
\(876\) 0 0
\(877\) 16890.8 0.650354 0.325177 0.945653i \(-0.394576\pi\)
0.325177 + 0.945653i \(0.394576\pi\)
\(878\) 0 0
\(879\) −3124.21 −0.119883
\(880\) 0 0
\(881\) −27802.3 −1.06321 −0.531603 0.846994i \(-0.678410\pi\)
−0.531603 + 0.846994i \(0.678410\pi\)
\(882\) 0 0
\(883\) −9794.15 −0.373272 −0.186636 0.982429i \(-0.559759\pi\)
−0.186636 + 0.982429i \(0.559759\pi\)
\(884\) 0 0
\(885\) −2742.85 −0.104181
\(886\) 0 0
\(887\) 41652.4 1.57672 0.788360 0.615214i \(-0.210930\pi\)
0.788360 + 0.615214i \(0.210930\pi\)
\(888\) 0 0
\(889\) −3763.93 −0.142000
\(890\) 0 0
\(891\) −5867.80 −0.220627
\(892\) 0 0
\(893\) 2446.34 0.0916724
\(894\) 0 0
\(895\) 16123.1 0.602164
\(896\) 0 0
\(897\) 824.898 0.0307052
\(898\) 0 0
\(899\) 31207.7 1.15777
\(900\) 0 0
\(901\) −44045.2 −1.62859
\(902\) 0 0
\(903\) −1213.67 −0.0447269
\(904\) 0 0
\(905\) −11121.1 −0.408485
\(906\) 0 0
\(907\) −5467.21 −0.200150 −0.100075 0.994980i \(-0.531908\pi\)
−0.100075 + 0.994980i \(0.531908\pi\)
\(908\) 0 0
\(909\) 2431.79 0.0887318
\(910\) 0 0
\(911\) −33520.9 −1.21910 −0.609549 0.792749i \(-0.708649\pi\)
−0.609549 + 0.792749i \(0.708649\pi\)
\(912\) 0 0
\(913\) −1390.52 −0.0504046
\(914\) 0 0
\(915\) 1471.26 0.0531565
\(916\) 0 0
\(917\) 1326.26 0.0477612
\(918\) 0 0
\(919\) −21423.9 −0.768996 −0.384498 0.923126i \(-0.625625\pi\)
−0.384498 + 0.923126i \(0.625625\pi\)
\(920\) 0 0
\(921\) −23623.2 −0.845182
\(922\) 0 0
\(923\) −8507.56 −0.303391
\(924\) 0 0
\(925\) −2698.90 −0.0959344
\(926\) 0 0
\(927\) −16733.3 −0.592873
\(928\) 0 0
\(929\) 25838.9 0.912538 0.456269 0.889842i \(-0.349186\pi\)
0.456269 + 0.889842i \(0.349186\pi\)
\(930\) 0 0
\(931\) −16199.7 −0.570271
\(932\) 0 0
\(933\) −27572.9 −0.967519
\(934\) 0 0
\(935\) −36443.2 −1.27467
\(936\) 0 0
\(937\) 48095.9 1.67687 0.838434 0.545003i \(-0.183472\pi\)
0.838434 + 0.545003i \(0.183472\pi\)
\(938\) 0 0
\(939\) −8105.38 −0.281692
\(940\) 0 0
\(941\) −32450.4 −1.12418 −0.562089 0.827077i \(-0.690002\pi\)
−0.562089 + 0.827077i \(0.690002\pi\)
\(942\) 0 0
\(943\) 9076.66 0.313443
\(944\) 0 0
\(945\) −2010.17 −0.0691966
\(946\) 0 0
\(947\) −39067.9 −1.34059 −0.670294 0.742096i \(-0.733832\pi\)
−0.670294 + 0.742096i \(0.733832\pi\)
\(948\) 0 0
\(949\) 8277.16 0.283127
\(950\) 0 0
\(951\) 10279.8 0.350522
\(952\) 0 0
\(953\) 17756.2 0.603546 0.301773 0.953380i \(-0.402422\pi\)
0.301773 + 0.953380i \(0.402422\pi\)
\(954\) 0 0
\(955\) −16367.6 −0.554601
\(956\) 0 0
\(957\) 27729.5 0.936642
\(958\) 0 0
\(959\) −1280.85 −0.0431292
\(960\) 0 0
\(961\) 23469.5 0.787805
\(962\) 0 0
\(963\) 33813.9 1.13150
\(964\) 0 0
\(965\) 25634.3 0.855127
\(966\) 0 0
\(967\) 10696.2 0.355705 0.177853 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(968\) 0 0
\(969\) −15388.3 −0.510160
\(970\) 0 0
\(971\) 53635.1 1.77264 0.886320 0.463074i \(-0.153254\pi\)
0.886320 + 0.463074i \(0.153254\pi\)
\(972\) 0 0
\(973\) −745.561 −0.0245649
\(974\) 0 0
\(975\) 896.629 0.0294514
\(976\) 0 0
\(977\) −21627.7 −0.708220 −0.354110 0.935204i \(-0.615216\pi\)
−0.354110 + 0.935204i \(0.615216\pi\)
\(978\) 0 0
\(979\) 74617.7 2.43595
\(980\) 0 0
\(981\) 17678.2 0.575353
\(982\) 0 0
\(983\) −12031.4 −0.390378 −0.195189 0.980766i \(-0.562532\pi\)
−0.195189 + 0.980766i \(0.562532\pi\)
\(984\) 0 0
\(985\) 16312.8 0.527686
\(986\) 0 0
\(987\) 450.021 0.0145130
\(988\) 0 0
\(989\) −3129.44 −0.100617
\(990\) 0 0
\(991\) 24573.0 0.787675 0.393838 0.919180i \(-0.371147\pi\)
0.393838 + 0.919180i \(0.371147\pi\)
\(992\) 0 0
\(993\) 9740.36 0.311280
\(994\) 0 0
\(995\) 841.410 0.0268085
\(996\) 0 0
\(997\) −14209.7 −0.451379 −0.225690 0.974199i \(-0.572464\pi\)
−0.225690 + 0.974199i \(0.572464\pi\)
\(998\) 0 0
\(999\) −14539.2 −0.460460
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 1840.4.a.k.1.3 4
4.3 odd 2 230.4.a.j.1.2 4
12.11 even 2 2070.4.a.bg.1.2 4
20.3 even 4 1150.4.b.o.599.2 8
20.7 even 4 1150.4.b.o.599.7 8
20.19 odd 2 1150.4.a.n.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.j.1.2 4 4.3 odd 2
1150.4.a.n.1.3 4 20.19 odd 2
1150.4.b.o.599.2 8 20.3 even 4
1150.4.b.o.599.7 8 20.7 even 4
1840.4.a.k.1.3 4 1.1 even 1 trivial
2070.4.a.bg.1.2 4 12.11 even 2