Properties

Label 2151.4.a.b.1.12
Level $2151$
Weight $4$
Character 2151.1
Self dual yes
Analytic conductor $126.913$
Analytic rank $1$
Dimension $28$
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] = [2151,4,Mod(1,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2151.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2151.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: \(126.913108422\)
Analytic rank: \(1\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 2151.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.915825 q^{2} -7.16127 q^{4} -7.54581 q^{5} +24.7091 q^{7} +13.8851 q^{8} +O(q^{10})\) \(q-0.915825 q^{2} -7.16127 q^{4} -7.54581 q^{5} +24.7091 q^{7} +13.8851 q^{8} +6.91064 q^{10} +49.4899 q^{11} -26.5926 q^{13} -22.6292 q^{14} +44.5738 q^{16} -21.6855 q^{17} -36.0776 q^{19} +54.0376 q^{20} -45.3241 q^{22} -137.325 q^{23} -68.0607 q^{25} +24.3541 q^{26} -176.948 q^{28} +0.350892 q^{29} +302.936 q^{31} -151.902 q^{32} +19.8601 q^{34} -186.450 q^{35} -340.352 q^{37} +33.0407 q^{38} -104.774 q^{40} +31.5267 q^{41} -291.827 q^{43} -354.410 q^{44} +125.766 q^{46} +576.423 q^{47} +267.538 q^{49} +62.3317 q^{50} +190.437 q^{52} -108.321 q^{53} -373.442 q^{55} +343.087 q^{56} -0.321356 q^{58} -138.023 q^{59} +622.364 q^{61} -277.436 q^{62} -217.475 q^{64} +200.663 q^{65} +348.715 q^{67} +155.295 q^{68} +170.755 q^{70} +322.287 q^{71} -1219.41 q^{73} +311.703 q^{74} +258.361 q^{76} +1222.85 q^{77} -1116.69 q^{79} -336.346 q^{80} -28.8729 q^{82} +379.080 q^{83} +163.634 q^{85} +267.262 q^{86} +687.170 q^{88} +1277.30 q^{89} -657.078 q^{91} +983.420 q^{92} -527.902 q^{94} +272.235 q^{95} -814.774 q^{97} -245.017 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 5 q^{2} + 103 q^{4} - 6 q^{5} - 68 q^{7} + 39 q^{8} - 88 q^{10} + 110 q^{11} - 82 q^{13} - 126 q^{14} + 271 q^{16} - 100 q^{17} - 292 q^{19} + 52 q^{20} - 351 q^{22} + 276 q^{23} + 386 q^{25} - 84 q^{26} - 1010 q^{28} + 38 q^{29} - 432 q^{31} + 452 q^{32} - 524 q^{34} + 166 q^{35} - 936 q^{37} + 41 q^{38} - 1183 q^{40} - 1054 q^{41} - 1804 q^{43} + 341 q^{44} - 888 q^{46} + 560 q^{47} + 1074 q^{49} + 1054 q^{50} - 632 q^{52} + 160 q^{53} - 842 q^{55} - 509 q^{56} - 1266 q^{58} - 846 q^{59} - 2220 q^{61} - 82 q^{62} - 1565 q^{64} - 296 q^{65} - 4752 q^{67} + 1719 q^{68} - 5601 q^{70} + 802 q^{71} - 2732 q^{73} + 4581 q^{74} - 5614 q^{76} + 1008 q^{77} - 3172 q^{79} + 732 q^{80} - 9709 q^{82} + 4780 q^{83} - 4624 q^{85} + 2009 q^{86} - 9331 q^{88} - 4372 q^{89} - 7398 q^{91} + 6138 q^{92} - 7068 q^{94} + 3160 q^{95} - 4846 q^{97} + 3772 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.915825 −0.323793 −0.161896 0.986808i \(-0.551761\pi\)
−0.161896 + 0.986808i \(0.551761\pi\)
\(3\) 0 0
\(4\) −7.16127 −0.895158
\(5\) −7.54581 −0.674918 −0.337459 0.941340i \(-0.609567\pi\)
−0.337459 + 0.941340i \(0.609567\pi\)
\(6\) 0 0
\(7\) 24.7091 1.33416 0.667082 0.744984i \(-0.267543\pi\)
0.667082 + 0.744984i \(0.267543\pi\)
\(8\) 13.8851 0.613639
\(9\) 0 0
\(10\) 6.91064 0.218534
\(11\) 49.4899 1.35652 0.678262 0.734820i \(-0.262733\pi\)
0.678262 + 0.734820i \(0.262733\pi\)
\(12\) 0 0
\(13\) −26.5926 −0.567343 −0.283671 0.958922i \(-0.591552\pi\)
−0.283671 + 0.958922i \(0.591552\pi\)
\(14\) −22.6292 −0.431993
\(15\) 0 0
\(16\) 44.5738 0.696466
\(17\) −21.6855 −0.309382 −0.154691 0.987963i \(-0.549438\pi\)
−0.154691 + 0.987963i \(0.549438\pi\)
\(18\) 0 0
\(19\) −36.0776 −0.435619 −0.217810 0.975991i \(-0.569891\pi\)
−0.217810 + 0.975991i \(0.569891\pi\)
\(20\) 54.0376 0.604158
\(21\) 0 0
\(22\) −45.3241 −0.439233
\(23\) −137.325 −1.24497 −0.622483 0.782633i \(-0.713876\pi\)
−0.622483 + 0.782633i \(0.713876\pi\)
\(24\) 0 0
\(25\) −68.0607 −0.544486
\(26\) 24.3541 0.183702
\(27\) 0 0
\(28\) −176.948 −1.19429
\(29\) 0.350892 0.00224686 0.00112343 0.999999i \(-0.499642\pi\)
0.00112343 + 0.999999i \(0.499642\pi\)
\(30\) 0 0
\(31\) 302.936 1.75512 0.877562 0.479464i \(-0.159169\pi\)
0.877562 + 0.479464i \(0.159169\pi\)
\(32\) −151.902 −0.839150
\(33\) 0 0
\(34\) 19.8601 0.100176
\(35\) −186.450 −0.900451
\(36\) 0 0
\(37\) −340.352 −1.51226 −0.756128 0.654424i \(-0.772911\pi\)
−0.756128 + 0.654424i \(0.772911\pi\)
\(38\) 33.0407 0.141050
\(39\) 0 0
\(40\) −104.774 −0.414156
\(41\) 31.5267 0.120089 0.0600444 0.998196i \(-0.480876\pi\)
0.0600444 + 0.998196i \(0.480876\pi\)
\(42\) 0 0
\(43\) −291.827 −1.03496 −0.517478 0.855696i \(-0.673129\pi\)
−0.517478 + 0.855696i \(0.673129\pi\)
\(44\) −354.410 −1.21430
\(45\) 0 0
\(46\) 125.766 0.403111
\(47\) 576.423 1.78893 0.894467 0.447134i \(-0.147555\pi\)
0.894467 + 0.447134i \(0.147555\pi\)
\(48\) 0 0
\(49\) 267.538 0.779993
\(50\) 62.3317 0.176301
\(51\) 0 0
\(52\) 190.437 0.507861
\(53\) −108.321 −0.280737 −0.140368 0.990099i \(-0.544829\pi\)
−0.140368 + 0.990099i \(0.544829\pi\)
\(54\) 0 0
\(55\) −373.442 −0.915543
\(56\) 343.087 0.818695
\(57\) 0 0
\(58\) −0.321356 −0.000727519 0
\(59\) −138.023 −0.304561 −0.152281 0.988337i \(-0.548662\pi\)
−0.152281 + 0.988337i \(0.548662\pi\)
\(60\) 0 0
\(61\) 622.364 1.30632 0.653160 0.757220i \(-0.273443\pi\)
0.653160 + 0.757220i \(0.273443\pi\)
\(62\) −277.436 −0.568297
\(63\) 0 0
\(64\) −217.475 −0.424755
\(65\) 200.663 0.382910
\(66\) 0 0
\(67\) 348.715 0.635856 0.317928 0.948115i \(-0.397013\pi\)
0.317928 + 0.948115i \(0.397013\pi\)
\(68\) 155.295 0.276946
\(69\) 0 0
\(70\) 170.755 0.291560
\(71\) 322.287 0.538711 0.269355 0.963041i \(-0.413189\pi\)
0.269355 + 0.963041i \(0.413189\pi\)
\(72\) 0 0
\(73\) −1219.41 −1.95509 −0.977543 0.210737i \(-0.932414\pi\)
−0.977543 + 0.210737i \(0.932414\pi\)
\(74\) 311.703 0.489658
\(75\) 0 0
\(76\) 258.361 0.389948
\(77\) 1222.85 1.80983
\(78\) 0 0
\(79\) −1116.69 −1.59035 −0.795177 0.606378i \(-0.792622\pi\)
−0.795177 + 0.606378i \(0.792622\pi\)
\(80\) −336.346 −0.470058
\(81\) 0 0
\(82\) −28.8729 −0.0388839
\(83\) 379.080 0.501319 0.250659 0.968075i \(-0.419353\pi\)
0.250659 + 0.968075i \(0.419353\pi\)
\(84\) 0 0
\(85\) 163.634 0.208808
\(86\) 267.262 0.335112
\(87\) 0 0
\(88\) 687.170 0.832416
\(89\) 1277.30 1.52127 0.760637 0.649177i \(-0.224887\pi\)
0.760637 + 0.649177i \(0.224887\pi\)
\(90\) 0 0
\(91\) −657.078 −0.756928
\(92\) 983.420 1.11444
\(93\) 0 0
\(94\) −527.902 −0.579244
\(95\) 272.235 0.294007
\(96\) 0 0
\(97\) −814.774 −0.852864 −0.426432 0.904520i \(-0.640230\pi\)
−0.426432 + 0.904520i \(0.640230\pi\)
\(98\) −245.017 −0.252556
\(99\) 0 0
\(100\) 487.401 0.487401
\(101\) 421.449 0.415206 0.207603 0.978213i \(-0.433434\pi\)
0.207603 + 0.978213i \(0.433434\pi\)
\(102\) 0 0
\(103\) 32.5268 0.0311161 0.0155581 0.999879i \(-0.495048\pi\)
0.0155581 + 0.999879i \(0.495048\pi\)
\(104\) −369.240 −0.348143
\(105\) 0 0
\(106\) 99.2032 0.0909007
\(107\) 1387.59 1.25368 0.626840 0.779148i \(-0.284348\pi\)
0.626840 + 0.779148i \(0.284348\pi\)
\(108\) 0 0
\(109\) −1069.15 −0.939503 −0.469752 0.882799i \(-0.655657\pi\)
−0.469752 + 0.882799i \(0.655657\pi\)
\(110\) 342.007 0.296446
\(111\) 0 0
\(112\) 1101.38 0.929200
\(113\) 519.470 0.432457 0.216228 0.976343i \(-0.430624\pi\)
0.216228 + 0.976343i \(0.430624\pi\)
\(114\) 0 0
\(115\) 1036.23 0.840250
\(116\) −2.51283 −0.00201130
\(117\) 0 0
\(118\) 126.405 0.0986147
\(119\) −535.827 −0.412767
\(120\) 0 0
\(121\) 1118.25 0.840158
\(122\) −569.976 −0.422977
\(123\) 0 0
\(124\) −2169.40 −1.57111
\(125\) 1456.80 1.04240
\(126\) 0 0
\(127\) 1420.46 0.992486 0.496243 0.868184i \(-0.334713\pi\)
0.496243 + 0.868184i \(0.334713\pi\)
\(128\) 1414.39 0.976682
\(129\) 0 0
\(130\) −183.772 −0.123983
\(131\) 1795.64 1.19760 0.598799 0.800899i \(-0.295645\pi\)
0.598799 + 0.800899i \(0.295645\pi\)
\(132\) 0 0
\(133\) −891.443 −0.581187
\(134\) −319.362 −0.205886
\(135\) 0 0
\(136\) −301.104 −0.189849
\(137\) −1581.74 −0.986405 −0.493203 0.869914i \(-0.664174\pi\)
−0.493203 + 0.869914i \(0.664174\pi\)
\(138\) 0 0
\(139\) −816.896 −0.498477 −0.249238 0.968442i \(-0.580180\pi\)
−0.249238 + 0.968442i \(0.580180\pi\)
\(140\) 1335.22 0.806046
\(141\) 0 0
\(142\) −295.159 −0.174431
\(143\) −1316.06 −0.769614
\(144\) 0 0
\(145\) −2.64777 −0.00151645
\(146\) 1116.77 0.633043
\(147\) 0 0
\(148\) 2437.35 1.35371
\(149\) −3569.30 −1.96247 −0.981236 0.192810i \(-0.938240\pi\)
−0.981236 + 0.192810i \(0.938240\pi\)
\(150\) 0 0
\(151\) −2320.85 −1.25078 −0.625391 0.780311i \(-0.715061\pi\)
−0.625391 + 0.780311i \(0.715061\pi\)
\(152\) −500.939 −0.267313
\(153\) 0 0
\(154\) −1119.92 −0.586009
\(155\) −2285.89 −1.18456
\(156\) 0 0
\(157\) 3209.19 1.63135 0.815673 0.578513i \(-0.196367\pi\)
0.815673 + 0.578513i \(0.196367\pi\)
\(158\) 1022.70 0.514945
\(159\) 0 0
\(160\) 1146.23 0.566357
\(161\) −3393.17 −1.66099
\(162\) 0 0
\(163\) −2042.28 −0.981370 −0.490685 0.871337i \(-0.663253\pi\)
−0.490685 + 0.871337i \(0.663253\pi\)
\(164\) −225.771 −0.107498
\(165\) 0 0
\(166\) −347.171 −0.162323
\(167\) −884.456 −0.409828 −0.204914 0.978780i \(-0.565691\pi\)
−0.204914 + 0.978780i \(0.565691\pi\)
\(168\) 0 0
\(169\) −1489.83 −0.678122
\(170\) −149.861 −0.0676105
\(171\) 0 0
\(172\) 2089.85 0.926450
\(173\) −410.066 −0.180212 −0.0901062 0.995932i \(-0.528721\pi\)
−0.0901062 + 0.995932i \(0.528721\pi\)
\(174\) 0 0
\(175\) −1681.72 −0.726433
\(176\) 2205.95 0.944773
\(177\) 0 0
\(178\) −1169.78 −0.492578
\(179\) 2521.04 1.05269 0.526346 0.850271i \(-0.323562\pi\)
0.526346 + 0.850271i \(0.323562\pi\)
\(180\) 0 0
\(181\) 947.700 0.389182 0.194591 0.980884i \(-0.437662\pi\)
0.194591 + 0.980884i \(0.437662\pi\)
\(182\) 601.768 0.245088
\(183\) 0 0
\(184\) −1906.76 −0.763959
\(185\) 2568.23 1.02065
\(186\) 0 0
\(187\) −1073.21 −0.419685
\(188\) −4127.92 −1.60138
\(189\) 0 0
\(190\) −249.319 −0.0951975
\(191\) −3979.72 −1.50766 −0.753829 0.657071i \(-0.771795\pi\)
−0.753829 + 0.657071i \(0.771795\pi\)
\(192\) 0 0
\(193\) 3640.34 1.35771 0.678853 0.734274i \(-0.262477\pi\)
0.678853 + 0.734274i \(0.262477\pi\)
\(194\) 746.191 0.276151
\(195\) 0 0
\(196\) −1915.91 −0.698217
\(197\) −1661.13 −0.600766 −0.300383 0.953819i \(-0.597115\pi\)
−0.300383 + 0.953819i \(0.597115\pi\)
\(198\) 0 0
\(199\) −1317.39 −0.469283 −0.234641 0.972082i \(-0.575392\pi\)
−0.234641 + 0.972082i \(0.575392\pi\)
\(200\) −945.027 −0.334117
\(201\) 0 0
\(202\) −385.974 −0.134441
\(203\) 8.67022 0.00299768
\(204\) 0 0
\(205\) −237.895 −0.0810501
\(206\) −29.7889 −0.0100752
\(207\) 0 0
\(208\) −1185.33 −0.395135
\(209\) −1785.48 −0.590928
\(210\) 0 0
\(211\) −4554.14 −1.48588 −0.742939 0.669359i \(-0.766569\pi\)
−0.742939 + 0.669359i \(0.766569\pi\)
\(212\) 775.717 0.251304
\(213\) 0 0
\(214\) −1270.79 −0.405933
\(215\) 2202.07 0.698511
\(216\) 0 0
\(217\) 7485.25 2.34162
\(218\) 979.153 0.304205
\(219\) 0 0
\(220\) 2674.31 0.819556
\(221\) 576.672 0.175526
\(222\) 0 0
\(223\) 1217.03 0.365462 0.182731 0.983163i \(-0.441506\pi\)
0.182731 + 0.983163i \(0.441506\pi\)
\(224\) −3753.36 −1.11956
\(225\) 0 0
\(226\) −475.743 −0.140026
\(227\) −5712.96 −1.67041 −0.835203 0.549942i \(-0.814650\pi\)
−0.835203 + 0.549942i \(0.814650\pi\)
\(228\) 0 0
\(229\) −5241.04 −1.51239 −0.756196 0.654345i \(-0.772944\pi\)
−0.756196 + 0.654345i \(0.772944\pi\)
\(230\) −949.003 −0.272067
\(231\) 0 0
\(232\) 4.87216 0.00137876
\(233\) −4672.74 −1.31382 −0.656912 0.753967i \(-0.728138\pi\)
−0.656912 + 0.753967i \(0.728138\pi\)
\(234\) 0 0
\(235\) −4349.58 −1.20738
\(236\) 988.422 0.272630
\(237\) 0 0
\(238\) 490.724 0.133651
\(239\) −239.000 −0.0646846
\(240\) 0 0
\(241\) 3779.64 1.01024 0.505120 0.863049i \(-0.331448\pi\)
0.505120 + 0.863049i \(0.331448\pi\)
\(242\) −1024.12 −0.272037
\(243\) 0 0
\(244\) −4456.91 −1.16936
\(245\) −2018.79 −0.526431
\(246\) 0 0
\(247\) 959.396 0.247145
\(248\) 4206.28 1.07701
\(249\) 0 0
\(250\) −1334.17 −0.337522
\(251\) −4718.47 −1.18656 −0.593281 0.804995i \(-0.702168\pi\)
−0.593281 + 0.804995i \(0.702168\pi\)
\(252\) 0 0
\(253\) −6796.19 −1.68883
\(254\) −1300.90 −0.321360
\(255\) 0 0
\(256\) 444.467 0.108513
\(257\) 1973.28 0.478949 0.239474 0.970903i \(-0.423025\pi\)
0.239474 + 0.970903i \(0.423025\pi\)
\(258\) 0 0
\(259\) −8409.77 −2.01760
\(260\) −1437.00 −0.342765
\(261\) 0 0
\(262\) −1644.49 −0.387774
\(263\) 3962.75 0.929102 0.464551 0.885546i \(-0.346216\pi\)
0.464551 + 0.885546i \(0.346216\pi\)
\(264\) 0 0
\(265\) 817.372 0.189474
\(266\) 816.406 0.188184
\(267\) 0 0
\(268\) −2497.24 −0.569191
\(269\) 3186.72 0.722296 0.361148 0.932508i \(-0.382385\pi\)
0.361148 + 0.932508i \(0.382385\pi\)
\(270\) 0 0
\(271\) −4140.92 −0.928203 −0.464101 0.885782i \(-0.653623\pi\)
−0.464101 + 0.885782i \(0.653623\pi\)
\(272\) −966.604 −0.215474
\(273\) 0 0
\(274\) 1448.60 0.319391
\(275\) −3368.32 −0.738608
\(276\) 0 0
\(277\) −4612.25 −1.00044 −0.500222 0.865897i \(-0.666748\pi\)
−0.500222 + 0.865897i \(0.666748\pi\)
\(278\) 748.134 0.161403
\(279\) 0 0
\(280\) −2588.87 −0.552552
\(281\) −3827.71 −0.812604 −0.406302 0.913739i \(-0.633182\pi\)
−0.406302 + 0.913739i \(0.633182\pi\)
\(282\) 0 0
\(283\) −8715.37 −1.83065 −0.915326 0.402713i \(-0.868067\pi\)
−0.915326 + 0.402713i \(0.868067\pi\)
\(284\) −2307.98 −0.482231
\(285\) 0 0
\(286\) 1205.28 0.249196
\(287\) 778.995 0.160218
\(288\) 0 0
\(289\) −4442.74 −0.904283
\(290\) 2.42489 0.000491016 0
\(291\) 0 0
\(292\) 8732.53 1.75011
\(293\) 5279.18 1.05260 0.526302 0.850298i \(-0.323578\pi\)
0.526302 + 0.850298i \(0.323578\pi\)
\(294\) 0 0
\(295\) 1041.50 0.205554
\(296\) −4725.80 −0.927979
\(297\) 0 0
\(298\) 3268.85 0.635435
\(299\) 3651.82 0.706322
\(300\) 0 0
\(301\) −7210.76 −1.38080
\(302\) 2125.49 0.404995
\(303\) 0 0
\(304\) −1608.12 −0.303394
\(305\) −4696.24 −0.881659
\(306\) 0 0
\(307\) 644.834 0.119878 0.0599391 0.998202i \(-0.480909\pi\)
0.0599391 + 0.998202i \(0.480909\pi\)
\(308\) −8757.14 −1.62008
\(309\) 0 0
\(310\) 2093.48 0.383554
\(311\) −7145.34 −1.30281 −0.651407 0.758728i \(-0.725821\pi\)
−0.651407 + 0.758728i \(0.725821\pi\)
\(312\) 0 0
\(313\) 5382.86 0.972069 0.486034 0.873940i \(-0.338443\pi\)
0.486034 + 0.873940i \(0.338443\pi\)
\(314\) −2939.06 −0.528218
\(315\) 0 0
\(316\) 7996.95 1.42362
\(317\) 832.861 0.147565 0.0737826 0.997274i \(-0.476493\pi\)
0.0737826 + 0.997274i \(0.476493\pi\)
\(318\) 0 0
\(319\) 17.3656 0.00304793
\(320\) 1641.02 0.286675
\(321\) 0 0
\(322\) 3107.55 0.537816
\(323\) 782.359 0.134773
\(324\) 0 0
\(325\) 1809.91 0.308910
\(326\) 1870.37 0.317761
\(327\) 0 0
\(328\) 437.750 0.0736912
\(329\) 14242.9 2.38673
\(330\) 0 0
\(331\) 5984.19 0.993718 0.496859 0.867831i \(-0.334487\pi\)
0.496859 + 0.867831i \(0.334487\pi\)
\(332\) −2714.69 −0.448759
\(333\) 0 0
\(334\) 810.007 0.132699
\(335\) −2631.34 −0.429150
\(336\) 0 0
\(337\) 9699.50 1.56785 0.783925 0.620855i \(-0.213215\pi\)
0.783925 + 0.620855i \(0.213215\pi\)
\(338\) 1364.43 0.219571
\(339\) 0 0
\(340\) −1171.83 −0.186916
\(341\) 14992.2 2.38087
\(342\) 0 0
\(343\) −1864.61 −0.293526
\(344\) −4052.03 −0.635090
\(345\) 0 0
\(346\) 375.549 0.0583515
\(347\) 11627.2 1.79879 0.899397 0.437132i \(-0.144006\pi\)
0.899397 + 0.437132i \(0.144006\pi\)
\(348\) 0 0
\(349\) −4284.59 −0.657160 −0.328580 0.944476i \(-0.606570\pi\)
−0.328580 + 0.944476i \(0.606570\pi\)
\(350\) 1540.16 0.235214
\(351\) 0 0
\(352\) −7517.63 −1.13833
\(353\) −1870.98 −0.282103 −0.141051 0.990002i \(-0.545048\pi\)
−0.141051 + 0.990002i \(0.545048\pi\)
\(354\) 0 0
\(355\) −2431.92 −0.363586
\(356\) −9147.08 −1.36178
\(357\) 0 0
\(358\) −2308.83 −0.340854
\(359\) −6816.46 −1.00211 −0.501057 0.865414i \(-0.667055\pi\)
−0.501057 + 0.865414i \(0.667055\pi\)
\(360\) 0 0
\(361\) −5557.41 −0.810236
\(362\) −867.927 −0.126014
\(363\) 0 0
\(364\) 4705.51 0.677570
\(365\) 9201.45 1.31952
\(366\) 0 0
\(367\) 785.509 0.111725 0.0558627 0.998438i \(-0.482209\pi\)
0.0558627 + 0.998438i \(0.482209\pi\)
\(368\) −6121.10 −0.867077
\(369\) 0 0
\(370\) −2352.05 −0.330479
\(371\) −2676.51 −0.374549
\(372\) 0 0
\(373\) −1641.96 −0.227929 −0.113965 0.993485i \(-0.536355\pi\)
−0.113965 + 0.993485i \(0.536355\pi\)
\(374\) 982.874 0.135891
\(375\) 0 0
\(376\) 8003.67 1.09776
\(377\) −9.33113 −0.00127474
\(378\) 0 0
\(379\) 7525.41 1.01993 0.509966 0.860194i \(-0.329658\pi\)
0.509966 + 0.860194i \(0.329658\pi\)
\(380\) −1949.54 −0.263183
\(381\) 0 0
\(382\) 3644.73 0.488169
\(383\) 2465.18 0.328890 0.164445 0.986386i \(-0.447417\pi\)
0.164445 + 0.986386i \(0.447417\pi\)
\(384\) 0 0
\(385\) −9227.39 −1.22148
\(386\) −3333.91 −0.439616
\(387\) 0 0
\(388\) 5834.82 0.763448
\(389\) −10287.3 −1.34085 −0.670423 0.741979i \(-0.733887\pi\)
−0.670423 + 0.741979i \(0.733887\pi\)
\(390\) 0 0
\(391\) 2977.95 0.385170
\(392\) 3714.78 0.478634
\(393\) 0 0
\(394\) 1521.31 0.194524
\(395\) 8426.37 1.07336
\(396\) 0 0
\(397\) −916.280 −0.115836 −0.0579179 0.998321i \(-0.518446\pi\)
−0.0579179 + 0.998321i \(0.518446\pi\)
\(398\) 1206.50 0.151950
\(399\) 0 0
\(400\) −3033.73 −0.379216
\(401\) −13110.9 −1.63274 −0.816369 0.577530i \(-0.804017\pi\)
−0.816369 + 0.577530i \(0.804017\pi\)
\(402\) 0 0
\(403\) −8055.84 −0.995756
\(404\) −3018.11 −0.371675
\(405\) 0 0
\(406\) −7.94040 −0.000970629 0
\(407\) −16844.0 −2.05141
\(408\) 0 0
\(409\) −9105.34 −1.10081 −0.550403 0.834899i \(-0.685526\pi\)
−0.550403 + 0.834899i \(0.685526\pi\)
\(410\) 217.870 0.0262435
\(411\) 0 0
\(412\) −232.933 −0.0278539
\(413\) −3410.43 −0.406334
\(414\) 0 0
\(415\) −2860.47 −0.338349
\(416\) 4039.47 0.476085
\(417\) 0 0
\(418\) 1635.18 0.191338
\(419\) 3712.52 0.432861 0.216430 0.976298i \(-0.430559\pi\)
0.216430 + 0.976298i \(0.430559\pi\)
\(420\) 0 0
\(421\) −471.976 −0.0546383 −0.0273191 0.999627i \(-0.508697\pi\)
−0.0273191 + 0.999627i \(0.508697\pi\)
\(422\) 4170.80 0.481117
\(423\) 0 0
\(424\) −1504.05 −0.172271
\(425\) 1475.93 0.168454
\(426\) 0 0
\(427\) 15378.0 1.74284
\(428\) −9936.93 −1.12224
\(429\) 0 0
\(430\) −2016.71 −0.226173
\(431\) 559.067 0.0624810 0.0312405 0.999512i \(-0.490054\pi\)
0.0312405 + 0.999512i \(0.490054\pi\)
\(432\) 0 0
\(433\) 9113.61 1.01148 0.505742 0.862685i \(-0.331219\pi\)
0.505742 + 0.862685i \(0.331219\pi\)
\(434\) −6855.18 −0.758201
\(435\) 0 0
\(436\) 7656.46 0.841004
\(437\) 4954.35 0.542331
\(438\) 0 0
\(439\) 6031.78 0.655766 0.327883 0.944718i \(-0.393665\pi\)
0.327883 + 0.944718i \(0.393665\pi\)
\(440\) −5185.26 −0.561813
\(441\) 0 0
\(442\) −528.131 −0.0568340
\(443\) 16257.9 1.74365 0.871823 0.489821i \(-0.162938\pi\)
0.871823 + 0.489821i \(0.162938\pi\)
\(444\) 0 0
\(445\) −9638.26 −1.02674
\(446\) −1114.58 −0.118334
\(447\) 0 0
\(448\) −5373.60 −0.566693
\(449\) −8313.62 −0.873817 −0.436909 0.899506i \(-0.643927\pi\)
−0.436909 + 0.899506i \(0.643927\pi\)
\(450\) 0 0
\(451\) 1560.25 0.162903
\(452\) −3720.06 −0.387117
\(453\) 0 0
\(454\) 5232.07 0.540866
\(455\) 4958.18 0.510864
\(456\) 0 0
\(457\) 17797.2 1.82170 0.910852 0.412733i \(-0.135426\pi\)
0.910852 + 0.412733i \(0.135426\pi\)
\(458\) 4799.87 0.489702
\(459\) 0 0
\(460\) −7420.70 −0.752157
\(461\) 1209.43 0.122188 0.0610940 0.998132i \(-0.480541\pi\)
0.0610940 + 0.998132i \(0.480541\pi\)
\(462\) 0 0
\(463\) −18160.0 −1.82282 −0.911410 0.411501i \(-0.865005\pi\)
−0.911410 + 0.411501i \(0.865005\pi\)
\(464\) 15.6406 0.00156486
\(465\) 0 0
\(466\) 4279.41 0.425407
\(467\) −3033.10 −0.300546 −0.150273 0.988645i \(-0.548015\pi\)
−0.150273 + 0.988645i \(0.548015\pi\)
\(468\) 0 0
\(469\) 8616.42 0.848336
\(470\) 3983.45 0.390942
\(471\) 0 0
\(472\) −1916.46 −0.186891
\(473\) −14442.5 −1.40394
\(474\) 0 0
\(475\) 2455.47 0.237188
\(476\) 3837.20 0.369491
\(477\) 0 0
\(478\) 218.882 0.0209444
\(479\) 1205.11 0.114953 0.0574767 0.998347i \(-0.481695\pi\)
0.0574767 + 0.998347i \(0.481695\pi\)
\(480\) 0 0
\(481\) 9050.83 0.857967
\(482\) −3461.48 −0.327108
\(483\) 0 0
\(484\) −8008.09 −0.752074
\(485\) 6148.14 0.575613
\(486\) 0 0
\(487\) −18893.4 −1.75799 −0.878995 0.476831i \(-0.841786\pi\)
−0.878995 + 0.476831i \(0.841786\pi\)
\(488\) 8641.56 0.801609
\(489\) 0 0
\(490\) 1848.86 0.170455
\(491\) −7189.48 −0.660808 −0.330404 0.943840i \(-0.607185\pi\)
−0.330404 + 0.943840i \(0.607185\pi\)
\(492\) 0 0
\(493\) −7.60926 −0.000695140 0
\(494\) −878.638 −0.0800239
\(495\) 0 0
\(496\) 13503.0 1.22238
\(497\) 7963.41 0.718728
\(498\) 0 0
\(499\) −8363.23 −0.750279 −0.375140 0.926968i \(-0.622405\pi\)
−0.375140 + 0.926968i \(0.622405\pi\)
\(500\) −10432.5 −0.933114
\(501\) 0 0
\(502\) 4321.29 0.384201
\(503\) −16512.4 −1.46372 −0.731859 0.681456i \(-0.761347\pi\)
−0.731859 + 0.681456i \(0.761347\pi\)
\(504\) 0 0
\(505\) −3180.18 −0.280230
\(506\) 6224.12 0.546830
\(507\) 0 0
\(508\) −10172.3 −0.888432
\(509\) 20223.3 1.76106 0.880532 0.473987i \(-0.157186\pi\)
0.880532 + 0.473987i \(0.157186\pi\)
\(510\) 0 0
\(511\) −30130.5 −2.60840
\(512\) −11722.2 −1.01182
\(513\) 0 0
\(514\) −1807.18 −0.155080
\(515\) −245.441 −0.0210009
\(516\) 0 0
\(517\) 28527.1 2.42673
\(518\) 7701.88 0.653284
\(519\) 0 0
\(520\) 2786.21 0.234968
\(521\) 5376.56 0.452114 0.226057 0.974114i \(-0.427416\pi\)
0.226057 + 0.974114i \(0.427416\pi\)
\(522\) 0 0
\(523\) 12438.5 1.03996 0.519981 0.854178i \(-0.325939\pi\)
0.519981 + 0.854178i \(0.325939\pi\)
\(524\) −12859.0 −1.07204
\(525\) 0 0
\(526\) −3629.19 −0.300837
\(527\) −6569.30 −0.543004
\(528\) 0 0
\(529\) 6691.12 0.549940
\(530\) −748.569 −0.0613505
\(531\) 0 0
\(532\) 6383.86 0.520255
\(533\) −838.376 −0.0681315
\(534\) 0 0
\(535\) −10470.5 −0.846131
\(536\) 4841.93 0.390186
\(537\) 0 0
\(538\) −2918.48 −0.233874
\(539\) 13240.4 1.05808
\(540\) 0 0
\(541\) 655.093 0.0520604 0.0260302 0.999661i \(-0.491713\pi\)
0.0260302 + 0.999661i \(0.491713\pi\)
\(542\) 3792.36 0.300545
\(543\) 0 0
\(544\) 3294.07 0.259618
\(545\) 8067.60 0.634088
\(546\) 0 0
\(547\) 761.634 0.0595341 0.0297670 0.999557i \(-0.490523\pi\)
0.0297670 + 0.999557i \(0.490523\pi\)
\(548\) 11327.3 0.882989
\(549\) 0 0
\(550\) 3084.79 0.239156
\(551\) −12.6593 −0.000978777 0
\(552\) 0 0
\(553\) −27592.5 −2.12179
\(554\) 4224.01 0.323937
\(555\) 0 0
\(556\) 5850.01 0.446215
\(557\) −12915.3 −0.982475 −0.491238 0.871026i \(-0.663455\pi\)
−0.491238 + 0.871026i \(0.663455\pi\)
\(558\) 0 0
\(559\) 7760.42 0.587175
\(560\) −8310.79 −0.627134
\(561\) 0 0
\(562\) 3505.51 0.263116
\(563\) −2721.26 −0.203707 −0.101854 0.994799i \(-0.532477\pi\)
−0.101854 + 0.994799i \(0.532477\pi\)
\(564\) 0 0
\(565\) −3919.82 −0.291873
\(566\) 7981.75 0.592753
\(567\) 0 0
\(568\) 4474.98 0.330574
\(569\) −17019.8 −1.25397 −0.626984 0.779032i \(-0.715711\pi\)
−0.626984 + 0.779032i \(0.715711\pi\)
\(570\) 0 0
\(571\) −8436.61 −0.618321 −0.309160 0.951010i \(-0.600048\pi\)
−0.309160 + 0.951010i \(0.600048\pi\)
\(572\) 9424.68 0.688926
\(573\) 0 0
\(574\) −713.423 −0.0518775
\(575\) 9346.43 0.677866
\(576\) 0 0
\(577\) −10300.6 −0.743191 −0.371595 0.928395i \(-0.621189\pi\)
−0.371595 + 0.928395i \(0.621189\pi\)
\(578\) 4068.77 0.292800
\(579\) 0 0
\(580\) 18.9614 0.00135746
\(581\) 9366.71 0.668841
\(582\) 0 0
\(583\) −5360.81 −0.380827
\(584\) −16931.6 −1.19972
\(585\) 0 0
\(586\) −4834.80 −0.340826
\(587\) 1605.22 0.112870 0.0564350 0.998406i \(-0.482027\pi\)
0.0564350 + 0.998406i \(0.482027\pi\)
\(588\) 0 0
\(589\) −10929.2 −0.764565
\(590\) −953.830 −0.0665569
\(591\) 0 0
\(592\) −15170.8 −1.05324
\(593\) 13691.9 0.948159 0.474079 0.880482i \(-0.342781\pi\)
0.474079 + 0.880482i \(0.342781\pi\)
\(594\) 0 0
\(595\) 4043.25 0.278584
\(596\) 25560.7 1.75672
\(597\) 0 0
\(598\) −3344.43 −0.228702
\(599\) −26216.9 −1.78830 −0.894151 0.447765i \(-0.852220\pi\)
−0.894151 + 0.447765i \(0.852220\pi\)
\(600\) 0 0
\(601\) −11341.8 −0.769787 −0.384894 0.922961i \(-0.625762\pi\)
−0.384894 + 0.922961i \(0.625762\pi\)
\(602\) 6603.79 0.447094
\(603\) 0 0
\(604\) 16620.2 1.11965
\(605\) −8438.11 −0.567038
\(606\) 0 0
\(607\) −11101.0 −0.742303 −0.371151 0.928572i \(-0.621037\pi\)
−0.371151 + 0.928572i \(0.621037\pi\)
\(608\) 5480.27 0.365550
\(609\) 0 0
\(610\) 4300.93 0.285475
\(611\) −15328.6 −1.01494
\(612\) 0 0
\(613\) 17017.3 1.12124 0.560620 0.828073i \(-0.310563\pi\)
0.560620 + 0.828073i \(0.310563\pi\)
\(614\) −590.555 −0.0388157
\(615\) 0 0
\(616\) 16979.3 1.11058
\(617\) −14467.0 −0.943956 −0.471978 0.881610i \(-0.656460\pi\)
−0.471978 + 0.881610i \(0.656460\pi\)
\(618\) 0 0
\(619\) −14990.7 −0.973385 −0.486693 0.873573i \(-0.661797\pi\)
−0.486693 + 0.873573i \(0.661797\pi\)
\(620\) 16369.9 1.06037
\(621\) 0 0
\(622\) 6543.88 0.421842
\(623\) 31560.9 2.02963
\(624\) 0 0
\(625\) −2485.15 −0.159050
\(626\) −4929.76 −0.314749
\(627\) 0 0
\(628\) −22981.9 −1.46031
\(629\) 7380.69 0.467865
\(630\) 0 0
\(631\) −16047.2 −1.01241 −0.506203 0.862414i \(-0.668951\pi\)
−0.506203 + 0.862414i \(0.668951\pi\)
\(632\) −15505.4 −0.975903
\(633\) 0 0
\(634\) −762.755 −0.0477806
\(635\) −10718.5 −0.669847
\(636\) 0 0
\(637\) −7114.51 −0.442523
\(638\) −15.9039 −0.000986897 0
\(639\) 0 0
\(640\) −10672.7 −0.659181
\(641\) 20880.9 1.28666 0.643328 0.765591i \(-0.277553\pi\)
0.643328 + 0.765591i \(0.277553\pi\)
\(642\) 0 0
\(643\) 10675.4 0.654738 0.327369 0.944897i \(-0.393838\pi\)
0.327369 + 0.944897i \(0.393838\pi\)
\(644\) 24299.4 1.48685
\(645\) 0 0
\(646\) −716.504 −0.0436385
\(647\) −23770.6 −1.44439 −0.722195 0.691690i \(-0.756867\pi\)
−0.722195 + 0.691690i \(0.756867\pi\)
\(648\) 0 0
\(649\) −6830.76 −0.413145
\(650\) −1657.56 −0.100023
\(651\) 0 0
\(652\) 14625.3 0.878482
\(653\) 7609.84 0.456043 0.228022 0.973656i \(-0.426774\pi\)
0.228022 + 0.973656i \(0.426774\pi\)
\(654\) 0 0
\(655\) −13549.5 −0.808281
\(656\) 1405.27 0.0836378
\(657\) 0 0
\(658\) −13044.0 −0.772807
\(659\) −5332.02 −0.315184 −0.157592 0.987504i \(-0.550373\pi\)
−0.157592 + 0.987504i \(0.550373\pi\)
\(660\) 0 0
\(661\) −9421.86 −0.554415 −0.277207 0.960810i \(-0.589409\pi\)
−0.277207 + 0.960810i \(0.589409\pi\)
\(662\) −5480.47 −0.321759
\(663\) 0 0
\(664\) 5263.55 0.307629
\(665\) 6726.66 0.392254
\(666\) 0 0
\(667\) −48.1862 −0.00279727
\(668\) 6333.83 0.366861
\(669\) 0 0
\(670\) 2409.85 0.138956
\(671\) 30800.7 1.77205
\(672\) 0 0
\(673\) 9723.05 0.556903 0.278452 0.960450i \(-0.410179\pi\)
0.278452 + 0.960450i \(0.410179\pi\)
\(674\) −8883.04 −0.507659
\(675\) 0 0
\(676\) 10669.1 0.607027
\(677\) −4577.67 −0.259873 −0.129937 0.991522i \(-0.541477\pi\)
−0.129937 + 0.991522i \(0.541477\pi\)
\(678\) 0 0
\(679\) −20132.3 −1.13786
\(680\) 2272.08 0.128132
\(681\) 0 0
\(682\) −13730.3 −0.770908
\(683\) −26280.5 −1.47232 −0.736160 0.676808i \(-0.763363\pi\)
−0.736160 + 0.676808i \(0.763363\pi\)
\(684\) 0 0
\(685\) 11935.5 0.665743
\(686\) 1707.65 0.0950415
\(687\) 0 0
\(688\) −13007.8 −0.720812
\(689\) 2880.54 0.159274
\(690\) 0 0
\(691\) −14534.8 −0.800190 −0.400095 0.916474i \(-0.631023\pi\)
−0.400095 + 0.916474i \(0.631023\pi\)
\(692\) 2936.59 0.161319
\(693\) 0 0
\(694\) −10648.5 −0.582437
\(695\) 6164.15 0.336431
\(696\) 0 0
\(697\) −683.671 −0.0371533
\(698\) 3923.93 0.212784
\(699\) 0 0
\(700\) 12043.2 0.650272
\(701\) −22046.9 −1.18788 −0.593938 0.804511i \(-0.702428\pi\)
−0.593938 + 0.804511i \(0.702428\pi\)
\(702\) 0 0
\(703\) 12279.1 0.658768
\(704\) −10762.8 −0.576191
\(705\) 0 0
\(706\) 1713.49 0.0913428
\(707\) 10413.6 0.553952
\(708\) 0 0
\(709\) −1160.25 −0.0614583 −0.0307292 0.999528i \(-0.509783\pi\)
−0.0307292 + 0.999528i \(0.509783\pi\)
\(710\) 2227.21 0.117726
\(711\) 0 0
\(712\) 17735.4 0.933513
\(713\) −41600.6 −2.18507
\(714\) 0 0
\(715\) 9930.77 0.519426
\(716\) −18053.9 −0.942325
\(717\) 0 0
\(718\) 6242.68 0.324477
\(719\) −26058.1 −1.35161 −0.675803 0.737082i \(-0.736203\pi\)
−0.675803 + 0.737082i \(0.736203\pi\)
\(720\) 0 0
\(721\) 803.707 0.0415140
\(722\) 5089.61 0.262349
\(723\) 0 0
\(724\) −6786.73 −0.348380
\(725\) −23.8820 −0.00122339
\(726\) 0 0
\(727\) −34483.7 −1.75919 −0.879594 0.475725i \(-0.842186\pi\)
−0.879594 + 0.475725i \(0.842186\pi\)
\(728\) −9123.56 −0.464480
\(729\) 0 0
\(730\) −8426.91 −0.427252
\(731\) 6328.40 0.320197
\(732\) 0 0
\(733\) 1508.37 0.0760067 0.0380034 0.999278i \(-0.487900\pi\)
0.0380034 + 0.999278i \(0.487900\pi\)
\(734\) −719.389 −0.0361759
\(735\) 0 0
\(736\) 20860.0 1.04471
\(737\) 17257.9 0.862554
\(738\) 0 0
\(739\) −13898.1 −0.691813 −0.345907 0.938269i \(-0.612429\pi\)
−0.345907 + 0.938269i \(0.612429\pi\)
\(740\) −18391.8 −0.913642
\(741\) 0 0
\(742\) 2451.22 0.121276
\(743\) −24303.8 −1.20003 −0.600013 0.799990i \(-0.704838\pi\)
−0.600013 + 0.799990i \(0.704838\pi\)
\(744\) 0 0
\(745\) 26933.3 1.32451
\(746\) 1503.75 0.0738018
\(747\) 0 0
\(748\) 7685.55 0.375684
\(749\) 34286.1 1.67261
\(750\) 0 0
\(751\) 29770.3 1.44652 0.723259 0.690577i \(-0.242643\pi\)
0.723259 + 0.690577i \(0.242643\pi\)
\(752\) 25693.4 1.24593
\(753\) 0 0
\(754\) 8.54568 0.000412752 0
\(755\) 17512.7 0.844176
\(756\) 0 0
\(757\) 1684.16 0.0808610 0.0404305 0.999182i \(-0.487127\pi\)
0.0404305 + 0.999182i \(0.487127\pi\)
\(758\) −6891.96 −0.330247
\(759\) 0 0
\(760\) 3780.00 0.180414
\(761\) 10307.3 0.490986 0.245493 0.969398i \(-0.421050\pi\)
0.245493 + 0.969398i \(0.421050\pi\)
\(762\) 0 0
\(763\) −26417.7 −1.25345
\(764\) 28499.8 1.34959
\(765\) 0 0
\(766\) −2257.67 −0.106492
\(767\) 3670.40 0.172791
\(768\) 0 0
\(769\) 33644.6 1.57771 0.788853 0.614582i \(-0.210675\pi\)
0.788853 + 0.614582i \(0.210675\pi\)
\(770\) 8450.67 0.395508
\(771\) 0 0
\(772\) −26069.4 −1.21536
\(773\) 3463.13 0.161139 0.0805693 0.996749i \(-0.474326\pi\)
0.0805693 + 0.996749i \(0.474326\pi\)
\(774\) 0 0
\(775\) −20618.0 −0.955639
\(776\) −11313.2 −0.523351
\(777\) 0 0
\(778\) 9421.41 0.434157
\(779\) −1137.41 −0.0523130
\(780\) 0 0
\(781\) 15950.0 0.730774
\(782\) −2727.28 −0.124715
\(783\) 0 0
\(784\) 11925.2 0.543239
\(785\) −24216.0 −1.10102
\(786\) 0 0
\(787\) −30239.6 −1.36966 −0.684832 0.728701i \(-0.740125\pi\)
−0.684832 + 0.728701i \(0.740125\pi\)
\(788\) 11895.8 0.537781
\(789\) 0 0
\(790\) −7717.08 −0.347546
\(791\) 12835.6 0.576968
\(792\) 0 0
\(793\) −16550.3 −0.741131
\(794\) 839.152 0.0375068
\(795\) 0 0
\(796\) 9434.17 0.420082
\(797\) 30267.7 1.34522 0.672609 0.739998i \(-0.265174\pi\)
0.672609 + 0.739998i \(0.265174\pi\)
\(798\) 0 0
\(799\) −12500.0 −0.553464
\(800\) 10338.6 0.456905
\(801\) 0 0
\(802\) 12007.3 0.528669
\(803\) −60348.5 −2.65212
\(804\) 0 0
\(805\) 25604.2 1.12103
\(806\) 7377.73 0.322419
\(807\) 0 0
\(808\) 5851.85 0.254786
\(809\) −21323.9 −0.926710 −0.463355 0.886173i \(-0.653355\pi\)
−0.463355 + 0.886173i \(0.653355\pi\)
\(810\) 0 0
\(811\) 4544.15 0.196753 0.0983766 0.995149i \(-0.468635\pi\)
0.0983766 + 0.995149i \(0.468635\pi\)
\(812\) −62.0897 −0.00268340
\(813\) 0 0
\(814\) 15426.1 0.664233
\(815\) 15410.6 0.662345
\(816\) 0 0
\(817\) 10528.4 0.450847
\(818\) 8338.89 0.356433
\(819\) 0 0
\(820\) 1703.63 0.0725527
\(821\) −15313.0 −0.650949 −0.325474 0.945551i \(-0.605524\pi\)
−0.325474 + 0.945551i \(0.605524\pi\)
\(822\) 0 0
\(823\) 10661.7 0.451573 0.225786 0.974177i \(-0.427505\pi\)
0.225786 + 0.974177i \(0.427505\pi\)
\(824\) 451.637 0.0190941
\(825\) 0 0
\(826\) 3123.35 0.131568
\(827\) 17341.6 0.729172 0.364586 0.931170i \(-0.381211\pi\)
0.364586 + 0.931170i \(0.381211\pi\)
\(828\) 0 0
\(829\) −20108.0 −0.842435 −0.421217 0.906960i \(-0.638397\pi\)
−0.421217 + 0.906960i \(0.638397\pi\)
\(830\) 2619.69 0.109555
\(831\) 0 0
\(832\) 5783.22 0.240982
\(833\) −5801.68 −0.241316
\(834\) 0 0
\(835\) 6673.94 0.276600
\(836\) 12786.3 0.528974
\(837\) 0 0
\(838\) −3400.02 −0.140157
\(839\) 30229.1 1.24389 0.621946 0.783060i \(-0.286342\pi\)
0.621946 + 0.783060i \(0.286342\pi\)
\(840\) 0 0
\(841\) −24388.9 −0.999995
\(842\) 432.248 0.0176915
\(843\) 0 0
\(844\) 32613.4 1.33010
\(845\) 11242.0 0.457677
\(846\) 0 0
\(847\) 27630.9 1.12091
\(848\) −4828.29 −0.195524
\(849\) 0 0
\(850\) −1351.69 −0.0545443
\(851\) 46738.8 1.88271
\(852\) 0 0
\(853\) −5756.55 −0.231067 −0.115534 0.993304i \(-0.536858\pi\)
−0.115534 + 0.993304i \(0.536858\pi\)
\(854\) −14083.6 −0.564321
\(855\) 0 0
\(856\) 19266.8 0.769307
\(857\) −7273.51 −0.289916 −0.144958 0.989438i \(-0.546305\pi\)
−0.144958 + 0.989438i \(0.546305\pi\)
\(858\) 0 0
\(859\) −20497.9 −0.814179 −0.407090 0.913388i \(-0.633456\pi\)
−0.407090 + 0.913388i \(0.633456\pi\)
\(860\) −15769.6 −0.625278
\(861\) 0 0
\(862\) −512.007 −0.0202309
\(863\) 34895.9 1.37644 0.688221 0.725501i \(-0.258392\pi\)
0.688221 + 0.725501i \(0.258392\pi\)
\(864\) 0 0
\(865\) 3094.28 0.121629
\(866\) −8346.47 −0.327511
\(867\) 0 0
\(868\) −53603.9 −2.09612
\(869\) −55265.1 −2.15735
\(870\) 0 0
\(871\) −9273.23 −0.360748
\(872\) −14845.2 −0.576516
\(873\) 0 0
\(874\) −4537.32 −0.175603
\(875\) 35996.2 1.39073
\(876\) 0 0
\(877\) 15900.6 0.612229 0.306114 0.951995i \(-0.400971\pi\)
0.306114 + 0.951995i \(0.400971\pi\)
\(878\) −5524.05 −0.212332
\(879\) 0 0
\(880\) −16645.7 −0.637645
\(881\) 25571.7 0.977904 0.488952 0.872311i \(-0.337379\pi\)
0.488952 + 0.872311i \(0.337379\pi\)
\(882\) 0 0
\(883\) −18259.8 −0.695912 −0.347956 0.937511i \(-0.613124\pi\)
−0.347956 + 0.937511i \(0.613124\pi\)
\(884\) −4129.70 −0.157123
\(885\) 0 0
\(886\) −14889.4 −0.564580
\(887\) 14766.5 0.558973 0.279487 0.960150i \(-0.409836\pi\)
0.279487 + 0.960150i \(0.409836\pi\)
\(888\) 0 0
\(889\) 35098.3 1.32414
\(890\) 8826.96 0.332450
\(891\) 0 0
\(892\) −8715.45 −0.327147
\(893\) −20795.9 −0.779294
\(894\) 0 0
\(895\) −19023.3 −0.710480
\(896\) 34948.2 1.30305
\(897\) 0 0
\(898\) 7613.82 0.282936
\(899\) 106.298 0.00394352
\(900\) 0 0
\(901\) 2349.00 0.0868550
\(902\) −1428.92 −0.0527470
\(903\) 0 0
\(904\) 7212.87 0.265372
\(905\) −7151.17 −0.262666
\(906\) 0 0
\(907\) 289.782 0.0106087 0.00530433 0.999986i \(-0.498312\pi\)
0.00530433 + 0.999986i \(0.498312\pi\)
\(908\) 40912.0 1.49528
\(909\) 0 0
\(910\) −4540.83 −0.165414
\(911\) −8011.03 −0.291347 −0.145674 0.989333i \(-0.546535\pi\)
−0.145674 + 0.989333i \(0.546535\pi\)
\(912\) 0 0
\(913\) 18760.6 0.680051
\(914\) −16299.1 −0.589855
\(915\) 0 0
\(916\) 37532.5 1.35383
\(917\) 44368.5 1.59779
\(918\) 0 0
\(919\) 40447.3 1.45183 0.725916 0.687783i \(-0.241416\pi\)
0.725916 + 0.687783i \(0.241416\pi\)
\(920\) 14388.1 0.515610
\(921\) 0 0
\(922\) −1107.62 −0.0395636
\(923\) −8570.45 −0.305633
\(924\) 0 0
\(925\) 23164.6 0.823402
\(926\) 16631.3 0.590216
\(927\) 0 0
\(928\) −53.3014 −0.00188546
\(929\) −4973.47 −0.175645 −0.0878225 0.996136i \(-0.527991\pi\)
−0.0878225 + 0.996136i \(0.527991\pi\)
\(930\) 0 0
\(931\) −9652.11 −0.339780
\(932\) 33462.7 1.17608
\(933\) 0 0
\(934\) 2777.79 0.0973147
\(935\) 8098.26 0.283253
\(936\) 0 0
\(937\) 30016.5 1.04653 0.523264 0.852171i \(-0.324714\pi\)
0.523264 + 0.852171i \(0.324714\pi\)
\(938\) −7891.13 −0.274685
\(939\) 0 0
\(940\) 31148.5 1.08080
\(941\) 37395.5 1.29549 0.647745 0.761857i \(-0.275712\pi\)
0.647745 + 0.761857i \(0.275712\pi\)
\(942\) 0 0
\(943\) −4329.40 −0.149506
\(944\) −6152.23 −0.212117
\(945\) 0 0
\(946\) 13226.8 0.454587
\(947\) 14566.8 0.499848 0.249924 0.968265i \(-0.419594\pi\)
0.249924 + 0.968265i \(0.419594\pi\)
\(948\) 0 0
\(949\) 32427.3 1.10920
\(950\) −2248.78 −0.0767999
\(951\) 0 0
\(952\) −7440.00 −0.253290
\(953\) 18307.3 0.622278 0.311139 0.950364i \(-0.399290\pi\)
0.311139 + 0.950364i \(0.399290\pi\)
\(954\) 0 0
\(955\) 30030.2 1.01755
\(956\) 1711.54 0.0579030
\(957\) 0 0
\(958\) −1103.67 −0.0372211
\(959\) −39083.4 −1.31603
\(960\) 0 0
\(961\) 61978.9 2.08046
\(962\) −8288.97 −0.277804
\(963\) 0 0
\(964\) −27067.0 −0.904324
\(965\) −27469.3 −0.916340
\(966\) 0 0
\(967\) −27458.7 −0.913145 −0.456572 0.889686i \(-0.650923\pi\)
−0.456572 + 0.889686i \(0.650923\pi\)
\(968\) 15527.0 0.515554
\(969\) 0 0
\(970\) −5630.62 −0.186380
\(971\) −52683.2 −1.74118 −0.870590 0.492010i \(-0.836262\pi\)
−0.870590 + 0.492010i \(0.836262\pi\)
\(972\) 0 0
\(973\) −20184.7 −0.665049
\(974\) 17303.0 0.569225
\(975\) 0 0
\(976\) 27741.1 0.909808
\(977\) −26057.2 −0.853270 −0.426635 0.904424i \(-0.640301\pi\)
−0.426635 + 0.904424i \(0.640301\pi\)
\(978\) 0 0
\(979\) 63213.4 2.06365
\(980\) 14457.1 0.471239
\(981\) 0 0
\(982\) 6584.30 0.213965
\(983\) −14618.3 −0.474314 −0.237157 0.971471i \(-0.576216\pi\)
−0.237157 + 0.971471i \(0.576216\pi\)
\(984\) 0 0
\(985\) 12534.6 0.405468
\(986\) 6.96875 0.000225081 0
\(987\) 0 0
\(988\) −6870.49 −0.221234
\(989\) 40075.0 1.28849
\(990\) 0 0
\(991\) 12768.4 0.409285 0.204642 0.978837i \(-0.434397\pi\)
0.204642 + 0.978837i \(0.434397\pi\)
\(992\) −46016.6 −1.47281
\(993\) 0 0
\(994\) −7293.09 −0.232719
\(995\) 9940.77 0.316727
\(996\) 0 0
\(997\) 42906.2 1.36294 0.681472 0.731845i \(-0.261340\pi\)
0.681472 + 0.731845i \(0.261340\pi\)
\(998\) 7659.25 0.242935
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2151.4.a.b.1.12 28
3.2 odd 2 717.4.a.b.1.17 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.4.a.b.1.17 28 3.2 odd 2
2151.4.a.b.1.12 28 1.1 even 1 trivial