Properties

Label 234.4.a.b.1.1
Level $234$
Weight $4$
Character 234.1
Self dual yes
Analytic conductor $13.806$
Analytic rank $0$
Dimension $1$
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] = [234,4,Mod(1,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, 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("234.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 234.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: \(13.8064469413\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +20.0000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} -6.00000 q^{5} +20.0000 q^{7} -8.00000 q^{8} +12.0000 q^{10} -24.0000 q^{11} +13.0000 q^{13} -40.0000 q^{14} +16.0000 q^{16} +30.0000 q^{17} -16.0000 q^{19} -24.0000 q^{20} +48.0000 q^{22} +72.0000 q^{23} -89.0000 q^{25} -26.0000 q^{26} +80.0000 q^{28} +282.000 q^{29} +164.000 q^{31} -32.0000 q^{32} -60.0000 q^{34} -120.000 q^{35} +110.000 q^{37} +32.0000 q^{38} +48.0000 q^{40} +126.000 q^{41} +164.000 q^{43} -96.0000 q^{44} -144.000 q^{46} +204.000 q^{47} +57.0000 q^{49} +178.000 q^{50} +52.0000 q^{52} +738.000 q^{53} +144.000 q^{55} -160.000 q^{56} -564.000 q^{58} -120.000 q^{59} +614.000 q^{61} -328.000 q^{62} +64.0000 q^{64} -78.0000 q^{65} +848.000 q^{67} +120.000 q^{68} +240.000 q^{70} -132.000 q^{71} +218.000 q^{73} -220.000 q^{74} -64.0000 q^{76} -480.000 q^{77} -1096.00 q^{79} -96.0000 q^{80} -252.000 q^{82} -552.000 q^{83} -180.000 q^{85} -328.000 q^{86} +192.000 q^{88} -210.000 q^{89} +260.000 q^{91} +288.000 q^{92} -408.000 q^{94} +96.0000 q^{95} -1726.00 q^{97} -114.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) 0 0
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 12.0000 0.379473
\(11\) −24.0000 −0.657843 −0.328921 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) −40.0000 −0.763604
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 30.0000 0.428004 0.214002 0.976833i \(-0.431350\pi\)
0.214002 + 0.976833i \(0.431350\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.193192 −0.0965961 0.995324i \(-0.530796\pi\)
−0.0965961 + 0.995324i \(0.530796\pi\)
\(20\) −24.0000 −0.268328
\(21\) 0 0
\(22\) 48.0000 0.465165
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) −26.0000 −0.196116
\(27\) 0 0
\(28\) 80.0000 0.539949
\(29\) 282.000 1.80573 0.902864 0.429927i \(-0.141461\pi\)
0.902864 + 0.429927i \(0.141461\pi\)
\(30\) 0 0
\(31\) 164.000 0.950170 0.475085 0.879940i \(-0.342417\pi\)
0.475085 + 0.879940i \(0.342417\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −60.0000 −0.302645
\(35\) −120.000 −0.579534
\(36\) 0 0
\(37\) 110.000 0.488754 0.244377 0.969680i \(-0.421417\pi\)
0.244377 + 0.969680i \(0.421417\pi\)
\(38\) 32.0000 0.136608
\(39\) 0 0
\(40\) 48.0000 0.189737
\(41\) 126.000 0.479949 0.239974 0.970779i \(-0.422861\pi\)
0.239974 + 0.970779i \(0.422861\pi\)
\(42\) 0 0
\(43\) 164.000 0.581622 0.290811 0.956780i \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) −96.0000 −0.328921
\(45\) 0 0
\(46\) −144.000 −0.461557
\(47\) 204.000 0.633116 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 178.000 0.503460
\(51\) 0 0
\(52\) 52.0000 0.138675
\(53\) 738.000 1.91268 0.956341 0.292255i \(-0.0944055\pi\)
0.956341 + 0.292255i \(0.0944055\pi\)
\(54\) 0 0
\(55\) 144.000 0.353036
\(56\) −160.000 −0.381802
\(57\) 0 0
\(58\) −564.000 −1.27684
\(59\) −120.000 −0.264791 −0.132396 0.991197i \(-0.542267\pi\)
−0.132396 + 0.991197i \(0.542267\pi\)
\(60\) 0 0
\(61\) 614.000 1.28876 0.644382 0.764703i \(-0.277115\pi\)
0.644382 + 0.764703i \(0.277115\pi\)
\(62\) −328.000 −0.671872
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −78.0000 −0.148842
\(66\) 0 0
\(67\) 848.000 1.54626 0.773132 0.634245i \(-0.218689\pi\)
0.773132 + 0.634245i \(0.218689\pi\)
\(68\) 120.000 0.214002
\(69\) 0 0
\(70\) 240.000 0.409793
\(71\) −132.000 −0.220641 −0.110321 0.993896i \(-0.535188\pi\)
−0.110321 + 0.993896i \(0.535188\pi\)
\(72\) 0 0
\(73\) 218.000 0.349520 0.174760 0.984611i \(-0.444085\pi\)
0.174760 + 0.984611i \(0.444085\pi\)
\(74\) −220.000 −0.345601
\(75\) 0 0
\(76\) −64.0000 −0.0965961
\(77\) −480.000 −0.710404
\(78\) 0 0
\(79\) −1096.00 −1.56088 −0.780441 0.625230i \(-0.785005\pi\)
−0.780441 + 0.625230i \(0.785005\pi\)
\(80\) −96.0000 −0.134164
\(81\) 0 0
\(82\) −252.000 −0.339375
\(83\) −552.000 −0.729998 −0.364999 0.931008i \(-0.618931\pi\)
−0.364999 + 0.931008i \(0.618931\pi\)
\(84\) 0 0
\(85\) −180.000 −0.229691
\(86\) −328.000 −0.411269
\(87\) 0 0
\(88\) 192.000 0.232583
\(89\) −210.000 −0.250112 −0.125056 0.992150i \(-0.539911\pi\)
−0.125056 + 0.992150i \(0.539911\pi\)
\(90\) 0 0
\(91\) 260.000 0.299510
\(92\) 288.000 0.326370
\(93\) 0 0
\(94\) −408.000 −0.447681
\(95\) 96.0000 0.103678
\(96\) 0 0
\(97\) −1726.00 −1.80669 −0.903344 0.428917i \(-0.858895\pi\)
−0.903344 + 0.428917i \(0.858895\pi\)
\(98\) −114.000 −0.117508
\(99\) 0 0
\(100\) −356.000 −0.356000
\(101\) −798.000 −0.786178 −0.393089 0.919500i \(-0.628594\pi\)
−0.393089 + 0.919500i \(0.628594\pi\)
\(102\) 0 0
\(103\) −520.000 −0.497448 −0.248724 0.968574i \(-0.580011\pi\)
−0.248724 + 0.968574i \(0.580011\pi\)
\(104\) −104.000 −0.0980581
\(105\) 0 0
\(106\) −1476.00 −1.35247
\(107\) −12.0000 −0.0108419 −0.00542095 0.999985i \(-0.501726\pi\)
−0.00542095 + 0.999985i \(0.501726\pi\)
\(108\) 0 0
\(109\) −1834.00 −1.61161 −0.805804 0.592182i \(-0.798267\pi\)
−0.805804 + 0.592182i \(0.798267\pi\)
\(110\) −288.000 −0.249634
\(111\) 0 0
\(112\) 320.000 0.269975
\(113\) 366.000 0.304694 0.152347 0.988327i \(-0.451317\pi\)
0.152347 + 0.988327i \(0.451317\pi\)
\(114\) 0 0
\(115\) −432.000 −0.350297
\(116\) 1128.00 0.902864
\(117\) 0 0
\(118\) 240.000 0.187236
\(119\) 600.000 0.462201
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) −1228.00 −0.911294
\(123\) 0 0
\(124\) 656.000 0.475085
\(125\) 1284.00 0.918756
\(126\) 0 0
\(127\) 2144.00 1.49803 0.749013 0.662556i \(-0.230528\pi\)
0.749013 + 0.662556i \(0.230528\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 156.000 0.105247
\(131\) 2748.00 1.83278 0.916389 0.400289i \(-0.131090\pi\)
0.916389 + 0.400289i \(0.131090\pi\)
\(132\) 0 0
\(133\) −320.000 −0.208628
\(134\) −1696.00 −1.09337
\(135\) 0 0
\(136\) −240.000 −0.151322
\(137\) −2754.00 −1.71745 −0.858723 0.512440i \(-0.828742\pi\)
−0.858723 + 0.512440i \(0.828742\pi\)
\(138\) 0 0
\(139\) 2252.00 1.37419 0.687094 0.726568i \(-0.258886\pi\)
0.687094 + 0.726568i \(0.258886\pi\)
\(140\) −480.000 −0.289767
\(141\) 0 0
\(142\) 264.000 0.156017
\(143\) −312.000 −0.182453
\(144\) 0 0
\(145\) −1692.00 −0.969055
\(146\) −436.000 −0.247148
\(147\) 0 0
\(148\) 440.000 0.244377
\(149\) 1770.00 0.973182 0.486591 0.873630i \(-0.338240\pi\)
0.486591 + 0.873630i \(0.338240\pi\)
\(150\) 0 0
\(151\) −988.000 −0.532466 −0.266233 0.963909i \(-0.585779\pi\)
−0.266233 + 0.963909i \(0.585779\pi\)
\(152\) 128.000 0.0683038
\(153\) 0 0
\(154\) 960.000 0.502331
\(155\) −984.000 −0.509915
\(156\) 0 0
\(157\) 326.000 0.165717 0.0828587 0.996561i \(-0.473595\pi\)
0.0828587 + 0.996561i \(0.473595\pi\)
\(158\) 2192.00 1.10371
\(159\) 0 0
\(160\) 192.000 0.0948683
\(161\) 1440.00 0.704894
\(162\) 0 0
\(163\) 1496.00 0.718870 0.359435 0.933170i \(-0.382969\pi\)
0.359435 + 0.933170i \(0.382969\pi\)
\(164\) 504.000 0.239974
\(165\) 0 0
\(166\) 1104.00 0.516187
\(167\) −1116.00 −0.517118 −0.258559 0.965995i \(-0.583248\pi\)
−0.258559 + 0.965995i \(0.583248\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 360.000 0.162416
\(171\) 0 0
\(172\) 656.000 0.290811
\(173\) −4374.00 −1.92225 −0.961124 0.276116i \(-0.910953\pi\)
−0.961124 + 0.276116i \(0.910953\pi\)
\(174\) 0 0
\(175\) −1780.00 −0.768888
\(176\) −384.000 −0.164461
\(177\) 0 0
\(178\) 420.000 0.176856
\(179\) −12.0000 −0.00501074 −0.00250537 0.999997i \(-0.500797\pi\)
−0.00250537 + 0.999997i \(0.500797\pi\)
\(180\) 0 0
\(181\) 4718.00 1.93749 0.968746 0.248053i \(-0.0797909\pi\)
0.968746 + 0.248053i \(0.0797909\pi\)
\(182\) −520.000 −0.211786
\(183\) 0 0
\(184\) −576.000 −0.230779
\(185\) −660.000 −0.262293
\(186\) 0 0
\(187\) −720.000 −0.281559
\(188\) 816.000 0.316558
\(189\) 0 0
\(190\) −192.000 −0.0733113
\(191\) 1368.00 0.518246 0.259123 0.965844i \(-0.416566\pi\)
0.259123 + 0.965844i \(0.416566\pi\)
\(192\) 0 0
\(193\) −3310.00 −1.23450 −0.617251 0.786766i \(-0.711754\pi\)
−0.617251 + 0.786766i \(0.711754\pi\)
\(194\) 3452.00 1.27752
\(195\) 0 0
\(196\) 228.000 0.0830904
\(197\) −3126.00 −1.13055 −0.565275 0.824903i \(-0.691230\pi\)
−0.565275 + 0.824903i \(0.691230\pi\)
\(198\) 0 0
\(199\) 4664.00 1.66142 0.830709 0.556707i \(-0.187935\pi\)
0.830709 + 0.556707i \(0.187935\pi\)
\(200\) 712.000 0.251730
\(201\) 0 0
\(202\) 1596.00 0.555912
\(203\) 5640.00 1.95000
\(204\) 0 0
\(205\) −756.000 −0.257567
\(206\) 1040.00 0.351749
\(207\) 0 0
\(208\) 208.000 0.0693375
\(209\) 384.000 0.127090
\(210\) 0 0
\(211\) −556.000 −0.181406 −0.0907029 0.995878i \(-0.528911\pi\)
−0.0907029 + 0.995878i \(0.528911\pi\)
\(212\) 2952.00 0.956341
\(213\) 0 0
\(214\) 24.0000 0.00766638
\(215\) −984.000 −0.312131
\(216\) 0 0
\(217\) 3280.00 1.02609
\(218\) 3668.00 1.13958
\(219\) 0 0
\(220\) 576.000 0.176518
\(221\) 390.000 0.118707
\(222\) 0 0
\(223\) −268.000 −0.0804781 −0.0402390 0.999190i \(-0.512812\pi\)
−0.0402390 + 0.999190i \(0.512812\pi\)
\(224\) −640.000 −0.190901
\(225\) 0 0
\(226\) −732.000 −0.215451
\(227\) −1800.00 −0.526300 −0.263150 0.964755i \(-0.584761\pi\)
−0.263150 + 0.964755i \(0.584761\pi\)
\(228\) 0 0
\(229\) 2990.00 0.862816 0.431408 0.902157i \(-0.358017\pi\)
0.431408 + 0.902157i \(0.358017\pi\)
\(230\) 864.000 0.247698
\(231\) 0 0
\(232\) −2256.00 −0.638421
\(233\) −2826.00 −0.794581 −0.397291 0.917693i \(-0.630049\pi\)
−0.397291 + 0.917693i \(0.630049\pi\)
\(234\) 0 0
\(235\) −1224.00 −0.339766
\(236\) −480.000 −0.132396
\(237\) 0 0
\(238\) −1200.00 −0.326825
\(239\) 1812.00 0.490412 0.245206 0.969471i \(-0.421144\pi\)
0.245206 + 0.969471i \(0.421144\pi\)
\(240\) 0 0
\(241\) −1582.00 −0.422845 −0.211422 0.977395i \(-0.567810\pi\)
−0.211422 + 0.977395i \(0.567810\pi\)
\(242\) 1510.00 0.401101
\(243\) 0 0
\(244\) 2456.00 0.644382
\(245\) −342.000 −0.0891820
\(246\) 0 0
\(247\) −208.000 −0.0535819
\(248\) −1312.00 −0.335936
\(249\) 0 0
\(250\) −2568.00 −0.649658
\(251\) −2148.00 −0.540162 −0.270081 0.962838i \(-0.587050\pi\)
−0.270081 + 0.962838i \(0.587050\pi\)
\(252\) 0 0
\(253\) −1728.00 −0.429401
\(254\) −4288.00 −1.05926
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 558.000 0.135436 0.0677181 0.997704i \(-0.478428\pi\)
0.0677181 + 0.997704i \(0.478428\pi\)
\(258\) 0 0
\(259\) 2200.00 0.527804
\(260\) −312.000 −0.0744208
\(261\) 0 0
\(262\) −5496.00 −1.29597
\(263\) −2112.00 −0.495177 −0.247588 0.968865i \(-0.579638\pi\)
−0.247588 + 0.968865i \(0.579638\pi\)
\(264\) 0 0
\(265\) −4428.00 −1.02645
\(266\) 640.000 0.147522
\(267\) 0 0
\(268\) 3392.00 0.773132
\(269\) −5046.00 −1.14372 −0.571859 0.820352i \(-0.693777\pi\)
−0.571859 + 0.820352i \(0.693777\pi\)
\(270\) 0 0
\(271\) −3796.00 −0.850888 −0.425444 0.904985i \(-0.639882\pi\)
−0.425444 + 0.904985i \(0.639882\pi\)
\(272\) 480.000 0.107001
\(273\) 0 0
\(274\) 5508.00 1.21442
\(275\) 2136.00 0.468384
\(276\) 0 0
\(277\) 5582.00 1.21079 0.605397 0.795924i \(-0.293014\pi\)
0.605397 + 0.795924i \(0.293014\pi\)
\(278\) −4504.00 −0.971698
\(279\) 0 0
\(280\) 960.000 0.204896
\(281\) 1950.00 0.413976 0.206988 0.978343i \(-0.433634\pi\)
0.206988 + 0.978343i \(0.433634\pi\)
\(282\) 0 0
\(283\) −4732.00 −0.993951 −0.496976 0.867765i \(-0.665556\pi\)
−0.496976 + 0.867765i \(0.665556\pi\)
\(284\) −528.000 −0.110321
\(285\) 0 0
\(286\) 624.000 0.129014
\(287\) 2520.00 0.518296
\(288\) 0 0
\(289\) −4013.00 −0.816813
\(290\) 3384.00 0.685225
\(291\) 0 0
\(292\) 872.000 0.174760
\(293\) −4998.00 −0.996540 −0.498270 0.867022i \(-0.666031\pi\)
−0.498270 + 0.867022i \(0.666031\pi\)
\(294\) 0 0
\(295\) 720.000 0.142102
\(296\) −880.000 −0.172801
\(297\) 0 0
\(298\) −3540.00 −0.688143
\(299\) 936.000 0.181038
\(300\) 0 0
\(301\) 3280.00 0.628093
\(302\) 1976.00 0.376510
\(303\) 0 0
\(304\) −256.000 −0.0482980
\(305\) −3684.00 −0.691624
\(306\) 0 0
\(307\) 6824.00 1.26862 0.634310 0.773079i \(-0.281284\pi\)
0.634310 + 0.773079i \(0.281284\pi\)
\(308\) −1920.00 −0.355202
\(309\) 0 0
\(310\) 1968.00 0.360564
\(311\) 8760.00 1.59722 0.798608 0.601852i \(-0.205570\pi\)
0.798608 + 0.601852i \(0.205570\pi\)
\(312\) 0 0
\(313\) 3962.00 0.715481 0.357740 0.933821i \(-0.383547\pi\)
0.357740 + 0.933821i \(0.383547\pi\)
\(314\) −652.000 −0.117180
\(315\) 0 0
\(316\) −4384.00 −0.780441
\(317\) −7086.00 −1.25549 −0.627744 0.778420i \(-0.716021\pi\)
−0.627744 + 0.778420i \(0.716021\pi\)
\(318\) 0 0
\(319\) −6768.00 −1.18788
\(320\) −384.000 −0.0670820
\(321\) 0 0
\(322\) −2880.00 −0.498435
\(323\) −480.000 −0.0826870
\(324\) 0 0
\(325\) −1157.00 −0.197473
\(326\) −2992.00 −0.508318
\(327\) 0 0
\(328\) −1008.00 −0.169687
\(329\) 4080.00 0.683701
\(330\) 0 0
\(331\) −9016.00 −1.49717 −0.748586 0.663037i \(-0.769267\pi\)
−0.748586 + 0.663037i \(0.769267\pi\)
\(332\) −2208.00 −0.364999
\(333\) 0 0
\(334\) 2232.00 0.365658
\(335\) −5088.00 −0.829812
\(336\) 0 0
\(337\) 2306.00 0.372747 0.186374 0.982479i \(-0.440327\pi\)
0.186374 + 0.982479i \(0.440327\pi\)
\(338\) −338.000 −0.0543928
\(339\) 0 0
\(340\) −720.000 −0.114846
\(341\) −3936.00 −0.625063
\(342\) 0 0
\(343\) −5720.00 −0.900440
\(344\) −1312.00 −0.205635
\(345\) 0 0
\(346\) 8748.00 1.35924
\(347\) 11076.0 1.71352 0.856759 0.515717i \(-0.172474\pi\)
0.856759 + 0.515717i \(0.172474\pi\)
\(348\) 0 0
\(349\) 2342.00 0.359210 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(350\) 3560.00 0.543686
\(351\) 0 0
\(352\) 768.000 0.116291
\(353\) −4650.00 −0.701118 −0.350559 0.936541i \(-0.614008\pi\)
−0.350559 + 0.936541i \(0.614008\pi\)
\(354\) 0 0
\(355\) 792.000 0.118408
\(356\) −840.000 −0.125056
\(357\) 0 0
\(358\) 24.0000 0.00354313
\(359\) 11268.0 1.65655 0.828276 0.560320i \(-0.189322\pi\)
0.828276 + 0.560320i \(0.189322\pi\)
\(360\) 0 0
\(361\) −6603.00 −0.962677
\(362\) −9436.00 −1.37001
\(363\) 0 0
\(364\) 1040.00 0.149755
\(365\) −1308.00 −0.187572
\(366\) 0 0
\(367\) −7288.00 −1.03660 −0.518298 0.855200i \(-0.673434\pi\)
−0.518298 + 0.855200i \(0.673434\pi\)
\(368\) 1152.00 0.163185
\(369\) 0 0
\(370\) 1320.00 0.185469
\(371\) 14760.0 2.06550
\(372\) 0 0
\(373\) −9970.00 −1.38399 −0.691993 0.721904i \(-0.743267\pi\)
−0.691993 + 0.721904i \(0.743267\pi\)
\(374\) 1440.00 0.199093
\(375\) 0 0
\(376\) −1632.00 −0.223840
\(377\) 3666.00 0.500819
\(378\) 0 0
\(379\) 13448.0 1.82263 0.911316 0.411708i \(-0.135068\pi\)
0.911316 + 0.411708i \(0.135068\pi\)
\(380\) 384.000 0.0518389
\(381\) 0 0
\(382\) −2736.00 −0.366455
\(383\) −11820.0 −1.57696 −0.788478 0.615064i \(-0.789130\pi\)
−0.788478 + 0.615064i \(0.789130\pi\)
\(384\) 0 0
\(385\) 2880.00 0.381243
\(386\) 6620.00 0.872925
\(387\) 0 0
\(388\) −6904.00 −0.903344
\(389\) −174.000 −0.0226790 −0.0113395 0.999936i \(-0.503610\pi\)
−0.0113395 + 0.999936i \(0.503610\pi\)
\(390\) 0 0
\(391\) 2160.00 0.279376
\(392\) −456.000 −0.0587538
\(393\) 0 0
\(394\) 6252.00 0.799419
\(395\) 6576.00 0.837657
\(396\) 0 0
\(397\) −2986.00 −0.377489 −0.188744 0.982026i \(-0.560442\pi\)
−0.188744 + 0.982026i \(0.560442\pi\)
\(398\) −9328.00 −1.17480
\(399\) 0 0
\(400\) −1424.00 −0.178000
\(401\) 10566.0 1.31581 0.657906 0.753100i \(-0.271442\pi\)
0.657906 + 0.753100i \(0.271442\pi\)
\(402\) 0 0
\(403\) 2132.00 0.263530
\(404\) −3192.00 −0.393089
\(405\) 0 0
\(406\) −11280.0 −1.37886
\(407\) −2640.00 −0.321523
\(408\) 0 0
\(409\) −7270.00 −0.878920 −0.439460 0.898262i \(-0.644830\pi\)
−0.439460 + 0.898262i \(0.644830\pi\)
\(410\) 1512.00 0.182128
\(411\) 0 0
\(412\) −2080.00 −0.248724
\(413\) −2400.00 −0.285947
\(414\) 0 0
\(415\) 3312.00 0.391758
\(416\) −416.000 −0.0490290
\(417\) 0 0
\(418\) −768.000 −0.0898663
\(419\) 7308.00 0.852074 0.426037 0.904706i \(-0.359909\pi\)
0.426037 + 0.904706i \(0.359909\pi\)
\(420\) 0 0
\(421\) −5938.00 −0.687412 −0.343706 0.939077i \(-0.611682\pi\)
−0.343706 + 0.939077i \(0.611682\pi\)
\(422\) 1112.00 0.128273
\(423\) 0 0
\(424\) −5904.00 −0.676235
\(425\) −2670.00 −0.304739
\(426\) 0 0
\(427\) 12280.0 1.39174
\(428\) −48.0000 −0.00542095
\(429\) 0 0
\(430\) 1968.00 0.220710
\(431\) −11532.0 −1.28881 −0.644405 0.764685i \(-0.722895\pi\)
−0.644405 + 0.764685i \(0.722895\pi\)
\(432\) 0 0
\(433\) −718.000 −0.0796879 −0.0398440 0.999206i \(-0.512686\pi\)
−0.0398440 + 0.999206i \(0.512686\pi\)
\(434\) −6560.00 −0.725553
\(435\) 0 0
\(436\) −7336.00 −0.805804
\(437\) −1152.00 −0.126104
\(438\) 0 0
\(439\) 8984.00 0.976726 0.488363 0.872640i \(-0.337594\pi\)
0.488363 + 0.872640i \(0.337594\pi\)
\(440\) −1152.00 −0.124817
\(441\) 0 0
\(442\) −780.000 −0.0839385
\(443\) −2604.00 −0.279277 −0.139639 0.990203i \(-0.544594\pi\)
−0.139639 + 0.990203i \(0.544594\pi\)
\(444\) 0 0
\(445\) 1260.00 0.134224
\(446\) 536.000 0.0569066
\(447\) 0 0
\(448\) 1280.00 0.134987
\(449\) 13206.0 1.38804 0.694020 0.719956i \(-0.255838\pi\)
0.694020 + 0.719956i \(0.255838\pi\)
\(450\) 0 0
\(451\) −3024.00 −0.315731
\(452\) 1464.00 0.152347
\(453\) 0 0
\(454\) 3600.00 0.372151
\(455\) −1560.00 −0.160734
\(456\) 0 0
\(457\) 8426.00 0.862476 0.431238 0.902238i \(-0.358077\pi\)
0.431238 + 0.902238i \(0.358077\pi\)
\(458\) −5980.00 −0.610103
\(459\) 0 0
\(460\) −1728.00 −0.175149
\(461\) −16686.0 −1.68578 −0.842890 0.538086i \(-0.819148\pi\)
−0.842890 + 0.538086i \(0.819148\pi\)
\(462\) 0 0
\(463\) 15932.0 1.59919 0.799593 0.600543i \(-0.205049\pi\)
0.799593 + 0.600543i \(0.205049\pi\)
\(464\) 4512.00 0.451432
\(465\) 0 0
\(466\) 5652.00 0.561854
\(467\) −18540.0 −1.83711 −0.918553 0.395297i \(-0.870642\pi\)
−0.918553 + 0.395297i \(0.870642\pi\)
\(468\) 0 0
\(469\) 16960.0 1.66981
\(470\) 2448.00 0.240251
\(471\) 0 0
\(472\) 960.000 0.0936178
\(473\) −3936.00 −0.382616
\(474\) 0 0
\(475\) 1424.00 0.137553
\(476\) 2400.00 0.231100
\(477\) 0 0
\(478\) −3624.00 −0.346774
\(479\) −6180.00 −0.589502 −0.294751 0.955574i \(-0.595237\pi\)
−0.294751 + 0.955574i \(0.595237\pi\)
\(480\) 0 0
\(481\) 1430.00 0.135556
\(482\) 3164.00 0.298996
\(483\) 0 0
\(484\) −3020.00 −0.283621
\(485\) 10356.0 0.969571
\(486\) 0 0
\(487\) 11756.0 1.09387 0.546936 0.837175i \(-0.315794\pi\)
0.546936 + 0.837175i \(0.315794\pi\)
\(488\) −4912.00 −0.455647
\(489\) 0 0
\(490\) 684.000 0.0630612
\(491\) −1908.00 −0.175370 −0.0876852 0.996148i \(-0.527947\pi\)
−0.0876852 + 0.996148i \(0.527947\pi\)
\(492\) 0 0
\(493\) 8460.00 0.772858
\(494\) 416.000 0.0378881
\(495\) 0 0
\(496\) 2624.00 0.237542
\(497\) −2640.00 −0.238270
\(498\) 0 0
\(499\) −8944.00 −0.802382 −0.401191 0.915995i \(-0.631404\pi\)
−0.401191 + 0.915995i \(0.631404\pi\)
\(500\) 5136.00 0.459378
\(501\) 0 0
\(502\) 4296.00 0.381952
\(503\) 6528.00 0.578666 0.289333 0.957228i \(-0.406566\pi\)
0.289333 + 0.957228i \(0.406566\pi\)
\(504\) 0 0
\(505\) 4788.00 0.421907
\(506\) 3456.00 0.303632
\(507\) 0 0
\(508\) 8576.00 0.749013
\(509\) 12114.0 1.05490 0.527450 0.849586i \(-0.323148\pi\)
0.527450 + 0.849586i \(0.323148\pi\)
\(510\) 0 0
\(511\) 4360.00 0.377446
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) −1116.00 −0.0957678
\(515\) 3120.00 0.266958
\(516\) 0 0
\(517\) −4896.00 −0.416491
\(518\) −4400.00 −0.373214
\(519\) 0 0
\(520\) 624.000 0.0526235
\(521\) 14310.0 1.20333 0.601663 0.798750i \(-0.294505\pi\)
0.601663 + 0.798750i \(0.294505\pi\)
\(522\) 0 0
\(523\) −18340.0 −1.53337 −0.766685 0.642024i \(-0.778095\pi\)
−0.766685 + 0.642024i \(0.778095\pi\)
\(524\) 10992.0 0.916389
\(525\) 0 0
\(526\) 4224.00 0.350143
\(527\) 4920.00 0.406677
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 8856.00 0.725811
\(531\) 0 0
\(532\) −1280.00 −0.104314
\(533\) 1638.00 0.133114
\(534\) 0 0
\(535\) 72.0000 0.00581838
\(536\) −6784.00 −0.546687
\(537\) 0 0
\(538\) 10092.0 0.808731
\(539\) −1368.00 −0.109321
\(540\) 0 0
\(541\) 9254.00 0.735417 0.367708 0.929941i \(-0.380142\pi\)
0.367708 + 0.929941i \(0.380142\pi\)
\(542\) 7592.00 0.601668
\(543\) 0 0
\(544\) −960.000 −0.0756611
\(545\) 11004.0 0.864880
\(546\) 0 0
\(547\) 17444.0 1.36353 0.681766 0.731571i \(-0.261212\pi\)
0.681766 + 0.731571i \(0.261212\pi\)
\(548\) −11016.0 −0.858723
\(549\) 0 0
\(550\) −4272.00 −0.331198
\(551\) −4512.00 −0.348852
\(552\) 0 0
\(553\) −21920.0 −1.68559
\(554\) −11164.0 −0.856160
\(555\) 0 0
\(556\) 9008.00 0.687094
\(557\) 3714.00 0.282526 0.141263 0.989972i \(-0.454884\pi\)
0.141263 + 0.989972i \(0.454884\pi\)
\(558\) 0 0
\(559\) 2132.00 0.161313
\(560\) −1920.00 −0.144884
\(561\) 0 0
\(562\) −3900.00 −0.292725
\(563\) 13812.0 1.03394 0.516968 0.856004i \(-0.327060\pi\)
0.516968 + 0.856004i \(0.327060\pi\)
\(564\) 0 0
\(565\) −2196.00 −0.163516
\(566\) 9464.00 0.702830
\(567\) 0 0
\(568\) 1056.00 0.0780084
\(569\) 15942.0 1.17456 0.587279 0.809385i \(-0.300199\pi\)
0.587279 + 0.809385i \(0.300199\pi\)
\(570\) 0 0
\(571\) 1604.00 0.117557 0.0587787 0.998271i \(-0.481279\pi\)
0.0587787 + 0.998271i \(0.481279\pi\)
\(572\) −1248.00 −0.0912264
\(573\) 0 0
\(574\) −5040.00 −0.366490
\(575\) −6408.00 −0.464751
\(576\) 0 0
\(577\) −10654.0 −0.768686 −0.384343 0.923190i \(-0.625572\pi\)
−0.384343 + 0.923190i \(0.625572\pi\)
\(578\) 8026.00 0.577574
\(579\) 0 0
\(580\) −6768.00 −0.484527
\(581\) −11040.0 −0.788324
\(582\) 0 0
\(583\) −17712.0 −1.25824
\(584\) −1744.00 −0.123574
\(585\) 0 0
\(586\) 9996.00 0.704660
\(587\) 9984.00 0.702017 0.351008 0.936372i \(-0.385839\pi\)
0.351008 + 0.936372i \(0.385839\pi\)
\(588\) 0 0
\(589\) −2624.00 −0.183565
\(590\) −1440.00 −0.100481
\(591\) 0 0
\(592\) 1760.00 0.122188
\(593\) −12618.0 −0.873793 −0.436896 0.899512i \(-0.643922\pi\)
−0.436896 + 0.899512i \(0.643922\pi\)
\(594\) 0 0
\(595\) −3600.00 −0.248043
\(596\) 7080.00 0.486591
\(597\) 0 0
\(598\) −1872.00 −0.128013
\(599\) −11184.0 −0.762881 −0.381441 0.924393i \(-0.624572\pi\)
−0.381441 + 0.924393i \(0.624572\pi\)
\(600\) 0 0
\(601\) 2810.00 0.190719 0.0953596 0.995443i \(-0.469600\pi\)
0.0953596 + 0.995443i \(0.469600\pi\)
\(602\) −6560.00 −0.444129
\(603\) 0 0
\(604\) −3952.00 −0.266233
\(605\) 4530.00 0.304414
\(606\) 0 0
\(607\) 1064.00 0.0711473 0.0355737 0.999367i \(-0.488674\pi\)
0.0355737 + 0.999367i \(0.488674\pi\)
\(608\) 512.000 0.0341519
\(609\) 0 0
\(610\) 7368.00 0.489052
\(611\) 2652.00 0.175595
\(612\) 0 0
\(613\) −20914.0 −1.37799 −0.688996 0.724766i \(-0.741948\pi\)
−0.688996 + 0.724766i \(0.741948\pi\)
\(614\) −13648.0 −0.897050
\(615\) 0 0
\(616\) 3840.00 0.251166
\(617\) −9714.00 −0.633826 −0.316913 0.948455i \(-0.602646\pi\)
−0.316913 + 0.948455i \(0.602646\pi\)
\(618\) 0 0
\(619\) −14848.0 −0.964122 −0.482061 0.876138i \(-0.660112\pi\)
−0.482061 + 0.876138i \(0.660112\pi\)
\(620\) −3936.00 −0.254957
\(621\) 0 0
\(622\) −17520.0 −1.12940
\(623\) −4200.00 −0.270095
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) −7924.00 −0.505921
\(627\) 0 0
\(628\) 1304.00 0.0828587
\(629\) 3300.00 0.209189
\(630\) 0 0
\(631\) 19172.0 1.20955 0.604774 0.796397i \(-0.293263\pi\)
0.604774 + 0.796397i \(0.293263\pi\)
\(632\) 8768.00 0.551855
\(633\) 0 0
\(634\) 14172.0 0.887763
\(635\) −12864.0 −0.803925
\(636\) 0 0
\(637\) 741.000 0.0460902
\(638\) 13536.0 0.839961
\(639\) 0 0
\(640\) 768.000 0.0474342
\(641\) 11502.0 0.708739 0.354369 0.935105i \(-0.384696\pi\)
0.354369 + 0.935105i \(0.384696\pi\)
\(642\) 0 0
\(643\) −15568.0 −0.954809 −0.477404 0.878684i \(-0.658422\pi\)
−0.477404 + 0.878684i \(0.658422\pi\)
\(644\) 5760.00 0.352447
\(645\) 0 0
\(646\) 960.000 0.0584686
\(647\) −1128.00 −0.0685414 −0.0342707 0.999413i \(-0.510911\pi\)
−0.0342707 + 0.999413i \(0.510911\pi\)
\(648\) 0 0
\(649\) 2880.00 0.174191
\(650\) 2314.00 0.139635
\(651\) 0 0
\(652\) 5984.00 0.359435
\(653\) −8118.00 −0.486496 −0.243248 0.969964i \(-0.578213\pi\)
−0.243248 + 0.969964i \(0.578213\pi\)
\(654\) 0 0
\(655\) −16488.0 −0.983572
\(656\) 2016.00 0.119987
\(657\) 0 0
\(658\) −8160.00 −0.483450
\(659\) −13572.0 −0.802261 −0.401131 0.916021i \(-0.631383\pi\)
−0.401131 + 0.916021i \(0.631383\pi\)
\(660\) 0 0
\(661\) −13138.0 −0.773085 −0.386542 0.922272i \(-0.626331\pi\)
−0.386542 + 0.922272i \(0.626331\pi\)
\(662\) 18032.0 1.05866
\(663\) 0 0
\(664\) 4416.00 0.258093
\(665\) 1920.00 0.111962
\(666\) 0 0
\(667\) 20304.0 1.17867
\(668\) −4464.00 −0.258559
\(669\) 0 0
\(670\) 10176.0 0.586766
\(671\) −14736.0 −0.847805
\(672\) 0 0
\(673\) −718.000 −0.0411246 −0.0205623 0.999789i \(-0.506546\pi\)
−0.0205623 + 0.999789i \(0.506546\pi\)
\(674\) −4612.00 −0.263572
\(675\) 0 0
\(676\) 676.000 0.0384615
\(677\) 2994.00 0.169969 0.0849843 0.996382i \(-0.472916\pi\)
0.0849843 + 0.996382i \(0.472916\pi\)
\(678\) 0 0
\(679\) −34520.0 −1.95104
\(680\) 1440.00 0.0812081
\(681\) 0 0
\(682\) 7872.00 0.441986
\(683\) −27384.0 −1.53414 −0.767071 0.641562i \(-0.778287\pi\)
−0.767071 + 0.641562i \(0.778287\pi\)
\(684\) 0 0
\(685\) 16524.0 0.921678
\(686\) 11440.0 0.636707
\(687\) 0 0
\(688\) 2624.00 0.145406
\(689\) 9594.00 0.530482
\(690\) 0 0
\(691\) 27632.0 1.52123 0.760616 0.649202i \(-0.224897\pi\)
0.760616 + 0.649202i \(0.224897\pi\)
\(692\) −17496.0 −0.961124
\(693\) 0 0
\(694\) −22152.0 −1.21164
\(695\) −13512.0 −0.737467
\(696\) 0 0
\(697\) 3780.00 0.205420
\(698\) −4684.00 −0.254000
\(699\) 0 0
\(700\) −7120.00 −0.384444
\(701\) −19062.0 −1.02705 −0.513525 0.858075i \(-0.671661\pi\)
−0.513525 + 0.858075i \(0.671661\pi\)
\(702\) 0 0
\(703\) −1760.00 −0.0944234
\(704\) −1536.00 −0.0822304
\(705\) 0 0
\(706\) 9300.00 0.495765
\(707\) −15960.0 −0.848992
\(708\) 0 0
\(709\) 3854.00 0.204147 0.102073 0.994777i \(-0.467452\pi\)
0.102073 + 0.994777i \(0.467452\pi\)
\(710\) −1584.00 −0.0837274
\(711\) 0 0
\(712\) 1680.00 0.0884279
\(713\) 11808.0 0.620215
\(714\) 0 0
\(715\) 1872.00 0.0979144
\(716\) −48.0000 −0.00250537
\(717\) 0 0
\(718\) −22536.0 −1.17136
\(719\) −20976.0 −1.08800 −0.544001 0.839085i \(-0.683091\pi\)
−0.544001 + 0.839085i \(0.683091\pi\)
\(720\) 0 0
\(721\) −10400.0 −0.537193
\(722\) 13206.0 0.680715
\(723\) 0 0
\(724\) 18872.0 0.968746
\(725\) −25098.0 −1.28568
\(726\) 0 0
\(727\) −29464.0 −1.50311 −0.751554 0.659672i \(-0.770695\pi\)
−0.751554 + 0.659672i \(0.770695\pi\)
\(728\) −2080.00 −0.105893
\(729\) 0 0
\(730\) 2616.00 0.132634
\(731\) 4920.00 0.248937
\(732\) 0 0
\(733\) −2698.00 −0.135952 −0.0679761 0.997687i \(-0.521654\pi\)
−0.0679761 + 0.997687i \(0.521654\pi\)
\(734\) 14576.0 0.732984
\(735\) 0 0
\(736\) −2304.00 −0.115389
\(737\) −20352.0 −1.01720
\(738\) 0 0
\(739\) 632.000 0.0314594 0.0157297 0.999876i \(-0.494993\pi\)
0.0157297 + 0.999876i \(0.494993\pi\)
\(740\) −2640.00 −0.131146
\(741\) 0 0
\(742\) −29520.0 −1.46053
\(743\) 20844.0 1.02920 0.514598 0.857432i \(-0.327941\pi\)
0.514598 + 0.857432i \(0.327941\pi\)
\(744\) 0 0
\(745\) −10620.0 −0.522264
\(746\) 19940.0 0.978626
\(747\) 0 0
\(748\) −2880.00 −0.140780
\(749\) −240.000 −0.0117082
\(750\) 0 0
\(751\) 272.000 0.0132163 0.00660814 0.999978i \(-0.497897\pi\)
0.00660814 + 0.999978i \(0.497897\pi\)
\(752\) 3264.00 0.158279
\(753\) 0 0
\(754\) −7332.00 −0.354132
\(755\) 5928.00 0.285751
\(756\) 0 0
\(757\) 37550.0 1.80288 0.901439 0.432907i \(-0.142512\pi\)
0.901439 + 0.432907i \(0.142512\pi\)
\(758\) −26896.0 −1.28880
\(759\) 0 0
\(760\) −768.000 −0.0366556
\(761\) −33330.0 −1.58766 −0.793832 0.608138i \(-0.791917\pi\)
−0.793832 + 0.608138i \(0.791917\pi\)
\(762\) 0 0
\(763\) −36680.0 −1.74037
\(764\) 5472.00 0.259123
\(765\) 0 0
\(766\) 23640.0 1.11508
\(767\) −1560.00 −0.0734398
\(768\) 0 0
\(769\) −15406.0 −0.722438 −0.361219 0.932481i \(-0.617639\pi\)
−0.361219 + 0.932481i \(0.617639\pi\)
\(770\) −5760.00 −0.269579
\(771\) 0 0
\(772\) −13240.0 −0.617251
\(773\) 29514.0 1.37328 0.686640 0.726998i \(-0.259085\pi\)
0.686640 + 0.726998i \(0.259085\pi\)
\(774\) 0 0
\(775\) −14596.0 −0.676521
\(776\) 13808.0 0.638761
\(777\) 0 0
\(778\) 348.000 0.0160365
\(779\) −2016.00 −0.0927223
\(780\) 0 0
\(781\) 3168.00 0.145147
\(782\) −4320.00 −0.197548
\(783\) 0 0
\(784\) 912.000 0.0415452
\(785\) −1956.00 −0.0889333
\(786\) 0 0
\(787\) 33176.0 1.50266 0.751332 0.659924i \(-0.229412\pi\)
0.751332 + 0.659924i \(0.229412\pi\)
\(788\) −12504.0 −0.565275
\(789\) 0 0
\(790\) −13152.0 −0.592313
\(791\) 7320.00 0.329038
\(792\) 0 0
\(793\) 7982.00 0.357439
\(794\) 5972.00 0.266925
\(795\) 0 0
\(796\) 18656.0 0.830709
\(797\) 16746.0 0.744258 0.372129 0.928181i \(-0.378628\pi\)
0.372129 + 0.928181i \(0.378628\pi\)
\(798\) 0 0
\(799\) 6120.00 0.270976
\(800\) 2848.00 0.125865
\(801\) 0 0
\(802\) −21132.0 −0.930420
\(803\) −5232.00 −0.229929
\(804\) 0 0
\(805\) −8640.00 −0.378286
\(806\) −4264.00 −0.186344
\(807\) 0 0
\(808\) 6384.00 0.277956
\(809\) 15846.0 0.688647 0.344324 0.938851i \(-0.388108\pi\)
0.344324 + 0.938851i \(0.388108\pi\)
\(810\) 0 0
\(811\) 22952.0 0.993778 0.496889 0.867814i \(-0.334476\pi\)
0.496889 + 0.867814i \(0.334476\pi\)
\(812\) 22560.0 0.975001
\(813\) 0 0
\(814\) 5280.00 0.227351
\(815\) −8976.00 −0.385786
\(816\) 0 0
\(817\) −2624.00 −0.112365
\(818\) 14540.0 0.621490
\(819\) 0 0
\(820\) −3024.00 −0.128784
\(821\) 37146.0 1.57906 0.789528 0.613715i \(-0.210326\pi\)
0.789528 + 0.613715i \(0.210326\pi\)
\(822\) 0 0
\(823\) −9592.00 −0.406265 −0.203133 0.979151i \(-0.565112\pi\)
−0.203133 + 0.979151i \(0.565112\pi\)
\(824\) 4160.00 0.175874
\(825\) 0 0
\(826\) 4800.00 0.202195
\(827\) 39960.0 1.68022 0.840112 0.542413i \(-0.182489\pi\)
0.840112 + 0.542413i \(0.182489\pi\)
\(828\) 0 0
\(829\) −3706.00 −0.155265 −0.0776325 0.996982i \(-0.524736\pi\)
−0.0776325 + 0.996982i \(0.524736\pi\)
\(830\) −6624.00 −0.277015
\(831\) 0 0
\(832\) 832.000 0.0346688
\(833\) 1710.00 0.0711260
\(834\) 0 0
\(835\) 6696.00 0.277515
\(836\) 1536.00 0.0635451
\(837\) 0 0
\(838\) −14616.0 −0.602508
\(839\) −9756.00 −0.401448 −0.200724 0.979648i \(-0.564329\pi\)
−0.200724 + 0.979648i \(0.564329\pi\)
\(840\) 0 0
\(841\) 55135.0 2.26065
\(842\) 11876.0 0.486074
\(843\) 0 0
\(844\) −2224.00 −0.0907029
\(845\) −1014.00 −0.0412813
\(846\) 0 0
\(847\) −15100.0 −0.612565
\(848\) 11808.0 0.478170
\(849\) 0 0
\(850\) 5340.00 0.215483
\(851\) 7920.00 0.319029
\(852\) 0 0
\(853\) 11342.0 0.455267 0.227633 0.973747i \(-0.426901\pi\)
0.227633 + 0.973747i \(0.426901\pi\)
\(854\) −24560.0 −0.984105
\(855\) 0 0
\(856\) 96.0000 0.00383319
\(857\) 16134.0 0.643089 0.321544 0.946895i \(-0.395798\pi\)
0.321544 + 0.946895i \(0.395798\pi\)
\(858\) 0 0
\(859\) −20932.0 −0.831421 −0.415710 0.909497i \(-0.636467\pi\)
−0.415710 + 0.909497i \(0.636467\pi\)
\(860\) −3936.00 −0.156066
\(861\) 0 0
\(862\) 23064.0 0.911326
\(863\) −10044.0 −0.396178 −0.198089 0.980184i \(-0.563474\pi\)
−0.198089 + 0.980184i \(0.563474\pi\)
\(864\) 0 0
\(865\) 26244.0 1.03159
\(866\) 1436.00 0.0563479
\(867\) 0 0
\(868\) 13120.0 0.513044
\(869\) 26304.0 1.02681
\(870\) 0 0
\(871\) 11024.0 0.428856
\(872\) 14672.0 0.569790
\(873\) 0 0
\(874\) 2304.00 0.0891693
\(875\) 25680.0 0.992163
\(876\) 0 0
\(877\) −26314.0 −1.01318 −0.506591 0.862186i \(-0.669095\pi\)
−0.506591 + 0.862186i \(0.669095\pi\)
\(878\) −17968.0 −0.690650
\(879\) 0 0
\(880\) 2304.00 0.0882589
\(881\) −37506.0 −1.43429 −0.717145 0.696924i \(-0.754551\pi\)
−0.717145 + 0.696924i \(0.754551\pi\)
\(882\) 0 0
\(883\) −6388.00 −0.243458 −0.121729 0.992563i \(-0.538844\pi\)
−0.121729 + 0.992563i \(0.538844\pi\)
\(884\) 1560.00 0.0593535
\(885\) 0 0
\(886\) 5208.00 0.197479
\(887\) 5472.00 0.207138 0.103569 0.994622i \(-0.466974\pi\)
0.103569 + 0.994622i \(0.466974\pi\)
\(888\) 0 0
\(889\) 42880.0 1.61772
\(890\) −2520.00 −0.0949108
\(891\) 0 0
\(892\) −1072.00 −0.0402390
\(893\) −3264.00 −0.122313
\(894\) 0 0
\(895\) 72.0000 0.00268904
\(896\) −2560.00 −0.0954504
\(897\) 0 0
\(898\) −26412.0 −0.981492
\(899\) 46248.0 1.71575
\(900\) 0 0
\(901\) 22140.0 0.818635
\(902\) 6048.00 0.223255
\(903\) 0 0
\(904\) −2928.00 −0.107725
\(905\) −28308.0 −1.03977
\(906\) 0 0
\(907\) −7180.00 −0.262853 −0.131427 0.991326i \(-0.541956\pi\)
−0.131427 + 0.991326i \(0.541956\pi\)
\(908\) −7200.00 −0.263150
\(909\) 0 0
\(910\) 3120.00 0.113656
\(911\) −27624.0 −1.00464 −0.502318 0.864683i \(-0.667519\pi\)
−0.502318 + 0.864683i \(0.667519\pi\)
\(912\) 0 0
\(913\) 13248.0 0.480224
\(914\) −16852.0 −0.609863
\(915\) 0 0
\(916\) 11960.0 0.431408
\(917\) 54960.0 1.97921
\(918\) 0 0
\(919\) −30256.0 −1.08602 −0.543011 0.839726i \(-0.682716\pi\)
−0.543011 + 0.839726i \(0.682716\pi\)
\(920\) 3456.00 0.123849
\(921\) 0 0
\(922\) 33372.0 1.19203
\(923\) −1716.00 −0.0611948
\(924\) 0 0
\(925\) −9790.00 −0.347993
\(926\) −31864.0 −1.13079
\(927\) 0 0
\(928\) −9024.00 −0.319210
\(929\) 1926.00 0.0680194 0.0340097 0.999422i \(-0.489172\pi\)
0.0340097 + 0.999422i \(0.489172\pi\)
\(930\) 0 0
\(931\) −912.000 −0.0321048
\(932\) −11304.0 −0.397291
\(933\) 0 0
\(934\) 37080.0 1.29903
\(935\) 4320.00 0.151101
\(936\) 0 0
\(937\) 3962.00 0.138135 0.0690677 0.997612i \(-0.477998\pi\)
0.0690677 + 0.997612i \(0.477998\pi\)
\(938\) −33920.0 −1.18073
\(939\) 0 0
\(940\) −4896.00 −0.169883
\(941\) 1074.00 0.0372066 0.0186033 0.999827i \(-0.494078\pi\)
0.0186033 + 0.999827i \(0.494078\pi\)
\(942\) 0 0
\(943\) 9072.00 0.313282
\(944\) −1920.00 −0.0661978
\(945\) 0 0
\(946\) 7872.00 0.270551
\(947\) −4848.00 −0.166356 −0.0831778 0.996535i \(-0.526507\pi\)
−0.0831778 + 0.996535i \(0.526507\pi\)
\(948\) 0 0
\(949\) 2834.00 0.0969394
\(950\) −2848.00 −0.0972645
\(951\) 0 0
\(952\) −4800.00 −0.163413
\(953\) −762.000 −0.0259009 −0.0129505 0.999916i \(-0.504122\pi\)
−0.0129505 + 0.999916i \(0.504122\pi\)
\(954\) 0 0
\(955\) −8208.00 −0.278120
\(956\) 7248.00 0.245206
\(957\) 0 0
\(958\) 12360.0 0.416841
\(959\) −55080.0 −1.85467
\(960\) 0 0
\(961\) −2895.00 −0.0971770
\(962\) −2860.00 −0.0958525
\(963\) 0 0
\(964\) −6328.00 −0.211422
\(965\) 19860.0 0.662504
\(966\) 0 0
\(967\) 35804.0 1.19067 0.595336 0.803477i \(-0.297019\pi\)
0.595336 + 0.803477i \(0.297019\pi\)
\(968\) 6040.00 0.200551
\(969\) 0 0
\(970\) −20712.0 −0.685590
\(971\) 4260.00 0.140793 0.0703964 0.997519i \(-0.477574\pi\)
0.0703964 + 0.997519i \(0.477574\pi\)
\(972\) 0 0
\(973\) 45040.0 1.48398
\(974\) −23512.0 −0.773484
\(975\) 0 0
\(976\) 9824.00 0.322191
\(977\) 28710.0 0.940137 0.470069 0.882630i \(-0.344229\pi\)
0.470069 + 0.882630i \(0.344229\pi\)
\(978\) 0 0
\(979\) 5040.00 0.164534
\(980\) −1368.00 −0.0445910
\(981\) 0 0
\(982\) 3816.00 0.124006
\(983\) 49524.0 1.60689 0.803444 0.595381i \(-0.202999\pi\)
0.803444 + 0.595381i \(0.202999\pi\)
\(984\) 0 0
\(985\) 18756.0 0.606717
\(986\) −16920.0 −0.546493
\(987\) 0 0
\(988\) −832.000 −0.0267909
\(989\) 11808.0 0.379649
\(990\) 0 0
\(991\) 44408.0 1.42348 0.711739 0.702444i \(-0.247908\pi\)
0.711739 + 0.702444i \(0.247908\pi\)
\(992\) −5248.00 −0.167968
\(993\) 0 0
\(994\) 5280.00 0.168482
\(995\) −27984.0 −0.891610
\(996\) 0 0
\(997\) 18398.0 0.584424 0.292212 0.956354i \(-0.405609\pi\)
0.292212 + 0.956354i \(0.405609\pi\)
\(998\) 17888.0 0.567369
\(999\) 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 234.4.a.b.1.1 1
3.2 odd 2 78.4.a.e.1.1 1
4.3 odd 2 1872.4.a.e.1.1 1
12.11 even 2 624.4.a.i.1.1 1
15.14 odd 2 1950.4.a.c.1.1 1
24.5 odd 2 2496.4.a.k.1.1 1
24.11 even 2 2496.4.a.b.1.1 1
39.5 even 4 1014.4.b.c.337.1 2
39.8 even 4 1014.4.b.c.337.2 2
39.38 odd 2 1014.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.e.1.1 1 3.2 odd 2
234.4.a.b.1.1 1 1.1 even 1 trivial
624.4.a.i.1.1 1 12.11 even 2
1014.4.a.b.1.1 1 39.38 odd 2
1014.4.b.c.337.1 2 39.5 even 4
1014.4.b.c.337.2 2 39.8 even 4
1872.4.a.e.1.1 1 4.3 odd 2
1950.4.a.c.1.1 1 15.14 odd 2
2496.4.a.b.1.1 1 24.11 even 2
2496.4.a.k.1.1 1 24.5 odd 2