Properties

Label 832.4.a.d.1.1
Level $832$
Weight $4$
Character 832.1
Self dual yes
Analytic conductor $49.090$
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] = [832,4,Mod(1,832)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(832, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("832.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 832.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: \(49.0895891248\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 832.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{3} +18.0000 q^{5} +20.0000 q^{7} -11.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{3} +18.0000 q^{5} +20.0000 q^{7} -11.0000 q^{9} +48.0000 q^{11} -13.0000 q^{13} -72.0000 q^{15} +66.0000 q^{17} +16.0000 q^{19} -80.0000 q^{21} +168.000 q^{23} +199.000 q^{25} +152.000 q^{27} -6.00000 q^{29} +20.0000 q^{31} -192.000 q^{33} +360.000 q^{35} -254.000 q^{37} +52.0000 q^{39} -390.000 q^{41} +124.000 q^{43} -198.000 q^{45} -468.000 q^{47} +57.0000 q^{49} -264.000 q^{51} -558.000 q^{53} +864.000 q^{55} -64.0000 q^{57} +96.0000 q^{59} +826.000 q^{61} -220.000 q^{63} -234.000 q^{65} +160.000 q^{67} -672.000 q^{69} -420.000 q^{71} +362.000 q^{73} -796.000 q^{75} +960.000 q^{77} +776.000 q^{79} -311.000 q^{81} +1188.00 q^{85} +24.0000 q^{87} +1626.00 q^{89} -260.000 q^{91} -80.0000 q^{93} +288.000 q^{95} -1294.00 q^{97} -528.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.00000 −0.769800 −0.384900 0.922958i \(-0.625764\pi\)
−0.384900 + 0.922958i \(0.625764\pi\)
\(4\) 0 0
\(5\) 18.0000 1.60997 0.804984 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) 0 0
\(7\) 20.0000 1.07990 0.539949 0.841698i \(-0.318443\pi\)
0.539949 + 0.841698i \(0.318443\pi\)
\(8\) 0 0
\(9\) −11.0000 −0.407407
\(10\) 0 0
\(11\) 48.0000 1.31569 0.657843 0.753155i \(-0.271469\pi\)
0.657843 + 0.753155i \(0.271469\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −72.0000 −1.23935
\(16\) 0 0
\(17\) 66.0000 0.941609 0.470804 0.882238i \(-0.343964\pi\)
0.470804 + 0.882238i \(0.343964\pi\)
\(18\) 0 0
\(19\) 16.0000 0.193192 0.0965961 0.995324i \(-0.469204\pi\)
0.0965961 + 0.995324i \(0.469204\pi\)
\(20\) 0 0
\(21\) −80.0000 −0.831306
\(22\) 0 0
\(23\) 168.000 1.52306 0.761531 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(24\) 0 0
\(25\) 199.000 1.59200
\(26\) 0 0
\(27\) 152.000 1.08342
\(28\) 0 0
\(29\) −6.00000 −0.0384197 −0.0192099 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) 20.0000 0.115874 0.0579372 0.998320i \(-0.481548\pi\)
0.0579372 + 0.998320i \(0.481548\pi\)
\(32\) 0 0
\(33\) −192.000 −1.01282
\(34\) 0 0
\(35\) 360.000 1.73860
\(36\) 0 0
\(37\) −254.000 −1.12858 −0.564288 0.825578i \(-0.690849\pi\)
−0.564288 + 0.825578i \(0.690849\pi\)
\(38\) 0 0
\(39\) 52.0000 0.213504
\(40\) 0 0
\(41\) −390.000 −1.48556 −0.742778 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(42\) 0 0
\(43\) 124.000 0.439763 0.219882 0.975527i \(-0.429433\pi\)
0.219882 + 0.975527i \(0.429433\pi\)
\(44\) 0 0
\(45\) −198.000 −0.655913
\(46\) 0 0
\(47\) −468.000 −1.45244 −0.726221 0.687461i \(-0.758725\pi\)
−0.726221 + 0.687461i \(0.758725\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) −264.000 −0.724851
\(52\) 0 0
\(53\) −558.000 −1.44617 −0.723087 0.690757i \(-0.757277\pi\)
−0.723087 + 0.690757i \(0.757277\pi\)
\(54\) 0 0
\(55\) 864.000 2.11821
\(56\) 0 0
\(57\) −64.0000 −0.148719
\(58\) 0 0
\(59\) 96.0000 0.211833 0.105916 0.994375i \(-0.466222\pi\)
0.105916 + 0.994375i \(0.466222\pi\)
\(60\) 0 0
\(61\) 826.000 1.73375 0.866873 0.498530i \(-0.166127\pi\)
0.866873 + 0.498530i \(0.166127\pi\)
\(62\) 0 0
\(63\) −220.000 −0.439959
\(64\) 0 0
\(65\) −234.000 −0.446525
\(66\) 0 0
\(67\) 160.000 0.291748 0.145874 0.989303i \(-0.453401\pi\)
0.145874 + 0.989303i \(0.453401\pi\)
\(68\) 0 0
\(69\) −672.000 −1.17245
\(70\) 0 0
\(71\) −420.000 −0.702040 −0.351020 0.936368i \(-0.614165\pi\)
−0.351020 + 0.936368i \(0.614165\pi\)
\(72\) 0 0
\(73\) 362.000 0.580396 0.290198 0.956967i \(-0.406279\pi\)
0.290198 + 0.956967i \(0.406279\pi\)
\(74\) 0 0
\(75\) −796.000 −1.22552
\(76\) 0 0
\(77\) 960.000 1.42081
\(78\) 0 0
\(79\) 776.000 1.10515 0.552575 0.833463i \(-0.313645\pi\)
0.552575 + 0.833463i \(0.313645\pi\)
\(80\) 0 0
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 1188.00 1.51596
\(86\) 0 0
\(87\) 24.0000 0.0295755
\(88\) 0 0
\(89\) 1626.00 1.93658 0.968290 0.249828i \(-0.0803741\pi\)
0.968290 + 0.249828i \(0.0803741\pi\)
\(90\) 0 0
\(91\) −260.000 −0.299510
\(92\) 0 0
\(93\) −80.0000 −0.0892001
\(94\) 0 0
\(95\) 288.000 0.311033
\(96\) 0 0
\(97\) −1294.00 −1.35449 −0.677246 0.735756i \(-0.736827\pi\)
−0.677246 + 0.735756i \(0.736827\pi\)
\(98\) 0 0
\(99\) −528.000 −0.536020
\(100\) 0 0
\(101\) −222.000 −0.218711 −0.109356 0.994003i \(-0.534879\pi\)
−0.109356 + 0.994003i \(0.534879\pi\)
\(102\) 0 0
\(103\) 632.000 0.604590 0.302295 0.953214i \(-0.402247\pi\)
0.302295 + 0.953214i \(0.402247\pi\)
\(104\) 0 0
\(105\) −1440.00 −1.33838
\(106\) 0 0
\(107\) 948.000 0.856510 0.428255 0.903658i \(-0.359128\pi\)
0.428255 + 0.903658i \(0.359128\pi\)
\(108\) 0 0
\(109\) −758.000 −0.666085 −0.333042 0.942912i \(-0.608075\pi\)
−0.333042 + 0.942912i \(0.608075\pi\)
\(110\) 0 0
\(111\) 1016.00 0.868779
\(112\) 0 0
\(113\) 642.000 0.534463 0.267231 0.963632i \(-0.413891\pi\)
0.267231 + 0.963632i \(0.413891\pi\)
\(114\) 0 0
\(115\) 3024.00 2.45208
\(116\) 0 0
\(117\) 143.000 0.112994
\(118\) 0 0
\(119\) 1320.00 1.01684
\(120\) 0 0
\(121\) 973.000 0.731029
\(122\) 0 0
\(123\) 1560.00 1.14358
\(124\) 0 0
\(125\) 1332.00 0.953102
\(126\) 0 0
\(127\) −880.000 −0.614861 −0.307431 0.951571i \(-0.599469\pi\)
−0.307431 + 0.951571i \(0.599469\pi\)
\(128\) 0 0
\(129\) −496.000 −0.338530
\(130\) 0 0
\(131\) −324.000 −0.216092 −0.108046 0.994146i \(-0.534459\pi\)
−0.108046 + 0.994146i \(0.534459\pi\)
\(132\) 0 0
\(133\) 320.000 0.208628
\(134\) 0 0
\(135\) 2736.00 1.74428
\(136\) 0 0
\(137\) 1722.00 1.07387 0.536936 0.843623i \(-0.319582\pi\)
0.536936 + 0.843623i \(0.319582\pi\)
\(138\) 0 0
\(139\) 340.000 0.207471 0.103735 0.994605i \(-0.466921\pi\)
0.103735 + 0.994605i \(0.466921\pi\)
\(140\) 0 0
\(141\) 1872.00 1.11809
\(142\) 0 0
\(143\) −624.000 −0.364906
\(144\) 0 0
\(145\) −108.000 −0.0618546
\(146\) 0 0
\(147\) −228.000 −0.127926
\(148\) 0 0
\(149\) −750.000 −0.412365 −0.206183 0.978514i \(-0.566104\pi\)
−0.206183 + 0.978514i \(0.566104\pi\)
\(150\) 0 0
\(151\) 1748.00 0.942054 0.471027 0.882119i \(-0.343883\pi\)
0.471027 + 0.882119i \(0.343883\pi\)
\(152\) 0 0
\(153\) −726.000 −0.383618
\(154\) 0 0
\(155\) 360.000 0.186554
\(156\) 0 0
\(157\) −614.000 −0.312118 −0.156059 0.987748i \(-0.549879\pi\)
−0.156059 + 0.987748i \(0.549879\pi\)
\(158\) 0 0
\(159\) 2232.00 1.11326
\(160\) 0 0
\(161\) 3360.00 1.64475
\(162\) 0 0
\(163\) 808.000 0.388267 0.194133 0.980975i \(-0.437811\pi\)
0.194133 + 0.980975i \(0.437811\pi\)
\(164\) 0 0
\(165\) −3456.00 −1.63060
\(166\) 0 0
\(167\) −2028.00 −0.939709 −0.469854 0.882744i \(-0.655694\pi\)
−0.469854 + 0.882744i \(0.655694\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −176.000 −0.0787079
\(172\) 0 0
\(173\) 1194.00 0.524729 0.262365 0.964969i \(-0.415498\pi\)
0.262365 + 0.964969i \(0.415498\pi\)
\(174\) 0 0
\(175\) 3980.00 1.71920
\(176\) 0 0
\(177\) −384.000 −0.163069
\(178\) 0 0
\(179\) 2820.00 1.17752 0.588762 0.808307i \(-0.299616\pi\)
0.588762 + 0.808307i \(0.299616\pi\)
\(180\) 0 0
\(181\) 754.000 0.309637 0.154819 0.987943i \(-0.450521\pi\)
0.154819 + 0.987943i \(0.450521\pi\)
\(182\) 0 0
\(183\) −3304.00 −1.33464
\(184\) 0 0
\(185\) −4572.00 −1.81697
\(186\) 0 0
\(187\) 3168.00 1.23886
\(188\) 0 0
\(189\) 3040.00 1.16999
\(190\) 0 0
\(191\) −2328.00 −0.881928 −0.440964 0.897525i \(-0.645363\pi\)
−0.440964 + 0.897525i \(0.645363\pi\)
\(192\) 0 0
\(193\) 2450.00 0.913756 0.456878 0.889529i \(-0.348968\pi\)
0.456878 + 0.889529i \(0.348968\pi\)
\(194\) 0 0
\(195\) 936.000 0.343735
\(196\) 0 0
\(197\) −4542.00 −1.64266 −0.821330 0.570453i \(-0.806768\pi\)
−0.821330 + 0.570453i \(0.806768\pi\)
\(198\) 0 0
\(199\) −664.000 −0.236531 −0.118266 0.992982i \(-0.537733\pi\)
−0.118266 + 0.992982i \(0.537733\pi\)
\(200\) 0 0
\(201\) −640.000 −0.224588
\(202\) 0 0
\(203\) −120.000 −0.0414894
\(204\) 0 0
\(205\) −7020.00 −2.39170
\(206\) 0 0
\(207\) −1848.00 −0.620507
\(208\) 0 0
\(209\) 768.000 0.254180
\(210\) 0 0
\(211\) 4156.00 1.35598 0.677988 0.735073i \(-0.262852\pi\)
0.677988 + 0.735073i \(0.262852\pi\)
\(212\) 0 0
\(213\) 1680.00 0.540431
\(214\) 0 0
\(215\) 2232.00 0.708005
\(216\) 0 0
\(217\) 400.000 0.125133
\(218\) 0 0
\(219\) −1448.00 −0.446789
\(220\) 0 0
\(221\) −858.000 −0.261155
\(222\) 0 0
\(223\) −3292.00 −0.988559 −0.494279 0.869303i \(-0.664568\pi\)
−0.494279 + 0.869303i \(0.664568\pi\)
\(224\) 0 0
\(225\) −2189.00 −0.648593
\(226\) 0 0
\(227\) −2352.00 −0.687699 −0.343850 0.939025i \(-0.611731\pi\)
−0.343850 + 0.939025i \(0.611731\pi\)
\(228\) 0 0
\(229\) −686.000 −0.197957 −0.0989785 0.995090i \(-0.531558\pi\)
−0.0989785 + 0.995090i \(0.531558\pi\)
\(230\) 0 0
\(231\) −3840.00 −1.09374
\(232\) 0 0
\(233\) 1818.00 0.511164 0.255582 0.966787i \(-0.417733\pi\)
0.255582 + 0.966787i \(0.417733\pi\)
\(234\) 0 0
\(235\) −8424.00 −2.33839
\(236\) 0 0
\(237\) −3104.00 −0.850745
\(238\) 0 0
\(239\) −540.000 −0.146149 −0.0730747 0.997326i \(-0.523281\pi\)
−0.0730747 + 0.997326i \(0.523281\pi\)
\(240\) 0 0
\(241\) −862.000 −0.230400 −0.115200 0.993342i \(-0.536751\pi\)
−0.115200 + 0.993342i \(0.536751\pi\)
\(242\) 0 0
\(243\) −2860.00 −0.755017
\(244\) 0 0
\(245\) 1026.00 0.267546
\(246\) 0 0
\(247\) −208.000 −0.0535819
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4836.00 −1.21612 −0.608059 0.793892i \(-0.708052\pi\)
−0.608059 + 0.793892i \(0.708052\pi\)
\(252\) 0 0
\(253\) 8064.00 2.00387
\(254\) 0 0
\(255\) −4752.00 −1.16699
\(256\) 0 0
\(257\) 1410.00 0.342231 0.171116 0.985251i \(-0.445263\pi\)
0.171116 + 0.985251i \(0.445263\pi\)
\(258\) 0 0
\(259\) −5080.00 −1.21875
\(260\) 0 0
\(261\) 66.0000 0.0156525
\(262\) 0 0
\(263\) 8304.00 1.94695 0.973473 0.228804i \(-0.0734814\pi\)
0.973473 + 0.228804i \(0.0734814\pi\)
\(264\) 0 0
\(265\) −10044.0 −2.32829
\(266\) 0 0
\(267\) −6504.00 −1.49078
\(268\) 0 0
\(269\) 2634.00 0.597018 0.298509 0.954407i \(-0.403511\pi\)
0.298509 + 0.954407i \(0.403511\pi\)
\(270\) 0 0
\(271\) 7436.00 1.66681 0.833404 0.552665i \(-0.186389\pi\)
0.833404 + 0.552665i \(0.186389\pi\)
\(272\) 0 0
\(273\) 1040.00 0.230563
\(274\) 0 0
\(275\) 9552.00 2.09457
\(276\) 0 0
\(277\) 5074.00 1.10060 0.550302 0.834966i \(-0.314513\pi\)
0.550302 + 0.834966i \(0.314513\pi\)
\(278\) 0 0
\(279\) −220.000 −0.0472081
\(280\) 0 0
\(281\) −1638.00 −0.347740 −0.173870 0.984769i \(-0.555627\pi\)
−0.173870 + 0.984769i \(0.555627\pi\)
\(282\) 0 0
\(283\) 4588.00 0.963704 0.481852 0.876253i \(-0.339964\pi\)
0.481852 + 0.876253i \(0.339964\pi\)
\(284\) 0 0
\(285\) −1152.00 −0.239434
\(286\) 0 0
\(287\) −7800.00 −1.60425
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) 0 0
\(291\) 5176.00 1.04269
\(292\) 0 0
\(293\) 2850.00 0.568255 0.284128 0.958786i \(-0.408296\pi\)
0.284128 + 0.958786i \(0.408296\pi\)
\(294\) 0 0
\(295\) 1728.00 0.341044
\(296\) 0 0
\(297\) 7296.00 1.42544
\(298\) 0 0
\(299\) −2184.00 −0.422421
\(300\) 0 0
\(301\) 2480.00 0.474900
\(302\) 0 0
\(303\) 888.000 0.168364
\(304\) 0 0
\(305\) 14868.0 2.79128
\(306\) 0 0
\(307\) −8120.00 −1.50955 −0.754777 0.655982i \(-0.772255\pi\)
−0.754777 + 0.655982i \(0.772255\pi\)
\(308\) 0 0
\(309\) −2528.00 −0.465414
\(310\) 0 0
\(311\) −3528.00 −0.643262 −0.321631 0.946865i \(-0.604231\pi\)
−0.321631 + 0.946865i \(0.604231\pi\)
\(312\) 0 0
\(313\) −6982.00 −1.26085 −0.630425 0.776250i \(-0.717119\pi\)
−0.630425 + 0.776250i \(0.717119\pi\)
\(314\) 0 0
\(315\) −3960.00 −0.708320
\(316\) 0 0
\(317\) −9270.00 −1.64245 −0.821223 0.570608i \(-0.806708\pi\)
−0.821223 + 0.570608i \(0.806708\pi\)
\(318\) 0 0
\(319\) −288.000 −0.0505483
\(320\) 0 0
\(321\) −3792.00 −0.659342
\(322\) 0 0
\(323\) 1056.00 0.181911
\(324\) 0 0
\(325\) −2587.00 −0.441541
\(326\) 0 0
\(327\) 3032.00 0.512752
\(328\) 0 0
\(329\) −9360.00 −1.56849
\(330\) 0 0
\(331\) 7720.00 1.28196 0.640981 0.767557i \(-0.278528\pi\)
0.640981 + 0.767557i \(0.278528\pi\)
\(332\) 0 0
\(333\) 2794.00 0.459791
\(334\) 0 0
\(335\) 2880.00 0.469705
\(336\) 0 0
\(337\) −1726.00 −0.278995 −0.139497 0.990222i \(-0.544549\pi\)
−0.139497 + 0.990222i \(0.544549\pi\)
\(338\) 0 0
\(339\) −2568.00 −0.411430
\(340\) 0 0
\(341\) 960.000 0.152454
\(342\) 0 0
\(343\) −5720.00 −0.900440
\(344\) 0 0
\(345\) −12096.0 −1.88761
\(346\) 0 0
\(347\) 4020.00 0.621916 0.310958 0.950424i \(-0.399350\pi\)
0.310958 + 0.950424i \(0.399350\pi\)
\(348\) 0 0
\(349\) −1910.00 −0.292951 −0.146476 0.989214i \(-0.546793\pi\)
−0.146476 + 0.989214i \(0.546793\pi\)
\(350\) 0 0
\(351\) −1976.00 −0.300487
\(352\) 0 0
\(353\) 5442.00 0.820534 0.410267 0.911965i \(-0.365436\pi\)
0.410267 + 0.911965i \(0.365436\pi\)
\(354\) 0 0
\(355\) −7560.00 −1.13026
\(356\) 0 0
\(357\) −5280.00 −0.782765
\(358\) 0 0
\(359\) −9324.00 −1.37076 −0.685379 0.728187i \(-0.740363\pi\)
−0.685379 + 0.728187i \(0.740363\pi\)
\(360\) 0 0
\(361\) −6603.00 −0.962677
\(362\) 0 0
\(363\) −3892.00 −0.562747
\(364\) 0 0
\(365\) 6516.00 0.934419
\(366\) 0 0
\(367\) 4520.00 0.642894 0.321447 0.946928i \(-0.395831\pi\)
0.321447 + 0.946928i \(0.395831\pi\)
\(368\) 0 0
\(369\) 4290.00 0.605226
\(370\) 0 0
\(371\) −11160.0 −1.56172
\(372\) 0 0
\(373\) 5938.00 0.824284 0.412142 0.911120i \(-0.364781\pi\)
0.412142 + 0.911120i \(0.364781\pi\)
\(374\) 0 0
\(375\) −5328.00 −0.733698
\(376\) 0 0
\(377\) 78.0000 0.0106557
\(378\) 0 0
\(379\) −2216.00 −0.300338 −0.150169 0.988660i \(-0.547982\pi\)
−0.150169 + 0.988660i \(0.547982\pi\)
\(380\) 0 0
\(381\) 3520.00 0.473320
\(382\) 0 0
\(383\) 3828.00 0.510709 0.255355 0.966847i \(-0.417808\pi\)
0.255355 + 0.966847i \(0.417808\pi\)
\(384\) 0 0
\(385\) 17280.0 2.28746
\(386\) 0 0
\(387\) −1364.00 −0.179163
\(388\) 0 0
\(389\) −5022.00 −0.654564 −0.327282 0.944927i \(-0.606133\pi\)
−0.327282 + 0.944927i \(0.606133\pi\)
\(390\) 0 0
\(391\) 11088.0 1.43413
\(392\) 0 0
\(393\) 1296.00 0.166347
\(394\) 0 0
\(395\) 13968.0 1.77926
\(396\) 0 0
\(397\) −6086.00 −0.769389 −0.384695 0.923044i \(-0.625693\pi\)
−0.384695 + 0.923044i \(0.625693\pi\)
\(398\) 0 0
\(399\) −1280.00 −0.160602
\(400\) 0 0
\(401\) 1122.00 0.139726 0.0698629 0.997557i \(-0.477744\pi\)
0.0698629 + 0.997557i \(0.477744\pi\)
\(402\) 0 0
\(403\) −260.000 −0.0321378
\(404\) 0 0
\(405\) −5598.00 −0.686832
\(406\) 0 0
\(407\) −12192.0 −1.48485
\(408\) 0 0
\(409\) 362.000 0.0437647 0.0218823 0.999761i \(-0.493034\pi\)
0.0218823 + 0.999761i \(0.493034\pi\)
\(410\) 0 0
\(411\) −6888.00 −0.826667
\(412\) 0 0
\(413\) 1920.00 0.228758
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1360.00 −0.159711
\(418\) 0 0
\(419\) 2316.00 0.270033 0.135017 0.990843i \(-0.456891\pi\)
0.135017 + 0.990843i \(0.456891\pi\)
\(420\) 0 0
\(421\) −5006.00 −0.579519 −0.289760 0.957099i \(-0.593575\pi\)
−0.289760 + 0.957099i \(0.593575\pi\)
\(422\) 0 0
\(423\) 5148.00 0.591736
\(424\) 0 0
\(425\) 13134.0 1.49904
\(426\) 0 0
\(427\) 16520.0 1.87227
\(428\) 0 0
\(429\) 2496.00 0.280904
\(430\) 0 0
\(431\) −11244.0 −1.25662 −0.628311 0.777962i \(-0.716254\pi\)
−0.628311 + 0.777962i \(0.716254\pi\)
\(432\) 0 0
\(433\) 13106.0 1.45458 0.727291 0.686329i \(-0.240779\pi\)
0.727291 + 0.686329i \(0.240779\pi\)
\(434\) 0 0
\(435\) 432.000 0.0476157
\(436\) 0 0
\(437\) 2688.00 0.294244
\(438\) 0 0
\(439\) −13480.0 −1.46552 −0.732762 0.680485i \(-0.761769\pi\)
−0.732762 + 0.680485i \(0.761769\pi\)
\(440\) 0 0
\(441\) −627.000 −0.0677033
\(442\) 0 0
\(443\) −14508.0 −1.55597 −0.777986 0.628281i \(-0.783759\pi\)
−0.777986 + 0.628281i \(0.783759\pi\)
\(444\) 0 0
\(445\) 29268.0 3.11783
\(446\) 0 0
\(447\) 3000.00 0.317439
\(448\) 0 0
\(449\) −7566.00 −0.795237 −0.397619 0.917551i \(-0.630163\pi\)
−0.397619 + 0.917551i \(0.630163\pi\)
\(450\) 0 0
\(451\) −18720.0 −1.95452
\(452\) 0 0
\(453\) −6992.00 −0.725194
\(454\) 0 0
\(455\) −4680.00 −0.482202
\(456\) 0 0
\(457\) 5402.00 0.552943 0.276471 0.961022i \(-0.410835\pi\)
0.276471 + 0.961022i \(0.410835\pi\)
\(458\) 0 0
\(459\) 10032.0 1.02016
\(460\) 0 0
\(461\) 4650.00 0.469788 0.234894 0.972021i \(-0.424526\pi\)
0.234894 + 0.972021i \(0.424526\pi\)
\(462\) 0 0
\(463\) −17188.0 −1.72526 −0.862629 0.505838i \(-0.831183\pi\)
−0.862629 + 0.505838i \(0.831183\pi\)
\(464\) 0 0
\(465\) −1440.00 −0.143609
\(466\) 0 0
\(467\) 14580.0 1.44472 0.722358 0.691520i \(-0.243059\pi\)
0.722358 + 0.691520i \(0.243059\pi\)
\(468\) 0 0
\(469\) 3200.00 0.315058
\(470\) 0 0
\(471\) 2456.00 0.240269
\(472\) 0 0
\(473\) 5952.00 0.578590
\(474\) 0 0
\(475\) 3184.00 0.307562
\(476\) 0 0
\(477\) 6138.00 0.589182
\(478\) 0 0
\(479\) −2100.00 −0.200316 −0.100158 0.994972i \(-0.531935\pi\)
−0.100158 + 0.994972i \(0.531935\pi\)
\(480\) 0 0
\(481\) 3302.00 0.313011
\(482\) 0 0
\(483\) −13440.0 −1.26613
\(484\) 0 0
\(485\) −23292.0 −2.18069
\(486\) 0 0
\(487\) −12004.0 −1.11695 −0.558473 0.829522i \(-0.688613\pi\)
−0.558473 + 0.829522i \(0.688613\pi\)
\(488\) 0 0
\(489\) −3232.00 −0.298888
\(490\) 0 0
\(491\) −13236.0 −1.21656 −0.608281 0.793721i \(-0.708141\pi\)
−0.608281 + 0.793721i \(0.708141\pi\)
\(492\) 0 0
\(493\) −396.000 −0.0361764
\(494\) 0 0
\(495\) −9504.00 −0.862976
\(496\) 0 0
\(497\) −8400.00 −0.758132
\(498\) 0 0
\(499\) −18560.0 −1.66505 −0.832525 0.553988i \(-0.813105\pi\)
−0.832525 + 0.553988i \(0.813105\pi\)
\(500\) 0 0
\(501\) 8112.00 0.723388
\(502\) 0 0
\(503\) 12432.0 1.10202 0.551009 0.834499i \(-0.314243\pi\)
0.551009 + 0.834499i \(0.314243\pi\)
\(504\) 0 0
\(505\) −3996.00 −0.352118
\(506\) 0 0
\(507\) −676.000 −0.0592154
\(508\) 0 0
\(509\) 7914.00 0.689159 0.344579 0.938757i \(-0.388022\pi\)
0.344579 + 0.938757i \(0.388022\pi\)
\(510\) 0 0
\(511\) 7240.00 0.626769
\(512\) 0 0
\(513\) 2432.00 0.209309
\(514\) 0 0
\(515\) 11376.0 0.973372
\(516\) 0 0
\(517\) −22464.0 −1.91096
\(518\) 0 0
\(519\) −4776.00 −0.403937
\(520\) 0 0
\(521\) −14742.0 −1.23965 −0.619826 0.784739i \(-0.712797\pi\)
−0.619826 + 0.784739i \(0.712797\pi\)
\(522\) 0 0
\(523\) 2500.00 0.209020 0.104510 0.994524i \(-0.466673\pi\)
0.104510 + 0.994524i \(0.466673\pi\)
\(524\) 0 0
\(525\) −15920.0 −1.32344
\(526\) 0 0
\(527\) 1320.00 0.109108
\(528\) 0 0
\(529\) 16057.0 1.31972
\(530\) 0 0
\(531\) −1056.00 −0.0863023
\(532\) 0 0
\(533\) 5070.00 0.412019
\(534\) 0 0
\(535\) 17064.0 1.37896
\(536\) 0 0
\(537\) −11280.0 −0.906458
\(538\) 0 0
\(539\) 2736.00 0.218642
\(540\) 0 0
\(541\) −17894.0 −1.42204 −0.711020 0.703172i \(-0.751766\pi\)
−0.711020 + 0.703172i \(0.751766\pi\)
\(542\) 0 0
\(543\) −3016.00 −0.238359
\(544\) 0 0
\(545\) −13644.0 −1.07238
\(546\) 0 0
\(547\) −17444.0 −1.36353 −0.681766 0.731571i \(-0.738788\pi\)
−0.681766 + 0.731571i \(0.738788\pi\)
\(548\) 0 0
\(549\) −9086.00 −0.706341
\(550\) 0 0
\(551\) −96.0000 −0.00742239
\(552\) 0 0
\(553\) 15520.0 1.19345
\(554\) 0 0
\(555\) 18288.0 1.39871
\(556\) 0 0
\(557\) 1002.00 0.0762228 0.0381114 0.999273i \(-0.487866\pi\)
0.0381114 + 0.999273i \(0.487866\pi\)
\(558\) 0 0
\(559\) −1612.00 −0.121968
\(560\) 0 0
\(561\) −12672.0 −0.953676
\(562\) 0 0
\(563\) 4740.00 0.354826 0.177413 0.984136i \(-0.443227\pi\)
0.177413 + 0.984136i \(0.443227\pi\)
\(564\) 0 0
\(565\) 11556.0 0.860468
\(566\) 0 0
\(567\) −6220.00 −0.460697
\(568\) 0 0
\(569\) 8682.00 0.639663 0.319832 0.947474i \(-0.396374\pi\)
0.319832 + 0.947474i \(0.396374\pi\)
\(570\) 0 0
\(571\) −5492.00 −0.402510 −0.201255 0.979539i \(-0.564502\pi\)
−0.201255 + 0.979539i \(0.564502\pi\)
\(572\) 0 0
\(573\) 9312.00 0.678908
\(574\) 0 0
\(575\) 33432.0 2.42471
\(576\) 0 0
\(577\) −17278.0 −1.24661 −0.623304 0.781980i \(-0.714210\pi\)
−0.623304 + 0.781980i \(0.714210\pi\)
\(578\) 0 0
\(579\) −9800.00 −0.703410
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −26784.0 −1.90271
\(584\) 0 0
\(585\) 2574.00 0.181918
\(586\) 0 0
\(587\) 15240.0 1.07159 0.535794 0.844349i \(-0.320012\pi\)
0.535794 + 0.844349i \(0.320012\pi\)
\(588\) 0 0
\(589\) 320.000 0.0223860
\(590\) 0 0
\(591\) 18168.0 1.26452
\(592\) 0 0
\(593\) −9198.00 −0.636959 −0.318479 0.947930i \(-0.603172\pi\)
−0.318479 + 0.947930i \(0.603172\pi\)
\(594\) 0 0
\(595\) 23760.0 1.63708
\(596\) 0 0
\(597\) 2656.00 0.182082
\(598\) 0 0
\(599\) 7200.00 0.491125 0.245563 0.969381i \(-0.421027\pi\)
0.245563 + 0.969381i \(0.421027\pi\)
\(600\) 0 0
\(601\) −14470.0 −0.982103 −0.491051 0.871131i \(-0.663387\pi\)
−0.491051 + 0.871131i \(0.663387\pi\)
\(602\) 0 0
\(603\) −1760.00 −0.118860
\(604\) 0 0
\(605\) 17514.0 1.17693
\(606\) 0 0
\(607\) −20824.0 −1.39245 −0.696227 0.717821i \(-0.745139\pi\)
−0.696227 + 0.717821i \(0.745139\pi\)
\(608\) 0 0
\(609\) 480.000 0.0319386
\(610\) 0 0
\(611\) 6084.00 0.402835
\(612\) 0 0
\(613\) −8606.00 −0.567036 −0.283518 0.958967i \(-0.591502\pi\)
−0.283518 + 0.958967i \(0.591502\pi\)
\(614\) 0 0
\(615\) 28080.0 1.84113
\(616\) 0 0
\(617\) −9654.00 −0.629912 −0.314956 0.949106i \(-0.601990\pi\)
−0.314956 + 0.949106i \(0.601990\pi\)
\(618\) 0 0
\(619\) −14384.0 −0.933993 −0.466997 0.884259i \(-0.654664\pi\)
−0.466997 + 0.884259i \(0.654664\pi\)
\(620\) 0 0
\(621\) 25536.0 1.65012
\(622\) 0 0
\(623\) 32520.0 2.09131
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 0 0
\(627\) −3072.00 −0.195668
\(628\) 0 0
\(629\) −16764.0 −1.06268
\(630\) 0 0
\(631\) 1460.00 0.0921104 0.0460552 0.998939i \(-0.485335\pi\)
0.0460552 + 0.998939i \(0.485335\pi\)
\(632\) 0 0
\(633\) −16624.0 −1.04383
\(634\) 0 0
\(635\) −15840.0 −0.989907
\(636\) 0 0
\(637\) −741.000 −0.0460902
\(638\) 0 0
\(639\) 4620.00 0.286016
\(640\) 0 0
\(641\) −12462.0 −0.767893 −0.383946 0.923355i \(-0.625435\pi\)
−0.383946 + 0.923355i \(0.625435\pi\)
\(642\) 0 0
\(643\) 9952.00 0.610371 0.305186 0.952293i \(-0.401282\pi\)
0.305186 + 0.952293i \(0.401282\pi\)
\(644\) 0 0
\(645\) −8928.00 −0.545023
\(646\) 0 0
\(647\) 26088.0 1.58520 0.792601 0.609741i \(-0.208727\pi\)
0.792601 + 0.609741i \(0.208727\pi\)
\(648\) 0 0
\(649\) 4608.00 0.278705
\(650\) 0 0
\(651\) −1600.00 −0.0963271
\(652\) 0 0
\(653\) −3894.00 −0.233360 −0.116680 0.993170i \(-0.537225\pi\)
−0.116680 + 0.993170i \(0.537225\pi\)
\(654\) 0 0
\(655\) −5832.00 −0.347901
\(656\) 0 0
\(657\) −3982.00 −0.236458
\(658\) 0 0
\(659\) 23820.0 1.40804 0.704018 0.710182i \(-0.251388\pi\)
0.704018 + 0.710182i \(0.251388\pi\)
\(660\) 0 0
\(661\) −7742.00 −0.455566 −0.227783 0.973712i \(-0.573148\pi\)
−0.227783 + 0.973712i \(0.573148\pi\)
\(662\) 0 0
\(663\) 3432.00 0.201037
\(664\) 0 0
\(665\) 5760.00 0.335885
\(666\) 0 0
\(667\) −1008.00 −0.0585156
\(668\) 0 0
\(669\) 13168.0 0.760993
\(670\) 0 0
\(671\) 39648.0 2.28106
\(672\) 0 0
\(673\) 21170.0 1.21255 0.606273 0.795257i \(-0.292664\pi\)
0.606273 + 0.795257i \(0.292664\pi\)
\(674\) 0 0
\(675\) 30248.0 1.72481
\(676\) 0 0
\(677\) −17982.0 −1.02083 −0.510417 0.859927i \(-0.670509\pi\)
−0.510417 + 0.859927i \(0.670509\pi\)
\(678\) 0 0
\(679\) −25880.0 −1.46271
\(680\) 0 0
\(681\) 9408.00 0.529391
\(682\) 0 0
\(683\) 17520.0 0.981529 0.490764 0.871292i \(-0.336718\pi\)
0.490764 + 0.871292i \(0.336718\pi\)
\(684\) 0 0
\(685\) 30996.0 1.72890
\(686\) 0 0
\(687\) 2744.00 0.152387
\(688\) 0 0
\(689\) 7254.00 0.401096
\(690\) 0 0
\(691\) 28096.0 1.54678 0.773388 0.633933i \(-0.218560\pi\)
0.773388 + 0.633933i \(0.218560\pi\)
\(692\) 0 0
\(693\) −10560.0 −0.578847
\(694\) 0 0
\(695\) 6120.00 0.334021
\(696\) 0 0
\(697\) −25740.0 −1.39881
\(698\) 0 0
\(699\) −7272.00 −0.393494
\(700\) 0 0
\(701\) −18342.0 −0.988256 −0.494128 0.869389i \(-0.664513\pi\)
−0.494128 + 0.869389i \(0.664513\pi\)
\(702\) 0 0
\(703\) −4064.00 −0.218032
\(704\) 0 0
\(705\) 33696.0 1.80009
\(706\) 0 0
\(707\) −4440.00 −0.236186
\(708\) 0 0
\(709\) 37330.0 1.97737 0.988687 0.149996i \(-0.0479261\pi\)
0.988687 + 0.149996i \(0.0479261\pi\)
\(710\) 0 0
\(711\) −8536.00 −0.450246
\(712\) 0 0
\(713\) 3360.00 0.176484
\(714\) 0 0
\(715\) −11232.0 −0.587487
\(716\) 0 0
\(717\) 2160.00 0.112506
\(718\) 0 0
\(719\) 4800.00 0.248971 0.124485 0.992221i \(-0.460272\pi\)
0.124485 + 0.992221i \(0.460272\pi\)
\(720\) 0 0
\(721\) 12640.0 0.652896
\(722\) 0 0
\(723\) 3448.00 0.177362
\(724\) 0 0
\(725\) −1194.00 −0.0611642
\(726\) 0 0
\(727\) 23960.0 1.22232 0.611160 0.791507i \(-0.290703\pi\)
0.611160 + 0.791507i \(0.290703\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 0 0
\(731\) 8184.00 0.414085
\(732\) 0 0
\(733\) 21418.0 1.07925 0.539626 0.841905i \(-0.318566\pi\)
0.539626 + 0.841905i \(0.318566\pi\)
\(734\) 0 0
\(735\) −4104.00 −0.205957
\(736\) 0 0
\(737\) 7680.00 0.383849
\(738\) 0 0
\(739\) −5384.00 −0.268002 −0.134001 0.990981i \(-0.542783\pi\)
−0.134001 + 0.990981i \(0.542783\pi\)
\(740\) 0 0
\(741\) 832.000 0.0412473
\(742\) 0 0
\(743\) −1524.00 −0.0752492 −0.0376246 0.999292i \(-0.511979\pi\)
−0.0376246 + 0.999292i \(0.511979\pi\)
\(744\) 0 0
\(745\) −13500.0 −0.663895
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18960.0 0.924944
\(750\) 0 0
\(751\) −19312.0 −0.938355 −0.469178 0.883104i \(-0.655450\pi\)
−0.469178 + 0.883104i \(0.655450\pi\)
\(752\) 0 0
\(753\) 19344.0 0.936168
\(754\) 0 0
\(755\) 31464.0 1.51668
\(756\) 0 0
\(757\) −35246.0 −1.69226 −0.846128 0.532980i \(-0.821072\pi\)
−0.846128 + 0.532980i \(0.821072\pi\)
\(758\) 0 0
\(759\) −32256.0 −1.54258
\(760\) 0 0
\(761\) 12522.0 0.596481 0.298241 0.954491i \(-0.403600\pi\)
0.298241 + 0.954491i \(0.403600\pi\)
\(762\) 0 0
\(763\) −15160.0 −0.719304
\(764\) 0 0
\(765\) −13068.0 −0.617614
\(766\) 0 0
\(767\) −1248.00 −0.0587518
\(768\) 0 0
\(769\) 24050.0 1.12778 0.563892 0.825849i \(-0.309304\pi\)
0.563892 + 0.825849i \(0.309304\pi\)
\(770\) 0 0
\(771\) −5640.00 −0.263450
\(772\) 0 0
\(773\) −25806.0 −1.20075 −0.600373 0.799720i \(-0.704981\pi\)
−0.600373 + 0.799720i \(0.704981\pi\)
\(774\) 0 0
\(775\) 3980.00 0.184472
\(776\) 0 0
\(777\) 20320.0 0.938193
\(778\) 0 0
\(779\) −6240.00 −0.286998
\(780\) 0 0
\(781\) −20160.0 −0.923664
\(782\) 0 0
\(783\) −912.000 −0.0416248
\(784\) 0 0
\(785\) −11052.0 −0.502500
\(786\) 0 0
\(787\) −18632.0 −0.843912 −0.421956 0.906616i \(-0.638656\pi\)
−0.421956 + 0.906616i \(0.638656\pi\)
\(788\) 0 0
\(789\) −33216.0 −1.49876
\(790\) 0 0
\(791\) 12840.0 0.577165
\(792\) 0 0
\(793\) −10738.0 −0.480854
\(794\) 0 0
\(795\) 40176.0 1.79232
\(796\) 0 0
\(797\) 16314.0 0.725058 0.362529 0.931972i \(-0.381913\pi\)
0.362529 + 0.931972i \(0.381913\pi\)
\(798\) 0 0
\(799\) −30888.0 −1.36763
\(800\) 0 0
\(801\) −17886.0 −0.788977
\(802\) 0 0
\(803\) 17376.0 0.763619
\(804\) 0 0
\(805\) 60480.0 2.64800
\(806\) 0 0
\(807\) −10536.0 −0.459585
\(808\) 0 0
\(809\) −4278.00 −0.185917 −0.0929583 0.995670i \(-0.529632\pi\)
−0.0929583 + 0.995670i \(0.529632\pi\)
\(810\) 0 0
\(811\) −18632.0 −0.806730 −0.403365 0.915039i \(-0.632159\pi\)
−0.403365 + 0.915039i \(0.632159\pi\)
\(812\) 0 0
\(813\) −29744.0 −1.28311
\(814\) 0 0
\(815\) 14544.0 0.625097
\(816\) 0 0
\(817\) 1984.00 0.0849588
\(818\) 0 0
\(819\) 2860.00 0.122023
\(820\) 0 0
\(821\) 46434.0 1.97388 0.986941 0.161080i \(-0.0514976\pi\)
0.986941 + 0.161080i \(0.0514976\pi\)
\(822\) 0 0
\(823\) 24968.0 1.05751 0.528754 0.848775i \(-0.322659\pi\)
0.528754 + 0.848775i \(0.322659\pi\)
\(824\) 0 0
\(825\) −38208.0 −1.61240
\(826\) 0 0
\(827\) 14112.0 0.593376 0.296688 0.954974i \(-0.404118\pi\)
0.296688 + 0.954974i \(0.404118\pi\)
\(828\) 0 0
\(829\) −37190.0 −1.55810 −0.779048 0.626964i \(-0.784297\pi\)
−0.779048 + 0.626964i \(0.784297\pi\)
\(830\) 0 0
\(831\) −20296.0 −0.847245
\(832\) 0 0
\(833\) 3762.00 0.156477
\(834\) 0 0
\(835\) −36504.0 −1.51290
\(836\) 0 0
\(837\) 3040.00 0.125541
\(838\) 0 0
\(839\) 1380.00 0.0567853 0.0283927 0.999597i \(-0.490961\pi\)
0.0283927 + 0.999597i \(0.490961\pi\)
\(840\) 0 0
\(841\) −24353.0 −0.998524
\(842\) 0 0
\(843\) 6552.00 0.267690
\(844\) 0 0
\(845\) 3042.00 0.123844
\(846\) 0 0
\(847\) 19460.0 0.789437
\(848\) 0 0
\(849\) −18352.0 −0.741860
\(850\) 0 0
\(851\) −42672.0 −1.71889
\(852\) 0 0
\(853\) −5150.00 −0.206721 −0.103360 0.994644i \(-0.532959\pi\)
−0.103360 + 0.994644i \(0.532959\pi\)
\(854\) 0 0
\(855\) −3168.00 −0.126717
\(856\) 0 0
\(857\) 23562.0 0.939163 0.469581 0.882889i \(-0.344405\pi\)
0.469581 + 0.882889i \(0.344405\pi\)
\(858\) 0 0
\(859\) 34612.0 1.37479 0.687396 0.726283i \(-0.258754\pi\)
0.687396 + 0.726283i \(0.258754\pi\)
\(860\) 0 0
\(861\) 31200.0 1.23495
\(862\) 0 0
\(863\) −14940.0 −0.589297 −0.294649 0.955606i \(-0.595203\pi\)
−0.294649 + 0.955606i \(0.595203\pi\)
\(864\) 0 0
\(865\) 21492.0 0.844798
\(866\) 0 0
\(867\) 2228.00 0.0872743
\(868\) 0 0
\(869\) 37248.0 1.45403
\(870\) 0 0
\(871\) −2080.00 −0.0809163
\(872\) 0 0
\(873\) 14234.0 0.551830
\(874\) 0 0
\(875\) 26640.0 1.02925
\(876\) 0 0
\(877\) −17030.0 −0.655715 −0.327858 0.944727i \(-0.606327\pi\)
−0.327858 + 0.944727i \(0.606327\pi\)
\(878\) 0 0
\(879\) −11400.0 −0.437443
\(880\) 0 0
\(881\) −27246.0 −1.04193 −0.520965 0.853578i \(-0.674428\pi\)
−0.520965 + 0.853578i \(0.674428\pi\)
\(882\) 0 0
\(883\) 8260.00 0.314803 0.157402 0.987535i \(-0.449688\pi\)
0.157402 + 0.987535i \(0.449688\pi\)
\(884\) 0 0
\(885\) −6912.00 −0.262536
\(886\) 0 0
\(887\) −43392.0 −1.64257 −0.821286 0.570517i \(-0.806743\pi\)
−0.821286 + 0.570517i \(0.806743\pi\)
\(888\) 0 0
\(889\) −17600.0 −0.663988
\(890\) 0 0
\(891\) −14928.0 −0.561287
\(892\) 0 0
\(893\) −7488.00 −0.280601
\(894\) 0 0
\(895\) 50760.0 1.89578
\(896\) 0 0
\(897\) 8736.00 0.325180
\(898\) 0 0
\(899\) −120.000 −0.00445186
\(900\) 0 0
\(901\) −36828.0 −1.36173
\(902\) 0 0
\(903\) −9920.00 −0.365578
\(904\) 0 0
\(905\) 13572.0 0.498507
\(906\) 0 0
\(907\) −1028.00 −0.0376342 −0.0188171 0.999823i \(-0.505990\pi\)
−0.0188171 + 0.999823i \(0.505990\pi\)
\(908\) 0 0
\(909\) 2442.00 0.0891045
\(910\) 0 0
\(911\) −21816.0 −0.793410 −0.396705 0.917946i \(-0.629846\pi\)
−0.396705 + 0.917946i \(0.629846\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −59472.0 −2.14873
\(916\) 0 0
\(917\) −6480.00 −0.233357
\(918\) 0 0
\(919\) 42752.0 1.53456 0.767279 0.641314i \(-0.221610\pi\)
0.767279 + 0.641314i \(0.221610\pi\)
\(920\) 0 0
\(921\) 32480.0 1.16205
\(922\) 0 0
\(923\) 5460.00 0.194711
\(924\) 0 0
\(925\) −50546.0 −1.79669
\(926\) 0 0
\(927\) −6952.00 −0.246315
\(928\) 0 0
\(929\) 24978.0 0.882133 0.441067 0.897474i \(-0.354600\pi\)
0.441067 + 0.897474i \(0.354600\pi\)
\(930\) 0 0
\(931\) 912.000 0.0321048
\(932\) 0 0
\(933\) 14112.0 0.495183
\(934\) 0 0
\(935\) 57024.0 1.99453
\(936\) 0 0
\(937\) 33914.0 1.18241 0.591207 0.806520i \(-0.298652\pi\)
0.591207 + 0.806520i \(0.298652\pi\)
\(938\) 0 0
\(939\) 27928.0 0.970603
\(940\) 0 0
\(941\) 8442.00 0.292456 0.146228 0.989251i \(-0.453287\pi\)
0.146228 + 0.989251i \(0.453287\pi\)
\(942\) 0 0
\(943\) −65520.0 −2.26259
\(944\) 0 0
\(945\) 54720.0 1.88364
\(946\) 0 0
\(947\) 43176.0 1.48155 0.740777 0.671751i \(-0.234458\pi\)
0.740777 + 0.671751i \(0.234458\pi\)
\(948\) 0 0
\(949\) −4706.00 −0.160973
\(950\) 0 0
\(951\) 37080.0 1.26435
\(952\) 0 0
\(953\) −43926.0 −1.49308 −0.746539 0.665342i \(-0.768286\pi\)
−0.746539 + 0.665342i \(0.768286\pi\)
\(954\) 0 0
\(955\) −41904.0 −1.41988
\(956\) 0 0
\(957\) 1152.00 0.0389121
\(958\) 0 0
\(959\) 34440.0 1.15967
\(960\) 0 0
\(961\) −29391.0 −0.986573
\(962\) 0 0
\(963\) −10428.0 −0.348949
\(964\) 0 0
\(965\) 44100.0 1.47112
\(966\) 0 0
\(967\) −11572.0 −0.384830 −0.192415 0.981314i \(-0.561632\pi\)
−0.192415 + 0.981314i \(0.561632\pi\)
\(968\) 0 0
\(969\) −4224.00 −0.140036
\(970\) 0 0
\(971\) −14412.0 −0.476316 −0.238158 0.971226i \(-0.576544\pi\)
−0.238158 + 0.971226i \(0.576544\pi\)
\(972\) 0 0
\(973\) 6800.00 0.224047
\(974\) 0 0
\(975\) 10348.0 0.339899
\(976\) 0 0
\(977\) 25602.0 0.838363 0.419181 0.907902i \(-0.362317\pi\)
0.419181 + 0.907902i \(0.362317\pi\)
\(978\) 0 0
\(979\) 78048.0 2.54793
\(980\) 0 0
\(981\) 8338.00 0.271368
\(982\) 0 0
\(983\) 32148.0 1.04309 0.521547 0.853222i \(-0.325355\pi\)
0.521547 + 0.853222i \(0.325355\pi\)
\(984\) 0 0
\(985\) −81756.0 −2.64463
\(986\) 0 0
\(987\) 37440.0 1.20742
\(988\) 0 0
\(989\) 20832.0 0.669787
\(990\) 0 0
\(991\) −9736.00 −0.312083 −0.156041 0.987751i \(-0.549873\pi\)
−0.156041 + 0.987751i \(0.549873\pi\)
\(992\) 0 0
\(993\) −30880.0 −0.986855
\(994\) 0 0
\(995\) −11952.0 −0.380808
\(996\) 0 0
\(997\) −6878.00 −0.218484 −0.109242 0.994015i \(-0.534842\pi\)
−0.109242 + 0.994015i \(0.534842\pi\)
\(998\) 0 0
\(999\) −38608.0 −1.22273
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 832.4.a.d.1.1 1
4.3 odd 2 832.4.a.o.1.1 1
8.3 odd 2 208.4.a.b.1.1 1
8.5 even 2 26.4.a.c.1.1 1
24.5 odd 2 234.4.a.e.1.1 1
24.11 even 2 1872.4.a.q.1.1 1
40.13 odd 4 650.4.b.f.599.1 2
40.29 even 2 650.4.a.b.1.1 1
40.37 odd 4 650.4.b.f.599.2 2
56.13 odd 2 1274.4.a.d.1.1 1
104.5 odd 4 338.4.b.d.337.1 2
104.21 odd 4 338.4.b.d.337.2 2
104.29 even 6 338.4.c.a.191.1 2
104.37 odd 12 338.4.e.a.147.1 4
104.45 odd 12 338.4.e.a.23.1 4
104.61 even 6 338.4.c.a.315.1 2
104.69 even 6 338.4.c.e.315.1 2
104.77 even 2 338.4.a.c.1.1 1
104.85 odd 12 338.4.e.a.23.2 4
104.93 odd 12 338.4.e.a.147.2 4
104.101 even 6 338.4.c.e.191.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.a.c.1.1 1 8.5 even 2
208.4.a.b.1.1 1 8.3 odd 2
234.4.a.e.1.1 1 24.5 odd 2
338.4.a.c.1.1 1 104.77 even 2
338.4.b.d.337.1 2 104.5 odd 4
338.4.b.d.337.2 2 104.21 odd 4
338.4.c.a.191.1 2 104.29 even 6
338.4.c.a.315.1 2 104.61 even 6
338.4.c.e.191.1 2 104.101 even 6
338.4.c.e.315.1 2 104.69 even 6
338.4.e.a.23.1 4 104.45 odd 12
338.4.e.a.23.2 4 104.85 odd 12
338.4.e.a.147.1 4 104.37 odd 12
338.4.e.a.147.2 4 104.93 odd 12
650.4.a.b.1.1 1 40.29 even 2
650.4.b.f.599.1 2 40.13 odd 4
650.4.b.f.599.2 2 40.37 odd 4
832.4.a.d.1.1 1 1.1 even 1 trivial
832.4.a.o.1.1 1 4.3 odd 2
1274.4.a.d.1.1 1 56.13 odd 2
1872.4.a.q.1.1 1 24.11 even 2