Properties

Label 960.4.a.c.1.1
Level $960$
Weight $4$
Character 960.1
Self dual yes
Analytic conductor $56.642$
Analytic rank $1$
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] = [960,4,Mod(1,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.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: \(56.6418336055\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} +24.0000 q^{11} +14.0000 q^{13} +15.0000 q^{15} -18.0000 q^{17} +36.0000 q^{19} +48.0000 q^{21} -104.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +250.000 q^{29} +28.0000 q^{31} -72.0000 q^{33} +80.0000 q^{35} +54.0000 q^{37} -42.0000 q^{39} +354.000 q^{41} +228.000 q^{43} -45.0000 q^{45} -408.000 q^{47} -87.0000 q^{49} +54.0000 q^{51} -262.000 q^{53} -120.000 q^{55} -108.000 q^{57} -64.0000 q^{59} -374.000 q^{61} -144.000 q^{63} -70.0000 q^{65} +300.000 q^{67} +312.000 q^{69} -1016.00 q^{71} +274.000 q^{73} -75.0000 q^{75} -384.000 q^{77} -788.000 q^{79} +81.0000 q^{81} -396.000 q^{83} +90.0000 q^{85} -750.000 q^{87} +786.000 q^{89} -224.000 q^{91} -84.0000 q^{93} -180.000 q^{95} -1086.00 q^{97} +216.000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −16.0000 −0.863919 −0.431959 0.901893i \(-0.642178\pi\)
−0.431959 + 0.901893i \(0.642178\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 24.0000 0.657843 0.328921 0.944357i \(-0.393315\pi\)
0.328921 + 0.944357i \(0.393315\pi\)
\(12\) 0 0
\(13\) 14.0000 0.298685 0.149342 0.988786i \(-0.452284\pi\)
0.149342 + 0.988786i \(0.452284\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) −18.0000 −0.256802 −0.128401 0.991722i \(-0.540985\pi\)
−0.128401 + 0.991722i \(0.540985\pi\)
\(18\) 0 0
\(19\) 36.0000 0.434682 0.217341 0.976096i \(-0.430262\pi\)
0.217341 + 0.976096i \(0.430262\pi\)
\(20\) 0 0
\(21\) 48.0000 0.498784
\(22\) 0 0
\(23\) −104.000 −0.942848 −0.471424 0.881907i \(-0.656260\pi\)
−0.471424 + 0.881907i \(0.656260\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 250.000 1.60082 0.800411 0.599452i \(-0.204615\pi\)
0.800411 + 0.599452i \(0.204615\pi\)
\(30\) 0 0
\(31\) 28.0000 0.162224 0.0811121 0.996705i \(-0.474153\pi\)
0.0811121 + 0.996705i \(0.474153\pi\)
\(32\) 0 0
\(33\) −72.0000 −0.379806
\(34\) 0 0
\(35\) 80.0000 0.386356
\(36\) 0 0
\(37\) 54.0000 0.239934 0.119967 0.992778i \(-0.461721\pi\)
0.119967 + 0.992778i \(0.461721\pi\)
\(38\) 0 0
\(39\) −42.0000 −0.172446
\(40\) 0 0
\(41\) 354.000 1.34843 0.674214 0.738536i \(-0.264483\pi\)
0.674214 + 0.738536i \(0.264483\pi\)
\(42\) 0 0
\(43\) 228.000 0.808597 0.404299 0.914627i \(-0.367516\pi\)
0.404299 + 0.914627i \(0.367516\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −408.000 −1.26623 −0.633116 0.774057i \(-0.718224\pi\)
−0.633116 + 0.774057i \(0.718224\pi\)
\(48\) 0 0
\(49\) −87.0000 −0.253644
\(50\) 0 0
\(51\) 54.0000 0.148265
\(52\) 0 0
\(53\) −262.000 −0.679028 −0.339514 0.940601i \(-0.610263\pi\)
−0.339514 + 0.940601i \(0.610263\pi\)
\(54\) 0 0
\(55\) −120.000 −0.294196
\(56\) 0 0
\(57\) −108.000 −0.250964
\(58\) 0 0
\(59\) −64.0000 −0.141222 −0.0706109 0.997504i \(-0.522495\pi\)
−0.0706109 + 0.997504i \(0.522495\pi\)
\(60\) 0 0
\(61\) −374.000 −0.785013 −0.392507 0.919749i \(-0.628392\pi\)
−0.392507 + 0.919749i \(0.628392\pi\)
\(62\) 0 0
\(63\) −144.000 −0.287973
\(64\) 0 0
\(65\) −70.0000 −0.133576
\(66\) 0 0
\(67\) 300.000 0.547027 0.273514 0.961868i \(-0.411814\pi\)
0.273514 + 0.961868i \(0.411814\pi\)
\(68\) 0 0
\(69\) 312.000 0.544353
\(70\) 0 0
\(71\) −1016.00 −1.69827 −0.849134 0.528178i \(-0.822876\pi\)
−0.849134 + 0.528178i \(0.822876\pi\)
\(72\) 0 0
\(73\) 274.000 0.439305 0.219653 0.975578i \(-0.429508\pi\)
0.219653 + 0.975578i \(0.429508\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) −384.000 −0.568323
\(78\) 0 0
\(79\) −788.000 −1.12224 −0.561120 0.827735i \(-0.689629\pi\)
−0.561120 + 0.827735i \(0.689629\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −396.000 −0.523695 −0.261847 0.965109i \(-0.584332\pi\)
−0.261847 + 0.965109i \(0.584332\pi\)
\(84\) 0 0
\(85\) 90.0000 0.114846
\(86\) 0 0
\(87\) −750.000 −0.924235
\(88\) 0 0
\(89\) 786.000 0.936133 0.468066 0.883693i \(-0.344951\pi\)
0.468066 + 0.883693i \(0.344951\pi\)
\(90\) 0 0
\(91\) −224.000 −0.258039
\(92\) 0 0
\(93\) −84.0000 −0.0936602
\(94\) 0 0
\(95\) −180.000 −0.194396
\(96\) 0 0
\(97\) −1086.00 −1.13677 −0.568385 0.822763i \(-0.692431\pi\)
−0.568385 + 0.822763i \(0.692431\pi\)
\(98\) 0 0
\(99\) 216.000 0.219281
\(100\) 0 0
\(101\) −78.0000 −0.0768445 −0.0384222 0.999262i \(-0.512233\pi\)
−0.0384222 + 0.999262i \(0.512233\pi\)
\(102\) 0 0
\(103\) −1208.00 −1.15561 −0.577805 0.816175i \(-0.696090\pi\)
−0.577805 + 0.816175i \(0.696090\pi\)
\(104\) 0 0
\(105\) −240.000 −0.223063
\(106\) 0 0
\(107\) −44.0000 −0.0397537 −0.0198768 0.999802i \(-0.506327\pi\)
−0.0198768 + 0.999802i \(0.506327\pi\)
\(108\) 0 0
\(109\) 1122.00 0.985946 0.492973 0.870045i \(-0.335910\pi\)
0.492973 + 0.870045i \(0.335910\pi\)
\(110\) 0 0
\(111\) −162.000 −0.138526
\(112\) 0 0
\(113\) 606.000 0.504493 0.252246 0.967663i \(-0.418831\pi\)
0.252246 + 0.967663i \(0.418831\pi\)
\(114\) 0 0
\(115\) 520.000 0.421654
\(116\) 0 0
\(117\) 126.000 0.0995616
\(118\) 0 0
\(119\) 288.000 0.221856
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) 0 0
\(123\) −1062.00 −0.778515
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1744.00 −1.21854 −0.609272 0.792962i \(-0.708538\pi\)
−0.609272 + 0.792962i \(0.708538\pi\)
\(128\) 0 0
\(129\) −684.000 −0.466844
\(130\) 0 0
\(131\) −480.000 −0.320136 −0.160068 0.987106i \(-0.551171\pi\)
−0.160068 + 0.987106i \(0.551171\pi\)
\(132\) 0 0
\(133\) −576.000 −0.375530
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) 1598.00 0.996543 0.498271 0.867021i \(-0.333968\pi\)
0.498271 + 0.867021i \(0.333968\pi\)
\(138\) 0 0
\(139\) −2964.00 −1.80866 −0.904328 0.426838i \(-0.859627\pi\)
−0.904328 + 0.426838i \(0.859627\pi\)
\(140\) 0 0
\(141\) 1224.00 0.731060
\(142\) 0 0
\(143\) 336.000 0.196488
\(144\) 0 0
\(145\) −1250.00 −0.715909
\(146\) 0 0
\(147\) 261.000 0.146442
\(148\) 0 0
\(149\) −334.000 −0.183640 −0.0918200 0.995776i \(-0.529268\pi\)
−0.0918200 + 0.995776i \(0.529268\pi\)
\(150\) 0 0
\(151\) 1148.00 0.618695 0.309347 0.950949i \(-0.399889\pi\)
0.309347 + 0.950949i \(0.399889\pi\)
\(152\) 0 0
\(153\) −162.000 −0.0856008
\(154\) 0 0
\(155\) −140.000 −0.0725488
\(156\) 0 0
\(157\) −906.000 −0.460552 −0.230276 0.973125i \(-0.573963\pi\)
−0.230276 + 0.973125i \(0.573963\pi\)
\(158\) 0 0
\(159\) 786.000 0.392037
\(160\) 0 0
\(161\) 1664.00 0.814544
\(162\) 0 0
\(163\) −1916.00 −0.920691 −0.460346 0.887740i \(-0.652275\pi\)
−0.460346 + 0.887740i \(0.652275\pi\)
\(164\) 0 0
\(165\) 360.000 0.169854
\(166\) 0 0
\(167\) −1152.00 −0.533799 −0.266900 0.963724i \(-0.585999\pi\)
−0.266900 + 0.963724i \(0.585999\pi\)
\(168\) 0 0
\(169\) −2001.00 −0.910787
\(170\) 0 0
\(171\) 324.000 0.144894
\(172\) 0 0
\(173\) −3142.00 −1.38082 −0.690410 0.723418i \(-0.742570\pi\)
−0.690410 + 0.723418i \(0.742570\pi\)
\(174\) 0 0
\(175\) −400.000 −0.172784
\(176\) 0 0
\(177\) 192.000 0.0815345
\(178\) 0 0
\(179\) −1032.00 −0.430923 −0.215462 0.976512i \(-0.569126\pi\)
−0.215462 + 0.976512i \(0.569126\pi\)
\(180\) 0 0
\(181\) 1562.00 0.641451 0.320725 0.947172i \(-0.396073\pi\)
0.320725 + 0.947172i \(0.396073\pi\)
\(182\) 0 0
\(183\) 1122.00 0.453227
\(184\) 0 0
\(185\) −270.000 −0.107302
\(186\) 0 0
\(187\) −432.000 −0.168936
\(188\) 0 0
\(189\) 432.000 0.166261
\(190\) 0 0
\(191\) −1960.00 −0.742516 −0.371258 0.928530i \(-0.621074\pi\)
−0.371258 + 0.928530i \(0.621074\pi\)
\(192\) 0 0
\(193\) −4006.00 −1.49408 −0.747042 0.664777i \(-0.768527\pi\)
−0.747042 + 0.664777i \(0.768527\pi\)
\(194\) 0 0
\(195\) 210.000 0.0771201
\(196\) 0 0
\(197\) −2118.00 −0.765996 −0.382998 0.923749i \(-0.625108\pi\)
−0.382998 + 0.923749i \(0.625108\pi\)
\(198\) 0 0
\(199\) 3748.00 1.33512 0.667559 0.744556i \(-0.267339\pi\)
0.667559 + 0.744556i \(0.267339\pi\)
\(200\) 0 0
\(201\) −900.000 −0.315826
\(202\) 0 0
\(203\) −4000.00 −1.38298
\(204\) 0 0
\(205\) −1770.00 −0.603035
\(206\) 0 0
\(207\) −936.000 −0.314283
\(208\) 0 0
\(209\) 864.000 0.285953
\(210\) 0 0
\(211\) −4796.00 −1.56479 −0.782394 0.622784i \(-0.786002\pi\)
−0.782394 + 0.622784i \(0.786002\pi\)
\(212\) 0 0
\(213\) 3048.00 0.980495
\(214\) 0 0
\(215\) −1140.00 −0.361616
\(216\) 0 0
\(217\) −448.000 −0.140148
\(218\) 0 0
\(219\) −822.000 −0.253633
\(220\) 0 0
\(221\) −252.000 −0.0767030
\(222\) 0 0
\(223\) −2560.00 −0.768746 −0.384373 0.923178i \(-0.625582\pi\)
−0.384373 + 0.923178i \(0.625582\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 3500.00 1.02336 0.511681 0.859176i \(-0.329023\pi\)
0.511681 + 0.859176i \(0.329023\pi\)
\(228\) 0 0
\(229\) −1966.00 −0.567323 −0.283661 0.958924i \(-0.591549\pi\)
−0.283661 + 0.958924i \(0.591549\pi\)
\(230\) 0 0
\(231\) 1152.00 0.328121
\(232\) 0 0
\(233\) 3246.00 0.912672 0.456336 0.889808i \(-0.349162\pi\)
0.456336 + 0.889808i \(0.349162\pi\)
\(234\) 0 0
\(235\) 2040.00 0.566276
\(236\) 0 0
\(237\) 2364.00 0.647925
\(238\) 0 0
\(239\) 7320.00 1.98114 0.990568 0.137023i \(-0.0437534\pi\)
0.990568 + 0.137023i \(0.0437534\pi\)
\(240\) 0 0
\(241\) 3490.00 0.932824 0.466412 0.884568i \(-0.345546\pi\)
0.466412 + 0.884568i \(0.345546\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 435.000 0.113433
\(246\) 0 0
\(247\) 504.000 0.129833
\(248\) 0 0
\(249\) 1188.00 0.302355
\(250\) 0 0
\(251\) −7456.00 −1.87497 −0.937487 0.348020i \(-0.886854\pi\)
−0.937487 + 0.348020i \(0.886854\pi\)
\(252\) 0 0
\(253\) −2496.00 −0.620246
\(254\) 0 0
\(255\) −270.000 −0.0663061
\(256\) 0 0
\(257\) 4558.00 1.10630 0.553152 0.833080i \(-0.313425\pi\)
0.553152 + 0.833080i \(0.313425\pi\)
\(258\) 0 0
\(259\) −864.000 −0.207283
\(260\) 0 0
\(261\) 2250.00 0.533607
\(262\) 0 0
\(263\) 2848.00 0.667738 0.333869 0.942619i \(-0.391646\pi\)
0.333869 + 0.942619i \(0.391646\pi\)
\(264\) 0 0
\(265\) 1310.00 0.303670
\(266\) 0 0
\(267\) −2358.00 −0.540477
\(268\) 0 0
\(269\) −3110.00 −0.704907 −0.352454 0.935829i \(-0.614653\pi\)
−0.352454 + 0.935829i \(0.614653\pi\)
\(270\) 0 0
\(271\) −1700.00 −0.381061 −0.190531 0.981681i \(-0.561021\pi\)
−0.190531 + 0.981681i \(0.561021\pi\)
\(272\) 0 0
\(273\) 672.000 0.148979
\(274\) 0 0
\(275\) 600.000 0.131569
\(276\) 0 0
\(277\) 6494.00 1.40862 0.704308 0.709895i \(-0.251257\pi\)
0.704308 + 0.709895i \(0.251257\pi\)
\(278\) 0 0
\(279\) 252.000 0.0540747
\(280\) 0 0
\(281\) 2498.00 0.530314 0.265157 0.964205i \(-0.414576\pi\)
0.265157 + 0.964205i \(0.414576\pi\)
\(282\) 0 0
\(283\) −5324.00 −1.11830 −0.559150 0.829066i \(-0.688872\pi\)
−0.559150 + 0.829066i \(0.688872\pi\)
\(284\) 0 0
\(285\) 540.000 0.112235
\(286\) 0 0
\(287\) −5664.00 −1.16493
\(288\) 0 0
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) 3258.00 0.656314
\(292\) 0 0
\(293\) 522.000 0.104080 0.0520402 0.998645i \(-0.483428\pi\)
0.0520402 + 0.998645i \(0.483428\pi\)
\(294\) 0 0
\(295\) 320.000 0.0631563
\(296\) 0 0
\(297\) −648.000 −0.126602
\(298\) 0 0
\(299\) −1456.00 −0.281614
\(300\) 0 0
\(301\) −3648.00 −0.698562
\(302\) 0 0
\(303\) 234.000 0.0443662
\(304\) 0 0
\(305\) 1870.00 0.351068
\(306\) 0 0
\(307\) 7844.00 1.45824 0.729122 0.684384i \(-0.239929\pi\)
0.729122 + 0.684384i \(0.239929\pi\)
\(308\) 0 0
\(309\) 3624.00 0.667191
\(310\) 0 0
\(311\) 3248.00 0.592210 0.296105 0.955155i \(-0.404312\pi\)
0.296105 + 0.955155i \(0.404312\pi\)
\(312\) 0 0
\(313\) −5374.00 −0.970468 −0.485234 0.874384i \(-0.661266\pi\)
−0.485234 + 0.874384i \(0.661266\pi\)
\(314\) 0 0
\(315\) 720.000 0.128785
\(316\) 0 0
\(317\) 6786.00 1.20233 0.601167 0.799124i \(-0.294703\pi\)
0.601167 + 0.799124i \(0.294703\pi\)
\(318\) 0 0
\(319\) 6000.00 1.05309
\(320\) 0 0
\(321\) 132.000 0.0229518
\(322\) 0 0
\(323\) −648.000 −0.111628
\(324\) 0 0
\(325\) 350.000 0.0597369
\(326\) 0 0
\(327\) −3366.00 −0.569236
\(328\) 0 0
\(329\) 6528.00 1.09392
\(330\) 0 0
\(331\) 6596.00 1.09531 0.547657 0.836703i \(-0.315520\pi\)
0.547657 + 0.836703i \(0.315520\pi\)
\(332\) 0 0
\(333\) 486.000 0.0799779
\(334\) 0 0
\(335\) −1500.00 −0.244638
\(336\) 0 0
\(337\) −5830.00 −0.942375 −0.471187 0.882033i \(-0.656174\pi\)
−0.471187 + 0.882033i \(0.656174\pi\)
\(338\) 0 0
\(339\) −1818.00 −0.291269
\(340\) 0 0
\(341\) 672.000 0.106718
\(342\) 0 0
\(343\) 6880.00 1.08305
\(344\) 0 0
\(345\) −1560.00 −0.243442
\(346\) 0 0
\(347\) 11732.0 1.81501 0.907503 0.420047i \(-0.137986\pi\)
0.907503 + 0.420047i \(0.137986\pi\)
\(348\) 0 0
\(349\) −1014.00 −0.155525 −0.0777624 0.996972i \(-0.524778\pi\)
−0.0777624 + 0.996972i \(0.524778\pi\)
\(350\) 0 0
\(351\) −378.000 −0.0574819
\(352\) 0 0
\(353\) −8202.00 −1.23668 −0.618341 0.785910i \(-0.712195\pi\)
−0.618341 + 0.785910i \(0.712195\pi\)
\(354\) 0 0
\(355\) 5080.00 0.759488
\(356\) 0 0
\(357\) −864.000 −0.128089
\(358\) 0 0
\(359\) 8160.00 1.19963 0.599817 0.800138i \(-0.295240\pi\)
0.599817 + 0.800138i \(0.295240\pi\)
\(360\) 0 0
\(361\) −5563.00 −0.811051
\(362\) 0 0
\(363\) 2265.00 0.327498
\(364\) 0 0
\(365\) −1370.00 −0.196463
\(366\) 0 0
\(367\) −12360.0 −1.75800 −0.879001 0.476820i \(-0.841789\pi\)
−0.879001 + 0.476820i \(0.841789\pi\)
\(368\) 0 0
\(369\) 3186.00 0.449476
\(370\) 0 0
\(371\) 4192.00 0.586625
\(372\) 0 0
\(373\) −930.000 −0.129098 −0.0645490 0.997915i \(-0.520561\pi\)
−0.0645490 + 0.997915i \(0.520561\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) 3500.00 0.478141
\(378\) 0 0
\(379\) 4228.00 0.573028 0.286514 0.958076i \(-0.407503\pi\)
0.286514 + 0.958076i \(0.407503\pi\)
\(380\) 0 0
\(381\) 5232.00 0.703526
\(382\) 0 0
\(383\) −8384.00 −1.11854 −0.559272 0.828984i \(-0.688919\pi\)
−0.559272 + 0.828984i \(0.688919\pi\)
\(384\) 0 0
\(385\) 1920.00 0.254162
\(386\) 0 0
\(387\) 2052.00 0.269532
\(388\) 0 0
\(389\) −5534.00 −0.721298 −0.360649 0.932702i \(-0.617445\pi\)
−0.360649 + 0.932702i \(0.617445\pi\)
\(390\) 0 0
\(391\) 1872.00 0.242126
\(392\) 0 0
\(393\) 1440.00 0.184831
\(394\) 0 0
\(395\) 3940.00 0.501881
\(396\) 0 0
\(397\) −5426.00 −0.685952 −0.342976 0.939344i \(-0.611435\pi\)
−0.342976 + 0.939344i \(0.611435\pi\)
\(398\) 0 0
\(399\) 1728.00 0.216813
\(400\) 0 0
\(401\) −78.0000 −0.00971355 −0.00485678 0.999988i \(-0.501546\pi\)
−0.00485678 + 0.999988i \(0.501546\pi\)
\(402\) 0 0
\(403\) 392.000 0.0484539
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 1296.00 0.157839
\(408\) 0 0
\(409\) −454.000 −0.0548872 −0.0274436 0.999623i \(-0.508737\pi\)
−0.0274436 + 0.999623i \(0.508737\pi\)
\(410\) 0 0
\(411\) −4794.00 −0.575354
\(412\) 0 0
\(413\) 1024.00 0.122004
\(414\) 0 0
\(415\) 1980.00 0.234203
\(416\) 0 0
\(417\) 8892.00 1.04423
\(418\) 0 0
\(419\) 12296.0 1.43365 0.716824 0.697254i \(-0.245595\pi\)
0.716824 + 0.697254i \(0.245595\pi\)
\(420\) 0 0
\(421\) −12798.0 −1.48156 −0.740780 0.671748i \(-0.765544\pi\)
−0.740780 + 0.671748i \(0.765544\pi\)
\(422\) 0 0
\(423\) −3672.00 −0.422077
\(424\) 0 0
\(425\) −450.000 −0.0513605
\(426\) 0 0
\(427\) 5984.00 0.678187
\(428\) 0 0
\(429\) −1008.00 −0.113442
\(430\) 0 0
\(431\) −9912.00 −1.10776 −0.553880 0.832597i \(-0.686853\pi\)
−0.553880 + 0.832597i \(0.686853\pi\)
\(432\) 0 0
\(433\) −6774.00 −0.751819 −0.375910 0.926656i \(-0.622670\pi\)
−0.375910 + 0.926656i \(0.622670\pi\)
\(434\) 0 0
\(435\) 3750.00 0.413330
\(436\) 0 0
\(437\) −3744.00 −0.409839
\(438\) 0 0
\(439\) 16628.0 1.80777 0.903885 0.427775i \(-0.140703\pi\)
0.903885 + 0.427775i \(0.140703\pi\)
\(440\) 0 0
\(441\) −783.000 −0.0845481
\(442\) 0 0
\(443\) −940.000 −0.100814 −0.0504072 0.998729i \(-0.516052\pi\)
−0.0504072 + 0.998729i \(0.516052\pi\)
\(444\) 0 0
\(445\) −3930.00 −0.418651
\(446\) 0 0
\(447\) 1002.00 0.106025
\(448\) 0 0
\(449\) −1662.00 −0.174687 −0.0873437 0.996178i \(-0.527838\pi\)
−0.0873437 + 0.996178i \(0.527838\pi\)
\(450\) 0 0
\(451\) 8496.00 0.887053
\(452\) 0 0
\(453\) −3444.00 −0.357204
\(454\) 0 0
\(455\) 1120.00 0.115399
\(456\) 0 0
\(457\) −13942.0 −1.42709 −0.713544 0.700610i \(-0.752911\pi\)
−0.713544 + 0.700610i \(0.752911\pi\)
\(458\) 0 0
\(459\) 486.000 0.0494217
\(460\) 0 0
\(461\) 16170.0 1.63365 0.816824 0.576887i \(-0.195733\pi\)
0.816824 + 0.576887i \(0.195733\pi\)
\(462\) 0 0
\(463\) −1048.00 −0.105194 −0.0525969 0.998616i \(-0.516750\pi\)
−0.0525969 + 0.998616i \(0.516750\pi\)
\(464\) 0 0
\(465\) 420.000 0.0418861
\(466\) 0 0
\(467\) 13716.0 1.35910 0.679551 0.733628i \(-0.262175\pi\)
0.679551 + 0.733628i \(0.262175\pi\)
\(468\) 0 0
\(469\) −4800.00 −0.472587
\(470\) 0 0
\(471\) 2718.00 0.265900
\(472\) 0 0
\(473\) 5472.00 0.531930
\(474\) 0 0
\(475\) 900.000 0.0869365
\(476\) 0 0
\(477\) −2358.00 −0.226343
\(478\) 0 0
\(479\) 8832.00 0.842473 0.421236 0.906951i \(-0.361596\pi\)
0.421236 + 0.906951i \(0.361596\pi\)
\(480\) 0 0
\(481\) 756.000 0.0716645
\(482\) 0 0
\(483\) −4992.00 −0.470277
\(484\) 0 0
\(485\) 5430.00 0.508379
\(486\) 0 0
\(487\) −10120.0 −0.941645 −0.470822 0.882228i \(-0.656043\pi\)
−0.470822 + 0.882228i \(0.656043\pi\)
\(488\) 0 0
\(489\) 5748.00 0.531561
\(490\) 0 0
\(491\) −4376.00 −0.402212 −0.201106 0.979569i \(-0.564454\pi\)
−0.201106 + 0.979569i \(0.564454\pi\)
\(492\) 0 0
\(493\) −4500.00 −0.411095
\(494\) 0 0
\(495\) −1080.00 −0.0980654
\(496\) 0 0
\(497\) 16256.0 1.46717
\(498\) 0 0
\(499\) 12364.0 1.10920 0.554598 0.832119i \(-0.312872\pi\)
0.554598 + 0.832119i \(0.312872\pi\)
\(500\) 0 0
\(501\) 3456.00 0.308189
\(502\) 0 0
\(503\) 1248.00 0.110627 0.0553137 0.998469i \(-0.482384\pi\)
0.0553137 + 0.998469i \(0.482384\pi\)
\(504\) 0 0
\(505\) 390.000 0.0343659
\(506\) 0 0
\(507\) 6003.00 0.525843
\(508\) 0 0
\(509\) 12730.0 1.10854 0.554270 0.832337i \(-0.312997\pi\)
0.554270 + 0.832337i \(0.312997\pi\)
\(510\) 0 0
\(511\) −4384.00 −0.379524
\(512\) 0 0
\(513\) −972.000 −0.0836547
\(514\) 0 0
\(515\) 6040.00 0.516804
\(516\) 0 0
\(517\) −9792.00 −0.832982
\(518\) 0 0
\(519\) 9426.00 0.797217
\(520\) 0 0
\(521\) −13286.0 −1.11722 −0.558609 0.829431i \(-0.688665\pi\)
−0.558609 + 0.829431i \(0.688665\pi\)
\(522\) 0 0
\(523\) −15892.0 −1.32870 −0.664349 0.747423i \(-0.731291\pi\)
−0.664349 + 0.747423i \(0.731291\pi\)
\(524\) 0 0
\(525\) 1200.00 0.0997567
\(526\) 0 0
\(527\) −504.000 −0.0416596
\(528\) 0 0
\(529\) −1351.00 −0.111038
\(530\) 0 0
\(531\) −576.000 −0.0470740
\(532\) 0 0
\(533\) 4956.00 0.402755
\(534\) 0 0
\(535\) 220.000 0.0177784
\(536\) 0 0
\(537\) 3096.00 0.248794
\(538\) 0 0
\(539\) −2088.00 −0.166858
\(540\) 0 0
\(541\) −9662.00 −0.767841 −0.383920 0.923366i \(-0.625426\pi\)
−0.383920 + 0.923366i \(0.625426\pi\)
\(542\) 0 0
\(543\) −4686.00 −0.370342
\(544\) 0 0
\(545\) −5610.00 −0.440928
\(546\) 0 0
\(547\) 9596.00 0.750083 0.375041 0.927008i \(-0.377628\pi\)
0.375041 + 0.927008i \(0.377628\pi\)
\(548\) 0 0
\(549\) −3366.00 −0.261671
\(550\) 0 0
\(551\) 9000.00 0.695849
\(552\) 0 0
\(553\) 12608.0 0.969524
\(554\) 0 0
\(555\) 810.000 0.0619506
\(556\) 0 0
\(557\) 4458.00 0.339123 0.169562 0.985520i \(-0.445765\pi\)
0.169562 + 0.985520i \(0.445765\pi\)
\(558\) 0 0
\(559\) 3192.00 0.241516
\(560\) 0 0
\(561\) 1296.00 0.0975350
\(562\) 0 0
\(563\) −4708.00 −0.352431 −0.176215 0.984352i \(-0.556386\pi\)
−0.176215 + 0.984352i \(0.556386\pi\)
\(564\) 0 0
\(565\) −3030.00 −0.225616
\(566\) 0 0
\(567\) −1296.00 −0.0959910
\(568\) 0 0
\(569\) −12358.0 −0.910500 −0.455250 0.890364i \(-0.650450\pi\)
−0.455250 + 0.890364i \(0.650450\pi\)
\(570\) 0 0
\(571\) −7532.00 −0.552022 −0.276011 0.961155i \(-0.589013\pi\)
−0.276011 + 0.961155i \(0.589013\pi\)
\(572\) 0 0
\(573\) 5880.00 0.428692
\(574\) 0 0
\(575\) −2600.00 −0.188570
\(576\) 0 0
\(577\) −18878.0 −1.36205 −0.681024 0.732261i \(-0.738465\pi\)
−0.681024 + 0.732261i \(0.738465\pi\)
\(578\) 0 0
\(579\) 12018.0 0.862610
\(580\) 0 0
\(581\) 6336.00 0.452430
\(582\) 0 0
\(583\) −6288.00 −0.446694
\(584\) 0 0
\(585\) −630.000 −0.0445253
\(586\) 0 0
\(587\) −22380.0 −1.57363 −0.786816 0.617188i \(-0.788272\pi\)
−0.786816 + 0.617188i \(0.788272\pi\)
\(588\) 0 0
\(589\) 1008.00 0.0705160
\(590\) 0 0
\(591\) 6354.00 0.442248
\(592\) 0 0
\(593\) 7726.00 0.535023 0.267512 0.963555i \(-0.413799\pi\)
0.267512 + 0.963555i \(0.413799\pi\)
\(594\) 0 0
\(595\) −1440.00 −0.0992172
\(596\) 0 0
\(597\) −11244.0 −0.770831
\(598\) 0 0
\(599\) 21232.0 1.44827 0.724137 0.689656i \(-0.242238\pi\)
0.724137 + 0.689656i \(0.242238\pi\)
\(600\) 0 0
\(601\) 18954.0 1.28644 0.643219 0.765682i \(-0.277598\pi\)
0.643219 + 0.765682i \(0.277598\pi\)
\(602\) 0 0
\(603\) 2700.00 0.182342
\(604\) 0 0
\(605\) 3775.00 0.253679
\(606\) 0 0
\(607\) −1896.00 −0.126781 −0.0633907 0.997989i \(-0.520191\pi\)
−0.0633907 + 0.997989i \(0.520191\pi\)
\(608\) 0 0
\(609\) 12000.0 0.798464
\(610\) 0 0
\(611\) −5712.00 −0.378204
\(612\) 0 0
\(613\) 9862.00 0.649792 0.324896 0.945750i \(-0.394671\pi\)
0.324896 + 0.945750i \(0.394671\pi\)
\(614\) 0 0
\(615\) 5310.00 0.348162
\(616\) 0 0
\(617\) −20434.0 −1.33329 −0.666647 0.745374i \(-0.732271\pi\)
−0.666647 + 0.745374i \(0.732271\pi\)
\(618\) 0 0
\(619\) −12644.0 −0.821010 −0.410505 0.911858i \(-0.634648\pi\)
−0.410505 + 0.911858i \(0.634648\pi\)
\(620\) 0 0
\(621\) 2808.00 0.181451
\(622\) 0 0
\(623\) −12576.0 −0.808743
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −2592.00 −0.165095
\(628\) 0 0
\(629\) −972.000 −0.0616155
\(630\) 0 0
\(631\) 4660.00 0.293996 0.146998 0.989137i \(-0.453039\pi\)
0.146998 + 0.989137i \(0.453039\pi\)
\(632\) 0 0
\(633\) 14388.0 0.903431
\(634\) 0 0
\(635\) 8720.00 0.544949
\(636\) 0 0
\(637\) −1218.00 −0.0757597
\(638\) 0 0
\(639\) −9144.00 −0.566089
\(640\) 0 0
\(641\) −8598.00 −0.529798 −0.264899 0.964276i \(-0.585339\pi\)
−0.264899 + 0.964276i \(0.585339\pi\)
\(642\) 0 0
\(643\) −1836.00 −0.112605 −0.0563023 0.998414i \(-0.517931\pi\)
−0.0563023 + 0.998414i \(0.517931\pi\)
\(644\) 0 0
\(645\) 3420.00 0.208779
\(646\) 0 0
\(647\) −1696.00 −0.103055 −0.0515275 0.998672i \(-0.516409\pi\)
−0.0515275 + 0.998672i \(0.516409\pi\)
\(648\) 0 0
\(649\) −1536.00 −0.0929018
\(650\) 0 0
\(651\) 1344.00 0.0809148
\(652\) 0 0
\(653\) 24730.0 1.48202 0.741010 0.671493i \(-0.234347\pi\)
0.741010 + 0.671493i \(0.234347\pi\)
\(654\) 0 0
\(655\) 2400.00 0.143169
\(656\) 0 0
\(657\) 2466.00 0.146435
\(658\) 0 0
\(659\) 4800.00 0.283735 0.141868 0.989886i \(-0.454689\pi\)
0.141868 + 0.989886i \(0.454689\pi\)
\(660\) 0 0
\(661\) −32174.0 −1.89323 −0.946614 0.322370i \(-0.895521\pi\)
−0.946614 + 0.322370i \(0.895521\pi\)
\(662\) 0 0
\(663\) 756.000 0.0442845
\(664\) 0 0
\(665\) 2880.00 0.167942
\(666\) 0 0
\(667\) −26000.0 −1.50933
\(668\) 0 0
\(669\) 7680.00 0.443836
\(670\) 0 0
\(671\) −8976.00 −0.516415
\(672\) 0 0
\(673\) 7114.00 0.407466 0.203733 0.979026i \(-0.434693\pi\)
0.203733 + 0.979026i \(0.434693\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) 20466.0 1.16185 0.580925 0.813957i \(-0.302691\pi\)
0.580925 + 0.813957i \(0.302691\pi\)
\(678\) 0 0
\(679\) 17376.0 0.982076
\(680\) 0 0
\(681\) −10500.0 −0.590838
\(682\) 0 0
\(683\) −34068.0 −1.90860 −0.954301 0.298846i \(-0.903398\pi\)
−0.954301 + 0.298846i \(0.903398\pi\)
\(684\) 0 0
\(685\) −7990.00 −0.445667
\(686\) 0 0
\(687\) 5898.00 0.327544
\(688\) 0 0
\(689\) −3668.00 −0.202815
\(690\) 0 0
\(691\) −21340.0 −1.17484 −0.587418 0.809284i \(-0.699856\pi\)
−0.587418 + 0.809284i \(0.699856\pi\)
\(692\) 0 0
\(693\) −3456.00 −0.189441
\(694\) 0 0
\(695\) 14820.0 0.808856
\(696\) 0 0
\(697\) −6372.00 −0.346279
\(698\) 0 0
\(699\) −9738.00 −0.526931
\(700\) 0 0
\(701\) 5370.00 0.289333 0.144666 0.989481i \(-0.453789\pi\)
0.144666 + 0.989481i \(0.453789\pi\)
\(702\) 0 0
\(703\) 1944.00 0.104295
\(704\) 0 0
\(705\) −6120.00 −0.326940
\(706\) 0 0
\(707\) 1248.00 0.0663874
\(708\) 0 0
\(709\) 18690.0 0.990011 0.495005 0.868890i \(-0.335166\pi\)
0.495005 + 0.868890i \(0.335166\pi\)
\(710\) 0 0
\(711\) −7092.00 −0.374080
\(712\) 0 0
\(713\) −2912.00 −0.152953
\(714\) 0 0
\(715\) −1680.00 −0.0878719
\(716\) 0 0
\(717\) −21960.0 −1.14381
\(718\) 0 0
\(719\) 14328.0 0.743177 0.371588 0.928398i \(-0.378813\pi\)
0.371588 + 0.928398i \(0.378813\pi\)
\(720\) 0 0
\(721\) 19328.0 0.998353
\(722\) 0 0
\(723\) −10470.0 −0.538566
\(724\) 0 0
\(725\) 6250.00 0.320164
\(726\) 0 0
\(727\) 14488.0 0.739106 0.369553 0.929210i \(-0.379511\pi\)
0.369553 + 0.929210i \(0.379511\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4104.00 −0.207650
\(732\) 0 0
\(733\) −25354.0 −1.27759 −0.638794 0.769378i \(-0.720566\pi\)
−0.638794 + 0.769378i \(0.720566\pi\)
\(734\) 0 0
\(735\) −1305.00 −0.0654907
\(736\) 0 0
\(737\) 7200.00 0.359858
\(738\) 0 0
\(739\) 33100.0 1.64764 0.823818 0.566854i \(-0.191840\pi\)
0.823818 + 0.566854i \(0.191840\pi\)
\(740\) 0 0
\(741\) −1512.00 −0.0749591
\(742\) 0 0
\(743\) 4456.00 0.220020 0.110010 0.993930i \(-0.464912\pi\)
0.110010 + 0.993930i \(0.464912\pi\)
\(744\) 0 0
\(745\) 1670.00 0.0821263
\(746\) 0 0
\(747\) −3564.00 −0.174565
\(748\) 0 0
\(749\) 704.000 0.0343439
\(750\) 0 0
\(751\) −23268.0 −1.13057 −0.565287 0.824894i \(-0.691235\pi\)
−0.565287 + 0.824894i \(0.691235\pi\)
\(752\) 0 0
\(753\) 22368.0 1.08252
\(754\) 0 0
\(755\) −5740.00 −0.276689
\(756\) 0 0
\(757\) 35726.0 1.71530 0.857651 0.514232i \(-0.171923\pi\)
0.857651 + 0.514232i \(0.171923\pi\)
\(758\) 0 0
\(759\) 7488.00 0.358099
\(760\) 0 0
\(761\) −12278.0 −0.584858 −0.292429 0.956287i \(-0.594464\pi\)
−0.292429 + 0.956287i \(0.594464\pi\)
\(762\) 0 0
\(763\) −17952.0 −0.851777
\(764\) 0 0
\(765\) 810.000 0.0382818
\(766\) 0 0
\(767\) −896.000 −0.0421808
\(768\) 0 0
\(769\) −26542.0 −1.24464 −0.622321 0.782763i \(-0.713810\pi\)
−0.622321 + 0.782763i \(0.713810\pi\)
\(770\) 0 0
\(771\) −13674.0 −0.638725
\(772\) 0 0
\(773\) −9942.00 −0.462599 −0.231299 0.972883i \(-0.574298\pi\)
−0.231299 + 0.972883i \(0.574298\pi\)
\(774\) 0 0
\(775\) 700.000 0.0324448
\(776\) 0 0
\(777\) 2592.00 0.119675
\(778\) 0 0
\(779\) 12744.0 0.586138
\(780\) 0 0
\(781\) −24384.0 −1.11719
\(782\) 0 0
\(783\) −6750.00 −0.308078
\(784\) 0 0
\(785\) 4530.00 0.205965
\(786\) 0 0
\(787\) −11132.0 −0.504210 −0.252105 0.967700i \(-0.581123\pi\)
−0.252105 + 0.967700i \(0.581123\pi\)
\(788\) 0 0
\(789\) −8544.00 −0.385519
\(790\) 0 0
\(791\) −9696.00 −0.435841
\(792\) 0 0
\(793\) −5236.00 −0.234471
\(794\) 0 0
\(795\) −3930.00 −0.175324
\(796\) 0 0
\(797\) −23910.0 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 7344.00 0.325172
\(800\) 0 0
\(801\) 7074.00 0.312044
\(802\) 0 0
\(803\) 6576.00 0.288994
\(804\) 0 0
\(805\) −8320.00 −0.364275
\(806\) 0 0
\(807\) 9330.00 0.406978
\(808\) 0 0
\(809\) −15934.0 −0.692472 −0.346236 0.938148i \(-0.612540\pi\)
−0.346236 + 0.938148i \(0.612540\pi\)
\(810\) 0 0
\(811\) −23756.0 −1.02859 −0.514295 0.857614i \(-0.671946\pi\)
−0.514295 + 0.857614i \(0.671946\pi\)
\(812\) 0 0
\(813\) 5100.00 0.220006
\(814\) 0 0
\(815\) 9580.00 0.411746
\(816\) 0 0
\(817\) 8208.00 0.351483
\(818\) 0 0
\(819\) −2016.00 −0.0860131
\(820\) 0 0
\(821\) 114.000 0.00484607 0.00242304 0.999997i \(-0.499229\pi\)
0.00242304 + 0.999997i \(0.499229\pi\)
\(822\) 0 0
\(823\) −43784.0 −1.85445 −0.927226 0.374502i \(-0.877814\pi\)
−0.927226 + 0.374502i \(0.877814\pi\)
\(824\) 0 0
\(825\) −1800.00 −0.0759612
\(826\) 0 0
\(827\) −17044.0 −0.716660 −0.358330 0.933595i \(-0.616654\pi\)
−0.358330 + 0.933595i \(0.616654\pi\)
\(828\) 0 0
\(829\) 21682.0 0.908380 0.454190 0.890905i \(-0.349929\pi\)
0.454190 + 0.890905i \(0.349929\pi\)
\(830\) 0 0
\(831\) −19482.0 −0.813265
\(832\) 0 0
\(833\) 1566.00 0.0651365
\(834\) 0 0
\(835\) 5760.00 0.238722
\(836\) 0 0
\(837\) −756.000 −0.0312201
\(838\) 0 0
\(839\) −39488.0 −1.62488 −0.812442 0.583042i \(-0.801862\pi\)
−0.812442 + 0.583042i \(0.801862\pi\)
\(840\) 0 0
\(841\) 38111.0 1.56263
\(842\) 0 0
\(843\) −7494.00 −0.306177
\(844\) 0 0
\(845\) 10005.0 0.407317
\(846\) 0 0
\(847\) 12080.0 0.490052
\(848\) 0 0
\(849\) 15972.0 0.645651
\(850\) 0 0
\(851\) −5616.00 −0.226221
\(852\) 0 0
\(853\) 14182.0 0.569264 0.284632 0.958637i \(-0.408129\pi\)
0.284632 + 0.958637i \(0.408129\pi\)
\(854\) 0 0
\(855\) −1620.00 −0.0647986
\(856\) 0 0
\(857\) 27094.0 1.07995 0.539973 0.841682i \(-0.318435\pi\)
0.539973 + 0.841682i \(0.318435\pi\)
\(858\) 0 0
\(859\) 26692.0 1.06021 0.530104 0.847932i \(-0.322153\pi\)
0.530104 + 0.847932i \(0.322153\pi\)
\(860\) 0 0
\(861\) 16992.0 0.672574
\(862\) 0 0
\(863\) −38872.0 −1.53328 −0.766639 0.642079i \(-0.778072\pi\)
−0.766639 + 0.642079i \(0.778072\pi\)
\(864\) 0 0
\(865\) 15710.0 0.617521
\(866\) 0 0
\(867\) 13767.0 0.539275
\(868\) 0 0
\(869\) −18912.0 −0.738257
\(870\) 0 0
\(871\) 4200.00 0.163389
\(872\) 0 0
\(873\) −9774.00 −0.378923
\(874\) 0 0
\(875\) 2000.00 0.0772712
\(876\) 0 0
\(877\) −6490.00 −0.249888 −0.124944 0.992164i \(-0.539875\pi\)
−0.124944 + 0.992164i \(0.539875\pi\)
\(878\) 0 0
\(879\) −1566.00 −0.0600909
\(880\) 0 0
\(881\) −35766.0 −1.36775 −0.683875 0.729600i \(-0.739706\pi\)
−0.683875 + 0.729600i \(0.739706\pi\)
\(882\) 0 0
\(883\) 1316.00 0.0501551 0.0250775 0.999686i \(-0.492017\pi\)
0.0250775 + 0.999686i \(0.492017\pi\)
\(884\) 0 0
\(885\) −960.000 −0.0364633
\(886\) 0 0
\(887\) −6656.00 −0.251958 −0.125979 0.992033i \(-0.540207\pi\)
−0.125979 + 0.992033i \(0.540207\pi\)
\(888\) 0 0
\(889\) 27904.0 1.05272
\(890\) 0 0
\(891\) 1944.00 0.0730937
\(892\) 0 0
\(893\) −14688.0 −0.550409
\(894\) 0 0
\(895\) 5160.00 0.192715
\(896\) 0 0
\(897\) 4368.00 0.162590
\(898\) 0 0
\(899\) 7000.00 0.259692
\(900\) 0 0
\(901\) 4716.00 0.174376
\(902\) 0 0
\(903\) 10944.0 0.403315
\(904\) 0 0
\(905\) −7810.00 −0.286865
\(906\) 0 0
\(907\) −15772.0 −0.577399 −0.288699 0.957420i \(-0.593223\pi\)
−0.288699 + 0.957420i \(0.593223\pi\)
\(908\) 0 0
\(909\) −702.000 −0.0256148
\(910\) 0 0
\(911\) −15168.0 −0.551634 −0.275817 0.961210i \(-0.588948\pi\)
−0.275817 + 0.961210i \(0.588948\pi\)
\(912\) 0 0
\(913\) −9504.00 −0.344509
\(914\) 0 0
\(915\) −5610.00 −0.202689
\(916\) 0 0
\(917\) 7680.00 0.276571
\(918\) 0 0
\(919\) −7148.00 −0.256573 −0.128287 0.991737i \(-0.540948\pi\)
−0.128287 + 0.991737i \(0.540948\pi\)
\(920\) 0 0
\(921\) −23532.0 −0.841917
\(922\) 0 0
\(923\) −14224.0 −0.507247
\(924\) 0 0
\(925\) 1350.00 0.0479867
\(926\) 0 0
\(927\) −10872.0 −0.385203
\(928\) 0 0
\(929\) −8206.00 −0.289806 −0.144903 0.989446i \(-0.546287\pi\)
−0.144903 + 0.989446i \(0.546287\pi\)
\(930\) 0 0
\(931\) −3132.00 −0.110255
\(932\) 0 0
\(933\) −9744.00 −0.341912
\(934\) 0 0
\(935\) 2160.00 0.0755503
\(936\) 0 0
\(937\) −55574.0 −1.93759 −0.968796 0.247860i \(-0.920273\pi\)
−0.968796 + 0.247860i \(0.920273\pi\)
\(938\) 0 0
\(939\) 16122.0 0.560300
\(940\) 0 0
\(941\) 3690.00 0.127833 0.0639163 0.997955i \(-0.479641\pi\)
0.0639163 + 0.997955i \(0.479641\pi\)
\(942\) 0 0
\(943\) −36816.0 −1.27136
\(944\) 0 0
\(945\) −2160.00 −0.0743543
\(946\) 0 0
\(947\) −46700.0 −1.60248 −0.801239 0.598345i \(-0.795825\pi\)
−0.801239 + 0.598345i \(0.795825\pi\)
\(948\) 0 0
\(949\) 3836.00 0.131214
\(950\) 0 0
\(951\) −20358.0 −0.694168
\(952\) 0 0
\(953\) −40018.0 −1.36024 −0.680121 0.733100i \(-0.738073\pi\)
−0.680121 + 0.733100i \(0.738073\pi\)
\(954\) 0 0
\(955\) 9800.00 0.332063
\(956\) 0 0
\(957\) −18000.0 −0.608001
\(958\) 0 0
\(959\) −25568.0 −0.860932
\(960\) 0 0
\(961\) −29007.0 −0.973683
\(962\) 0 0
\(963\) −396.000 −0.0132512
\(964\) 0 0
\(965\) 20030.0 0.668175
\(966\) 0 0
\(967\) 1064.00 0.0353836 0.0176918 0.999843i \(-0.494368\pi\)
0.0176918 + 0.999843i \(0.494368\pi\)
\(968\) 0 0
\(969\) 1944.00 0.0644482
\(970\) 0 0
\(971\) 5664.00 0.187195 0.0935975 0.995610i \(-0.470163\pi\)
0.0935975 + 0.995610i \(0.470163\pi\)
\(972\) 0 0
\(973\) 47424.0 1.56253
\(974\) 0 0
\(975\) −1050.00 −0.0344891
\(976\) 0 0
\(977\) 33870.0 1.10911 0.554553 0.832148i \(-0.312889\pi\)
0.554553 + 0.832148i \(0.312889\pi\)
\(978\) 0 0
\(979\) 18864.0 0.615828
\(980\) 0 0
\(981\) 10098.0 0.328649
\(982\) 0 0
\(983\) −19976.0 −0.648154 −0.324077 0.946031i \(-0.605054\pi\)
−0.324077 + 0.946031i \(0.605054\pi\)
\(984\) 0 0
\(985\) 10590.0 0.342564
\(986\) 0 0
\(987\) −19584.0 −0.631576
\(988\) 0 0
\(989\) −23712.0 −0.762384
\(990\) 0 0
\(991\) −28748.0 −0.921504 −0.460752 0.887529i \(-0.652420\pi\)
−0.460752 + 0.887529i \(0.652420\pi\)
\(992\) 0 0
\(993\) −19788.0 −0.632380
\(994\) 0 0
\(995\) −18740.0 −0.597083
\(996\) 0 0
\(997\) 16830.0 0.534615 0.267308 0.963611i \(-0.413866\pi\)
0.267308 + 0.963611i \(0.413866\pi\)
\(998\) 0 0
\(999\) −1458.00 −0.0461753
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 960.4.a.c.1.1 1
4.3 odd 2 960.4.a.z.1.1 1
8.3 odd 2 480.4.a.f.1.1 1
8.5 even 2 480.4.a.i.1.1 yes 1
24.5 odd 2 1440.4.a.c.1.1 1
24.11 even 2 1440.4.a.h.1.1 1
40.19 odd 2 2400.4.a.o.1.1 1
40.29 even 2 2400.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.f.1.1 1 8.3 odd 2
480.4.a.i.1.1 yes 1 8.5 even 2
960.4.a.c.1.1 1 1.1 even 1 trivial
960.4.a.z.1.1 1 4.3 odd 2
1440.4.a.c.1.1 1 24.5 odd 2
1440.4.a.h.1.1 1 24.11 even 2
2400.4.a.h.1.1 1 40.29 even 2
2400.4.a.o.1.1 1 40.19 odd 2