Properties

Label 1872.4.a.v.1.1
Level $1872$
Weight $4$
Character 1872.1
Self dual yes
Analytic conductor $110.452$
Analytic rank $1$
Dimension $2$
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] = [1872,4,Mod(1,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, 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("1872.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1872.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: \(110.451575531\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{43}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.55744\) of defining polynomial
Character \(\chi\) \(=\) 1872.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-19.1149 q^{5} -35.1149 q^{7} +O(q^{10})\) \(q-19.1149 q^{5} -35.1149 q^{7} +26.0000 q^{11} -13.0000 q^{13} +36.2298 q^{17} +95.5744 q^{19} -161.379 q^{23} +240.379 q^{25} +91.3785 q^{29} +266.723 q^{31} +671.217 q^{35} -149.608 q^{37} +77.8041 q^{41} -183.608 q^{43} -60.6893 q^{47} +890.055 q^{49} -281.540 q^{53} -496.987 q^{55} -542.527 q^{59} +65.0810 q^{61} +248.493 q^{65} +1033.94 q^{67} +1041.81 q^{71} +483.311 q^{73} -912.987 q^{77} +1337.05 q^{79} +812.825 q^{83} -692.527 q^{85} -936.885 q^{89} +456.493 q^{91} -1826.89 q^{95} -954.068 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{5} - 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{5} - 44 q^{7} + 52 q^{11} - 26 q^{13} + 20 q^{17} + 60 q^{19} - 8 q^{23} + 166 q^{25} - 132 q^{29} + 140 q^{31} + 608 q^{35} + 68 q^{37} - 28 q^{41} + 36 q^{47} + 626 q^{49} - 668 q^{53} - 312 q^{55} - 508 q^{59} + 340 q^{61} + 156 q^{65} + 940 q^{67} + 300 q^{71} + 1124 q^{73} - 1144 q^{77} + 1520 q^{79} + 524 q^{83} - 808 q^{85} - 1900 q^{89} + 572 q^{91} - 2080 q^{95} - 1436 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −19.1149 −1.70969 −0.854843 0.518886i \(-0.826347\pi\)
−0.854843 + 0.518886i \(0.826347\pi\)
\(6\) 0 0
\(7\) −35.1149 −1.89603 −0.948013 0.318233i \(-0.896911\pi\)
−0.948013 + 0.318233i \(0.896911\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 26.0000 0.712663 0.356332 0.934360i \(-0.384027\pi\)
0.356332 + 0.934360i \(0.384027\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 36.2298 0.516883 0.258441 0.966027i \(-0.416791\pi\)
0.258441 + 0.966027i \(0.416791\pi\)
\(18\) 0 0
\(19\) 95.5744 1.15401 0.577007 0.816739i \(-0.304221\pi\)
0.577007 + 0.816739i \(0.304221\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −161.379 −1.46303 −0.731516 0.681824i \(-0.761187\pi\)
−0.731516 + 0.681824i \(0.761187\pi\)
\(24\) 0 0
\(25\) 240.379 1.92303
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 91.3785 0.585123 0.292561 0.956247i \(-0.405492\pi\)
0.292561 + 0.956247i \(0.405492\pi\)
\(30\) 0 0
\(31\) 266.723 1.54532 0.772660 0.634821i \(-0.218926\pi\)
0.772660 + 0.634821i \(0.218926\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 671.217 3.24161
\(36\) 0 0
\(37\) −149.608 −0.664742 −0.332371 0.943149i \(-0.607849\pi\)
−0.332371 + 0.943149i \(0.607849\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 77.8041 0.296365 0.148183 0.988960i \(-0.452658\pi\)
0.148183 + 0.988960i \(0.452658\pi\)
\(42\) 0 0
\(43\) −183.608 −0.651163 −0.325581 0.945514i \(-0.605560\pi\)
−0.325581 + 0.945514i \(0.605560\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −60.6893 −0.188350 −0.0941749 0.995556i \(-0.530021\pi\)
−0.0941749 + 0.995556i \(0.530021\pi\)
\(48\) 0 0
\(49\) 890.055 2.59491
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −281.540 −0.729671 −0.364835 0.931072i \(-0.618875\pi\)
−0.364835 + 0.931072i \(0.618875\pi\)
\(54\) 0 0
\(55\) −496.987 −1.21843
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −542.527 −1.19714 −0.598568 0.801072i \(-0.704263\pi\)
−0.598568 + 0.801072i \(0.704263\pi\)
\(60\) 0 0
\(61\) 65.0810 0.136603 0.0683014 0.997665i \(-0.478242\pi\)
0.0683014 + 0.997665i \(0.478242\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 248.493 0.474182
\(66\) 0 0
\(67\) 1033.94 1.88531 0.942656 0.333767i \(-0.108320\pi\)
0.942656 + 0.333767i \(0.108320\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1041.81 1.74141 0.870706 0.491803i \(-0.163662\pi\)
0.870706 + 0.491803i \(0.163662\pi\)
\(72\) 0 0
\(73\) 483.311 0.774894 0.387447 0.921892i \(-0.373357\pi\)
0.387447 + 0.921892i \(0.373357\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −912.987 −1.35123
\(78\) 0 0
\(79\) 1337.05 1.90418 0.952091 0.305815i \(-0.0989288\pi\)
0.952091 + 0.305815i \(0.0989288\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 812.825 1.07493 0.537465 0.843286i \(-0.319382\pi\)
0.537465 + 0.843286i \(0.319382\pi\)
\(84\) 0 0
\(85\) −692.527 −0.883707
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −936.885 −1.11584 −0.557919 0.829895i \(-0.688400\pi\)
−0.557919 + 0.829895i \(0.688400\pi\)
\(90\) 0 0
\(91\) 456.493 0.525863
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1826.89 −1.97300
\(96\) 0 0
\(97\) −954.068 −0.998669 −0.499335 0.866409i \(-0.666422\pi\)
−0.499335 + 0.866409i \(0.666422\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1401.15 −1.38039 −0.690196 0.723623i \(-0.742476\pi\)
−0.690196 + 0.723623i \(0.742476\pi\)
\(102\) 0 0
\(103\) −954.960 −0.913544 −0.456772 0.889584i \(-0.650995\pi\)
−0.456772 + 0.889584i \(0.650995\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −924.595 −0.835364 −0.417682 0.908593i \(-0.637157\pi\)
−0.417682 + 0.908593i \(0.637157\pi\)
\(108\) 0 0
\(109\) 172.893 0.151928 0.0759638 0.997111i \(-0.475797\pi\)
0.0759638 + 0.997111i \(0.475797\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1241.81 −1.03380 −0.516902 0.856045i \(-0.672915\pi\)
−0.516902 + 0.856045i \(0.672915\pi\)
\(114\) 0 0
\(115\) 3084.73 2.50133
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1272.20 −0.980023
\(120\) 0 0
\(121\) −655.000 −0.492111
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2205.45 −1.57809
\(126\) 0 0
\(127\) 1079.93 0.754555 0.377278 0.926100i \(-0.376860\pi\)
0.377278 + 0.926100i \(0.376860\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −918.026 −0.612277 −0.306139 0.951987i \(-0.599037\pi\)
−0.306139 + 0.951987i \(0.599037\pi\)
\(132\) 0 0
\(133\) −3356.08 −2.18804
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1311.06 −0.817603 −0.408801 0.912623i \(-0.634053\pi\)
−0.408801 + 0.912623i \(0.634053\pi\)
\(138\) 0 0
\(139\) 227.593 0.138879 0.0694396 0.997586i \(-0.477879\pi\)
0.0694396 + 0.997586i \(0.477879\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −338.000 −0.197657
\(144\) 0 0
\(145\) −1746.69 −1.00038
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1325.48 −0.728776 −0.364388 0.931247i \(-0.618722\pi\)
−0.364388 + 0.931247i \(0.618722\pi\)
\(150\) 0 0
\(151\) −252.426 −0.136040 −0.0680202 0.997684i \(-0.521668\pi\)
−0.0680202 + 0.997684i \(0.521668\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5098.38 −2.64201
\(156\) 0 0
\(157\) −2268.62 −1.15322 −0.576611 0.817019i \(-0.695625\pi\)
−0.576611 + 0.817019i \(0.695625\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5666.79 2.77395
\(162\) 0 0
\(163\) 1083.45 0.520630 0.260315 0.965524i \(-0.416174\pi\)
0.260315 + 0.965524i \(0.416174\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −796.041 −0.368859 −0.184430 0.982846i \(-0.559044\pi\)
−0.184430 + 0.982846i \(0.559044\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.09598 0.000481654 0 0.000240827 1.00000i \(-0.499923\pi\)
0.000240827 1.00000i \(0.499923\pi\)
\(174\) 0 0
\(175\) −8440.86 −3.64611
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3250.06 1.35710 0.678549 0.734555i \(-0.262609\pi\)
0.678549 + 0.734555i \(0.262609\pi\)
\(180\) 0 0
\(181\) 2980.84 1.22411 0.612055 0.790815i \(-0.290343\pi\)
0.612055 + 0.790815i \(0.290343\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2859.74 1.13650
\(186\) 0 0
\(187\) 941.974 0.368363
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 612.595 0.232072 0.116036 0.993245i \(-0.462981\pi\)
0.116036 + 0.993245i \(0.462981\pi\)
\(192\) 0 0
\(193\) −4185.81 −1.56115 −0.780573 0.625064i \(-0.785073\pi\)
−0.780573 + 0.625064i \(0.785073\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.06029 −0.000745124 0 −0.000372562 1.00000i \(-0.500119\pi\)
−0.000372562 1.00000i \(0.500119\pi\)
\(198\) 0 0
\(199\) −1492.39 −0.531622 −0.265811 0.964025i \(-0.585640\pi\)
−0.265811 + 0.964025i \(0.585640\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3208.75 −1.10941
\(204\) 0 0
\(205\) −1487.22 −0.506691
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2484.93 0.822423
\(210\) 0 0
\(211\) −2748.27 −0.896677 −0.448338 0.893864i \(-0.647984\pi\)
−0.448338 + 0.893864i \(0.647984\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3509.65 1.11328
\(216\) 0 0
\(217\) −9365.95 −2.92996
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −470.987 −0.143357
\(222\) 0 0
\(223\) −186.655 −0.0560510 −0.0280255 0.999607i \(-0.508922\pi\)
−0.0280255 + 0.999607i \(0.508922\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3932.57 1.14984 0.574920 0.818210i \(-0.305033\pi\)
0.574920 + 0.818210i \(0.305033\pi\)
\(228\) 0 0
\(229\) 5951.00 1.71726 0.858632 0.512593i \(-0.171315\pi\)
0.858632 + 0.512593i \(0.171315\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6548.94 −1.84135 −0.920676 0.390327i \(-0.872362\pi\)
−0.920676 + 0.390327i \(0.872362\pi\)
\(234\) 0 0
\(235\) 1160.07 0.322019
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3462.35 −0.937076 −0.468538 0.883443i \(-0.655219\pi\)
−0.468538 + 0.883443i \(0.655219\pi\)
\(240\) 0 0
\(241\) 4581.96 1.22469 0.612345 0.790591i \(-0.290226\pi\)
0.612345 + 0.790591i \(0.290226\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −17013.3 −4.43649
\(246\) 0 0
\(247\) −1242.47 −0.320066
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4746.76 1.19368 0.596838 0.802362i \(-0.296424\pi\)
0.596838 + 0.802362i \(0.296424\pi\)
\(252\) 0 0
\(253\) −4195.84 −1.04265
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3373.64 0.818840 0.409420 0.912346i \(-0.365731\pi\)
0.409420 + 0.912346i \(0.365731\pi\)
\(258\) 0 0
\(259\) 5253.48 1.26037
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3527.19 0.826981 0.413491 0.910508i \(-0.364309\pi\)
0.413491 + 0.910508i \(0.364309\pi\)
\(264\) 0 0
\(265\) 5381.61 1.24751
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1144.30 −0.259365 −0.129682 0.991556i \(-0.541396\pi\)
−0.129682 + 0.991556i \(0.541396\pi\)
\(270\) 0 0
\(271\) −2254.25 −0.505298 −0.252649 0.967558i \(-0.581302\pi\)
−0.252649 + 0.967558i \(0.581302\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6249.84 1.37047
\(276\) 0 0
\(277\) 7101.62 1.54042 0.770208 0.637793i \(-0.220152\pi\)
0.770208 + 0.637793i \(0.220152\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3831.81 0.813475 0.406738 0.913545i \(-0.366666\pi\)
0.406738 + 0.913545i \(0.366666\pi\)
\(282\) 0 0
\(283\) 3032.56 0.636985 0.318493 0.947925i \(-0.396823\pi\)
0.318493 + 0.947925i \(0.396823\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2732.08 −0.561915
\(288\) 0 0
\(289\) −3600.40 −0.732832
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2296.44 0.457882 0.228941 0.973440i \(-0.426474\pi\)
0.228941 + 0.973440i \(0.426474\pi\)
\(294\) 0 0
\(295\) 10370.3 2.04673
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2097.92 0.405772
\(300\) 0 0
\(301\) 6447.38 1.23462
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1244.01 −0.233548
\(306\) 0 0
\(307\) −5174.26 −0.961925 −0.480962 0.876741i \(-0.659713\pi\)
−0.480962 + 0.876741i \(0.659713\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7481.54 −1.36411 −0.682057 0.731299i \(-0.738914\pi\)
−0.682057 + 0.731299i \(0.738914\pi\)
\(312\) 0 0
\(313\) −4158.97 −0.751051 −0.375526 0.926812i \(-0.622538\pi\)
−0.375526 + 0.926812i \(0.622538\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 370.535 0.0656508 0.0328254 0.999461i \(-0.489549\pi\)
0.0328254 + 0.999461i \(0.489549\pi\)
\(318\) 0 0
\(319\) 2375.84 0.416996
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3462.64 0.596490
\(324\) 0 0
\(325\) −3124.92 −0.533352
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2131.10 0.357116
\(330\) 0 0
\(331\) −1820.84 −0.302364 −0.151182 0.988506i \(-0.548308\pi\)
−0.151182 + 0.988506i \(0.548308\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −19763.6 −3.22329
\(336\) 0 0
\(337\) −11838.2 −1.91356 −0.956781 0.290809i \(-0.906076\pi\)
−0.956781 + 0.290809i \(0.906076\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6934.80 1.10129
\(342\) 0 0
\(343\) −19209.8 −3.02399
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10603.5 1.64043 0.820214 0.572057i \(-0.193855\pi\)
0.820214 + 0.572057i \(0.193855\pi\)
\(348\) 0 0
\(349\) −6398.72 −0.981421 −0.490710 0.871323i \(-0.663263\pi\)
−0.490710 + 0.871323i \(0.663263\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7031.27 1.06016 0.530080 0.847948i \(-0.322162\pi\)
0.530080 + 0.847948i \(0.322162\pi\)
\(354\) 0 0
\(355\) −19914.1 −2.97727
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5053.75 0.742972 0.371486 0.928439i \(-0.378848\pi\)
0.371486 + 0.928439i \(0.378848\pi\)
\(360\) 0 0
\(361\) 2275.46 0.331749
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9238.43 −1.32483
\(366\) 0 0
\(367\) 81.1902 0.0115479 0.00577397 0.999983i \(-0.498162\pi\)
0.00577397 + 0.999983i \(0.498162\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9886.26 1.38347
\(372\) 0 0
\(373\) 8295.79 1.15158 0.575790 0.817597i \(-0.304695\pi\)
0.575790 + 0.817597i \(0.304695\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1187.92 −0.162284
\(378\) 0 0
\(379\) 6592.90 0.893548 0.446774 0.894647i \(-0.352573\pi\)
0.446774 + 0.894647i \(0.352573\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7472.96 −0.996999 −0.498499 0.866890i \(-0.666115\pi\)
−0.498499 + 0.866890i \(0.666115\pi\)
\(384\) 0 0
\(385\) 17451.6 2.31018
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3175.88 0.413943 0.206971 0.978347i \(-0.433639\pi\)
0.206971 + 0.978347i \(0.433639\pi\)
\(390\) 0 0
\(391\) −5846.70 −0.756216
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −25557.6 −3.25555
\(396\) 0 0
\(397\) 11394.6 1.44050 0.720251 0.693713i \(-0.244027\pi\)
0.720251 + 0.693713i \(0.244027\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13046.7 −1.62474 −0.812370 0.583142i \(-0.801823\pi\)
−0.812370 + 0.583142i \(0.801823\pi\)
\(402\) 0 0
\(403\) −3467.40 −0.428594
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3889.82 −0.473737
\(408\) 0 0
\(409\) −2888.77 −0.349243 −0.174622 0.984636i \(-0.555870\pi\)
−0.174622 + 0.984636i \(0.555870\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 19050.8 2.26980
\(414\) 0 0
\(415\) −15537.0 −1.83779
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14260.8 −1.66273 −0.831367 0.555724i \(-0.812441\pi\)
−0.831367 + 0.555724i \(0.812441\pi\)
\(420\) 0 0
\(421\) 890.723 0.103114 0.0515572 0.998670i \(-0.483582\pi\)
0.0515572 + 0.998670i \(0.483582\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8708.85 0.993980
\(426\) 0 0
\(427\) −2285.31 −0.259002
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1323.12 −0.147871 −0.0739353 0.997263i \(-0.523556\pi\)
−0.0739353 + 0.997263i \(0.523556\pi\)
\(432\) 0 0
\(433\) −9399.47 −1.04321 −0.521605 0.853187i \(-0.674666\pi\)
−0.521605 + 0.853187i \(0.674666\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −15423.7 −1.68836
\(438\) 0 0
\(439\) 6115.04 0.664817 0.332409 0.943135i \(-0.392139\pi\)
0.332409 + 0.943135i \(0.392139\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12931.1 −1.38685 −0.693425 0.720529i \(-0.743899\pi\)
−0.693425 + 0.720529i \(0.743899\pi\)
\(444\) 0 0
\(445\) 17908.4 1.90773
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2050.73 0.215546 0.107773 0.994176i \(-0.465628\pi\)
0.107773 + 0.994176i \(0.465628\pi\)
\(450\) 0 0
\(451\) 2022.91 0.211208
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8725.82 −0.899061
\(456\) 0 0
\(457\) −11923.7 −1.22049 −0.610247 0.792211i \(-0.708930\pi\)
−0.610247 + 0.792211i \(0.708930\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1324.14 −0.133777 −0.0668886 0.997760i \(-0.521307\pi\)
−0.0668886 + 0.997760i \(0.521307\pi\)
\(462\) 0 0
\(463\) 6840.65 0.686635 0.343318 0.939219i \(-0.388449\pi\)
0.343318 + 0.939219i \(0.388449\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9041.73 −0.895935 −0.447967 0.894050i \(-0.647852\pi\)
−0.447967 + 0.894050i \(0.647852\pi\)
\(468\) 0 0
\(469\) −36306.7 −3.57460
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4773.82 −0.464060
\(474\) 0 0
\(475\) 22974.0 2.21920
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4650.71 0.443625 0.221812 0.975089i \(-0.428803\pi\)
0.221812 + 0.975089i \(0.428803\pi\)
\(480\) 0 0
\(481\) 1944.91 0.184366
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18236.9 1.70741
\(486\) 0 0
\(487\) 19632.4 1.82675 0.913375 0.407120i \(-0.133467\pi\)
0.913375 + 0.407120i \(0.133467\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −14472.0 −1.33017 −0.665084 0.746769i \(-0.731604\pi\)
−0.665084 + 0.746769i \(0.731604\pi\)
\(492\) 0 0
\(493\) 3310.62 0.302440
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −36583.1 −3.30176
\(498\) 0 0
\(499\) 7065.91 0.633895 0.316947 0.948443i \(-0.397342\pi\)
0.316947 + 0.948443i \(0.397342\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11955.1 1.05974 0.529871 0.848078i \(-0.322240\pi\)
0.529871 + 0.848078i \(0.322240\pi\)
\(504\) 0 0
\(505\) 26782.8 2.36004
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8321.15 0.724614 0.362307 0.932059i \(-0.381989\pi\)
0.362307 + 0.932059i \(0.381989\pi\)
\(510\) 0 0
\(511\) −16971.4 −1.46922
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18254.0 1.56187
\(516\) 0 0
\(517\) −1577.92 −0.134230
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20130.2 1.69274 0.846372 0.532592i \(-0.178782\pi\)
0.846372 + 0.532592i \(0.178782\pi\)
\(522\) 0 0
\(523\) −8810.52 −0.736630 −0.368315 0.929701i \(-0.620065\pi\)
−0.368315 + 0.929701i \(0.620065\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9663.31 0.798749
\(528\) 0 0
\(529\) 13876.0 1.14046
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1011.45 −0.0821969
\(534\) 0 0
\(535\) 17673.5 1.42821
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 23141.4 1.84930
\(540\) 0 0
\(541\) 7765.62 0.617135 0.308567 0.951202i \(-0.400150\pi\)
0.308567 + 0.951202i \(0.400150\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3304.82 −0.259749
\(546\) 0 0
\(547\) −19782.4 −1.54631 −0.773156 0.634216i \(-0.781323\pi\)
−0.773156 + 0.634216i \(0.781323\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8733.45 0.675240
\(552\) 0 0
\(553\) −46950.5 −3.61038
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10153.9 0.772413 0.386206 0.922412i \(-0.373785\pi\)
0.386206 + 0.922412i \(0.373785\pi\)
\(558\) 0 0
\(559\) 2386.91 0.180600
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18388.2 1.37650 0.688252 0.725472i \(-0.258378\pi\)
0.688252 + 0.725472i \(0.258378\pi\)
\(564\) 0 0
\(565\) 23737.1 1.76748
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −16365.0 −1.20573 −0.602863 0.797845i \(-0.705973\pi\)
−0.602863 + 0.797845i \(0.705973\pi\)
\(570\) 0 0
\(571\) −544.241 −0.0398875 −0.0199437 0.999801i \(-0.506349\pi\)
−0.0199437 + 0.999801i \(0.506349\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −38791.9 −2.81345
\(576\) 0 0
\(577\) −798.738 −0.0576289 −0.0288145 0.999585i \(-0.509173\pi\)
−0.0288145 + 0.999585i \(0.509173\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −28542.2 −2.03809
\(582\) 0 0
\(583\) −7320.05 −0.520010
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9187.53 0.646013 0.323007 0.946397i \(-0.395306\pi\)
0.323007 + 0.946397i \(0.395306\pi\)
\(588\) 0 0
\(589\) 25491.9 1.78332
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4371.80 0.302746 0.151373 0.988477i \(-0.451631\pi\)
0.151373 + 0.988477i \(0.451631\pi\)
\(594\) 0 0
\(595\) 24318.0 1.67553
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6780.78 −0.462530 −0.231265 0.972891i \(-0.574286\pi\)
−0.231265 + 0.972891i \(0.574286\pi\)
\(600\) 0 0
\(601\) −21348.5 −1.44895 −0.724477 0.689299i \(-0.757919\pi\)
−0.724477 + 0.689299i \(0.757919\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12520.2 0.841356
\(606\) 0 0
\(607\) −11161.8 −0.746362 −0.373181 0.927758i \(-0.621733\pi\)
−0.373181 + 0.927758i \(0.621733\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 788.960 0.0522388
\(612\) 0 0
\(613\) 12407.8 0.817528 0.408764 0.912640i \(-0.365960\pi\)
0.408764 + 0.912640i \(0.365960\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5035.01 −0.328528 −0.164264 0.986416i \(-0.552525\pi\)
−0.164264 + 0.986416i \(0.552525\pi\)
\(618\) 0 0
\(619\) 1623.67 0.105430 0.0527148 0.998610i \(-0.483213\pi\)
0.0527148 + 0.998610i \(0.483213\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 32898.6 2.11566
\(624\) 0 0
\(625\) 12109.5 0.775009
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5420.27 −0.343594
\(630\) 0 0
\(631\) −24809.0 −1.56519 −0.782593 0.622534i \(-0.786103\pi\)
−0.782593 + 0.622534i \(0.786103\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −20642.8 −1.29005
\(636\) 0 0
\(637\) −11570.7 −0.719699
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18076.6 −1.11386 −0.556928 0.830561i \(-0.688020\pi\)
−0.556928 + 0.830561i \(0.688020\pi\)
\(642\) 0 0
\(643\) −19954.2 −1.22382 −0.611910 0.790928i \(-0.709598\pi\)
−0.611910 + 0.790928i \(0.709598\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −11206.1 −0.680921 −0.340461 0.940259i \(-0.610583\pi\)
−0.340461 + 0.940259i \(0.610583\pi\)
\(648\) 0 0
\(649\) −14105.7 −0.853155
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2456.10 −0.147189 −0.0735947 0.997288i \(-0.523447\pi\)
−0.0735947 + 0.997288i \(0.523447\pi\)
\(654\) 0 0
\(655\) 17548.0 1.04680
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −23280.8 −1.37616 −0.688081 0.725633i \(-0.741547\pi\)
−0.688081 + 0.725633i \(0.741547\pi\)
\(660\) 0 0
\(661\) −6999.49 −0.411874 −0.205937 0.978565i \(-0.566024\pi\)
−0.205937 + 0.978565i \(0.566024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 64151.1 3.74086
\(666\) 0 0
\(667\) −14746.5 −0.856054
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1692.11 0.0973517
\(672\) 0 0
\(673\) 33166.1 1.89964 0.949820 0.312798i \(-0.101266\pi\)
0.949820 + 0.312798i \(0.101266\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19998.6 −1.13532 −0.567658 0.823265i \(-0.692150\pi\)
−0.567658 + 0.823265i \(0.692150\pi\)
\(678\) 0 0
\(679\) 33502.0 1.89350
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22857.9 1.28058 0.640289 0.768134i \(-0.278815\pi\)
0.640289 + 0.768134i \(0.278815\pi\)
\(684\) 0 0
\(685\) 25060.8 1.39784
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3660.03 0.202374
\(690\) 0 0
\(691\) 2866.56 0.157813 0.0789066 0.996882i \(-0.474857\pi\)
0.0789066 + 0.996882i \(0.474857\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4350.42 −0.237440
\(696\) 0 0
\(697\) 2818.82 0.153186
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30788.1 −1.65885 −0.829424 0.558620i \(-0.811331\pi\)
−0.829424 + 0.558620i \(0.811331\pi\)
\(702\) 0 0
\(703\) −14298.7 −0.767121
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 49201.2 2.61726
\(708\) 0 0
\(709\) −4389.07 −0.232489 −0.116245 0.993221i \(-0.537086\pi\)
−0.116245 + 0.993221i \(0.537086\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −43043.4 −2.26085
\(714\) 0 0
\(715\) 6460.83 0.337932
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −8729.09 −0.452768 −0.226384 0.974038i \(-0.572690\pi\)
−0.226384 + 0.974038i \(0.572690\pi\)
\(720\) 0 0
\(721\) 33533.3 1.73210
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21965.4 1.12521
\(726\) 0 0
\(727\) −1471.81 −0.0750845 −0.0375423 0.999295i \(-0.511953\pi\)
−0.0375423 + 0.999295i \(0.511953\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6652.08 −0.336575
\(732\) 0 0
\(733\) −14082.8 −0.709632 −0.354816 0.934936i \(-0.615457\pi\)
−0.354816 + 0.934936i \(0.615457\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 26882.4 1.34359
\(738\) 0 0
\(739\) −12529.5 −0.623687 −0.311843 0.950133i \(-0.600946\pi\)
−0.311843 + 0.950133i \(0.600946\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −19044.9 −0.940363 −0.470182 0.882570i \(-0.655812\pi\)
−0.470182 + 0.882570i \(0.655812\pi\)
\(744\) 0 0
\(745\) 25336.4 1.24598
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 32467.0 1.58387
\(750\) 0 0
\(751\) 22009.9 1.06944 0.534722 0.845028i \(-0.320416\pi\)
0.534722 + 0.845028i \(0.320416\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4825.08 0.232587
\(756\) 0 0
\(757\) −8198.54 −0.393634 −0.196817 0.980440i \(-0.563061\pi\)
−0.196817 + 0.980440i \(0.563061\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 31929.6 1.52096 0.760478 0.649363i \(-0.224964\pi\)
0.760478 + 0.649363i \(0.224964\pi\)
\(762\) 0 0
\(763\) −6071.10 −0.288059
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7052.85 0.332026
\(768\) 0 0
\(769\) −26725.8 −1.25326 −0.626630 0.779317i \(-0.715566\pi\)
−0.626630 + 0.779317i \(0.715566\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −23760.8 −1.10558 −0.552792 0.833320i \(-0.686437\pi\)
−0.552792 + 0.833320i \(0.686437\pi\)
\(774\) 0 0
\(775\) 64114.5 2.97169
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7436.08 0.342009
\(780\) 0 0
\(781\) 27087.1 1.24104
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 43364.4 1.97165
\(786\) 0 0
\(787\) 23647.2 1.07107 0.535536 0.844513i \(-0.320110\pi\)
0.535536 + 0.844513i \(0.320110\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 43606.1 1.96012
\(792\) 0 0
\(793\) −846.053 −0.0378868
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21067.5 −0.936324 −0.468162 0.883643i \(-0.655084\pi\)
−0.468162 + 0.883643i \(0.655084\pi\)
\(798\) 0 0
\(799\) −2198.76 −0.0973547
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12566.1 0.552238
\(804\) 0 0
\(805\) −108320. −4.74258
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1877.67 −0.0816011 −0.0408005 0.999167i \(-0.512991\pi\)
−0.0408005 + 0.999167i \(0.512991\pi\)
\(810\) 0 0
\(811\) −2178.85 −0.0943401 −0.0471700 0.998887i \(-0.515020\pi\)
−0.0471700 + 0.998887i \(0.515020\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20710.1 −0.890114
\(816\) 0 0
\(817\) −17548.2 −0.751451
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −11731.4 −0.498696 −0.249348 0.968414i \(-0.580216\pi\)
−0.249348 + 0.968414i \(0.580216\pi\)
\(822\) 0 0
\(823\) 6925.69 0.293334 0.146667 0.989186i \(-0.453145\pi\)
0.146667 + 0.989186i \(0.453145\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −35912.7 −1.51004 −0.755022 0.655700i \(-0.772374\pi\)
−0.755022 + 0.655700i \(0.772374\pi\)
\(828\) 0 0
\(829\) −7107.67 −0.297780 −0.148890 0.988854i \(-0.547570\pi\)
−0.148890 + 0.988854i \(0.547570\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 32246.5 1.34126
\(834\) 0 0
\(835\) 15216.2 0.630634
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 37844.0 1.55724 0.778618 0.627498i \(-0.215921\pi\)
0.778618 + 0.627498i \(0.215921\pi\)
\(840\) 0 0
\(841\) −16039.0 −0.657631
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3230.41 −0.131514
\(846\) 0 0
\(847\) 23000.2 0.933055
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24143.6 0.972539
\(852\) 0 0
\(853\) 23280.9 0.934492 0.467246 0.884127i \(-0.345246\pi\)
0.467246 + 0.884127i \(0.345246\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10395.1 −0.414342 −0.207171 0.978305i \(-0.566426\pi\)
−0.207171 + 0.978305i \(0.566426\pi\)
\(858\) 0 0
\(859\) 6698.89 0.266080 0.133040 0.991111i \(-0.457526\pi\)
0.133040 + 0.991111i \(0.457526\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31844.8 1.25609 0.628047 0.778175i \(-0.283854\pi\)
0.628047 + 0.778175i \(0.283854\pi\)
\(864\) 0 0
\(865\) −20.9496 −0.000823477 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 34763.4 1.35704
\(870\) 0 0
\(871\) −13441.2 −0.522891
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 77444.0 2.99210
\(876\) 0 0
\(877\) −20114.1 −0.774465 −0.387233 0.921982i \(-0.626569\pi\)
−0.387233 + 0.921982i \(0.626569\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2596.61 0.0992987 0.0496493 0.998767i \(-0.484190\pi\)
0.0496493 + 0.998767i \(0.484190\pi\)
\(882\) 0 0
\(883\) 46374.7 1.76742 0.883710 0.468035i \(-0.155038\pi\)
0.883710 + 0.468035i \(0.155038\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −31863.1 −1.20615 −0.603077 0.797683i \(-0.706059\pi\)
−0.603077 + 0.797683i \(0.706059\pi\)
\(888\) 0 0
\(889\) −37921.7 −1.43066
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −5800.34 −0.217358
\(894\) 0 0
\(895\) −62124.4 −2.32021
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24372.8 0.904202
\(900\) 0 0
\(901\) −10200.1 −0.377154
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −56978.4 −2.09285
\(906\) 0 0
\(907\) −3004.29 −0.109984 −0.0549922 0.998487i \(-0.517513\pi\)
−0.0549922 + 0.998487i \(0.517513\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −9985.05 −0.363139 −0.181569 0.983378i \(-0.558118\pi\)
−0.181569 + 0.983378i \(0.558118\pi\)
\(912\) 0 0
\(913\) 21133.4 0.766062
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 32236.4 1.16089
\(918\) 0 0
\(919\) 8216.29 0.294919 0.147459 0.989068i \(-0.452890\pi\)
0.147459 + 0.989068i \(0.452890\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −13543.6 −0.482981
\(924\) 0 0
\(925\) −35962.6 −1.27832
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 17832.8 0.629791 0.314896 0.949126i \(-0.398030\pi\)
0.314896 + 0.949126i \(0.398030\pi\)
\(930\) 0 0
\(931\) 85066.4 2.99456
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −18005.7 −0.629786
\(936\) 0 0
\(937\) −444.289 −0.0154902 −0.00774509 0.999970i \(-0.502465\pi\)
−0.00774509 + 0.999970i \(0.502465\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −30762.5 −1.06571 −0.532853 0.846208i \(-0.678880\pi\)
−0.532853 + 0.846208i \(0.678880\pi\)
\(942\) 0 0
\(943\) −12555.9 −0.433592
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8791.45 −0.301672 −0.150836 0.988559i \(-0.548197\pi\)
−0.150836 + 0.988559i \(0.548197\pi\)
\(948\) 0 0
\(949\) −6283.04 −0.214917
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1754.41 −0.0596336 −0.0298168 0.999555i \(-0.509492\pi\)
−0.0298168 + 0.999555i \(0.509492\pi\)
\(954\) 0 0
\(955\) −11709.7 −0.396771
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 46037.8 1.55020
\(960\) 0 0
\(961\) 41350.2 1.38801
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 80011.3 2.66907
\(966\) 0 0
\(967\) −26655.9 −0.886447 −0.443224 0.896411i \(-0.646165\pi\)
−0.443224 + 0.896411i \(0.646165\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −42151.7 −1.39311 −0.696556 0.717502i \(-0.745285\pi\)
−0.696556 + 0.717502i \(0.745285\pi\)
\(972\) 0 0
\(973\) −7991.91 −0.263318
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −14594.7 −0.477919 −0.238959 0.971030i \(-0.576806\pi\)
−0.238959 + 0.971030i \(0.576806\pi\)
\(978\) 0 0
\(979\) −24359.0 −0.795217
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 57077.8 1.85198 0.925991 0.377546i \(-0.123232\pi\)
0.925991 + 0.377546i \(0.123232\pi\)
\(984\) 0 0
\(985\) 39.3821 0.00127393
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 29630.4 0.952672
\(990\) 0 0
\(991\) −11986.6 −0.384224 −0.192112 0.981373i \(-0.561534\pi\)
−0.192112 + 0.981373i \(0.561534\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 28526.9 0.908908
\(996\) 0 0
\(997\) 32210.1 1.02317 0.511587 0.859231i \(-0.329058\pi\)
0.511587 + 0.859231i \(0.329058\pi\)
\(998\) 0 0
\(999\) 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 1872.4.a.v.1.1 2
3.2 odd 2 624.4.a.l.1.2 2
4.3 odd 2 936.4.a.c.1.1 2
12.11 even 2 312.4.a.f.1.2 2
24.5 odd 2 2496.4.a.bd.1.1 2
24.11 even 2 2496.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.f.1.2 2 12.11 even 2
624.4.a.l.1.2 2 3.2 odd 2
936.4.a.c.1.1 2 4.3 odd 2
1872.4.a.v.1.1 2 1.1 even 1 trivial
2496.4.a.u.1.1 2 24.11 even 2
2496.4.a.bd.1.1 2 24.5 odd 2