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Properties

Label 1764.4.a.k.1.1
Level $1764$
Weight $4$
Character 1764.1
Self dual yes
Analytic conductor $104.079$
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] = [1764,4,Mod(1,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, 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("1764.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.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: \(104.079369250\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1764.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.00000 q^{5} +O(q^{10})\) \(q+6.00000 q^{5} +12.0000 q^{11} +82.0000 q^{13} -30.0000 q^{17} -68.0000 q^{19} -216.000 q^{23} -89.0000 q^{25} -246.000 q^{29} +112.000 q^{31} +110.000 q^{37} -246.000 q^{41} -172.000 q^{43} +192.000 q^{47} -558.000 q^{53} +72.0000 q^{55} +540.000 q^{59} -110.000 q^{61} +492.000 q^{65} +140.000 q^{67} +840.000 q^{71} +550.000 q^{73} -208.000 q^{79} +516.000 q^{83} -180.000 q^{85} -1398.00 q^{89} -408.000 q^{95} -1586.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 6.00000 0.536656 0.268328 − 0.963328i \(-0.413529\pi\)
0.268328 + 0.963328i \(0.413529\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 12.0000 0.328921 0.164461 − 0.986384i \(-0.447412\pi\)
0.164461 + 0.986384i \(0.447412\pi\)
\(12\) 0 0
\(13\) 82.0000 1.74944 0.874720 − 0.484629i \(-0.161046\pi\)
0.874720 + 0.484629i \(0.161046\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −30.0000 −0.428004 −0.214002 − 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 0 0
\(19\) −68.0000 −0.821067 −0.410533 − 0.911846i \(-0.634657\pi\)
−0.410533 + 0.911846i \(0.634657\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −216.000 −1.95822 −0.979111 − 0.203326i \(-0.934825\pi\)
−0.979111 + 0.203326i \(0.934825\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −246.000 −1.57521 −0.787604 − 0.616181i \(-0.788679\pi\)
−0.787604 + 0.616181i \(0.788679\pi\)
\(30\) 0 0
\(31\) 112.000 0.648897 0.324448 − 0.945903i \(-0.394821\pi\)
0.324448 + 0.945903i \(0.394821\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 110.000 0.488754 0.244377 − 0.969680i \(-0.421417\pi\)
0.244377 + 0.969680i \(0.421417\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −246.000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −172.000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 192.000 0.595874 0.297937 − 0.954586i \(-0.403701\pi\)
0.297937 + 0.954586i \(0.403701\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −558.000 −1.44617 −0.723087 − 0.690757i \(-0.757277\pi\)
−0.723087 + 0.690757i \(0.757277\pi\)
\(54\) 0 0
\(55\) 72.0000 0.176518
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 540.000 1.19156 0.595780 − 0.803148i \(-0.296843\pi\)
0.595780 + 0.803148i \(0.296843\pi\)
\(60\) 0 0
\(61\) −110.000 −0.230886 −0.115443 − 0.993314i \(-0.536829\pi\)
−0.115443 + 0.993314i \(0.536829\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 492.000 0.938848
\(66\) 0 0
\(67\) 140.000 0.255279 0.127640 − 0.991821i \(-0.459260\pi\)
0.127640 + 0.991821i \(0.459260\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 840.000 1.40408 0.702040 − 0.712138i \(-0.252273\pi\)
0.702040 + 0.712138i \(0.252273\pi\)
\(72\) 0 0
\(73\) 550.000 0.881817 0.440908 − 0.897552i \(-0.354656\pi\)
0.440908 + 0.897552i \(0.354656\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −208.000 −0.296226 −0.148113 − 0.988970i \(-0.547320\pi\)
−0.148113 + 0.988970i \(0.547320\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 516.000 0.682390 0.341195 − 0.939993i \(-0.389168\pi\)
0.341195 + 0.939993i \(0.389168\pi\)
\(84\) 0 0
\(85\) −180.000 −0.229691
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1398.00 −1.66503 −0.832515 − 0.554002i \(-0.813100\pi\)
−0.832515 + 0.554002i \(0.813100\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −408.000 −0.440631
\(96\) 0 0
\(97\) −1586.00 −1.66014 −0.830072 − 0.557657i \(-0.811701\pi\)
−0.830072 + 0.557657i \(0.811701\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1242.00 −1.22360 −0.611800 − 0.791012i \(-0.709554\pi\)
−0.611800 + 0.791012i \(0.709554\pi\)
\(102\) 0 0
\(103\) −680.000 −0.650509 −0.325254 − 0.945627i \(-0.605450\pi\)
−0.325254 + 0.945627i \(0.605450\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −996.000 −0.899878 −0.449939 − 0.893059i \(-0.648554\pi\)
−0.449939 + 0.893059i \(0.648554\pi\)
\(108\) 0 0
\(109\) 1382.00 1.21442 0.607209 − 0.794542i \(-0.292289\pi\)
0.607209 + 0.794542i \(0.292289\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 750.000 0.624372 0.312186 − 0.950021i \(-0.398939\pi\)
0.312186 + 0.950021i \(0.398939\pi\)
\(114\) 0 0
\(115\) −1296.00 −1.05089
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1284.00 −0.918756
\(126\) 0 0
\(127\) 176.000 0.122972 0.0614861 − 0.998108i \(-0.480416\pi\)
0.0614861 + 0.998108i \(0.480416\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1548.00 −1.03244 −0.516219 − 0.856457i \(-0.672661\pi\)
−0.516219 + 0.856457i \(0.672661\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −378.000 −0.235728 −0.117864 − 0.993030i \(-0.537605\pi\)
−0.117864 + 0.993030i \(0.537605\pi\)
\(138\) 0 0
\(139\) 2500.00 1.52552 0.762760 − 0.646682i \(-0.223844\pi\)
0.762760 + 0.646682i \(0.223844\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 984.000 0.575428
\(144\) 0 0
\(145\) −1476.00 −0.845346
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −846.000 −0.465148 −0.232574 − 0.972579i \(-0.574715\pi\)
−0.232574 + 0.972579i \(0.574715\pi\)
\(150\) 0 0
\(151\) −2536.00 −1.36673 −0.683367 − 0.730075i \(-0.739485\pi\)
−0.683367 + 0.730075i \(0.739485\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 672.000 0.348234
\(156\) 0 0
\(157\) 1186.00 0.602886 0.301443 − 0.953484i \(-0.402532\pi\)
0.301443 + 0.953484i \(0.402532\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2108.00 1.01295 0.506476 − 0.862254i \(-0.330948\pi\)
0.506476 + 0.862254i \(0.330948\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1944.00 −0.900786 −0.450393 − 0.892830i \(-0.648716\pi\)
−0.450393 + 0.892830i \(0.648716\pi\)
\(168\) 0 0
\(169\) 4527.00 2.06054
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1362.00 −0.598560 −0.299280 − 0.954165i \(-0.596747\pi\)
−0.299280 + 0.954165i \(0.596747\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1596.00 −0.666428 −0.333214 − 0.942851i \(-0.608133\pi\)
−0.333214 + 0.942851i \(0.608133\pi\)
\(180\) 0 0
\(181\) 1690.00 0.694015 0.347007 − 0.937862i \(-0.387198\pi\)
0.347007 + 0.937862i \(0.387198\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 660.000 0.262293
\(186\) 0 0
\(187\) −360.000 −0.140780
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3552.00 −1.34562 −0.672811 − 0.739815i \(-0.734913\pi\)
−0.672811 + 0.739815i \(0.734913\pi\)
\(192\) 0 0
\(193\) −2686.00 −1.00177 −0.500887 − 0.865512i \(-0.666993\pi\)
−0.500887 + 0.865512i \(0.666993\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1410.00 0.509941 0.254970 − 0.966949i \(-0.417934\pi\)
0.254970 + 0.966949i \(0.417934\pi\)
\(198\) 0 0
\(199\) 2968.00 1.05727 0.528633 − 0.848850i \(-0.322705\pi\)
0.528633 + 0.848850i \(0.322705\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1476.00 −0.502870
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −816.000 −0.270067
\(210\) 0 0
\(211\) −1348.00 −0.439811 −0.219906 − 0.975521i \(-0.570575\pi\)
−0.219906 + 0.975521i \(0.570575\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1032.00 −0.327357
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2460.00 −0.748767
\(222\) 0 0
\(223\) −3872.00 −1.16273 −0.581364 − 0.813644i \(-0.697481\pi\)
−0.581364 + 0.813644i \(0.697481\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5364.00 1.56838 0.784188 − 0.620524i \(-0.213080\pi\)
0.784188 + 0.620524i \(0.213080\pi\)
\(228\) 0 0
\(229\) 874.000 0.252208 0.126104 − 0.992017i \(-0.459753\pi\)
0.126104 + 0.992017i \(0.459753\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −378.000 −0.106282 −0.0531408 − 0.998587i \(-0.516923\pi\)
−0.0531408 + 0.998587i \(0.516923\pi\)
\(234\) 0 0
\(235\) 1152.00 0.319780
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1920.00 −0.519642 −0.259821 − 0.965657i \(-0.583664\pi\)
−0.259821 + 0.965657i \(0.583664\pi\)
\(240\) 0 0
\(241\) −4322.00 −1.15521 −0.577603 − 0.816318i \(-0.696012\pi\)
−0.577603 + 0.816318i \(0.696012\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5576.00 −1.43641
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5292.00 1.33079 0.665395 − 0.746492i \(-0.268263\pi\)
0.665395 + 0.746492i \(0.268263\pi\)
\(252\) 0 0
\(253\) −2592.00 −0.644101
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5118.00 −1.24223 −0.621113 − 0.783721i \(-0.713319\pi\)
−0.621113 + 0.783721i \(0.713319\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3768.00 −0.883440 −0.441720 − 0.897153i \(-0.645632\pi\)
−0.441720 + 0.897153i \(0.645632\pi\)
\(264\) 0 0
\(265\) −3348.00 −0.776098
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3918.00 0.888047 0.444024 − 0.896015i \(-0.353551\pi\)
0.444024 + 0.896015i \(0.353551\pi\)
\(270\) 0 0
\(271\) −4880.00 −1.09387 −0.546935 − 0.837175i \(-0.684206\pi\)
−0.546935 + 0.837175i \(0.684206\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1068.00 −0.234192
\(276\) 0 0
\(277\) −3538.00 −0.767429 −0.383714 − 0.923452i \(-0.625355\pi\)
−0.383714 + 0.923452i \(0.625355\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5430.00 1.15276 0.576382 − 0.817180i \(-0.304464\pi\)
0.576382 + 0.817180i \(0.304464\pi\)
\(282\) 0 0
\(283\) 6436.00 1.35187 0.675937 − 0.736959i \(-0.263739\pi\)
0.675937 + 0.736959i \(0.263739\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4013.00 −0.816813
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1350.00 0.269174 0.134587 − 0.990902i \(-0.457029\pi\)
0.134587 + 0.990902i \(0.457029\pi\)
\(294\) 0 0
\(295\) 3240.00 0.639458
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −17712.0 −3.42579
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −660.000 −0.123907
\(306\) 0 0
\(307\) −3332.00 −0.619437 −0.309719 − 0.950828i \(-0.600235\pi\)
−0.309719 + 0.950828i \(0.600235\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4728.00 −0.862059 −0.431029 − 0.902338i \(-0.641849\pi\)
−0.431029 + 0.902338i \(0.641849\pi\)
\(312\) 0 0
\(313\) −5114.00 −0.923516 −0.461758 − 0.887006i \(-0.652781\pi\)
−0.461758 + 0.887006i \(0.652781\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7206.00 −1.27675 −0.638374 − 0.769726i \(-0.720393\pi\)
−0.638374 + 0.769726i \(0.720393\pi\)
\(318\) 0 0
\(319\) −2952.00 −0.518120
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2040.00 0.351420
\(324\) 0 0
\(325\) −7298.00 −1.24560
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6260.00 1.03952 0.519759 − 0.854313i \(-0.326022\pi\)
0.519759 + 0.854313i \(0.326022\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 840.000 0.136997
\(336\) 0 0
\(337\) −5326.00 −0.860907 −0.430454 − 0.902613i \(-0.641646\pi\)
−0.430454 + 0.902613i \(0.641646\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1344.00 0.213436
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −36.0000 −0.00556940 −0.00278470 − 0.999996i \(-0.500886\pi\)
−0.00278470 + 0.999996i \(0.500886\pi\)
\(348\) 0 0
\(349\) −3134.00 −0.480685 −0.240343 − 0.970688i \(-0.577260\pi\)
−0.240343 + 0.970688i \(0.577260\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1218.00 0.183648 0.0918238 − 0.995775i \(-0.470730\pi\)
0.0918238 + 0.995775i \(0.470730\pi\)
\(354\) 0 0
\(355\) 5040.00 0.753508
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10008.0 1.47131 0.735657 − 0.677354i \(-0.236873\pi\)
0.735657 + 0.677354i \(0.236873\pi\)
\(360\) 0 0
\(361\) −2235.00 −0.325849
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3300.00 0.473233
\(366\) 0 0
\(367\) 1072.00 0.152474 0.0762370 − 0.997090i \(-0.475709\pi\)
0.0762370 + 0.997090i \(0.475709\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −274.000 −0.0380353 −0.0190177 − 0.999819i \(-0.506054\pi\)
−0.0190177 + 0.999819i \(0.506054\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20172.0 −2.75573
\(378\) 0 0
\(379\) 7652.00 1.03709 0.518545 − 0.855051i \(-0.326474\pi\)
0.518545 + 0.855051i \(0.326474\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2160.00 0.288175 0.144087 − 0.989565i \(-0.453975\pi\)
0.144087 + 0.989565i \(0.453975\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1074.00 0.139984 0.0699922 − 0.997548i \(-0.477703\pi\)
0.0699922 + 0.997548i \(0.477703\pi\)
\(390\) 0 0
\(391\) 6480.00 0.838127
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1248.00 −0.158971
\(396\) 0 0
\(397\) −6926.00 −0.875582 −0.437791 − 0.899077i \(-0.644239\pi\)
−0.437791 + 0.899077i \(0.644239\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1938.00 −0.241344 −0.120672 − 0.992692i \(-0.538505\pi\)
−0.120672 + 0.992692i \(0.538505\pi\)
\(402\) 0 0
\(403\) 9184.00 1.13521
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1320.00 0.160762
\(408\) 0 0
\(409\) 9574.00 1.15747 0.578733 − 0.815517i \(-0.303547\pi\)
0.578733 + 0.815517i \(0.303547\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3096.00 0.366209
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5052.00 −0.589037 −0.294518 − 0.955646i \(-0.595159\pi\)
−0.294518 + 0.955646i \(0.595159\pi\)
\(420\) 0 0
\(421\) 3422.00 0.396147 0.198074 − 0.980187i \(-0.436531\pi\)
0.198074 + 0.980187i \(0.436531\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2670.00 0.304739
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2208.00 −0.246765 −0.123382 − 0.992359i \(-0.539374\pi\)
−0.123382 + 0.992359i \(0.539374\pi\)
\(432\) 0 0
\(433\) 6814.00 0.756259 0.378129 − 0.925753i \(-0.376567\pi\)
0.378129 + 0.925753i \(0.376567\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14688.0 1.60783
\(438\) 0 0
\(439\) −12584.0 −1.36811 −0.684056 − 0.729429i \(-0.739786\pi\)
−0.684056 + 0.729429i \(0.739786\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6996.00 −0.750316 −0.375158 − 0.926961i \(-0.622412\pi\)
−0.375158 + 0.926961i \(0.622412\pi\)
\(444\) 0 0
\(445\) −8388.00 −0.893549
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9474.00 −0.995781 −0.497891 − 0.867240i \(-0.665892\pi\)
−0.497891 + 0.867240i \(0.665892\pi\)
\(450\) 0 0
\(451\) −2952.00 −0.308213
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5786.00 0.592249 0.296124 − 0.955149i \(-0.404306\pi\)
0.296124 + 0.955149i \(0.404306\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3438.00 0.347340 0.173670 − 0.984804i \(-0.444437\pi\)
0.173670 + 0.984804i \(0.444437\pi\)
\(462\) 0 0
\(463\) 9392.00 0.942728 0.471364 − 0.881939i \(-0.343762\pi\)
0.471364 + 0.881939i \(0.343762\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4956.00 −0.491084 −0.245542 − 0.969386i \(-0.578966\pi\)
−0.245542 + 0.969386i \(0.578966\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2064.00 −0.200640
\(474\) 0 0
\(475\) 6052.00 0.584600
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20592.0 −1.96424 −0.982122 − 0.188248i \(-0.939719\pi\)
−0.982122 + 0.188248i \(0.939719\pi\)
\(480\) 0 0
\(481\) 9020.00 0.855045
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9516.00 −0.890926
\(486\) 0 0
\(487\) −13432.0 −1.24982 −0.624910 − 0.780697i \(-0.714864\pi\)
−0.624910 + 0.780697i \(0.714864\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14172.0 1.30259 0.651297 − 0.758823i \(-0.274225\pi\)
0.651297 + 0.758823i \(0.274225\pi\)
\(492\) 0 0
\(493\) 7380.00 0.674196
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −5956.00 −0.534323 −0.267162 − 0.963652i \(-0.586086\pi\)
−0.267162 + 0.963652i \(0.586086\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16968.0 1.50411 0.752053 − 0.659102i \(-0.229064\pi\)
0.752053 + 0.659102i \(0.229064\pi\)
\(504\) 0 0
\(505\) −7452.00 −0.656653
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5214.00 0.454040 0.227020 − 0.973890i \(-0.427102\pi\)
0.227020 + 0.973890i \(0.427102\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4080.00 −0.349100
\(516\) 0 0
\(517\) 2304.00 0.195996
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1398.00 −0.117558 −0.0587788 − 0.998271i \(-0.518721\pi\)
−0.0587788 + 0.998271i \(0.518721\pi\)
\(522\) 0 0
\(523\) 18580.0 1.55344 0.776718 − 0.629849i \(-0.216883\pi\)
0.776718 + 0.629849i \(0.216883\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3360.00 −0.277730
\(528\) 0 0
\(529\) 34489.0 2.83463
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −20172.0 −1.63930
\(534\) 0 0
\(535\) −5976.00 −0.482925
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18970.0 −1.50755 −0.753774 − 0.657133i \(-0.771769\pi\)
−0.753774 + 0.657133i \(0.771769\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8292.00 0.651725
\(546\) 0 0
\(547\) −16036.0 −1.25347 −0.626737 − 0.779231i \(-0.715610\pi\)
−0.626737 + 0.779231i \(0.715610\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 16728.0 1.29335
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8310.00 −0.632147 −0.316074 − 0.948735i \(-0.602365\pi\)
−0.316074 + 0.948735i \(0.602365\pi\)
\(558\) 0 0
\(559\) −14104.0 −1.06715
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7092.00 0.530892 0.265446 − 0.964126i \(-0.414481\pi\)
0.265446 + 0.964126i \(0.414481\pi\)
\(564\) 0 0
\(565\) 4500.00 0.335073
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7158.00 0.527380 0.263690 − 0.964608i \(-0.415060\pi\)
0.263690 + 0.964608i \(0.415060\pi\)
\(570\) 0 0
\(571\) 6500.00 0.476386 0.238193 − 0.971218i \(-0.423445\pi\)
0.238193 + 0.971218i \(0.423445\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 19224.0 1.39425
\(576\) 0 0
\(577\) −21794.0 −1.57244 −0.786218 − 0.617949i \(-0.787964\pi\)
−0.786218 + 0.617949i \(0.787964\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6696.00 −0.475678
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9756.00 0.685985 0.342993 − 0.939338i \(-0.388559\pi\)
0.342993 + 0.939338i \(0.388559\pi\)
\(588\) 0 0
\(589\) −7616.00 −0.532787
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5586.00 0.386829 0.193414 − 0.981117i \(-0.438044\pi\)
0.193414 + 0.981117i \(0.438044\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.00163708 0.000818542 − 1.00000i \(-0.499739\pi\)
0.000818542 1.00000i \(0.499739\pi\)
\(600\) 0 0
\(601\) −4298.00 −0.291712 −0.145856 − 0.989306i \(-0.546594\pi\)
−0.145856 + 0.989306i \(0.546594\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7122.00 −0.478596
\(606\) 0 0
\(607\) −8480.00 −0.567039 −0.283519 − 0.958966i \(-0.591502\pi\)
−0.283519 + 0.958966i \(0.591502\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15744.0 1.04245
\(612\) 0 0
\(613\) −1906.00 −0.125583 −0.0627917 − 0.998027i \(-0.520000\pi\)
−0.0627917 + 0.998027i \(0.520000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7482.00 −0.488191 −0.244096 − 0.969751i \(-0.578491\pi\)
−0.244096 + 0.969751i \(0.578491\pi\)
\(618\) 0 0
\(619\) 7348.00 0.477126 0.238563 − 0.971127i \(-0.423324\pi\)
0.238563 + 0.971127i \(0.423324\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3300.00 −0.209189
\(630\) 0 0
\(631\) 4520.00 0.285164 0.142582 − 0.989783i \(-0.454460\pi\)
0.142582 + 0.989783i \(0.454460\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1056.00 0.0659938
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19806.0 1.22042 0.610211 − 0.792239i \(-0.291085\pi\)
0.610211 + 0.792239i \(0.291085\pi\)
\(642\) 0 0
\(643\) 5020.00 0.307884 0.153942 − 0.988080i \(-0.450803\pi\)
0.153942 + 0.988080i \(0.450803\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28392.0 1.72520 0.862600 − 0.505886i \(-0.168834\pi\)
0.862600 + 0.505886i \(0.168834\pi\)
\(648\) 0 0
\(649\) 6480.00 0.391930
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17562.0 1.05246 0.526228 − 0.850343i \(-0.323606\pi\)
0.526228 + 0.850343i \(0.323606\pi\)
\(654\) 0 0
\(655\) −9288.00 −0.554064
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4716.00 −0.278770 −0.139385 − 0.990238i \(-0.544513\pi\)
−0.139385 + 0.990238i \(0.544513\pi\)
\(660\) 0 0
\(661\) 22762.0 1.33939 0.669697 − 0.742635i \(-0.266424\pi\)
0.669697 + 0.742635i \(0.266424\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 53136.0 3.08461
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1320.00 −0.0759434
\(672\) 0 0
\(673\) 4802.00 0.275042 0.137521 − 0.990499i \(-0.456086\pi\)
0.137521 + 0.990499i \(0.456086\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21558.0 1.22384 0.611921 − 0.790919i \(-0.290397\pi\)
0.611921 + 0.790919i \(0.290397\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −3780.00 −0.211768 −0.105884 − 0.994378i \(-0.533767\pi\)
−0.105884 + 0.994378i \(0.533767\pi\)
\(684\) 0 0
\(685\) −2268.00 −0.126505
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −45756.0 −2.52999
\(690\) 0 0
\(691\) 5500.00 0.302793 0.151396 − 0.988473i \(-0.451623\pi\)
0.151396 + 0.988473i \(0.451623\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15000.0 0.818680
\(696\) 0 0
\(697\) 7380.00 0.401058
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10230.0 −0.551187 −0.275593 − 0.961274i \(-0.588874\pi\)
−0.275593 + 0.961274i \(0.588874\pi\)
\(702\) 0 0
\(703\) −7480.00 −0.401299
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10190.0 0.539765 0.269883 − 0.962893i \(-0.413015\pi\)
0.269883 + 0.962893i \(0.413015\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24192.0 −1.27068
\(714\) 0 0
\(715\) 5904.00 0.308807
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9408.00 0.487982 0.243991 − 0.969777i \(-0.421543\pi\)
0.243991 + 0.969777i \(0.421543\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21894.0 1.12155
\(726\) 0 0
\(727\) 33064.0 1.68676 0.843381 − 0.537316i \(-0.180562\pi\)
0.843381 + 0.537316i \(0.180562\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5160.00 0.261080
\(732\) 0 0
\(733\) 6322.00 0.318565 0.159283 − 0.987233i \(-0.449082\pi\)
0.159283 + 0.987233i \(0.449082\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1680.00 0.0839669
\(738\) 0 0
\(739\) −20740.0 −1.03239 −0.516193 − 0.856472i \(-0.672651\pi\)
−0.516193 + 0.856472i \(0.672651\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32040.0 −1.58201 −0.791005 − 0.611810i \(-0.790442\pi\)
−0.791005 + 0.611810i \(0.790442\pi\)
\(744\) 0 0
\(745\) −5076.00 −0.249624
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −12832.0 −0.623497 −0.311749 − 0.950165i \(-0.600915\pi\)
−0.311749 + 0.950165i \(0.600915\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −15216.0 −0.733466
\(756\) 0 0
\(757\) −19906.0 −0.955741 −0.477870 − 0.878430i \(-0.658591\pi\)
−0.477870 + 0.878430i \(0.658591\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10842.0 0.516455 0.258227 − 0.966084i \(-0.416862\pi\)
0.258227 + 0.966084i \(0.416862\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 44280.0 2.08456
\(768\) 0 0
\(769\) −28274.0 −1.32586 −0.662930 − 0.748681i \(-0.730687\pi\)
−0.662930 + 0.748681i \(0.730687\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −32346.0 −1.50505 −0.752526 − 0.658563i \(-0.771165\pi\)
−0.752526 + 0.658563i \(0.771165\pi\)
\(774\) 0 0
\(775\) −9968.00 −0.462014
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 16728.0 0.769375
\(780\) 0 0
\(781\) 10080.0 0.461832
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7116.00 0.323543
\(786\) 0 0
\(787\) −30116.0 −1.36407 −0.682033 − 0.731322i \(-0.738904\pi\)
−0.682033 + 0.731322i \(0.738904\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9020.00 −0.403921
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6594.00 −0.293063 −0.146532 − 0.989206i \(-0.546811\pi\)
−0.146532 + 0.989206i \(0.546811\pi\)
\(798\) 0 0
\(799\) −5760.00 −0.255036
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6600.00 0.290048
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 43014.0 1.86933 0.934667 − 0.355524i \(-0.115697\pi\)
0.934667 + 0.355524i \(0.115697\pi\)
\(810\) 0 0
\(811\) 14164.0 0.613274 0.306637 − 0.951827i \(-0.400796\pi\)
0.306637 + 0.951827i \(0.400796\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12648.0 0.543608
\(816\) 0 0
\(817\) 11696.0 0.500846
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −34830.0 −1.48060 −0.740302 − 0.672275i \(-0.765317\pi\)
−0.740302 + 0.672275i \(0.765317\pi\)
\(822\) 0 0
\(823\) 31016.0 1.31367 0.656835 − 0.754035i \(-0.271895\pi\)
0.656835 + 0.754035i \(0.271895\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9876.00 −0.415263 −0.207631 − 0.978207i \(-0.566575\pi\)
−0.207631 + 0.978207i \(0.566575\pi\)
\(828\) 0 0
\(829\) 3154.00 0.132139 0.0660693 − 0.997815i \(-0.478954\pi\)
0.0660693 + 0.997815i \(0.478954\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −11664.0 −0.483412
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36936.0 1.51987 0.759936 − 0.649998i \(-0.225230\pi\)
0.759936 + 0.649998i \(0.225230\pi\)
\(840\) 0 0
\(841\) 36127.0 1.48128
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 27162.0 1.10580
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −23760.0 −0.957088
\(852\) 0 0
\(853\) −9638.00 −0.386869 −0.193434 − 0.981113i \(-0.561963\pi\)
−0.193434 + 0.981113i \(0.561963\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10266.0 0.409195 0.204597 − 0.978846i \(-0.434411\pi\)
0.204597 + 0.978846i \(0.434411\pi\)
\(858\) 0 0
\(859\) 4084.00 0.162217 0.0811084 − 0.996705i \(-0.474154\pi\)
0.0811084 + 0.996705i \(0.474154\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 192.000 0.00757330 0.00378665 − 0.999993i \(-0.498795\pi\)
0.00378665 + 0.999993i \(0.498795\pi\)
\(864\) 0 0
\(865\) −8172.00 −0.321221
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2496.00 −0.0974350
\(870\) 0 0
\(871\) 11480.0 0.446596
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19910.0 0.766605 0.383303 − 0.923623i \(-0.374787\pi\)
0.383303 + 0.923623i \(0.374787\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 14802.0 0.566052 0.283026 − 0.959112i \(-0.408662\pi\)
0.283026 + 0.959112i \(0.408662\pi\)
\(882\) 0 0
\(883\) −32548.0 −1.24046 −0.620231 − 0.784419i \(-0.712961\pi\)
−0.620231 + 0.784419i \(0.712961\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1464.00 0.0554186 0.0277093 − 0.999616i \(-0.491179\pi\)
0.0277093 + 0.999616i \(0.491179\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −13056.0 −0.489252
\(894\) 0 0
\(895\) −9576.00 −0.357643
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −27552.0 −1.02215
\(900\) 0 0
\(901\) 16740.0 0.618968
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10140.0 0.372448
\(906\) 0 0
\(907\) −49564.0 −1.81449 −0.907247 − 0.420599i \(-0.861820\pi\)
−0.907247 + 0.420599i \(0.861820\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8448.00 −0.307239 −0.153619 − 0.988130i \(-0.549093\pi\)
−0.153619 + 0.988130i \(0.549093\pi\)
\(912\) 0 0
\(913\) 6192.00 0.224453
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 14600.0 0.524058 0.262029 − 0.965060i \(-0.415608\pi\)
0.262029 + 0.965060i \(0.415608\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 68880.0 2.45635
\(924\) 0 0
\(925\) −9790.00 −0.347993
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −21102.0 −0.745247 −0.372623 − 0.927983i \(-0.621542\pi\)
−0.372623 + 0.927983i \(0.621542\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2160.00 −0.0755503
\(936\) 0 0
\(937\) 20806.0 0.725403 0.362701 − 0.931905i \(-0.381854\pi\)
0.362701 + 0.931905i \(0.381854\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 24510.0 0.849100 0.424550 − 0.905404i \(-0.360432\pi\)
0.424550 + 0.905404i \(0.360432\pi\)
\(942\) 0 0
\(943\) 53136.0 1.83494
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 44148.0 1.51491 0.757454 − 0.652889i \(-0.226443\pi\)
0.757454 + 0.652889i \(0.226443\pi\)
\(948\) 0 0
\(949\) 45100.0 1.54268
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −27114.0 −0.921625 −0.460812 − 0.887498i \(-0.652442\pi\)
−0.460812 + 0.887498i \(0.652442\pi\)
\(954\) 0 0
\(955\) −21312.0 −0.722136
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −17247.0 −0.578933
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −16116.0 −0.537609
\(966\) 0 0
\(967\) −10264.0 −0.341332 −0.170666 − 0.985329i \(-0.554592\pi\)
−0.170666 + 0.985329i \(0.554592\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 51468.0 1.70102 0.850508 − 0.525962i \(-0.176295\pi\)
0.850508 + 0.525962i \(0.176295\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23790.0 0.779027 0.389514 − 0.921021i \(-0.372643\pi\)
0.389514 + 0.921021i \(0.372643\pi\)
\(978\) 0 0
\(979\) −16776.0 −0.547664
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 26424.0 0.857370 0.428685 − 0.903454i \(-0.358977\pi\)
0.428685 + 0.903454i \(0.358977\pi\)
\(984\) 0 0
\(985\) 8460.00 0.273663
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 37152.0 1.19450
\(990\) 0 0
\(991\) 39488.0 1.26577 0.632885 − 0.774246i \(-0.281871\pi\)
0.632885 + 0.774246i \(0.281871\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17808.0 0.567388
\(996\) 0 0
\(997\) −30854.0 −0.980096 −0.490048 − 0.871695i \(-0.663021\pi\)
−0.490048 + 0.871695i \(0.663021\pi\)
\(998\) 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 1764.4.a.k.1.1 1
3.2 odd 2 196.4.a.b.1.1 1
7.2 even 3 1764.4.k.e.361.1 2
7.3 odd 6 1764.4.k.k.1549.1 2
7.4 even 3 1764.4.k.e.1549.1 2
7.5 odd 6 1764.4.k.k.361.1 2
7.6 odd 2 252.4.a.c.1.1 1
12.11 even 2 784.4.a.n.1.1 1
21.2 odd 6 196.4.e.d.165.1 2
21.5 even 6 196.4.e.c.165.1 2
21.11 odd 6 196.4.e.d.177.1 2
21.17 even 6 196.4.e.c.177.1 2
21.20 even 2 28.4.a.b.1.1 ✓ 1
28.27 even 2 1008.4.a.f.1.1 1
84.83 odd 2 112.4.a.c.1.1 1
105.62 odd 4 700.4.e.f.449.1 2
105.83 odd 4 700.4.e.f.449.2 2
105.104 even 2 700.4.a.e.1.1 1
168.83 odd 2 448.4.a.m.1.1 1
168.125 even 2 448.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.4.a.b.1.1 ✓ 1 21.20 even 2
112.4.a.c.1.1 1 84.83 odd 2
196.4.a.b.1.1 1 3.2 odd 2
196.4.e.c.165.1 2 21.5 even 6
196.4.e.c.177.1 2 21.17 even 6
196.4.e.d.165.1 2 21.2 odd 6
196.4.e.d.177.1 2 21.11 odd 6
252.4.a.c.1.1 1 7.6 odd 2
448.4.a.d.1.1 1 168.125 even 2
448.4.a.m.1.1 1 168.83 odd 2
700.4.a.e.1.1 1 105.104 even 2
700.4.e.f.449.1 2 105.62 odd 4
700.4.e.f.449.2 2 105.83 odd 4
784.4.a.n.1.1 1 12.11 even 2
1008.4.a.f.1.1 1 28.27 even 2
1764.4.a.k.1.1 1 1.1 even 1 trivial
1764.4.k.e.361.1 2 7.2 even 3
1764.4.k.e.1549.1 2 7.4 even 3
1764.4.k.k.361.1 2 7.5 odd 6
1764.4.k.k.1549.1 2 7.3 odd 6