↑

Properties

Label 1296.4.a.g.1.1
Level $1296$
Weight $4$
Character 1296.1
Self dual yes
Analytic conductor $76.466$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,4,Mod(1,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, 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("1296.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.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: \(76.4664753674\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1296.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.00000 q^{5} +31.0000 q^{7} +O(q^{10})\) \(q+9.00000 q^{5} +31.0000 q^{7} -15.0000 q^{11} -37.0000 q^{13} +42.0000 q^{17} +28.0000 q^{19} +195.000 q^{23} -44.0000 q^{25} -111.000 q^{29} +205.000 q^{31} +279.000 q^{35} -166.000 q^{37} +261.000 q^{41} +43.0000 q^{43} +177.000 q^{47} +618.000 q^{49} -114.000 q^{53} -135.000 q^{55} +159.000 q^{59} +191.000 q^{61} -333.000 q^{65} +421.000 q^{67} +156.000 q^{71} +182.000 q^{73} -465.000 q^{77} -1133.00 q^{79} -1083.00 q^{83} +378.000 q^{85} +1050.00 q^{89} -1147.00 q^{91} +252.000 q^{95} -901.000 q^{97} +O(q^{100})\)

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
\(5\) 9.00000 0.804984 0.402492 − 0.915423i \(-0.368144\pi\)
0.402492 + 0.915423i \(0.368144\pi\)
\(6\) 0 0
\(7\) 31.0000 1.67384 0.836921 − 0.547323i \(-0.184353\pi\)
0.836921 + 0.547323i \(0.184353\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −15.0000 −0.411152 −0.205576 − 0.978641i \(-0.565907\pi\)
−0.205576 + 0.978641i \(0.565907\pi\)
\(12\) 0 0
\(13\) −37.0000 −0.789381 −0.394691 − 0.918814i \(-0.629148\pi\)
−0.394691 + 0.918814i \(0.629148\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 42.0000 0.599206 0.299603 − 0.954064i \(-0.403146\pi\)
0.299603 + 0.954064i \(0.403146\pi\)
\(18\) 0 0
\(19\) 28.0000 0.338086 0.169043 − 0.985609i \(-0.445932\pi\)
0.169043 + 0.985609i \(0.445932\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 195.000 1.76784 0.883920 − 0.467639i \(-0.154895\pi\)
0.883920 + 0.467639i \(0.154895\pi\)
\(24\) 0 0
\(25\) −44.0000 −0.352000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −111.000 −0.710765 −0.355382 − 0.934721i \(-0.615649\pi\)
−0.355382 + 0.934721i \(0.615649\pi\)
\(30\) 0 0
\(31\) 205.000 1.18771 0.593856 − 0.804571i \(-0.297605\pi\)
0.593856 + 0.804571i \(0.297605\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 279.000 1.34742
\(36\) 0 0
\(37\) −166.000 −0.737574 −0.368787 − 0.929514i \(-0.620227\pi\)
−0.368787 + 0.929514i \(0.620227\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 261.000 0.994179 0.497090 − 0.867699i \(-0.334402\pi\)
0.497090 + 0.867699i \(0.334402\pi\)
\(42\) 0 0
\(43\) 43.0000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 177.000 0.549321 0.274661 − 0.961541i \(-0.411434\pi\)
0.274661 + 0.961541i \(0.411434\pi\)
\(48\) 0 0
\(49\) 618.000 1.80175
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −114.000 −0.295455 −0.147727 − 0.989028i \(-0.547196\pi\)
−0.147727 + 0.989028i \(0.547196\pi\)
\(54\) 0 0
\(55\) −135.000 −0.330971
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 159.000 0.350848 0.175424 − 0.984493i \(-0.443870\pi\)
0.175424 + 0.984493i \(0.443870\pi\)
\(60\) 0 0
\(61\) 191.000 0.400902 0.200451 − 0.979704i \(-0.435759\pi\)
0.200451 + 0.979704i \(0.435759\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −333.000 −0.635439
\(66\) 0 0
\(67\) 421.000 0.767662 0.383831 − 0.923403i \(-0.374605\pi\)
0.383831 + 0.923403i \(0.374605\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 156.000 0.260758 0.130379 − 0.991464i \(-0.458381\pi\)
0.130379 + 0.991464i \(0.458381\pi\)
\(72\) 0 0
\(73\) 182.000 0.291801 0.145901 − 0.989299i \(-0.453392\pi\)
0.145901 + 0.989299i \(0.453392\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −465.000 −0.688203
\(78\) 0 0
\(79\) −1133.00 −1.61358 −0.806788 − 0.590841i \(-0.798796\pi\)
−0.806788 + 0.590841i \(0.798796\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1083.00 −1.43223 −0.716113 − 0.697985i \(-0.754080\pi\)
−0.716113 + 0.697985i \(0.754080\pi\)
\(84\) 0 0
\(85\) 378.000 0.482351
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1050.00 1.25056 0.625280 − 0.780401i \(-0.284985\pi\)
0.625280 + 0.780401i \(0.284985\pi\)
\(90\) 0 0
\(91\) −1147.00 −1.32130
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 252.000 0.272154
\(96\) 0 0
\(97\) −901.000 −0.943121 −0.471560 − 0.881834i \(-0.656309\pi\)
−0.471560 + 0.881834i \(0.656309\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −387.000 −0.381267 −0.190633 − 0.981661i \(-0.561054\pi\)
−0.190633 + 0.981661i \(0.561054\pi\)
\(102\) 0 0
\(103\) −551.000 −0.527103 −0.263552 − 0.964645i \(-0.584894\pi\)
−0.263552 + 0.964645i \(0.584894\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −0.0108419 −0.00542095 − 0.999985i \(-0.501726\pi\)
−0.00542095 + 0.999985i \(0.501726\pi\)
\(108\) 0 0
\(109\) −502.000 −0.441127 −0.220564 − 0.975373i \(-0.570790\pi\)
−0.220564 + 0.975373i \(0.570790\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1401.00 1.16633 0.583164 − 0.812355i \(-0.301815\pi\)
0.583164 + 0.812355i \(0.301815\pi\)
\(114\) 0 0
\(115\) 1755.00 1.42308
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1302.00 1.00298
\(120\) 0 0
\(121\) −1106.00 −0.830954
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1521.00 −1.08834
\(126\) 0 0
\(127\) 880.000 0.614861 0.307431 − 0.951571i \(-0.400531\pi\)
0.307431 + 0.951571i \(0.400531\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1503.00 1.00243 0.501213 − 0.865324i \(-0.332887\pi\)
0.501213 + 0.865324i \(0.332887\pi\)
\(132\) 0 0
\(133\) 868.000 0.565903
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2661.00 1.65945 0.829725 − 0.558173i \(-0.188497\pi\)
0.829725 + 0.558173i \(0.188497\pi\)
\(138\) 0 0
\(139\) 121.000 0.0738352 0.0369176 − 0.999318i \(-0.488246\pi\)
0.0369176 + 0.999318i \(0.488246\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 555.000 0.324555
\(144\) 0 0
\(145\) −999.000 −0.572155
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2829.00 1.55544 0.777721 − 0.628610i \(-0.216376\pi\)
0.777721 + 0.628610i \(0.216376\pi\)
\(150\) 0 0
\(151\) −461.000 −0.248448 −0.124224 − 0.992254i \(-0.539644\pi\)
−0.124224 + 0.992254i \(0.539644\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1845.00 0.956090
\(156\) 0 0
\(157\) −2977.00 −1.51332 −0.756658 − 0.653811i \(-0.773169\pi\)
−0.756658 + 0.653811i \(0.773169\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6045.00 2.95909
\(162\) 0 0
\(163\) 3316.00 1.59343 0.796715 − 0.604355i \(-0.206569\pi\)
0.796715 + 0.604355i \(0.206569\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −681.000 −0.315553 −0.157777 − 0.987475i \(-0.550433\pi\)
−0.157777 + 0.987475i \(0.550433\pi\)
\(168\) 0 0
\(169\) −828.000 −0.376878
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3981.00 1.74954 0.874768 − 0.484541i \(-0.161014\pi\)
0.874768 + 0.484541i \(0.161014\pi\)
\(174\) 0 0
\(175\) −1364.00 −0.589193
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2004.00 0.836793 0.418397 − 0.908264i \(-0.362592\pi\)
0.418397 + 0.908264i \(0.362592\pi\)
\(180\) 0 0
\(181\) 1274.00 0.523181 0.261590 − 0.965179i \(-0.415753\pi\)
0.261590 + 0.965179i \(0.415753\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1494.00 −0.593735
\(186\) 0 0
\(187\) −630.000 −0.246365
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1161.00 0.439827 0.219914 − 0.975519i \(-0.429422\pi\)
0.219914 + 0.975519i \(0.429422\pi\)
\(192\) 0 0
\(193\) 3611.00 1.34676 0.673382 − 0.739295i \(-0.264841\pi\)
0.673382 + 0.739295i \(0.264841\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2046.00 −0.739957 −0.369978 − 0.929040i \(-0.620635\pi\)
−0.369978 + 0.929040i \(0.620635\pi\)
\(198\) 0 0
\(199\) −2996.00 −1.06724 −0.533620 − 0.845724i \(-0.679169\pi\)
−0.533620 + 0.845724i \(0.679169\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3441.00 −1.18971
\(204\) 0 0
\(205\) 2349.00 0.800299
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −420.000 −0.139005
\(210\) 0 0
\(211\) −755.000 −0.246333 −0.123167 − 0.992386i \(-0.539305\pi\)
−0.123167 + 0.992386i \(0.539305\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 387.000 0.122759
\(216\) 0 0
\(217\) 6355.00 1.98804
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1554.00 −0.473002
\(222\) 0 0
\(223\) 3463.00 1.03991 0.519954 − 0.854194i \(-0.325949\pi\)
0.519954 + 0.854194i \(0.325949\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6225.00 1.82012 0.910061 − 0.414474i \(-0.136034\pi\)
0.910061 + 0.414474i \(0.136034\pi\)
\(228\) 0 0
\(229\) −1465.00 −0.422751 −0.211375 − 0.977405i \(-0.567794\pi\)
−0.211375 + 0.977405i \(0.567794\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2634.00 −0.740597 −0.370298 − 0.928913i \(-0.620745\pi\)
−0.370298 + 0.928913i \(0.620745\pi\)
\(234\) 0 0
\(235\) 1593.00 0.442195
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6915.00 1.87152 0.935762 − 0.352633i \(-0.114713\pi\)
0.935762 + 0.352633i \(0.114713\pi\)
\(240\) 0 0
\(241\) −1489.00 −0.397987 −0.198994 − 0.980001i \(-0.563767\pi\)
−0.198994 + 0.980001i \(0.563767\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5562.00 1.45038
\(246\) 0 0
\(247\) −1036.00 −0.266879
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4620.00 −1.16180 −0.580900 − 0.813975i \(-0.697299\pi\)
−0.580900 + 0.813975i \(0.697299\pi\)
\(252\) 0 0
\(253\) −2925.00 −0.726850
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3351.00 −0.813345 −0.406672 − 0.913574i \(-0.633311\pi\)
−0.406672 + 0.913574i \(0.633311\pi\)
\(258\) 0 0
\(259\) −5146.00 −1.23458
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −603.000 −0.141379 −0.0706893 − 0.997498i \(-0.522520\pi\)
−0.0706893 + 0.997498i \(0.522520\pi\)
\(264\) 0 0
\(265\) −1026.00 −0.237837
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1470.00 0.333188 0.166594 − 0.986026i \(-0.446723\pi\)
0.166594 + 0.986026i \(0.446723\pi\)
\(270\) 0 0
\(271\) −2072.00 −0.464447 −0.232223 − 0.972662i \(-0.574600\pi\)
−0.232223 + 0.972662i \(0.574600\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 660.000 0.144725
\(276\) 0 0
\(277\) 7139.00 1.54852 0.774262 − 0.632866i \(-0.218121\pi\)
0.774262 + 0.632866i \(0.218121\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8427.00 −1.78901 −0.894507 − 0.447055i \(-0.852473\pi\)
−0.894507 + 0.447055i \(0.852473\pi\)
\(282\) 0 0
\(283\) 457.000 0.0959923 0.0479962 − 0.998848i \(-0.484716\pi\)
0.0479962 + 0.998848i \(0.484716\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8091.00 1.66410
\(288\) 0 0
\(289\) −3149.00 −0.640953
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5889.00 1.17419 0.587097 − 0.809516i \(-0.300271\pi\)
0.587097 + 0.809516i \(0.300271\pi\)
\(294\) 0 0
\(295\) 1431.00 0.282427
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7215.00 −1.39550
\(300\) 0 0
\(301\) 1333.00 0.255259
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1719.00 0.322720
\(306\) 0 0
\(307\) 1204.00 0.223830 0.111915 − 0.993718i \(-0.464302\pi\)
0.111915 + 0.993718i \(0.464302\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3285.00 −0.598956 −0.299478 − 0.954103i \(-0.596812\pi\)
−0.299478 + 0.954103i \(0.596812\pi\)
\(312\) 0 0
\(313\) −10057.0 −1.81615 −0.908075 − 0.418807i \(-0.862449\pi\)
−0.908075 + 0.418807i \(0.862449\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2295.00 −0.406625 −0.203312 − 0.979114i \(-0.565171\pi\)
−0.203312 + 0.979114i \(0.565171\pi\)
\(318\) 0 0
\(319\) 1665.00 0.292232
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1176.00 0.202583
\(324\) 0 0
\(325\) 1628.00 0.277862
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5487.00 0.919478
\(330\) 0 0
\(331\) 6679.00 1.10910 0.554548 − 0.832151i \(-0.312891\pi\)
0.554548 + 0.832151i \(0.312891\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3789.00 0.617956
\(336\) 0 0
\(337\) 2183.00 0.352865 0.176433 − 0.984313i \(-0.443544\pi\)
0.176433 + 0.984313i \(0.443544\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3075.00 −0.488330
\(342\) 0 0
\(343\) 8525.00 1.34200
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3891.00 0.601959 0.300980 − 0.953631i \(-0.402686\pi\)
0.300980 + 0.953631i \(0.402686\pi\)
\(348\) 0 0
\(349\) 2795.00 0.428690 0.214345 − 0.976758i \(-0.431238\pi\)
0.214345 + 0.976758i \(0.431238\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4755.00 −0.716949 −0.358475 − 0.933539i \(-0.616703\pi\)
−0.358475 + 0.933539i \(0.616703\pi\)
\(354\) 0 0
\(355\) 1404.00 0.209906
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4608.00 0.677440 0.338720 − 0.940887i \(-0.390006\pi\)
0.338720 + 0.940887i \(0.390006\pi\)
\(360\) 0 0
\(361\) −6075.00 −0.885698
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1638.00 0.234895
\(366\) 0 0
\(367\) −3845.00 −0.546887 −0.273443 − 0.961888i \(-0.588163\pi\)
−0.273443 + 0.961888i \(0.588163\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3534.00 −0.494545
\(372\) 0 0
\(373\) −8317.00 −1.15453 −0.577263 − 0.816559i \(-0.695879\pi\)
−0.577263 + 0.816559i \(0.695879\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4107.00 0.561064
\(378\) 0 0
\(379\) −12560.0 −1.70228 −0.851140 − 0.524939i \(-0.824088\pi\)
−0.851140 + 0.524939i \(0.824088\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12087.0 1.61258 0.806288 − 0.591523i \(-0.201473\pi\)
0.806288 + 0.591523i \(0.201473\pi\)
\(384\) 0 0
\(385\) −4185.00 −0.553993
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8541.00 1.11323 0.556614 − 0.830771i \(-0.312100\pi\)
0.556614 + 0.830771i \(0.312100\pi\)
\(390\) 0 0
\(391\) 8190.00 1.05930
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10197.0 −1.29890
\(396\) 0 0
\(397\) −13174.0 −1.66545 −0.832726 − 0.553686i \(-0.813221\pi\)
−0.832726 + 0.553686i \(0.813221\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9603.00 −1.19589 −0.597944 − 0.801538i \(-0.704015\pi\)
−0.597944 + 0.801538i \(0.704015\pi\)
\(402\) 0 0
\(403\) −7585.00 −0.937558
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2490.00 0.303255
\(408\) 0 0
\(409\) 11471.0 1.38681 0.693404 − 0.720549i \(-0.256110\pi\)
0.693404 + 0.720549i \(0.256110\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4929.00 0.587264
\(414\) 0 0
\(415\) −9747.00 −1.15292
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5973.00 −0.696420 −0.348210 − 0.937416i \(-0.613210\pi\)
−0.348210 + 0.937416i \(0.613210\pi\)
\(420\) 0 0
\(421\) −8905.00 −1.03089 −0.515443 − 0.856924i \(-0.672373\pi\)
−0.515443 + 0.856924i \(0.672373\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1848.00 −0.210920
\(426\) 0 0
\(427\) 5921.00 0.671047
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1416.00 0.158251 0.0791257 − 0.996865i \(-0.474787\pi\)
0.0791257 + 0.996865i \(0.474787\pi\)
\(432\) 0 0
\(433\) 10766.0 1.19488 0.597438 − 0.801915i \(-0.296186\pi\)
0.597438 + 0.801915i \(0.296186\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5460.00 0.597682
\(438\) 0 0
\(439\) −4349.00 −0.472817 −0.236408 − 0.971654i \(-0.575970\pi\)
−0.236408 + 0.971654i \(0.575970\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14547.0 −1.56016 −0.780078 − 0.625683i \(-0.784820\pi\)
−0.780078 + 0.625683i \(0.784820\pi\)
\(444\) 0 0
\(445\) 9450.00 1.00668
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3330.00 0.350005 0.175003 − 0.984568i \(-0.444007\pi\)
0.175003 + 0.984568i \(0.444007\pi\)
\(450\) 0 0
\(451\) −3915.00 −0.408759
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10323.0 −1.06363
\(456\) 0 0
\(457\) 8147.00 0.833918 0.416959 − 0.908925i \(-0.363096\pi\)
0.416959 + 0.908925i \(0.363096\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8031.00 −0.811369 −0.405684 − 0.914013i \(-0.632967\pi\)
−0.405684 + 0.914013i \(0.632967\pi\)
\(462\) 0 0
\(463\) −4283.00 −0.429909 −0.214955 − 0.976624i \(-0.568960\pi\)
−0.214955 + 0.976624i \(0.568960\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5460.00 0.541025 0.270512 − 0.962716i \(-0.412807\pi\)
0.270512 + 0.962716i \(0.412807\pi\)
\(468\) 0 0
\(469\) 13051.0 1.28494
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −645.000 −0.0627001
\(474\) 0 0
\(475\) −1232.00 −0.119006
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 429.000 0.0409217 0.0204609 − 0.999791i \(-0.493487\pi\)
0.0204609 + 0.999791i \(0.493487\pi\)
\(480\) 0 0
\(481\) 6142.00 0.582227
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8109.00 −0.759197
\(486\) 0 0
\(487\) 11296.0 1.05107 0.525535 − 0.850772i \(-0.323865\pi\)
0.525535 + 0.850772i \(0.323865\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −14673.0 −1.34864 −0.674321 − 0.738438i \(-0.735564\pi\)
−0.674321 + 0.738438i \(0.735564\pi\)
\(492\) 0 0
\(493\) −4662.00 −0.425894
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4836.00 0.436467
\(498\) 0 0
\(499\) −13439.0 −1.20564 −0.602818 − 0.797879i \(-0.705955\pi\)
−0.602818 + 0.797879i \(0.705955\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17388.0 −1.54134 −0.770669 − 0.637236i \(-0.780078\pi\)
−0.770669 + 0.637236i \(0.780078\pi\)
\(504\) 0 0
\(505\) −3483.00 −0.306914
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3789.00 0.329950 0.164975 − 0.986298i \(-0.447246\pi\)
0.164975 + 0.986298i \(0.447246\pi\)
\(510\) 0 0
\(511\) 5642.00 0.488429
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4959.00 −0.424310
\(516\) 0 0
\(517\) −2655.00 −0.225854
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9786.00 −0.822903 −0.411451 − 0.911432i \(-0.634978\pi\)
−0.411451 + 0.911432i \(0.634978\pi\)
\(522\) 0 0
\(523\) 8008.00 0.669532 0.334766 − 0.942301i \(-0.391343\pi\)
0.334766 + 0.942301i \(0.391343\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8610.00 0.711684
\(528\) 0 0
\(529\) 25858.0 2.12526
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −9657.00 −0.784786
\(534\) 0 0
\(535\) −108.000 −0.00872756
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9270.00 −0.740793
\(540\) 0 0
\(541\) −2938.00 −0.233483 −0.116742 − 0.993162i \(-0.537245\pi\)
−0.116742 + 0.993162i \(0.537245\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4518.00 −0.355101
\(546\) 0 0
\(547\) 10375.0 0.810974 0.405487 − 0.914101i \(-0.367102\pi\)
0.405487 + 0.914101i \(0.367102\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3108.00 −0.240300
\(552\) 0 0
\(553\) −35123.0 −2.70087
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3306.00 −0.251490 −0.125745 − 0.992063i \(-0.540132\pi\)
−0.125745 + 0.992063i \(0.540132\pi\)
\(558\) 0 0
\(559\) −1591.00 −0.120379
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21093.0 −1.57898 −0.789488 − 0.613765i \(-0.789654\pi\)
−0.789488 + 0.613765i \(0.789654\pi\)
\(564\) 0 0
\(565\) 12609.0 0.938875
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1287.00 −0.0948222 −0.0474111 − 0.998875i \(-0.515097\pi\)
−0.0474111 + 0.998875i \(0.515097\pi\)
\(570\) 0 0
\(571\) −15035.0 −1.10192 −0.550959 − 0.834532i \(-0.685738\pi\)
−0.550959 + 0.834532i \(0.685738\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8580.00 −0.622280
\(576\) 0 0
\(577\) 1190.00 0.0858585 0.0429292 − 0.999078i \(-0.486331\pi\)
0.0429292 + 0.999078i \(0.486331\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −33573.0 −2.39732
\(582\) 0 0
\(583\) 1710.00 0.121477
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17883.0 −1.25743 −0.628714 − 0.777637i \(-0.716418\pi\)
−0.628714 + 0.777637i \(0.716418\pi\)
\(588\) 0 0
\(589\) 5740.00 0.401549
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20118.0 −1.39317 −0.696583 − 0.717476i \(-0.745297\pi\)
−0.696583 + 0.717476i \(0.745297\pi\)
\(594\) 0 0
\(595\) 11718.0 0.807380
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1065.00 −0.0726456 −0.0363228 − 0.999340i \(-0.511564\pi\)
−0.0363228 + 0.999340i \(0.511564\pi\)
\(600\) 0 0
\(601\) −20725.0 −1.40664 −0.703320 − 0.710874i \(-0.748300\pi\)
−0.703320 + 0.710874i \(0.748300\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9954.00 −0.668905
\(606\) 0 0
\(607\) 15745.0 1.05283 0.526417 − 0.850227i \(-0.323535\pi\)
0.526417 + 0.850227i \(0.323535\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6549.00 −0.433624
\(612\) 0 0
\(613\) 5042.00 0.332210 0.166105 − 0.986108i \(-0.446881\pi\)
0.166105 + 0.986108i \(0.446881\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10053.0 0.655946 0.327973 − 0.944687i \(-0.393635\pi\)
0.327973 + 0.944687i \(0.393635\pi\)
\(618\) 0 0
\(619\) 5983.00 0.388493 0.194246 − 0.980953i \(-0.437774\pi\)
0.194246 + 0.980953i \(0.437774\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 32550.0 2.09324
\(624\) 0 0
\(625\) −8189.00 −0.524096
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6972.00 −0.441958
\(630\) 0 0
\(631\) 19696.0 1.24261 0.621304 − 0.783570i \(-0.286603\pi\)
0.621304 + 0.783570i \(0.286603\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7920.00 0.494954
\(636\) 0 0
\(637\) −22866.0 −1.42227
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10977.0 0.676389 0.338195 − 0.941076i \(-0.390184\pi\)
0.338195 + 0.941076i \(0.390184\pi\)
\(642\) 0 0
\(643\) 15829.0 0.970816 0.485408 − 0.874288i \(-0.338671\pi\)
0.485408 + 0.874288i \(0.338671\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28224.0 1.71499 0.857496 − 0.514490i \(-0.172019\pi\)
0.857496 + 0.514490i \(0.172019\pi\)
\(648\) 0 0
\(649\) −2385.00 −0.144252
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −28167.0 −1.68799 −0.843997 − 0.536348i \(-0.819803\pi\)
−0.843997 + 0.536348i \(0.819803\pi\)
\(654\) 0 0
\(655\) 13527.0 0.806937
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10737.0 0.634680 0.317340 − 0.948312i \(-0.397210\pi\)
0.317340 + 0.948312i \(0.397210\pi\)
\(660\) 0 0
\(661\) 10127.0 0.595907 0.297954 − 0.954580i \(-0.403696\pi\)
0.297954 + 0.954580i \(0.403696\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7812.00 0.455543
\(666\) 0 0
\(667\) −21645.0 −1.25652
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2865.00 −0.164832
\(672\) 0 0
\(673\) 251.000 0.0143764 0.00718822 − 0.999974i \(-0.497712\pi\)
0.00718822 + 0.999974i \(0.497712\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8451.00 −0.479761 −0.239881 − 0.970802i \(-0.577108\pi\)
−0.239881 + 0.970802i \(0.577108\pi\)
\(678\) 0 0
\(679\) −27931.0 −1.57864
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −25884.0 −1.45011 −0.725054 − 0.688692i \(-0.758185\pi\)
−0.725054 + 0.688692i \(0.758185\pi\)
\(684\) 0 0
\(685\) 23949.0 1.33583
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4218.00 0.233226
\(690\) 0 0
\(691\) −6365.00 −0.350414 −0.175207 − 0.984532i \(-0.556059\pi\)
−0.175207 + 0.984532i \(0.556059\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1089.00 0.0594362
\(696\) 0 0
\(697\) 10962.0 0.595718
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1122.00 −0.0604527 −0.0302264 − 0.999543i \(-0.509623\pi\)
−0.0302264 + 0.999543i \(0.509623\pi\)
\(702\) 0 0
\(703\) −4648.00 −0.249364
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −11997.0 −0.638181
\(708\) 0 0
\(709\) 4283.00 0.226871 0.113435 − 0.993545i \(-0.463814\pi\)
0.113435 + 0.993545i \(0.463814\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 39975.0 2.09969
\(714\) 0 0
\(715\) 4995.00 0.261262
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4032.00 0.209135 0.104568 − 0.994518i \(-0.466654\pi\)
0.104568 + 0.994518i \(0.466654\pi\)
\(720\) 0 0
\(721\) −17081.0 −0.882288
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4884.00 0.250189
\(726\) 0 0
\(727\) −24005.0 −1.22462 −0.612308 − 0.790619i \(-0.709759\pi\)
−0.612308 + 0.790619i \(0.709759\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1806.00 0.0913780
\(732\) 0 0
\(733\) −37501.0 −1.88967 −0.944837 − 0.327541i \(-0.893780\pi\)
−0.944837 + 0.327541i \(0.893780\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6315.00 −0.315626
\(738\) 0 0
\(739\) 880.000 0.0438042 0.0219021 − 0.999760i \(-0.493028\pi\)
0.0219021 + 0.999760i \(0.493028\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1623.00 0.0801374 0.0400687 − 0.999197i \(-0.487242\pi\)
0.0400687 + 0.999197i \(0.487242\pi\)
\(744\) 0 0
\(745\) 25461.0 1.25211
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −372.000 −0.0181476
\(750\) 0 0
\(751\) 6889.00 0.334731 0.167366 − 0.985895i \(-0.446474\pi\)
0.167366 + 0.985895i \(0.446474\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4149.00 −0.199997
\(756\) 0 0
\(757\) −12850.0 −0.616963 −0.308482 − 0.951230i \(-0.599821\pi\)
−0.308482 + 0.951230i \(0.599821\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4611.00 −0.219643 −0.109822 − 0.993951i \(-0.535028\pi\)
−0.109822 + 0.993951i \(0.535028\pi\)
\(762\) 0 0
\(763\) −15562.0 −0.738378
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5883.00 −0.276953
\(768\) 0 0
\(769\) −2305.00 −0.108089 −0.0540445 − 0.998539i \(-0.517211\pi\)
−0.0540445 + 0.998539i \(0.517211\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34902.0 1.62398 0.811991 − 0.583670i \(-0.198384\pi\)
0.811991 + 0.583670i \(0.198384\pi\)
\(774\) 0 0
\(775\) −9020.00 −0.418075
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7308.00 0.336118
\(780\) 0 0
\(781\) −2340.00 −0.107211
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −26793.0 −1.21820
\(786\) 0 0
\(787\) −26255.0 −1.18919 −0.594593 − 0.804027i \(-0.702687\pi\)
−0.594593 + 0.804027i \(0.702687\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 43431.0 1.95225
\(792\) 0 0
\(793\) −7067.00 −0.316465
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1437.00 0.0638659 0.0319330 − 0.999490i \(-0.489834\pi\)
0.0319330 + 0.999490i \(0.489834\pi\)
\(798\) 0 0
\(799\) 7434.00 0.329156
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2730.00 −0.119975
\(804\) 0 0
\(805\) 54405.0 2.38202
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −25734.0 −1.11837 −0.559184 − 0.829044i \(-0.688885\pi\)
−0.559184 + 0.829044i \(0.688885\pi\)
\(810\) 0 0
\(811\) 3220.00 0.139420 0.0697099 − 0.997567i \(-0.477793\pi\)
0.0697099 + 0.997567i \(0.477793\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 29844.0 1.28269
\(816\) 0 0
\(817\) 1204.00 0.0515577
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −44631.0 −1.89724 −0.948619 − 0.316420i \(-0.897519\pi\)
−0.948619 + 0.316420i \(0.897519\pi\)
\(822\) 0 0
\(823\) −24947.0 −1.05662 −0.528310 − 0.849052i \(-0.677174\pi\)
−0.528310 + 0.849052i \(0.677174\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14964.0 −0.629201 −0.314601 − 0.949224i \(-0.601871\pi\)
−0.314601 + 0.949224i \(0.601871\pi\)
\(828\) 0 0
\(829\) −7462.00 −0.312625 −0.156312 − 0.987708i \(-0.549961\pi\)
−0.156312 + 0.987708i \(0.549961\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 25956.0 1.07962
\(834\) 0 0
\(835\) −6129.00 −0.254015
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2697.00 0.110978 0.0554891 − 0.998459i \(-0.482328\pi\)
0.0554891 + 0.998459i \(0.482328\pi\)
\(840\) 0 0
\(841\) −12068.0 −0.494813
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7452.00 −0.303381
\(846\) 0 0
\(847\) −34286.0 −1.39089
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −32370.0 −1.30391
\(852\) 0 0
\(853\) −39625.0 −1.59054 −0.795272 − 0.606253i \(-0.792672\pi\)
−0.795272 + 0.606253i \(0.792672\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2973.00 0.118501 0.0592507 − 0.998243i \(-0.481129\pi\)
0.0592507 + 0.998243i \(0.481129\pi\)
\(858\) 0 0
\(859\) 45229.0 1.79650 0.898250 − 0.439485i \(-0.144839\pi\)
0.898250 + 0.439485i \(0.144839\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1416.00 0.0558531 0.0279265 − 0.999610i \(-0.491110\pi\)
0.0279265 + 0.999610i \(0.491110\pi\)
\(864\) 0 0
\(865\) 35829.0 1.40835
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 16995.0 0.663424
\(870\) 0 0
\(871\) −15577.0 −0.605978
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −47151.0 −1.82171
\(876\) 0 0
\(877\) 17555.0 0.675930 0.337965 − 0.941159i \(-0.390262\pi\)
0.337965 + 0.941159i \(0.390262\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30030.0 −1.14840 −0.574198 − 0.818717i \(-0.694686\pi\)
−0.574198 + 0.818717i \(0.694686\pi\)
\(882\) 0 0
\(883\) 35740.0 1.36211 0.681057 − 0.732230i \(-0.261521\pi\)
0.681057 + 0.732230i \(0.261521\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 747.000 0.0282771 0.0141386 − 0.999900i \(-0.495499\pi\)
0.0141386 + 0.999900i \(0.495499\pi\)
\(888\) 0 0
\(889\) 27280.0 1.02918
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4956.00 0.185718
\(894\) 0 0
\(895\) 18036.0 0.673606
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −22755.0 −0.844184
\(900\) 0 0
\(901\) −4788.00 −0.177038
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11466.0 0.421152
\(906\) 0 0
\(907\) 20539.0 0.751914 0.375957 − 0.926637i \(-0.377314\pi\)
0.375957 + 0.926637i \(0.377314\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24123.0 −0.877311 −0.438656 − 0.898655i \(-0.644545\pi\)
−0.438656 + 0.898655i \(0.644545\pi\)
\(912\) 0 0
\(913\) 16245.0 0.588862
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 46593.0 1.67790
\(918\) 0 0
\(919\) −23312.0 −0.836770 −0.418385 − 0.908270i \(-0.637404\pi\)
−0.418385 + 0.908270i \(0.637404\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −5772.00 −0.205837
\(924\) 0 0
\(925\) 7304.00 0.259626
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −18699.0 −0.660381 −0.330191 − 0.943914i \(-0.607113\pi\)
−0.330191 + 0.943914i \(0.607113\pi\)
\(930\) 0 0
\(931\) 17304.0 0.609147
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5670.00 −0.198320
\(936\) 0 0
\(937\) −31234.0 −1.08898 −0.544488 − 0.838769i \(-0.683276\pi\)
−0.544488 + 0.838769i \(0.683276\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11751.0 −0.407090 −0.203545 − 0.979066i \(-0.565246\pi\)
−0.203545 + 0.979066i \(0.565246\pi\)
\(942\) 0 0
\(943\) 50895.0 1.75755
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25887.0 −0.888294 −0.444147 − 0.895954i \(-0.646493\pi\)
−0.444147 + 0.895954i \(0.646493\pi\)
\(948\) 0 0
\(949\) −6734.00 −0.230342
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 41130.0 1.39804 0.699020 − 0.715102i \(-0.253620\pi\)
0.699020 + 0.715102i \(0.253620\pi\)
\(954\) 0 0
\(955\) 10449.0 0.354054
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 82491.0 2.77766
\(960\) 0 0
\(961\) 12234.0 0.410661
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 32499.0 1.08412
\(966\) 0 0
\(967\) 28645.0 0.952597 0.476298 − 0.879284i \(-0.341978\pi\)
0.476298 + 0.879284i \(0.341978\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −32256.0 −1.06606 −0.533030 − 0.846096i \(-0.678947\pi\)
−0.533030 + 0.846096i \(0.678947\pi\)
\(972\) 0 0
\(973\) 3751.00 0.123588
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9015.00 −0.295205 −0.147603 − 0.989047i \(-0.547156\pi\)
−0.147603 + 0.989047i \(0.547156\pi\)
\(978\) 0 0
\(979\) −15750.0 −0.514170
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 15885.0 0.515415 0.257707 − 0.966223i \(-0.417033\pi\)
0.257707 + 0.966223i \(0.417033\pi\)
\(984\) 0 0
\(985\) −18414.0 −0.595654
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8385.00 0.269593
\(990\) 0 0
\(991\) −34904.0 −1.11883 −0.559416 − 0.828887i \(-0.688974\pi\)
−0.559416 + 0.828887i \(0.688974\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −26964.0 −0.859112
\(996\) 0 0
\(997\) −35341.0 −1.12263 −0.561314 − 0.827603i \(-0.689704\pi\)
−0.561314 + 0.827603i \(0.689704\pi\)
\(998\) 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 1296.4.a.g.1.1 1
3.2 odd 2 1296.4.a.b.1.1 1
4.3 odd 2 162.4.a.a.1.1 1
9.2 odd 6 144.4.i.a.49.1 2
9.4 even 3 432.4.i.a.289.1 2
9.5 odd 6 144.4.i.a.97.1 2
9.7 even 3 432.4.i.a.145.1 2
12.11 even 2 162.4.a.d.1.1 1
36.7 odd 6 54.4.c.a.37.1 2
36.11 even 6 18.4.c.a.13.1 yes 2
36.23 even 6 18.4.c.a.7.1 ✓ 2
36.31 odd 6 54.4.c.a.19.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.4.c.a.7.1 ✓ 2 36.23 even 6
18.4.c.a.13.1 yes 2 36.11 even 6
54.4.c.a.19.1 2 36.31 odd 6
54.4.c.a.37.1 2 36.7 odd 6
144.4.i.a.49.1 2 9.2 odd 6
144.4.i.a.97.1 2 9.5 odd 6
162.4.a.a.1.1 1 4.3 odd 2
162.4.a.d.1.1 1 12.11 even 2
432.4.i.a.145.1 2 9.7 even 3
432.4.i.a.289.1 2 9.4 even 3
1296.4.a.b.1.1 1 3.2 odd 2
1296.4.a.g.1.1 1 1.1 even 1 trivial