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Properties

Label 1872.2.a.k.1.1
Level $1872$
Weight $2$
Character 1872.1
Self dual yes
Analytic conductor $14.948$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{7} +O(q^{10})\) \(q+4.00000 q^{7} -2.00000 q^{11} -1.00000 q^{13} +6.00000 q^{17} +4.00000 q^{19} +4.00000 q^{23} -5.00000 q^{25} -10.0000 q^{29} +8.00000 q^{31} -2.00000 q^{37} +4.00000 q^{43} +2.00000 q^{47} +9.00000 q^{49} +2.00000 q^{53} +10.0000 q^{59} +10.0000 q^{61} -8.00000 q^{67} +2.00000 q^{71} -10.0000 q^{73} -8.00000 q^{77} -8.00000 q^{79} +6.00000 q^{83} +12.0000 q^{89} -4.00000 q^{91} -2.00000 q^{97} +O(q^{100})\)

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
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.0000 −1.54678 −0.773389 − 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 − 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 24.0000 1.95309 0.976546 − 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 −0.464294 −0.232147 − 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) −20.0000 −1.51186
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 − 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 − 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −40.0000 −2.80745
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 32.0000 2.17230
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −22.0000 −1.46019 −0.730096 − 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 − 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 − 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 − 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 − 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 − 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 − 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 − 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.0000 0.603023
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 − 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 − 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −32.0000 −1.79730 −0.898650 − 0.438667i \(-0.855451\pi\)
−0.898650 + 0.438667i \(0.855451\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 5.00000 0.277350
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 − 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.0000 −1.50312 −0.751559 − 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 − 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.0000 0.851594 0.425797 − 0.904819i \(-0.359994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −22.0000 −1.16112 −0.580558 − 0.814219i \(-0.697165\pi\)
−0.580558 + 0.814219i \(0.697165\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 − 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 − 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.00000 −0.199750 −0.0998752 − 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 40.0000 1.96827
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 − 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 40.0000 1.93574
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 − 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 − 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 − 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −32.0000 −1.48078 −0.740392 − 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −20.0000 −0.917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 30.0000 1.37073 0.685367 − 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −60.0000 −2.70226
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −40.0000 −1.77297 −0.886484 − 0.462758i \(-0.846860\pi\)
−0.886484 + 0.462758i \(0.846860\pi\)
\(510\) 0 0
\(511\) −40.0000 −1.76950
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 − 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.00000 0.169485 0.0847427 − 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −38.0000 −1.59304 −0.796521 − 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 − 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −20.0000 −0.834058
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18.0000 −0.742940 −0.371470 − 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −28.0000 −1.14982 −0.574911 − 0.818216i \(-0.694963\pi\)
−0.574911 + 0.818216i \(0.694963\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 − 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 − 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.00000 −0.0809113
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −24.0000 −0.966204 −0.483102 − 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 − 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 48.0000 1.92308
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.00000 −0.356593
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 22.0000 0.868948 0.434474 − 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) −40.0000 −1.57745 −0.788723 − 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.0000 −0.629025 −0.314512 − 0.949253i \(-0.601841\pi\)
−0.314512 + 0.949253i \(0.601841\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 − 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −40.0000 −1.54881
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.0000 0.384331 0.192166 − 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18.0000 −0.688751 −0.344375 − 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −16.0000 −0.608669 −0.304334 − 0.952565i \(-0.598434\pi\)
−0.304334 + 0.952565i \(0.598434\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 − 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.00000 −0.300871
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 − 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 28.0000 1.04422 0.522112 − 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 0 0
\(721\) 64.0000 2.38348
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 50.0000 1.85695
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 − 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 − 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.00000 0.0733729 0.0366864 − 0.999327i \(-0.488320\pi\)
0.0366864 + 0.999327i \(0.488320\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −64.0000 −2.33851
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 − 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −48.0000 −1.74000 −0.869999 − 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(762\) 0 0
\(763\) −24.0000 −0.868858
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.0000 −0.361079
\(768\) 0 0
\(769\) −38.0000 −1.37032 −0.685158 − 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 28.0000 1.00709 0.503545 − 0.863969i \(-0.332029\pi\)
0.503545 + 0.863969i \(0.332029\pi\)
\(774\) 0 0
\(775\) −40.0000 −1.43684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 − 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 72.0000 2.56003
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.00000 −0.212531 −0.106265 − 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 42.0000 1.47664 0.738321 − 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 − 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 − 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 − 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 54.0000 1.87776 0.938882 − 0.344239i \(-0.111863\pi\)
0.938882 + 0.344239i \(0.111863\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.0000 −1.17381 −0.586905 − 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −28.0000 −0.962091
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.0000 −0.340404 −0.170202 − 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 − 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 − 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.0000 −1.07445 −0.537227 − 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) −64.0000 −2.14649
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −80.0000 −2.66815
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 − 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 32.0000 1.05673
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.00000 −0.0658308
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −32.0000 −1.04989 −0.524943 − 0.851137i \(-0.675913\pi\)
−0.524943 + 0.851137i \(0.675913\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 − 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −36.0000 −1.17357 −0.586783 − 0.809744i \(-0.699606\pi\)
−0.586783 + 0.809744i \(0.699606\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.0000 0.584921 0.292461 − 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 − 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 48.0000 1.53881
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.00000 −0.127971 −0.0639857 − 0.997951i \(-0.520381\pi\)
−0.0639857 + 0.997951i \(0.520381\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −18.0000 −0.574111 −0.287055 − 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 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 1872.2.a.k.1.1 1
3.2 odd 2 624.2.a.g.1.1 1
4.3 odd 2 936.2.a.d.1.1 1
8.3 odd 2 7488.2.a.y.1.1 1
8.5 even 2 7488.2.a.bh.1.1 1
12.11 even 2 312.2.a.b.1.1 ✓ 1
24.5 odd 2 2496.2.a.j.1.1 1
24.11 even 2 2496.2.a.u.1.1 1
39.38 odd 2 8112.2.a.y.1.1 1
60.59 even 2 7800.2.a.w.1.1 1
156.47 odd 4 4056.2.c.f.337.2 2
156.83 odd 4 4056.2.c.f.337.1 2
156.155 even 2 4056.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.a.b.1.1 ✓ 1 12.11 even 2
624.2.a.g.1.1 1 3.2 odd 2
936.2.a.d.1.1 1 4.3 odd 2
1872.2.a.k.1.1 1 1.1 even 1 trivial
2496.2.a.j.1.1 1 24.5 odd 2
2496.2.a.u.1.1 1 24.11 even 2
4056.2.a.f.1.1 1 156.155 even 2
4056.2.c.f.337.1 2 156.83 odd 4
4056.2.c.f.337.2 2 156.47 odd 4
7488.2.a.y.1.1 1 8.3 odd 2
7488.2.a.bh.1.1 1 8.5 even 2
7800.2.a.w.1.1 1 60.59 even 2
8112.2.a.y.1.1 1 39.38 odd 2