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Properties

Label 1584.2.a.l.1.1
Level $1584$
Weight $2$
Character 1584.1
Self dual yes
Analytic conductor $12.648$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1584,2,Mod(1,1584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1584, 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("1584.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1584.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: \(12.6483036802\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 792)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1584.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{7} +O(q^{10})\) \(q+2.00000 q^{7} -1.00000 q^{11} -6.00000 q^{13} -6.00000 q^{17} +2.00000 q^{19} -8.00000 q^{23} -5.00000 q^{25} +2.00000 q^{29} +4.00000 q^{31} +2.00000 q^{37} -10.0000 q^{41} +6.00000 q^{43} +4.00000 q^{47} -3.00000 q^{49} +4.00000 q^{53} -4.00000 q^{59} -2.00000 q^{61} +8.00000 q^{67} -12.0000 q^{71} -2.00000 q^{73} -2.00000 q^{77} -14.0000 q^{79} +4.00000 q^{83} -12.0000 q^{91} +2.00000 q^{97} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 − 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 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
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −20.0000 −1.88144 −0.940721 − 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 − 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 − 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −10.0000 −0.755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 − 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 − 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 − 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 − 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 − 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 − 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.00000 −0.499026 −0.249513 − 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 − 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.00000 0.243884 0.121942 − 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) 6.00000 0.364474 0.182237 − 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 − 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.0000 −1.18056
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 30.0000 1.75262 0.876309 − 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 48.0000 2.77591
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 − 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −18.0000 −1.01742 −0.508710 − 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.0000 1.34797 0.673987 − 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 30.0000 1.66410
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 − 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 24.0000 1.25279 0.626395 − 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) −34.0000 −1.76045 −0.880227 − 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 − 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.00000 0.405616 0.202808 − 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 − 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 − 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −40.0000 −1.95413 −0.977064 − 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 30.0000 1.45521
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 − 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 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
\(439\) −10.0000 −0.477274 −0.238637 − 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 − 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 − 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.0000 1.02464 0.512321 − 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −10.0000 −0.458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 − 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 − 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 − 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.0000 −0.886484 −0.443242 − 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 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
\(521\) −4.00000 −0.175243 −0.0876216 − 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 − 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 60.0000 2.59889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 − 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18.0000 −0.769624 −0.384812 − 0.922995i \(-0.625734\pi\)
−0.384812 + 0.922995i \(0.625734\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) −28.0000 −1.19068
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 − 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 40.0000 1.66812
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.00000 −0.0821302 −0.0410651 − 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −30.0000 −1.21766 −0.608831 − 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 − 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) −36.0000 −1.44696 −0.723481 − 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 − 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 44.0000 1.73519 0.867595 − 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −48.0000 −1.88707 −0.943537 − 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28.0000 1.09572 0.547862 − 0.836569i \(-0.315442\pi\)
0.547862 + 0.836569i \(0.315442\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 − 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.00000 0.306111 0.153056 − 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 60.0000 2.27266
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 − 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −28.0000 −1.05305
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 − 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 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
\(719\) −20.0000 −0.745874 −0.372937 − 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −10.0000 −0.371391
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 − 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 − 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 − 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 − 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 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
\(773\) −36.0000 −1.29483 −0.647415 − 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 0 0
\(775\) −20.0000 −0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −42.0000 −1.49714 −0.748569 − 0.663057i \(-0.769259\pi\)
−0.748569 + 0.663057i \(0.769259\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −40.0000 −1.42224
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.0000 −0.566749 −0.283375 − 0.959009i \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −46.0000 −1.61528 −0.807639 − 0.589677i \(-0.799255\pi\)
−0.807639 + 0.589677i \(0.799255\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 − 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 − 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 46.0000 1.59765 0.798823 − 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 32.0000 1.08929 0.544646 − 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 − 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 28.0000 0.943344 0.471672 − 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 − 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40.0000 −1.34307 −0.671534 − 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 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
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 − 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −40.0000 −1.32092
\(918\) 0 0
\(919\) 38.0000 1.25350 0.626752 − 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −20.0000 −0.656179 −0.328089 − 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 − 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 − 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 80.0000 2.60516
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.0000 −0.519930 −0.259965 − 0.965618i \(-0.583711\pi\)
−0.259965 + 0.965618i \(0.583711\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 10.0000 0.323932 0.161966 − 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 − 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −52.0000 −1.66876 −0.834380 − 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 0 0
\(973\) −28.0000 −0.897639
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20.0000 0.639857 0.319928 − 0.947442i \(-0.396341\pi\)
0.319928 + 0.947442i \(0.396341\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −32.0000 −1.02064 −0.510321 − 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 − 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −34.0000 −1.07679 −0.538395 − 0.842692i \(-0.680969\pi\)
−0.538395 + 0.842692i \(0.680969\pi\)
\(998\) 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 1584.2.a.l.1.1 1
3.2 odd 2 1584.2.a.m.1.1 1
4.3 odd 2 792.2.a.d.1.1 yes 1
8.3 odd 2 6336.2.a.bc.1.1 1
8.5 even 2 6336.2.a.bp.1.1 1
12.11 even 2 792.2.a.c.1.1 ✓ 1
24.5 odd 2 6336.2.a.bk.1.1 1
24.11 even 2 6336.2.a.bg.1.1 1
44.43 even 2 8712.2.a.o.1.1 1
132.131 odd 2 8712.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
792.2.a.c.1.1 ✓ 1 12.11 even 2
792.2.a.d.1.1 yes 1 4.3 odd 2
1584.2.a.l.1.1 1 1.1 even 1 trivial
1584.2.a.m.1.1 1 3.2 odd 2
6336.2.a.bc.1.1 1 8.3 odd 2
6336.2.a.bg.1.1 1 24.11 even 2
6336.2.a.bk.1.1 1 24.5 odd 2
6336.2.a.bp.1.1 1 8.5 even 2
8712.2.a.n.1.1 1 132.131 odd 2
8712.2.a.o.1.1 1 44.43 even 2