↑

Properties

Label 5040.2.a.bb.1.1
Level $5040$
Weight $2$
Character 5040.1
Self dual yes
Analytic conductor $40.245$
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] = [5040,2,Mod(1,5040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.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: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2520)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -1.00000 q^{7} +2.00000 q^{11} -2.00000 q^{13} +2.00000 q^{17} +8.00000 q^{19} +2.00000 q^{23} +1.00000 q^{25} -8.00000 q^{29} +8.00000 q^{31} -1.00000 q^{35} -6.00000 q^{37} +6.00000 q^{41} -4.00000 q^{43} -4.00000 q^{47} +1.00000 q^{49} +2.00000 q^{55} +4.00000 q^{59} -2.00000 q^{61} -2.00000 q^{65} +4.00000 q^{67} -6.00000 q^{71} -6.00000 q^{73} -2.00000 q^{77} +16.0000 q^{83} +2.00000 q^{85} +6.00000 q^{89} +2.00000 q^{91} +8.00000 q^{95} -14.0000 q^{97} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 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
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 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
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 − 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.0000 1.36697 0.683486 − 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 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
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.0000 1.63846 0.819232 − 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −22.0000 −1.67263 −0.836315 − 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.00000 −0.149487 −0.0747435 − 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 − 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 − 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 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
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 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
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.0000 1.94054 0.970269 − 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 − 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 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
\(271\) 24.0000 1.45790 0.728948 − 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.0000 1.67034 0.835170 − 0.549992i \(-0.185369\pi\)
0.835170 + 0.549992i \(0.185369\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 − 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 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
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 − 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −14.0000 −0.751559 −0.375780 − 0.926709i \(-0.622625\pi\)
−0.375780 + 0.926709i \(0.622625\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 − 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −22.0000 −1.17094 −0.585471 − 0.810693i \(-0.699090\pi\)
−0.585471 + 0.810693i \(0.699090\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 − 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 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
\(371\) 0 0
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 − 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 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
\(401\) 36.0000 1.79775 0.898877 − 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 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
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.0000 −0.674356 −0.337178 − 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 − 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 42.0000 1.99548 0.997740 − 0.0671913i \(-0.0214038\pi\)
0.997740 + 0.0671913i \(0.0214038\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 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
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 − 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 − 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 − 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 − 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 − 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 − 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 − 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 − 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 − 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 − 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −64.0000 −2.72649
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 − 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 − 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 64.0000 2.63707
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 − 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 − 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 30.0000 1.21169 0.605844 − 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 − 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 − 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.00000 −0.156532 −0.0782660 − 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 − 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 − 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 30.0000 1.15642 0.578208 − 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −38.0000 −1.46046 −0.730229 − 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.00000 −0.229584 −0.114792 − 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.00000 0.302156 0.151078 − 0.988522i \(-0.451726\pi\)
0.151078 + 0.988522i \(0.451726\pi\)
\(702\) 0 0
\(703\) −48.0000 −1.81035
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.0000 −0.676960
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 − 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32.0000 1.19340 0.596699 − 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) 4.00000 0.148352 0.0741759 − 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −42.0000 −1.55131 −0.775653 − 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 − 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 − 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −46.0000 −1.67190 −0.835949 − 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 − 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −54.0000 −1.94225 −0.971123 − 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 − 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 20.0000 0.700569
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.0000 −0.698005 −0.349002 − 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 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
\(827\) 6.00000 0.208640 0.104320 − 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 − 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 28.0000 0.966667 0.483334 − 0.875436i \(-0.339426\pi\)
0.483334 + 0.875436i \(0.339426\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 − 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 − 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 − 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) −22.0000 −0.748022
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 − 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −52.0000 −1.74599 −0.872995 − 0.487730i \(-0.837825\pi\)
−0.872995 + 0.487730i \(0.837825\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) −2.00000 −0.0668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −64.0000 −2.13452
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −36.0000 −1.19536 −0.597680 − 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26.0000 0.861418 0.430709 − 0.902491i \(-0.358263\pi\)
0.430709 + 0.902491i \(0.358263\pi\)
\(912\) 0 0
\(913\) 32.0000 1.05905
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −50.0000 −1.64045 −0.820223 − 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −50.0000 −1.63343 −0.816714 − 0.577042i \(-0.804207\pi\)
−0.816714 + 0.577042i \(0.804207\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 − 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −58.0000 −1.88475 −0.942373 − 0.334563i \(-0.891411\pi\)
−0.942373 + 0.334563i \(0.891411\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −60.0000 −1.94359 −0.971795 − 0.235826i \(-0.924220\pi\)
−0.971795 + 0.235826i \(0.924220\pi\)
\(954\) 0 0
\(955\) 10.0000 0.323592
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 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
\(971\) −56.0000 −1.79713 −0.898563 − 0.438845i \(-0.855388\pi\)
−0.898563 + 0.438845i \(0.855388\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.0000 1.02377 0.511885 − 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −8.00000 −0.254901
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −30.0000 −0.950110 −0.475055 − 0.879956i \(-0.657572\pi\)
−0.475055 + 0.879956i \(0.657572\pi\)
\(998\) 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 5040.2.a.bb.1.1 1
3.2 odd 2 5040.2.a.b.1.1 1
4.3 odd 2 2520.2.a.q.1.1 yes 1
12.11 even 2 2520.2.a.h.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2520.2.a.h.1.1 ✓ 1 12.11 even 2
2520.2.a.q.1.1 yes 1 4.3 odd 2
5040.2.a.b.1.1 1 3.2 odd 2
5040.2.a.bb.1.1 1 1.1 even 1 trivial