Properties

Label 4410.2.a.w.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $0$
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] = [4410,2,Mod(1,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4410.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.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: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4410.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} -5.00000 q^{13} +1.00000 q^{16} -5.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} +9.00000 q^{23} +1.00000 q^{25} -5.00000 q^{26} +10.0000 q^{31} +1.00000 q^{32} -1.00000 q^{37} -5.00000 q^{38} -1.00000 q^{40} +9.00000 q^{41} +8.00000 q^{43} -3.00000 q^{44} +9.00000 q^{46} +3.00000 q^{47} +1.00000 q^{50} -5.00000 q^{52} +3.00000 q^{53} +3.00000 q^{55} +12.0000 q^{59} -8.00000 q^{61} +10.0000 q^{62} +1.00000 q^{64} +5.00000 q^{65} +8.00000 q^{67} +6.00000 q^{71} -2.00000 q^{73} -1.00000 q^{74} -5.00000 q^{76} +8.00000 q^{79} -1.00000 q^{80} +9.00000 q^{82} +8.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} +9.00000 q^{92} +3.00000 q^{94} +5.00000 q^{95} -8.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 9.00000 1.87663 0.938315 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −5.00000 −0.980581
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −5.00000 −0.811107
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −5.00000 −0.693375
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.00000 0.620174
\(66\) 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
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.00000 0.938315
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) 5.00000 0.512989
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 3.00000 0.286039
\(111\) 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
\(115\) −9.00000 −0.839254
\(116\) 0 0
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) 10.0000 0.898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 5.00000 0.438529
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 15.0000 1.25436
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −5.00000 −0.405554
\(153\) 0 0
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 15.0000 1.12115 0.560576 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 9.00000 0.663489
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 5.00000 0.362738
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 0 0
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −5.00000 −0.346688
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 3.00000 0.206041
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) 0 0
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) 1.00000 0.0644157 0.0322078 0.999481i \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) 25.0000 1.59071
\(248\) 10.0000 0.635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 9.00000 0.568075 0.284037 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 0 0
\(253\) −27.0000 −1.69748
\(254\) −13.0000 −0.815693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 5.00000 0.310087
\(261\) 0 0
\(262\) −9.00000 −0.556022
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −3.00000 −0.184289
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 0 0
\(281\) 21.0000 1.25275 0.626377 0.779520i \(-0.284537\pi\)
0.626377 + 0.779520i \(0.284537\pi\)
\(282\) 0 0
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 15.0000 0.886969
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) 0 0
\(299\) −45.0000 −2.60242
\(300\) 0 0
\(301\) 0 0
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −5.00000 −0.286770
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −10.0000 −0.567962
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) −5.00000 −0.282166
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −5.00000 −0.277350
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) 11.0000 0.604615 0.302307 0.953211i \(-0.402243\pi\)
0.302307 + 0.953211i \(0.402243\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −3.00000 −0.164153
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) 0 0
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 15.0000 0.792775
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −8.00000 −0.420471
\(363\) 0 0
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 19.0000 0.991792 0.495896 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) 9.00000 0.469157
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 0 0
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) 5.00000 0.256495
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) 27.0000 1.37964 0.689818 0.723983i \(-0.257691\pi\)
0.689818 + 0.723983i \(0.257691\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −15.0000 −0.755689
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) −50.0000 −2.49068
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) −9.00000 −0.444478
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 15.0000 0.733674
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 5.00000 0.243396
\(423\) 0 0
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) −45.0000 −2.15264
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 3.00000 0.143019
\(441\) 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
\(445\) −6.00000 −0.284427
\(446\) 28.0000 1.32584
\(447\) 0 0
\(448\) 0 0
\(449\) 33.0000 1.55737 0.778683 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) 18.0000 0.846649
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) −9.00000 −0.419627
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −1.00000 −0.0464739 −0.0232370 0.999730i \(-0.507397\pi\)
−0.0232370 + 0.999730i \(0.507397\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −3.00000 −0.138380
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) −30.0000 −1.37217
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) 5.00000 0.227980
\(482\) 1.00000 0.0455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −8.00000 −0.362143
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 25.0000 1.12480
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 9.00000 0.401690
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −27.0000 −1.20030
\(507\) 0 0
\(508\) −13.0000 −0.576782
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) 0 0
\(520\) 5.00000 0.219265
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) −9.00000 −0.393167
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) −3.00000 −0.130312
\(531\) 0 0
\(532\) 0 0
\(533\) −45.0000 −1.94917
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 0 0
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −39.0000 −1.65248 −0.826242 0.563316i \(-0.809525\pi\)
−0.826242 + 0.563316i \(0.809525\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 21.0000 0.885832
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 22.0000 0.924729
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 15.0000 0.627182
\(573\) 0 0
\(574\) 0 0
\(575\) 9.00000 0.375326
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −9.00000 −0.372742
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −21.0000 −0.867502
\(587\) −42.0000 −1.73353 −0.866763 0.498721i \(-0.833803\pi\)
−0.866763 + 0.498721i \(0.833803\pi\)
\(588\) 0 0
\(589\) −50.0000 −2.06021
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −45.0000 −1.84019
\(599\) −18.0000 −0.735460 −0.367730 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 25.0000 1.01472 0.507359 0.861735i \(-0.330622\pi\)
0.507359 + 0.861735i \(0.330622\pi\)
\(608\) −5.00000 −0.202777
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) −15.0000 −0.606835
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 28.0000 1.12999
\(615\) 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
\(619\) −29.0000 −1.16561 −0.582804 0.812613i \(-0.698045\pi\)
−0.582804 + 0.812613i \(0.698045\pi\)
\(620\) −10.0000 −0.401610
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 4.00000 0.159872
\(627\) 0 0
\(628\) −5.00000 −0.199522
\(629\) 0 0
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 13.0000 0.515889
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.00000 −0.353827 −0.176913 0.984226i \(-0.556611\pi\)
−0.176913 + 0.984226i \(0.556611\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) −5.00000 −0.196116
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −39.0000 −1.52619 −0.763094 0.646288i \(-0.776321\pi\)
−0.763094 + 0.646288i \(0.776321\pi\)
\(654\) 0 0
\(655\) 9.00000 0.351659
\(656\) 9.00000 0.351391
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) 11.0000 0.427527
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −3.00000 −0.116073
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 20.0000 0.770943 0.385472 0.922720i \(-0.374039\pi\)
0.385472 + 0.922720i \(0.374039\pi\)
\(674\) 20.0000 0.770371
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 3.00000 0.115299 0.0576497 0.998337i \(-0.481639\pi\)
0.0576497 + 0.998337i \(0.481639\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −30.0000 −1.14876
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −15.0000 −0.571454
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 9.00000 0.342129
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 0 0
\(698\) 28.0000 1.05982
\(699\) 0 0
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 5.00000 0.188579
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 0 0
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) −6.00000 −0.225176
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 90.0000 3.37053
\(714\) 0 0
\(715\) −15.0000 −0.560968
\(716\) 15.0000 0.560576
\(717\) 0 0
\(718\) −12.0000 −0.447836
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6.00000 0.223297
\(723\) 0 0
\(724\) −8.00000 −0.297318
\(725\) 0 0
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.00000 0.0740233
\(731\) 0 0
\(732\) 0 0
\(733\) 1.00000 0.0369358 0.0184679 0.999829i \(-0.494121\pi\)
0.0184679 + 0.999829i \(0.494121\pi\)
\(734\) 19.0000 0.701303
\(735\) 0 0
\(736\) 9.00000 0.331744
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 0 0
\(743\) 39.0000 1.43077 0.715386 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 3.00000 0.109399
\(753\) 0 0
\(754\) 0 0
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −19.0000 −0.690111
\(759\) 0 0
\(760\) 5.00000 0.181369
\(761\) −33.0000 −1.19625 −0.598125 0.801403i \(-0.704087\pi\)
−0.598125 + 0.801403i \(0.704087\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 27.0000 0.975550
\(767\) −60.0000 −2.16647
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 33.0000 1.18693 0.593464 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −45.0000 −1.61229
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.00000 0.178458
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) −15.0000 −0.534353
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −15.0000 −0.529668
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) −50.0000 −1.76117
\(807\) 0 0
\(808\) 0 0
\(809\) 27.0000 0.949269 0.474635 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(810\) 0 0
\(811\) 7.00000 0.245803 0.122902 0.992419i \(-0.460780\pi\)
0.122902 + 0.992419i \(0.460780\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −40.0000 −1.39942
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −6.00000 −0.208640 −0.104320 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) 0 0
\(829\) 40.0000 1.38926 0.694629 0.719368i \(-0.255569\pi\)
0.694629 + 0.719368i \(0.255569\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.00000 −0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 15.0000 0.518786
\(837\) 0 0
\(838\) −9.00000 −0.310900
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 14.0000 0.482472
\(843\) 0 0
\(844\) 5.00000 0.172107
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) 3.00000 0.103020
\(849\) 0 0
\(850\) 0 0
\(851\) −9.00000 −0.308516
\(852\) 0 0
\(853\) 37.0000 1.26686 0.633428 0.773802i \(-0.281647\pi\)
0.633428 + 0.773802i \(0.281647\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) 21.0000 0.714848 0.357424 0.933942i \(-0.383655\pi\)
0.357424 + 0.933942i \(0.383655\pi\)
\(864\) 0 0
\(865\) −9.00000 −0.306009
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −40.0000 −1.35535
\(872\) 14.0000 0.474100
\(873\) 0 0
\(874\) −45.0000 −1.52215
\(875\) 0 0
\(876\) 0 0
\(877\) 5.00000 0.168838 0.0844190 0.996430i \(-0.473097\pi\)
0.0844190 + 0.996430i \(0.473097\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) 3.00000 0.101130
\(881\) −33.0000 −1.11180 −0.555899 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 0 0
\(883\) 14.0000 0.471138 0.235569 0.971858i \(-0.424305\pi\)
0.235569 + 0.971858i \(0.424305\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) −15.0000 −0.501956
\(894\) 0 0
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) 0 0
\(898\) 33.0000 1.10122
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −27.0000 −0.899002
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 8.00000 0.265929
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 0 0
\(919\) −28.0000 −0.923635 −0.461817 0.886975i \(-0.652802\pi\)
−0.461817 + 0.886975i \(0.652802\pi\)
\(920\) −9.00000 −0.296721
\(921\) 0 0
\(922\) 12.0000 0.395199
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) −1.00000 −0.0328620
\(927\) 0 0
\(928\) 0 0
\(929\) 39.0000 1.27955 0.639774 0.768563i \(-0.279028\pi\)
0.639774 + 0.768563i \(0.279028\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −44.0000 −1.43742 −0.718709 0.695311i \(-0.755266\pi\)
−0.718709 + 0.695311i \(0.755266\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −3.00000 −0.0978492
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 81.0000 2.63772
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 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
\(950\) −5.00000 −0.162221
\(951\) 0 0
\(952\) 0 0
\(953\) −12.0000 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) 0 0
\(955\) 18.0000 0.582466
\(956\) −30.0000 −0.970269
\(957\) 0 0
\(958\) −18.0000 −0.581554
\(959\) 0 0
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 5.00000 0.161206
\(963\) 0 0
\(964\) 1.00000 0.0322078
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 9.00000 0.287055 0.143528 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) 0 0
\(985\) 15.0000 0.477940
\(986\) 0 0
\(987\) 0 0
\(988\) 25.0000 0.795356
\(989\) 72.0000 2.28947
\(990\) 0 0
\(991\) −58.0000 −1.84243 −0.921215 0.389053i \(-0.872802\pi\)
−0.921215 + 0.389053i \(0.872802\pi\)
\(992\) 10.0000 0.317500
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) −16.0000 −0.506471
\(999\) 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 4410.2.a.w.1.1 1
3.2 odd 2 1470.2.a.e.1.1 1
7.3 odd 6 630.2.k.a.541.1 2
7.5 odd 6 630.2.k.a.361.1 2
7.6 odd 2 4410.2.a.bh.1.1 1
15.14 odd 2 7350.2.a.cx.1.1 1
21.2 odd 6 1470.2.i.p.361.1 2
21.5 even 6 210.2.i.c.151.1 yes 2
21.11 odd 6 1470.2.i.p.961.1 2
21.17 even 6 210.2.i.c.121.1 2
21.20 even 2 1470.2.a.f.1.1 1
84.47 odd 6 1680.2.bg.n.1201.1 2
84.59 odd 6 1680.2.bg.n.961.1 2
105.17 odd 12 1050.2.o.c.499.2 4
105.38 odd 12 1050.2.o.c.499.1 4
105.47 odd 12 1050.2.o.c.949.1 4
105.59 even 6 1050.2.i.i.751.1 2
105.68 odd 12 1050.2.o.c.949.2 4
105.89 even 6 1050.2.i.i.151.1 2
105.104 even 2 7350.2.a.cd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.i.c.121.1 2 21.17 even 6
210.2.i.c.151.1 yes 2 21.5 even 6
630.2.k.a.361.1 2 7.5 odd 6
630.2.k.a.541.1 2 7.3 odd 6
1050.2.i.i.151.1 2 105.89 even 6
1050.2.i.i.751.1 2 105.59 even 6
1050.2.o.c.499.1 4 105.38 odd 12
1050.2.o.c.499.2 4 105.17 odd 12
1050.2.o.c.949.1 4 105.47 odd 12
1050.2.o.c.949.2 4 105.68 odd 12
1470.2.a.e.1.1 1 3.2 odd 2
1470.2.a.f.1.1 1 21.20 even 2
1470.2.i.p.361.1 2 21.2 odd 6
1470.2.i.p.961.1 2 21.11 odd 6
1680.2.bg.n.961.1 2 84.59 odd 6
1680.2.bg.n.1201.1 2 84.47 odd 6
4410.2.a.w.1.1 1 1.1 even 1 trivial
4410.2.a.bh.1.1 1 7.6 odd 2
7350.2.a.cd.1.1 1 105.104 even 2
7350.2.a.cx.1.1 1 15.14 odd 2