Properties

Label 5400.2.f.r.649.2
Level $5400$
Weight $2$
Character 5400.649
Analytic conductor $43.119$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5400,2,Mod(649,5400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5400.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5400.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(43.1192170915\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1080)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5400.649
Dual form 5400.2.f.r.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+O(q^{10})\) \(q+2.00000 q^{11} +3.00000i q^{17} +1.00000 q^{19} -3.00000i q^{23} -4.00000 q^{29} -5.00000 q^{31} +10.0000i q^{37} +6.00000 q^{41} +6.00000i q^{43} +8.00000i q^{47} +7.00000 q^{49} -3.00000i q^{53} +5.00000 q^{61} -2.00000i q^{67} +2.00000 q^{71} -6.00000i q^{73} +11.0000 q^{79} -9.00000i q^{83} -10.0000 q^{89} +8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{11} + 2 q^{19} - 8 q^{29} - 10 q^{31} + 12 q^{41} + 14 q^{49} + 10 q^{61} + 4 q^{71} + 22 q^{79} - 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\) \(2701\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 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
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 3.00000i − 0.625543i −0.949828 0.312772i \(-0.898743\pi\)
0.949828 0.312772i \(-0.101257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 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
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 3.00000i − 0.412082i −0.978543 0.206041i \(-0.933942\pi\)
0.978543 0.206041i \(-0.0660580\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 12.0000i 1.18240i 0.806527 + 0.591198i \(0.201345\pi\)
−0.806527 + 0.591198i \(0.798655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 22.0000i 1.95218i 0.217357 + 0.976092i \(0.430256\pi\)
−0.217357 + 0.976092i \(0.569744\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 7.00000i − 0.598050i −0.954245 0.299025i \(-0.903339\pi\)
0.954245 0.299025i \(-0.0966615\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000i 0.156652i 0.996928 + 0.0783260i \(0.0249575\pi\)
−0.996928 + 0.0783260i \(0.975042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 17.0000i − 1.31550i −0.753237 0.657750i \(-0.771508\pi\)
0.753237 0.657750i \(-0.228492\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 19.0000i 1.44454i 0.691609 + 0.722272i \(0.256902\pi\)
−0.691609 + 0.722272i \(0.743098\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000i 0.438763i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0 0
\(193\) 20.0000i 1.43963i 0.694165 + 0.719816i \(0.255774\pi\)
−0.694165 + 0.719816i \(0.744226\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.00000i 0.356235i 0.984009 + 0.178118i \(0.0570008\pi\)
−0.984009 + 0.178118i \(0.942999\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.0000i 1.12833i 0.825662 + 0.564165i \(0.190802\pi\)
−0.825662 + 0.564165i \(0.809198\pi\)
\(228\) 0 0
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(241\) −17.0000 −1.09507 −0.547533 0.836784i \(-0.684433\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) − 6.00000i − 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 23.0000i − 1.43470i −0.696713 0.717350i \(-0.745355\pi\)
0.696713 0.717350i \(-0.254645\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 3.00000 0.182237 0.0911185 0.995840i \(-0.470956\pi\)
0.0911185 + 0.995840i \(0.470956\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −28.0000 −1.67034 −0.835170 0.549992i \(-0.814631\pi\)
−0.835170 + 0.549992i \(0.814631\pi\)
\(282\) 0 0
\(283\) − 2.00000i − 0.118888i −0.998232 0.0594438i \(-0.981067\pi\)
0.998232 0.0594438i \(-0.0189327\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.0000i 0.642627i 0.946973 + 0.321313i \(0.104124\pi\)
−0.946973 + 0.321313i \(0.895876\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.0000i 1.48390i 0.670456 + 0.741949i \(0.266098\pi\)
−0.670456 + 0.741949i \(0.733902\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) 20.0000i 1.13047i 0.824931 + 0.565233i \(0.191214\pi\)
−0.824931 + 0.565233i \(0.808786\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.0000i 0.617822i 0.951091 + 0.308911i \(0.0999645\pi\)
−0.951091 + 0.308911i \(0.900036\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000i 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 7.00000 0.374701 0.187351 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.0000i 0.532246i 0.963939 + 0.266123i \(0.0857428\pi\)
−0.963939 + 0.266123i \(0.914257\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 3.00000i − 0.153293i −0.997058 0.0766464i \(-0.975579\pi\)
0.997058 0.0766464i \(-0.0244213\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 4.00000i − 0.200754i −0.994949 0.100377i \(-0.967995\pi\)
0.994949 0.100377i \(-0.0320049\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000i 0.991363i
\(408\) 0 0
\(409\) 31.0000 1.53285 0.766426 0.642333i \(-0.222033\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) 24.0000i 1.15337i 0.816968 + 0.576683i \(0.195653\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.00000i − 0.143509i
\(438\) 0 0
\(439\) −7.00000 −0.334092 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 29.0000i − 1.37783i −0.724841 0.688916i \(-0.758087\pi\)
0.724841 0.688916i \(-0.241913\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 16.0000i − 0.748448i −0.927338 0.374224i \(-0.877909\pi\)
0.927338 0.374224i \(-0.122091\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) − 14.0000i − 0.650635i −0.945605 0.325318i \(-0.894529\pi\)
0.945605 0.325318i \(-0.105471\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 7.00000i − 0.323921i −0.986797 0.161961i \(-0.948218\pi\)
0.986797 0.161961i \(-0.0517818\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −10.0000 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 42.0000 1.89543 0.947717 0.319113i \(-0.103385\pi\)
0.947717 + 0.319113i \(0.103385\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −43.0000 −1.92494 −0.962472 0.271380i \(-0.912520\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 35.0000i − 1.56057i −0.625422 0.780286i \(-0.715073\pi\)
0.625422 0.780286i \(-0.284927\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 15.0000i − 0.653410i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14.0000 0.603023
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18.0000i 0.769624i 0.922995 + 0.384812i \(0.125734\pi\)
−0.922995 + 0.384812i \(0.874266\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 28.0000i − 1.18006i −0.807382 0.590030i \(-0.799116\pi\)
0.807382 0.590030i \(-0.200884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −9.00000 −0.376638 −0.188319 0.982108i \(-0.560304\pi\)
−0.188319 + 0.982108i \(0.560304\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 46.0000i − 1.91501i −0.288425 0.957503i \(-0.593132\pi\)
0.288425 0.957503i \(-0.406868\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 6.00000i − 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.0000i 0.949312i 0.880172 + 0.474656i \(0.157427\pi\)
−0.880172 + 0.474656i \(0.842573\pi\)
\(588\) 0 0
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.00000i 0.123195i 0.998101 + 0.0615976i \(0.0196196\pi\)
−0.998101 + 0.0615976i \(0.980380\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) 17.0000 0.693444 0.346722 0.937968i \(-0.387295\pi\)
0.346722 + 0.937968i \(0.387295\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0000i 0.405887i 0.979190 + 0.202944i \(0.0650509\pi\)
−0.979190 + 0.202944i \(0.934949\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 16.0000i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 25.0000i − 1.00646i −0.864152 0.503231i \(-0.832144\pi\)
0.864152 0.503231i \(-0.167856\pi\)
\(618\) 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
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −30.0000 −1.19618
\(630\) 0 0
\(631\) −19.0000 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 39.0000i 1.53325i 0.642096 + 0.766624i \(0.278065\pi\)
−0.642096 + 0.766624i \(0.721935\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 11.0000i − 0.430463i −0.976563 0.215232i \(-0.930949\pi\)
0.976563 0.215232i \(-0.0690506\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) 8.00000i 0.308377i 0.988041 + 0.154189i \(0.0492764\pi\)
−0.988041 + 0.154189i \(0.950724\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 50.0000i 1.92166i 0.277145 + 0.960828i \(0.410612\pi\)
−0.277145 + 0.960828i \(0.589388\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23.0000i 0.880071i 0.897980 + 0.440035i \(0.145034\pi\)
−0.897980 + 0.440035i \(0.854966\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 13.0000 0.494543 0.247272 0.968946i \(-0.420466\pi\)
0.247272 + 0.968946i \(0.420466\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 18.0000i 0.681799i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 10.0000i 0.377157i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 15.0000i 0.561754i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 44.0000i − 1.63187i −0.578144 0.815935i \(-0.696223\pi\)
0.578144 0.815935i \(-0.303777\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) 0 0
\(733\) 8.00000i 0.295487i 0.989026 + 0.147743i \(0.0472010\pi\)
−0.989026 + 0.147743i \(0.952799\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 4.00000i − 0.147342i
\(738\) 0 0
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 53.0000 1.93400 0.966999 0.254781i \(-0.0820034\pi\)
0.966999 + 0.254781i \(0.0820034\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 50.0000i − 1.81728i −0.417579 0.908640i \(-0.637121\pi\)
0.417579 0.908640i \(-0.362879\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 44.0000 1.59500 0.797499 0.603320i \(-0.206156\pi\)
0.797499 + 0.603320i \(0.206156\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −21.0000 −0.757279 −0.378640 0.925544i \(-0.623608\pi\)
−0.378640 + 0.925544i \(0.623608\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 3.00000i − 0.107903i −0.998544 0.0539513i \(-0.982818\pi\)
0.998544 0.0539513i \(-0.0171816\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 38.0000i − 1.35455i −0.735728 0.677277i \(-0.763160\pi\)
0.735728 0.677277i \(-0.236840\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 23.0000i − 0.814702i −0.913272 0.407351i \(-0.866453\pi\)
0.913272 0.407351i \(-0.133547\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 12.0000i − 0.423471i
\(804\) 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
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.00000i 0.209913i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −34.0000 −1.18661 −0.593304 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) 0 0
\(823\) 6.00000i 0.209147i 0.994517 + 0.104573i \(0.0333477\pi\)
−0.994517 + 0.104573i \(0.966652\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 21.0000i − 0.730242i −0.930960 0.365121i \(-0.881028\pi\)
0.930960 0.365121i \(-0.118972\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 21.0000i 0.727607i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 30.0000 1.02839
\(852\) 0 0
\(853\) − 8.00000i − 0.273915i −0.990577 0.136957i \(-0.956268\pi\)
0.990577 0.136957i \(-0.0437323\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.00000i − 0.0341593i −0.999854 0.0170797i \(-0.994563\pi\)
0.999854 0.0170797i \(-0.00543689\pi\)
\(858\) 0 0
\(859\) −37.0000 −1.26242 −0.631212 0.775610i \(-0.717442\pi\)
−0.631212 + 0.775610i \(0.717442\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 23.0000i − 0.782929i −0.920193 0.391465i \(-0.871969\pi\)
0.920193 0.391465i \(-0.128031\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 22.0000 0.746299
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.00000 −0.134763 −0.0673817 0.997727i \(-0.521465\pi\)
−0.0673817 + 0.997727i \(0.521465\pi\)
\(882\) 0 0
\(883\) − 16.0000i − 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 33.0000i − 1.10803i −0.832506 0.554016i \(-0.813095\pi\)
0.832506 0.554016i \(-0.186905\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.00000i 0.267710i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 44.0000i − 1.46100i −0.682915 0.730498i \(-0.739288\pi\)
0.682915 0.730498i \(-0.260712\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) − 18.0000i − 0.595713i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 12.0000i − 0.392023i −0.980602 0.196011i \(-0.937201\pi\)
0.980602 0.196011i \(-0.0627990\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) − 18.0000i − 0.586161i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.00000i − 0.0974869i −0.998811 0.0487435i \(-0.984478\pi\)
0.998811 0.0487435i \(-0.0155217\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000i 0.842223i 0.907009 + 0.421111i \(0.138360\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 42.0000i − 1.35063i −0.737530 0.675314i \(-0.764008\pi\)
0.737530 0.675314i \(-0.235992\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 50.0000i − 1.59964i −0.600239 0.799821i \(-0.704928\pi\)
0.600239 0.799821i \(-0.295072\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 11.0000i − 0.350846i −0.984493 0.175423i \(-0.943871\pi\)
0.984493 0.175423i \(-0.0561292\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) 1.00000 0.0317660 0.0158830 0.999874i \(-0.494944\pi\)
0.0158830 + 0.999874i \(0.494944\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 18.0000i − 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) 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 5400.2.f.r.649.2 2
3.2 odd 2 5400.2.f.j.649.1 2
5.2 odd 4 5400.2.a.x.1.1 1
5.3 odd 4 1080.2.a.j.1.1 yes 1
5.4 even 2 inner 5400.2.f.r.649.1 2
15.2 even 4 5400.2.a.w.1.1 1
15.8 even 4 1080.2.a.d.1.1 1
15.14 odd 2 5400.2.f.j.649.2 2
20.3 even 4 2160.2.a.r.1.1 1
40.3 even 4 8640.2.a.p.1.1 1
40.13 odd 4 8640.2.a.o.1.1 1
45.13 odd 12 3240.2.q.e.2161.1 2
45.23 even 12 3240.2.q.s.2161.1 2
45.38 even 12 3240.2.q.s.1081.1 2
45.43 odd 12 3240.2.q.e.1081.1 2
60.23 odd 4 2160.2.a.f.1.1 1
120.53 even 4 8640.2.a.bt.1.1 1
120.83 odd 4 8640.2.a.bs.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.a.d.1.1 1 15.8 even 4
1080.2.a.j.1.1 yes 1 5.3 odd 4
2160.2.a.f.1.1 1 60.23 odd 4
2160.2.a.r.1.1 1 20.3 even 4
3240.2.q.e.1081.1 2 45.43 odd 12
3240.2.q.e.2161.1 2 45.13 odd 12
3240.2.q.s.1081.1 2 45.38 even 12
3240.2.q.s.2161.1 2 45.23 even 12
5400.2.a.w.1.1 1 15.2 even 4
5400.2.a.x.1.1 1 5.2 odd 4
5400.2.f.j.649.1 2 3.2 odd 2
5400.2.f.j.649.2 2 15.14 odd 2
5400.2.f.r.649.1 2 5.4 even 2 inner
5400.2.f.r.649.2 2 1.1 even 1 trivial
8640.2.a.o.1.1 1 40.13 odd 4
8640.2.a.p.1.1 1 40.3 even 4
8640.2.a.bs.1.1 1 120.83 odd 4
8640.2.a.bt.1.1 1 120.53 even 4