Properties

Label 2448.2.a.v.1.1
Level $2448$
Weight $2$
Character 2448.1
Self dual yes
Analytic conductor $19.547$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2448,2,Mod(1,2448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2448.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2448.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: \(19.5473784148\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2448.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.56155 q^{5} +O(q^{10})\) \(q-3.56155 q^{5} +1.56155 q^{11} +0.438447 q^{13} -1.00000 q^{17} +4.68466 q^{19} -2.43845 q^{23} +7.68466 q^{25} +8.24621 q^{29} -3.12311 q^{31} -5.12311 q^{37} +3.56155 q^{41} -4.68466 q^{43} -11.1231 q^{47} -7.00000 q^{49} -12.2462 q^{53} -5.56155 q^{55} +7.12311 q^{59} +9.12311 q^{61} -1.56155 q^{65} -4.00000 q^{67} -6.24621 q^{71} -12.2462 q^{73} +9.36932 q^{79} -0.876894 q^{83} +3.56155 q^{85} +1.12311 q^{89} -16.6847 q^{95} -2.87689 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} - q^{11} + 5 q^{13} - 2 q^{17} - 3 q^{19} - 9 q^{23} + 3 q^{25} + 2 q^{31} - 2 q^{37} + 3 q^{41} + 3 q^{43} - 14 q^{47} - 14 q^{49} - 8 q^{53} - 7 q^{55} + 6 q^{59} + 10 q^{61} + q^{65} - 8 q^{67} + 4 q^{71} - 8 q^{73} - 6 q^{79} - 10 q^{83} + 3 q^{85} - 6 q^{89} - 21 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.56155 0.470826 0.235413 0.971895i \(-0.424356\pi\)
0.235413 + 0.971895i \(0.424356\pi\)
\(12\) 0 0
\(13\) 0.438447 0.121603 0.0608017 0.998150i \(-0.480634\pi\)
0.0608017 + 0.998150i \(0.480634\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 4.68466 1.07473 0.537367 0.843348i \(-0.319419\pi\)
0.537367 + 0.843348i \(0.319419\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.43845 −0.508451 −0.254226 0.967145i \(-0.581821\pi\)
−0.254226 + 0.967145i \(0.581821\pi\)
\(24\) 0 0
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.24621 1.53128 0.765641 0.643268i \(-0.222422\pi\)
0.765641 + 0.643268i \(0.222422\pi\)
\(30\) 0 0
\(31\) −3.12311 −0.560926 −0.280463 0.959865i \(-0.590488\pi\)
−0.280463 + 0.959865i \(0.590488\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.12311 −0.842233 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.56155 0.556221 0.278111 0.960549i \(-0.410292\pi\)
0.278111 + 0.960549i \(0.410292\pi\)
\(42\) 0 0
\(43\) −4.68466 −0.714404 −0.357202 0.934027i \(-0.616269\pi\)
−0.357202 + 0.934027i \(0.616269\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.1231 −1.62247 −0.811236 0.584719i \(-0.801205\pi\)
−0.811236 + 0.584719i \(0.801205\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.2462 −1.68215 −0.841073 0.540921i \(-0.818076\pi\)
−0.841073 + 0.540921i \(0.818076\pi\)
\(54\) 0 0
\(55\) −5.56155 −0.749920
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.12311 0.927349 0.463675 0.886006i \(-0.346531\pi\)
0.463675 + 0.886006i \(0.346531\pi\)
\(60\) 0 0
\(61\) 9.12311 1.16809 0.584047 0.811720i \(-0.301468\pi\)
0.584047 + 0.811720i \(0.301468\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.56155 −0.193687
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.24621 −0.741289 −0.370644 0.928775i \(-0.620863\pi\)
−0.370644 + 0.928775i \(0.620863\pi\)
\(72\) 0 0
\(73\) −12.2462 −1.43331 −0.716655 0.697428i \(-0.754328\pi\)
−0.716655 + 0.697428i \(0.754328\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.36932 1.05413 0.527065 0.849825i \(-0.323292\pi\)
0.527065 + 0.849825i \(0.323292\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.876894 −0.0962517 −0.0481258 0.998841i \(-0.515325\pi\)
−0.0481258 + 0.998841i \(0.515325\pi\)
\(84\) 0 0
\(85\) 3.56155 0.386305
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.12311 0.119049 0.0595245 0.998227i \(-0.481042\pi\)
0.0595245 + 0.998227i \(0.481042\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.6847 −1.71181
\(96\) 0 0
\(97\) −2.87689 −0.292104 −0.146052 0.989277i \(-0.546657\pi\)
−0.146052 + 0.989277i \(0.546657\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.8769 −1.08229 −0.541146 0.840929i \(-0.682009\pi\)
−0.541146 + 0.840929i \(0.682009\pi\)
\(102\) 0 0
\(103\) −16.6847 −1.64399 −0.821994 0.569496i \(-0.807138\pi\)
−0.821994 + 0.569496i \(0.807138\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.68466 −0.452883 −0.226442 0.974025i \(-0.572709\pi\)
−0.226442 + 0.974025i \(0.572709\pi\)
\(108\) 0 0
\(109\) −6.87689 −0.658687 −0.329344 0.944210i \(-0.606827\pi\)
−0.329344 + 0.944210i \(0.606827\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.438447 0.0412456 0.0206228 0.999787i \(-0.493435\pi\)
0.0206228 + 0.999787i \(0.493435\pi\)
\(114\) 0 0
\(115\) 8.68466 0.809849
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.56155 −0.855211
\(126\) 0 0
\(127\) −19.8078 −1.75765 −0.878827 0.477140i \(-0.841674\pi\)
−0.878827 + 0.477140i \(0.841674\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 14.4384 1.26149 0.630746 0.775989i \(-0.282749\pi\)
0.630746 + 0.775989i \(0.282749\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.246211 −0.0210352 −0.0105176 0.999945i \(-0.503348\pi\)
−0.0105176 + 0.999945i \(0.503348\pi\)
\(138\) 0 0
\(139\) 0.876894 0.0743772 0.0371886 0.999308i \(-0.488160\pi\)
0.0371886 + 0.999308i \(0.488160\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.684658 0.0572540
\(144\) 0 0
\(145\) −29.3693 −2.43899
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.2462 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 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
\(155\) 11.1231 0.893429
\(156\) 0 0
\(157\) 6.68466 0.533494 0.266747 0.963767i \(-0.414051\pi\)
0.266747 + 0.963767i \(0.414051\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.1231 1.18453 0.592267 0.805742i \(-0.298233\pi\)
0.592267 + 0.805742i \(0.298233\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.8078 1.53277 0.766385 0.642381i \(-0.222053\pi\)
0.766385 + 0.642381i \(0.222053\pi\)
\(168\) 0 0
\(169\) −12.8078 −0.985213
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.80776 −0.137442 −0.0687209 0.997636i \(-0.521892\pi\)
−0.0687209 + 0.997636i \(0.521892\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −0.876894 −0.0655422 −0.0327711 0.999463i \(-0.510433\pi\)
−0.0327711 + 0.999463i \(0.510433\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 18.2462 1.34149
\(186\) 0 0
\(187\) −1.56155 −0.114192
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.87689 −0.352880 −0.176440 0.984311i \(-0.556458\pi\)
−0.176440 + 0.984311i \(0.556458\pi\)
\(192\) 0 0
\(193\) −7.75379 −0.558130 −0.279065 0.960272i \(-0.590024\pi\)
−0.279065 + 0.960272i \(0.590024\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.93087 0.636298 0.318149 0.948041i \(-0.396939\pi\)
0.318149 + 0.948041i \(0.396939\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −12.6847 −0.885935
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.31534 0.506013
\(210\) 0 0
\(211\) −13.3693 −0.920382 −0.460191 0.887820i \(-0.652219\pi\)
−0.460191 + 0.887820i \(0.652219\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.6847 1.13788
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.438447 −0.0294931
\(222\) 0 0
\(223\) 14.9309 0.999845 0.499922 0.866070i \(-0.333362\pi\)
0.499922 + 0.866070i \(0.333362\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.0540 −0.932795 −0.466398 0.884575i \(-0.654448\pi\)
−0.466398 + 0.884575i \(0.654448\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.56155 0.233325 0.116663 0.993172i \(-0.462780\pi\)
0.116663 + 0.993172i \(0.462780\pi\)
\(234\) 0 0
\(235\) 39.6155 2.58423
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.24621 −0.404034 −0.202017 0.979382i \(-0.564750\pi\)
−0.202017 + 0.979382i \(0.564750\pi\)
\(240\) 0 0
\(241\) 3.36932 0.217037 0.108518 0.994094i \(-0.465389\pi\)
0.108518 + 0.994094i \(0.465389\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 24.9309 1.59277
\(246\) 0 0
\(247\) 2.05398 0.130691
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.49242 0.536037 0.268018 0.963414i \(-0.413631\pi\)
0.268018 + 0.963414i \(0.413631\pi\)
\(252\) 0 0
\(253\) −3.80776 −0.239392
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.3693 0.958712 0.479356 0.877621i \(-0.340870\pi\)
0.479356 + 0.877621i \(0.340870\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.4924 −1.26362 −0.631808 0.775125i \(-0.717687\pi\)
−0.631808 + 0.775125i \(0.717687\pi\)
\(264\) 0 0
\(265\) 43.6155 2.67928
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.4384 −1.00227 −0.501135 0.865369i \(-0.667084\pi\)
−0.501135 + 0.865369i \(0.667084\pi\)
\(270\) 0 0
\(271\) −19.8078 −1.20324 −0.601618 0.798784i \(-0.705477\pi\)
−0.601618 + 0.798784i \(0.705477\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.8769 0.648861 0.324431 0.945910i \(-0.394827\pi\)
0.324431 + 0.945910i \(0.394827\pi\)
\(282\) 0 0
\(283\) −21.3693 −1.27027 −0.635137 0.772399i \(-0.719056\pi\)
−0.635137 + 0.772399i \(0.719056\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.12311 −0.0656125 −0.0328063 0.999462i \(-0.510444\pi\)
−0.0328063 + 0.999462i \(0.510444\pi\)
\(294\) 0 0
\(295\) −25.3693 −1.47706
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.06913 −0.0618294
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −32.4924 −1.86051
\(306\) 0 0
\(307\) −32.4924 −1.85444 −0.927220 0.374516i \(-0.877809\pi\)
−0.927220 + 0.374516i \(0.877809\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 33.6155 1.90006 0.950031 0.312156i \(-0.101051\pi\)
0.950031 + 0.312156i \(0.101051\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 12.8769 0.720968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.68466 −0.260661
\(324\) 0 0
\(325\) 3.36932 0.186896
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 34.9309 1.91997 0.959987 0.280044i \(-0.0903491\pi\)
0.959987 + 0.280044i \(0.0903491\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 14.2462 0.778354
\(336\) 0 0
\(337\) −16.7386 −0.911811 −0.455906 0.890028i \(-0.650685\pi\)
−0.455906 + 0.890028i \(0.650685\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.87689 −0.264099
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.49242 −0.455897 −0.227949 0.973673i \(-0.573202\pi\)
−0.227949 + 0.973673i \(0.573202\pi\)
\(348\) 0 0
\(349\) 11.5616 0.618876 0.309438 0.950920i \(-0.399859\pi\)
0.309438 + 0.950920i \(0.399859\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.4924 0.558455 0.279228 0.960225i \(-0.409922\pi\)
0.279228 + 0.960225i \(0.409922\pi\)
\(354\) 0 0
\(355\) 22.2462 1.18071
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.2462 0.751886 0.375943 0.926643i \(-0.377319\pi\)
0.375943 + 0.926643i \(0.377319\pi\)
\(360\) 0 0
\(361\) 2.94602 0.155054
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 43.6155 2.28294
\(366\) 0 0
\(367\) 1.75379 0.0915470 0.0457735 0.998952i \(-0.485425\pi\)
0.0457735 + 0.998952i \(0.485425\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.246211 −0.0127483 −0.00637417 0.999980i \(-0.502029\pi\)
−0.00637417 + 0.999980i \(0.502029\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.61553 0.186209
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.24621 0.319166 0.159583 0.987184i \(-0.448985\pi\)
0.159583 + 0.987184i \(0.448985\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −35.8617 −1.81826 −0.909131 0.416510i \(-0.863253\pi\)
−0.909131 + 0.416510i \(0.863253\pi\)
\(390\) 0 0
\(391\) 2.43845 0.123318
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −33.3693 −1.67899
\(396\) 0 0
\(397\) −19.3693 −0.972118 −0.486059 0.873926i \(-0.661566\pi\)
−0.486059 + 0.873926i \(0.661566\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −39.1771 −1.95641 −0.978205 0.207641i \(-0.933421\pi\)
−0.978205 + 0.207641i \(0.933421\pi\)
\(402\) 0 0
\(403\) −1.36932 −0.0682105
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −14.6847 −0.726110 −0.363055 0.931768i \(-0.618266\pi\)
−0.363055 + 0.931768i \(0.618266\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3.12311 0.153307
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.492423 −0.0240564 −0.0120282 0.999928i \(-0.503829\pi\)
−0.0120282 + 0.999928i \(0.503829\pi\)
\(420\) 0 0
\(421\) 24.4384 1.19106 0.595529 0.803334i \(-0.296943\pi\)
0.595529 + 0.803334i \(0.296943\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.68466 −0.372761
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 26.6847 1.28238 0.641191 0.767381i \(-0.278440\pi\)
0.641191 + 0.767381i \(0.278440\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.4233 −0.546450
\(438\) 0 0
\(439\) 22.2462 1.06175 0.530877 0.847449i \(-0.321863\pi\)
0.530877 + 0.847449i \(0.321863\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −31.1231 −1.47870 −0.739352 0.673319i \(-0.764868\pi\)
−0.739352 + 0.673319i \(0.764868\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −36.7386 −1.73380 −0.866902 0.498479i \(-0.833892\pi\)
−0.866902 + 0.498479i \(0.833892\pi\)
\(450\) 0 0
\(451\) 5.56155 0.261883
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13.8078 0.645900 0.322950 0.946416i \(-0.395325\pi\)
0.322950 + 0.946416i \(0.395325\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.24621 0.384064 0.192032 0.981389i \(-0.438492\pi\)
0.192032 + 0.981389i \(0.438492\pi\)
\(462\) 0 0
\(463\) 40.9848 1.90473 0.952364 0.304965i \(-0.0986447\pi\)
0.952364 + 0.304965i \(0.0986447\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.3693 0.988854 0.494427 0.869219i \(-0.335378\pi\)
0.494427 + 0.869219i \(0.335378\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.31534 −0.336360
\(474\) 0 0
\(475\) 36.0000 1.65179
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.3002 1.11030 0.555152 0.831749i \(-0.312660\pi\)
0.555152 + 0.831749i \(0.312660\pi\)
\(480\) 0 0
\(481\) −2.24621 −0.102418
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.2462 0.465256
\(486\) 0 0
\(487\) −17.3693 −0.787079 −0.393539 0.919308i \(-0.628750\pi\)
−0.393539 + 0.919308i \(0.628750\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −21.3693 −0.964384 −0.482192 0.876066i \(-0.660159\pi\)
−0.482192 + 0.876066i \(0.660159\pi\)
\(492\) 0 0
\(493\) −8.24621 −0.371391
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −13.3693 −0.598493 −0.299246 0.954176i \(-0.596735\pi\)
−0.299246 + 0.954176i \(0.596735\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −29.5616 −1.31808 −0.659042 0.752106i \(-0.729038\pi\)
−0.659042 + 0.752106i \(0.729038\pi\)
\(504\) 0 0
\(505\) 38.7386 1.72385
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −25.1231 −1.11356 −0.556781 0.830659i \(-0.687964\pi\)
−0.556781 + 0.830659i \(0.687964\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 59.4233 2.61850
\(516\) 0 0
\(517\) −17.3693 −0.763902
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 35.5616 1.55798 0.778990 0.627036i \(-0.215732\pi\)
0.778990 + 0.627036i \(0.215732\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.12311 0.136045
\(528\) 0 0
\(529\) −17.0540 −0.741477
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.56155 0.0676384
\(534\) 0 0
\(535\) 16.6847 0.721341
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10.9309 −0.470826
\(540\) 0 0
\(541\) 34.1080 1.46642 0.733208 0.680005i \(-0.238022\pi\)
0.733208 + 0.680005i \(0.238022\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.4924 1.04914
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 38.6307 1.64572
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −26.4924 −1.12252 −0.561260 0.827640i \(-0.689683\pi\)
−0.561260 + 0.827640i \(0.689683\pi\)
\(558\) 0 0
\(559\) −2.05398 −0.0868739
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.1231 1.31168 0.655841 0.754899i \(-0.272314\pi\)
0.655841 + 0.754899i \(0.272314\pi\)
\(564\) 0 0
\(565\) −1.56155 −0.0656950
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −21.1231 −0.885527 −0.442763 0.896639i \(-0.646002\pi\)
−0.442763 + 0.896639i \(0.646002\pi\)
\(570\) 0 0
\(571\) 30.7386 1.28637 0.643186 0.765710i \(-0.277612\pi\)
0.643186 + 0.765710i \(0.277612\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −18.7386 −0.781455
\(576\) 0 0
\(577\) −3.94602 −0.164275 −0.0821376 0.996621i \(-0.526175\pi\)
−0.0821376 + 0.996621i \(0.526175\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −19.1231 −0.791998
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.9848 −1.19633 −0.598166 0.801372i \(-0.704104\pi\)
−0.598166 + 0.801372i \(0.704104\pi\)
\(588\) 0 0
\(589\) −14.6307 −0.602847
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −27.7538 −1.13971 −0.569856 0.821745i \(-0.693001\pi\)
−0.569856 + 0.821745i \(0.693001\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −0.384472 −0.0157091 −0.00785455 0.999969i \(-0.502500\pi\)
−0.00785455 + 0.999969i \(0.502500\pi\)
\(600\) 0 0
\(601\) −30.9848 −1.26390 −0.631949 0.775010i \(-0.717745\pi\)
−0.631949 + 0.775010i \(0.717745\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 30.4924 1.23969
\(606\) 0 0
\(607\) 9.36932 0.380289 0.190144 0.981756i \(-0.439104\pi\)
0.190144 + 0.981756i \(0.439104\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.87689 −0.197298
\(612\) 0 0
\(613\) 14.6847 0.593108 0.296554 0.955016i \(-0.404163\pi\)
0.296554 + 0.955016i \(0.404163\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 44.2462 1.78129 0.890643 0.454704i \(-0.150255\pi\)
0.890643 + 0.454704i \(0.150255\pi\)
\(618\) 0 0
\(619\) −5.36932 −0.215811 −0.107906 0.994161i \(-0.534414\pi\)
−0.107906 + 0.994161i \(0.534414\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.12311 0.204272
\(630\) 0 0
\(631\) 0.684658 0.0272558 0.0136279 0.999907i \(-0.495662\pi\)
0.0136279 + 0.999907i \(0.495662\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 70.5464 2.79955
\(636\) 0 0
\(637\) −3.06913 −0.121603
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 28.9309 1.14270 0.571350 0.820706i \(-0.306420\pi\)
0.571350 + 0.820706i \(0.306420\pi\)
\(642\) 0 0
\(643\) 13.7538 0.542396 0.271198 0.962524i \(-0.412580\pi\)
0.271198 + 0.962524i \(0.412580\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.36932 −0.368346 −0.184173 0.982894i \(-0.558961\pi\)
−0.184173 + 0.982894i \(0.558961\pi\)
\(648\) 0 0
\(649\) 11.1231 0.436620
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 32.9309 1.28868 0.644342 0.764737i \(-0.277131\pi\)
0.644342 + 0.764737i \(0.277131\pi\)
\(654\) 0 0
\(655\) −51.4233 −2.00927
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.86174 −0.384159 −0.192079 0.981379i \(-0.561523\pi\)
−0.192079 + 0.981379i \(0.561523\pi\)
\(660\) 0 0
\(661\) 13.3153 0.517907 0.258953 0.965890i \(-0.416622\pi\)
0.258953 + 0.965890i \(0.416622\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.1080 −0.778583
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.2462 0.549969
\(672\) 0 0
\(673\) −0.738634 −0.0284722 −0.0142361 0.999899i \(-0.504532\pi\)
−0.0142361 + 0.999899i \(0.504532\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.31534 0.0505527 0.0252763 0.999681i \(-0.491953\pi\)
0.0252763 + 0.999681i \(0.491953\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.56155 −0.365863 −0.182931 0.983126i \(-0.558559\pi\)
−0.182931 + 0.983126i \(0.558559\pi\)
\(684\) 0 0
\(685\) 0.876894 0.0335044
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.36932 −0.204555
\(690\) 0 0
\(691\) 28.9848 1.10264 0.551318 0.834295i \(-0.314125\pi\)
0.551318 + 0.834295i \(0.314125\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.12311 −0.118466
\(696\) 0 0
\(697\) −3.56155 −0.134903
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −15.3693 −0.580491 −0.290246 0.956952i \(-0.593737\pi\)
−0.290246 + 0.956952i \(0.593737\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −44.7386 −1.68019 −0.840097 0.542436i \(-0.817502\pi\)
−0.840097 + 0.542436i \(0.817502\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.61553 0.285204
\(714\) 0 0
\(715\) −2.43845 −0.0911928
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −11.8078 −0.440355 −0.220178 0.975460i \(-0.570664\pi\)
−0.220178 + 0.975460i \(0.570664\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 63.3693 2.35348
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.68466 0.173268
\(732\) 0 0
\(733\) −11.7538 −0.434136 −0.217068 0.976156i \(-0.569649\pi\)
−0.217068 + 0.976156i \(0.569649\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.24621 −0.230082
\(738\) 0 0
\(739\) 20.6847 0.760897 0.380449 0.924802i \(-0.375770\pi\)
0.380449 + 0.924802i \(0.375770\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.4924 1.04529 0.522643 0.852552i \(-0.324946\pi\)
0.522643 + 0.852552i \(0.324946\pi\)
\(744\) 0 0
\(745\) 43.6155 1.59795
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 25.3693 0.925740 0.462870 0.886426i \(-0.346820\pi\)
0.462870 + 0.886426i \(0.346820\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 28.4924 1.03695
\(756\) 0 0
\(757\) 16.0540 0.583492 0.291746 0.956496i \(-0.405764\pi\)
0.291746 + 0.956496i \(0.405764\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.7538 0.571074 0.285537 0.958368i \(-0.407828\pi\)
0.285537 + 0.958368i \(0.407828\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.12311 0.112769
\(768\) 0 0
\(769\) 40.5464 1.46214 0.731070 0.682302i \(-0.239021\pi\)
0.731070 + 0.682302i \(0.239021\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.63068 0.310424 0.155212 0.987881i \(-0.450394\pi\)
0.155212 + 0.987881i \(0.450394\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 16.6847 0.597790
\(780\) 0 0
\(781\) −9.75379 −0.349018
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −23.8078 −0.849736
\(786\) 0 0
\(787\) 10.2462 0.365238 0.182619 0.983184i \(-0.441543\pi\)
0.182619 + 0.983184i \(0.441543\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.61553 0.340599 0.170300 0.985392i \(-0.445526\pi\)
0.170300 + 0.985392i \(0.445526\pi\)
\(798\) 0 0
\(799\) 11.1231 0.393507
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −19.1231 −0.674840
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15.9460 −0.560632 −0.280316 0.959908i \(-0.590439\pi\)
−0.280316 + 0.959908i \(0.590439\pi\)
\(810\) 0 0
\(811\) 45.3693 1.59313 0.796566 0.604551i \(-0.206648\pi\)
0.796566 + 0.604551i \(0.206648\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −53.8617 −1.88669
\(816\) 0 0
\(817\) −21.9460 −0.767794
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.4384 0.434105 0.217052 0.976160i \(-0.430356\pi\)
0.217052 + 0.976160i \(0.430356\pi\)
\(822\) 0 0
\(823\) −3.50758 −0.122266 −0.0611332 0.998130i \(-0.519471\pi\)
−0.0611332 + 0.998130i \(0.519471\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 47.4233 1.64907 0.824535 0.565811i \(-0.191437\pi\)
0.824535 + 0.565811i \(0.191437\pi\)
\(828\) 0 0
\(829\) 17.5076 0.608063 0.304032 0.952662i \(-0.401667\pi\)
0.304032 + 0.952662i \(0.401667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.00000 0.242536
\(834\) 0 0
\(835\) −70.5464 −2.44136
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26.0540 −0.899483 −0.449742 0.893159i \(-0.648484\pi\)
−0.449742 + 0.893159i \(0.648484\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 45.6155 1.56922
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.4924 0.428235
\(852\) 0 0
\(853\) −28.7386 −0.983992 −0.491996 0.870597i \(-0.663733\pi\)
−0.491996 + 0.870597i \(0.663733\pi\)
\(854\) 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
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9.75379 −0.332023 −0.166011 0.986124i \(-0.553089\pi\)
−0.166011 + 0.986124i \(0.553089\pi\)
\(864\) 0 0
\(865\) 6.43845 0.218914
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14.6307 0.496312
\(870\) 0 0
\(871\) −1.75379 −0.0594249
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −40.2462 −1.35593 −0.677965 0.735095i \(-0.737138\pi\)
−0.677965 + 0.735095i \(0.737138\pi\)
\(882\) 0 0
\(883\) 23.4233 0.788257 0.394128 0.919055i \(-0.371047\pi\)
0.394128 + 0.919055i \(0.371047\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.4384 0.619102 0.309551 0.950883i \(-0.399821\pi\)
0.309551 + 0.950883i \(0.399821\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −52.1080 −1.74373
\(894\) 0 0
\(895\) 3.12311 0.104394
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −25.7538 −0.858937
\(900\) 0 0
\(901\) 12.2462 0.407980
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −21.3693 −0.710340
\(906\) 0 0
\(907\) 9.86174 0.327454 0.163727 0.986506i \(-0.447648\pi\)
0.163727 + 0.986506i \(0.447648\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.3002 0.805101 0.402551 0.915398i \(-0.368124\pi\)
0.402551 + 0.915398i \(0.368124\pi\)
\(912\) 0 0
\(913\) −1.36932 −0.0453178
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16.6847 0.550376 0.275188 0.961390i \(-0.411260\pi\)
0.275188 + 0.961390i \(0.411260\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.73863 −0.0901432
\(924\) 0 0
\(925\) −39.3693 −1.29446
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.06913 −0.100695 −0.0503474 0.998732i \(-0.516033\pi\)
−0.0503474 + 0.998732i \(0.516033\pi\)
\(930\) 0 0
\(931\) −32.7926 −1.07473
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.56155 0.181882
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) −8.68466 −0.282811
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −5.36932 −0.174295
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −36.3542 −1.17763 −0.588813 0.808269i \(-0.700405\pi\)
−0.588813 + 0.808269i \(0.700405\pi\)
\(954\) 0 0
\(955\) 17.3693 0.562058
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −21.2462 −0.685362
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 27.6155 0.888975
\(966\) 0 0
\(967\) −42.4384 −1.36473 −0.682364 0.731012i \(-0.739048\pi\)
−0.682364 + 0.731012i \(0.739048\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −43.6155 −1.39969 −0.699844 0.714295i \(-0.746747\pi\)
−0.699844 + 0.714295i \(0.746747\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.24621 −0.263820 −0.131910 0.991262i \(-0.542111\pi\)
−0.131910 + 0.991262i \(0.542111\pi\)
\(978\) 0 0
\(979\) 1.75379 0.0560513
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 30.9309 0.986542 0.493271 0.869876i \(-0.335801\pi\)
0.493271 + 0.869876i \(0.335801\pi\)
\(984\) 0 0
\(985\) −31.8078 −1.01348
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.4233 0.363240
\(990\) 0 0
\(991\) −42.7386 −1.35764 −0.678819 0.734306i \(-0.737508\pi\)
−0.678819 + 0.734306i \(0.737508\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 56.9848 1.80654
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 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 2448.2.a.v.1.1 2
3.2 odd 2 816.2.a.m.1.2 2
4.3 odd 2 153.2.a.e.1.2 2
8.3 odd 2 9792.2.a.cy.1.2 2
8.5 even 2 9792.2.a.cz.1.2 2
12.11 even 2 51.2.a.b.1.1 2
20.19 odd 2 3825.2.a.s.1.1 2
24.5 odd 2 3264.2.a.bg.1.1 2
24.11 even 2 3264.2.a.bl.1.1 2
28.27 even 2 7497.2.a.v.1.2 2
60.23 odd 4 1275.2.b.d.1174.4 4
60.47 odd 4 1275.2.b.d.1174.1 4
60.59 even 2 1275.2.a.n.1.2 2
68.67 odd 2 2601.2.a.t.1.2 2
84.83 odd 2 2499.2.a.o.1.1 2
132.131 odd 2 6171.2.a.p.1.2 2
156.155 even 2 8619.2.a.q.1.2 2
204.11 odd 16 867.2.h.j.733.2 16
204.23 odd 16 867.2.h.j.733.1 16
204.47 even 4 867.2.d.c.577.3 4
204.59 even 8 867.2.e.f.829.4 8
204.71 odd 16 867.2.h.j.757.1 16
204.83 even 8 867.2.e.f.616.2 8
204.95 odd 16 867.2.h.j.712.3 16
204.107 odd 16 867.2.h.j.688.4 16
204.131 odd 16 867.2.h.j.688.3 16
204.143 odd 16 867.2.h.j.712.4 16
204.155 even 8 867.2.e.f.616.1 8
204.167 odd 16 867.2.h.j.757.2 16
204.179 even 8 867.2.e.f.829.3 8
204.191 even 4 867.2.d.c.577.4 4
204.203 even 2 867.2.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.2.a.b.1.1 2 12.11 even 2
153.2.a.e.1.2 2 4.3 odd 2
816.2.a.m.1.2 2 3.2 odd 2
867.2.a.f.1.1 2 204.203 even 2
867.2.d.c.577.3 4 204.47 even 4
867.2.d.c.577.4 4 204.191 even 4
867.2.e.f.616.1 8 204.155 even 8
867.2.e.f.616.2 8 204.83 even 8
867.2.e.f.829.3 8 204.179 even 8
867.2.e.f.829.4 8 204.59 even 8
867.2.h.j.688.3 16 204.131 odd 16
867.2.h.j.688.4 16 204.107 odd 16
867.2.h.j.712.3 16 204.95 odd 16
867.2.h.j.712.4 16 204.143 odd 16
867.2.h.j.733.1 16 204.23 odd 16
867.2.h.j.733.2 16 204.11 odd 16
867.2.h.j.757.1 16 204.71 odd 16
867.2.h.j.757.2 16 204.167 odd 16
1275.2.a.n.1.2 2 60.59 even 2
1275.2.b.d.1174.1 4 60.47 odd 4
1275.2.b.d.1174.4 4 60.23 odd 4
2448.2.a.v.1.1 2 1.1 even 1 trivial
2499.2.a.o.1.1 2 84.83 odd 2
2601.2.a.t.1.2 2 68.67 odd 2
3264.2.a.bg.1.1 2 24.5 odd 2
3264.2.a.bl.1.1 2 24.11 even 2
3825.2.a.s.1.1 2 20.19 odd 2
6171.2.a.p.1.2 2 132.131 odd 2
7497.2.a.v.1.2 2 28.27 even 2
8619.2.a.q.1.2 2 156.155 even 2
9792.2.a.cy.1.2 2 8.3 odd 2
9792.2.a.cz.1.2 2 8.5 even 2