Properties

Label 7488.2.a.cj.1.1
Level $7488$
Weight $2$
Character 7488.1
Self dual yes
Analytic conductor $59.792$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7488,2,Mod(1,7488)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7488, 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("7488.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7488.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: \(59.7919810335\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7488.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{7} -3.46410 q^{11} -1.00000 q^{13} -6.92820 q^{17} +2.00000 q^{19} -6.92820 q^{23} -5.00000 q^{25} -6.92820 q^{29} -2.00000 q^{31} -2.00000 q^{37} +6.92820 q^{41} +8.00000 q^{43} +10.3923 q^{47} -3.00000 q^{49} +3.46410 q^{59} +10.0000 q^{61} +14.0000 q^{67} -3.46410 q^{71} -10.0000 q^{73} +6.92820 q^{77} +4.00000 q^{79} +10.3923 q^{83} +6.92820 q^{89} +2.00000 q^{91} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 2 q^{13} + 4 q^{19} - 10 q^{25} - 4 q^{31} - 4 q^{37} + 16 q^{43} - 6 q^{49} + 20 q^{61} + 28 q^{67} - 20 q^{73} + 8 q^{79} + 4 q^{91} - 20 q^{97}+O(q^{100}) \) 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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.92820 −1.68034 −0.840168 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.3923 1.51587 0.757937 0.652328i \(-0.226208\pi\)
0.757937 + 0.652328i \(0.226208\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.46410 0.450988 0.225494 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.3923 1.14070 0.570352 0.821401i \(-0.306807\pi\)
0.570352 + 0.821401i \(0.306807\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.92820 0.734388 0.367194 0.930144i \(-0.380318\pi\)
0.367194 + 0.930144i \(0.380318\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.8564 1.37876 0.689382 0.724398i \(-0.257882\pi\)
0.689382 + 0.724398i \(0.257882\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.92820 −0.651751 −0.325875 0.945413i \(-0.605659\pi\)
−0.325875 + 0.945413i \(0.605659\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 13.8564 1.27021
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −20.7846 −1.81596 −0.907980 0.419014i \(-0.862376\pi\)
−0.907980 + 0.419014i \(0.862376\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.92820 −0.591916 −0.295958 0.955201i \(-0.595639\pi\)
−0.295958 + 0.955201i \(0.595639\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.46410 0.289683
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.8564 −1.13516 −0.567581 0.823318i \(-0.692120\pi\)
−0.567581 + 0.823318i \(0.692120\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 13.8564 1.09204
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −13.8564 −1.05348 −0.526742 0.850026i \(-0.676586\pi\)
−0.526742 + 0.850026i \(0.676586\pi\)
\(174\) 0 0
\(175\) 10.0000 0.755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.8564 −1.03568 −0.517838 0.855479i \(-0.673263\pi\)
−0.517838 + 0.855479i \(0.673263\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.8564 −1.00261 −0.501307 0.865269i \(-0.667147\pi\)
−0.501307 + 0.865269i \(0.667147\pi\)
\(192\) 0 0
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.8564 0.987228 0.493614 0.869681i \(-0.335676\pi\)
0.493614 + 0.869681i \(0.335676\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 13.8564 0.972529
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.92820 −0.479234
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.92820 0.466041
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.46410 0.229920 0.114960 0.993370i \(-0.463326\pi\)
0.114960 + 0.993370i \(0.463326\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.92820 −0.437304 −0.218652 0.975803i \(-0.570166\pi\)
−0.218652 + 0.975803i \(0.570166\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.7128 1.72868 0.864339 0.502910i \(-0.167737\pi\)
0.864339 + 0.502910i \(0.167737\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.92820 −0.427211 −0.213606 0.976920i \(-0.568521\pi\)
−0.213606 + 0.976920i \(0.568521\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.92820 0.422420 0.211210 0.977441i \(-0.432260\pi\)
0.211210 + 0.977441i \(0.432260\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.3205 1.04447
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −20.7846 −1.23991 −0.619953 0.784639i \(-0.712848\pi\)
−0.619953 + 0.784639i \(0.712848\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.8564 −0.817918
\(288\) 0 0
\(289\) 31.0000 1.82353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.92820 0.400668
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.7846 1.17859 0.589294 0.807919i \(-0.299406\pi\)
0.589294 + 0.807919i \(0.299406\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −13.8564 −0.770991
\(324\) 0 0
\(325\) 5.00000 0.277350
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −20.7846 −1.14589
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.92820 0.375183
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −27.7128 −1.48770 −0.743851 0.668346i \(-0.767003\pi\)
−0.743851 + 0.668346i \(0.767003\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.92820 0.368751 0.184376 0.982856i \(-0.440974\pi\)
0.184376 + 0.982856i \(0.440974\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.92820 0.356821
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.3205 0.885037 0.442518 0.896759i \(-0.354085\pi\)
0.442518 + 0.896759i \(0.354085\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −20.7846 −1.05382 −0.526911 0.849921i \(-0.676650\pi\)
−0.526911 + 0.849921i \(0.676650\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −34.6410 −1.72989 −0.864945 0.501867i \(-0.832647\pi\)
−0.864945 + 0.501867i \(0.832647\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.92820 0.343418
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.92820 0.338465 0.169232 0.985576i \(-0.445871\pi\)
0.169232 + 0.985576i \(0.445871\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 34.6410 1.68034
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −38.1051 −1.83546 −0.917729 0.397206i \(-0.869980\pi\)
−0.917729 + 0.397206i \(0.869980\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13.8564 −0.662842
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.7846 0.987507 0.493753 0.869602i \(-0.335625\pi\)
0.493753 + 0.869602i \(0.335625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.92820 −0.326962 −0.163481 0.986546i \(-0.552272\pi\)
−0.163481 + 0.986546i \(0.552272\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) 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 \(0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 27.7128 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(462\) 0 0
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −27.7128 −1.27424
\(474\) 0 0
\(475\) −10.0000 −0.458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.46410 0.158279 0.0791394 0.996864i \(-0.474783\pi\)
0.0791394 + 0.996864i \(0.474783\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.92820 0.312665 0.156333 0.987704i \(-0.450033\pi\)
0.156333 + 0.987704i \(0.450033\pi\)
\(492\) 0 0
\(493\) 48.0000 2.16181
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.92820 0.310772
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.92820 0.308913 0.154457 0.988000i \(-0.450637\pi\)
0.154457 + 0.988000i \(0.450637\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.8564 0.614174 0.307087 0.951681i \(-0.400646\pi\)
0.307087 + 0.951681i \(0.400646\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.7846 0.910590 0.455295 0.890341i \(-0.349534\pi\)
0.455295 + 0.890341i \(0.349534\pi\)
\(522\) 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
\(527\) 13.8564 0.603595
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.92820 −0.300094
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.3923 0.447628
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −13.8564 −0.590303
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −27.7128 −1.17423 −0.587115 0.809504i \(-0.699736\pi\)
−0.587115 + 0.809504i \(0.699736\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 13.8564 0.583978 0.291989 0.956422i \(-0.405683\pi\)
0.291989 + 0.956422i \(0.405683\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.8564 −0.580891 −0.290445 0.956892i \(-0.593803\pi\)
−0.290445 + 0.956892i \(0.593803\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 34.6410 1.44463
\(576\) 0 0
\(577\) 26.0000 1.08239 0.541197 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −20.7846 −0.862291
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −45.0333 −1.85872 −0.929362 0.369170i \(-0.879642\pi\)
−0.929362 + 0.369170i \(0.879642\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.7846 0.853522 0.426761 0.904365i \(-0.359655\pi\)
0.426761 + 0.904365i \(0.359655\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −20.7846 −0.849236 −0.424618 0.905373i \(-0.639592\pi\)
−0.424618 + 0.905373i \(0.639592\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.3923 −0.420428
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.92820 −0.278919 −0.139459 0.990228i \(-0.544536\pi\)
−0.139459 + 0.990228i \(0.544536\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.8564 −0.555145
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 34.6410 1.36824 0.684119 0.729370i \(-0.260187\pi\)
0.684119 + 0.729370i \(0.260187\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.8564 0.544752 0.272376 0.962191i \(-0.412191\pi\)
0.272376 + 0.962191i \(0.412191\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.6410 1.35561 0.677804 0.735243i \(-0.262932\pi\)
0.677804 + 0.735243i \(0.262932\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13.8564 0.539769 0.269884 0.962893i \(-0.413014\pi\)
0.269884 + 0.962893i \(0.413014\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −34.6410 −1.33730
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 41.5692 1.59763 0.798817 0.601574i \(-0.205459\pi\)
0.798817 + 0.601574i \(0.205459\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −17.3205 −0.662751 −0.331375 0.943499i \(-0.607513\pi\)
−0.331375 + 0.943499i \(0.607513\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −48.0000 −1.81813
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 20.7846 0.785024 0.392512 0.919747i \(-0.371606\pi\)
0.392512 + 0.919747i \(0.371606\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −27.7128 −1.04225
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.8564 0.518927
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34.6410 −1.29189 −0.645946 0.763383i \(-0.723537\pi\)
−0.645946 + 0.763383i \(0.723537\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 34.6410 1.28654
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −55.4256 −2.04999
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −48.4974 −1.78643
\(738\) 0 0
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.2487 0.889599 0.444799 0.895630i \(-0.353275\pi\)
0.444799 + 0.895630i \(0.353275\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −13.8564 −0.506302
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.6410 1.25574 0.627868 0.778320i \(-0.283928\pi\)
0.627868 + 0.778320i \(0.283928\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3.46410 −0.125081
\(768\) 0 0
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −13.8564 −0.498380 −0.249190 0.968455i \(-0.580164\pi\)
−0.249190 + 0.968455i \(0.580164\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.8564 0.496457
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 13.8564 0.492677
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34.6410 −1.22705 −0.613524 0.789676i \(-0.710249\pi\)
−0.613524 + 0.789676i \(0.710249\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 34.6410 1.22245
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.92820 0.243583 0.121791 0.992556i \(-0.461136\pi\)
0.121791 + 0.992556i \(0.461136\pi\)
\(810\) 0 0
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.7128 −0.967184 −0.483592 0.875294i \(-0.660668\pi\)
−0.483592 + 0.875294i \(0.660668\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.3923 0.361376 0.180688 0.983540i \(-0.442168\pi\)
0.180688 + 0.983540i \(0.442168\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 20.7846 0.720144
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.2487 −0.837158 −0.418579 0.908180i \(-0.637472\pi\)
−0.418579 + 0.908180i \(0.637472\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.8564 0.474991
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 31.1769 1.06127 0.530637 0.847599i \(-0.321953\pi\)
0.530637 + 0.847599i \(0.321953\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13.8564 −0.470046
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −13.8564 −0.466834 −0.233417 0.972377i \(-0.574991\pi\)
−0.233417 + 0.972377i \(0.574991\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −27.7128 −0.930505 −0.465253 0.885178i \(-0.654037\pi\)
−0.465253 + 0.885178i \(0.654037\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 20.7846 0.695530
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.8564 0.462137
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 41.5692 1.37725 0.688625 0.725118i \(-0.258215\pi\)
0.688625 + 0.725118i \(0.258215\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 41.5692 1.37274
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.46410 0.114022
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −48.4974 −1.59115 −0.795574 0.605856i \(-0.792831\pi\)
−0.795574 + 0.605856i \(0.792831\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −41.5692 −1.35512 −0.677559 0.735469i \(-0.736962\pi\)
−0.677559 + 0.735469i \(0.736962\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.3205 0.562841 0.281420 0.959585i \(-0.409194\pi\)
0.281420 + 0.959585i \(0.409194\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −27.7128 −0.897706 −0.448853 0.893606i \(-0.648167\pi\)
−0.448853 + 0.893606i \(0.648167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 13.8564 0.447447
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 55.4256 1.77869 0.889346 0.457234i \(-0.151160\pi\)
0.889346 + 0.457234i \(0.151160\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.92820 0.221653 0.110826 0.993840i \(-0.464650\pi\)
0.110826 + 0.993840i \(0.464650\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −10.3923 −0.331463 −0.165732 0.986171i \(-0.552999\pi\)
−0.165732 + 0.986171i \(0.552999\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −55.4256 −1.76243
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 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 7488.2.a.cj.1.1 2
3.2 odd 2 inner 7488.2.a.cj.1.2 2
4.3 odd 2 7488.2.a.cq.1.2 2
8.3 odd 2 117.2.a.b.1.2 yes 2
8.5 even 2 1872.2.a.v.1.2 2
12.11 even 2 7488.2.a.cq.1.1 2
24.5 odd 2 1872.2.a.v.1.1 2
24.11 even 2 117.2.a.b.1.1 2
40.3 even 4 2925.2.c.s.2224.1 4
40.19 odd 2 2925.2.a.y.1.1 2
40.27 even 4 2925.2.c.s.2224.4 4
56.27 even 2 5733.2.a.t.1.2 2
72.11 even 6 1053.2.e.i.352.2 4
72.43 odd 6 1053.2.e.i.352.1 4
72.59 even 6 1053.2.e.i.703.2 4
72.67 odd 6 1053.2.e.i.703.1 4
104.51 odd 2 1521.2.a.j.1.1 2
104.83 even 4 1521.2.b.i.1351.2 4
104.99 even 4 1521.2.b.i.1351.3 4
120.59 even 2 2925.2.a.y.1.2 2
120.83 odd 4 2925.2.c.s.2224.3 4
120.107 odd 4 2925.2.c.s.2224.2 4
168.83 odd 2 5733.2.a.t.1.1 2
312.83 odd 4 1521.2.b.i.1351.4 4
312.155 even 2 1521.2.a.j.1.2 2
312.203 odd 4 1521.2.b.i.1351.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.a.b.1.1 2 24.11 even 2
117.2.a.b.1.2 yes 2 8.3 odd 2
1053.2.e.i.352.1 4 72.43 odd 6
1053.2.e.i.352.2 4 72.11 even 6
1053.2.e.i.703.1 4 72.67 odd 6
1053.2.e.i.703.2 4 72.59 even 6
1521.2.a.j.1.1 2 104.51 odd 2
1521.2.a.j.1.2 2 312.155 even 2
1521.2.b.i.1351.1 4 312.203 odd 4
1521.2.b.i.1351.2 4 104.83 even 4
1521.2.b.i.1351.3 4 104.99 even 4
1521.2.b.i.1351.4 4 312.83 odd 4
1872.2.a.v.1.1 2 24.5 odd 2
1872.2.a.v.1.2 2 8.5 even 2
2925.2.a.y.1.1 2 40.19 odd 2
2925.2.a.y.1.2 2 120.59 even 2
2925.2.c.s.2224.1 4 40.3 even 4
2925.2.c.s.2224.2 4 120.107 odd 4
2925.2.c.s.2224.3 4 120.83 odd 4
2925.2.c.s.2224.4 4 40.27 even 4
5733.2.a.t.1.1 2 168.83 odd 2
5733.2.a.t.1.2 2 56.27 even 2
7488.2.a.cj.1.1 2 1.1 even 1 trivial
7488.2.a.cj.1.2 2 3.2 odd 2 inner
7488.2.a.cq.1.1 2 12.11 even 2
7488.2.a.cq.1.2 2 4.3 odd 2