Properties

Label 2304.2.c.j.2303.8
Level $2304$
Weight $2$
Character 2304.2303
Analytic conductor $18.398$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,2,Mod(2303,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("2304.2303");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.c (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: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2303.8
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2303
Dual form 2304.2.c.j.2303.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.44949i q^{5} +2.00000i q^{7} +O(q^{10})\) \(q+2.44949i q^{5} +2.00000i q^{7} +4.89898 q^{11} +3.46410 q^{13} -4.24264i q^{17} -6.92820i q^{19} +8.48528 q^{23} -1.00000 q^{25} -7.34847i q^{29} +2.00000i q^{31} -4.89898 q^{35} -6.92820 q^{37} +4.24264i q^{41} +8.48528 q^{47} +3.00000 q^{49} +2.44949i q^{53} +12.0000i q^{55} -6.92820 q^{61} +8.48528i q^{65} +6.92820i q^{67} +8.48528 q^{71} -4.00000 q^{73} +9.79796i q^{77} +14.0000i q^{79} +4.89898 q^{83} +10.3923 q^{85} -4.24264i q^{89} +6.92820i q^{91} +16.9706 q^{95} -16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{25} + 24 q^{49} - 32 q^{73} - 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(5\) 2.44949i 1.09545i 0.836660 + 0.547723i \(0.184505\pi\)
−0.836660 + 0.547723i \(0.815495\pi\)
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.24264i − 1.02899i −0.857493 0.514496i \(-0.827979\pi\)
0.857493 0.514496i \(-0.172021\pi\)
\(18\) 0 0
\(19\) − 6.92820i − 1.58944i −0.606977 0.794719i \(-0.707618\pi\)
0.606977 0.794719i \(-0.292382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.48528 1.76930 0.884652 0.466252i \(-0.154396\pi\)
0.884652 + 0.466252i \(0.154396\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 7.34847i − 1.36458i −0.731083 0.682288i \(-0.760985\pi\)
0.731083 0.682288i \(-0.239015\pi\)
\(30\) 0 0
\(31\) 2.00000i 0.359211i 0.983739 + 0.179605i \(0.0574821\pi\)
−0.983739 + 0.179605i \(0.942518\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.89898 −0.828079
\(36\) 0 0
\(37\) −6.92820 −1.13899 −0.569495 0.821995i \(-0.692861\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.24264i 0.662589i 0.943527 + 0.331295i \(0.107485\pi\)
−0.943527 + 0.331295i \(0.892515\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.48528 1.23771 0.618853 0.785507i \(-0.287598\pi\)
0.618853 + 0.785507i \(0.287598\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.44949i 0.336463i 0.985747 + 0.168232i \(0.0538057\pi\)
−0.985747 + 0.168232i \(0.946194\pi\)
\(54\) 0 0
\(55\) 12.0000i 1.61808i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.92820 −0.887066 −0.443533 0.896258i \(-0.646275\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.48528i 1.05247i
\(66\) 0 0
\(67\) 6.92820i 0.846415i 0.906033 + 0.423207i \(0.139096\pi\)
−0.906033 + 0.423207i \(0.860904\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.48528 1.00702 0.503509 0.863990i \(-0.332042\pi\)
0.503509 + 0.863990i \(0.332042\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.79796i 1.11658i
\(78\) 0 0
\(79\) 14.0000i 1.57512i 0.616236 + 0.787562i \(0.288657\pi\)
−0.616236 + 0.787562i \(0.711343\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.89898 0.537733 0.268866 0.963177i \(-0.413351\pi\)
0.268866 + 0.963177i \(0.413351\pi\)
\(84\) 0 0
\(85\) 10.3923 1.12720
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4.24264i − 0.449719i −0.974391 0.224860i \(-0.927808\pi\)
0.974391 0.224860i \(-0.0721923\pi\)
\(90\) 0 0
\(91\) 6.92820i 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.9706 1.74114
\(96\) 0 0
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 12.2474i − 1.21867i −0.792914 0.609333i \(-0.791437\pi\)
0.792914 0.609333i \(-0.208563\pi\)
\(102\) 0 0
\(103\) − 2.00000i − 0.197066i −0.995134 0.0985329i \(-0.968585\pi\)
0.995134 0.0985329i \(-0.0314150\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −19.5959 −1.89441 −0.947204 0.320630i \(-0.896105\pi\)
−0.947204 + 0.320630i \(0.896105\pi\)
\(108\) 0 0
\(109\) 10.3923 0.995402 0.497701 0.867349i \(-0.334178\pi\)
0.497701 + 0.867349i \(0.334178\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 4.24264i − 0.399114i −0.979886 0.199557i \(-0.936050\pi\)
0.979886 0.199557i \(-0.0639503\pi\)
\(114\) 0 0
\(115\) 20.7846i 1.93817i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.48528 0.777844
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.79796i 0.876356i
\(126\) 0 0
\(127\) 10.0000i 0.887357i 0.896186 + 0.443678i \(0.146327\pi\)
−0.896186 + 0.443678i \(0.853673\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.79796 0.856052 0.428026 0.903767i \(-0.359209\pi\)
0.428026 + 0.903767i \(0.359209\pi\)
\(132\) 0 0
\(133\) 13.8564 1.20150
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.24264i − 0.362473i −0.983440 0.181237i \(-0.941990\pi\)
0.983440 0.181237i \(-0.0580100\pi\)
\(138\) 0 0
\(139\) − 6.92820i − 0.587643i −0.955860 0.293821i \(-0.905073\pi\)
0.955860 0.293821i \(-0.0949270\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.9706 1.41915
\(144\) 0 0
\(145\) 18.0000 1.49482
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2.44949i − 0.200670i −0.994954 0.100335i \(-0.968009\pi\)
0.994954 0.100335i \(-0.0319915\pi\)
\(150\) 0 0
\(151\) 2.00000i 0.162758i 0.996683 + 0.0813788i \(0.0259324\pi\)
−0.996683 + 0.0813788i \(0.974068\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.89898 −0.393496
\(156\) 0 0
\(157\) −20.7846 −1.65879 −0.829396 0.558661i \(-0.811315\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.9706i 1.33747i
\(162\) 0 0
\(163\) − 13.8564i − 1.08532i −0.839953 0.542659i \(-0.817418\pi\)
0.839953 0.542659i \(-0.182582\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.44949i 0.186231i 0.995655 + 0.0931156i \(0.0296826\pi\)
−0.995655 + 0.0931156i \(0.970317\pi\)
\(174\) 0 0
\(175\) − 2.00000i − 0.151186i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.79796 −0.732334 −0.366167 0.930549i \(-0.619330\pi\)
−0.366167 + 0.930549i \(0.619330\pi\)
\(180\) 0 0
\(181\) −17.3205 −1.28742 −0.643712 0.765268i \(-0.722606\pi\)
−0.643712 + 0.765268i \(0.722606\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 16.9706i − 1.24770i
\(186\) 0 0
\(187\) − 20.7846i − 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2.44949i − 0.174519i −0.996186 0.0872595i \(-0.972189\pi\)
0.996186 0.0872595i \(-0.0278109\pi\)
\(198\) 0 0
\(199\) 22.0000i 1.55954i 0.626067 + 0.779769i \(0.284664\pi\)
−0.626067 + 0.779769i \(0.715336\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 14.6969 1.03152
\(204\) 0 0
\(205\) −10.3923 −0.725830
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 33.9411i − 2.34776i
\(210\) 0 0
\(211\) − 6.92820i − 0.476957i −0.971148 0.238479i \(-0.923351\pi\)
0.971148 0.238479i \(-0.0766487\pi\)
\(212\) 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
\(221\) − 14.6969i − 0.988623i
\(222\) 0 0
\(223\) 10.0000i 0.669650i 0.942280 + 0.334825i \(0.108677\pi\)
−0.942280 + 0.334825i \(0.891323\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.6969 0.975470 0.487735 0.872992i \(-0.337823\pi\)
0.487735 + 0.872992i \(0.337823\pi\)
\(228\) 0 0
\(229\) −10.3923 −0.686743 −0.343371 0.939200i \(-0.611569\pi\)
−0.343371 + 0.939200i \(0.611569\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.2132i 1.38972i 0.719144 + 0.694862i \(0.244534\pi\)
−0.719144 + 0.694862i \(0.755466\pi\)
\(234\) 0 0
\(235\) 20.7846i 1.35584i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.34847i 0.469476i
\(246\) 0 0
\(247\) − 24.0000i − 1.52708i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 14.6969 0.927663 0.463831 0.885924i \(-0.346474\pi\)
0.463831 + 0.885924i \(0.346474\pi\)
\(252\) 0 0
\(253\) 41.5692 2.61343
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.2132i 1.32324i 0.749838 + 0.661622i \(0.230131\pi\)
−0.749838 + 0.661622i \(0.769869\pi\)
\(258\) 0 0
\(259\) − 13.8564i − 0.860995i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.9706 1.04645 0.523225 0.852195i \(-0.324729\pi\)
0.523225 + 0.852195i \(0.324729\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 22.0454i − 1.34413i −0.740491 0.672066i \(-0.765407\pi\)
0.740491 0.672066i \(-0.234593\pi\)
\(270\) 0 0
\(271\) − 14.0000i − 0.850439i −0.905090 0.425220i \(-0.860197\pi\)
0.905090 0.425220i \(-0.139803\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.89898 −0.295420
\(276\) 0 0
\(277\) 10.3923 0.624413 0.312207 0.950014i \(-0.398932\pi\)
0.312207 + 0.950014i \(0.398932\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.7279i 0.759284i 0.925133 + 0.379642i \(0.123953\pi\)
−0.925133 + 0.379642i \(0.876047\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.48528 −0.500870
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.0454i 1.28791i 0.765065 + 0.643953i \(0.222707\pi\)
−0.765065 + 0.643953i \(0.777293\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 29.3939 1.69989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 16.9706i − 0.971732i
\(306\) 0 0
\(307\) − 20.7846i − 1.18624i −0.805114 0.593120i \(-0.797896\pi\)
0.805114 0.593120i \(-0.202104\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.9706 −0.962312 −0.481156 0.876635i \(-0.659783\pi\)
−0.481156 + 0.876635i \(0.659783\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.1464i 0.963039i 0.876435 + 0.481520i \(0.159915\pi\)
−0.876435 + 0.481520i \(0.840085\pi\)
\(318\) 0 0
\(319\) − 36.0000i − 2.01561i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −29.3939 −1.63552
\(324\) 0 0
\(325\) −3.46410 −0.192154
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16.9706i 0.935617i
\(330\) 0 0
\(331\) − 6.92820i − 0.380808i −0.981706 0.190404i \(-0.939020\pi\)
0.981706 0.190404i \(-0.0609799\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −16.9706 −0.927201
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.79796i 0.530589i
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.89898 −0.262991 −0.131495 0.991317i \(-0.541978\pi\)
−0.131495 + 0.991317i \(0.541978\pi\)
\(348\) 0 0
\(349\) 6.92820 0.370858 0.185429 0.982658i \(-0.440632\pi\)
0.185429 + 0.982658i \(0.440632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 29.6985i 1.58069i 0.612661 + 0.790345i \(0.290099\pi\)
−0.612661 + 0.790345i \(0.709901\pi\)
\(354\) 0 0
\(355\) 20.7846i 1.10313i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.48528 −0.447836 −0.223918 0.974608i \(-0.571885\pi\)
−0.223918 + 0.974608i \(0.571885\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 9.79796i − 0.512849i
\(366\) 0 0
\(367\) 2.00000i 0.104399i 0.998637 + 0.0521996i \(0.0166232\pi\)
−0.998637 + 0.0521996i \(0.983377\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.89898 −0.254342
\(372\) 0 0
\(373\) −6.92820 −0.358729 −0.179364 0.983783i \(-0.557404\pi\)
−0.179364 + 0.983783i \(0.557404\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 25.4558i − 1.31104i
\(378\) 0 0
\(379\) 34.6410i 1.77939i 0.456556 + 0.889695i \(0.349083\pi\)
−0.456556 + 0.889695i \(0.650917\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.9706 −0.867155 −0.433578 0.901116i \(-0.642749\pi\)
−0.433578 + 0.901116i \(0.642749\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 26.9444i 1.36613i 0.730355 + 0.683067i \(0.239354\pi\)
−0.730355 + 0.683067i \(0.760646\pi\)
\(390\) 0 0
\(391\) − 36.0000i − 1.82060i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −34.2929 −1.72546
\(396\) 0 0
\(397\) −20.7846 −1.04315 −0.521575 0.853206i \(-0.674655\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 38.1838i − 1.90681i −0.301699 0.953403i \(-0.597554\pi\)
0.301699 0.953403i \(-0.402446\pi\)
\(402\) 0 0
\(403\) 6.92820i 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −33.9411 −1.68240
\(408\) 0 0
\(409\) 8.00000 0.395575 0.197787 0.980245i \(-0.436624\pi\)
0.197787 + 0.980245i \(0.436624\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000i 0.589057i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.6969 −0.717992 −0.358996 0.933339i \(-0.616881\pi\)
−0.358996 + 0.933339i \(0.616881\pi\)
\(420\) 0 0
\(421\) −3.46410 −0.168830 −0.0844150 0.996431i \(-0.526902\pi\)
−0.0844150 + 0.996431i \(0.526902\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.24264i 0.205798i
\(426\) 0 0
\(427\) − 13.8564i − 0.670559i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.4558 −1.22616 −0.613082 0.790019i \(-0.710071\pi\)
−0.613082 + 0.790019i \(0.710071\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 58.7878i − 2.81220i
\(438\) 0 0
\(439\) − 14.0000i − 0.668184i −0.942541 0.334092i \(-0.891570\pi\)
0.942541 0.334092i \(-0.108430\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14.6969 −0.698273 −0.349136 0.937072i \(-0.613525\pi\)
−0.349136 + 0.937072i \(0.613525\pi\)
\(444\) 0 0
\(445\) 10.3923 0.492642
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.24264i 0.200223i 0.994976 + 0.100111i \(0.0319199\pi\)
−0.994976 + 0.100111i \(0.968080\pi\)
\(450\) 0 0
\(451\) 20.7846i 0.978709i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −16.9706 −0.795592
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 2.44949i − 0.114084i −0.998372 0.0570421i \(-0.981833\pi\)
0.998372 0.0570421i \(-0.0181669\pi\)
\(462\) 0 0
\(463\) − 26.0000i − 1.20832i −0.796862 0.604161i \(-0.793508\pi\)
0.796862 0.604161i \(-0.206492\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 34.2929 1.58688 0.793442 0.608646i \(-0.208287\pi\)
0.793442 + 0.608646i \(0.208287\pi\)
\(468\) 0 0
\(469\) −13.8564 −0.639829
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.92820i 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.48528 −0.387702 −0.193851 0.981031i \(-0.562098\pi\)
−0.193851 + 0.981031i \(0.562098\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 39.1918i − 1.77961i
\(486\) 0 0
\(487\) − 10.0000i − 0.453143i −0.973995 0.226572i \(-0.927248\pi\)
0.973995 0.226572i \(-0.0727517\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.79796 −0.442176 −0.221088 0.975254i \(-0.570961\pi\)
−0.221088 + 0.975254i \(0.570961\pi\)
\(492\) 0 0
\(493\) −31.1769 −1.40414
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.9706i 0.761234i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.48528 −0.378340 −0.189170 0.981944i \(-0.560580\pi\)
−0.189170 + 0.981944i \(0.560580\pi\)
\(504\) 0 0
\(505\) 30.0000 1.33498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.2474i 0.542859i 0.962458 + 0.271429i \(0.0874963\pi\)
−0.962458 + 0.271429i \(0.912504\pi\)
\(510\) 0 0
\(511\) − 8.00000i − 0.353899i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.89898 0.215875
\(516\) 0 0
\(517\) 41.5692 1.82821
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 21.2132i − 0.929367i −0.885477 0.464684i \(-0.846168\pi\)
0.885477 0.464684i \(-0.153832\pi\)
\(522\) 0 0
\(523\) − 6.92820i − 0.302949i −0.988461 0.151475i \(-0.951598\pi\)
0.988461 0.151475i \(-0.0484022\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.48528 0.369625
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.6969i 0.636595i
\(534\) 0 0
\(535\) − 48.0000i − 2.07522i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14.6969 0.633042
\(540\) 0 0
\(541\) −17.3205 −0.744667 −0.372333 0.928099i \(-0.621442\pi\)
−0.372333 + 0.928099i \(0.621442\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 25.4558i 1.09041i
\(546\) 0 0
\(547\) 27.7128i 1.18491i 0.805602 + 0.592457i \(0.201842\pi\)
−0.805602 + 0.592457i \(0.798158\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −50.9117 −2.16891
\(552\) 0 0
\(553\) −28.0000 −1.19068
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.9444i 1.14167i 0.821065 + 0.570835i \(0.193380\pi\)
−0.821065 + 0.570835i \(0.806620\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.4949 1.03234 0.516168 0.856487i \(-0.327358\pi\)
0.516168 + 0.856487i \(0.327358\pi\)
\(564\) 0 0
\(565\) 10.3923 0.437208
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 38.1838i − 1.60075i −0.599502 0.800373i \(-0.704635\pi\)
0.599502 0.800373i \(-0.295365\pi\)
\(570\) 0 0
\(571\) − 13.8564i − 0.579873i −0.957046 0.289936i \(-0.906366\pi\)
0.957046 0.289936i \(-0.0936341\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.79796i 0.406488i
\(582\) 0 0
\(583\) 12.0000i 0.496989i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.5959 −0.808810 −0.404405 0.914580i \(-0.632521\pi\)
−0.404405 + 0.914580i \(0.632521\pi\)
\(588\) 0 0
\(589\) 13.8564 0.570943
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 29.6985i − 1.21957i −0.792567 0.609785i \(-0.791256\pi\)
0.792567 0.609785i \(-0.208744\pi\)
\(594\) 0 0
\(595\) 20.7846i 0.852086i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.48528 0.346699 0.173350 0.984860i \(-0.444541\pi\)
0.173350 + 0.984860i \(0.444541\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.8434i 1.29462i
\(606\) 0 0
\(607\) − 38.0000i − 1.54237i −0.636610 0.771186i \(-0.719664\pi\)
0.636610 0.771186i \(-0.280336\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 29.3939 1.18915
\(612\) 0 0
\(613\) 6.92820 0.279827 0.139914 0.990164i \(-0.455317\pi\)
0.139914 + 0.990164i \(0.455317\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.7279i 0.512407i 0.966623 + 0.256203i \(0.0824717\pi\)
−0.966623 + 0.256203i \(0.917528\pi\)
\(618\) 0 0
\(619\) − 13.8564i − 0.556936i −0.960446 0.278468i \(-0.910173\pi\)
0.960446 0.278468i \(-0.0898266\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.48528 0.339956
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 29.3939i 1.17201i
\(630\) 0 0
\(631\) 14.0000i 0.557331i 0.960388 + 0.278666i \(0.0898921\pi\)
−0.960388 + 0.278666i \(0.910108\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.4949 −0.972050
\(636\) 0 0
\(637\) 10.3923 0.411758
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.7279i 0.502723i 0.967893 + 0.251361i \(0.0808782\pi\)
−0.967893 + 0.251361i \(0.919122\pi\)
\(642\) 0 0
\(643\) − 41.5692i − 1.63933i −0.572843 0.819665i \(-0.694160\pi\)
0.572843 0.819665i \(-0.305840\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −42.4264 −1.66795 −0.833977 0.551799i \(-0.813942\pi\)
−0.833977 + 0.551799i \(0.813942\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.44949i 0.0958559i 0.998851 + 0.0479280i \(0.0152618\pi\)
−0.998851 + 0.0479280i \(0.984738\pi\)
\(654\) 0 0
\(655\) 24.0000i 0.937758i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −39.1918 −1.52670 −0.763349 0.645987i \(-0.776446\pi\)
−0.763349 + 0.645987i \(0.776446\pi\)
\(660\) 0 0
\(661\) 6.92820 0.269476 0.134738 0.990881i \(-0.456981\pi\)
0.134738 + 0.990881i \(0.456981\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 33.9411i 1.31618i
\(666\) 0 0
\(667\) − 62.3538i − 2.41435i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −33.9411 −1.31028
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 36.7423i − 1.41212i −0.708150 0.706062i \(-0.750470\pi\)
0.708150 0.706062i \(-0.249530\pi\)
\(678\) 0 0
\(679\) − 32.0000i − 1.22805i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0908 1.68709 0.843544 0.537060i \(-0.180465\pi\)
0.843544 + 0.537060i \(0.180465\pi\)
\(684\) 0 0
\(685\) 10.3923 0.397070
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.48528i 0.323263i
\(690\) 0 0
\(691\) − 41.5692i − 1.58137i −0.612225 0.790684i \(-0.709725\pi\)
0.612225 0.790684i \(-0.290275\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.9706 0.643730
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 17.1464i − 0.647612i −0.946124 0.323806i \(-0.895038\pi\)
0.946124 0.323806i \(-0.104962\pi\)
\(702\) 0 0
\(703\) 48.0000i 1.81035i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.4949 0.921225
\(708\) 0 0
\(709\) −24.2487 −0.910679 −0.455340 0.890318i \(-0.650482\pi\)
−0.455340 + 0.890318i \(0.650482\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.9706i 0.635553i
\(714\) 0 0
\(715\) 41.5692i 1.55460i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −25.4558 −0.949343 −0.474671 0.880163i \(-0.657433\pi\)
−0.474671 + 0.880163i \(0.657433\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.34847i 0.272915i
\(726\) 0 0
\(727\) 14.0000i 0.519231i 0.965712 + 0.259616i \(0.0835959\pi\)
−0.965712 + 0.259616i \(0.916404\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −10.3923 −0.383849 −0.191924 0.981410i \(-0.561473\pi\)
−0.191924 + 0.981410i \(0.561473\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.9411i 1.25024i
\(738\) 0 0
\(739\) − 41.5692i − 1.52915i −0.644536 0.764574i \(-0.722949\pi\)
0.644536 0.764574i \(-0.277051\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.9411 1.24518 0.622590 0.782549i \(-0.286081\pi\)
0.622590 + 0.782549i \(0.286081\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 39.1918i − 1.43204i
\(750\) 0 0
\(751\) − 46.0000i − 1.67856i −0.543696 0.839282i \(-0.682976\pi\)
0.543696 0.839282i \(-0.317024\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.89898 −0.178292
\(756\) 0 0
\(757\) 17.3205 0.629525 0.314762 0.949171i \(-0.398075\pi\)
0.314762 + 0.949171i \(0.398075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 12.7279i − 0.461387i −0.973026 0.230693i \(-0.925901\pi\)
0.973026 0.230693i \(-0.0740994\pi\)
\(762\) 0 0
\(763\) 20.7846i 0.752453i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 7.34847i − 0.264306i −0.991229 0.132153i \(-0.957811\pi\)
0.991229 0.132153i \(-0.0421890\pi\)
\(774\) 0 0
\(775\) − 2.00000i − 0.0718421i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 29.3939 1.05314
\(780\) 0 0
\(781\) 41.5692 1.48746
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 50.9117i − 1.81712i
\(786\) 0 0
\(787\) 34.6410i 1.23482i 0.786642 + 0.617409i \(0.211818\pi\)
−0.786642 + 0.617409i \(0.788182\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.48528 0.301702
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 26.9444i − 0.954419i −0.878790 0.477210i \(-0.841648\pi\)
0.878790 0.477210i \(-0.158352\pi\)
\(798\) 0 0
\(799\) − 36.0000i − 1.27359i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −19.5959 −0.691525
\(804\) 0 0
\(805\) −41.5692 −1.46512
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 38.1838i 1.34247i 0.741245 + 0.671235i \(0.234236\pi\)
−0.741245 + 0.671235i \(0.765764\pi\)
\(810\) 0 0
\(811\) − 20.7846i − 0.729846i −0.931038 0.364923i \(-0.881095\pi\)
0.931038 0.364923i \(-0.118905\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 33.9411 1.18891
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.34847i 0.256463i 0.991744 + 0.128232i \(0.0409301\pi\)
−0.991744 + 0.128232i \(0.959070\pi\)
\(822\) 0 0
\(823\) 26.0000i 0.906303i 0.891434 + 0.453152i \(0.149700\pi\)
−0.891434 + 0.453152i \(0.850300\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.3939 1.02213 0.511063 0.859543i \(-0.329252\pi\)
0.511063 + 0.859543i \(0.329252\pi\)
\(828\) 0 0
\(829\) −45.0333 −1.56407 −0.782036 0.623233i \(-0.785819\pi\)
−0.782036 + 0.623233i \(0.785819\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 12.7279i − 0.440996i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 25.4558 0.878833 0.439417 0.898283i \(-0.355185\pi\)
0.439417 + 0.898283i \(0.355185\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 2.44949i − 0.0842650i
\(846\) 0 0
\(847\) 26.0000i 0.893371i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −58.7878 −2.01522
\(852\) 0 0
\(853\) −48.4974 −1.66052 −0.830260 0.557376i \(-0.811808\pi\)
−0.830260 + 0.557376i \(0.811808\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.24264i 0.144926i 0.997371 + 0.0724629i \(0.0230859\pi\)
−0.997371 + 0.0724629i \(0.976914\pi\)
\(858\) 0 0
\(859\) 27.7128i 0.945549i 0.881183 + 0.472774i \(0.156747\pi\)
−0.881183 + 0.472774i \(0.843253\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 33.9411 1.15537 0.577685 0.816260i \(-0.303956\pi\)
0.577685 + 0.816260i \(0.303956\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 68.5857i 2.32661i
\(870\) 0 0
\(871\) 24.0000i 0.813209i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −19.5959 −0.662463
\(876\) 0 0
\(877\) 6.92820 0.233949 0.116974 0.993135i \(-0.462680\pi\)
0.116974 + 0.993135i \(0.462680\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 21.2132i 0.714691i 0.933972 + 0.357345i \(0.116318\pi\)
−0.933972 + 0.357345i \(0.883682\pi\)
\(882\) 0 0
\(883\) 27.7128i 0.932610i 0.884624 + 0.466305i \(0.154415\pi\)
−0.884624 + 0.466305i \(0.845585\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.9706 −0.569816 −0.284908 0.958555i \(-0.591963\pi\)
−0.284908 + 0.958555i \(0.591963\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 58.7878i − 1.96726i
\(894\) 0 0
\(895\) − 24.0000i − 0.802232i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 14.6969 0.490170
\(900\) 0 0
\(901\) 10.3923 0.346218
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 42.4264i − 1.41030i
\(906\) 0 0
\(907\) 55.4256i 1.84038i 0.391475 + 0.920189i \(0.371965\pi\)
−0.391475 + 0.920189i \(0.628035\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −33.9411 −1.12452 −0.562260 0.826961i \(-0.690068\pi\)
−0.562260 + 0.826961i \(0.690068\pi\)
\(912\) 0 0
\(913\) 24.0000 0.794284
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 19.5959i 0.647114i
\(918\) 0 0
\(919\) 26.0000i 0.857661i 0.903385 + 0.428830i \(0.141074\pi\)
−0.903385 + 0.428830i \(0.858926\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 29.3939 0.967511
\(924\) 0 0
\(925\) 6.92820 0.227798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 38.1838i − 1.25277i −0.779514 0.626384i \(-0.784534\pi\)
0.779514 0.626384i \(-0.215466\pi\)
\(930\) 0 0
\(931\) − 20.7846i − 0.681188i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 50.9117 1.66499
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 56.3383i 1.83657i 0.395914 + 0.918287i \(0.370428\pi\)
−0.395914 + 0.918287i \(0.629572\pi\)
\(942\) 0 0
\(943\) 36.0000i 1.17232i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19.5959 0.636782 0.318391 0.947960i \(-0.396858\pi\)
0.318391 + 0.947960i \(0.396858\pi\)
\(948\) 0 0
\(949\) −13.8564 −0.449798
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 21.2132i − 0.687163i −0.939123 0.343582i \(-0.888360\pi\)
0.939123 0.343582i \(-0.111640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.48528 0.274004
\(960\) 0 0
\(961\) 27.0000 0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 24.4949i − 0.788519i
\(966\) 0 0
\(967\) − 26.0000i − 0.836104i −0.908423 0.418052i \(-0.862713\pi\)
0.908423 0.418052i \(-0.137287\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.89898 0.157216 0.0786079 0.996906i \(-0.474952\pi\)
0.0786079 + 0.996906i \(0.474952\pi\)
\(972\) 0 0
\(973\) 13.8564 0.444216
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 29.6985i − 0.950139i −0.879948 0.475069i \(-0.842423\pi\)
0.879948 0.475069i \(-0.157577\pi\)
\(978\) 0 0
\(979\) − 20.7846i − 0.664279i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 50.9117 1.62383 0.811915 0.583775i \(-0.198425\pi\)
0.811915 + 0.583775i \(0.198425\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 50.0000i 1.58830i 0.607720 + 0.794151i \(0.292084\pi\)
−0.607720 + 0.794151i \(0.707916\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −53.8888 −1.70839
\(996\) 0 0
\(997\) 20.7846 0.658255 0.329128 0.944285i \(-0.393245\pi\)
0.329128 + 0.944285i \(0.393245\pi\)
\(998\) 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 2304.2.c.j.2303.8 8
3.2 odd 2 inner 2304.2.c.j.2303.4 8
4.3 odd 2 inner 2304.2.c.j.2303.6 8
8.3 odd 2 inner 2304.2.c.j.2303.1 8
8.5 even 2 inner 2304.2.c.j.2303.3 8
12.11 even 2 inner 2304.2.c.j.2303.2 8
16.3 odd 4 576.2.f.a.287.8 yes 8
16.5 even 4 576.2.f.a.287.1 8
16.11 odd 4 576.2.f.a.287.3 yes 8
16.13 even 4 576.2.f.a.287.6 yes 8
24.5 odd 2 inner 2304.2.c.j.2303.7 8
24.11 even 2 inner 2304.2.c.j.2303.5 8
48.5 odd 4 576.2.f.a.287.5 yes 8
48.11 even 4 576.2.f.a.287.7 yes 8
48.29 odd 4 576.2.f.a.287.2 yes 8
48.35 even 4 576.2.f.a.287.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.f.a.287.1 8 16.5 even 4
576.2.f.a.287.2 yes 8 48.29 odd 4
576.2.f.a.287.3 yes 8 16.11 odd 4
576.2.f.a.287.4 yes 8 48.35 even 4
576.2.f.a.287.5 yes 8 48.5 odd 4
576.2.f.a.287.6 yes 8 16.13 even 4
576.2.f.a.287.7 yes 8 48.11 even 4
576.2.f.a.287.8 yes 8 16.3 odd 4
2304.2.c.j.2303.1 8 8.3 odd 2 inner
2304.2.c.j.2303.2 8 12.11 even 2 inner
2304.2.c.j.2303.3 8 8.5 even 2 inner
2304.2.c.j.2303.4 8 3.2 odd 2 inner
2304.2.c.j.2303.5 8 24.11 even 2 inner
2304.2.c.j.2303.6 8 4.3 odd 2 inner
2304.2.c.j.2303.7 8 24.5 odd 2 inner
2304.2.c.j.2303.8 8 1.1 even 1 trivial