Properties

Label 2400.2.f.d.1249.1
Level $2400$
Weight $2$
Character 2400.1249
Analytic conductor $19.164$
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] = [2400,2,Mod(1249,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2400.1249
Dual form 2400.2.f.d.1249.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -6.00000i q^{13} +2.00000i q^{17} +4.00000 q^{19} +4.00000 q^{21} +1.00000i q^{27} -10.0000 q^{29} +4.00000 q^{31} +4.00000i q^{33} -10.0000i q^{37} -6.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} -8.00000i q^{47} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{53} -4.00000i q^{57} +12.0000 q^{59} -10.0000 q^{61} -4.00000i q^{63} -12.0000i q^{67} -10.0000i q^{73} -16.0000i q^{77} -4.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +10.0000i q^{87} +6.00000 q^{89} +24.0000 q^{91} -4.00000i q^{93} -14.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 8 q^{11} + 8 q^{19} + 8 q^{21} - 20 q^{29} + 8 q^{31} - 12 q^{39} + 4 q^{41} - 18 q^{49} + 4 q^{51} + 24 q^{59} - 20 q^{61} - 8 q^{79} + 2 q^{81} + 12 q^{89} + 48 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 16.0000i − 1.82337i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.0000i 1.07211i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 16.0000i 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 22.0000i − 1.87959i −0.341743 0.939793i \(-0.611017\pi\)
0.341743 0.939793i \(-0.388983\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 24.0000i 2.00698i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 10.0000i 0.739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) − 40.0000i − 2.80745i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 28.0000i 1.87502i 0.347960 + 0.937509i \(0.386874\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 0 0
\(233\) − 2.00000i − 0.131024i −0.997852 0.0655122i \(-0.979132\pi\)
0.997852 0.0655122i \(-0.0208681\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 24.0000i − 1.52708i
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 26.0000i 1.62184i 0.585160 + 0.810918i \(0.301032\pi\)
−0.585160 + 0.810918i \(0.698968\pi\)
\(258\) 0 0
\(259\) 40.0000 2.48548
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) − 24.0000i − 1.45255i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.0000i 0.841178i 0.907251 + 0.420589i \(0.138177\pi\)
−0.907251 + 0.420589i \(0.861823\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000i 0.472225i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 4.00000i − 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) − 2.00000i − 0.114897i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 22.0000i − 1.23564i −0.786318 0.617822i \(-0.788015\pi\)
0.786318 0.617822i \(-0.211985\pi\)
\(318\) 0 0
\(319\) 40.0000 2.23957
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.00000i − 0.110600i
\(328\) 0 0
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 10.0000i 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) − 8.00000i − 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 8.00000i 0.423405i
\(358\) 0 0
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 36.0000i − 1.87918i −0.342296 0.939592i \(-0.611204\pi\)
0.342296 0.939592i \(-0.388796\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) 26.0000i 1.34623i 0.739538 + 0.673114i \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 60.0000i 3.09016i
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) − 16.0000i − 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 20.0000i 1.00887i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) − 24.0000i − 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 40.0000i 1.98273i
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −22.0000 −1.08518
\(412\) 0 0
\(413\) 48.0000i 2.36193i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 40.0000i − 1.93574i
\(428\) 0 0
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) 38.0000 1.79333 0.896665 0.442709i \(-0.145982\pi\)
0.896665 + 0.442709i \(0.145982\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 4.00000i 0.187936i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 6.00000i − 0.280668i −0.990104 0.140334i \(-0.955182\pi\)
0.990104 0.140334i \(-0.0448177\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) 48.0000 2.21643
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000i 0.906287i 0.891438 + 0.453143i \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 0 0
\(493\) − 20.0000i − 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) 0 0
\(503\) − 16.0000i − 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 40.0000 1.76950
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 32.0000i 1.40736i
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) − 12.0000i − 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000i 0.517838i
\(538\) 0 0
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) − 6.00000i − 0.257485i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) − 16.0000i − 0.680389i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.00000i 0.167984i
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 16.0000i 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) 0 0
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 20.0000i − 0.825488i −0.910847 0.412744i \(-0.864570\pi\)
0.910847 0.412744i \(-0.135430\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.0000i 0.818546i
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.00000i 0.162355i 0.996700 + 0.0811775i \(0.0258681\pi\)
−0.996700 + 0.0811775i \(0.974132\pi\)
\(608\) 0 0
\(609\) −40.0000 −1.62088
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) − 38.0000i − 1.53481i −0.641165 0.767403i \(-0.721549\pi\)
0.641165 0.767403i \(-0.278451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000i 0.961540i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 16.0000i 0.638978i
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 54.0000i 2.13956i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 0 0
\(653\) 30.0000i 1.17399i 0.809590 + 0.586995i \(0.199689\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 0 0
\(663\) − 12.0000i − 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 28.0000 1.08254
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 0 0
\(679\) 56.0000 2.14908
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 28.0000i 1.07139i 0.844411 + 0.535695i \(0.179950\pi\)
−0.844411 + 0.535695i \(0.820050\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 10.0000i − 0.381524i
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 0 0
\(693\) 16.0000i 0.607790i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) 0 0
\(699\) −2.00000 −0.0756469
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) − 40.0000i − 1.50863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.00000i 0.300871i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.0000i 0.597531i
\(718\) 0 0
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) − 18.0000i − 0.669427i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 4.00000i 0.148352i 0.997245 + 0.0741759i \(0.0236326\pi\)
−0.997245 + 0.0741759i \(0.976367\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000i 1.76810i
\(738\) 0 0
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) − 28.0000i − 1.02038i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) 8.00000i 0.289619i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 72.0000i − 2.59977i
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 26.0000 0.936367
\(772\) 0 0
\(773\) − 34.0000i − 1.22290i −0.791285 0.611448i \(-0.790588\pi\)
0.791285 0.611448i \(-0.209412\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 40.0000i − 1.43499i
\(778\) 0 0
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 10.0000i − 0.357371i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) 0 0
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 60.0000i 2.13066i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 40.0000i 1.41157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.00000i − 0.211210i
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 0 0
\(813\) − 28.0000i − 0.982003i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 16.0000i − 0.559769i
\(818\) 0 0
\(819\) −24.0000 −0.838628
\(820\) 0 0
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 0 0
\(823\) 12.0000i 0.418294i 0.977884 + 0.209147i \(0.0670687\pi\)
−0.977884 + 0.209147i \(0.932931\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 52.0000i − 1.80822i −0.427303 0.904109i \(-0.640536\pi\)
0.427303 0.904109i \(-0.359464\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) 0 0
\(833\) − 18.0000i − 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000i 0.138260i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 22.0000i 0.757720i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 20.0000i 0.687208i
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) − 32.0000i − 1.08929i −0.838666 0.544646i \(-0.816664\pi\)
0.838666 0.544646i \(-0.183336\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −72.0000 −2.43963
\(872\) 0 0
\(873\) 14.0000i 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) − 28.0000i − 0.942275i −0.882060 0.471138i \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 0 0
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) − 32.0000i − 1.07084i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000i 0.664089i 0.943264 + 0.332045i \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) − 16.0000i − 0.529523i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 80.0000i − 2.64183i
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4.00000i − 0.131377i
\(928\) 0 0
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 0 0
\(933\) 8.00000i 0.261908i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 10.0000i 0.326686i 0.986569 + 0.163343i \(0.0522277\pi\)
−0.986569 + 0.163343i \(0.947772\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) − 10.0000i − 0.323932i −0.986796 0.161966i \(-0.948217\pi\)
0.986796 0.161966i \(-0.0517835\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 40.0000i − 1.29302i
\(958\) 0 0
\(959\) 88.0000 2.84167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 12.0000i − 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 28.0000i 0.900419i 0.892923 + 0.450210i \(0.148651\pi\)
−0.892923 + 0.450210i \(0.851349\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 16.0000i 0.512936i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) − 8.00000i − 0.255160i −0.991828 0.127580i \(-0.959279\pi\)
0.991828 0.127580i \(-0.0407210\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 32.0000i − 1.01857i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 0 0
\(993\) − 28.0000i − 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 42.0000i − 1.33015i −0.746775 0.665077i \(-0.768399\pi\)
0.746775 0.665077i \(-0.231601\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 2400.2.f.d.1249.1 2
3.2 odd 2 7200.2.f.ba.6049.2 2
4.3 odd 2 2400.2.f.o.1249.2 2
5.2 odd 4 2400.2.a.a.1.1 1
5.3 odd 4 480.2.a.f.1.1 yes 1
5.4 even 2 inner 2400.2.f.d.1249.2 2
8.3 odd 2 4800.2.f.h.3649.1 2
8.5 even 2 4800.2.f.bb.3649.2 2
12.11 even 2 7200.2.f.c.6049.1 2
15.2 even 4 7200.2.a.d.1.1 1
15.8 even 4 1440.2.a.n.1.1 1
15.14 odd 2 7200.2.f.ba.6049.1 2
20.3 even 4 480.2.a.a.1.1 1
20.7 even 4 2400.2.a.bh.1.1 1
20.19 odd 2 2400.2.f.o.1249.1 2
40.3 even 4 960.2.a.m.1.1 1
40.13 odd 4 960.2.a.h.1.1 1
40.19 odd 2 4800.2.f.h.3649.2 2
40.27 even 4 4800.2.a.bg.1.1 1
40.29 even 2 4800.2.f.bb.3649.1 2
40.37 odd 4 4800.2.a.bo.1.1 1
60.23 odd 4 1440.2.a.g.1.1 1
60.47 odd 4 7200.2.a.bw.1.1 1
60.59 even 2 7200.2.f.c.6049.2 2
80.3 even 4 3840.2.k.bb.1921.1 2
80.13 odd 4 3840.2.k.c.1921.2 2
80.43 even 4 3840.2.k.bb.1921.2 2
80.53 odd 4 3840.2.k.c.1921.1 2
120.53 even 4 2880.2.a.p.1.1 1
120.83 odd 4 2880.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.2.a.a.1.1 1 20.3 even 4
480.2.a.f.1.1 yes 1 5.3 odd 4
960.2.a.h.1.1 1 40.13 odd 4
960.2.a.m.1.1 1 40.3 even 4
1440.2.a.g.1.1 1 60.23 odd 4
1440.2.a.n.1.1 1 15.8 even 4
2400.2.a.a.1.1 1 5.2 odd 4
2400.2.a.bh.1.1 1 20.7 even 4
2400.2.f.d.1249.1 2 1.1 even 1 trivial
2400.2.f.d.1249.2 2 5.4 even 2 inner
2400.2.f.o.1249.1 2 20.19 odd 2
2400.2.f.o.1249.2 2 4.3 odd 2
2880.2.a.c.1.1 1 120.83 odd 4
2880.2.a.p.1.1 1 120.53 even 4
3840.2.k.c.1921.1 2 80.53 odd 4
3840.2.k.c.1921.2 2 80.13 odd 4
3840.2.k.bb.1921.1 2 80.3 even 4
3840.2.k.bb.1921.2 2 80.43 even 4
4800.2.a.bg.1.1 1 40.27 even 4
4800.2.a.bo.1.1 1 40.37 odd 4
4800.2.f.h.3649.1 2 8.3 odd 2
4800.2.f.h.3649.2 2 40.19 odd 2
4800.2.f.bb.3649.1 2 40.29 even 2
4800.2.f.bb.3649.2 2 8.5 even 2
7200.2.a.d.1.1 1 15.2 even 4
7200.2.a.bw.1.1 1 60.47 odd 4
7200.2.f.c.6049.1 2 12.11 even 2
7200.2.f.c.6049.2 2 60.59 even 2
7200.2.f.ba.6049.1 2 15.14 odd 2
7200.2.f.ba.6049.2 2 3.2 odd 2