Properties

Label 2400.2.f.c.1249.2
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -3.00000i q^{13} +4.00000i q^{17} +1.00000 q^{19} +1.00000 q^{21} -1.00000i q^{27} +8.00000 q^{29} +1.00000 q^{31} -4.00000i q^{33} -2.00000i q^{37} +3.00000 q^{39} +2.00000 q^{41} -11.0000i q^{43} +2.00000i q^{47} +6.00000 q^{49} -4.00000 q^{51} -10.0000i q^{53} +1.00000i q^{57} +6.00000 q^{59} +11.0000 q^{61} +1.00000i q^{63} -9.00000i q^{67} +6.00000 q^{71} -14.0000i q^{73} +4.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} +2.00000i q^{83} +8.00000i q^{87} -3.00000 q^{91} +1.00000i q^{93} +11.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 8 q^{11} + 2 q^{19} + 2 q^{21} + 16 q^{29} + 2 q^{31} + 6 q^{39} + 4 q^{41} + 12 q^{49} - 8 q^{51} + 12 q^{59} + 22 q^{61} + 12 q^{71} - 32 q^{79} + 2 q^{81} - 6 q^{91} + 8 q^{99}+O(q^{100}) \) 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
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 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
\(13\) − 3.00000i − 0.832050i −0.909353 0.416025i \(-0.863423\pi\)
0.909353 0.416025i \(-0.136577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 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
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) − 4.00000i − 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 3.00000 0.480384
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 11.0000i − 1.67748i −0.544529 0.838742i \(-0.683292\pi\)
0.544529 0.838742i \(-0.316708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 9.00000i − 1.09952i −0.835321 0.549762i \(-0.814718\pi\)
0.835321 0.549762i \(-0.185282\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.00000i 0.857690i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0 0
\(93\) 1.00000i 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.0000i 1.11688i 0.829545 + 0.558440i \(0.188600\pi\)
−0.829545 + 0.558440i \(0.811400\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) − 12.0000i − 1.12887i −0.825479 0.564433i \(-0.809095\pi\)
0.825479 0.564433i \(-0.190905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.00000i 0.277350i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 2.00000i 0.180334i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 0 0
\(133\) − 1.00000i − 0.0867110i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2.00000i − 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 23.0000 1.87171 0.935857 0.352381i \(-0.114628\pi\)
0.935857 + 0.352381i \(0.114628\pi\)
\(152\) 0 0
\(153\) − 4.00000i − 0.323381i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.00000i 0.399043i 0.979893 + 0.199522i \(0.0639388\pi\)
−0.979893 + 0.199522i \(0.936061\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 9.00000i − 0.704934i −0.935824 0.352467i \(-0.885343\pi\)
0.935824 0.352467i \(-0.114657\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.0000i 1.70241i 0.524832 + 0.851206i \(0.324128\pi\)
−0.524832 + 0.851206i \(0.675872\pi\)
\(168\) 0 0
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) − 24.0000i − 1.82469i −0.409426 0.912343i \(-0.634271\pi\)
0.409426 0.912343i \(-0.365729\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −15.0000 −1.11494 −0.557471 0.830197i \(-0.688228\pi\)
−0.557471 + 0.830197i \(0.688228\pi\)
\(182\) 0 0
\(183\) 11.0000i 0.813143i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 16.0000i − 1.17004i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) − 13.0000i − 0.935760i −0.883792 0.467880i \(-0.845018\pi\)
0.883792 0.467880i \(-0.154982\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 0 0
\(203\) − 8.00000i − 0.561490i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 6.00000i 0.411113i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.00000i − 0.0678844i
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) − 19.0000i − 1.27233i −0.771551 0.636167i \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) 21.0000 1.35273 0.676364 0.736567i \(-0.263554\pi\)
0.676364 + 0.736567i \(0.263554\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.00000i − 0.190885i
\(248\) 0 0
\(249\) −2.00000 −0.126745
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 8.00000i − 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) 26.0000i 1.60323i 0.597841 + 0.801614i \(0.296025\pi\)
−0.597841 + 0.801614i \(0.703975\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) − 3.00000i − 0.181568i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.00000i 0.420589i 0.977638 + 0.210295i \(0.0674423\pi\)
−0.977638 + 0.210295i \(0.932558\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) − 17.0000i − 1.01055i −0.862960 0.505273i \(-0.831392\pi\)
0.862960 0.505273i \(-0.168608\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.00000i − 0.118056i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −11.0000 −0.644831
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) 0 0
\(303\) 14.0000i 0.804279i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 23.0000i − 1.31268i −0.754466 0.656340i \(-0.772104\pi\)
0.754466 0.656340i \(-0.227896\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −26.0000 −1.47432 −0.737162 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) 0 0
\(313\) 15.0000i 0.847850i 0.905697 + 0.423925i \(0.139348\pi\)
−0.905697 + 0.423925i \(0.860652\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 28.0000i 1.57264i 0.617822 + 0.786318i \(0.288015\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) −32.0000 −1.79166
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 4.00000i 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.0000i 0.608301i
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 13.0000i − 0.708155i −0.935216 0.354078i \(-0.884795\pi\)
0.935216 0.354078i \(-0.115205\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) − 13.0000i − 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 30.0000i − 1.61048i −0.592946 0.805242i \(-0.702035\pi\)
0.592946 0.805242i \(-0.297965\pi\)
\(348\) 0 0
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 0 0
\(353\) 24.0000i 1.27739i 0.769460 + 0.638696i \(0.220526\pi\)
−0.769460 + 0.638696i \(0.779474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 0 0
\(359\) 26.0000 1.37223 0.686114 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3.00000i − 0.156599i −0.996930 0.0782994i \(-0.975051\pi\)
0.996930 0.0782994i \(-0.0249490\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 19.0000i 0.983783i 0.870657 + 0.491891i \(0.163694\pi\)
−0.870657 + 0.491891i \(0.836306\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 24.0000i − 1.23606i
\(378\) 0 0
\(379\) −33.0000 −1.69510 −0.847548 0.530719i \(-0.821922\pi\)
−0.847548 + 0.530719i \(0.821922\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.0000i 0.559161i
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) − 14.0000i − 0.706207i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 23.0000i − 1.15434i −0.816625 0.577168i \(-0.804158\pi\)
0.816625 0.577168i \(-0.195842\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) − 3.00000i − 0.149441i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000i 0.396545i
\(408\) 0 0
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) − 6.00000i − 0.295241i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 20.0000i − 0.979404i
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) − 2.00000i − 0.0972433i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 11.0000i − 0.532327i
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) − 1.00000i − 0.0480569i −0.999711 0.0240285i \(-0.992351\pi\)
0.999711 0.0240285i \(-0.00764923\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.00000i 0.0945968i
\(448\) 0 0
\(449\) −40.0000 −1.88772 −0.943858 0.330350i \(-0.892833\pi\)
−0.943858 + 0.330350i \(0.892833\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 23.0000i 1.08063i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000i 0.280668i 0.990104 + 0.140334i \(0.0448177\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) − 20.0000i − 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) 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
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) −5.00000 −0.230388
\(472\) 0 0
\(473\) 44.0000i 2.02312i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 11.0000i − 0.498458i −0.968445 0.249229i \(-0.919823\pi\)
0.968445 0.249229i \(-0.0801771\pi\)
\(488\) 0 0
\(489\) 9.00000 0.406994
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 32.0000i 1.44121i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.00000i − 0.269137i
\(498\) 0 0
\(499\) 13.0000 0.581960 0.290980 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) 0 0
\(501\) −22.0000 −0.982888
\(502\) 0 0
\(503\) 10.0000i 0.445878i 0.974832 + 0.222939i \(0.0715651\pi\)
−0.974832 + 0.222939i \(0.928435\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.00000i 0.177646i
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) − 1.00000i − 0.0441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 8.00000i − 0.351840i
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) − 17.0000i − 0.743358i −0.928361 0.371679i \(-0.878782\pi\)
0.928361 0.371679i \(-0.121218\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) − 6.00000i − 0.259889i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18.0000i 0.776757i
\(538\) 0 0
\(539\) −24.0000 −1.03375
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) − 15.0000i − 0.643712i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 0 0
\(549\) −11.0000 −0.469469
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 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 \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) −33.0000 −1.39575
\(560\) 0 0
\(561\) 16.0000 0.675521
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 31.0000 1.29731 0.648655 0.761083i \(-0.275332\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) 0 0
\(573\) 20.0000i 0.835512i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 7.00000i − 0.291414i −0.989328 0.145707i \(-0.953454\pi\)
0.989328 0.145707i \(-0.0465456\pi\)
\(578\) 0 0
\(579\) 13.0000 0.540262
\(580\) 0 0
\(581\) 2.00000 0.0829740
\(582\) 0 0
\(583\) 40.0000i 1.65663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.0000i 1.07313i 0.843857 + 0.536567i \(0.180279\pi\)
−0.843857 + 0.536567i \(0.819721\pi\)
\(588\) 0 0
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 11.0000i − 0.450200i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 0 0
\(603\) 9.00000i 0.366508i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) 38.0000i 1.53481i 0.641165 + 0.767403i \(0.278451\pi\)
−0.641165 + 0.767403i \(0.721549\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) −23.0000 −0.924448 −0.462224 0.886763i \(-0.652948\pi\)
−0.462224 + 0.886763i \(0.652948\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 4.00000i − 0.159745i
\(628\) 0 0
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 0 0
\(633\) − 5.00000i − 0.198732i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 18.0000i − 0.713186i
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) 0 0
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000i 0.546192i
\(658\) 0 0
\(659\) −14.0000 −0.545363 −0.272681 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) −44.0000 −1.69860
\(672\) 0 0
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 44.0000i − 1.69106i −0.533930 0.845529i \(-0.679285\pi\)
0.533930 0.845529i \(-0.320715\pi\)
\(678\) 0 0
\(679\) 11.0000 0.422141
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) 8.00000i 0.306111i 0.988218 + 0.153056i \(0.0489114\pi\)
−0.988218 + 0.153056i \(0.951089\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 5.00000i − 0.190762i
\(688\) 0 0
\(689\) −30.0000 −1.14291
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 0 0
\(693\) − 4.00000i − 0.151947i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.00000i 0.303022i
\(698\) 0 0
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) − 2.00000i − 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 14.0000i − 0.526524i
\(708\) 0 0
\(709\) −17.0000 −0.638448 −0.319224 0.947679i \(-0.603422\pi\)
−0.319224 + 0.947679i \(0.603422\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 10.0000i − 0.373457i
\(718\) 0 0
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 21.0000i 0.780998i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.0000i 0.853023i 0.904482 + 0.426511i \(0.140258\pi\)
−0.904482 + 0.426511i \(0.859742\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 44.0000 1.62740
\(732\) 0 0
\(733\) 6.00000i 0.221615i 0.993842 + 0.110808i \(0.0353437\pi\)
−0.993842 + 0.110808i \(0.964656\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.0000i 1.32608i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 3.00000 0.110208
\(742\) 0 0
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 2.00000i − 0.0731762i
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) − 8.00000i − 0.291536i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.0000i 0.399802i 0.979816 + 0.199901i \(0.0640620\pi\)
−0.979816 + 0.199901i \(0.935938\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20.0000 −0.724999 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(762\) 0 0
\(763\) − 11.0000i − 0.398227i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 18.0000i − 0.649942i
\(768\) 0 0
\(769\) 9.00000 0.324548 0.162274 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) 0 0
\(773\) − 2.00000i − 0.0719350i −0.999353 0.0359675i \(-0.988549\pi\)
0.999353 0.0359675i \(-0.0114513\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.00000i − 0.0717496i
\(778\) 0 0
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 0 0
\(783\) − 8.00000i − 0.285897i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 53.0000i 1.88925i 0.328158 + 0.944623i \(0.393572\pi\)
−0.328158 + 0.944623i \(0.606428\pi\)
\(788\) 0 0
\(789\) −26.0000 −0.925625
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) − 33.0000i − 1.17186i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.0000i 0.850124i 0.905164 + 0.425062i \(0.139748\pi\)
−0.905164 + 0.425062i \(0.860252\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 56.0000i 1.97620i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12.0000i 0.422420i
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 43.0000 1.50993 0.754967 0.655763i \(-0.227653\pi\)
0.754967 + 0.655763i \(0.227653\pi\)
\(812\) 0 0
\(813\) 28.0000i 0.982003i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 11.0000i − 0.384841i
\(818\) 0 0
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 0 0
\(823\) − 39.0000i − 1.35945i −0.733465 0.679727i \(-0.762098\pi\)
0.733465 0.679727i \(-0.237902\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.0000i 0.347734i 0.984769 + 0.173867i \(0.0556263\pi\)
−0.984769 + 0.173867i \(0.944374\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −7.00000 −0.242827
\(832\) 0 0
\(833\) 24.0000i 0.831551i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 1.00000i − 0.0345651i
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) − 16.0000i − 0.551069i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 5.00000i − 0.171802i
\(848\) 0 0
\(849\) 17.0000 0.583438
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 17.0000i − 0.582069i −0.956713 0.291034i \(-0.906001\pi\)
0.956713 0.291034i \(-0.0939994\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 24.0000i − 0.819824i −0.912125 0.409912i \(-0.865559\pi\)
0.912125 0.409912i \(-0.134441\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 2.00000 0.0681598
\(862\) 0 0
\(863\) − 28.0000i − 0.953131i −0.879139 0.476566i \(-0.841881\pi\)
0.879139 0.476566i \(-0.158119\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000i 0.0339618i
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) −27.0000 −0.914860
\(872\) 0 0
\(873\) − 11.0000i − 0.372294i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 17.0000i 0.574049i 0.957923 + 0.287025i \(0.0926662\pi\)
−0.957923 + 0.287025i \(0.907334\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 0 0
\(883\) 25.0000i 0.841317i 0.907219 + 0.420658i \(0.138201\pi\)
−0.907219 + 0.420658i \(0.861799\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 34.0000i − 1.14161i −0.821086 0.570804i \(-0.806632\pi\)
0.821086 0.570804i \(-0.193368\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 2.00000i 0.0669274i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 40.0000 1.33259
\(902\) 0 0
\(903\) − 11.0000i − 0.366057i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 20.0000i − 0.664089i −0.943264 0.332045i \(-0.892262\pi\)
0.943264 0.332045i \(-0.107738\pi\)
\(908\) 0 0
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 50.0000 1.65657 0.828287 0.560304i \(-0.189316\pi\)
0.828287 + 0.560304i \(0.189316\pi\)
\(912\) 0 0
\(913\) − 8.00000i − 0.264761i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 14.0000i 0.462321i
\(918\) 0 0
\(919\) 49.0000 1.61636 0.808180 0.588935i \(-0.200453\pi\)
0.808180 + 0.588935i \(0.200453\pi\)
\(920\) 0 0
\(921\) 23.0000 0.757876
\(922\) 0 0
\(923\) − 18.0000i − 0.592477i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 8.00000i − 0.262754i
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) − 26.0000i − 0.851202i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 53.0000i 1.73143i 0.500533 + 0.865717i \(0.333137\pi\)
−0.500533 + 0.865717i \(0.666863\pi\)
\(938\) 0 0
\(939\) −15.0000 −0.489506
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.0000i 1.03986i 0.854209 + 0.519930i \(0.174042\pi\)
−0.854209 + 0.519930i \(0.825958\pi\)
\(948\) 0 0
\(949\) −42.0000 −1.36338
\(950\) 0 0
\(951\) −28.0000 −0.907962
\(952\) 0 0
\(953\) − 8.00000i − 0.259145i −0.991570 0.129573i \(-0.958639\pi\)
0.991570 0.129573i \(-0.0413606\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 32.0000i − 1.03441i
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 0 0
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 20.0000i 0.641171i
\(974\) 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
\(979\) 0 0
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) 0 0
\(983\) 44.0000i 1.40338i 0.712481 + 0.701691i \(0.247571\pi\)
−0.712481 + 0.701691i \(0.752429\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2.00000i 0.0636607i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −47.0000 −1.49300 −0.746502 0.665383i \(-0.768268\pi\)
−0.746502 + 0.665383i \(0.768268\pi\)
\(992\) 0 0
\(993\) − 8.00000i − 0.253872i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 18.0000i − 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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.c.1249.2 2
3.2 odd 2 7200.2.f.z.6049.1 2
4.3 odd 2 2400.2.f.p.1249.1 2
5.2 odd 4 2400.2.a.ba.1.1 yes 1
5.3 odd 4 2400.2.a.e.1.1 1
5.4 even 2 inner 2400.2.f.c.1249.1 2
8.3 odd 2 4800.2.f.g.3649.2 2
8.5 even 2 4800.2.f.bd.3649.1 2
12.11 even 2 7200.2.f.d.6049.2 2
15.2 even 4 7200.2.a.bk.1.1 1
15.8 even 4 7200.2.a.t.1.1 1
15.14 odd 2 7200.2.f.z.6049.2 2
20.3 even 4 2400.2.a.bd.1.1 yes 1
20.7 even 4 2400.2.a.h.1.1 yes 1
20.19 odd 2 2400.2.f.p.1249.2 2
40.3 even 4 4800.2.a.v.1.1 1
40.13 odd 4 4800.2.a.by.1.1 1
40.19 odd 2 4800.2.f.g.3649.1 2
40.27 even 4 4800.2.a.bv.1.1 1
40.29 even 2 4800.2.f.bd.3649.2 2
40.37 odd 4 4800.2.a.y.1.1 1
60.23 odd 4 7200.2.a.bh.1.1 1
60.47 odd 4 7200.2.a.q.1.1 1
60.59 even 2 7200.2.f.d.6049.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2400.2.a.e.1.1 1 5.3 odd 4
2400.2.a.h.1.1 yes 1 20.7 even 4
2400.2.a.ba.1.1 yes 1 5.2 odd 4
2400.2.a.bd.1.1 yes 1 20.3 even 4
2400.2.f.c.1249.1 2 5.4 even 2 inner
2400.2.f.c.1249.2 2 1.1 even 1 trivial
2400.2.f.p.1249.1 2 4.3 odd 2
2400.2.f.p.1249.2 2 20.19 odd 2
4800.2.a.v.1.1 1 40.3 even 4
4800.2.a.y.1.1 1 40.37 odd 4
4800.2.a.bv.1.1 1 40.27 even 4
4800.2.a.by.1.1 1 40.13 odd 4
4800.2.f.g.3649.1 2 40.19 odd 2
4800.2.f.g.3649.2 2 8.3 odd 2
4800.2.f.bd.3649.1 2 8.5 even 2
4800.2.f.bd.3649.2 2 40.29 even 2
7200.2.a.q.1.1 1 60.47 odd 4
7200.2.a.t.1.1 1 15.8 even 4
7200.2.a.bh.1.1 1 60.23 odd 4
7200.2.a.bk.1.1 1 15.2 even 4
7200.2.f.d.6049.1 2 60.59 even 2
7200.2.f.d.6049.2 2 12.11 even 2
7200.2.f.z.6049.1 2 3.2 odd 2
7200.2.f.z.6049.2 2 15.14 odd 2