Properties

Label 4800.2.f.t.3649.1
Level $4800$
Weight $2$
Character 4800.3649
Analytic conductor $38.328$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4800,2,Mod(3649,4800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4800, 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("4800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.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: \(38.3281929702\)
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 2400)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.3649
Dual form 4800.2.f.t.3649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -1.00000i q^{13} +3.00000 q^{19} -1.00000 q^{21} +4.00000i q^{23} +1.00000i q^{27} +4.00000 q^{29} -7.00000 q^{31} -6.00000i q^{37} -1.00000 q^{39} +6.00000 q^{41} -9.00000i q^{43} -6.00000i q^{47} +6.00000 q^{49} -2.00000i q^{53} -3.00000i q^{57} +10.0000 q^{59} +1.00000 q^{61} +1.00000i q^{63} -3.00000i q^{67} +4.00000 q^{69} -14.0000 q^{71} +10.0000i q^{73} -8.00000 q^{79} +1.00000 q^{81} -18.0000i q^{83} -4.00000i q^{87} -1.00000 q^{91} +7.00000i q^{93} +3.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 6 q^{19} - 2 q^{21} + 8 q^{29} - 14 q^{31} - 2 q^{39} + 12 q^{41} + 12 q^{49} + 20 q^{59} + 2 q^{61} + 8 q^{69} - 28 q^{71} - 16 q^{79} + 2 q^{81} - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 9.00000i − 1.37249i −0.727372 0.686244i \(-0.759258\pi\)
0.727372 0.686244i \(-0.240742\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 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
\(57\) − 3.00000i − 0.397360i
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 18.0000i − 1.97576i −0.155230 0.987878i \(-0.549612\pi\)
0.155230 0.987878i \(-0.450388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.00000i − 0.428845i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 7.00000i 0.725866i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.00000i 0.304604i 0.988334 + 0.152302i \(0.0486686\pi\)
−0.988334 + 0.152302i \(0.951331\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 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
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) −15.0000 −1.43674 −0.718370 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 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
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) − 3.00000i − 0.260133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 10.0000i − 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 6.00000i − 0.494872i
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 9.00000i − 0.718278i −0.933284 0.359139i \(-0.883070\pi\)
0.933284 0.359139i \(-0.116930\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) − 19.0000i − 1.48819i −0.668071 0.744097i \(-0.732880\pi\)
0.668071 0.744097i \(-0.267120\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 6.00000i − 0.464294i −0.972681 0.232147i \(-0.925425\pi\)
0.972681 0.232147i \(-0.0745750\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) 0 0
\(173\) 4.00000i 0.304114i 0.988372 + 0.152057i \(0.0485898\pi\)
−0.988372 + 0.152057i \(0.951410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 10.0000i − 0.751646i
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) − 1.00000i − 0.0739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 11.0000i 0.791797i 0.918294 + 0.395899i \(0.129567\pi\)
−0.918294 + 0.395899i \(0.870433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) 13.0000 0.921546 0.460773 0.887518i \(-0.347572\pi\)
0.460773 + 0.887518i \(0.347572\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 0 0
\(203\) − 4.00000i − 0.280745i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 4.00000i − 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −7.00000 −0.481900 −0.240950 0.970538i \(-0.577459\pi\)
−0.240950 + 0.970538i \(0.577459\pi\)
\(212\) 0 0
\(213\) 14.0000i 0.959264i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.00000i 0.475191i
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 11.0000i − 0.736614i −0.929704 0.368307i \(-0.879937\pi\)
0.929704 0.368307i \(-0.120063\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 14.0000i − 0.929213i −0.885517 0.464606i \(-0.846196\pi\)
0.885517 0.464606i \(-0.153804\pi\)
\(228\) 0 0
\(229\) −23.0000 −1.51988 −0.759941 0.649992i \(-0.774772\pi\)
−0.759941 + 0.649992i \(0.774772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.00000i − 0.190885i
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.0000i 0.998053i 0.866587 + 0.499026i \(0.166309\pi\)
−0.866587 + 0.499026i \(0.833691\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 0 0
\(263\) − 2.00000i − 0.123325i −0.998097 0.0616626i \(-0.980360\pi\)
0.998097 0.0616626i \(-0.0196403\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 1.00000i 0.0605228i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 11.0000i − 0.660926i −0.943819 0.330463i \(-0.892795\pi\)
0.943819 0.330463i \(-0.107205\pi\)
\(278\) 0 0
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 4.00000 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 0 0
\(283\) 5.00000i 0.297219i 0.988896 + 0.148610i \(0.0474798\pi\)
−0.988896 + 0.148610i \(0.952520\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 6.00000i − 0.354169i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 3.00000 0.175863
\(292\) 0 0
\(293\) − 30.0000i − 1.75262i −0.481749 0.876309i \(-0.659998\pi\)
0.481749 0.876309i \(-0.340002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) 0 0
\(303\) − 6.00000i − 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 13.0000i − 0.741949i −0.928643 0.370975i \(-0.879024\pi\)
0.928643 0.370975i \(-0.120976\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) 23.0000i 1.30004i 0.759918 + 0.650018i \(0.225239\pi\)
−0.759918 + 0.650018i \(0.774761\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 15.0000i 0.829502i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 19.0000i 1.03500i 0.855684 + 0.517498i \(0.173136\pi\)
−0.855684 + 0.517498i \(0.826864\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 13.0000i − 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) − 36.0000i − 1.91609i −0.286623 0.958043i \(-0.592533\pi\)
0.286623 0.958043i \(-0.407467\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 11.0000i 0.577350i
\(364\) 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
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) − 15.0000i − 0.776671i −0.921518 0.388335i \(-0.873050\pi\)
0.921518 0.388335i \(-0.126950\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.00000i − 0.206010i
\(378\) 0 0
\(379\) 13.0000 0.667765 0.333883 0.942615i \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) − 32.0000i − 1.63512i −0.575841 0.817562i \(-0.695325\pi\)
0.575841 0.817562i \(-0.304675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.00000i 0.457496i
\(388\) 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
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 13.0000i − 0.652451i −0.945292 0.326226i \(-0.894223\pi\)
0.945292 0.326226i \(-0.105777\pi\)
\(398\) 0 0
\(399\) −3.00000 −0.150188
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 7.00000i 0.348695i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) − 10.0000i − 0.492068i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 12.0000i − 0.587643i
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.00000i − 0.0483934i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) 15.0000i 0.720854i 0.932787 + 0.360427i \(0.117369\pi\)
−0.932787 + 0.360427i \(0.882631\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) 23.0000 1.09773 0.548865 0.835911i \(-0.315060\pi\)
0.548865 + 0.835911i \(0.315060\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 20.0000i 0.950229i 0.879924 + 0.475114i \(0.157593\pi\)
−0.879924 + 0.475114i \(0.842407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 14.0000i − 0.662177i
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 17.0000i 0.798730i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000i 1.02912i 0.857455 + 0.514558i \(0.172044\pi\)
−0.857455 + 0.514558i \(0.827956\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) − 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
\(467\) − 24.0000i − 1.11059i −0.831654 0.555294i \(-0.812606\pi\)
0.831654 0.555294i \(-0.187394\pi\)
\(468\) 0 0
\(469\) −3.00000 −0.138527
\(470\) 0 0
\(471\) −9.00000 −0.414698
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) − 4.00000i − 0.182006i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 29.0000i 1.31412i 0.753840 + 0.657058i \(0.228199\pi\)
−0.753840 + 0.657058i \(0.771801\pi\)
\(488\) 0 0
\(489\) −19.0000 −0.859210
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.0000i 0.627986i
\(498\) 0 0
\(499\) −41.0000 −1.83541 −0.917706 0.397260i \(-0.869961\pi\)
−0.917706 + 0.397260i \(0.869961\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) − 18.0000i − 0.802580i −0.915951 0.401290i \(-0.868562\pi\)
0.915951 0.401290i \(-0.131438\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 12.0000i − 0.532939i
\(508\) 0 0
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 3.00000i 0.132453i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 20.0000 0.876216 0.438108 0.898922i \(-0.355649\pi\)
0.438108 + 0.898922i \(0.355649\pi\)
\(522\) 0 0
\(523\) 37.0000i 1.61790i 0.587879 + 0.808949i \(0.299963\pi\)
−0.587879 + 0.808949i \(0.700037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) − 6.00000i − 0.259889i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 6.00000i − 0.258919i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.00000 −0.214967 −0.107483 0.994207i \(-0.534279\pi\)
−0.107483 + 0.994207i \(0.534279\pi\)
\(542\) 0 0
\(543\) − 11.0000i − 0.472055i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 16.0000i − 0.684111i −0.939680 0.342055i \(-0.888877\pi\)
0.939680 0.342055i \(-0.111123\pi\)
\(548\) 0 0
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 6.00000i − 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) 0 0
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.00000i 0.337160i 0.985688 + 0.168580i \(0.0539181\pi\)
−0.985688 + 0.168580i \(0.946082\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) 0 0
\(573\) 16.0000i 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 31.0000i − 1.29055i −0.763952 0.645273i \(-0.776743\pi\)
0.763952 0.645273i \(-0.223257\pi\)
\(578\) 0 0
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 0 0
\(589\) −21.0000 −0.865290
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 13.0000i − 0.532055i
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −21.0000 −0.856608 −0.428304 0.903635i \(-0.640889\pi\)
−0.428304 + 0.903635i \(0.640889\pi\)
\(602\) 0 0
\(603\) 3.00000i 0.122169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 0 0
\(619\) 19.0000 0.763674 0.381837 0.924230i \(-0.375291\pi\)
0.381837 + 0.924230i \(0.375291\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) 0 0
\(633\) 7.00000i 0.278225i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 0 0
\(639\) 14.0000 0.553831
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 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
\(647\) 28.0000i 1.10079i 0.834903 + 0.550397i \(0.185524\pi\)
−0.834903 + 0.550397i \(0.814476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 7.00000 0.274352
\(652\) 0 0
\(653\) − 10.0000i − 0.391330i −0.980671 0.195665i \(-0.937313\pi\)
0.980671 0.195665i \(-0.0626866\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000i 0.619522i
\(668\) 0 0
\(669\) −11.0000 −0.425285
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 22.0000i 0.848038i 0.905653 + 0.424019i \(0.139381\pi\)
−0.905653 + 0.424019i \(0.860619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 48.0000i 1.84479i 0.386248 + 0.922395i \(0.373771\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(678\) 0 0
\(679\) 3.00000 0.115129
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 23.0000i 0.877505i
\(688\) 0 0
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) − 18.0000i − 0.678883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) −3.00000 −0.112667 −0.0563337 0.998412i \(-0.517941\pi\)
−0.0563337 + 0.998412i \(0.517941\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) − 28.0000i − 1.04861i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000i 0.224074i
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 11.0000i 0.409094i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7.00000i 0.259616i 0.991539 + 0.129808i \(0.0414360\pi\)
−0.991539 + 0.129808i \(0.958564\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −3.00000 −0.110208
\(742\) 0 0
\(743\) 28.0000i 1.02722i 0.858024 + 0.513610i \(0.171692\pi\)
−0.858024 + 0.513610i \(0.828308\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 18.0000i 0.658586i
\(748\) 0 0
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 16.0000i 0.583072i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 47.0000i − 1.70824i −0.520073 0.854122i \(-0.674095\pi\)
0.520073 0.854122i \(-0.325905\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 44.0000 1.59500 0.797499 0.603320i \(-0.206156\pi\)
0.797499 + 0.603320i \(0.206156\pi\)
\(762\) 0 0
\(763\) 15.0000i 0.543036i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 10.0000i − 0.361079i
\(768\) 0 0
\(769\) 1.00000 0.0360609 0.0180305 0.999837i \(-0.494260\pi\)
0.0180305 + 0.999837i \(0.494260\pi\)
\(770\) 0 0
\(771\) 16.0000 0.576226
\(772\) 0 0
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.00000i 0.215249i
\(778\) 0 0
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 4.00000i 0.142948i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.00000i 0.249523i 0.992187 + 0.124762i \(0.0398166\pi\)
−0.992187 + 0.124762i \(0.960183\pi\)
\(788\) 0 0
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) − 1.00000i − 0.0355110i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 48.0000i − 1.70025i −0.526583 0.850124i \(-0.676527\pi\)
0.526583 0.850124i \(-0.323473\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 16.0000i − 0.563227i
\(808\) 0 0
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) 4.00000i 0.140286i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 27.0000i − 0.944610i
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) 41.0000i 1.42917i 0.699549 + 0.714585i \(0.253384\pi\)
−0.699549 + 0.714585i \(0.746616\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 34.0000i 1.18230i 0.806563 + 0.591148i \(0.201325\pi\)
−0.806563 + 0.591148i \(0.798675\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −11.0000 −0.381586
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 7.00000i − 0.241955i
\(838\) 0 0
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) − 4.00000i − 0.137767i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 11.0000i 0.377964i
\(848\) 0 0
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 37.0000i 1.26686i 0.773802 + 0.633428i \(0.218353\pi\)
−0.773802 + 0.633428i \(0.781647\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 32.0000i − 1.09310i −0.837427 0.546550i \(-0.815941\pi\)
0.837427 0.546550i \(-0.184059\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) − 4.00000i − 0.136162i −0.997680 0.0680808i \(-0.978312\pi\)
0.997680 0.0680808i \(-0.0216876\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 17.0000i − 0.577350i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −3.00000 −0.101651
\(872\) 0 0
\(873\) − 3.00000i − 0.101535i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 43.0000i 1.45201i 0.687691 + 0.726003i \(0.258624\pi\)
−0.687691 + 0.726003i \(0.741376\pi\)
\(878\) 0 0
\(879\) −30.0000 −1.01187
\(880\) 0 0
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 0 0
\(883\) − 29.0000i − 0.975928i −0.872864 0.487964i \(-0.837740\pi\)
0.872864 0.487964i \(-0.162260\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 38.0000i 1.27592i 0.770072 + 0.637958i \(0.220220\pi\)
−0.770072 + 0.637958i \(0.779780\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 18.0000i − 0.602347i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 4.00000i − 0.133556i
\(898\) 0 0
\(899\) −28.0000 −0.933852
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 9.00000i 0.299501i
\(904\) 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
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 42.0000 1.39152 0.695761 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.00000i 0.198137i
\(918\) 0 0
\(919\) 1.00000 0.0329870 0.0164935 0.999864i \(-0.494750\pi\)
0.0164935 + 0.999864i \(0.494750\pi\)
\(920\) 0 0
\(921\) −13.0000 −0.428365
\(922\) 0 0
\(923\) 14.0000i 0.460816i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 8.00000i − 0.262754i
\(928\) 0 0
\(929\) 8.00000 0.262471 0.131236 0.991351i \(-0.458106\pi\)
0.131236 + 0.991351i \(0.458106\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 0 0
\(933\) 10.0000i 0.327385i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 19.0000i − 0.620703i −0.950622 0.310351i \(-0.899553\pi\)
0.950622 0.310351i \(-0.100447\pi\)
\(938\) 0 0
\(939\) 23.0000 0.750577
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) 24.0000i 0.781548i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.0000i 0.649913i 0.945729 + 0.324956i \(0.105350\pi\)
−0.945729 + 0.324956i \(0.894650\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 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
\(957\) 0 0
\(958\) 0 0
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) − 2.00000i − 0.0644491i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 54.0000 1.73294 0.866471 0.499227i \(-0.166383\pi\)
0.866471 + 0.499227i \(0.166383\pi\)
\(972\) 0 0
\(973\) − 12.0000i − 0.384702i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 15.0000 0.478913
\(982\) 0 0
\(983\) 28.0000i 0.893061i 0.894768 + 0.446531i \(0.147341\pi\)
−0.894768 + 0.446531i \(0.852659\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000i 0.190982i
\(988\) 0 0
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) −31.0000 −0.984747 −0.492374 0.870384i \(-0.663871\pi\)
−0.492374 + 0.870384i \(0.663871\pi\)
\(992\) 0 0
\(993\) 32.0000i 1.01549i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 18.0000i 0.570066i 0.958518 + 0.285033i \(0.0920045\pi\)
−0.958518 + 0.285033i \(0.907995\pi\)
\(998\) 0 0
\(999\) 6.00000 0.189832
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 4800.2.f.t.3649.1 2
4.3 odd 2 4800.2.f.q.3649.2 2
5.2 odd 4 4800.2.a.w.1.1 1
5.3 odd 4 4800.2.a.bx.1.1 1
5.4 even 2 inner 4800.2.f.t.3649.2 2
8.3 odd 2 2400.2.f.k.1249.1 2
8.5 even 2 2400.2.f.h.1249.2 2
20.3 even 4 4800.2.a.x.1.1 1
20.7 even 4 4800.2.a.bw.1.1 1
20.19 odd 2 4800.2.f.q.3649.1 2
24.5 odd 2 7200.2.f.l.6049.1 2
24.11 even 2 7200.2.f.r.6049.2 2
40.3 even 4 2400.2.a.bb.1.1 yes 1
40.13 odd 4 2400.2.a.f.1.1 1
40.19 odd 2 2400.2.f.k.1249.2 2
40.27 even 4 2400.2.a.g.1.1 yes 1
40.29 even 2 2400.2.f.h.1249.1 2
40.37 odd 4 2400.2.a.bc.1.1 yes 1
120.29 odd 2 7200.2.f.l.6049.2 2
120.53 even 4 7200.2.a.r.1.1 1
120.59 even 2 7200.2.f.r.6049.1 2
120.77 even 4 7200.2.a.bj.1.1 1
120.83 odd 4 7200.2.a.bi.1.1 1
120.107 odd 4 7200.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2400.2.a.f.1.1 1 40.13 odd 4
2400.2.a.g.1.1 yes 1 40.27 even 4
2400.2.a.bb.1.1 yes 1 40.3 even 4
2400.2.a.bc.1.1 yes 1 40.37 odd 4
2400.2.f.h.1249.1 2 40.29 even 2
2400.2.f.h.1249.2 2 8.5 even 2
2400.2.f.k.1249.1 2 8.3 odd 2
2400.2.f.k.1249.2 2 40.19 odd 2
4800.2.a.w.1.1 1 5.2 odd 4
4800.2.a.x.1.1 1 20.3 even 4
4800.2.a.bw.1.1 1 20.7 even 4
4800.2.a.bx.1.1 1 5.3 odd 4
4800.2.f.q.3649.1 2 20.19 odd 2
4800.2.f.q.3649.2 2 4.3 odd 2
4800.2.f.t.3649.1 2 1.1 even 1 trivial
4800.2.f.t.3649.2 2 5.4 even 2 inner
7200.2.a.r.1.1 1 120.53 even 4
7200.2.a.s.1.1 1 120.107 odd 4
7200.2.a.bi.1.1 1 120.83 odd 4
7200.2.a.bj.1.1 1 120.77 even 4
7200.2.f.l.6049.1 2 24.5 odd 2
7200.2.f.l.6049.2 2 120.29 odd 2
7200.2.f.r.6049.1 2 120.59 even 2
7200.2.f.r.6049.2 2 24.11 even 2