Properties

Label 2450.2.a.r.1.1
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2450,2,Mod(1,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -6.00000 q^{18} -2.00000 q^{19} +1.00000 q^{23} -3.00000 q^{24} -2.00000 q^{26} +9.00000 q^{27} -1.00000 q^{29} +10.0000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +6.00000 q^{36} +8.00000 q^{37} +2.00000 q^{38} +6.00000 q^{39} -3.00000 q^{41} -5.00000 q^{43} -1.00000 q^{46} -8.00000 q^{47} +3.00000 q^{48} +6.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -9.00000 q^{54} -6.00000 q^{57} +1.00000 q^{58} +2.00000 q^{59} -9.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +7.00000 q^{67} +2.00000 q^{68} +3.00000 q^{69} +6.00000 q^{71} -6.00000 q^{72} +10.0000 q^{73} -8.00000 q^{74} -2.00000 q^{76} -6.00000 q^{78} -10.0000 q^{79} +9.00000 q^{81} +3.00000 q^{82} +9.00000 q^{83} +5.00000 q^{86} -3.00000 q^{87} -7.00000 q^{89} +1.00000 q^{92} +30.0000 q^{93} +8.00000 q^{94} -3.00000 q^{96} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 3.00000 0.866025
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −6.00000 −1.41421
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 2.00000 0.324443
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 3.00000 0.433013
\(49\) 0 0
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 1.00000 0.131306
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −9.00000 −1.15233 −0.576166 0.817333i \(-0.695452\pi\)
−0.576166 + 0.817333i \(0.695452\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 2.00000 0.242536
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −6.00000 −0.707107
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 3.00000 0.331295
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.00000 0.539164
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −7.00000 −0.741999 −0.370999 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 30.0000 3.11086
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) −6.00000 −0.594089
\(103\) 11.0000 1.08386 0.541931 0.840423i \(-0.317693\pi\)
0.541931 + 0.840423i \(0.317693\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 7.00000 0.676716 0.338358 0.941018i \(-0.390129\pi\)
0.338358 + 0.941018i \(0.390129\pi\)
\(108\) 9.00000 0.866025
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 12.0000 1.10940
\(118\) −2.00000 −0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 9.00000 0.814822
\(123\) −9.00000 −0.811503
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −15.0000 −1.32068
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) −3.00000 −0.255377
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 2.00000 0.162221
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 12.0000 0.957704 0.478852 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 10.0000 0.795557
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −12.0000 −0.917663
\(172\) −5.00000 −0.381246
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 7.00000 0.524672
\(179\) −26.0000 −1.94333 −0.971666 0.236360i \(-0.924046\pi\)
−0.971666 + 0.236360i \(0.924046\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) −27.0000 −1.99590
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −30.0000 −2.19971
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 3.00000 0.216506
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) 21.0000 1.48123
\(202\) 15.0000 1.05540
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) 6.00000 0.417029
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −18.0000 −1.23917 −0.619586 0.784929i \(-0.712699\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(212\) 6.00000 0.412082
\(213\) 18.0000 1.23334
\(214\) −7.00000 −0.478510
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 0 0
\(218\) −5.00000 −0.338643
\(219\) 30.0000 2.02721
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −24.0000 −1.61077
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −6.00000 −0.397360
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) −12.0000 −0.784465
\(235\) 0 0
\(236\) 2.00000 0.130189
\(237\) −30.0000 −1.94871
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −9.00000 −0.576166
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) −4.00000 −0.254514
\(248\) −10.0000 −0.635001
\(249\) 27.0000 1.71106
\(250\) 0 0
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 15.0000 0.933859
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) −20.0000 −1.23560
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −21.0000 −1.28518
\(268\) 7.00000 0.427593
\(269\) 5.00000 0.304855 0.152428 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 0 0
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 8.00000 0.479808
\(279\) 60.0000 3.59211
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 24.0000 1.42918
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −6.00000 −0.353553
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 15.0000 0.868927
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) −6.00000 −0.345261
\(303\) −45.0000 −2.58518
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −12.0000 −0.685994
\(307\) 19.0000 1.08439 0.542194 0.840254i \(-0.317594\pi\)
0.542194 + 0.840254i \(0.317594\pi\)
\(308\) 0 0
\(309\) 33.0000 1.87730
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) −6.00000 −0.339683
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −12.0000 −0.677199
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −18.0000 −1.00939
\(319\) 0 0
\(320\) 0 0
\(321\) 21.0000 1.17211
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 15.0000 0.829502
\(328\) 3.00000 0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) 9.00000 0.493939
\(333\) 48.0000 2.63038
\(334\) −9.00000 −0.492458
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 9.00000 0.489535
\(339\) 30.0000 1.62938
\(340\) 0 0
\(341\) 0 0
\(342\) 12.0000 0.648886
\(343\) 0 0
\(344\) 5.00000 0.269582
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) −21.0000 −1.12734 −0.563670 0.826000i \(-0.690611\pi\)
−0.563670 + 0.826000i \(0.690611\pi\)
\(348\) −3.00000 −0.160817
\(349\) −9.00000 −0.481759 −0.240879 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(350\) 0 0
\(351\) 18.0000 0.960769
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −7.00000 −0.370999
\(357\) 0 0
\(358\) 26.0000 1.37414
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −5.00000 −0.262794
\(363\) −33.0000 −1.73205
\(364\) 0 0
\(365\) 0 0
\(366\) 27.0000 1.41131
\(367\) 23.0000 1.20059 0.600295 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 1.00000 0.0521286
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 30.0000 1.55543
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) 24.0000 1.22956
\(382\) 20.0000 1.02329
\(383\) 9.00000 0.459879 0.229939 0.973205i \(-0.426147\pi\)
0.229939 + 0.973205i \(0.426147\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) −30.0000 −1.52499
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 60.0000 3.02660
\(394\) −8.00000 −0.403034
\(395\) 0 0
\(396\) 0 0
\(397\) 32.0000 1.60603 0.803017 0.595956i \(-0.203227\pi\)
0.803017 + 0.595956i \(0.203227\pi\)
\(398\) −12.0000 −0.601506
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) −21.0000 −1.04738
\(403\) 20.0000 0.996271
\(404\) −15.0000 −0.746278
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) −17.0000 −0.840596 −0.420298 0.907386i \(-0.638074\pi\)
−0.420298 + 0.907386i \(0.638074\pi\)
\(410\) 0 0
\(411\) −48.0000 −2.36767
\(412\) 11.0000 0.541931
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −24.0000 −1.17529
\(418\) 0 0
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) 31.0000 1.51085 0.755424 0.655237i \(-0.227431\pi\)
0.755424 + 0.655237i \(0.227431\pi\)
\(422\) 18.0000 0.876226
\(423\) −48.0000 −2.33384
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −18.0000 −0.872103
\(427\) 0 0
\(428\) 7.00000 0.338358
\(429\) 0 0
\(430\) 0 0
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 9.00000 0.433013
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5.00000 0.239457
\(437\) −2.00000 −0.0956730
\(438\) −30.0000 −1.43346
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 3.00000 0.142534 0.0712672 0.997457i \(-0.477296\pi\)
0.0712672 + 0.997457i \(0.477296\pi\)
\(444\) 24.0000 1.13899
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −45.0000 −2.12843
\(448\) 0 0
\(449\) 23.0000 1.08544 0.542719 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 10.0000 0.470360
\(453\) 18.0000 0.845714
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) −10.0000 −0.467269
\(459\) 18.0000 0.840168
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −25.0000 −1.16185 −0.580924 0.813958i \(-0.697309\pi\)
−0.580924 + 0.813958i \(0.697309\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) −1.00000 −0.0462745 −0.0231372 0.999732i \(-0.507365\pi\)
−0.0231372 + 0.999732i \(0.507365\pi\)
\(468\) 12.0000 0.554700
\(469\) 0 0
\(470\) 0 0
\(471\) 36.0000 1.65879
\(472\) −2.00000 −0.0920575
\(473\) 0 0
\(474\) 30.0000 1.37795
\(475\) 0 0
\(476\) 0 0
\(477\) 36.0000 1.64833
\(478\) 10.0000 0.457389
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 9.00000 0.407411
\(489\) −36.0000 −1.62798
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) −9.00000 −0.405751
\(493\) −2.00000 −0.0900755
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) −27.0000 −1.20990
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 27.0000 1.20627
\(502\) 10.0000 0.446322
\(503\) 5.00000 0.222939 0.111469 0.993768i \(-0.464444\pi\)
0.111469 + 0.993768i \(0.464444\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −27.0000 −1.19911
\(508\) 8.00000 0.354943
\(509\) 35.0000 1.55135 0.775674 0.631134i \(-0.217410\pi\)
0.775674 + 0.631134i \(0.217410\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −18.0000 −0.794719
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) −15.0000 −0.660338
\(517\) 0 0
\(518\) 0 0
\(519\) −36.0000 −1.58022
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 6.00000 0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 21.0000 0.915644
\(527\) 20.0000 0.871214
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 21.0000 0.908759
\(535\) 0 0
\(536\) −7.00000 −0.302354
\(537\) −78.0000 −3.36595
\(538\) −5.00000 −0.215565
\(539\) 0 0
\(540\) 0 0
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) −6.00000 −0.257722
\(543\) 15.0000 0.643712
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −37.0000 −1.58201 −0.791003 0.611812i \(-0.790441\pi\)
−0.791003 + 0.611812i \(0.790441\pi\)
\(548\) −16.0000 −0.683486
\(549\) −54.0000 −2.30466
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) −3.00000 −0.127688
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 4.00000 0.169485 0.0847427 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) −60.0000 −2.54000
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) −24.0000 −1.01058
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 0 0
\(573\) −60.0000 −2.50654
\(574\) 0 0
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 13.0000 0.540729
\(579\) −60.0000 −2.49351
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 8.00000 0.328798
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −15.0000 −0.614424
\(597\) 36.0000 1.47338
\(598\) −2.00000 −0.0817861
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 42.0000 1.71037
\(604\) 6.00000 0.244137
\(605\) 0 0
\(606\) 45.0000 1.82800
\(607\) −5.00000 −0.202944 −0.101472 0.994838i \(-0.532355\pi\)
−0.101472 + 0.994838i \(0.532355\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 12.0000 0.485071
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) −19.0000 −0.766778
\(615\) 0 0
\(616\) 0 0
\(617\) −20.0000 −0.805170 −0.402585 0.915383i \(-0.631888\pi\)
−0.402585 + 0.915383i \(0.631888\pi\)
\(618\) −33.0000 −1.32745
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) −6.00000 −0.240578
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) 12.0000 0.478852
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) 10.0000 0.397779
\(633\) −54.0000 −2.14631
\(634\) 22.0000 0.873732
\(635\) 0 0
\(636\) 18.0000 0.713746
\(637\) 0 0
\(638\) 0 0
\(639\) 36.0000 1.42414
\(640\) 0 0
\(641\) −35.0000 −1.38242 −0.691208 0.722655i \(-0.742921\pi\)
−0.691208 + 0.722655i \(0.742921\pi\)
\(642\) −21.0000 −0.828804
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) −21.0000 −0.825595 −0.412798 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(648\) −9.00000 −0.353553
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) −15.0000 −0.586546
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 60.0000 2.34082
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) −41.0000 −1.59472 −0.797358 0.603507i \(-0.793769\pi\)
−0.797358 + 0.603507i \(0.793769\pi\)
\(662\) −14.0000 −0.544125
\(663\) 12.0000 0.466041
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −48.0000 −1.85996
\(667\) −1.00000 −0.0387202
\(668\) 9.00000 0.348220
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) −30.0000 −1.15214
\(679\) 0 0
\(680\) 0 0
\(681\) −36.0000 −1.37952
\(682\) 0 0
\(683\) −25.0000 −0.956598 −0.478299 0.878197i \(-0.658747\pi\)
−0.478299 + 0.878197i \(0.658747\pi\)
\(684\) −12.0000 −0.458831
\(685\) 0 0
\(686\) 0 0
\(687\) 30.0000 1.14457
\(688\) −5.00000 −0.190623
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 21.0000 0.797149
\(695\) 0 0
\(696\) 3.00000 0.113715
\(697\) −6.00000 −0.227266
\(698\) 9.00000 0.340655
\(699\) 42.0000 1.58859
\(700\) 0 0
\(701\) −1.00000 −0.0377695 −0.0188847 0.999822i \(-0.506012\pi\)
−0.0188847 + 0.999822i \(0.506012\pi\)
\(702\) −18.0000 −0.679366
\(703\) −16.0000 −0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 6.00000 0.225494
\(709\) −9.00000 −0.338002 −0.169001 0.985616i \(-0.554054\pi\)
−0.169001 + 0.985616i \(0.554054\pi\)
\(710\) 0 0
\(711\) −60.0000 −2.25018
\(712\) 7.00000 0.262336
\(713\) 10.0000 0.374503
\(714\) 0 0
\(715\) 0 0
\(716\) −26.0000 −0.971666
\(717\) −30.0000 −1.12037
\(718\) 14.0000 0.522475
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) 54.0000 2.00828
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) 33.0000 1.22474
\(727\) 29.0000 1.07555 0.537775 0.843088i \(-0.319265\pi\)
0.537775 + 0.843088i \(0.319265\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −10.0000 −0.369863
\(732\) −27.0000 −0.997949
\(733\) 16.0000 0.590973 0.295487 0.955347i \(-0.404518\pi\)
0.295487 + 0.955347i \(0.404518\pi\)
\(734\) −23.0000 −0.848945
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 18.0000 0.662589
\(739\) −32.0000 −1.17714 −0.588570 0.808447i \(-0.700309\pi\)
−0.588570 + 0.808447i \(0.700309\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) −30.0000 −1.09985
\(745\) 0 0
\(746\) 8.00000 0.292901
\(747\) 54.0000 1.97576
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) −8.00000 −0.291730
\(753\) −30.0000 −1.09326
\(754\) 2.00000 0.0728357
\(755\) 0 0
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 2.00000 0.0726433
\(759\) 0 0
\(760\) 0 0
\(761\) −50.0000 −1.81250 −0.906249 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) −24.0000 −0.869428
\(763\) 0 0
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) −9.00000 −0.325183
\(767\) 4.00000 0.144432
\(768\) 3.00000 0.108253
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −36.0000 −1.29651
\(772\) −20.0000 −0.719816
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 30.0000 1.07833
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) −2.00000 −0.0715199
\(783\) −9.00000 −0.321634
\(784\) 0 0
\(785\) 0 0
\(786\) −60.0000 −2.14013
\(787\) −21.0000 −0.748569 −0.374285 0.927314i \(-0.622112\pi\)
−0.374285 + 0.927314i \(0.622112\pi\)
\(788\) 8.00000 0.284988
\(789\) −63.0000 −2.24286
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) −32.0000 −1.13564
\(795\) 0 0
\(796\) 12.0000 0.425329
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −42.0000 −1.48400
\(802\) −3.00000 −0.105934
\(803\) 0 0
\(804\) 21.0000 0.740613
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) 15.0000 0.528025
\(808\) 15.0000 0.527698
\(809\) 25.0000 0.878953 0.439477 0.898254i \(-0.355164\pi\)
0.439477 + 0.898254i \(0.355164\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 0 0
\(813\) 18.0000 0.631288
\(814\) 0 0
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 10.0000 0.349856
\(818\) 17.0000 0.594391
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 48.0000 1.67419
\(823\) 45.0000 1.56860 0.784301 0.620381i \(-0.213022\pi\)
0.784301 + 0.620381i \(0.213022\pi\)
\(824\) −11.0000 −0.383203
\(825\) 0 0
\(826\) 0 0
\(827\) 37.0000 1.28662 0.643308 0.765607i \(-0.277561\pi\)
0.643308 + 0.765607i \(0.277561\pi\)
\(828\) 6.00000 0.208514
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 78.0000 2.70579
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 24.0000 0.831052
\(835\) 0 0
\(836\) 0 0
\(837\) 90.0000 3.11086
\(838\) 40.0000 1.38178
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −31.0000 −1.06833
\(843\) 54.0000 1.85986
\(844\) −18.0000 −0.619586
\(845\) 0 0
\(846\) 48.0000 1.65027
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 18.0000 0.616670
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.00000 −0.239255
\(857\) 4.00000 0.136637 0.0683187 0.997664i \(-0.478237\pi\)
0.0683187 + 0.997664i \(0.478237\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) 11.0000 0.374444 0.187222 0.982318i \(-0.440052\pi\)
0.187222 + 0.982318i \(0.440052\pi\)
\(864\) −9.00000 −0.306186
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) −39.0000 −1.32451
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) −5.00000 −0.169321
\(873\) 0 0
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 30.0000 1.01361
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −8.00000 −0.269987
\(879\) −72.0000 −2.42850
\(880\) 0 0
\(881\) −3.00000 −0.101073 −0.0505363 0.998722i \(-0.516093\pi\)
−0.0505363 + 0.998722i \(0.516093\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −3.00000 −0.100787
\(887\) −9.00000 −0.302190 −0.151095 0.988519i \(-0.548280\pi\)
−0.151095 + 0.988519i \(0.548280\pi\)
\(888\) −24.0000 −0.805387
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 16.0000 0.535420
\(894\) 45.0000 1.50503
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) −23.0000 −0.767520
\(899\) −10.0000 −0.333519
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) −18.0000 −0.598010
\(907\) 25.0000 0.830111 0.415056 0.909796i \(-0.363762\pi\)
0.415056 + 0.909796i \(0.363762\pi\)
\(908\) −12.0000 −0.398234
\(909\) −90.0000 −2.98511
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) −6.00000 −0.198680
\(913\) 0 0
\(914\) 32.0000 1.05847
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) −18.0000 −0.594089
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) 57.0000 1.87821
\(922\) 14.0000 0.461065
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 0 0
\(926\) 25.0000 0.821551
\(927\) 66.0000 2.16772
\(928\) 1.00000 0.0328266
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 18.0000 0.589294
\(934\) 1.00000 0.0327210
\(935\) 0 0
\(936\) −12.0000 −0.392232
\(937\) 56.0000 1.82944 0.914720 0.404088i \(-0.132411\pi\)
0.914720 + 0.404088i \(0.132411\pi\)
\(938\) 0 0
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −36.0000 −1.17294
\(943\) −3.00000 −0.0976934
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 0 0
\(947\) 3.00000 0.0974869 0.0487435 0.998811i \(-0.484478\pi\)
0.0487435 + 0.998811i \(0.484478\pi\)
\(948\) −30.0000 −0.974355
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −66.0000 −2.14020
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) −36.0000 −1.16554
\(955\) 0 0
\(956\) −10.0000 −0.323423
\(957\) 0 0
\(958\) −18.0000 −0.581554
\(959\) 0 0
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −16.0000 −0.515861
\(963\) 42.0000 1.35343
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) −17.0000 −0.546683 −0.273342 0.961917i \(-0.588129\pi\)
−0.273342 + 0.961917i \(0.588129\pi\)
\(968\) 11.0000 0.353553
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) −9.00000 −0.288083
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 36.0000 1.15115
\(979\) 0 0
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 18.0000 0.574403
\(983\) −25.0000 −0.797376 −0.398688 0.917087i \(-0.630534\pi\)
−0.398688 + 0.917087i \(0.630534\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) −5.00000 −0.158991
\(990\) 0 0
\(991\) −26.0000 −0.825917 −0.412959 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) −10.0000 −0.317500
\(993\) 42.0000 1.33283
\(994\) 0 0
\(995\) 0 0
\(996\) 27.0000 0.855528
\(997\) −34.0000 −1.07679 −0.538395 0.842692i \(-0.680969\pi\)
−0.538395 + 0.842692i \(0.680969\pi\)
\(998\) −16.0000 −0.506471
\(999\) 72.0000 2.27798
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 2450.2.a.r.1.1 1
5.2 odd 4 490.2.c.c.99.1 2
5.3 odd 4 490.2.c.c.99.2 2
5.4 even 2 2450.2.a.s.1.1 1
7.2 even 3 350.2.e.g.151.1 2
7.4 even 3 350.2.e.g.51.1 2
7.6 odd 2 2450.2.a.c.1.1 1
35.2 odd 12 70.2.i.a.39.1 yes 4
35.3 even 12 490.2.i.b.79.1 4
35.4 even 6 350.2.e.f.51.1 2
35.9 even 6 350.2.e.f.151.1 2
35.12 even 12 490.2.i.b.459.1 4
35.13 even 4 490.2.c.b.99.2 2
35.17 even 12 490.2.i.b.79.2 4
35.18 odd 12 70.2.i.a.9.1 4
35.23 odd 12 70.2.i.a.39.2 yes 4
35.27 even 4 490.2.c.b.99.1 2
35.32 odd 12 70.2.i.a.9.2 yes 4
35.33 even 12 490.2.i.b.459.2 4
35.34 odd 2 2450.2.a.bh.1.1 1
105.2 even 12 630.2.u.b.109.2 4
105.23 even 12 630.2.u.b.109.1 4
105.32 even 12 630.2.u.b.289.1 4
105.53 even 12 630.2.u.b.289.2 4
140.23 even 12 560.2.bw.c.529.2 4
140.67 even 12 560.2.bw.c.289.2 4
140.107 even 12 560.2.bw.c.529.1 4
140.123 even 12 560.2.bw.c.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.i.a.9.1 4 35.18 odd 12
70.2.i.a.9.2 yes 4 35.32 odd 12
70.2.i.a.39.1 yes 4 35.2 odd 12
70.2.i.a.39.2 yes 4 35.23 odd 12
350.2.e.f.51.1 2 35.4 even 6
350.2.e.f.151.1 2 35.9 even 6
350.2.e.g.51.1 2 7.4 even 3
350.2.e.g.151.1 2 7.2 even 3
490.2.c.b.99.1 2 35.27 even 4
490.2.c.b.99.2 2 35.13 even 4
490.2.c.c.99.1 2 5.2 odd 4
490.2.c.c.99.2 2 5.3 odd 4
490.2.i.b.79.1 4 35.3 even 12
490.2.i.b.79.2 4 35.17 even 12
490.2.i.b.459.1 4 35.12 even 12
490.2.i.b.459.2 4 35.33 even 12
560.2.bw.c.289.1 4 140.123 even 12
560.2.bw.c.289.2 4 140.67 even 12
560.2.bw.c.529.1 4 140.107 even 12
560.2.bw.c.529.2 4 140.23 even 12
630.2.u.b.109.1 4 105.23 even 12
630.2.u.b.109.2 4 105.2 even 12
630.2.u.b.289.1 4 105.32 even 12
630.2.u.b.289.2 4 105.53 even 12
2450.2.a.c.1.1 1 7.6 odd 2
2450.2.a.r.1.1 1 1.1 even 1 trivial
2450.2.a.s.1.1 1 5.4 even 2
2450.2.a.bh.1.1 1 35.34 odd 2