Properties

Label 7942.2.a.e.1.1
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
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] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, 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("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7942.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +3.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +3.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +3.00000 q^{18} -3.00000 q^{20} +1.00000 q^{22} +2.00000 q^{23} +4.00000 q^{25} -2.00000 q^{26} +3.00000 q^{28} +2.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -9.00000 q^{35} -3.00000 q^{36} -9.00000 q^{37} +3.00000 q^{40} +2.00000 q^{41} -8.00000 q^{43} -1.00000 q^{44} +9.00000 q^{45} -2.00000 q^{46} -12.0000 q^{47} +2.00000 q^{49} -4.00000 q^{50} +2.00000 q^{52} -13.0000 q^{53} +3.00000 q^{55} -3.00000 q^{56} +10.0000 q^{59} -2.00000 q^{62} -9.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} +12.0000 q^{67} +4.00000 q^{68} +9.00000 q^{70} +2.00000 q^{71} +3.00000 q^{72} +12.0000 q^{73} +9.00000 q^{74} -3.00000 q^{77} -1.00000 q^{79} -3.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +9.00000 q^{83} -12.0000 q^{85} +8.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -9.00000 q^{90} +6.00000 q^{91} +2.00000 q^{92} +12.0000 q^{94} +5.00000 q^{97} -2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 3.00000 0.948683
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 3.00000 0.707107
\(19\) 0 0
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 3.00000 0.566947
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −9.00000 −1.52128
\(36\) −3.00000 −0.500000
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) 9.00000 1.34164
\(46\) −2.00000 −0.294884
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −13.0000 −1.78569 −0.892844 0.450367i \(-0.851293\pi\)
−0.892844 + 0.450367i \(0.851293\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −2.00000 −0.254000
\(63\) −9.00000 −1.13389
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 9.00000 1.07571
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 3.00000 0.353553
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 9.00000 1.04623
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −3.00000 −0.335410
\(81\) 9.00000 1.00000
\(82\) −2.00000 −0.220863
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −9.00000 −0.948683
\(91\) 6.00000 0.628971
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) −2.00000 −0.202031
\(99\) 3.00000 0.301511
\(100\) 4.00000 0.400000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 13.0000 1.26267
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −3.00000 −0.286039
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) −10.0000 −0.920575
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 3.00000 0.268328
\(126\) 9.00000 0.801784
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −9.00000 −0.760639
\(141\) 0 0
\(142\) −2.00000 −0.167836
\(143\) −2.00000 −0.167248
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) −9.00000 −0.739795
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) −3.00000 −0.244137 −0.122068 0.992522i \(-0.538953\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(152\) 0 0
\(153\) −12.0000 −0.970143
\(154\) 3.00000 0.241747
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 9.00000 0.718278 0.359139 0.933284i \(-0.383070\pi\)
0.359139 + 0.933284i \(0.383070\pi\)
\(158\) 1.00000 0.0795557
\(159\) 0 0
\(160\) 3.00000 0.237171
\(161\) 6.00000 0.472866
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 12.0000 0.907115
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 9.00000 0.670820
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −2.00000 −0.147442
\(185\) 27.0000 1.98508
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −5.00000 −0.358979
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −3.00000 −0.213201
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −14.0000 −0.975426
\(207\) −6.00000 −0.417029
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −13.0000 −0.892844
\(213\) 0 0
\(214\) 9.00000 0.615227
\(215\) 24.0000 1.63679
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −3.00000 −0.200446
\(225\) −12.0000 −0.800000
\(226\) −10.0000 −0.665190
\(227\) −25.0000 −1.65931 −0.829654 0.558278i \(-0.811462\pi\)
−0.829654 + 0.558278i \(0.811462\pi\)
\(228\) 0 0
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 6.00000 0.392232
\(235\) 36.0000 2.34838
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) −12.0000 −0.777844
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −9.00000 −0.566947
\(253\) −2.00000 −0.125739
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.0000 0.686161 0.343081 0.939306i \(-0.388530\pi\)
0.343081 + 0.939306i \(0.388530\pi\)
\(258\) 0 0
\(259\) −27.0000 −1.67770
\(260\) −6.00000 −0.372104
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −17.0000 −1.04826 −0.524132 0.851637i \(-0.675610\pi\)
−0.524132 + 0.851637i \(0.675610\pi\)
\(264\) 0 0
\(265\) 39.0000 2.39575
\(266\) 0 0
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) −11.0000 −0.668202 −0.334101 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 15.0000 0.906183
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −5.00000 −0.299880
\(279\) −6.00000 −0.359211
\(280\) 9.00000 0.537853
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −1.00000 −0.0594438 −0.0297219 0.999558i \(-0.509462\pi\)
−0.0297219 + 0.999558i \(0.509462\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 6.00000 0.354169
\(288\) 3.00000 0.176777
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 12.0000 0.702247
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) −30.0000 −1.74667
\(296\) 9.00000 0.523114
\(297\) 0 0
\(298\) 20.0000 1.15857
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 3.00000 0.172631
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 12.0000 0.685994
\(307\) −25.0000 −1.42683 −0.713413 0.700744i \(-0.752851\pi\)
−0.713413 + 0.700744i \(0.752851\pi\)
\(308\) −3.00000 −0.170941
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) 22.0000 1.24751 0.623753 0.781622i \(-0.285607\pi\)
0.623753 + 0.781622i \(0.285607\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) −9.00000 −0.507899
\(315\) 27.0000 1.52128
\(316\) −1.00000 −0.0562544
\(317\) −26.0000 −1.46031 −0.730153 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.00000 −0.167705
\(321\) 0 0
\(322\) −6.00000 −0.334367
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 8.00000 0.443760
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −2.00000 −0.110432
\(329\) −36.0000 −1.98474
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 9.00000 0.493939
\(333\) 27.0000 1.47959
\(334\) −19.0000 −1.03963
\(335\) −36.0000 −1.96689
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) −12.0000 −0.650791
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 19.0000 1.01997 0.509987 0.860182i \(-0.329650\pi\)
0.509987 + 0.860182i \(0.329650\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) −12.0000 −0.641427
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −23.0000 −1.21389 −0.606947 0.794742i \(-0.707606\pi\)
−0.606947 + 0.794742i \(0.707606\pi\)
\(360\) −9.00000 −0.474342
\(361\) 0 0
\(362\) −7.00000 −0.367912
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) −36.0000 −1.88433
\(366\) 0 0
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) 2.00000 0.104257
\(369\) −6.00000 −0.312348
\(370\) −27.0000 −1.40366
\(371\) −39.0000 −2.02478
\(372\) 0 0
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) 34.0000 1.74646 0.873231 0.487306i \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 18.0000 0.920960
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) 16.0000 0.814379
\(387\) 24.0000 1.21999
\(388\) 5.00000 0.253837
\(389\) 37.0000 1.87597 0.937987 0.346670i \(-0.112688\pi\)
0.937987 + 0.346670i \(0.112688\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) −2.00000 −0.101015
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) 3.00000 0.150946
\(396\) 3.00000 0.150756
\(397\) 33.0000 1.65622 0.828111 0.560564i \(-0.189416\pi\)
0.828111 + 0.560564i \(0.189416\pi\)
\(398\) 14.0000 0.701757
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −10.0000 −0.497519
\(405\) −27.0000 −1.34164
\(406\) 0 0
\(407\) 9.00000 0.446113
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) 14.0000 0.689730
\(413\) 30.0000 1.47620
\(414\) 6.00000 0.294884
\(415\) −27.0000 −1.32538
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 13.0000 0.632830
\(423\) 36.0000 1.75038
\(424\) 13.0000 0.631336
\(425\) 16.0000 0.776114
\(426\) 0 0
\(427\) 0 0
\(428\) −9.00000 −0.435031
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) −9.00000 −0.433515 −0.216757 0.976226i \(-0.569548\pi\)
−0.216757 + 0.976226i \(0.569548\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) −3.00000 −0.143019
\(441\) −6.00000 −0.285714
\(442\) −8.00000 −0.380521
\(443\) −26.0000 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 3.00000 0.141737
\(449\) 11.0000 0.519122 0.259561 0.965727i \(-0.416422\pi\)
0.259561 + 0.965727i \(0.416422\pi\)
\(450\) 12.0000 0.565685
\(451\) −2.00000 −0.0941763
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) 25.0000 1.17331
\(455\) −18.0000 −0.843853
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 5.00000 0.233635
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 2.00000 0.0926482
\(467\) −26.0000 −1.20314 −0.601568 0.798821i \(-0.705457\pi\)
−0.601568 + 0.798821i \(0.705457\pi\)
\(468\) −6.00000 −0.277350
\(469\) 36.0000 1.66233
\(470\) −36.0000 −1.66056
\(471\) 0 0
\(472\) −10.0000 −0.460287
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 39.0000 1.78569
\(478\) 15.0000 0.686084
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) −18.0000 −0.820729
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −15.0000 −0.681115
\(486\) 0 0
\(487\) −42.0000 −1.90320 −0.951601 0.307337i \(-0.900562\pi\)
−0.951601 + 0.307337i \(0.900562\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) −25.0000 −1.12823 −0.564117 0.825695i \(-0.690783\pi\)
−0.564117 + 0.825695i \(0.690783\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −9.00000 −0.404520
\(496\) 2.00000 0.0898027
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 9.00000 0.400892
\(505\) 30.0000 1.33498
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −17.0000 −0.753512 −0.376756 0.926313i \(-0.622960\pi\)
−0.376756 + 0.926313i \(0.622960\pi\)
\(510\) 0 0
\(511\) 36.0000 1.59255
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −11.0000 −0.485189
\(515\) −42.0000 −1.85074
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 27.0000 1.18631
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) 15.0000 0.655904 0.327952 0.944694i \(-0.393642\pi\)
0.327952 + 0.944694i \(0.393642\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 17.0000 0.741235
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −39.0000 −1.69405
\(531\) −30.0000 −1.30189
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) 27.0000 1.16731
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −15.0000 −0.646696
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) −20.0000 −0.859867 −0.429934 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 11.0000 0.472490
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −31.0000 −1.32546 −0.662732 0.748857i \(-0.730603\pi\)
−0.662732 + 0.748857i \(0.730603\pi\)
\(548\) −15.0000 −0.640768
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) −3.00000 −0.127573
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 6.00000 0.254000
\(559\) −16.0000 −0.676728
\(560\) −9.00000 −0.380319
\(561\) 0 0
\(562\) 0 0
\(563\) 41.0000 1.72794 0.863972 0.503540i \(-0.167969\pi\)
0.863972 + 0.503540i \(0.167969\pi\)
\(564\) 0 0
\(565\) −30.0000 −1.26211
\(566\) 1.00000 0.0420331
\(567\) 27.0000 1.13389
\(568\) −2.00000 −0.0839181
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 13.0000 0.544033 0.272017 0.962293i \(-0.412309\pi\)
0.272017 + 0.962293i \(0.412309\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 8.00000 0.333623
\(576\) −3.00000 −0.125000
\(577\) 41.0000 1.70685 0.853426 0.521214i \(-0.174521\pi\)
0.853426 + 0.521214i \(0.174521\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) 27.0000 1.12015
\(582\) 0 0
\(583\) 13.0000 0.538405
\(584\) −12.0000 −0.496564
\(585\) 18.0000 0.744208
\(586\) 24.0000 0.991431
\(587\) −2.00000 −0.0825488 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 30.0000 1.23508
\(591\) 0 0
\(592\) −9.00000 −0.369898
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) −4.00000 −0.163572
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 6.00000 0.244745 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 24.0000 0.978167
\(603\) −36.0000 −1.46603
\(604\) −3.00000 −0.122068
\(605\) −3.00000 −0.121967
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) −12.0000 −0.485071
\(613\) −44.0000 −1.77714 −0.888572 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) 25.0000 1.00892
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) 33.0000 1.32853 0.664265 0.747497i \(-0.268745\pi\)
0.664265 + 0.747497i \(0.268745\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) −22.0000 −0.882120
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −19.0000 −0.759393
\(627\) 0 0
\(628\) 9.00000 0.359139
\(629\) −36.0000 −1.43541
\(630\) −27.0000 −1.07571
\(631\) −10.0000 −0.398094 −0.199047 0.979990i \(-0.563785\pi\)
−0.199047 + 0.979990i \(0.563785\pi\)
\(632\) 1.00000 0.0397779
\(633\) 0 0
\(634\) 26.0000 1.03259
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 3.00000 0.118585
\(641\) −25.0000 −0.987441 −0.493720 0.869621i \(-0.664363\pi\)
−0.493720 + 0.869621i \(0.664363\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) −9.00000 −0.353553
\(649\) −10.0000 −0.392534
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 36.0000 1.40664
\(656\) 2.00000 0.0780869
\(657\) −36.0000 −1.40449
\(658\) 36.0000 1.40343
\(659\) −1.00000 −0.0389545 −0.0194772 0.999810i \(-0.506200\pi\)
−0.0194772 + 0.999810i \(0.506200\pi\)
\(660\) 0 0
\(661\) 1.00000 0.0388955 0.0194477 0.999811i \(-0.493809\pi\)
0.0194477 + 0.999811i \(0.493809\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) −27.0000 −1.04623
\(667\) 0 0
\(668\) 19.0000 0.735132
\(669\) 0 0
\(670\) 36.0000 1.39080
\(671\) 0 0
\(672\) 0 0
\(673\) −36.0000 −1.38770 −0.693849 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 8.00000 0.307465 0.153732 0.988113i \(-0.450871\pi\)
0.153732 + 0.988113i \(0.450871\pi\)
\(678\) 0 0
\(679\) 15.0000 0.575647
\(680\) 12.0000 0.460179
\(681\) 0 0
\(682\) 2.00000 0.0765840
\(683\) 30.0000 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 0 0
\(685\) 45.0000 1.71936
\(686\) 15.0000 0.572703
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) −26.0000 −0.990521
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 18.0000 0.684257
\(693\) 9.00000 0.341882
\(694\) −19.0000 −0.721230
\(695\) −15.0000 −0.568982
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 8.00000 0.302804
\(699\) 0 0
\(700\) 12.0000 0.453557
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 21.0000 0.790345
\(707\) −30.0000 −1.12827
\(708\) 0 0
\(709\) 31.0000 1.16423 0.582115 0.813107i \(-0.302225\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 6.00000 0.225176
\(711\) 3.00000 0.112509
\(712\) −6.00000 −0.224860
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 23.0000 0.858352
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 9.00000 0.335410
\(721\) 42.0000 1.56416
\(722\) 0 0
\(723\) 0 0
\(724\) 7.00000 0.260153
\(725\) 0 0
\(726\) 0 0
\(727\) 22.0000 0.815935 0.407967 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) −6.00000 −0.222375
\(729\) −27.0000 −1.00000
\(730\) 36.0000 1.33242
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) −24.0000 −0.886460 −0.443230 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) −12.0000 −0.442026
\(738\) 6.00000 0.220863
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 27.0000 0.992540
\(741\) 0 0
\(742\) 39.0000 1.43174
\(743\) 19.0000 0.697042 0.348521 0.937301i \(-0.386684\pi\)
0.348521 + 0.937301i \(0.386684\pi\)
\(744\) 0 0
\(745\) 60.0000 2.19823
\(746\) 8.00000 0.292901
\(747\) −27.0000 −0.987878
\(748\) −4.00000 −0.146254
\(749\) −27.0000 −0.986559
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 9.00000 0.327544
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −34.0000 −1.23494
\(759\) 0 0
\(760\) 0 0
\(761\) −44.0000 −1.59500 −0.797499 0.603320i \(-0.793844\pi\)
−0.797499 + 0.603320i \(0.793844\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) −18.0000 −0.651217
\(765\) 36.0000 1.30158
\(766\) 24.0000 0.867155
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −9.00000 −0.324337
\(771\) 0 0
\(772\) −16.0000 −0.575853
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −24.0000 −0.862662
\(775\) 8.00000 0.287368
\(776\) −5.00000 −0.179490
\(777\) 0 0
\(778\) −37.0000 −1.32651
\(779\) 0 0
\(780\) 0 0
\(781\) −2.00000 −0.0715656
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) −27.0000 −0.963671
\(786\) 0 0
\(787\) −11.0000 −0.392108 −0.196054 0.980593i \(-0.562813\pi\)
−0.196054 + 0.980593i \(0.562813\pi\)
\(788\) −22.0000 −0.783718
\(789\) 0 0
\(790\) −3.00000 −0.106735
\(791\) 30.0000 1.06668
\(792\) −3.00000 −0.106600
\(793\) 0 0
\(794\) −33.0000 −1.17113
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) −23.0000 −0.814702 −0.407351 0.913272i \(-0.633547\pi\)
−0.407351 + 0.913272i \(0.633547\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) −4.00000 −0.141421
\(801\) −18.0000 −0.635999
\(802\) 30.0000 1.05934
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) −18.0000 −0.634417
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 27.0000 0.948683
\(811\) 1.00000 0.0351147 0.0175574 0.999846i \(-0.494411\pi\)
0.0175574 + 0.999846i \(0.494411\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −9.00000 −0.315450
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) −18.0000 −0.628971
\(820\) −6.00000 −0.209529
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −30.0000 −1.04383
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) −6.00000 −0.208514
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 27.0000 0.937184
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 8.00000 0.277184
\(834\) 0 0
\(835\) −57.0000 −1.97257
\(836\) 0 0
\(837\) 0 0
\(838\) 4.00000 0.138178
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 19.0000 0.654783
\(843\) 0 0
\(844\) −13.0000 −0.447478
\(845\) 27.0000 0.928828
\(846\) −36.0000 −1.23771
\(847\) 3.00000 0.103081
\(848\) −13.0000 −0.446422
\(849\) 0 0
\(850\) −16.0000 −0.548795
\(851\) −18.0000 −0.617032
\(852\) 0 0
\(853\) 18.0000 0.616308 0.308154 0.951336i \(-0.400289\pi\)
0.308154 + 0.951336i \(0.400289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 9.00000 0.307614
\(857\) 48.0000 1.63965 0.819824 0.572615i \(-0.194071\pi\)
0.819824 + 0.572615i \(0.194071\pi\)
\(858\) 0 0
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) 9.00000 0.306541
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 0 0
\(865\) −54.0000 −1.83606
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) 1.00000 0.0339227
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) −2.00000 −0.0677285
\(873\) −15.0000 −0.507673
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −7.00000 −0.236239
\(879\) 0 0
\(880\) 3.00000 0.101130
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 6.00000 0.202031
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 26.0000 0.873487
\(887\) −9.00000 −0.302190 −0.151095 0.988519i \(-0.548280\pi\)
−0.151095 + 0.988519i \(0.548280\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 18.0000 0.603361
\(891\) −9.00000 −0.301511
\(892\) 4.00000 0.133930
\(893\) 0 0
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −11.0000 −0.367075
\(899\) 0 0
\(900\) −12.0000 −0.400000
\(901\) −52.0000 −1.73237
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) −21.0000 −0.698064
\(906\) 0 0
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) −25.0000 −0.829654
\(909\) 30.0000 0.995037
\(910\) 18.0000 0.596694
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) −9.00000 −0.297857
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −5.00000 −0.165205
\(917\) −36.0000 −1.18882
\(918\) 0 0
\(919\) 51.0000 1.68233 0.841167 0.540775i \(-0.181869\pi\)
0.841167 + 0.540775i \(0.181869\pi\)
\(920\) 6.00000 0.197814
\(921\) 0 0
\(922\) 42.0000 1.38320
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) −36.0000 −1.18367
\(926\) 16.0000 0.525793
\(927\) −42.0000 −1.37946
\(928\) 0 0
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.00000 −0.0655122
\(933\) 0 0
\(934\) 26.0000 0.850746
\(935\) 12.0000 0.392442
\(936\) 6.00000 0.196116
\(937\) −40.0000 −1.30674 −0.653372 0.757037i \(-0.726646\pi\)
−0.653372 + 0.757037i \(0.726646\pi\)
\(938\) −36.0000 −1.17544
\(939\) 0 0
\(940\) 36.0000 1.17419
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 34.0000 1.10485 0.552426 0.833562i \(-0.313702\pi\)
0.552426 + 0.833562i \(0.313702\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) −12.0000 −0.388922
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) −39.0000 −1.26267
\(955\) 54.0000 1.74740
\(956\) −15.0000 −0.485135
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) −45.0000 −1.45313
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 18.0000 0.580343
\(963\) 27.0000 0.870063
\(964\) 22.0000 0.708572
\(965\) 48.0000 1.54517
\(966\) 0 0
\(967\) 53.0000 1.70437 0.852183 0.523245i \(-0.175279\pi\)
0.852183 + 0.523245i \(0.175279\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 15.0000 0.481621
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) 15.0000 0.480878
\(974\) 42.0000 1.34577
\(975\) 0 0
\(976\) 0 0
\(977\) 9.00000 0.287936 0.143968 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) −6.00000 −0.191663
\(981\) −6.00000 −0.191565
\(982\) 25.0000 0.797782
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 0 0
\(985\) 66.0000 2.10293
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 9.00000 0.286039
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) −6.00000 −0.190308
\(995\) 42.0000 1.33149
\(996\) 0 0
\(997\) 52.0000 1.64686 0.823428 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) 28.0000 0.886325
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7942.2.a.e.1.1 1
19.7 even 3 418.2.e.e.353.1 yes 2
19.11 even 3 418.2.e.e.45.1 2
19.18 odd 2 7942.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.e.e.45.1 2 19.11 even 3
418.2.e.e.353.1 yes 2 19.7 even 3
7942.2.a.e.1.1 1 1.1 even 1 trivial
7942.2.a.p.1.1 1 19.18 odd 2