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Properties

Label 7942.2.a.q.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{18} +1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} -4.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -8.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} +1.00000 q^{35} -3.00000 q^{36} +3.00000 q^{37} +1.00000 q^{40} -2.00000 q^{41} -8.00000 q^{43} +1.00000 q^{44} -3.00000 q^{45} -6.00000 q^{46} -6.00000 q^{49} -4.00000 q^{50} +2.00000 q^{52} -9.00000 q^{53} +1.00000 q^{55} +1.00000 q^{56} -8.00000 q^{58} -14.0000 q^{59} -10.0000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} +1.00000 q^{70} +6.00000 q^{71} -3.00000 q^{72} +3.00000 q^{74} +1.00000 q^{77} -11.0000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +7.00000 q^{83} -8.00000 q^{86} +1.00000 q^{88} +14.0000 q^{89} -3.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} +5.00000 q^{97} -6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −3.00000 −0.707107
\(19\) 0 0
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −3.00000 −0.500000
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 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
\(45\) −3.00000 −0.447214
\(46\) −6.00000 −0.884652
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −10.0000 −1.27000
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −3.00000 −0.353553
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) −2.00000 −0.220863
\(83\) 7.00000 0.768350 0.384175 − 0.923260i \(-0.374486\pi\)
0.384175 + 0.923260i \(0.374486\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −3.00000 −0.316228
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) −6.00000 −0.606092
\(99\) −3.00000 −0.301511
\(100\) −4.00000 −0.400000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 − 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 17.0000 1.64345 0.821726 − 0.569883i \(-0.193011\pi\)
0.821726 + 0.569883i \(0.193011\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) −8.00000 −0.742781
\(117\) −6.00000 −0.554700
\(118\) −14.0000 −1.28880
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) −9.00000 −0.804984
\(126\) −3.00000 −0.267261
\(127\) 20.0000 1.77471 0.887357 − 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −20.0000 −1.74741 −0.873704 − 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) −23.0000 −1.96502 −0.982511 − 0.186203i \(-0.940382\pi\)
−0.982511 + 0.186203i \(0.940382\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 2.00000 0.167248
\(144\) −3.00000 −0.250000
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 3.00000 0.246598
\(149\) −4.00000 −0.327693 −0.163846 − 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 − 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) 21.0000 1.67598 0.837991 − 0.545684i \(-0.183730\pi\)
0.837991 + 0.545684i \(0.183730\pi\)
\(158\) −11.0000 −0.875113
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −6.00000 −0.472866
\(162\) 9.00000 0.707107
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) 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
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) −16.0000 −1.19590 −0.597948 − 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) −3.00000 −0.223607
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 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
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 5.00000 0.358979
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 2.00000 0.142494 0.0712470 − 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −3.00000 −0.213201
\(199\) 10.0000 0.708881 0.354441 − 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 6.00000 0.418040
\(207\) 18.0000 1.25109
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 − 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) 17.0000 1.16210
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 1.00000 0.0668153
\(225\) 12.0000 0.800000
\(226\) 2.00000 0.133038
\(227\) 25.0000 1.65931 0.829654 − 0.558278i \(-0.188538\pi\)
0.829654 + 0.558278i \(0.188538\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 − 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) 0 0
\(238\) 0 0
\(239\) −5.00000 −0.323423 −0.161712 − 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) −10.0000 −0.635001
\(249\) 0 0
\(250\) −9.00000 −0.569210
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −3.00000 −0.188982
\(253\) −6.00000 −0.377217
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −21.0000 −1.30994 −0.654972 − 0.755653i \(-0.727320\pi\)
−0.654972 + 0.755653i \(0.727320\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 2.00000 0.124035
\(261\) 24.0000 1.48556
\(262\) −20.0000 −1.23560
\(263\) −3.00000 −0.184988 −0.0924940 − 0.995713i \(-0.529484\pi\)
−0.0924940 + 0.995713i \(0.529484\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) −21.0000 −1.28039 −0.640196 − 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) 15.0000 0.911185 0.455593 − 0.890188i \(-0.349427\pi\)
0.455593 + 0.890188i \(0.349427\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −23.0000 −1.38948
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −5.00000 −0.299880
\(279\) 30.0000 1.79605
\(280\) 1.00000 0.0597614
\(281\) 20.0000 1.19310 0.596550 − 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) −7.00000 −0.416107 −0.208053 − 0.978117i \(-0.566713\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −2.00000 −0.118056
\(288\) −3.00000 −0.176777
\(289\) −17.0000 −1.00000
\(290\) −8.00000 −0.469776
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) 3.00000 0.174371
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −9.00000 −0.517892
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 − 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −10.0000 −0.567962
\(311\) −10.0000 −0.567048 −0.283524 − 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) −21.0000 −1.18699 −0.593495 − 0.804838i \(-0.702252\pi\)
−0.593495 + 0.804838i \(0.702252\pi\)
\(314\) 21.0000 1.18510
\(315\) −3.00000 −0.169031
\(316\) −11.0000 −0.618798
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 1.00000 0.0559017
\(321\) 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
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 7.00000 0.384175
\(333\) −9.00000 −0.493197
\(334\) 9.00000 0.492458
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) −3.00000 −0.161048 −0.0805242 − 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 − 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 11.0000 0.585471 0.292735 − 0.956193i \(-0.405434\pi\)
0.292735 + 0.956193i \(0.405434\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −16.0000 −0.845626
\(359\) 3.00000 0.158334 0.0791670 − 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) −3.00000 −0.158114
\(361\) 0 0
\(362\) −5.00000 −0.262794
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 10.0000 0.521996 0.260998 − 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −6.00000 −0.312772
\(369\) 6.00000 0.312348
\(370\) 3.00000 0.155963
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.0000 −0.824042
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 − 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) 8.00000 0.408781 0.204390 − 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 4.00000 0.203595
\(387\) 24.0000 1.21999
\(388\) 5.00000 0.253837
\(389\) −23.0000 −1.16615 −0.583073 − 0.812420i \(-0.698150\pi\)
−0.583073 + 0.812420i \(0.698150\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) −11.0000 −0.553470
\(396\) −3.00000 −0.150756
\(397\) −27.0000 −1.35509 −0.677546 − 0.735481i \(-0.736956\pi\)
−0.677546 + 0.735481i \(0.736956\pi\)
\(398\) 10.0000 0.501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) 10.0000 0.497519
\(405\) 9.00000 0.447214
\(406\) −8.00000 −0.397033
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) 6.00000 0.295599
\(413\) −14.0000 −0.688895
\(414\) 18.0000 0.884652
\(415\) 7.00000 0.343616
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −15.0000 −0.731055 −0.365528 − 0.930800i \(-0.619111\pi\)
−0.365528 + 0.930800i \(0.619111\pi\)
\(422\) 13.0000 0.632830
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 17.0000 0.821726
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 5.00000 0.240842 0.120421 − 0.992723i \(-0.461576\pi\)
0.120421 + 0.992723i \(0.461576\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 − 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) −10.0000 −0.480015
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 0 0
\(439\) 29.0000 1.38409 0.692047 − 0.721852i \(-0.256709\pi\)
0.692047 + 0.721852i \(0.256709\pi\)
\(440\) 1.00000 0.0476731
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 22.0000 1.04525 0.522626 − 0.852562i \(-0.324953\pi\)
0.522626 + 0.852562i \(0.324953\pi\)
\(444\) 0 0
\(445\) 14.0000 0.663664
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −5.00000 −0.235965 −0.117982 − 0.993016i \(-0.537643\pi\)
−0.117982 + 0.993016i \(0.537643\pi\)
\(450\) 12.0000 0.565685
\(451\) −2.00000 −0.0941763
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) 25.0000 1.17331
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 − 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −1.00000 −0.0467269
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 26.0000 1.21094 0.605470 − 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 6.00000 0.277647 0.138823 − 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) −6.00000 −0.277350
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) −14.0000 −0.644402
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 27.0000 1.23625
\(478\) −5.00000 −0.228695
\(479\) −32.0000 −1.46212 −0.731059 − 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 5.00000 0.227038
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 − 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) −15.0000 −0.676941 −0.338470 − 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) −10.0000 −0.449013
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) −9.00000 −0.402492
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −3.00000 −0.133631
\(505\) 10.0000 0.444994
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) −21.0000 −0.930809 −0.465404 − 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −21.0000 −0.926270
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) 0 0
\(518\) 3.00000 0.131812
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) −31.0000 −1.35813 −0.679067 − 0.734076i \(-0.737616\pi\)
−0.679067 + 0.734076i \(0.737616\pi\)
\(522\) 24.0000 1.05045
\(523\) −7.00000 −0.306089 −0.153044 − 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) −3.00000 −0.130806
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −9.00000 −0.390935
\(531\) 42.0000 1.82264
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 17.0000 0.734974
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) −21.0000 −0.905374
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 − 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 15.0000 0.644305
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 − 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) −23.0000 −0.982511
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) −11.0000 −0.467768
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −5.00000 −0.212047
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 30.0000 1.27000
\(559\) −16.0000 −0.676728
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 20.0000 0.843649
\(563\) −17.0000 −0.716465 −0.358232 − 0.933632i \(-0.616620\pi\)
−0.358232 + 0.933632i \(0.616620\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) −7.00000 −0.294232
\(567\) 9.00000 0.377964
\(568\) 6.00000 0.251754
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −37.0000 −1.54840 −0.774201 − 0.632940i \(-0.781848\pi\)
−0.774201 + 0.632940i \(0.781848\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 24.0000 1.00087
\(576\) −3.00000 −0.125000
\(577\) 25.0000 1.04076 0.520382 − 0.853934i \(-0.325790\pi\)
0.520382 + 0.853934i \(0.325790\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) −8.00000 −0.332182
\(581\) 7.00000 0.290409
\(582\) 0 0
\(583\) −9.00000 −0.372742
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) 26.0000 1.07313 0.536567 − 0.843857i \(-0.319721\pi\)
0.536567 + 0.843857i \(0.319721\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −14.0000 −0.576371
\(591\) 0 0
\(592\) 3.00000 0.123299
\(593\) 14.0000 0.574911 0.287456 − 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 − 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −8.00000 −0.326056
\(603\) −36.0000 −1.46603
\(604\) −9.00000 −0.366205
\(605\) 1.00000 0.0406558
\(606\) 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
\(611\) 0 0
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 − 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) −23.0000 −0.928204
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −15.0000 −0.603877 −0.301939 − 0.953327i \(-0.597634\pi\)
−0.301939 + 0.953327i \(0.597634\pi\)
\(618\) 0 0
\(619\) 46.0000 1.84890 0.924448 − 0.381308i \(-0.124526\pi\)
0.924448 + 0.381308i \(0.124526\pi\)
\(620\) −10.0000 −0.401610
\(621\) 0 0
\(622\) −10.0000 −0.400963
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −21.0000 −0.839329
\(627\) 0 0
\(628\) 21.0000 0.837991
\(629\) 0 0
\(630\) −3.00000 −0.119523
\(631\) −22.0000 −0.875806 −0.437903 − 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) −11.0000 −0.437557
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) 20.0000 0.793676
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) −8.00000 −0.316723
\(639\) −18.0000 −0.712069
\(640\) 1.00000 0.0395285
\(641\) 39.0000 1.54041 0.770204 − 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 − 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 9.00000 0.353553
\(649\) −14.0000 −0.549548
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) −20.0000 −0.781465
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 − 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) −11.0000 −0.427850 −0.213925 − 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) −9.00000 −0.348743
\(667\) 48.0000 1.85857
\(668\) 9.00000 0.348220
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) 44.0000 1.69608 0.848038 − 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −36.0000 −1.38359 −0.691796 − 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 0 0
\(679\) 5.00000 0.191882
\(680\) 0 0
\(681\) 0 0
\(682\) −10.0000 −0.382920
\(683\) 26.0000 0.994862 0.497431 − 0.867503i \(-0.334277\pi\)
0.497431 + 0.867503i \(0.334277\pi\)
\(684\) 0 0
\(685\) −23.0000 −0.878785
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) −50.0000 −1.90209 −0.951045 − 0.309053i \(-0.899988\pi\)
−0.951045 + 0.309053i \(0.899988\pi\)
\(692\) 22.0000 0.836315
\(693\) −3.00000 −0.113961
\(694\) −3.00000 −0.113878
\(695\) −5.00000 −0.189661
\(696\) 0 0
\(697\) 0 0
\(698\) 28.0000 1.05982
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −36.0000 −1.35970 −0.679851 − 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 11.0000 0.413990
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) 43.0000 1.61490 0.807449 − 0.589937i \(-0.200847\pi\)
0.807449 + 0.589937i \(0.200847\pi\)
\(710\) 6.00000 0.225176
\(711\) 33.0000 1.23760
\(712\) 14.0000 0.524672
\(713\) 60.0000 2.24702
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −16.0000 −0.597948
\(717\) 0 0
\(718\) 3.00000 0.111959
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −3.00000 −0.111803
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) 0 0
\(724\) −5.00000 −0.185824
\(725\) 32.0000 1.18845
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 − 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 2.00000 0.0741249
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −36.0000 −1.32969 −0.664845 − 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) 12.0000 0.442026
\(738\) 6.00000 0.220863
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 3.00000 0.110282
\(741\) 0 0
\(742\) −9.00000 −0.330400
\(743\) 41.0000 1.50414 0.752072 − 0.659081i \(-0.229055\pi\)
0.752072 + 0.659081i \(0.229055\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) −4.00000 −0.146450
\(747\) −21.0000 −0.768350
\(748\) 0 0
\(749\) 17.0000 0.621166
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 − 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −16.0000 −0.582686
\(755\) −9.00000 −0.327544
\(756\) 0 0
\(757\) −50.0000 −1.81728 −0.908640 − 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 2.00000 0.0726433
\(759\) 0 0
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 − 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) −28.0000 −1.01102
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 − 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 4.00000 0.143963
\(773\) 38.0000 1.36677 0.683383 − 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 24.0000 0.862662
\(775\) 40.0000 1.43684
\(776\) 5.00000 0.179490
\(777\) 0 0
\(778\) −23.0000 −0.824590
\(779\) 0 0
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 21.0000 0.749522
\(786\) 0 0
\(787\) 11.0000 0.392108 0.196054 − 0.980593i \(-0.437187\pi\)
0.196054 + 0.980593i \(0.437187\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) −11.0000 −0.391362
\(791\) 2.00000 0.0711118
\(792\) −3.00000 −0.106600
\(793\) 0 0
\(794\) −27.0000 −0.958194
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) 21.0000 0.743858 0.371929 − 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) −42.0000 −1.48400
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) 0 0
\(805\) −6.00000 −0.211472
\(806\) −20.0000 −0.704470
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) −44.0000 −1.54696 −0.773479 − 0.633822i \(-0.781485\pi\)
−0.773479 + 0.633822i \(0.781485\pi\)
\(810\) 9.00000 0.316228
\(811\) 23.0000 0.807639 0.403820 − 0.914839i \(-0.367682\pi\)
0.403820 + 0.914839i \(0.367682\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 0 0
\(818\) 10.0000 0.349642
\(819\) −6.00000 −0.209657
\(820\) −2.00000 −0.0698430
\(821\) −20.0000 −0.698005 −0.349002 − 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −14.0000 −0.487122
\(827\) −8.00000 −0.278187 −0.139094 − 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 18.0000 0.625543
\(829\) −10.0000 −0.347314 −0.173657 − 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 7.00000 0.242974
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 0 0
\(835\) 9.00000 0.311458
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.0000 −0.552381 −0.276191 − 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −15.0000 −0.516934
\(843\) 0 0
\(844\) 13.0000 0.447478
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −9.00000 −0.309061
\(849\) 0 0
\(850\) 0 0
\(851\) −18.0000 −0.617032
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 − 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 17.0000 0.581048
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 − 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 5.00000 0.170301
\(863\) −32.0000 −1.08929 −0.544646 − 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 30.0000 1.01944
\(867\) 0 0
\(868\) −10.0000 −0.339422
\(869\) −11.0000 −0.373149
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) −10.0000 −0.338643
\(873\) −15.0000 −0.507673
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 29.0000 0.978703
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 18.0000 0.606092
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 22.0000 0.739104
\(887\) −35.0000 −1.17518 −0.587592 − 0.809157i \(-0.699924\pi\)
−0.587592 + 0.809157i \(0.699924\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 14.0000 0.469281
\(891\) 9.00000 0.301511
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −5.00000 −0.166852
\(899\) 80.0000 2.66815
\(900\) 12.0000 0.400000
\(901\) 0 0
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −5.00000 −0.166206
\(906\) 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
\(910\) 2.00000 0.0662994
\(911\) −12.0000 −0.397578 −0.198789 − 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 7.00000 0.231666
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) −1.00000 −0.0330409
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) −39.0000 −1.28649 −0.643246 − 0.765660i \(-0.722413\pi\)
−0.643246 + 0.765660i \(0.722413\pi\)
\(920\) −6.00000 −0.197814
\(921\) 0 0
\(922\) 26.0000 0.856264
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 16.0000 0.525793
\(927\) −18.0000 −0.591198
\(928\) −8.00000 −0.262613
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 16.0000 0.522697 0.261349 − 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 − 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) −14.0000 −0.455661
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 10.0000 0.324956 0.162478 − 0.986712i \(-0.448051\pi\)
0.162478 + 0.986712i \(0.448051\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 − 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 27.0000 0.874157
\(955\) −18.0000 −0.582466
\(956\) −5.00000 −0.161712
\(957\) 0 0
\(958\) −32.0000 −1.03387
\(959\) −23.0000 −0.742709
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 6.00000 0.193448
\(963\) −51.0000 −1.64345
\(964\) 10.0000 0.322078
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −1.00000 −0.0321578 −0.0160789 − 0.999871i \(-0.505118\pi\)
−0.0160789 + 0.999871i \(0.505118\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 5.00000 0.160540
\(971\) −20.0000 −0.641831 −0.320915 − 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) −5.00000 −0.160293
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 0 0
\(977\) 33.0000 1.05576 0.527882 − 0.849318i \(-0.322986\pi\)
0.527882 + 0.849318i \(0.322986\pi\)
\(978\) 0 0
\(979\) 14.0000 0.447442
\(980\) −6.00000 −0.191663
\(981\) 30.0000 0.957826
\(982\) −15.0000 −0.478669
\(983\) 14.0000 0.446531 0.223265 − 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) −3.00000 −0.0953463
\(991\) 4.00000 0.127064 0.0635321 − 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −10.0000 −0.317500
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) 10.0000 0.317021
\(996\) 0 0
\(997\) 52.0000 1.64686 0.823428 − 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) −32.0000 −1.01294
\(999\) 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.q.1.1 1
19.7 even 3 418.2.e.b.353.1 yes 2
19.11 even 3 418.2.e.b.45.1 ✓ 2
19.18 odd 2 7942.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.e.b.45.1 ✓ 2 19.11 even 3
418.2.e.b.353.1 yes 2 19.7 even 3
7942.2.a.f.1.1 1 19.18 odd 2
7942.2.a.q.1.1 1 1.1 even 1 trivial