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Properties

Label 197.2.a.a.1.1
Level $197$
Weight $2$
Character 197.1
Self dual yes
Analytic conductor $1.573$
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] = [197,2,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 197.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: \(1.57305291982\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 197.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{7} -3.00000 q^{9} +4.00000 q^{11} -2.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} -8.00000 q^{17} +6.00000 q^{18} -3.00000 q^{19} -8.00000 q^{22} -3.00000 q^{23} -5.00000 q^{25} +4.00000 q^{26} -6.00000 q^{28} +7.00000 q^{29} -10.0000 q^{31} +8.00000 q^{32} +16.0000 q^{34} -6.00000 q^{36} +7.00000 q^{37} +6.00000 q^{38} +9.00000 q^{41} +1.00000 q^{43} +8.00000 q^{44} +6.00000 q^{46} -11.0000 q^{47} +2.00000 q^{49} +10.0000 q^{50} -4.00000 q^{52} +10.0000 q^{53} -14.0000 q^{58} +5.00000 q^{61} +20.0000 q^{62} +9.00000 q^{63} -8.00000 q^{64} -10.0000 q^{67} -16.0000 q^{68} +8.00000 q^{71} +6.00000 q^{73} -14.0000 q^{74} -6.00000 q^{76} -12.0000 q^{77} +2.00000 q^{79} +9.00000 q^{81} -18.0000 q^{82} -7.00000 q^{83} -2.00000 q^{86} -8.00000 q^{89} +6.00000 q^{91} -6.00000 q^{92} +22.0000 q^{94} -2.00000 q^{97} -4.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 − 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −8.00000 −1.94029 −0.970143 − 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 6.00000 1.41421
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.00000 −1.70561
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −6.00000 −1.13389
\(29\) 7.00000 1.29987 0.649934 − 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 16.0000 2.74398
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 8.00000 1.20605
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −11.0000 −1.60451 −0.802257 − 0.596978i \(-0.796368\pi\)
−0.802257 + 0.596978i \(0.796368\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 10.0000 1.41421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −14.0000 −1.83829
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 20.0000 2.54000
\(63\) 9.00000 1.13389
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −16.0000 −1.94029
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −14.0000 −1.62747
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 − 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −18.0000 −1.98777
\(83\) −7.00000 −0.768350 −0.384175 − 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 22.0000 2.26913
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −4.00000 −0.404061
\(99\) −12.0000 −1.20605
\(100\) −10.0000 −1.00000
\(101\) −15.0000 −1.49256 −0.746278 − 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −20.0000 −1.94257
\(107\) −9.00000 −0.870063 −0.435031 − 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) 13.0000 1.24517 0.622587 − 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12.0000 1.13389
\(113\) −4.00000 −0.376288 −0.188144 − 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 14.0000 1.29987
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −20.0000 −1.79605
\(125\) 0 0
\(126\) −18.0000 −1.60357
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 − 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 9.00000 0.780399
\(134\) 20.0000 1.72774
\(135\) 0 0
\(136\) 0 0
\(137\) −21.0000 −1.79415 −0.897076 − 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 − 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) −8.00000 −0.668994
\(144\) 12.0000 1.00000
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) 14.0000 1.15079
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 24.0000 1.94029
\(154\) 24.0000 1.93398
\(155\) 0 0
\(156\) 0 0
\(157\) 11.0000 0.877896 0.438948 − 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) −18.0000 −1.41421
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 18.0000 1.40556
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 9.00000 0.688247
\(172\) 2.00000 0.152499
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 15.0000 1.13389
\(176\) −16.0000 −1.20605
\(177\) 0 0
\(178\) 16.0000 1.19925
\(179\) 10.0000 0.747435 0.373718 − 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −12.0000 −0.889499
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −32.0000 −2.34007
\(188\) −22.0000 −1.60451
\(189\) 0 0
\(190\) 0 0
\(191\) −13.0000 −0.940647 −0.470323 − 0.882494i \(-0.655863\pi\)
−0.470323 + 0.882494i \(0.655863\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 − 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) −1.00000 −0.0712470
\(198\) 24.0000 1.70561
\(199\) 2.00000 0.141776 0.0708881 − 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 30.0000 2.11079
\(203\) −21.0000 −1.47391
\(204\) 0 0
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 9.00000 0.625543
\(208\) 8.00000 0.554700
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 20.0000 1.37361
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) 30.0000 2.03653
\(218\) −26.0000 −1.76094
\(219\) 0 0
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) −9.00000 −0.602685 −0.301342 − 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) −24.0000 −1.60357
\(225\) 15.0000 1.00000
\(226\) 8.00000 0.532152
\(227\) 8.00000 0.530979 0.265489 − 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −24.0000 −1.58596 −0.792982 − 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 −0.196537 −0.0982683 − 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) −12.0000 −0.784465
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) −48.0000 −3.11138
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 − 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 − 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 18.0000 1.13389
\(253\) −12.0000 −0.754434
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −1.00000 −0.0623783 −0.0311891 − 0.999514i \(-0.509929\pi\)
−0.0311891 + 0.999514i \(0.509929\pi\)
\(258\) 0 0
\(259\) −21.0000 −1.30488
\(260\) 0 0
\(261\) −21.0000 −1.29987
\(262\) −36.0000 −2.22409
\(263\) 26.0000 1.60323 0.801614 − 0.597841i \(-0.203975\pi\)
0.801614 + 0.597841i \(0.203975\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −18.0000 −1.10365
\(267\) 0 0
\(268\) −20.0000 −1.22169
\(269\) 28.0000 1.70719 0.853595 − 0.520937i \(-0.174417\pi\)
0.853595 + 0.520937i \(0.174417\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 32.0000 1.94029
\(273\) 0 0
\(274\) 42.0000 2.53731
\(275\) −20.0000 −1.20605
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 20.0000 1.19952
\(279\) 30.0000 1.79605
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 6.00000 0.356663 0.178331 − 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 16.0000 0.946100
\(287\) −27.0000 −1.59376
\(288\) −24.0000 −1.41421
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 0 0
\(292\) 12.0000 0.702247
\(293\) −9.00000 −0.525786 −0.262893 − 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) 12.0000 0.688247
\(305\) 0 0
\(306\) −48.0000 −2.74398
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −24.0000 −1.36753
\(309\) 0 0
\(310\) 0 0
\(311\) −27.0000 −1.53103 −0.765515 − 0.643418i \(-0.777516\pi\)
−0.765515 + 0.643418i \(0.777516\pi\)
\(312\) 0 0
\(313\) 17.0000 0.960897 0.480448 − 0.877023i \(-0.340474\pi\)
0.480448 + 0.877023i \(0.340474\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 28.0000 1.56770
\(320\) 0 0
\(321\) 0 0
\(322\) −18.0000 −1.00310
\(323\) 24.0000 1.33540
\(324\) 18.0000 1.00000
\(325\) 10.0000 0.554700
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 0 0
\(329\) 33.0000 1.81935
\(330\) 0 0
\(331\) −11.0000 −0.604615 −0.302307 − 0.953211i \(-0.597757\pi\)
−0.302307 + 0.953211i \(0.597757\pi\)
\(332\) −14.0000 −0.768350
\(333\) −21.0000 −1.15079
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 18.0000 0.979071
\(339\) 0 0
\(340\) 0 0
\(341\) −40.0000 −2.16612
\(342\) −18.0000 −0.973329
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 4.00000 0.215041
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 − 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) −30.0000 −1.60357
\(351\) 0 0
\(352\) 32.0000 1.70561
\(353\) 9.00000 0.479022 0.239511 − 0.970894i \(-0.423013\pi\)
0.239511 + 0.970894i \(0.423013\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −16.0000 −0.847998
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −14.0000 −0.738892 −0.369446 − 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −4.00000 −0.210235
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 − 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 12.0000 0.625543
\(369\) −27.0000 −1.40556
\(370\) 0 0
\(371\) −30.0000 −1.55752
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 64.0000 3.30936
\(375\) 0 0
\(376\) 0 0
\(377\) −14.0000 −0.721037
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 26.0000 1.33028
\(383\) −2.00000 −0.102195 −0.0510976 − 0.998694i \(-0.516272\pi\)
−0.0510976 + 0.998694i \(0.516272\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) −3.00000 −0.152499
\(388\) −4.00000 −0.203069
\(389\) −4.00000 −0.202808 −0.101404 − 0.994845i \(-0.532333\pi\)
−0.101404 + 0.994845i \(0.532333\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −24.0000 −1.20605
\(397\) 16.0000 0.803017 0.401508 − 0.915855i \(-0.368486\pi\)
0.401508 + 0.915855i \(0.368486\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 20.0000 1.00000
\(401\) −33.0000 −1.64794 −0.823971 − 0.566632i \(-0.808246\pi\)
−0.823971 + 0.566632i \(0.808246\pi\)
\(402\) 0 0
\(403\) 20.0000 0.996271
\(404\) −30.0000 −1.49256
\(405\) 0 0
\(406\) 42.0000 2.08443
\(407\) 28.0000 1.38791
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 − 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) −18.0000 −0.884652
\(415\) 0 0
\(416\) −16.0000 −0.784465
\(417\) 0 0
\(418\) 24.0000 1.17388
\(419\) 9.00000 0.439679 0.219839 − 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 40.0000 1.94717
\(423\) 33.0000 1.60451
\(424\) 0 0
\(425\) 40.0000 1.94029
\(426\) 0 0
\(427\) −15.0000 −0.725901
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 0 0
\(431\) 33.0000 1.58955 0.794777 − 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 − 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) −60.0000 −2.88009
\(435\) 0 0
\(436\) 26.0000 1.24517
\(437\) 9.00000 0.430528
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −32.0000 −1.52208
\(443\) −39.0000 −1.85295 −0.926473 − 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 18.0000 0.852325
\(447\) 0 0
\(448\) 24.0000 1.13389
\(449\) −15.0000 −0.707894 −0.353947 − 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) −30.0000 −1.41421
\(451\) 36.0000 1.69517
\(452\) −8.00000 −0.376288
\(453\) 0 0
\(454\) −16.0000 −0.750917
\(455\) 0 0
\(456\) 0 0
\(457\) 13.0000 0.608114 0.304057 − 0.952654i \(-0.401659\pi\)
0.304057 + 0.952654i \(0.401659\pi\)
\(458\) 48.0000 2.24289
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 − 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −28.0000 −1.29987
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 12.0000 0.554700
\(469\) 30.0000 1.38527
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 15.0000 0.688247
\(476\) 48.0000 2.20008
\(477\) −30.0000 −1.37361
\(478\) 42.0000 1.92104
\(479\) −4.00000 −0.182765 −0.0913823 − 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) 0 0
\(486\) 0 0
\(487\) −5.00000 −0.226572 −0.113286 − 0.993562i \(-0.536138\pi\)
−0.113286 + 0.993562i \(0.536138\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.00000 0.225647 0.112823 − 0.993615i \(-0.464011\pi\)
0.112823 + 0.993615i \(0.464011\pi\)
\(492\) 0 0
\(493\) −56.0000 −2.52211
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) 40.0000 1.79605
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −44.0000 −1.96971 −0.984855 − 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −10.0000 −0.446322
\(503\) −7.00000 −0.312115 −0.156057 − 0.987748i \(-0.549878\pi\)
−0.156057 + 0.987748i \(0.549878\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) 4.00000 0.177297 0.0886484 − 0.996063i \(-0.471745\pi\)
0.0886484 + 0.996063i \(0.471745\pi\)
\(510\) 0 0
\(511\) −18.0000 −0.796273
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 0 0
\(517\) −44.0000 −1.93512
\(518\) 42.0000 1.84537
\(519\) 0 0
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 − 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 42.0000 1.83829
\(523\) −38.0000 −1.66162 −0.830812 − 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) 36.0000 1.57267
\(525\) 0 0
\(526\) −52.0000 −2.26731
\(527\) 80.0000 3.48485
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 18.0000 0.780399
\(533\) −18.0000 −0.779667
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −56.0000 −2.41433
\(539\) 8.00000 0.344584
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 40.0000 1.71815
\(543\) 0 0
\(544\) −64.0000 −2.74398
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −42.0000 −1.79415
\(549\) −15.0000 −0.640184
\(550\) 40.0000 1.70561
\(551\) −21.0000 −0.894630
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) −16.0000 −0.679775
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −17.0000 −0.720313 −0.360157 − 0.932892i \(-0.617277\pi\)
−0.360157 + 0.932892i \(0.617277\pi\)
\(558\) −60.0000 −2.54000
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) −20.0000 −0.843649
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) −27.0000 −1.13389
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −14.0000 −0.585882 −0.292941 − 0.956131i \(-0.594634\pi\)
−0.292941 + 0.956131i \(0.594634\pi\)
\(572\) −16.0000 −0.668994
\(573\) 0 0
\(574\) 54.0000 2.25392
\(575\) 15.0000 0.625543
\(576\) 24.0000 1.00000
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −94.0000 −3.90988
\(579\) 0 0
\(580\) 0 0
\(581\) 21.0000 0.871227
\(582\) 0 0
\(583\) 40.0000 1.65663
\(584\) 0 0
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 32.0000 1.32078 0.660391 − 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 0 0
\(589\) 30.0000 1.23613
\(590\) 0 0
\(591\) 0 0
\(592\) −28.0000 −1.15079
\(593\) −16.0000 −0.657041 −0.328521 − 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 24.0000 0.983078
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) −30.0000 −1.22577 −0.612883 − 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 − 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 6.00000 0.244542
\(603\) 30.0000 1.22169
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 25.0000 1.01472 0.507359 − 0.861735i \(-0.330622\pi\)
0.507359 + 0.861735i \(0.330622\pi\)
\(608\) −24.0000 −0.973329
\(609\) 0 0
\(610\) 0 0
\(611\) 22.0000 0.890025
\(612\) 48.0000 1.94029
\(613\) −9.00000 −0.363507 −0.181753 − 0.983344i \(-0.558177\pi\)
−0.181753 + 0.983344i \(0.558177\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 54.0000 2.16520
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) −56.0000 −2.23287
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 − 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) −56.0000 −2.21706
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 − 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 0 0
\(643\) −32.0000 −1.26196 −0.630978 − 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) 18.0000 0.709299
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −20.0000 −0.784465
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 39.0000 1.52619 0.763094 − 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −36.0000 −1.40556
\(657\) −18.0000 −0.702247
\(658\) −66.0000 −2.57295
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 31.0000 1.20576 0.602880 − 0.797832i \(-0.294020\pi\)
0.602880 + 0.797832i \(0.294020\pi\)
\(662\) 22.0000 0.855054
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 42.0000 1.62747
\(667\) −21.0000 −0.813123
\(668\) 24.0000 0.928588
\(669\) 0 0
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 − 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 28.0000 1.07852
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 80.0000 3.06336
\(683\) 9.00000 0.344375 0.172188 − 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 18.0000 0.688247
\(685\) 0 0
\(686\) −30.0000 −1.14541
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) 5.00000 0.190209 0.0951045 − 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) −4.00000 −0.152057
\(693\) 36.0000 1.36753
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) −72.0000 −2.72719
\(698\) 40.0000 1.51402
\(699\) 0 0
\(700\) 30.0000 1.13389
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −21.0000 −0.792030
\(704\) −32.0000 −1.20605
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 45.0000 1.69240
\(708\) 0 0
\(709\) −12.0000 −0.450669 −0.225335 − 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 28.0000 1.04495
\(719\) −18.0000 −0.671287 −0.335643 − 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 20.0000 0.744323
\(723\) 0 0
\(724\) 4.00000 0.148659
\(725\) −35.0000 −1.29987
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 − 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 − 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 20.0000 0.738213
\(735\) 0 0
\(736\) −24.0000 −0.884652
\(737\) −40.0000 −1.47342
\(738\) 54.0000 1.98777
\(739\) 28.0000 1.03000 0.514998 − 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 60.0000 2.20267
\(743\) 26.0000 0.953847 0.476924 − 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −12.0000 −0.439351
\(747\) 21.0000 0.768350
\(748\) −64.0000 −2.34007
\(749\) 27.0000 0.986559
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 − 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 44.0000 1.60451
\(753\) 0 0
\(754\) 28.0000 1.01970
\(755\) 0 0
\(756\) 0 0
\(757\) 48.0000 1.74459 0.872295 − 0.488980i \(-0.162631\pi\)
0.872295 + 0.488980i \(0.162631\pi\)
\(758\) 56.0000 2.03401
\(759\) 0 0
\(760\) 0 0
\(761\) 40.0000 1.45000 0.724999 − 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) 0 0
\(763\) −39.0000 −1.41189
\(764\) −26.0000 −0.940647
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) 0 0
\(768\) 0 0
\(769\) 23.0000 0.829401 0.414701 − 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −20.0000 −0.719816
\(773\) 17.0000 0.611448 0.305724 − 0.952120i \(-0.401102\pi\)
0.305724 + 0.952120i \(0.401102\pi\)
\(774\) 6.00000 0.215666
\(775\) 50.0000 1.79605
\(776\) 0 0
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) −27.0000 −0.967375
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) −8.00000 −0.285714
\(785\) 0 0
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) −32.0000 −1.13564
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 5.00000 0.177109 0.0885545 − 0.996071i \(-0.471775\pi\)
0.0885545 + 0.996071i \(0.471775\pi\)
\(798\) 0 0
\(799\) 88.0000 3.11322
\(800\) −40.0000 −1.41421
\(801\) 24.0000 0.847998
\(802\) 66.0000 2.33054
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) −40.0000 −1.40894
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −13.0000 −0.456492 −0.228246 − 0.973604i \(-0.573299\pi\)
−0.228246 + 0.973604i \(0.573299\pi\)
\(812\) −42.0000 −1.47391
\(813\) 0 0
\(814\) −56.0000 −1.96280
\(815\) 0 0
\(816\) 0 0
\(817\) −3.00000 −0.104957
\(818\) 50.0000 1.74821
\(819\) −18.0000 −0.628971
\(820\) 0 0
\(821\) −31.0000 −1.08191 −0.540954 − 0.841052i \(-0.681937\pi\)
−0.540954 + 0.841052i \(0.681937\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 − 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 18.0000 0.625543
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 16.0000 0.554700
\(833\) −16.0000 −0.554367
\(834\) 0 0
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) −18.0000 −0.621800
\(839\) −13.0000 −0.448810 −0.224405 − 0.974496i \(-0.572044\pi\)
−0.224405 + 0.974496i \(0.572044\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 20.0000 0.689246
\(843\) 0 0
\(844\) −40.0000 −1.37686
\(845\) 0 0
\(846\) −66.0000 −2.26913
\(847\) −15.0000 −0.515406
\(848\) −40.0000 −1.37361
\(849\) 0 0
\(850\) −80.0000 −2.74398
\(851\) −21.0000 −0.719871
\(852\) 0 0
\(853\) −37.0000 −1.26686 −0.633428 − 0.773802i \(-0.718353\pi\)
−0.633428 + 0.773802i \(0.718353\pi\)
\(854\) 30.0000 1.02658
\(855\) 0 0
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 − 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −66.0000 −2.24797
\(863\) 4.00000 0.136162 0.0680808 − 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −22.0000 −0.747590
\(867\) 0 0
\(868\) 60.0000 2.03653
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) −18.0000 −0.608859
\(875\) 0 0
\(876\) 0 0
\(877\) −8.00000 −0.270141 −0.135070 − 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) −40.0000 −1.34993
\(879\) 0 0
\(880\) 0 0
\(881\) −33.0000 −1.11180 −0.555899 − 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 12.0000 0.404061
\(883\) 46.0000 1.54802 0.774012 − 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) 32.0000 1.07628
\(885\) 0 0
\(886\) 78.0000 2.62046
\(887\) 36.0000 1.20876 0.604381 − 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 36.0000 1.20605
\(892\) −18.0000 −0.602685
\(893\) 33.0000 1.10430
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) −70.0000 −2.33463
\(900\) 30.0000 1.00000
\(901\) −80.0000 −2.66519
\(902\) −72.0000 −2.39734
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 34.0000 1.12895 0.564476 − 0.825450i \(-0.309078\pi\)
0.564476 + 0.825450i \(0.309078\pi\)
\(908\) 16.0000 0.530979
\(909\) 45.0000 1.49256
\(910\) 0 0
\(911\) 34.0000 1.12647 0.563235 − 0.826297i \(-0.309557\pi\)
0.563235 + 0.826297i \(0.309557\pi\)
\(912\) 0 0
\(913\) −28.0000 −0.926665
\(914\) −26.0000 −0.860004
\(915\) 0 0
\(916\) −48.0000 −1.58596
\(917\) −54.0000 −1.78324
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −36.0000 −1.18560
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −35.0000 −1.15079
\(926\) −12.0000 −0.394344
\(927\) 12.0000 0.394132
\(928\) 56.0000 1.83829
\(929\) −40.0000 −1.31236 −0.656179 − 0.754606i \(-0.727828\pi\)
−0.656179 + 0.754606i \(0.727828\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 − 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) −60.0000 −1.95907
\(939\) 0 0
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 − 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 0 0
\(943\) −27.0000 −0.879241
\(944\) 0 0
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 58.0000 1.88475 0.942373 − 0.334563i \(-0.108589\pi\)
0.942373 + 0.334563i \(0.108589\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) −30.0000 −0.973329
\(951\) 0 0
\(952\) 0 0
\(953\) −14.0000 −0.453504 −0.226752 − 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 60.0000 1.94257
\(955\) 0 0
\(956\) −42.0000 −1.35838
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) 63.0000 2.03438
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 28.0000 0.902756
\(963\) 27.0000 0.870063
\(964\) −24.0000 −0.772988
\(965\) 0 0
\(966\) 0 0
\(967\) 44.0000 1.41494 0.707472 − 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 − 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 30.0000 0.961756
\(974\) 10.0000 0.320421
\(975\) 0 0
\(976\) −20.0000 −0.640184
\(977\) −38.0000 −1.21573 −0.607864 − 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 0 0
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) −39.0000 −1.24517
\(982\) −10.0000 −0.319113
\(983\) −24.0000 −0.765481 −0.382741 − 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 112.000 3.56681
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) −15.0000 −0.476491 −0.238245 − 0.971205i \(-0.576572\pi\)
−0.238245 + 0.971205i \(0.576572\pi\)
\(992\) −80.0000 −2.54000
\(993\) 0 0
\(994\) 48.0000 1.52247
\(995\) 0 0
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 − 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 88.0000 2.78559
\(999\) 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 197.2.a.a.1.1 ✓ 1
3.2 odd 2 1773.2.a.b.1.1 1
4.3 odd 2 3152.2.a.a.1.1 1
5.4 even 2 4925.2.a.d.1.1 1
7.6 odd 2 9653.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
1773.2.a.b.1.1 1 3.2 odd 2
3152.2.a.a.1.1 1 4.3 odd 2
4925.2.a.d.1.1 1 5.4 even 2
9653.2.a.a.1.1 1 7.6 odd 2