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Properties

Label 8325.2.a.ba.1.1
Level $8325$
Weight $2$
Character 8325.1
Self dual yes
Analytic conductor $66.475$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8325,2,Mod(1,8325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8325.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.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: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1665)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} -4.00000 q^{7} -1.00000 q^{13} -8.00000 q^{14} -4.00000 q^{16} +4.00000 q^{17} -4.00000 q^{19} +8.00000 q^{23} -2.00000 q^{26} -8.00000 q^{28} -1.00000 q^{29} -6.00000 q^{31} -8.00000 q^{32} +8.00000 q^{34} +1.00000 q^{37} -8.00000 q^{38} +10.0000 q^{41} -5.00000 q^{43} +16.0000 q^{46} +7.00000 q^{47} +9.00000 q^{49} -2.00000 q^{52} +9.00000 q^{53} -2.00000 q^{58} +3.00000 q^{59} -4.00000 q^{61} -12.0000 q^{62} -8.00000 q^{64} +8.00000 q^{67} +8.00000 q^{68} +6.00000 q^{71} +4.00000 q^{73} +2.00000 q^{74} -8.00000 q^{76} +6.00000 q^{79} +20.0000 q^{82} +9.00000 q^{83} -10.0000 q^{86} -7.00000 q^{89} +4.00000 q^{91} +16.0000 q^{92} +14.0000 q^{94} -2.00000 q^{97} +18.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −8.00000 −2.13809
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −8.00000 −1.51186
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 16.0000 2.35907
\(47\) 7.00000 1.02105 0.510527 − 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 3.00000 0.390567 0.195283 − 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −12.0000 −1.52400
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 20.0000 2.20863
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 0 0
\(88\) 0 0
\(89\) −7.00000 −0.741999 −0.370999 − 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 16.0000 1.66812
\(93\) 0 0
\(94\) 14.0000 1.44399
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 18.0000 1.81827
\(99\) 0 0
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 − 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 − 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) −9.00000 −0.870063 −0.435031 − 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 16.0000 1.51186
\(113\) 10.0000 0.940721 0.470360 − 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) −12.0000 −1.07763
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 − 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 23.0000 1.95083 0.975417 − 0.220366i \(-0.0707252\pi\)
0.975417 + 0.220366i \(0.0707252\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 − 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) 0 0
\(161\) −32.0000 −2.52195
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 20.0000 1.56174
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −10.0000 −0.762493
\(173\) 7.00000 0.532200 0.266100 − 0.963945i \(-0.414265\pi\)
0.266100 + 0.963945i \(0.414265\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) −13.0000 −0.971666 −0.485833 − 0.874052i \(-0.661484\pi\)
−0.485833 + 0.874052i \(0.661484\pi\)
\(180\) 0 0
\(181\) −17.0000 −1.26360 −0.631800 − 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 14.0000 1.02105
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 − 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.00000 0.562878
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 0 0
\(206\) 26.0000 1.81151
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 − 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 18.0000 1.23625
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) −32.0000 −2.16731
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 − 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 32.0000 2.13809
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) 8.00000 0.530979 0.265489 − 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −21.0000 −1.38772 −0.693860 − 0.720110i \(-0.744091\pi\)
−0.693860 + 0.720110i \(0.744091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 − 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) −32.0000 −2.07425
\(239\) −15.0000 −0.970269 −0.485135 − 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −22.0000 −1.41421
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.0000 0.820553 0.410276 − 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 24.0000 1.48272
\(263\) 23.0000 1.41824 0.709120 − 0.705087i \(-0.249092\pi\)
0.709120 + 0.705087i \(0.249092\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 32.0000 1.96205
\(267\) 0 0
\(268\) 16.0000 0.977356
\(269\) −12.0000 −0.731653 −0.365826 − 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) 15.0000 0.911185 0.455593 − 0.890188i \(-0.349427\pi\)
0.455593 + 0.890188i \(0.349427\pi\)
\(272\) −16.0000 −0.970143
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 46.0000 2.75890
\(279\) 0 0
\(280\) 0 0
\(281\) 29.0000 1.72999 0.864997 − 0.501776i \(-0.167320\pi\)
0.864997 + 0.501776i \(0.167320\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −40.0000 −2.36113
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 8.00000 0.468165
\(293\) −22.0000 −1.28525 −0.642627 − 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 36.0000 2.08542
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 20.0000 1.15278
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) 16.0000 0.917663
\(305\) 0 0
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 − 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 − 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) −25.0000 −1.41308 −0.706542 − 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(314\) −36.0000 −2.03160
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 17.0000 0.954815 0.477408 − 0.878682i \(-0.341577\pi\)
0.477408 + 0.878682i \(0.341577\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) −64.0000 −3.56658
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) 0 0
\(326\) 40.0000 2.21540
\(327\) 0 0
\(328\) 0 0
\(329\) −28.0000 −1.54369
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 18.0000 0.987878
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −24.0000 −1.30543
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 − 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −26.0000 −1.37414
\(359\) −6.00000 −0.316668 −0.158334 − 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −34.0000 −1.78700
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 − 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) −32.0000 −1.66812
\(369\) 0 0
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) 0 0
\(373\) −30.0000 −1.55334 −0.776671 − 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 − 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 32.0000 1.63726
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) −4.00000 −0.203069
\(389\) 15.0000 0.760530 0.380265 − 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) 0 0
\(393\) 0 0
\(394\) −20.0000 −1.00759
\(395\) 0 0
\(396\) 0 0
\(397\) −12.0000 −0.602263 −0.301131 − 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 − 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 0 0
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 − 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 26.0000 1.28093
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 8.00000 0.392232
\(417\) 0 0
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 − 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 34.0000 1.65509
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 48.0000 2.30407
\(435\) 0 0
\(436\) −32.0000 −1.53252
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 − 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) 27.0000 1.28281 0.641404 − 0.767203i \(-0.278352\pi\)
0.641404 + 0.767203i \(0.278352\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 52.0000 2.46227
\(447\) 0 0
\(448\) 32.0000 1.51186
\(449\) −10.0000 −0.471929 −0.235965 − 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 20.0000 0.940721
\(453\) 0 0
\(454\) 16.0000 0.750917
\(455\) 0 0
\(456\) 0 0
\(457\) −33.0000 −1.54367 −0.771837 − 0.635820i \(-0.780662\pi\)
−0.771837 + 0.635820i \(0.780662\pi\)
\(458\) −42.0000 −1.96253
\(459\) 0 0
\(460\) 0 0
\(461\) 27.0000 1.25752 0.628758 − 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(462\) 0 0
\(463\) −25.0000 −1.16185 −0.580924 − 0.813958i \(-0.697309\pi\)
−0.580924 + 0.813958i \(0.697309\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 28.0000 1.29707
\(467\) 10.0000 0.462745 0.231372 − 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −32.0000 −1.46672
\(477\) 0 0
\(478\) −30.0000 −1.37217
\(479\) 3.00000 0.137073 0.0685367 − 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) 0 0
\(481\) −1.00000 −0.0455961
\(482\) −16.0000 −0.728780
\(483\) 0 0
\(484\) −22.0000 −1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 3.00000 0.135943 0.0679715 − 0.997687i \(-0.478347\pi\)
0.0679715 + 0.997687i \(0.478347\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −34.0000 −1.53440 −0.767199 − 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 24.0000 1.07763
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 26.0000 1.16044
\(503\) 30.0000 1.33763 0.668817 − 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 − 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 24.0000 1.04844
\(525\) 0 0
\(526\) 46.0000 2.00570
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 32.0000 1.38738
\(533\) −10.0000 −0.433148
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −24.0000 −1.03471
\(539\) 0 0
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 30.0000 1.28861
\(543\) 0 0
\(544\) −32.0000 −1.37199
\(545\) 0 0
\(546\) 0 0
\(547\) −27.0000 −1.15444 −0.577218 − 0.816590i \(-0.695862\pi\)
−0.577218 + 0.816590i \(0.695862\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 44.0000 1.86938
\(555\) 0 0
\(556\) 46.0000 1.95083
\(557\) 12.0000 0.508456 0.254228 − 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 58.0000 2.44658
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 56.0000 2.35386
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 − 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −80.0000 −3.33914
\(575\) 0 0
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −2.00000 −0.0831890
\(579\) 0 0
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −44.0000 −1.81762
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −31.0000 −1.27302 −0.636509 − 0.771270i \(-0.719622\pi\)
−0.636509 + 0.771270i \(0.719622\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 36.0000 1.47462
\(597\) 0 0
\(598\) −16.0000 −0.654289
\(599\) −48.0000 −1.96123 −0.980613 − 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 − 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 40.0000 1.63028
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 32.0000 1.29777
\(609\) 0 0
\(610\) 0 0
\(611\) −7.00000 −0.283190
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 − 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 − 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) −12.0000 −0.482321 −0.241160 − 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.00000 0.240578
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) 0 0
\(626\) −50.0000 −1.99840
\(627\) 0 0
\(628\) −36.0000 −1.43656
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 − 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 34.0000 1.35031
\(635\) 0 0
\(636\) 0 0
\(637\) −9.00000 −0.356593
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 − 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 49.0000 1.93237 0.966186 − 0.257847i \(-0.0830131\pi\)
0.966186 + 0.257847i \(0.0830131\pi\)
\(644\) −64.0000 −2.52195
\(645\) 0 0
\(646\) −32.0000 −1.25902
\(647\) 10.0000 0.393141 0.196570 − 0.980490i \(-0.437020\pi\)
0.196570 + 0.980490i \(0.437020\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 40.0000 1.56652
\(653\) 10.0000 0.391330 0.195665 − 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −40.0000 −1.56174
\(657\) 0 0
\(658\) −56.0000 −2.18311
\(659\) 14.0000 0.545363 0.272681 − 0.962104i \(-0.412090\pi\)
0.272681 + 0.962104i \(0.412090\pi\)
\(660\) 0 0
\(661\) −6.00000 −0.233373 −0.116686 − 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) −35.0000 −1.34516 −0.672580 − 0.740025i \(-0.734814\pi\)
−0.672580 + 0.740025i \(0.734814\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.0000 0.994862 0.497431 − 0.867503i \(-0.334277\pi\)
0.497431 + 0.867503i \(0.334277\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −16.0000 −0.610883
\(687\) 0 0
\(688\) 20.0000 0.762493
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) 1.00000 0.0380418 0.0190209 − 0.999819i \(-0.493945\pi\)
0.0190209 + 0.999819i \(0.493945\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −40.0000 −1.51838
\(695\) 0 0
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) 0 0
\(701\) 33.0000 1.24639 0.623196 − 0.782065i \(-0.285834\pi\)
0.623196 + 0.782065i \(0.285834\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 4.00000 0.150542
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 − 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) 0 0
\(716\) −26.0000 −0.971666
\(717\) 0 0
\(718\) −12.0000 −0.447836
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −52.0000 −1.93658
\(722\) −6.00000 −0.223297
\(723\) 0 0
\(724\) −34.0000 −1.26360
\(725\) 0 0
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 − 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −56.0000 −2.06700
\(735\) 0 0
\(736\) −64.0000 −2.35907
\(737\) 0 0
\(738\) 0 0
\(739\) 15.0000 0.551784 0.275892 − 0.961189i \(-0.411027\pi\)
0.275892 + 0.961189i \(0.411027\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −72.0000 −2.64320
\(743\) −12.0000 −0.440237 −0.220119 − 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −60.0000 −2.19676
\(747\) 0 0
\(748\) 0 0
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −28.0000 −1.02105
\(753\) 0 0
\(754\) 2.00000 0.0728357
\(755\) 0 0
\(756\) 0 0
\(757\) 15.0000 0.545184 0.272592 − 0.962130i \(-0.412119\pi\)
0.272592 + 0.962130i \(0.412119\pi\)
\(758\) −58.0000 −2.10665
\(759\) 0 0
\(760\) 0 0
\(761\) −28.0000 −1.01500 −0.507500 − 0.861652i \(-0.669430\pi\)
−0.507500 + 0.861652i \(0.669430\pi\)
\(762\) 0 0
\(763\) 64.0000 2.31696
\(764\) 32.0000 1.15772
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) −3.00000 −0.108324
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 − 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −30.0000 −1.07903 −0.539513 − 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 0 0
\(782\) 64.0000 2.28864
\(783\) 0 0
\(784\) −36.0000 −1.28571
\(785\) 0 0
\(786\) 0 0
\(787\) −2.00000 −0.0712923 −0.0356462 − 0.999364i \(-0.511349\pi\)
−0.0356462 + 0.999364i \(0.511349\pi\)
\(788\) −20.0000 −0.712470
\(789\) 0 0
\(790\) 0 0
\(791\) −40.0000 −1.42224
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −20.0000 −0.708436 −0.354218 − 0.935163i \(-0.615253\pi\)
−0.354218 + 0.935163i \(0.615253\pi\)
\(798\) 0 0
\(799\) 28.0000 0.990569
\(800\) 0 0
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 12.0000 0.422682
\(807\) 0 0
\(808\) 0 0
\(809\) −17.0000 −0.597688 −0.298844 − 0.954302i \(-0.596601\pi\)
−0.298844 + 0.954302i \(0.596601\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 − 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 8.00000 0.280745
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 20.0000 0.699711
\(818\) 52.0000 1.81814
\(819\) 0 0
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 − 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −50.0000 −1.74289 −0.871445 − 0.490493i \(-0.836817\pi\)
−0.871445 + 0.490493i \(0.836817\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) 54.0000 1.87776 0.938882 − 0.344239i \(-0.111863\pi\)
0.938882 + 0.344239i \(0.111863\pi\)
\(828\) 0 0
\(829\) 32.0000 1.11141 0.555703 − 0.831381i \(-0.312449\pi\)
0.555703 + 0.831381i \(0.312449\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 8.00000 0.277350
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) −42.0000 −1.45000 −0.725001 − 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 4.00000 0.137849
\(843\) 0 0
\(844\) 34.0000 1.17033
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0000 1.51186
\(848\) −36.0000 −1.23625
\(849\) 0 0
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 21.0000 0.719026 0.359513 − 0.933140i \(-0.382943\pi\)
0.359513 + 0.933140i \(0.382943\pi\)
\(854\) 32.0000 1.09502
\(855\) 0 0
\(856\) 0 0
\(857\) −32.0000 −1.09310 −0.546550 − 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 − 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −48.0000 −1.63489
\(863\) −51.0000 −1.73606 −0.868030 − 0.496512i \(-0.834614\pi\)
−0.868030 + 0.496512i \(0.834614\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) 0 0
\(868\) 48.0000 1.62923
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 0 0
\(874\) −64.0000 −2.16483
\(875\) 0 0
\(876\) 0 0
\(877\) −4.00000 −0.135070 −0.0675352 − 0.997717i \(-0.521513\pi\)
−0.0675352 + 0.997717i \(0.521513\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 − 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) −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
\(886\) 54.0000 1.81417
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 52.0000 1.74109
\(893\) −28.0000 −0.936984
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −20.0000 −0.667409
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −41.0000 −1.36138 −0.680691 − 0.732570i \(-0.738320\pi\)
−0.680691 + 0.732570i \(0.738320\pi\)
\(908\) 16.0000 0.530979
\(909\) 0 0
\(910\) 0 0
\(911\) −13.0000 −0.430709 −0.215355 − 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −66.0000 −2.18309
\(915\) 0 0
\(916\) −42.0000 −1.38772
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 − 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 54.0000 1.77840
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 0 0
\(926\) −50.0000 −1.64310
\(927\) 0 0
\(928\) 8.00000 0.262613
\(929\) 52.0000 1.70606 0.853032 − 0.521858i \(-0.174761\pi\)
0.853032 + 0.521858i \(0.174761\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 28.0000 0.917170
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) −24.0000 −0.784046 −0.392023 − 0.919955i \(-0.628225\pi\)
−0.392023 + 0.919955i \(0.628225\pi\)
\(938\) −64.0000 −2.08967
\(939\) 0 0
\(940\) 0 0
\(941\) 58.0000 1.89075 0.945373 − 0.325991i \(-0.105698\pi\)
0.945373 + 0.325991i \(0.105698\pi\)
\(942\) 0 0
\(943\) 80.0000 2.60516
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −2.00000 −0.0649913 −0.0324956 − 0.999472i \(-0.510346\pi\)
−0.0324956 + 0.999472i \(0.510346\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −30.0000 −0.970269
\(957\) 0 0
\(958\) 6.00000 0.193851
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) −16.0000 −0.515325
\(965\) 0 0
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 − 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) −92.0000 −2.94938
\(974\) 6.00000 0.192252
\(975\) 0 0
\(976\) 16.0000 0.512148
\(977\) 32.0000 1.02377 0.511885 − 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −68.0000 −2.16997
\(983\) −56.0000 −1.78612 −0.893061 − 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 − 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 48.0000 1.52400
\(993\) 0 0
\(994\) −48.0000 −1.52247
\(995\) 0 0
\(996\) 0 0
\(997\) −18.0000 −0.570066 −0.285033 − 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) −8.00000 −0.253236
\(999\) 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 8325.2.a.ba.1.1 1
3.2 odd 2 8325.2.a.a.1.1 1
5.4 even 2 1665.2.a.a.1.1 ✓ 1
15.14 odd 2 1665.2.a.g.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1665.2.a.a.1.1 ✓ 1 5.4 even 2
1665.2.a.g.1.1 yes 1 15.14 odd 2
8325.2.a.a.1.1 1 3.2 odd 2
8325.2.a.ba.1.1 1 1.1 even 1 trivial