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Properties

Label 663.2.a.b.1.1
Level $663$
Weight $2$
Character 663.1
Self dual yes
Analytic conductor $5.294$
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] = [663,2,Mod(1,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, 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("663.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.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: \(5.29408165401\)
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 \(\chi\) \(=\) 663.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -2.00000 q^{21} +2.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} -5.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -6.00000 q^{29} +6.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} -1.00000 q^{34} -1.00000 q^{36} +4.00000 q^{37} -1.00000 q^{39} -12.0000 q^{41} +2.00000 q^{42} -4.00000 q^{43} +2.00000 q^{44} +8.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} +5.00000 q^{50} +1.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -6.00000 q^{56} +6.00000 q^{58} -2.00000 q^{61} -6.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} -1.00000 q^{68} -8.00000 q^{69} +10.0000 q^{71} +3.00000 q^{72} -4.00000 q^{73} -4.00000 q^{74} -5.00000 q^{75} +4.00000 q^{77} +1.00000 q^{78} +12.0000 q^{79} +1.00000 q^{81} +12.0000 q^{82} -4.00000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -6.00000 q^{87} -6.00000 q^{88} -10.0000 q^{89} +2.00000 q^{91} +8.00000 q^{92} +6.00000 q^{93} -5.00000 q^{96} +16.0000 q^{97} +3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 2.00000 0.426401
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000 0.612372
\(25\) −5.00000 −1.00000
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −5.00000 −0.883883
\(33\) −2.00000 −0.348155
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 − 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 2.00000 0.308607
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 5.00000 0.707107
\(51\) 1.00000 0.140028
\(52\) 1.00000 0.138675
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −6.00000 −0.762001
\(63\) −2.00000 −0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −1.00000 −0.121268
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 3.00000 0.353553
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −4.00000 −0.464991
\(75\) −5.00000 −0.577350
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 1.00000 0.113228
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) −6.00000 −0.639602
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 8.00000 0.834058
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 16.0000 1.62455 0.812277 − 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 3.00000 0.303046
\(99\) −2.00000 −0.201008
\(100\) 5.00000 0.500000
\(101\) −18.0000 −1.79107 −0.895533 − 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 2.00000 0.188982
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) −12.0000 −1.08200
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 3.00000 0.265165
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 8.00000 0.681005
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) 2.00000 0.167248
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) −3.00000 −0.247436
\(148\) −4.00000 −0.328798
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 5.00000 0.408248
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −12.0000 −0.954669
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) −1.00000 −0.0785674
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) −6.00000 −0.462910
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 2.00000 0.152057 0.0760286 − 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 6.00000 0.454859
\(175\) 10.0000 0.755929
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −2.00000 −0.148250
\(183\) −2.00000 −0.147844
\(184\) −24.0000 −1.76930
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 7.00000 0.505181
\(193\) −20.0000 −1.43963 −0.719816 − 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 24.0000 1.70993 0.854965 − 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 2.00000 0.142134
\(199\) −12.0000 −0.850657 −0.425329 − 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) −15.0000 −1.06066
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) 12.0000 0.842235
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) −8.00000 −0.556038
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 6.00000 0.412082
\(213\) 10.0000 0.685189
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −1.00000 −0.0672673
\(222\) −4.00000 −0.268462
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 10.0000 0.668153
\(225\) −5.00000 −0.333333
\(226\) −6.00000 −0.399114
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) −18.0000 −1.18176
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 2.00000 0.129641
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) 18.0000 1.14300
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 − 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 2.00000 0.125988
\(253\) 16.0000 1.00591
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −2.00000 −0.124757 −0.0623783 − 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 4.00000 0.249029
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 − 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 2.00000 0.121046
\(274\) 2.00000 0.120824
\(275\) 10.0000 0.603023
\(276\) 8.00000 0.481543
\(277\) −30.0000 −1.80253 −0.901263 − 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 24.0000 1.41668
\(288\) −5.00000 −0.294628
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 4.00000 0.234082
\(293\) 14.0000 0.817889 0.408944 − 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) −2.00000 −0.116052
\(298\) −14.0000 −0.810998
\(299\) 8.00000 0.462652
\(300\) 5.00000 0.288675
\(301\) 8.00000 0.461112
\(302\) −16.0000 −0.920697
\(303\) −18.0000 −1.03407
\(304\) 0 0
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) −4.00000 −0.227921
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 − 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) −3.00000 −0.169842
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 6.00000 0.336463
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) −16.0000 −0.891645
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 5.00000 0.277350
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) −36.0000 −1.98777
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 4.00000 0.219529
\(333\) 4.00000 0.219199
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −24.0000 −1.28839 −0.644194 − 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 6.00000 0.321634
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) −10.0000 −0.534522
\(351\) −1.00000 −0.0533761
\(352\) 10.0000 0.533002
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −2.00000 −0.105851
\(358\) −12.0000 −0.634220
\(359\) 20.0000 1.05556 0.527780 − 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −14.0000 −0.735824
\(363\) −7.00000 −0.367405
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −24.0000 −1.25279 −0.626395 − 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 8.00000 0.417029
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −6.00000 −0.311086
\(373\) −34.0000 −1.76045 −0.880227 − 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 2.00000 0.102869
\(379\) −10.0000 −0.513665 −0.256833 − 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) −28.0000 −1.43073 −0.715367 − 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) −4.00000 −0.203331
\(388\) −16.0000 −0.812277
\(389\) 2.00000 0.101404 0.0507020 − 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −9.00000 −0.454569
\(393\) 12.0000 0.605320
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −12.0000 −0.602263 −0.301131 − 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) 5.00000 0.250000
\(401\) 24.0000 1.19850 0.599251 − 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) −8.00000 −0.396545
\(408\) 3.00000 0.148522
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 − 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) −5.00000 −0.242536
\(426\) −10.0000 −0.484502
\(427\) 4.00000 0.193574
\(428\) 8.00000 0.386695
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 − 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 40.0000 1.90910 0.954548 − 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 1.00000 0.0475651
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) 14.0000 0.662177
\(448\) −14.0000 −0.661438
\(449\) 8.00000 0.377543 0.188772 − 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 5.00000 0.235702
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) 16.0000 0.751746
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 0 0
\(457\) −30.0000 −1.40334 −0.701670 − 0.712502i \(-0.747562\pi\)
−0.701670 + 0.712502i \(0.747562\pi\)
\(458\) 10.0000 0.467269
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 − 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) −4.00000 −0.186097
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 4.00000 0.185098 0.0925490 − 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) −12.0000 −0.551178
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) −6.00000 −0.274721
\(478\) 12.0000 0.548867
\(479\) −6.00000 −0.274147 −0.137073 − 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 26.0000 1.17817 0.589086 − 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) −6.00000 −0.271607
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 12.0000 0.541002
\(493\) −6.00000 −0.270226
\(494\) 0 0
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) −20.0000 −0.897123
\(498\) 4.00000 0.179244
\(499\) 34.0000 1.52205 0.761025 − 0.648723i \(-0.224697\pi\)
0.761025 + 0.648723i \(0.224697\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 28.0000 1.24970
\(503\) 8.00000 0.356702 0.178351 − 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 1.00000 0.0444116
\(508\) −16.0000 −0.709885
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 6.00000 0.262613
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 10.0000 0.436436
\(526\) 16.0000 0.697633
\(527\) 6.00000 0.261364
\(528\) 2.00000 0.0870388
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 6.00000 0.258678
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 − 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) −28.0000 −1.20270
\(543\) 14.0000 0.600798
\(544\) −5.00000 −0.214373
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) 36.0000 1.53925 0.769624 − 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 2.00000 0.0854358
\(549\) −2.00000 −0.0853579
\(550\) −10.0000 −0.426401
\(551\) 0 0
\(552\) −24.0000 −1.02151
\(553\) −24.0000 −1.02058
\(554\) 30.0000 1.27458
\(555\) 0 0
\(556\) 0 0
\(557\) −22.0000 −0.932170 −0.466085 − 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) −6.00000 −0.254000
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) −6.00000 −0.253095
\(563\) 4.00000 0.168580 0.0842900 − 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) −2.00000 −0.0839921
\(568\) 30.0000 1.25877
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 − 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 40.0000 1.66812
\(576\) 7.00000 0.291667
\(577\) −10.0000 −0.416305 −0.208153 − 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −20.0000 −0.831172
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) −16.0000 −0.663221
\(583\) 12.0000 0.496989
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −8.00000 −0.330195 −0.165098 − 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) −4.00000 −0.164399
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) −12.0000 −0.491127
\(598\) −8.00000 −0.327144
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −15.0000 −0.612372
\(601\) 30.0000 1.22373 0.611863 − 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) −28.0000 −1.13648 −0.568242 − 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) −18.0000 −0.727013 −0.363507 − 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 16.0000 0.643614
\(619\) −26.0000 −1.04503 −0.522514 − 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 4.00000 0.160385
\(623\) 20.0000 0.801283
\(624\) 1.00000 0.0400320
\(625\) 25.0000 1.00000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 36.0000 1.43200
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 3.00000 0.118864
\(638\) −12.0000 −0.475085
\(639\) 10.0000 0.395594
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 − 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 8.00000 0.315735
\(643\) −2.00000 −0.0788723 −0.0394362 − 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) −5.00000 −0.196116
\(651\) −12.0000 −0.470317
\(652\) −2.00000 −0.0783260
\(653\) −14.0000 −0.547862 −0.273931 − 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 − 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −28.0000 −1.08825
\(663\) −1.00000 −0.0388368
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 48.0000 1.85857
\(668\) −18.0000 −0.696441
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 10.0000 0.385758
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 14.0000 0.539260
\(675\) −5.00000 −0.192450
\(676\) −1.00000 −0.0384615
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −6.00000 −0.230429
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 12.0000 0.459504
\(683\) 30.0000 1.14792 0.573959 − 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) −10.0000 −0.381524
\(688\) 4.00000 0.152499
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 − 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 4.00000 0.151947
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) −18.0000 −0.682288
\(697\) −12.0000 −0.454532
\(698\) 14.0000 0.529908
\(699\) −18.0000 −0.680823
\(700\) −10.0000 −0.377964
\(701\) −2.00000 −0.0755390 −0.0377695 − 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 1.00000 0.0377426
\(703\) 0 0
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 36.0000 1.35392
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 − 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) −30.0000 −1.12430
\(713\) −48.0000 −1.79761
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −12.0000 −0.448148
\(718\) −20.0000 −0.746393
\(719\) 48.0000 1.79010 0.895049 − 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −14.0000 −0.520306
\(725\) 30.0000 1.11417
\(726\) 7.00000 0.259794
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 2.00000 0.0739221
\(733\) −22.0000 −0.812589 −0.406294 − 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) 0 0
\(738\) 12.0000 0.441726
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 38.0000 1.39408 0.697042 − 0.717030i \(-0.254499\pi\)
0.697042 + 0.717030i \(0.254499\pi\)
\(744\) 18.0000 0.659912
\(745\) 0 0
\(746\) 34.0000 1.24483
\(747\) −4.00000 −0.146352
\(748\) 2.00000 0.0731272
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 − 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) −28.0000 −1.02038
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −42.0000 −1.52652 −0.763258 − 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 10.0000 0.363216
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) 0 0
\(768\) −17.0000 −0.613435
\(769\) −26.0000 −0.937584 −0.468792 − 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 20.0000 0.719816
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 4.00000 0.143777
\(775\) −30.0000 −1.07763
\(776\) 48.0000 1.72310
\(777\) −8.00000 −0.286998
\(778\) −2.00000 −0.0717035
\(779\) 0 0
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 8.00000 0.286079
\(783\) −6.00000 −0.214423
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 18.0000 0.641631 0.320815 − 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) −24.0000 −0.854965
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) −6.00000 −0.213201
\(793\) 2.00000 0.0710221
\(794\) 12.0000 0.425864
\(795\) 0 0
\(796\) 12.0000 0.425329
\(797\) 46.0000 1.62940 0.814702 − 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 25.0000 0.883883
\(801\) −10.0000 −0.353333
\(802\) −24.0000 −0.847469
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) −6.00000 −0.211210
\(808\) −54.0000 −1.89971
\(809\) −50.0000 −1.75791 −0.878953 − 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 0 0
\(811\) 50.0000 1.75574 0.877869 − 0.478901i \(-0.158965\pi\)
0.877869 + 0.478901i \(0.158965\pi\)
\(812\) −12.0000 −0.421117
\(813\) 28.0000 0.982003
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 0 0
\(818\) 22.0000 0.769212
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 − 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 2.00000 0.0697580
\(823\) −44.0000 −1.53374 −0.766872 − 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) −48.0000 −1.67216
\(825\) 10.0000 0.348155
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 − 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 8.00000 0.278019
\(829\) 34.0000 1.18087 0.590434 − 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −30.0000 −1.04069
\(832\) −7.00000 −0.242681
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 0 0
\(839\) 18.0000 0.621429 0.310715 − 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −18.0000 −0.620321
\(843\) 6.00000 0.206651
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) 5.00000 0.171499
\(851\) −32.0000 −1.09695
\(852\) −10.0000 −0.342594
\(853\) −20.0000 −0.684787 −0.342393 − 0.939557i \(-0.611238\pi\)
−0.342393 + 0.939557i \(0.611238\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −24.0000 −0.820303
\(857\) −26.0000 −0.888143 −0.444072 − 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 30.0000 1.02180
\(863\) −28.0000 −0.953131 −0.476566 − 0.879139i \(-0.658119\pi\)
−0.476566 + 0.879139i \(0.658119\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 1.00000 0.0339618
\(868\) 12.0000 0.407307
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 16.0000 0.541518
\(874\) 0 0
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) −12.0000 −0.405211 −0.202606 − 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) −40.0000 −1.34993
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 − 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) 3.00000 0.101015
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 1.00000 0.0336336
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 12.0000 0.402694
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) 8.00000 0.267112
\(898\) −8.00000 −0.266963
\(899\) −36.0000 −1.20067
\(900\) 5.00000 0.166667
\(901\) −6.00000 −0.199889
\(902\) −24.0000 −0.799113
\(903\) 8.00000 0.266223
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) 52.0000 1.72663 0.863316 − 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 18.0000 0.597351
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 − 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) −24.0000 −0.792550
\(918\) −1.00000 −0.0330049
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 14.0000 0.461065
\(923\) −10.0000 −0.329154
\(924\) −4.00000 −0.131590
\(925\) −20.0000 −0.657596
\(926\) 8.00000 0.262896
\(927\) −16.0000 −0.525509
\(928\) 30.0000 0.984798
\(929\) −24.0000 −0.787414 −0.393707 − 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −4.00000 −0.130954
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) −18.0000 −0.586472
\(943\) 96.0000 3.12619
\(944\) 0 0
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) −30.0000 −0.974869 −0.487435 − 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) −12.0000 −0.389742
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 12.0000 0.387905
\(958\) 6.00000 0.193851
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 4.00000 0.128965
\(963\) −8.00000 −0.257796
\(964\) 0 0
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) −52.0000 −1.67221 −0.836104 − 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) −21.0000 −0.674966
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −26.0000 −0.833094
\(975\) 5.00000 0.160128
\(976\) 2.00000 0.0640184
\(977\) −34.0000 −1.08776 −0.543878 − 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) −58.0000 −1.84991 −0.924956 − 0.380073i \(-0.875899\pi\)
−0.924956 + 0.380073i \(0.875899\pi\)
\(984\) −36.0000 −1.14764
\(985\) 0 0
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 − 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −30.0000 −0.952501
\(993\) 28.0000 0.888553
\(994\) 20.0000 0.634361
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −2.00000 −0.0633406 −0.0316703 − 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −34.0000 −1.07625
\(999\) 4.00000 0.126554
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 663.2.a.b.1.1 ✓ 1
3.2 odd 2 1989.2.a.d.1.1 1
13.12 even 2 8619.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
663.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
1989.2.a.d.1.1 1 3.2 odd 2
8619.2.a.j.1.1 1 13.12 even 2