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Properties

Label 4830.2.a.bl.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
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] = [4830,2,Mod(1,4830)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4830.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.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: \(38.5677441763\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 4830.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -8.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +4.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +4.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +2.00000 q^{52} -4.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} -8.00000 q^{58} -8.00000 q^{59} +1.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} +16.0000 q^{67} +4.00000 q^{68} +1.00000 q^{69} +1.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +4.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -12.0000 q^{83} +1.00000 q^{84} +4.00000 q^{85} -8.00000 q^{86} -8.00000 q^{87} +4.00000 q^{88} +6.00000 q^{89} +1.00000 q^{90} +2.00000 q^{91} +1.00000 q^{92} +8.00000 q^{93} -4.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 4.00000 0.685994
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 1.00000 0.154303
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) 1.00000 0.147442
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 2.00000 0.277350
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 1.00000 0.133631
\(57\) −4.00000 −0.529813
\(58\) −8.00000 −1.05045
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000 1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 4.00000 0.492366
\(67\) 16.0000 1.95471 0.977356 − 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.00000 0.120386
\(70\) 1.00000 0.119523
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 1.00000 0.109109
\(85\) 4.00000 0.433861
\(86\) −8.00000 −0.862662
\(87\) −8.00000 −0.857690
\(88\) 4.00000 0.426401
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 4.00000 0.396059
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) 1.00000 0.0975900
\(106\) −4.00000 −0.388514
\(107\) 2.00000 0.193347 0.0966736 − 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.0000 1.53252 0.766261 − 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 4.00000 0.381385
\(111\) −2.00000 −0.189832
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −4.00000 −0.374634
\(115\) 1.00000 0.0932505
\(116\) −8.00000 −0.742781
\(117\) 2.00000 0.184900
\(118\) −8.00000 −0.736460
\(119\) 4.00000 0.366679
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 4.00000 0.360668
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 2.00000 0.175412
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 4.00000 0.348155
\(133\) −4.00000 −0.346844
\(134\) 16.0000 1.38219
\(135\) 1.00000 0.0860663
\(136\) 4.00000 0.342997
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 1.00000 0.0851257
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −4.00000 −0.324443
\(153\) 4.00000 0.323381
\(154\) 4.00000 0.322329
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −4.00000 −0.318223
\(159\) −4.00000 −0.317221
\(160\) 1.00000 0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 4.00000 0.312348
\(165\) 4.00000 0.311400
\(166\) −12.0000 −0.931381
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) −4.00000 −0.305888
\(172\) −8.00000 −0.609994
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −8.00000 −0.606478
\(175\) 1.00000 0.0755929
\(176\) 4.00000 0.301511
\(177\) −8.00000 −0.601317
\(178\) 6.00000 0.449719
\(179\) 10.0000 0.747435 0.373718 − 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 1.00000 0.0745356
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 2.00000 0.148250
\(183\) −2.00000 −0.147844
\(184\) 1.00000 0.0737210
\(185\) −2.00000 −0.147043
\(186\) 8.00000 0.586588
\(187\) 16.0000 1.17004
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) −4.00000 −0.290191
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −14.0000 −1.00514
\(195\) 2.00000 0.143223
\(196\) 1.00000 0.0714286
\(197\) 22.0000 1.56744 0.783718 − 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 4.00000 0.284268
\(199\) −2.00000 −0.141776 −0.0708881 − 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 1.00000 0.0707107
\(201\) 16.0000 1.12855
\(202\) 6.00000 0.422159
\(203\) −8.00000 −0.561490
\(204\) 4.00000 0.280056
\(205\) 4.00000 0.279372
\(206\) −4.00000 −0.278693
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) −16.0000 −1.10674
\(210\) 1.00000 0.0690066
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −4.00000 −0.274721
\(213\) −4.00000 −0.274075
\(214\) 2.00000 0.136717
\(215\) −8.00000 −0.545595
\(216\) 1.00000 0.0680414
\(217\) 8.00000 0.543075
\(218\) 16.0000 1.08366
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) 8.00000 0.538138
\(222\) −2.00000 −0.134231
\(223\) −26.0000 −1.74109 −0.870544 − 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) 24.0000 1.59294 0.796468 − 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −4.00000 −0.264906
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 1.00000 0.0659380
\(231\) 4.00000 0.263181
\(232\) −8.00000 −0.525226
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) −4.00000 −0.259828
\(238\) 4.00000 0.259281
\(239\) 28.0000 1.81117 0.905585 − 0.424165i \(-0.139432\pi\)
0.905585 + 0.424165i \(0.139432\pi\)
\(240\) 1.00000 0.0645497
\(241\) −20.0000 −1.28831 −0.644157 − 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) 4.00000 0.255031
\(247\) −8.00000 −0.509028
\(248\) 8.00000 0.508001
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) 4.00000 0.252478 0.126239 − 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 1.00000 0.0629941
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −8.00000 −0.498058
\(259\) −2.00000 −0.124274
\(260\) 2.00000 0.124035
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 4.00000 0.246183
\(265\) −4.00000 −0.245718
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) 16.0000 0.977356
\(269\) 2.00000 0.121942 0.0609711 − 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 4.00000 0.242536
\(273\) 2.00000 0.121046
\(274\) −14.0000 −0.845771
\(275\) 4.00000 0.241209
\(276\) 1.00000 0.0601929
\(277\) −8.00000 −0.480673 −0.240337 − 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −4.00000 −0.239904
\(279\) 8.00000 0.478947
\(280\) 1.00000 0.0597614
\(281\) −26.0000 −1.55103 −0.775515 − 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −4.00000 −0.237356
\(285\) −4.00000 −0.236940
\(286\) 8.00000 0.473050
\(287\) 4.00000 0.236113
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −8.00000 −0.469776
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 22.0000 1.28525 0.642627 − 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 1.00000 0.0583212
\(295\) −8.00000 −0.465778
\(296\) −2.00000 −0.116248
\(297\) 4.00000 0.232104
\(298\) −10.0000 −0.579284
\(299\) 2.00000 0.115663
\(300\) 1.00000 0.0577350
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 4.00000 0.228665
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 4.00000 0.227921
\(309\) −4.00000 −0.227552
\(310\) 8.00000 0.454369
\(311\) −34.0000 −1.92796 −0.963982 − 0.265969i \(-0.914308\pi\)
−0.963982 + 0.265969i \(0.914308\pi\)
\(312\) 2.00000 0.113228
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 2.00000 0.112867
\(315\) 1.00000 0.0563436
\(316\) −4.00000 −0.225018
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −4.00000 −0.224309
\(319\) −32.0000 −1.79166
\(320\) 1.00000 0.0559017
\(321\) 2.00000 0.111629
\(322\) 1.00000 0.0557278
\(323\) −16.0000 −0.890264
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 14.0000 0.775388
\(327\) 16.0000 0.884802
\(328\) 4.00000 0.220863
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −12.0000 −0.658586
\(333\) −2.00000 −0.109599
\(334\) −12.0000 −0.656611
\(335\) 16.0000 0.874173
\(336\) 1.00000 0.0545545
\(337\) 14.0000 0.762629 0.381314 − 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 2.00000 0.108625
\(340\) 4.00000 0.216930
\(341\) 32.0000 1.73290
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) −8.00000 −0.431331
\(345\) 1.00000 0.0538382
\(346\) −2.00000 −0.107521
\(347\) 20.0000 1.07366 0.536828 − 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −8.00000 −0.428845
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) −8.00000 −0.425195
\(355\) −4.00000 −0.212298
\(356\) 6.00000 0.317999
\(357\) 4.00000 0.211702
\(358\) 10.0000 0.528516
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 10.0000 0.525588
\(363\) 5.00000 0.262432
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −32.0000 −1.67039 −0.835193 − 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 1.00000 0.0521286
\(369\) 4.00000 0.208232
\(370\) −2.00000 −0.103975
\(371\) −4.00000 −0.207670
\(372\) 8.00000 0.414781
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 16.0000 0.827340
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −16.0000 −0.824042
\(378\) 1.00000 0.0514344
\(379\) −30.0000 −1.54100 −0.770498 − 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 24.0000 1.22795
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 1.00000 0.0510310
\(385\) 4.00000 0.203859
\(386\) −2.00000 −0.101797
\(387\) −8.00000 −0.406663
\(388\) −14.0000 −0.710742
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 2.00000 0.101274
\(391\) 4.00000 0.202289
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) −4.00000 −0.201262
\(396\) 4.00000 0.201008
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −2.00000 −0.100251
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 16.0000 0.798007
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) −8.00000 −0.397033
\(407\) −8.00000 −0.396545
\(408\) 4.00000 0.198030
\(409\) 34.0000 1.68119 0.840596 − 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 4.00000 0.197546
\(411\) −14.0000 −0.690569
\(412\) −4.00000 −0.197066
\(413\) −8.00000 −0.393654
\(414\) 1.00000 0.0491473
\(415\) −12.0000 −0.589057
\(416\) 2.00000 0.0980581
\(417\) −4.00000 −0.195881
\(418\) −16.0000 −0.782586
\(419\) −20.0000 −0.977064 −0.488532 − 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 1.00000 0.0487950
\(421\) 20.0000 0.974740 0.487370 − 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) 4.00000 0.194029
\(426\) −4.00000 −0.193801
\(427\) −2.00000 −0.0967868
\(428\) 2.00000 0.0966736
\(429\) 8.00000 0.386244
\(430\) −8.00000 −0.385794
\(431\) −40.0000 −1.92673 −0.963366 − 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) 1.00000 0.0481125
\(433\) −38.0000 −1.82616 −0.913082 − 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 8.00000 0.384012
\(435\) −8.00000 −0.383571
\(436\) 16.0000 0.766261
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 4.00000 0.190693
\(441\) 1.00000 0.0476190
\(442\) 8.00000 0.380521
\(443\) −20.0000 −0.950229 −0.475114 − 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 6.00000 0.284427
\(446\) −26.0000 −1.23114
\(447\) −10.0000 −0.472984
\(448\) 1.00000 0.0472456
\(449\) 38.0000 1.79333 0.896665 − 0.442709i \(-0.145982\pi\)
0.896665 + 0.442709i \(0.145982\pi\)
\(450\) 1.00000 0.0471405
\(451\) 16.0000 0.753411
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 2.00000 0.0937614
\(456\) −4.00000 −0.187317
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 2.00000 0.0934539
\(459\) 4.00000 0.186704
\(460\) 1.00000 0.0466252
\(461\) 42.0000 1.95614 0.978068 − 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 4.00000 0.186097
\(463\) −8.00000 −0.371792 −0.185896 − 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −8.00000 −0.371391
\(465\) 8.00000 0.370991
\(466\) 6.00000 0.277945
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 2.00000 0.0924500
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) −8.00000 −0.368230
\(473\) −32.0000 −1.47136
\(474\) −4.00000 −0.183726
\(475\) −4.00000 −0.183533
\(476\) 4.00000 0.183340
\(477\) −4.00000 −0.183147
\(478\) 28.0000 1.28069
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 1.00000 0.0456435
\(481\) −4.00000 −0.182384
\(482\) −20.0000 −0.910975
\(483\) 1.00000 0.0455016
\(484\) 5.00000 0.227273
\(485\) −14.0000 −0.635707
\(486\) 1.00000 0.0453609
\(487\) 12.0000 0.543772 0.271886 − 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 14.0000 0.633102
\(490\) 1.00000 0.0451754
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 4.00000 0.180334
\(493\) −32.0000 −1.44121
\(494\) −8.00000 −0.359937
\(495\) 4.00000 0.179787
\(496\) 8.00000 0.359211
\(497\) −4.00000 −0.179425
\(498\) −12.0000 −0.537733
\(499\) −16.0000 −0.716258 −0.358129 − 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 1.00000 0.0447214
\(501\) −12.0000 −0.536120
\(502\) 4.00000 0.178529
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 1.00000 0.0445435
\(505\) 6.00000 0.266996
\(506\) 4.00000 0.177822
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) −4.00000 −0.176261
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) −2.00000 −0.0877903
\(520\) 2.00000 0.0877058
\(521\) 26.0000 1.13908 0.569540 − 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) −8.00000 −0.350150
\(523\) −12.0000 −0.524723 −0.262362 − 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 24.0000 1.04645
\(527\) 32.0000 1.39394
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) −4.00000 −0.173749
\(531\) −8.00000 −0.347170
\(532\) −4.00000 −0.173422
\(533\) 8.00000 0.346518
\(534\) 6.00000 0.259645
\(535\) 2.00000 0.0864675
\(536\) 16.0000 0.691095
\(537\) 10.0000 0.431532
\(538\) 2.00000 0.0862261
\(539\) 4.00000 0.172292
\(540\) 1.00000 0.0430331
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −16.0000 −0.687259
\(543\) 10.0000 0.429141
\(544\) 4.00000 0.171499
\(545\) 16.0000 0.685365
\(546\) 2.00000 0.0855921
\(547\) 26.0000 1.11168 0.555840 − 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) −14.0000 −0.598050
\(549\) −2.00000 −0.0853579
\(550\) 4.00000 0.170561
\(551\) 32.0000 1.36325
\(552\) 1.00000 0.0425628
\(553\) −4.00000 −0.170097
\(554\) −8.00000 −0.339887
\(555\) −2.00000 −0.0848953
\(556\) −4.00000 −0.169638
\(557\) 8.00000 0.338971 0.169485 − 0.985533i \(-0.445789\pi\)
0.169485 + 0.985533i \(0.445789\pi\)
\(558\) 8.00000 0.338667
\(559\) −16.0000 −0.676728
\(560\) 1.00000 0.0422577
\(561\) 16.0000 0.675521
\(562\) −26.0000 −1.09674
\(563\) 8.00000 0.337160 0.168580 − 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −4.00000 −0.167836
\(569\) −38.0000 −1.59304 −0.796521 − 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) −4.00000 −0.167542
\(571\) −6.00000 −0.251092 −0.125546 − 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(572\) 8.00000 0.334497
\(573\) 24.0000 1.00261
\(574\) 4.00000 0.166957
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −12.0000 −0.499567 −0.249783 − 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −2.00000 −0.0831172
\(580\) −8.00000 −0.332182
\(581\) −12.0000 −0.497844
\(582\) −14.0000 −0.580319
\(583\) −16.0000 −0.662652
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 22.0000 0.908812
\(587\) −20.0000 −0.825488 −0.412744 − 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 1.00000 0.0412393
\(589\) −32.0000 −1.31854
\(590\) −8.00000 −0.329355
\(591\) 22.0000 0.904959
\(592\) −2.00000 −0.0821995
\(593\) −42.0000 −1.72473 −0.862367 − 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 4.00000 0.164122
\(595\) 4.00000 0.163984
\(596\) −10.0000 −0.409616
\(597\) −2.00000 −0.0818546
\(598\) 2.00000 0.0817861
\(599\) 16.0000 0.653742 0.326871 − 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 1.00000 0.0408248
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −8.00000 −0.326056
\(603\) 16.0000 0.651570
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 6.00000 0.243733
\(607\) 2.00000 0.0811775 0.0405887 − 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) −4.00000 −0.162221
\(609\) −8.00000 −0.324176
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) −30.0000 −1.21169 −0.605844 − 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 8.00000 0.322854
\(615\) 4.00000 0.161296
\(616\) 4.00000 0.161165
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −4.00000 −0.160904
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 8.00000 0.321288
\(621\) 1.00000 0.0401286
\(622\) −34.0000 −1.36328
\(623\) 6.00000 0.240385
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 14.0000 0.559553
\(627\) −16.0000 −0.638978
\(628\) 2.00000 0.0798087
\(629\) −8.00000 −0.318981
\(630\) 1.00000 0.0398410
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −4.00000 −0.159111
\(633\) −8.00000 −0.317971
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) −4.00000 −0.158610
\(637\) 2.00000 0.0792429
\(638\) −32.0000 −1.26689
\(639\) −4.00000 −0.158238
\(640\) 1.00000 0.0395285
\(641\) 22.0000 0.868948 0.434474 − 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 2.00000 0.0789337
\(643\) −44.0000 −1.73519 −0.867595 − 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 1.00000 0.0394055
\(645\) −8.00000 −0.315000
\(646\) −16.0000 −0.629512
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 1.00000 0.0392837
\(649\) −32.0000 −1.25611
\(650\) 2.00000 0.0784465
\(651\) 8.00000 0.313545
\(652\) 14.0000 0.548282
\(653\) 30.0000 1.17399 0.586995 − 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 16.0000 0.625650
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) 0 0
\(658\) 0 0
\(659\) 16.0000 0.623272 0.311636 − 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 4.00000 0.155700
\(661\) −42.0000 −1.63361 −0.816805 − 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 4.00000 0.155464
\(663\) 8.00000 0.310694
\(664\) −12.0000 −0.465690
\(665\) −4.00000 −0.155113
\(666\) −2.00000 −0.0774984
\(667\) −8.00000 −0.309761
\(668\) −12.0000 −0.464294
\(669\) −26.0000 −1.00522
\(670\) 16.0000 0.618134
\(671\) −8.00000 −0.308837
\(672\) 1.00000 0.0385758
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 2.00000 0.0768095
\(679\) −14.0000 −0.537271
\(680\) 4.00000 0.153393
\(681\) 24.0000 0.919682
\(682\) 32.0000 1.22534
\(683\) 28.0000 1.07139 0.535695 − 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) −4.00000 −0.152944
\(685\) −14.0000 −0.534913
\(686\) 1.00000 0.0381802
\(687\) 2.00000 0.0763048
\(688\) −8.00000 −0.304997
\(689\) −8.00000 −0.304776
\(690\) 1.00000 0.0380693
\(691\) −12.0000 −0.456502 −0.228251 − 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 4.00000 0.151947
\(694\) 20.0000 0.759190
\(695\) −4.00000 −0.151729
\(696\) −8.00000 −0.303239
\(697\) 16.0000 0.606043
\(698\) −2.00000 −0.0757011
\(699\) 6.00000 0.226941
\(700\) 1.00000 0.0377964
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 2.00000 0.0754851
\(703\) 8.00000 0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) 6.00000 0.225653
\(708\) −8.00000 −0.300658
\(709\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) −4.00000 −0.150117
\(711\) −4.00000 −0.150012
\(712\) 6.00000 0.224860
\(713\) 8.00000 0.299602
\(714\) 4.00000 0.149696
\(715\) 8.00000 0.299183
\(716\) 10.0000 0.373718
\(717\) 28.0000 1.04568
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 − 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 1.00000 0.0372678
\(721\) −4.00000 −0.148968
\(722\) −3.00000 −0.111648
\(723\) −20.0000 −0.743808
\(724\) 10.0000 0.371647
\(725\) −8.00000 −0.297113
\(726\) 5.00000 0.185567
\(727\) −12.0000 −0.445055 −0.222528 − 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) −2.00000 −0.0739221
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −32.0000 −1.18114
\(735\) 1.00000 0.0368856
\(736\) 1.00000 0.0368605
\(737\) 64.0000 2.35747
\(738\) 4.00000 0.147242
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −8.00000 −0.293887
\(742\) −4.00000 −0.146845
\(743\) 40.0000 1.46746 0.733729 − 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 8.00000 0.293294
\(745\) −10.0000 −0.366372
\(746\) −14.0000 −0.512576
\(747\) −12.0000 −0.439057
\(748\) 16.0000 0.585018
\(749\) 2.00000 0.0730784
\(750\) 1.00000 0.0365148
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) −16.0000 −0.582686
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −30.0000 −1.08965
\(759\) 4.00000 0.145191
\(760\) −4.00000 −0.145095
\(761\) 28.0000 1.01500 0.507500 − 0.861652i \(-0.330570\pi\)
0.507500 + 0.861652i \(0.330570\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 24.0000 0.868290
\(765\) 4.00000 0.144620
\(766\) 6.00000 0.216789
\(767\) −16.0000 −0.577727
\(768\) 1.00000 0.0360844
\(769\) −32.0000 −1.15395 −0.576975 − 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 4.00000 0.144150
\(771\) −18.0000 −0.648254
\(772\) −2.00000 −0.0719816
\(773\) −10.0000 −0.359675 −0.179838 − 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) −8.00000 −0.287554
\(775\) 8.00000 0.287368
\(776\) −14.0000 −0.502571
\(777\) −2.00000 −0.0717496
\(778\) −30.0000 −1.07555
\(779\) −16.0000 −0.573259
\(780\) 2.00000 0.0716115
\(781\) −16.0000 −0.572525
\(782\) 4.00000 0.143040
\(783\) −8.00000 −0.285897
\(784\) 1.00000 0.0357143
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −44.0000 −1.56843 −0.784215 − 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) 22.0000 0.783718
\(789\) 24.0000 0.854423
\(790\) −4.00000 −0.142314
\(791\) 2.00000 0.0711118
\(792\) 4.00000 0.142134
\(793\) −4.00000 −0.142044
\(794\) 2.00000 0.0709773
\(795\) −4.00000 −0.141865
\(796\) −2.00000 −0.0708881
\(797\) 34.0000 1.20434 0.602171 − 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) 16.0000 0.564276
\(805\) 1.00000 0.0352454
\(806\) 16.0000 0.563576
\(807\) 2.00000 0.0704033
\(808\) 6.00000 0.211079
\(809\) −26.0000 −0.914111 −0.457056 − 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 1.00000 0.0351364
\(811\) −4.00000 −0.140459 −0.0702295 − 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) −8.00000 −0.280745
\(813\) −16.0000 −0.561144
\(814\) −8.00000 −0.280400
\(815\) 14.0000 0.490399
\(816\) 4.00000 0.140028
\(817\) 32.0000 1.11954
\(818\) 34.0000 1.18878
\(819\) 2.00000 0.0698857
\(820\) 4.00000 0.139686
\(821\) −8.00000 −0.279202 −0.139601 − 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) −14.0000 −0.488306
\(823\) 24.0000 0.836587 0.418294 − 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −4.00000 −0.139347
\(825\) 4.00000 0.139262
\(826\) −8.00000 −0.278356
\(827\) 30.0000 1.04320 0.521601 − 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 1.00000 0.0347524
\(829\) −6.00000 −0.208389 −0.104194 − 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) −12.0000 −0.416526
\(831\) −8.00000 −0.277517
\(832\) 2.00000 0.0693375
\(833\) 4.00000 0.138592
\(834\) −4.00000 −0.138509
\(835\) −12.0000 −0.415277
\(836\) −16.0000 −0.553372
\(837\) 8.00000 0.276520
\(838\) −20.0000 −0.690889
\(839\) −44.0000 −1.51905 −0.759524 − 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 1.00000 0.0345033
\(841\) 35.0000 1.20690
\(842\) 20.0000 0.689246
\(843\) −26.0000 −0.895488
\(844\) −8.00000 −0.275371
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) −4.00000 −0.137361
\(849\) 4.00000 0.137280
\(850\) 4.00000 0.137199
\(851\) −2.00000 −0.0685591
\(852\) −4.00000 −0.137038
\(853\) −30.0000 −1.02718 −0.513590 − 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −4.00000 −0.136797
\(856\) 2.00000 0.0683586
\(857\) −14.0000 −0.478231 −0.239115 − 0.970991i \(-0.576857\pi\)
−0.239115 + 0.970991i \(0.576857\pi\)
\(858\) 8.00000 0.273115
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −8.00000 −0.272798
\(861\) 4.00000 0.136320
\(862\) −40.0000 −1.36241
\(863\) −8.00000 −0.272323 −0.136162 − 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.00000 −0.0680020
\(866\) −38.0000 −1.29129
\(867\) −1.00000 −0.0339618
\(868\) 8.00000 0.271538
\(869\) −16.0000 −0.542763
\(870\) −8.00000 −0.271225
\(871\) 32.0000 1.08428
\(872\) 16.0000 0.541828
\(873\) −14.0000 −0.473828
\(874\) −4.00000 −0.135302
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −52.0000 −1.75592 −0.877958 − 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) 0 0
\(879\) 22.0000 0.742042
\(880\) 4.00000 0.134840
\(881\) −50.0000 −1.68454 −0.842271 − 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 1.00000 0.0336718
\(883\) −34.0000 −1.14419 −0.572096 − 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 8.00000 0.269069
\(885\) −8.00000 −0.268917
\(886\) −20.0000 −0.671913
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 4.00000 0.134005
\(892\) −26.0000 −0.870544
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 10.0000 0.334263
\(896\) 1.00000 0.0334077
\(897\) 2.00000 0.0667781
\(898\) 38.0000 1.26808
\(899\) −64.0000 −2.13452
\(900\) 1.00000 0.0333333
\(901\) −16.0000 −0.533037
\(902\) 16.0000 0.532742
\(903\) −8.00000 −0.266223
\(904\) 2.00000 0.0665190
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) 16.0000 0.531271 0.265636 − 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 24.0000 0.796468
\(909\) 6.00000 0.199007
\(910\) 2.00000 0.0662994
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −4.00000 −0.132453
\(913\) −48.0000 −1.58857
\(914\) −10.0000 −0.330771
\(915\) −2.00000 −0.0661180
\(916\) 2.00000 0.0660819
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 1.00000 0.0329690
\(921\) 8.00000 0.263609
\(922\) 42.0000 1.38320
\(923\) −8.00000 −0.263323
\(924\) 4.00000 0.131590
\(925\) −2.00000 −0.0657596
\(926\) −8.00000 −0.262896
\(927\) −4.00000 −0.131377
\(928\) −8.00000 −0.262613
\(929\) −24.0000 −0.787414 −0.393707 − 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 8.00000 0.262330
\(931\) −4.00000 −0.131095
\(932\) 6.00000 0.196537
\(933\) −34.0000 −1.11311
\(934\) 0 0
\(935\) 16.0000 0.523256
\(936\) 2.00000 0.0653720
\(937\) −34.0000 −1.11073 −0.555366 − 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 16.0000 0.522419
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 2.00000 0.0651635
\(943\) 4.00000 0.130258
\(944\) −8.00000 −0.260378
\(945\) 1.00000 0.0325300
\(946\) −32.0000 −1.04041
\(947\) 48.0000 1.55979 0.779895 − 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) −4.00000 −0.129914
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) 2.00000 0.0648544
\(952\) 4.00000 0.129641
\(953\) 30.0000 0.971795 0.485898 − 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) −4.00000 −0.129505
\(955\) 24.0000 0.776622
\(956\) 28.0000 0.905585
\(957\) −32.0000 −1.03441
\(958\) 0 0
\(959\) −14.0000 −0.452084
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) −4.00000 −0.128965
\(963\) 2.00000 0.0644491
\(964\) −20.0000 −0.644157
\(965\) −2.00000 −0.0643823
\(966\) 1.00000 0.0321745
\(967\) 28.0000 0.900419 0.450210 − 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) 5.00000 0.160706
\(969\) −16.0000 −0.513994
\(970\) −14.0000 −0.449513
\(971\) 36.0000 1.15529 0.577647 − 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) 12.0000 0.384505
\(975\) 2.00000 0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −14.0000 −0.447900 −0.223950 − 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) 14.0000 0.447671
\(979\) 24.0000 0.767043
\(980\) 1.00000 0.0319438
\(981\) 16.0000 0.510841
\(982\) 30.0000 0.957338
\(983\) −2.00000 −0.0637901 −0.0318950 − 0.999491i \(-0.510154\pi\)
−0.0318950 + 0.999491i \(0.510154\pi\)
\(984\) 4.00000 0.127515
\(985\) 22.0000 0.700978
\(986\) −32.0000 −1.01909
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −8.00000 −0.254385
\(990\) 4.00000 0.127128
\(991\) 48.0000 1.52477 0.762385 − 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) 8.00000 0.254000
\(993\) 4.00000 0.126936
\(994\) −4.00000 −0.126872
\(995\) −2.00000 −0.0634043
\(996\) −12.0000 −0.380235
\(997\) 22.0000 0.696747 0.348373 − 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) −16.0000 −0.506471
\(999\) −2.00000 −0.0632772
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 4830.2.a.bl.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bl.1.1 ✓ 1 1.1 even 1 trivial