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Properties

Label 1914.2.a.l.1.1
Level $1914$
Weight $2$
Character 1914.1
Self dual yes
Analytic conductor $15.283$
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] = [1914,2,Mod(1,1914)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1914, 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("1914.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1914 = 2 \cdot 3 \cdot 11 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1914.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: \(15.2833669469\)
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 \(\chi\) \(=\) 1914.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +4.00000 q^{20} -1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{29} -4.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} +2.00000 q^{38} -2.00000 q^{39} +4.00000 q^{40} +10.0000 q^{41} -6.00000 q^{43} -1.00000 q^{44} +4.00000 q^{45} -6.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +11.0000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} -2.00000 q^{57} -1.00000 q^{58} +4.00000 q^{59} -4.00000 q^{60} -2.00000 q^{61} +10.0000 q^{62} +1.00000 q^{64} +8.00000 q^{65} +1.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +6.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -8.00000 q^{74} -11.0000 q^{75} +2.00000 q^{76} -2.00000 q^{78} +4.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -6.00000 q^{83} +8.00000 q^{85} -6.00000 q^{86} +1.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +4.00000 q^{90} -6.00000 q^{92} -10.0000 q^{93} +8.00000 q^{95} -1.00000 q^{96} +18.0000 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 − 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.0000 2.20000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) −4.00000 −0.730297
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 2.00000 0.324443
\(39\) −2.00000 −0.320256
\(40\) 4.00000 0.632456
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −1.00000 −0.150756
\(45\) 4.00000 0.596285
\(46\) −6.00000 −0.884652
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 11.0000 1.55563
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) −1.00000 −0.131306
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −4.00000 −0.516398
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) 1.00000 0.123091
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −8.00000 −0.929981
\(75\) −11.0000 −1.27017
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) −6.00000 −0.646997
\(87\) 1.00000 0.107211
\(88\) −1.00000 −0.106600
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 4.00000 0.421637
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) 18.0000 1.82762 0.913812 − 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) −7.00000 −0.707107
\(99\) −1.00000 −0.100504
\(100\) 11.0000 1.10000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.00000 −0.574696 −0.287348 − 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −4.00000 −0.381385
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) −2.00000 −0.187317
\(115\) −24.0000 −2.23801
\(116\) −1.00000 −0.0928477
\(117\) 2.00000 0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −4.00000 −0.365148
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) −10.0000 −0.901670
\(124\) 10.0000 0.898027
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.00000 0.528271
\(130\) 8.00000 0.701646
\(131\) −8.00000 −0.698963 −0.349482 − 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) −4.00000 −0.344265
\(136\) 2.00000 0.171499
\(137\) −22.0000 −1.87959 −0.939793 − 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 6.00000 0.510754
\(139\) 8.00000 0.678551 0.339276 − 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) −4.00000 −0.331042
\(147\) 7.00000 0.577350
\(148\) −8.00000 −0.657596
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −11.0000 −0.898146
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 2.00000 0.162221
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 40.0000 3.21288
\(156\) −2.00000 −0.160128
\(157\) 12.0000 0.957704 0.478852 − 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 4.00000 0.318223
\(159\) −4.00000 −0.317221
\(160\) 4.00000 0.316228
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 10.0000 0.780869
\(165\) 4.00000 0.311400
\(166\) −6.00000 −0.465690
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 8.00000 0.613572
\(171\) 2.00000 0.152944
\(172\) −6.00000 −0.457496
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −4.00000 −0.300658
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 4.00000 0.298142
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −6.00000 −0.442326
\(185\) −32.0000 −2.35269
\(186\) −10.0000 −0.733236
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −12.0000 −0.863779 −0.431889 − 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 18.0000 1.29232
\(195\) −8.00000 −0.572892
\(196\) −7.00000 −0.500000
\(197\) −22.0000 −1.56744 −0.783718 − 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 11.0000 0.777817
\(201\) 12.0000 0.846415
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 40.0000 2.79372
\(206\) 8.00000 0.557386
\(207\) −6.00000 −0.417029
\(208\) 2.00000 0.138675
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 18.0000 1.23917 0.619586 − 0.784929i \(-0.287301\pi\)
0.619586 + 0.784929i \(0.287301\pi\)
\(212\) 4.00000 0.274721
\(213\) 6.00000 0.411113
\(214\) −6.00000 −0.410152
\(215\) −24.0000 −1.63679
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 4.00000 0.270295
\(220\) −4.00000 −0.269680
\(221\) 4.00000 0.269069
\(222\) 8.00000 0.536925
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) −2.00000 −0.133038
\(227\) −6.00000 −0.398234 −0.199117 − 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) −2.00000 −0.132453
\(229\) −20.0000 −1.32164 −0.660819 − 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) −24.0000 −1.58251
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 4.00000 0.262049 0.131024 − 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) −4.00000 −0.258199
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −28.0000 −1.78885
\(246\) −10.0000 −0.637577
\(247\) 4.00000 0.254514
\(248\) 10.0000 0.635001
\(249\) 6.00000 0.380235
\(250\) 24.0000 1.51789
\(251\) −4.00000 −0.252478 −0.126239 − 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 16.0000 1.00393
\(255\) −8.00000 −0.500979
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 6.00000 0.373544
\(259\) 0 0
\(260\) 8.00000 0.496139
\(261\) −1.00000 −0.0618984
\(262\) −8.00000 −0.494242
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 1.00000 0.0615457
\(265\) 16.0000 0.982872
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) −12.0000 −0.733017
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −4.00000 −0.243432
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −22.0000 −1.32907
\(275\) −11.0000 −0.663325
\(276\) 6.00000 0.361158
\(277\) 6.00000 0.360505 0.180253 − 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 8.00000 0.479808
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 − 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −6.00000 −0.356034
\(285\) −8.00000 −0.473879
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) −18.0000 −1.05518
\(292\) −4.00000 −0.234082
\(293\) −22.0000 −1.28525 −0.642627 − 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 7.00000 0.408248
\(295\) 16.0000 0.931556
\(296\) −8.00000 −0.464991
\(297\) 1.00000 0.0580259
\(298\) 18.0000 1.04271
\(299\) −12.0000 −0.693978
\(300\) −11.0000 −0.635085
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) −10.0000 −0.574485
\(304\) 2.00000 0.114708
\(305\) −8.00000 −0.458079
\(306\) 2.00000 0.114332
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 40.0000 2.27185
\(311\) −32.0000 −1.81455 −0.907277 − 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) −2.00000 −0.113228
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 10.0000 0.561656 0.280828 − 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) −4.00000 −0.224309
\(319\) 1.00000 0.0559893
\(320\) 4.00000 0.223607
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 22.0000 1.22034
\(326\) 16.0000 0.886158
\(327\) 6.00000 0.331801
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −6.00000 −0.329293
\(333\) −8.00000 −0.438397
\(334\) −24.0000 −1.31322
\(335\) −48.0000 −2.62252
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 − 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) −9.00000 −0.489535
\(339\) 2.00000 0.108625
\(340\) 8.00000 0.433861
\(341\) −10.0000 −0.541530
\(342\) 2.00000 0.108148
\(343\) 0 0
\(344\) −6.00000 −0.323498
\(345\) 24.0000 1.29212
\(346\) 14.0000 0.752645
\(347\) −18.0000 −0.966291 −0.483145 − 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 1.00000 0.0536056
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −1.00000 −0.0533002
\(353\) 10.0000 0.532246 0.266123 − 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) −4.00000 −0.212598
\(355\) −24.0000 −1.27379
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 8.00000 0.422224 0.211112 − 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 4.00000 0.210819
\(361\) −15.0000 −0.789474
\(362\) −6.00000 −0.315353
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) 2.00000 0.104542
\(367\) −22.0000 −1.14839 −0.574195 − 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) −6.00000 −0.312772
\(369\) 10.0000 0.520579
\(370\) −32.0000 −1.66360
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −2.00000 −0.103418
\(375\) −24.0000 −1.23935
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 − 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 8.00000 0.410391
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −12.0000 −0.610784
\(387\) −6.00000 −0.304997
\(388\) 18.0000 0.913812
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) −8.00000 −0.405096
\(391\) −12.0000 −0.606866
\(392\) −7.00000 −0.353553
\(393\) 8.00000 0.403547
\(394\) −22.0000 −1.10834
\(395\) 16.0000 0.805047
\(396\) −1.00000 −0.0502519
\(397\) −26.0000 −1.30490 −0.652451 − 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −14.0000 −0.699127 −0.349563 − 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 12.0000 0.598506
\(403\) 20.0000 0.996271
\(404\) 10.0000 0.497519
\(405\) 4.00000 0.198762
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) −2.00000 −0.0990148
\(409\) 32.0000 1.58230 0.791149 − 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 40.0000 1.97546
\(411\) 22.0000 1.08518
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −24.0000 −1.17811
\(416\) 2.00000 0.0980581
\(417\) −8.00000 −0.391762
\(418\) −2.00000 −0.0978232
\(419\) −20.0000 −0.977064 −0.488532 − 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −40.0000 −1.94948 −0.974740 − 0.223341i \(-0.928304\pi\)
−0.974740 + 0.223341i \(0.928304\pi\)
\(422\) 18.0000 0.876226
\(423\) 0 0
\(424\) 4.00000 0.194257
\(425\) 22.0000 1.06716
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) 2.00000 0.0965609
\(430\) −24.0000 −1.15738
\(431\) −40.0000 −1.92673 −0.963366 − 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) −6.00000 −0.287348
\(437\) −12.0000 −0.574038
\(438\) 4.00000 0.191127
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) 4.00000 0.190261
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 8.00000 0.379663
\(445\) 24.0000 1.13771
\(446\) −8.00000 −0.378811
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 11.0000 0.518545
\(451\) −10.0000 −0.470882
\(452\) −2.00000 −0.0940721
\(453\) 16.0000 0.751746
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −26.0000 −1.21623 −0.608114 − 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −20.0000 −0.934539
\(459\) −2.00000 −0.0933520
\(460\) −24.0000 −1.11901
\(461\) 38.0000 1.76984 0.884918 − 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −40.0000 −1.85496
\(466\) 4.00000 0.185296
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) 4.00000 0.184115
\(473\) 6.00000 0.275880
\(474\) −4.00000 −0.183726
\(475\) 22.0000 1.00943
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) −12.0000 −0.548867
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −4.00000 −0.182574
\(481\) −16.0000 −0.729537
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 72.0000 3.26935
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −16.0000 −0.723545
\(490\) −28.0000 −1.26491
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −10.0000 −0.450835
\(493\) −2.00000 −0.0900755
\(494\) 4.00000 0.179969
\(495\) −4.00000 −0.179787
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 24.0000 1.07331
\(501\) 24.0000 1.07224
\(502\) −4.00000 −0.178529
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 40.0000 1.77998
\(506\) 6.00000 0.266733
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) 4.00000 0.177297 0.0886484 − 0.996063i \(-0.471745\pi\)
0.0886484 + 0.996063i \(0.471745\pi\)
\(510\) −8.00000 −0.354246
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −14.0000 −0.617514
\(515\) 32.0000 1.41009
\(516\) 6.00000 0.264135
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 8.00000 0.350823
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −4.00000 −0.174908 −0.0874539 − 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 20.0000 0.871214
\(528\) 1.00000 0.0435194
\(529\) 13.0000 0.565217
\(530\) 16.0000 0.694996
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) −6.00000 −0.259645
\(535\) −24.0000 −1.03761
\(536\) −12.0000 −0.518321
\(537\) 12.0000 0.517838
\(538\) −18.0000 −0.776035
\(539\) 7.00000 0.301511
\(540\) −4.00000 −0.172133
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) 6.00000 0.257485
\(544\) 2.00000 0.0857493
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 − 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −22.0000 −0.939793
\(549\) −2.00000 −0.0853579
\(550\) −11.0000 −0.469042
\(551\) −2.00000 −0.0852029
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 6.00000 0.254916
\(555\) 32.0000 1.35832
\(556\) 8.00000 0.339276
\(557\) −6.00000 −0.254228 −0.127114 − 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 10.0000 0.423334
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) 12.0000 0.506189
\(563\) 32.0000 1.34864 0.674320 − 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) −8.00000 −0.336563
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 14.0000 0.586911 0.293455 − 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) −8.00000 −0.335083
\(571\) 32.0000 1.33916 0.669579 − 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) −66.0000 −2.75239
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −13.0000 −0.540729
\(579\) 12.0000 0.498703
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) −18.0000 −0.746124
\(583\) −4.00000 −0.165663
\(584\) −4.00000 −0.165521
\(585\) 8.00000 0.330759
\(586\) −22.0000 −0.908812
\(587\) 4.00000 0.165098 0.0825488 − 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 7.00000 0.288675
\(589\) 20.0000 0.824086
\(590\) 16.0000 0.658710
\(591\) 22.0000 0.904959
\(592\) −8.00000 −0.328798
\(593\) −44.0000 −1.80686 −0.903432 − 0.428732i \(-0.858960\pi\)
−0.903432 + 0.428732i \(0.858960\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 4.00000 0.163709
\(598\) −12.0000 −0.490716
\(599\) −4.00000 −0.163436 −0.0817178 − 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) −11.0000 −0.449073
\(601\) 44.0000 1.79480 0.897399 − 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) −16.0000 −0.651031
\(605\) 4.00000 0.162623
\(606\) −10.0000 −0.406222
\(607\) 36.0000 1.46119 0.730597 − 0.682808i \(-0.239242\pi\)
0.730597 + 0.682808i \(0.239242\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −10.0000 −0.403896 −0.201948 − 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) −2.00000 −0.0807134
\(615\) −40.0000 −1.61296
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −8.00000 −0.321807
\(619\) 28.0000 1.12542 0.562708 − 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 40.0000 1.60644
\(621\) 6.00000 0.240772
\(622\) −32.0000 −1.28308
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 41.0000 1.64000
\(626\) 6.00000 0.239808
\(627\) 2.00000 0.0798723
\(628\) 12.0000 0.478852
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 36.0000 1.43314 0.716569 − 0.697517i \(-0.245712\pi\)
0.716569 + 0.697517i \(0.245712\pi\)
\(632\) 4.00000 0.159111
\(633\) −18.0000 −0.715436
\(634\) 10.0000 0.397151
\(635\) 64.0000 2.53976
\(636\) −4.00000 −0.158610
\(637\) −14.0000 −0.554700
\(638\) 1.00000 0.0395904
\(639\) −6.00000 −0.237356
\(640\) 4.00000 0.158114
\(641\) 42.0000 1.65890 0.829450 − 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 6.00000 0.236801
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 4.00000 0.157378
\(647\) −22.0000 −0.864909 −0.432455 − 0.901656i \(-0.642352\pi\)
−0.432455 + 0.901656i \(0.642352\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.00000 −0.157014
\(650\) 22.0000 0.862911
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 6.00000 0.234619
\(655\) −32.0000 −1.25034
\(656\) 10.0000 0.390434
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −44.0000 −1.71400 −0.856998 − 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 4.00000 0.155700
\(661\) 42.0000 1.63361 0.816805 − 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) −8.00000 −0.310929
\(663\) −4.00000 −0.155347
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 6.00000 0.232321
\(668\) −24.0000 −0.928588
\(669\) 8.00000 0.309298
\(670\) −48.0000 −1.85440
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 8.00000 0.308148
\(675\) −11.0000 −0.423390
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 2.00000 0.0768095
\(679\) 0 0
\(680\) 8.00000 0.306786
\(681\) 6.00000 0.229920
\(682\) −10.0000 −0.382920
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 2.00000 0.0764719
\(685\) −88.0000 −3.36231
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) −6.00000 −0.228748
\(689\) 8.00000 0.304776
\(690\) 24.0000 0.913664
\(691\) 52.0000 1.97817 0.989087 − 0.147335i \(-0.0470696\pi\)
0.989087 + 0.147335i \(0.0470696\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) 32.0000 1.21383
\(696\) 1.00000 0.0379049
\(697\) 20.0000 0.757554
\(698\) 22.0000 0.832712
\(699\) −4.00000 −0.151294
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −16.0000 −0.603451
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −38.0000 −1.42712 −0.713560 − 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) −24.0000 −0.900704
\(711\) 4.00000 0.150012
\(712\) 6.00000 0.224860
\(713\) −60.0000 −2.24702
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) 8.00000 0.298557
\(719\) 34.0000 1.26799 0.633993 − 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 4.00000 0.149071
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) −10.0000 −0.371904
\(724\) −6.00000 −0.222988
\(725\) −11.0000 −0.408530
\(726\) −1.00000 −0.0371135
\(727\) −22.0000 −0.815935 −0.407967 − 0.912996i \(-0.633762\pi\)
−0.407967 + 0.912996i \(0.633762\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −16.0000 −0.592187
\(731\) −12.0000 −0.443836
\(732\) 2.00000 0.0739221
\(733\) 14.0000 0.517102 0.258551 − 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −22.0000 −0.812035
\(735\) 28.0000 1.03280
\(736\) −6.00000 −0.221163
\(737\) 12.0000 0.442026
\(738\) 10.0000 0.368105
\(739\) 38.0000 1.39785 0.698926 − 0.715194i \(-0.253662\pi\)
0.698926 + 0.715194i \(0.253662\pi\)
\(740\) −32.0000 −1.17634
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −8.00000 −0.293492 −0.146746 − 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −10.0000 −0.366618
\(745\) 72.0000 2.63788
\(746\) −26.0000 −0.951928
\(747\) −6.00000 −0.219529
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) −24.0000 −0.876356
\(751\) 38.0000 1.38664 0.693320 − 0.720630i \(-0.256147\pi\)
0.693320 + 0.720630i \(0.256147\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) −2.00000 −0.0728357
\(755\) −64.0000 −2.32920
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 − 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −28.0000 −1.01701
\(759\) −6.00000 −0.217786
\(760\) 8.00000 0.290191
\(761\) 8.00000 0.290000 0.145000 − 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) 8.00000 0.289241
\(766\) 6.00000 0.216789
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) −12.0000 −0.432731 −0.216366 − 0.976312i \(-0.569420\pi\)
−0.216366 + 0.976312i \(0.569420\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) −12.0000 −0.431889
\(773\) −38.0000 −1.36677 −0.683383 − 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) −6.00000 −0.215666
\(775\) 110.000 3.95132
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) 20.0000 0.716574
\(780\) −8.00000 −0.286446
\(781\) 6.00000 0.214697
\(782\) −12.0000 −0.429119
\(783\) 1.00000 0.0357371
\(784\) −7.00000 −0.250000
\(785\) 48.0000 1.71319
\(786\) 8.00000 0.285351
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −22.0000 −0.783718
\(789\) −16.0000 −0.569615
\(790\) 16.0000 0.569254
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −4.00000 −0.142044
\(794\) −26.0000 −0.922705
\(795\) −16.0000 −0.567462
\(796\) −4.00000 −0.141776
\(797\) −18.0000 −0.637593 −0.318796 − 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 11.0000 0.388909
\(801\) 6.00000 0.212000
\(802\) −14.0000 −0.494357
\(803\) 4.00000 0.141157
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 18.0000 0.633630
\(808\) 10.0000 0.351799
\(809\) 26.0000 0.914111 0.457056 − 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 4.00000 0.140546
\(811\) −48.0000 −1.68551 −0.842754 − 0.538299i \(-0.819067\pi\)
−0.842754 + 0.538299i \(0.819067\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 8.00000 0.280400
\(815\) 64.0000 2.24182
\(816\) −2.00000 −0.0700140
\(817\) −12.0000 −0.419827
\(818\) 32.0000 1.11885
\(819\) 0 0
\(820\) 40.0000 1.39686
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 22.0000 0.767338
\(823\) 26.0000 0.906303 0.453152 − 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 8.00000 0.278693
\(825\) 11.0000 0.382971
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −6.00000 −0.208514
\(829\) −20.0000 −0.694629 −0.347314 − 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) −24.0000 −0.833052
\(831\) −6.00000 −0.208138
\(832\) 2.00000 0.0693375
\(833\) −14.0000 −0.485071
\(834\) −8.00000 −0.277017
\(835\) −96.0000 −3.32222
\(836\) −2.00000 −0.0691714
\(837\) −10.0000 −0.345651
\(838\) −20.0000 −0.690889
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −40.0000 −1.37849
\(843\) −12.0000 −0.413302
\(844\) 18.0000 0.619586
\(845\) −36.0000 −1.23844
\(846\) 0 0
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) −16.0000 −0.549119
\(850\) 22.0000 0.754594
\(851\) 48.0000 1.64542
\(852\) 6.00000 0.205557
\(853\) −34.0000 −1.16414 −0.582069 − 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) −6.00000 −0.205076
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 2.00000 0.0682789
\(859\) −24.0000 −0.818869 −0.409435 − 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) −40.0000 −1.36241
\(863\) −6.00000 −0.204242 −0.102121 − 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 56.0000 1.90406
\(866\) −14.0000 −0.475739
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 4.00000 0.135613
\(871\) −24.0000 −0.813209
\(872\) −6.00000 −0.203186
\(873\) 18.0000 0.609208
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) −26.0000 −0.877958 −0.438979 − 0.898497i \(-0.644660\pi\)
−0.438979 + 0.898497i \(0.644660\pi\)
\(878\) 0 0
\(879\) 22.0000 0.742042
\(880\) −4.00000 −0.134840
\(881\) −54.0000 −1.81931 −0.909653 − 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) −7.00000 −0.235702
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 4.00000 0.134535
\(885\) −16.0000 −0.537834
\(886\) −4.00000 −0.134383
\(887\) −8.00000 −0.268614 −0.134307 − 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) 24.0000 0.804482
\(891\) −1.00000 −0.0335013
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) −48.0000 −1.60446
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) −18.0000 −0.600668
\(899\) −10.0000 −0.333519
\(900\) 11.0000 0.366667
\(901\) 8.00000 0.266519
\(902\) −10.0000 −0.332964
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) −24.0000 −0.797787
\(906\) 16.0000 0.531564
\(907\) −8.00000 −0.265636 −0.132818 − 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −6.00000 −0.199117
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 − 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 6.00000 0.198571
\(914\) −26.0000 −0.860004
\(915\) 8.00000 0.264472
\(916\) −20.0000 −0.660819
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −24.0000 −0.791257
\(921\) 2.00000 0.0659022
\(922\) 38.0000 1.25146
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −88.0000 −2.89342
\(926\) 4.00000 0.131448
\(927\) 8.00000 0.262754
\(928\) −1.00000 −0.0328266
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −40.0000 −1.31165
\(931\) −14.0000 −0.458831
\(932\) 4.00000 0.131024
\(933\) 32.0000 1.04763
\(934\) −20.0000 −0.654420
\(935\) −8.00000 −0.261628
\(936\) 2.00000 0.0653720
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 − 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) −12.0000 −0.390981
\(943\) −60.0000 −1.95387
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −4.00000 −0.129914
\(949\) −8.00000 −0.259691
\(950\) 22.0000 0.713774
\(951\) −10.0000 −0.324272
\(952\) 0 0
\(953\) 20.0000 0.647864 0.323932 − 0.946080i \(-0.394995\pi\)
0.323932 + 0.946080i \(0.394995\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) −1.00000 −0.0323254
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) −4.00000 −0.129099
\(961\) 69.0000 2.22581
\(962\) −16.0000 −0.515861
\(963\) −6.00000 −0.193347
\(964\) 10.0000 0.322078
\(965\) −48.0000 −1.54517
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 1.00000 0.0321412
\(969\) −4.00000 −0.128499
\(970\) 72.0000 2.31178
\(971\) −28.0000 −0.898563 −0.449281 − 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) −22.0000 −0.704564
\(976\) −2.00000 −0.0640184
\(977\) −30.0000 −0.959785 −0.479893 − 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −16.0000 −0.511624
\(979\) −6.00000 −0.191761
\(980\) −28.0000 −0.894427
\(981\) −6.00000 −0.191565
\(982\) −24.0000 −0.765871
\(983\) 4.00000 0.127580 0.0637901 − 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) −10.0000 −0.318788
\(985\) −88.0000 −2.80391
\(986\) −2.00000 −0.0636930
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 36.0000 1.14473
\(990\) −4.00000 −0.127128
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 10.0000 0.317500
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 6.00000 0.190117
\(997\) 58.0000 1.83688 0.918439 − 0.395562i \(-0.129450\pi\)
0.918439 + 0.395562i \(0.129450\pi\)
\(998\) 36.0000 1.13956
\(999\) 8.00000 0.253109
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 1914.2.a.l.1.1 ✓ 1
3.2 odd 2 5742.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1914.2.a.l.1.1 ✓ 1 1.1 even 1 trivial
5742.2.a.b.1.1 1 3.2 odd 2