↑

Properties

Label 4002.2.a.d.1.1
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
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 \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +3.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -3.00000 q^{20} -3.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{27} -3.00000 q^{28} -1.00000 q^{29} +3.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -3.00000 q^{34} +9.00000 q^{35} +1.00000 q^{36} -5.00000 q^{37} +1.00000 q^{38} +3.00000 q^{40} +3.00000 q^{41} +3.00000 q^{42} +1.00000 q^{43} +2.00000 q^{44} -3.00000 q^{45} -1.00000 q^{46} -11.0000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} +3.00000 q^{51} -6.00000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +3.00000 q^{56} -1.00000 q^{57} +1.00000 q^{58} -5.00000 q^{59} -3.00000 q^{60} +14.0000 q^{61} -8.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +12.0000 q^{67} +3.00000 q^{68} +1.00000 q^{69} -9.00000 q^{70} -1.00000 q^{72} -6.00000 q^{73} +5.00000 q^{74} +4.00000 q^{75} -1.00000 q^{76} -6.00000 q^{77} +10.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -3.00000 q^{82} -8.00000 q^{83} -3.00000 q^{84} -9.00000 q^{85} -1.00000 q^{86} -1.00000 q^{87} -2.00000 q^{88} +10.0000 q^{89} +3.00000 q^{90} +1.00000 q^{92} +8.00000 q^{93} +11.0000 q^{94} +3.00000 q^{95} -1.00000 q^{96} -8.00000 q^{97} -2.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 − 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 3.00000 0.801784
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −3.00000 −0.670820
\(21\) −3.00000 −0.654654
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −3.00000 −0.566947
\(29\) −1.00000 −0.185695
\(30\) 3.00000 0.547723
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −3.00000 −0.514496
\(35\) 9.00000 1.52128
\(36\) 1.00000 0.166667
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 3.00000 0.462910
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 2.00000 0.301511
\(45\) −3.00000 −0.447214
\(46\) −1.00000 −0.147442
\(47\) −11.0000 −1.60451 −0.802257 − 0.596978i \(-0.796368\pi\)
−0.802257 + 0.596978i \(0.796368\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) −4.00000 −0.565685
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.00000 −0.809040
\(56\) 3.00000 0.400892
\(57\) −1.00000 −0.132453
\(58\) 1.00000 0.131306
\(59\) −5.00000 −0.650945 −0.325472 − 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) −3.00000 −0.387298
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −8.00000 −1.01600
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 3.00000 0.363803
\(69\) 1.00000 0.120386
\(70\) −9.00000 −1.07571
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 5.00000 0.581238
\(75\) 4.00000 0.461880
\(76\) −1.00000 −0.114708
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) −3.00000 −0.327327
\(85\) −9.00000 −0.976187
\(86\) −1.00000 −0.107833
\(87\) −1.00000 −0.107211
\(88\) −2.00000 −0.213201
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 11.0000 1.13456
\(95\) 3.00000 0.307794
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −2.00000 −0.202031
\(99\) 2.00000 0.201008
\(100\) 4.00000 0.400000
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −3.00000 −0.297044
\(103\) −5.00000 −0.492665 −0.246332 − 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 9.00000 0.878310
\(106\) 6.00000 0.582772
\(107\) −5.00000 −0.483368 −0.241684 − 0.970355i \(-0.577700\pi\)
−0.241684 + 0.970355i \(0.577700\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.0000 1.53252 0.766261 − 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 6.00000 0.572078
\(111\) −5.00000 −0.474579
\(112\) −3.00000 −0.283473
\(113\) −11.0000 −1.03479 −0.517396 − 0.855746i \(-0.673099\pi\)
−0.517396 + 0.855746i \(0.673099\pi\)
\(114\) 1.00000 0.0936586
\(115\) −3.00000 −0.279751
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 5.00000 0.460287
\(119\) −9.00000 −0.825029
\(120\) 3.00000 0.273861
\(121\) −7.00000 −0.636364
\(122\) −14.0000 −1.26750
\(123\) 3.00000 0.270501
\(124\) 8.00000 0.718421
\(125\) 3.00000 0.268328
\(126\) 3.00000 0.267261
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 − 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 2.00000 0.174078
\(133\) 3.00000 0.260133
\(134\) −12.0000 −1.03664
\(135\) −3.00000 −0.258199
\(136\) −3.00000 −0.257248
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 6.00000 0.508913 0.254457 − 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 9.00000 0.760639
\(141\) −11.0000 −0.926367
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.00000 0.249136
\(146\) 6.00000 0.496564
\(147\) 2.00000 0.164957
\(148\) −5.00000 −0.410997
\(149\) −13.0000 −1.06500 −0.532501 − 0.846430i \(-0.678748\pi\)
−0.532501 + 0.846430i \(0.678748\pi\)
\(150\) −4.00000 −0.326599
\(151\) −19.0000 −1.54620 −0.773099 − 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 1.00000 0.0811107
\(153\) 3.00000 0.242536
\(154\) 6.00000 0.483494
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 − 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) −10.0000 −0.795557
\(159\) −6.00000 −0.475831
\(160\) 3.00000 0.237171
\(161\) −3.00000 −0.236433
\(162\) −1.00000 −0.0785674
\(163\) −7.00000 −0.548282 −0.274141 − 0.961689i \(-0.588394\pi\)
−0.274141 + 0.961689i \(0.588394\pi\)
\(164\) 3.00000 0.234261
\(165\) −6.00000 −0.467099
\(166\) 8.00000 0.620920
\(167\) −18.0000 −1.39288 −0.696441 − 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 3.00000 0.231455
\(169\) −13.0000 −1.00000
\(170\) 9.00000 0.690268
\(171\) −1.00000 −0.0764719
\(172\) 1.00000 0.0762493
\(173\) −17.0000 −1.29249 −0.646243 − 0.763132i \(-0.723661\pi\)
−0.646243 + 0.763132i \(0.723661\pi\)
\(174\) 1.00000 0.0758098
\(175\) −12.0000 −0.907115
\(176\) 2.00000 0.150756
\(177\) −5.00000 −0.375823
\(178\) −10.0000 −0.749532
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −3.00000 −0.223607
\(181\) −20.0000 −1.48659 −0.743294 − 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) −1.00000 −0.0737210
\(185\) 15.0000 1.10282
\(186\) −8.00000 −0.586588
\(187\) 6.00000 0.438763
\(188\) −11.0000 −0.802257
\(189\) −3.00000 −0.218218
\(190\) −3.00000 −0.217643
\(191\) −17.0000 −1.23008 −0.615038 − 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 1.00000 0.0721688
\(193\) −12.0000 −0.863779 −0.431889 − 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −11.0000 −0.783718 −0.391859 − 0.920025i \(-0.628168\pi\)
−0.391859 + 0.920025i \(0.628168\pi\)
\(198\) −2.00000 −0.142134
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −4.00000 −0.282843
\(201\) 12.0000 0.846415
\(202\) 14.0000 0.985037
\(203\) 3.00000 0.210559
\(204\) 3.00000 0.210042
\(205\) −9.00000 −0.628587
\(206\) 5.00000 0.348367
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) −9.00000 −0.621059
\(211\) 3.00000 0.206529 0.103264 − 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 5.00000 0.341793
\(215\) −3.00000 −0.204598
\(216\) −1.00000 −0.0680414
\(217\) −24.0000 −1.62923
\(218\) −16.0000 −1.08366
\(219\) −6.00000 −0.405442
\(220\) −6.00000 −0.404520
\(221\) 0 0
\(222\) 5.00000 0.335578
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 3.00000 0.200446
\(225\) 4.00000 0.266667
\(226\) 11.0000 0.731709
\(227\) 23.0000 1.52656 0.763282 − 0.646066i \(-0.223587\pi\)
0.763282 + 0.646066i \(0.223587\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 5.00000 0.330409 0.165205 − 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 3.00000 0.197814
\(231\) −6.00000 −0.394771
\(232\) 1.00000 0.0656532
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 33.0000 2.15268
\(236\) −5.00000 −0.325472
\(237\) 10.0000 0.649570
\(238\) 9.00000 0.583383
\(239\) −18.0000 −1.16432 −0.582162 − 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) −3.00000 −0.193649
\(241\) 21.0000 1.35273 0.676364 − 0.736567i \(-0.263554\pi\)
0.676364 + 0.736567i \(0.263554\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) −6.00000 −0.383326
\(246\) −3.00000 −0.191273
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −8.00000 −0.506979
\(250\) −3.00000 −0.189737
\(251\) −8.00000 −0.504956 −0.252478 − 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) −3.00000 −0.188982
\(253\) 2.00000 0.125739
\(254\) −2.00000 −0.125491
\(255\) −9.00000 −0.563602
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) −1.00000 −0.0622573
\(259\) 15.0000 0.932055
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 16.0000 0.988483
\(263\) 1.00000 0.0616626 0.0308313 − 0.999525i \(-0.490185\pi\)
0.0308313 + 0.999525i \(0.490185\pi\)
\(264\) −2.00000 −0.123091
\(265\) 18.0000 1.10573
\(266\) −3.00000 −0.183942
\(267\) 10.0000 0.611990
\(268\) 12.0000 0.733017
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 3.00000 0.182574
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 8.00000 0.482418
\(276\) 1.00000 0.0601929
\(277\) −14.0000 −0.841178 −0.420589 − 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −6.00000 −0.359856
\(279\) 8.00000 0.478947
\(280\) −9.00000 −0.537853
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 11.0000 0.655040
\(283\) 26.0000 1.54554 0.772770 − 0.634686i \(-0.218871\pi\)
0.772770 + 0.634686i \(0.218871\pi\)
\(284\) 0 0
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) −3.00000 −0.176166
\(291\) −8.00000 −0.468968
\(292\) −6.00000 −0.351123
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −2.00000 −0.116642
\(295\) 15.0000 0.873334
\(296\) 5.00000 0.290619
\(297\) 2.00000 0.116052
\(298\) 13.0000 0.753070
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) −3.00000 −0.172917
\(302\) 19.0000 1.09333
\(303\) −14.0000 −0.804279
\(304\) −1.00000 −0.0573539
\(305\) −42.0000 −2.40491
\(306\) −3.00000 −0.171499
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) −6.00000 −0.341882
\(309\) −5.00000 −0.284440
\(310\) 24.0000 1.36311
\(311\) 5.00000 0.283524 0.141762 − 0.989901i \(-0.454723\pi\)
0.141762 + 0.989901i \(0.454723\pi\)
\(312\) 0 0
\(313\) 31.0000 1.75222 0.876112 − 0.482108i \(-0.160129\pi\)
0.876112 + 0.482108i \(0.160129\pi\)
\(314\) −7.00000 −0.395033
\(315\) 9.00000 0.507093
\(316\) 10.0000 0.562544
\(317\) −20.0000 −1.12331 −0.561656 − 0.827371i \(-0.689836\pi\)
−0.561656 + 0.827371i \(0.689836\pi\)
\(318\) 6.00000 0.336463
\(319\) −2.00000 −0.111979
\(320\) −3.00000 −0.167705
\(321\) −5.00000 −0.279073
\(322\) 3.00000 0.167183
\(323\) −3.00000 −0.166924
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 7.00000 0.387694
\(327\) 16.0000 0.884802
\(328\) −3.00000 −0.165647
\(329\) 33.0000 1.81935
\(330\) 6.00000 0.330289
\(331\) 21.0000 1.15426 0.577132 − 0.816651i \(-0.304172\pi\)
0.577132 + 0.816651i \(0.304172\pi\)
\(332\) −8.00000 −0.439057
\(333\) −5.00000 −0.273998
\(334\) 18.0000 0.984916
\(335\) −36.0000 −1.96689
\(336\) −3.00000 −0.163663
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 13.0000 0.707107
\(339\) −11.0000 −0.597438
\(340\) −9.00000 −0.488094
\(341\) 16.0000 0.866449
\(342\) 1.00000 0.0540738
\(343\) 15.0000 0.809924
\(344\) −1.00000 −0.0539164
\(345\) −3.00000 −0.161515
\(346\) 17.0000 0.913926
\(347\) 5.00000 0.268414 0.134207 − 0.990953i \(-0.457151\pi\)
0.134207 + 0.990953i \(0.457151\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 12.0000 0.641427
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 10.0000 0.532246 0.266123 − 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 5.00000 0.265747
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −9.00000 −0.476331
\(358\) 0 0
\(359\) −27.0000 −1.42501 −0.712503 − 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 3.00000 0.158114
\(361\) −18.0000 −0.947368
\(362\) 20.0000 1.05118
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 18.0000 0.942163
\(366\) −14.0000 −0.731792
\(367\) −10.0000 −0.521996 −0.260998 − 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 1.00000 0.0521286
\(369\) 3.00000 0.156174
\(370\) −15.0000 −0.779813
\(371\) 18.0000 0.934513
\(372\) 8.00000 0.414781
\(373\) 12.0000 0.621336 0.310668 − 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) −6.00000 −0.310253
\(375\) 3.00000 0.154919
\(376\) 11.0000 0.567282
\(377\) 0 0
\(378\) 3.00000 0.154303
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 3.00000 0.153897
\(381\) 2.00000 0.102463
\(382\) 17.0000 0.869796
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 18.0000 0.917365
\(386\) 12.0000 0.610784
\(387\) 1.00000 0.0508329
\(388\) −8.00000 −0.406138
\(389\) −12.0000 −0.608424 −0.304212 − 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) −2.00000 −0.101015
\(393\) −16.0000 −0.807093
\(394\) 11.0000 0.554172
\(395\) −30.0000 −1.50946
\(396\) 2.00000 0.100504
\(397\) 6.00000 0.301131 0.150566 − 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 4.00000 0.200000
\(401\) −10.0000 −0.499376 −0.249688 − 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) −3.00000 −0.149071
\(406\) −3.00000 −0.148888
\(407\) −10.0000 −0.495682
\(408\) −3.00000 −0.148522
\(409\) −20.0000 −0.988936 −0.494468 − 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 9.00000 0.444478
\(411\) −6.00000 −0.295958
\(412\) −5.00000 −0.246332
\(413\) 15.0000 0.738102
\(414\) −1.00000 −0.0491473
\(415\) 24.0000 1.17811
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 2.00000 0.0978232
\(419\) 33.0000 1.61216 0.806078 − 0.591810i \(-0.201586\pi\)
0.806078 + 0.591810i \(0.201586\pi\)
\(420\) 9.00000 0.439155
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −3.00000 −0.146038
\(423\) −11.0000 −0.534838
\(424\) 6.00000 0.291386
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) −42.0000 −2.03252
\(428\) −5.00000 −0.241684
\(429\) 0 0
\(430\) 3.00000 0.144673
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 24.0000 1.15204
\(435\) 3.00000 0.143839
\(436\) 16.0000 0.766261
\(437\) −1.00000 −0.0478365
\(438\) 6.00000 0.286691
\(439\) 19.0000 0.906821 0.453410 − 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 6.00000 0.286039
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −5.00000 −0.237289
\(445\) −30.0000 −1.42214
\(446\) 16.0000 0.757622
\(447\) −13.0000 −0.614879
\(448\) −3.00000 −0.141737
\(449\) 3.00000 0.141579 0.0707894 − 0.997491i \(-0.477448\pi\)
0.0707894 + 0.997491i \(0.477448\pi\)
\(450\) −4.00000 −0.188562
\(451\) 6.00000 0.282529
\(452\) −11.0000 −0.517396
\(453\) −19.0000 −0.892698
\(454\) −23.0000 −1.07944
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −17.0000 −0.795226 −0.397613 − 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) −5.00000 −0.233635
\(459\) 3.00000 0.140028
\(460\) −3.00000 −0.139876
\(461\) 32.0000 1.49039 0.745194 − 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 6.00000 0.279145
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −24.0000 −1.11297
\(466\) −24.0000 −1.11178
\(467\) −16.0000 −0.740392 −0.370196 − 0.928954i \(-0.620709\pi\)
−0.370196 + 0.928954i \(0.620709\pi\)
\(468\) 0 0
\(469\) −36.0000 −1.66233
\(470\) −33.0000 −1.52218
\(471\) 7.00000 0.322543
\(472\) 5.00000 0.230144
\(473\) 2.00000 0.0919601
\(474\) −10.0000 −0.459315
\(475\) −4.00000 −0.183533
\(476\) −9.00000 −0.412514
\(477\) −6.00000 −0.274721
\(478\) 18.0000 0.823301
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 3.00000 0.136931
\(481\) 0 0
\(482\) −21.0000 −0.956524
\(483\) −3.00000 −0.136505
\(484\) −7.00000 −0.318182
\(485\) 24.0000 1.08978
\(486\) −1.00000 −0.0453609
\(487\) −13.0000 −0.589086 −0.294543 − 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) −14.0000 −0.633750
\(489\) −7.00000 −0.316551
\(490\) 6.00000 0.271052
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 3.00000 0.135250
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 8.00000 0.358489
\(499\) −2.00000 −0.0895323 −0.0447661 − 0.998997i \(-0.514254\pi\)
−0.0447661 + 0.998997i \(0.514254\pi\)
\(500\) 3.00000 0.134164
\(501\) −18.0000 −0.804181
\(502\) 8.00000 0.357057
\(503\) 27.0000 1.20387 0.601935 − 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 3.00000 0.133631
\(505\) 42.0000 1.86898
\(506\) −2.00000 −0.0889108
\(507\) −13.0000 −0.577350
\(508\) 2.00000 0.0887357
\(509\) 23.0000 1.01946 0.509729 − 0.860335i \(-0.329746\pi\)
0.509729 + 0.860335i \(0.329746\pi\)
\(510\) 9.00000 0.398527
\(511\) 18.0000 0.796273
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −12.0000 −0.529297
\(515\) 15.0000 0.660979
\(516\) 1.00000 0.0440225
\(517\) −22.0000 −0.967559
\(518\) −15.0000 −0.659062
\(519\) −17.0000 −0.746217
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 1.00000 0.0437688
\(523\) 34.0000 1.48672 0.743358 − 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) −16.0000 −0.698963
\(525\) −12.0000 −0.523723
\(526\) −1.00000 −0.0436021
\(527\) 24.0000 1.04546
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) −18.0000 −0.781870
\(531\) −5.00000 −0.216982
\(532\) 3.00000 0.130066
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) 15.0000 0.648507
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 4.00000 0.172292
\(540\) −3.00000 −0.129099
\(541\) −1.00000 −0.0429934 −0.0214967 − 0.999769i \(-0.506843\pi\)
−0.0214967 + 0.999769i \(0.506843\pi\)
\(542\) 20.0000 0.859074
\(543\) −20.0000 −0.858282
\(544\) −3.00000 −0.128624
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 − 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −6.00000 −0.256307
\(549\) 14.0000 0.597505
\(550\) −8.00000 −0.341121
\(551\) 1.00000 0.0426014
\(552\) −1.00000 −0.0425628
\(553\) −30.0000 −1.27573
\(554\) 14.0000 0.594803
\(555\) 15.0000 0.636715
\(556\) 6.00000 0.254457
\(557\) −19.0000 −0.805056 −0.402528 − 0.915408i \(-0.631868\pi\)
−0.402528 + 0.915408i \(0.631868\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) 9.00000 0.380319
\(561\) 6.00000 0.253320
\(562\) −18.0000 −0.759284
\(563\) 2.00000 0.0842900 0.0421450 − 0.999112i \(-0.486581\pi\)
0.0421450 + 0.999112i \(0.486581\pi\)
\(564\) −11.0000 −0.463184
\(565\) 33.0000 1.38832
\(566\) −26.0000 −1.09286
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) −3.00000 −0.125656
\(571\) 30.0000 1.25546 0.627730 − 0.778431i \(-0.283984\pi\)
0.627730 + 0.778431i \(0.283984\pi\)
\(572\) 0 0
\(573\) −17.0000 −0.710185
\(574\) 9.00000 0.375653
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −20.0000 −0.832611 −0.416305 − 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 8.00000 0.332756
\(579\) −12.0000 −0.498703
\(580\) 3.00000 0.124568
\(581\) 24.0000 0.995688
\(582\) 8.00000 0.331611
\(583\) −12.0000 −0.496989
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −3.00000 −0.123823 −0.0619116 − 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 2.00000 0.0824786
\(589\) −8.00000 −0.329634
\(590\) −15.0000 −0.617540
\(591\) −11.0000 −0.452480
\(592\) −5.00000 −0.205499
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 27.0000 1.10689
\(596\) −13.0000 −0.532501
\(597\) 0 0
\(598\) 0 0
\(599\) 28.0000 1.14405 0.572024 − 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) −4.00000 −0.163299
\(601\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 3.00000 0.122271
\(603\) 12.0000 0.488678
\(604\) −19.0000 −0.773099
\(605\) 21.0000 0.853771
\(606\) 14.0000 0.568711
\(607\) −20.0000 −0.811775 −0.405887 − 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 1.00000 0.0405554
\(609\) 3.00000 0.121566
\(610\) 42.0000 1.70053
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) −26.0000 −1.05013 −0.525065 − 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 28.0000 1.12999
\(615\) −9.00000 −0.362915
\(616\) 6.00000 0.241747
\(617\) 25.0000 1.00646 0.503231 − 0.864152i \(-0.332144\pi\)
0.503231 + 0.864152i \(0.332144\pi\)
\(618\) 5.00000 0.201129
\(619\) −31.0000 −1.24600 −0.622998 − 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) −24.0000 −0.963863
\(621\) 1.00000 0.0401286
\(622\) −5.00000 −0.200482
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −31.0000 −1.23901
\(627\) −2.00000 −0.0798723
\(628\) 7.00000 0.279330
\(629\) −15.0000 −0.598089
\(630\) −9.00000 −0.358569
\(631\) 19.0000 0.756378 0.378189 − 0.925728i \(-0.376547\pi\)
0.378189 + 0.925728i \(0.376547\pi\)
\(632\) −10.0000 −0.397779
\(633\) 3.00000 0.119239
\(634\) 20.0000 0.794301
\(635\) −6.00000 −0.238103
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) 29.0000 1.14543 0.572716 − 0.819754i \(-0.305890\pi\)
0.572716 + 0.819754i \(0.305890\pi\)
\(642\) 5.00000 0.197334
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) −3.00000 −0.118217
\(645\) −3.00000 −0.118125
\(646\) 3.00000 0.118033
\(647\) −34.0000 −1.33668 −0.668339 − 0.743857i \(-0.732994\pi\)
−0.668339 + 0.743857i \(0.732994\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) −7.00000 −0.274141
\(653\) 34.0000 1.33052 0.665261 − 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) −16.0000 −0.625650
\(655\) 48.0000 1.87552
\(656\) 3.00000 0.117130
\(657\) −6.00000 −0.234082
\(658\) −33.0000 −1.28647
\(659\) −16.0000 −0.623272 −0.311636 − 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) −6.00000 −0.233550
\(661\) 8.00000 0.311164 0.155582 − 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) −21.0000 −0.816188
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) −9.00000 −0.349005
\(666\) 5.00000 0.193746
\(667\) −1.00000 −0.0387202
\(668\) −18.0000 −0.696441
\(669\) −16.0000 −0.618596
\(670\) 36.0000 1.39080
\(671\) 28.0000 1.08093
\(672\) 3.00000 0.115728
\(673\) 19.0000 0.732396 0.366198 − 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 2.00000 0.0770371
\(675\) 4.00000 0.153960
\(676\) −13.0000 −0.500000
\(677\) −26.0000 −0.999261 −0.499631 − 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 11.0000 0.422452
\(679\) 24.0000 0.921035
\(680\) 9.00000 0.345134
\(681\) 23.0000 0.881362
\(682\) −16.0000 −0.612672
\(683\) −49.0000 −1.87493 −0.937466 − 0.348076i \(-0.886835\pi\)
−0.937466 + 0.348076i \(0.886835\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 18.0000 0.687745
\(686\) −15.0000 −0.572703
\(687\) 5.00000 0.190762
\(688\) 1.00000 0.0381246
\(689\) 0 0
\(690\) 3.00000 0.114208
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −17.0000 −0.646243
\(693\) −6.00000 −0.227921
\(694\) −5.00000 −0.189797
\(695\) −18.0000 −0.682779
\(696\) 1.00000 0.0379049
\(697\) 9.00000 0.340899
\(698\) 14.0000 0.529908
\(699\) 24.0000 0.907763
\(700\) −12.0000 −0.453557
\(701\) −27.0000 −1.01978 −0.509888 − 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) 5.00000 0.188579
\(704\) 2.00000 0.0753778
\(705\) 33.0000 1.24285
\(706\) −10.0000 −0.376355
\(707\) 42.0000 1.57957
\(708\) −5.00000 −0.187912
\(709\) 34.0000 1.27690 0.638448 − 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) −10.0000 −0.374766
\(713\) 8.00000 0.299602
\(714\) 9.00000 0.336817
\(715\) 0 0
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) 27.0000 1.00763
\(719\) 12.0000 0.447524 0.223762 − 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) −3.00000 −0.111803
\(721\) 15.0000 0.558629
\(722\) 18.0000 0.669891
\(723\) 21.0000 0.780998
\(724\) −20.0000 −0.743294
\(725\) −4.00000 −0.148556
\(726\) 7.00000 0.259794
\(727\) −50.0000 −1.85440 −0.927199 − 0.374570i \(-0.877790\pi\)
−0.927199 + 0.374570i \(0.877790\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −18.0000 −0.666210
\(731\) 3.00000 0.110959
\(732\) 14.0000 0.517455
\(733\) −14.0000 −0.517102 −0.258551 − 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 10.0000 0.369107
\(735\) −6.00000 −0.221313
\(736\) −1.00000 −0.0368605
\(737\) 24.0000 0.884051
\(738\) −3.00000 −0.110432
\(739\) 16.0000 0.588570 0.294285 − 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 15.0000 0.551411
\(741\) 0 0
\(742\) −18.0000 −0.660801
\(743\) −33.0000 −1.21065 −0.605326 − 0.795977i \(-0.706957\pi\)
−0.605326 + 0.795977i \(0.706957\pi\)
\(744\) −8.00000 −0.293294
\(745\) 39.0000 1.42885
\(746\) −12.0000 −0.439351
\(747\) −8.00000 −0.292705
\(748\) 6.00000 0.219382
\(749\) 15.0000 0.548088
\(750\) −3.00000 −0.109545
\(751\) −18.0000 −0.656829 −0.328415 − 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) −11.0000 −0.401129
\(753\) −8.00000 −0.291536
\(754\) 0 0
\(755\) 57.0000 2.07444
\(756\) −3.00000 −0.109109
\(757\) 13.0000 0.472493 0.236247 − 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) 4.00000 0.145287
\(759\) 2.00000 0.0725954
\(760\) −3.00000 −0.108821
\(761\) 8.00000 0.290000 0.145000 − 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) −2.00000 −0.0724524
\(763\) −48.0000 −1.73772
\(764\) −17.0000 −0.615038
\(765\) −9.00000 −0.325396
\(766\) −6.00000 −0.216789
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −18.0000 −0.649097 −0.324548 − 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) −18.0000 −0.648675
\(771\) 12.0000 0.432169
\(772\) −12.0000 −0.431889
\(773\) −34.0000 −1.22290 −0.611448 − 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 32.0000 1.14947
\(776\) 8.00000 0.287183
\(777\) 15.0000 0.538122
\(778\) 12.0000 0.430221
\(779\) −3.00000 −0.107486
\(780\) 0 0
\(781\) 0 0
\(782\) −3.00000 −0.107280
\(783\) −1.00000 −0.0357371
\(784\) 2.00000 0.0714286
\(785\) −21.0000 −0.749522
\(786\) 16.0000 0.570701
\(787\) −20.0000 −0.712923 −0.356462 − 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) −11.0000 −0.391859
\(789\) 1.00000 0.0356009
\(790\) 30.0000 1.06735
\(791\) 33.0000 1.17334
\(792\) −2.00000 −0.0710669
\(793\) 0 0
\(794\) −6.00000 −0.212932
\(795\) 18.0000 0.638394
\(796\) 0 0
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) −3.00000 −0.106199
\(799\) −33.0000 −1.16746
\(800\) −4.00000 −0.141421
\(801\) 10.0000 0.353333
\(802\) 10.0000 0.353112
\(803\) −12.0000 −0.423471
\(804\) 12.0000 0.423207
\(805\) 9.00000 0.317208
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 14.0000 0.492518
\(809\) 30.0000 1.05474 0.527372 − 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 3.00000 0.105409
\(811\) 12.0000 0.421377 0.210688 − 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 3.00000 0.105279
\(813\) −20.0000 −0.701431
\(814\) 10.0000 0.350500
\(815\) 21.0000 0.735598
\(816\) 3.00000 0.105021
\(817\) −1.00000 −0.0349856
\(818\) 20.0000 0.699284
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) −2.00000 −0.0698005 −0.0349002 − 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 6.00000 0.209274
\(823\) 54.0000 1.88232 0.941161 − 0.337959i \(-0.109737\pi\)
0.941161 + 0.337959i \(0.109737\pi\)
\(824\) 5.00000 0.174183
\(825\) 8.00000 0.278524
\(826\) −15.0000 −0.521917
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 1.00000 0.0347524
\(829\) 47.0000 1.63238 0.816189 − 0.577785i \(-0.196083\pi\)
0.816189 + 0.577785i \(0.196083\pi\)
\(830\) −24.0000 −0.833052
\(831\) −14.0000 −0.485655
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) −6.00000 −0.207763
\(835\) 54.0000 1.86875
\(836\) −2.00000 −0.0691714
\(837\) 8.00000 0.276520
\(838\) −33.0000 −1.13997
\(839\) −47.0000 −1.62262 −0.811310 − 0.584616i \(-0.801245\pi\)
−0.811310 + 0.584616i \(0.801245\pi\)
\(840\) −9.00000 −0.310530
\(841\) 1.00000 0.0344828
\(842\) 10.0000 0.344623
\(843\) 18.0000 0.619953
\(844\) 3.00000 0.103264
\(845\) 39.0000 1.34164
\(846\) 11.0000 0.378188
\(847\) 21.0000 0.721569
\(848\) −6.00000 −0.206041
\(849\) 26.0000 0.892318
\(850\) −12.0000 −0.411597
\(851\) −5.00000 −0.171398
\(852\) 0 0
\(853\) 5.00000 0.171197 0.0855984 − 0.996330i \(-0.472720\pi\)
0.0855984 + 0.996330i \(0.472720\pi\)
\(854\) 42.0000 1.43721
\(855\) 3.00000 0.102598
\(856\) 5.00000 0.170896
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 41.0000 1.39890 0.699451 − 0.714681i \(-0.253428\pi\)
0.699451 + 0.714681i \(0.253428\pi\)
\(860\) −3.00000 −0.102299
\(861\) −9.00000 −0.306719
\(862\) 32.0000 1.08992
\(863\) −8.00000 −0.272323 −0.136162 − 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 51.0000 1.73405
\(866\) 26.0000 0.883516
\(867\) −8.00000 −0.271694
\(868\) −24.0000 −0.814613
\(869\) 20.0000 0.678454
\(870\) −3.00000 −0.101710
\(871\) 0 0
\(872\) −16.0000 −0.541828
\(873\) −8.00000 −0.270759
\(874\) 1.00000 0.0338255
\(875\) −9.00000 −0.304256
\(876\) −6.00000 −0.202721
\(877\) 20.0000 0.675352 0.337676 − 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) −19.0000 −0.641219
\(879\) −6.00000 −0.202375
\(880\) −6.00000 −0.202260
\(881\) 14.0000 0.471672 0.235836 − 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 15.0000 0.504219
\(886\) −36.0000 −1.20944
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 5.00000 0.167789
\(889\) −6.00000 −0.201234
\(890\) 30.0000 1.00560
\(891\) 2.00000 0.0670025
\(892\) −16.0000 −0.535720
\(893\) 11.0000 0.368101
\(894\) 13.0000 0.434785
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −3.00000 −0.100111
\(899\) −8.00000 −0.266815
\(900\) 4.00000 0.133333
\(901\) −18.0000 −0.599667
\(902\) −6.00000 −0.199778
\(903\) −3.00000 −0.0998337
\(904\) 11.0000 0.365855
\(905\) 60.0000 1.99447
\(906\) 19.0000 0.631233
\(907\) −20.0000 −0.664089 −0.332045 − 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 23.0000 0.763282
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −16.0000 −0.529523
\(914\) 17.0000 0.562310
\(915\) −42.0000 −1.38848
\(916\) 5.00000 0.165205
\(917\) 48.0000 1.58510
\(918\) −3.00000 −0.0990148
\(919\) −11.0000 −0.362857 −0.181428 − 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 3.00000 0.0989071
\(921\) −28.0000 −0.922631
\(922\) −32.0000 −1.05386
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) −20.0000 −0.657596
\(926\) −24.0000 −0.788689
\(927\) −5.00000 −0.164222
\(928\) 1.00000 0.0328266
\(929\) 42.0000 1.37798 0.688988 − 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 24.0000 0.786991
\(931\) −2.00000 −0.0655474
\(932\) 24.0000 0.786146
\(933\) 5.00000 0.163693
\(934\) 16.0000 0.523536
\(935\) −18.0000 −0.588663
\(936\) 0 0
\(937\) 11.0000 0.359354 0.179677 − 0.983726i \(-0.442495\pi\)
0.179677 + 0.983726i \(0.442495\pi\)
\(938\) 36.0000 1.17544
\(939\) 31.0000 1.01165
\(940\) 33.0000 1.07634
\(941\) 6.00000 0.195594 0.0977972 − 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −7.00000 −0.228072
\(943\) 3.00000 0.0976934
\(944\) −5.00000 −0.162736
\(945\) 9.00000 0.292770
\(946\) −2.00000 −0.0650256
\(947\) −48.0000 −1.55979 −0.779895 − 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 10.0000 0.324785
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) −20.0000 −0.648544
\(952\) 9.00000 0.291692
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 6.00000 0.194257
\(955\) 51.0000 1.65032
\(956\) −18.0000 −0.582162
\(957\) −2.00000 −0.0646508
\(958\) 24.0000 0.775405
\(959\) 18.0000 0.581250
\(960\) −3.00000 −0.0968246
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −5.00000 −0.161123
\(964\) 21.0000 0.676364
\(965\) 36.0000 1.15888
\(966\) 3.00000 0.0965234
\(967\) −48.0000 −1.54358 −0.771788 − 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 7.00000 0.224989
\(969\) −3.00000 −0.0963739
\(970\) −24.0000 −0.770594
\(971\) −10.0000 −0.320915 −0.160458 − 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(972\) 1.00000 0.0320750
\(973\) −18.0000 −0.577054
\(974\) 13.0000 0.416547
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) 8.00000 0.255943 0.127971 − 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 7.00000 0.223835
\(979\) 20.0000 0.639203
\(980\) −6.00000 −0.191663
\(981\) 16.0000 0.510841
\(982\) −30.0000 −0.957338
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 33.0000 1.05147
\(986\) 3.00000 0.0955395
\(987\) 33.0000 1.05040
\(988\) 0 0
\(989\) 1.00000 0.0317982
\(990\) 6.00000 0.190693
\(991\) −5.00000 −0.158830 −0.0794151 − 0.996842i \(-0.525305\pi\)
−0.0794151 + 0.996842i \(0.525305\pi\)
\(992\) −8.00000 −0.254000
\(993\) 21.0000 0.666415
\(994\) 0 0
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −11.0000 −0.348373 −0.174187 − 0.984713i \(-0.555730\pi\)
−0.174187 + 0.984713i \(0.555730\pi\)
\(998\) 2.00000 0.0633089
\(999\) −5.00000 −0.158193
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 4002.2.a.d.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.d.1.1 ✓ 1 1.1 even 1 trivial