Properties

Label 4998.2.a.i.1.1
Level $4998$
Weight $2$
Character 4998.1
Self dual yes
Analytic conductor $39.909$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, 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("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.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: \(39.9092309302\)
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 \(\chi\) \(=\) 4998.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -5.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -4.00000 q^{29} +1.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} -5.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} -11.0000 q^{43} +5.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} -2.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} -1.00000 q^{51} +1.00000 q^{52} -7.00000 q^{53} +1.00000 q^{54} +5.00000 q^{55} -4.00000 q^{57} +4.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +14.0000 q^{61} +10.0000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +5.00000 q^{66} -5.00000 q^{67} +1.00000 q^{68} +6.00000 q^{69} -10.0000 q^{71} -1.00000 q^{72} -1.00000 q^{73} +3.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} +7.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +9.00000 q^{83} +1.00000 q^{85} +11.0000 q^{86} +4.00000 q^{87} -5.00000 q^{88} -13.0000 q^{89} -1.00000 q^{90} -6.00000 q^{92} +10.0000 q^{93} +2.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} -13.0000 q^{97} +5.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 1.00000 0.182574
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.00000 −0.870388
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −4.00000 −0.648886
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 5.00000 0.753778
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) −1.00000 −0.140028
\(52\) 1.00000 0.138675
\(53\) −7.00000 −0.961524 −0.480762 0.876851i \(-0.659640\pi\)
−0.480762 + 0.876851i \(0.659640\pi\)
\(54\) 1.00000 0.136083
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 4.00000 0.525226
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 5.00000 0.615457
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 1.00000 0.121268
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −1.00000 −0.117851
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 3.00000 0.348743
\(75\) 4.00000 0.461880
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 11.0000 1.18616
\(87\) 4.00000 0.428845
\(88\) −5.00000 −0.533002
\(89\) −13.0000 −1.37800 −0.688999 0.724763i \(-0.741949\pi\)
−0.688999 + 0.724763i \(0.741949\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 10.0000 1.03695
\(94\) 2.00000 0.206284
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) −4.00000 −0.400000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 1.00000 0.0990148
\(103\) 7.00000 0.689730 0.344865 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 7.00000 0.679900
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −5.00000 −0.476731
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 4.00000 0.374634
\(115\) −6.00000 −0.559503
\(116\) −4.00000 −0.371391
\(117\) 1.00000 0.0924500
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 14.0000 1.27273
\(122\) −14.0000 −1.26750
\(123\) 6.00000 0.541002
\(124\) −10.0000 −0.898027
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −22.0000 −1.95218 −0.976092 0.217357i \(-0.930256\pi\)
−0.976092 + 0.217357i \(0.930256\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.0000 0.968496
\(130\) −1.00000 −0.0877058
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −5.00000 −0.435194
\(133\) 0 0
\(134\) 5.00000 0.431934
\(135\) −1.00000 −0.0860663
\(136\) −1.00000 −0.0857493
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) −6.00000 −0.510754
\(139\) 21.0000 1.78120 0.890598 0.454791i \(-0.150286\pi\)
0.890598 + 0.454791i \(0.150286\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 10.0000 0.839181
\(143\) 5.00000 0.418121
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) 1.00000 0.0827606
\(147\) 0 0
\(148\) −3.00000 −0.246598
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) −4.00000 −0.326599
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −4.00000 −0.324443
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) −1.00000 −0.0800641
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −7.00000 −0.556890
\(159\) 7.00000 0.555136
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −6.00000 −0.468521
\(165\) −5.00000 −0.389249
\(166\) −9.00000 −0.698535
\(167\) −19.0000 −1.47026 −0.735132 0.677924i \(-0.762880\pi\)
−0.735132 + 0.677924i \(0.762880\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −1.00000 −0.0766965
\(171\) 4.00000 0.305888
\(172\) −11.0000 −0.838742
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) −4.00000 −0.300658
\(178\) 13.0000 0.974391
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 6.00000 0.442326
\(185\) −3.00000 −0.220564
\(186\) −10.0000 −0.733236
\(187\) 5.00000 0.365636
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 13.0000 0.933346
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −5.00000 −0.355335
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 4.00000 0.282843
\(201\) 5.00000 0.352673
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) −6.00000 −0.419058
\(206\) −7.00000 −0.487713
\(207\) −6.00000 −0.417029
\(208\) 1.00000 0.0693375
\(209\) 20.0000 1.38343
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −7.00000 −0.480762
\(213\) 10.0000 0.685189
\(214\) −16.0000 −1.09374
\(215\) −11.0000 −0.750194
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 1.00000 0.0675737
\(220\) 5.00000 0.337100
\(221\) 1.00000 0.0672673
\(222\) −3.00000 −0.201347
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) −1.00000 −0.0665190
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −4.00000 −0.264906
\(229\) 19.0000 1.25556 0.627778 0.778393i \(-0.283965\pi\)
0.627778 + 0.778393i \(0.283965\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) 17.0000 1.11371 0.556854 0.830611i \(-0.312008\pi\)
0.556854 + 0.830611i \(0.312008\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −2.00000 −0.130466
\(236\) 4.00000 0.260378
\(237\) −7.00000 −0.454699
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −14.0000 −0.899954
\(243\) −1.00000 −0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 4.00000 0.254514
\(248\) 10.0000 0.635001
\(249\) −9.00000 −0.570352
\(250\) 9.00000 0.569210
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) 0 0
\(253\) −30.0000 −1.88608
\(254\) 22.0000 1.38040
\(255\) −1.00000 −0.0626224
\(256\) 1.00000 0.0625000
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) −11.0000 −0.684830
\(259\) 0 0
\(260\) 1.00000 0.0620174
\(261\) −4.00000 −0.247594
\(262\) 6.00000 0.370681
\(263\) −1.00000 −0.0616626 −0.0308313 0.999525i \(-0.509815\pi\)
−0.0308313 + 0.999525i \(0.509815\pi\)
\(264\) 5.00000 0.307729
\(265\) −7.00000 −0.430007
\(266\) 0 0
\(267\) 13.0000 0.795587
\(268\) −5.00000 −0.305424
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 1.00000 0.0608581
\(271\) 25.0000 1.51864 0.759321 0.650716i \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 22.0000 1.32907
\(275\) −20.0000 −1.20605
\(276\) 6.00000 0.361158
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −21.0000 −1.25950
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −2.00000 −0.119098
\(283\) 29.0000 1.72387 0.861936 0.507018i \(-0.169252\pi\)
0.861936 + 0.507018i \(0.169252\pi\)
\(284\) −10.0000 −0.593391
\(285\) −4.00000 −0.236940
\(286\) −5.00000 −0.295656
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 4.00000 0.234888
\(291\) 13.0000 0.762073
\(292\) −1.00000 −0.0585206
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 3.00000 0.174371
\(297\) −5.00000 −0.290129
\(298\) −5.00000 −0.289642
\(299\) −6.00000 −0.346989
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) −10.0000 −0.575435
\(303\) 14.0000 0.804279
\(304\) 4.00000 0.229416
\(305\) 14.0000 0.801638
\(306\) −1.00000 −0.0571662
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 10.0000 0.567962
\(311\) 17.0000 0.963982 0.481991 0.876176i \(-0.339914\pi\)
0.481991 + 0.876176i \(0.339914\pi\)
\(312\) 1.00000 0.0566139
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 7.00000 0.393781
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −7.00000 −0.392541
\(319\) −20.0000 −1.11979
\(320\) 1.00000 0.0559017
\(321\) −16.0000 −0.893033
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 12.0000 0.664619
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 5.00000 0.275241
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 9.00000 0.493939
\(333\) −3.00000 −0.164399
\(334\) 19.0000 1.03963
\(335\) −5.00000 −0.273179
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 12.0000 0.652714
\(339\) −1.00000 −0.0543125
\(340\) 1.00000 0.0542326
\(341\) −50.0000 −2.70765
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 11.0000 0.593080
\(345\) 6.00000 0.323029
\(346\) −14.0000 −0.752645
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 4.00000 0.214423
\(349\) 27.0000 1.44528 0.722638 0.691226i \(-0.242929\pi\)
0.722638 + 0.691226i \(0.242929\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −5.00000 −0.266501
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 4.00000 0.212598
\(355\) −10.0000 −0.530745
\(356\) −13.0000 −0.688999
\(357\) 0 0
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) −1.00000 −0.0523424
\(366\) 14.0000 0.731792
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −6.00000 −0.312772
\(369\) −6.00000 −0.312348
\(370\) 3.00000 0.155963
\(371\) 0 0
\(372\) 10.0000 0.518476
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) −5.00000 −0.258544
\(375\) 9.00000 0.464758
\(376\) 2.00000 0.103142
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 36.0000 1.84920 0.924598 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 4.00000 0.205196
\(381\) 22.0000 1.12709
\(382\) −3.00000 −0.153493
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −11.0000 −0.559161
\(388\) −13.0000 −0.659975
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 1.00000 0.0506370
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 6.00000 0.302276
\(395\) 7.00000 0.352208
\(396\) 5.00000 0.251259
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) −2.00000 −0.100251
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −5.00000 −0.249377
\(403\) −10.0000 −0.498135
\(404\) −14.0000 −0.696526
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −15.0000 −0.743522
\(408\) 1.00000 0.0495074
\(409\) 12.0000 0.593362 0.296681 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) 6.00000 0.296319
\(411\) 22.0000 1.08518
\(412\) 7.00000 0.344865
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 9.00000 0.441793
\(416\) −1.00000 −0.0490290
\(417\) −21.0000 −1.02837
\(418\) −20.0000 −0.978232
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) 4.00000 0.194717
\(423\) −2.00000 −0.0972433
\(424\) 7.00000 0.339950
\(425\) −4.00000 −0.194029
\(426\) −10.0000 −0.484502
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) −5.00000 −0.241402
\(430\) 11.0000 0.530467
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) −2.00000 −0.0957826
\(437\) −24.0000 −1.14808
\(438\) −1.00000 −0.0477818
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) −5.00000 −0.238366
\(441\) 0 0
\(442\) −1.00000 −0.0475651
\(443\) −30.0000 −1.42534 −0.712672 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) 3.00000 0.142374
\(445\) −13.0000 −0.616259
\(446\) 16.0000 0.757622
\(447\) −5.00000 −0.236492
\(448\) 0 0
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 4.00000 0.188562
\(451\) −30.0000 −1.41264
\(452\) 1.00000 0.0470360
\(453\) −10.0000 −0.469841
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −13.0000 −0.608114 −0.304057 0.952654i \(-0.598341\pi\)
−0.304057 + 0.952654i \(0.598341\pi\)
\(458\) −19.0000 −0.887812
\(459\) −1.00000 −0.0466760
\(460\) −6.00000 −0.279751
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −4.00000 −0.185695
\(465\) 10.0000 0.463739
\(466\) −17.0000 −0.787510
\(467\) 32.0000 1.48078 0.740392 0.672176i \(-0.234640\pi\)
0.740392 + 0.672176i \(0.234640\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 2.00000 0.0922531
\(471\) −22.0000 −1.01371
\(472\) −4.00000 −0.184115
\(473\) −55.0000 −2.52890
\(474\) 7.00000 0.321521
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) −7.00000 −0.320508
\(478\) 15.0000 0.686084
\(479\) −9.00000 −0.411220 −0.205610 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(480\) 1.00000 0.0456435
\(481\) −3.00000 −0.136788
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −13.0000 −0.590300
\(486\) 1.00000 0.0453609
\(487\) 1.00000 0.0453143 0.0226572 0.999743i \(-0.492787\pi\)
0.0226572 + 0.999743i \(0.492787\pi\)
\(488\) −14.0000 −0.633750
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −4.00000 −0.180151
\(494\) −4.00000 −0.179969
\(495\) 5.00000 0.224733
\(496\) −10.0000 −0.449013
\(497\) 0 0
\(498\) 9.00000 0.403300
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) −9.00000 −0.402492
\(501\) 19.0000 0.848857
\(502\) 5.00000 0.223161
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 30.0000 1.33366
\(507\) 12.0000 0.532939
\(508\) −22.0000 −0.976092
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 1.00000 0.0442807
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 7.00000 0.308757
\(515\) 7.00000 0.308457
\(516\) 11.0000 0.484248
\(517\) −10.0000 −0.439799
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) −1.00000 −0.0438529
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 4.00000 0.175075
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 1.00000 0.0436021
\(527\) −10.0000 −0.435607
\(528\) −5.00000 −0.217597
\(529\) 13.0000 0.565217
\(530\) 7.00000 0.304061
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) −13.0000 −0.562565
\(535\) 16.0000 0.691740
\(536\) 5.00000 0.215967
\(537\) 0 0
\(538\) 26.0000 1.12094
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −11.0000 −0.472927 −0.236463 0.971640i \(-0.575988\pi\)
−0.236463 + 0.971640i \(0.575988\pi\)
\(542\) −25.0000 −1.07384
\(543\) −2.00000 −0.0858282
\(544\) −1.00000 −0.0428746
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −22.0000 −0.939793
\(549\) 14.0000 0.597505
\(550\) 20.0000 0.852803
\(551\) −16.0000 −0.681623
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 3.00000 0.127343
\(556\) 21.0000 0.890598
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 10.0000 0.423334
\(559\) −11.0000 −0.465250
\(560\) 0 0
\(561\) −5.00000 −0.211100
\(562\) −6.00000 −0.253095
\(563\) 5.00000 0.210725 0.105362 0.994434i \(-0.466400\pi\)
0.105362 + 0.994434i \(0.466400\pi\)
\(564\) 2.00000 0.0842152
\(565\) 1.00000 0.0420703
\(566\) −29.0000 −1.21896
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 4.00000 0.167542
\(571\) 30.0000 1.25546 0.627730 0.778431i \(-0.283984\pi\)
0.627730 + 0.778431i \(0.283984\pi\)
\(572\) 5.00000 0.209061
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −14.0000 −0.581820
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) −13.0000 −0.538867
\(583\) −35.0000 −1.44955
\(584\) 1.00000 0.0413803
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) 37.0000 1.52715 0.763577 0.645717i \(-0.223441\pi\)
0.763577 + 0.645717i \(0.223441\pi\)
\(588\) 0 0
\(589\) −40.0000 −1.64817
\(590\) −4.00000 −0.164677
\(591\) 6.00000 0.246807
\(592\) −3.00000 −0.123299
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) 5.00000 0.204808
\(597\) −2.00000 −0.0818546
\(598\) 6.00000 0.245358
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −4.00000 −0.163299
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) −5.00000 −0.203616
\(604\) 10.0000 0.406894
\(605\) 14.0000 0.569181
\(606\) −14.0000 −0.568711
\(607\) −6.00000 −0.243532 −0.121766 0.992559i \(-0.538856\pi\)
−0.121766 + 0.992559i \(0.538856\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) −2.00000 −0.0809113
\(612\) 1.00000 0.0404226
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −20.0000 −0.807134
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −11.0000 −0.442843 −0.221422 0.975178i \(-0.571070\pi\)
−0.221422 + 0.975178i \(0.571070\pi\)
\(618\) 7.00000 0.281581
\(619\) −49.0000 −1.96948 −0.984738 0.174042i \(-0.944317\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) −10.0000 −0.401610
\(621\) 6.00000 0.240772
\(622\) −17.0000 −0.681638
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 11.0000 0.440000
\(626\) 6.00000 0.239808
\(627\) −20.0000 −0.798723
\(628\) 22.0000 0.877896
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) 18.0000 0.716569 0.358284 0.933613i \(-0.383362\pi\)
0.358284 + 0.933613i \(0.383362\pi\)
\(632\) −7.00000 −0.278445
\(633\) 4.00000 0.158986
\(634\) 18.0000 0.714871
\(635\) −22.0000 −0.873043
\(636\) 7.00000 0.277568
\(637\) 0 0
\(638\) 20.0000 0.791808
\(639\) −10.0000 −0.395594
\(640\) −1.00000 −0.0395285
\(641\) 37.0000 1.46141 0.730706 0.682692i \(-0.239191\pi\)
0.730706 + 0.682692i \(0.239191\pi\)
\(642\) 16.0000 0.631470
\(643\) −39.0000 −1.53801 −0.769005 0.639243i \(-0.779248\pi\)
−0.769005 + 0.639243i \(0.779248\pi\)
\(644\) 0 0
\(645\) 11.0000 0.433125
\(646\) −4.00000 −0.157378
\(647\) −30.0000 −1.17942 −0.589711 0.807614i \(-0.700758\pi\)
−0.589711 + 0.807614i \(0.700758\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 20.0000 0.785069
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −6.00000 −0.234439
\(656\) −6.00000 −0.234261
\(657\) −1.00000 −0.0390137
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) −5.00000 −0.194625
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 1.00000 0.0388661
\(663\) −1.00000 −0.0388368
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) 24.0000 0.929284
\(668\) −19.0000 −0.735132
\(669\) 16.0000 0.618596
\(670\) 5.00000 0.193167
\(671\) 70.0000 2.70232
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 22.0000 0.847408
\(675\) 4.00000 0.153960
\(676\) −12.0000 −0.461538
\(677\) −15.0000 −0.576497 −0.288248 0.957556i \(-0.593073\pi\)
−0.288248 + 0.957556i \(0.593073\pi\)
\(678\) 1.00000 0.0384048
\(679\) 0 0
\(680\) −1.00000 −0.0383482
\(681\) 4.00000 0.153280
\(682\) 50.0000 1.91460
\(683\) 52.0000 1.98972 0.994862 0.101237i \(-0.0322800\pi\)
0.994862 + 0.101237i \(0.0322800\pi\)
\(684\) 4.00000 0.152944
\(685\) −22.0000 −0.840577
\(686\) 0 0
\(687\) −19.0000 −0.724895
\(688\) −11.0000 −0.419371
\(689\) −7.00000 −0.266679
\(690\) −6.00000 −0.228416
\(691\) 23.0000 0.874961 0.437481 0.899228i \(-0.355871\pi\)
0.437481 + 0.899228i \(0.355871\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) 21.0000 0.796575
\(696\) −4.00000 −0.151620
\(697\) −6.00000 −0.227266
\(698\) −27.0000 −1.02197
\(699\) −17.0000 −0.642999
\(700\) 0 0
\(701\) −29.0000 −1.09531 −0.547657 0.836703i \(-0.684480\pi\)
−0.547657 + 0.836703i \(0.684480\pi\)
\(702\) 1.00000 0.0377426
\(703\) −12.0000 −0.452589
\(704\) 5.00000 0.188445
\(705\) 2.00000 0.0753244
\(706\) 21.0000 0.790345
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 10.0000 0.375293
\(711\) 7.00000 0.262521
\(712\) 13.0000 0.487196
\(713\) 60.0000 2.24702
\(714\) 0 0
\(715\) 5.00000 0.186989
\(716\) 0 0
\(717\) 15.0000 0.560185
\(718\) 20.0000 0.746393
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 10.0000 0.371904
\(724\) 2.00000 0.0743294
\(725\) 16.0000 0.594225
\(726\) 14.0000 0.519589
\(727\) 19.0000 0.704671 0.352335 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.00000 0.0370117
\(731\) −11.0000 −0.406850
\(732\) −14.0000 −0.517455
\(733\) 50.0000 1.84679 0.923396 0.383849i \(-0.125402\pi\)
0.923396 + 0.383849i \(0.125402\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −25.0000 −0.920887
\(738\) 6.00000 0.220863
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) −3.00000 −0.110282
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) −10.0000 −0.366618
\(745\) 5.00000 0.183186
\(746\) 2.00000 0.0732252
\(747\) 9.00000 0.329293
\(748\) 5.00000 0.182818
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 5.00000 0.182210
\(754\) 4.00000 0.145671
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) −36.0000 −1.30758
\(759\) 30.0000 1.08893
\(760\) −4.00000 −0.145095
\(761\) 13.0000 0.471250 0.235625 0.971844i \(-0.424286\pi\)
0.235625 + 0.971844i \(0.424286\pi\)
\(762\) −22.0000 −0.796976
\(763\) 0 0
\(764\) 3.00000 0.108536
\(765\) 1.00000 0.0361551
\(766\) 18.0000 0.650366
\(767\) 4.00000 0.144432
\(768\) −1.00000 −0.0360844
\(769\) −32.0000 −1.15395 −0.576975 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 7.00000 0.252099
\(772\) 14.0000 0.503871
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 11.0000 0.395387
\(775\) 40.0000 1.43684
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −24.0000 −0.859889
\(780\) −1.00000 −0.0358057
\(781\) −50.0000 −1.78914
\(782\) 6.00000 0.214560
\(783\) 4.00000 0.142948
\(784\) 0 0
\(785\) 22.0000 0.785214
\(786\) −6.00000 −0.214013
\(787\) −37.0000 −1.31891 −0.659454 0.751745i \(-0.729212\pi\)
−0.659454 + 0.751745i \(0.729212\pi\)
\(788\) −6.00000 −0.213741
\(789\) 1.00000 0.0356009
\(790\) −7.00000 −0.249049
\(791\) 0 0
\(792\) −5.00000 −0.177667
\(793\) 14.0000 0.497155
\(794\) 8.00000 0.283909
\(795\) 7.00000 0.248264
\(796\) 2.00000 0.0708881
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −2.00000 −0.0707549
\(800\) 4.00000 0.141421
\(801\) −13.0000 −0.459332
\(802\) −30.0000 −1.05934
\(803\) −5.00000 −0.176446
\(804\) 5.00000 0.176336
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) 26.0000 0.915243
\(808\) 14.0000 0.492518
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 25.0000 0.877869 0.438934 0.898519i \(-0.355356\pi\)
0.438934 + 0.898519i \(0.355356\pi\)
\(812\) 0 0
\(813\) −25.0000 −0.876788
\(814\) 15.0000 0.525750
\(815\) −12.0000 −0.420342
\(816\) −1.00000 −0.0350070
\(817\) −44.0000 −1.53937
\(818\) −12.0000 −0.419570
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) −16.0000 −0.558404 −0.279202 0.960232i \(-0.590070\pi\)
−0.279202 + 0.960232i \(0.590070\pi\)
\(822\) −22.0000 −0.767338
\(823\) 7.00000 0.244005 0.122002 0.992530i \(-0.461068\pi\)
0.122002 + 0.992530i \(0.461068\pi\)
\(824\) −7.00000 −0.243857
\(825\) 20.0000 0.696311
\(826\) 0 0
\(827\) 45.0000 1.56480 0.782402 0.622774i \(-0.213994\pi\)
0.782402 + 0.622774i \(0.213994\pi\)
\(828\) −6.00000 −0.208514
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) −9.00000 −0.312395
\(831\) 14.0000 0.485655
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 21.0000 0.727171
\(835\) −19.0000 −0.657522
\(836\) 20.0000 0.691714
\(837\) 10.0000 0.345651
\(838\) −4.00000 −0.138178
\(839\) 51.0000 1.76072 0.880358 0.474310i \(-0.157302\pi\)
0.880358 + 0.474310i \(0.157302\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −24.0000 −0.827095
\(843\) −6.00000 −0.206651
\(844\) −4.00000 −0.137686
\(845\) −12.0000 −0.412813
\(846\) 2.00000 0.0687614
\(847\) 0 0
\(848\) −7.00000 −0.240381
\(849\) −29.0000 −0.995277
\(850\) 4.00000 0.137199
\(851\) 18.0000 0.617032
\(852\) 10.0000 0.342594
\(853\) 16.0000 0.547830 0.273915 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) −16.0000 −0.546869
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 5.00000 0.170697
\(859\) 6.00000 0.204717 0.102359 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) −11.0000 −0.375097
\(861\) 0 0
\(862\) 14.0000 0.476842
\(863\) −9.00000 −0.306364 −0.153182 0.988198i \(-0.548952\pi\)
−0.153182 + 0.988198i \(0.548952\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) 2.00000 0.0679628
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 35.0000 1.18729
\(870\) −4.00000 −0.135613
\(871\) −5.00000 −0.169419
\(872\) 2.00000 0.0677285
\(873\) −13.0000 −0.439983
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 1.00000 0.0337869
\(877\) −33.0000 −1.11433 −0.557165 0.830402i \(-0.688111\pi\)
−0.557165 + 0.830402i \(0.688111\pi\)
\(878\) −6.00000 −0.202490
\(879\) 0 0
\(880\) 5.00000 0.168550
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 1.00000 0.0336336
\(885\) −4.00000 −0.134459
\(886\) 30.0000 1.00787
\(887\) −33.0000 −1.10803 −0.554016 0.832506i \(-0.686905\pi\)
−0.554016 + 0.832506i \(0.686905\pi\)
\(888\) −3.00000 −0.100673
\(889\) 0 0
\(890\) 13.0000 0.435761
\(891\) 5.00000 0.167506
\(892\) −16.0000 −0.535720
\(893\) −8.00000 −0.267710
\(894\) 5.00000 0.167225
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 27.0000 0.901002
\(899\) 40.0000 1.33407
\(900\) −4.00000 −0.133333
\(901\) −7.00000 −0.233204
\(902\) 30.0000 0.998891
\(903\) 0 0
\(904\) −1.00000 −0.0332595
\(905\) 2.00000 0.0664822
\(906\) 10.0000 0.332228
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) −4.00000 −0.132745
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) −4.00000 −0.132453
\(913\) 45.0000 1.48928
\(914\) 13.0000 0.430002
\(915\) −14.0000 −0.462826
\(916\) 19.0000 0.627778
\(917\) 0 0
\(918\) 1.00000 0.0330049
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 6.00000 0.197814
\(921\) −20.0000 −0.659022
\(922\) 6.00000 0.197599
\(923\) −10.0000 −0.329154
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 4.00000 0.131448
\(927\) 7.00000 0.229910
\(928\) 4.00000 0.131306
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) −10.0000 −0.327913
\(931\) 0 0
\(932\) 17.0000 0.556854
\(933\) −17.0000 −0.556555
\(934\) −32.0000 −1.04707
\(935\) 5.00000 0.163517
\(936\) −1.00000 −0.0326860
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) −2.00000 −0.0652328
\(941\) −9.00000 −0.293392 −0.146696 0.989182i \(-0.546864\pi\)
−0.146696 + 0.989182i \(0.546864\pi\)
\(942\) 22.0000 0.716799
\(943\) 36.0000 1.17232
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 55.0000 1.78820
\(947\) 56.0000 1.81976 0.909878 0.414876i \(-0.136175\pi\)
0.909878 + 0.414876i \(0.136175\pi\)
\(948\) −7.00000 −0.227349
\(949\) −1.00000 −0.0324614
\(950\) 16.0000 0.519109
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 7.00000 0.226633
\(955\) 3.00000 0.0970777
\(956\) −15.0000 −0.485135
\(957\) 20.0000 0.646508
\(958\) 9.00000 0.290777
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 69.0000 2.22581
\(962\) 3.00000 0.0967239
\(963\) 16.0000 0.515593
\(964\) −10.0000 −0.322078
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 46.0000 1.47926 0.739630 0.673014i \(-0.235000\pi\)
0.739630 + 0.673014i \(0.235000\pi\)
\(968\) −14.0000 −0.449977
\(969\) −4.00000 −0.128499
\(970\) 13.0000 0.417405
\(971\) 49.0000 1.57248 0.786242 0.617918i \(-0.212024\pi\)
0.786242 + 0.617918i \(0.212024\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −1.00000 −0.0320421
\(975\) 4.00000 0.128103
\(976\) 14.0000 0.448129
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) −12.0000 −0.383718
\(979\) −65.0000 −2.07741
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) 40.0000 1.27580 0.637901 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) −6.00000 −0.191273
\(985\) −6.00000 −0.191176
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 66.0000 2.09868
\(990\) −5.00000 −0.158910
\(991\) 11.0000 0.349427 0.174713 0.984619i \(-0.444100\pi\)
0.174713 + 0.984619i \(0.444100\pi\)
\(992\) 10.0000 0.317500
\(993\) 1.00000 0.0317340
\(994\) 0 0
\(995\) 2.00000 0.0634043
\(996\) −9.00000 −0.285176
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −6.00000 −0.189927
\(999\) 3.00000 0.0949158
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 4998.2.a.i.1.1 1
7.6 odd 2 4998.2.a.r.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4998.2.a.i.1.1 1 1.1 even 1 trivial
4998.2.a.r.1.1 yes 1 7.6 odd 2