Properties

Label 1001.2.a.b.1.1
Level $1001$
Weight $2$
Character 1001.1
Self dual yes
Analytic conductor $7.993$
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] = [1001,2,Mod(1,1001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, 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("1001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1001.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: \(7.99302524233\)
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 \(\chi\) \(=\) 1001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -1.00000 q^{22} -1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} +2.00000 q^{35} +3.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} -6.00000 q^{40} +6.00000 q^{41} +12.0000 q^{43} -1.00000 q^{44} +6.00000 q^{45} +4.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -10.0000 q^{53} -2.00000 q^{55} -3.00000 q^{56} -6.00000 q^{58} +10.0000 q^{61} +4.00000 q^{62} +3.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} +2.00000 q^{68} -2.00000 q^{70} -9.00000 q^{72} +14.0000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -1.00000 q^{77} +2.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{85} -12.0000 q^{86} +3.00000 q^{88} -14.0000 q^{89} -6.00000 q^{90} +1.00000 q^{91} -4.00000 q^{94} +8.00000 q^{95} -14.0000 q^{97} -1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) −3.00000 −1.00000
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 3.00000 0.707107
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 2.00000 0.338062
\(36\) 3.00000 0.500000
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −6.00000 −0.948683
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −1.00000 −0.150756
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) 3.00000 0.377964
\(64\) 7.00000 0.875000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −9.00000 −1.06066
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 9.00000 1.00000
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −6.00000 −0.632456
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −1.00000 −0.101015
\(99\) −3.00000 −0.301511
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 3.00000 0.277350
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 12.0000 1.07331
\(126\) −3.00000 −0.267261
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 3.00000 0.250000
\(145\) −12.0000 −0.996546
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −12.0000 −0.973329
\(153\) 6.00000 0.485071
\(154\) 1.00000 0.0805823
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) 12.0000 0.917663
\(172\) −12.0000 −0.914991
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −6.00000 −0.447214
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 3.00000 0.213201
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 5.00000 0.334077
\(225\) 3.00000 0.200000
\(226\) −18.0000 −1.19734
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −3.00000 −0.196116
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −12.0000 −0.762001
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) −2.00000 −0.124035
\(261\) −18.0000 −1.11417
\(262\) 12.0000 0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 20.0000 1.22859
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 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
\(274\) −10.0000 −0.604122
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −4.00000 −0.239904
\(279\) 12.0000 0.718421
\(280\) 6.00000 0.358569
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 1.00000 0.0591312
\(287\) −6.00000 −0.354169
\(288\) 15.0000 0.883883
\(289\) −13.0000 −0.764706
\(290\) 12.0000 0.704664
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −20.0000 −1.14520
\(306\) −6.00000 −0.342997
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −14.0000 −0.790066
\(315\) −6.00000 −0.338062
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −14.0000 −0.782624
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) −9.00000 −0.500000
\(325\) 1.00000 0.0554700
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) 18.0000 0.993884
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 4.00000 0.219529
\(333\) −18.0000 −0.986394
\(334\) 8.00000 0.437741
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) −4.00000 −0.216612
\(342\) −12.0000 −0.648886
\(343\) −1.00000 −0.0539949
\(344\) 36.0000 1.94099
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 18.0000 0.948683
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) −28.0000 −1.46559
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 12.0000 0.623850
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 0 0
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 22.0000 1.11977
\(387\) −36.0000 −1.82998
\(388\) 14.0000 0.710742
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −10.0000 −0.497519
\(405\) −18.0000 −0.894427
\(406\) 6.00000 0.297775
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 32.0000 1.56330 0.781651 0.623716i \(-0.214378\pi\)
0.781651 + 0.623716i \(0.214378\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −28.0000 −1.36302
\(423\) −12.0000 −0.583460
\(424\) −30.0000 −1.45693
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −6.00000 −0.286039
\(441\) −3.00000 −0.142857
\(442\) −2.00000 −0.0951303
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 28.0000 1.32733
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −7.00000 −0.330719
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −3.00000 −0.141421
\(451\) 6.00000 0.282529
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) 28.0000 1.31411
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −3.00000 −0.138675
\(469\) −12.0000 −0.554109
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −2.00000 −0.0916698
\(477\) 30.0000 1.37361
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 28.0000 1.27141
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 30.0000 1.35804
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) −4.00000 −0.179969
\(495\) 6.00000 0.269680
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 9.00000 0.400892
\(505\) −20.0000 −0.889988
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 18.0000 0.787839
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −20.0000 −0.868744
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 36.0000 1.55496
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −10.0000 −0.427179
\(549\) −30.0000 −1.28037
\(550\) 1.00000 0.0426401
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) −12.0000 −0.508001
\(559\) −12.0000 −0.507546
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) −36.0000 −1.51453
\(566\) −28.0000 −1.17693
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 1.00000 0.0418121
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) −21.0000 −0.875000
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 12.0000 0.498273
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 42.0000 1.73797
\(585\) −6.00000 −0.248069
\(586\) 30.0000 1.23929
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 2.00000 0.0819232
\(597\) 0 0
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 12.0000 0.489083
\(603\) −36.0000 −1.46603
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) 20.0000 0.809776
\(611\) −4.00000 −0.161823
\(612\) −6.00000 −0.242536
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −20.0000 −0.801927
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) −12.0000 −0.478471
\(630\) 6.00000 0.239046
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) −6.00000 −0.237171
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 27.0000 1.06066
\(649\) 0 0
\(650\) −1.00000 −0.0392232
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) −6.00000 −0.234261
\(657\) −42.0000 −1.63858
\(658\) 4.00000 0.155936
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) −8.00000 −0.310227
\(666\) 18.0000 0.697486
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 24.0000 0.927201
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 12.0000 0.460179
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −12.0000 −0.458831
\(685\) −20.0000 −0.764161
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 3.00000 0.113961
\(694\) 4.00000 0.151838
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −42.0000 −1.57402
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −6.00000 −0.223607
\(721\) −4.00000 −0.148968
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 3.00000 0.111187
\(729\) −27.0000 −1.00000
\(730\) 28.0000 1.03633
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 18.0000 0.662589
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 4.00000 0.146549
\(746\) −22.0000 −0.805477
\(747\) 12.0000 0.439057
\(748\) 2.00000 0.0731272
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) −28.0000 −1.01168
\(767\) 0 0
\(768\) 0 0
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 0 0
\(772\) 22.0000 0.791797
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 36.0000 1.29399
\(775\) 4.00000 0.143684
\(776\) −42.0000 −1.50771
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) −28.0000 −0.999363
\(786\) 0 0
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) −9.00000 −0.319801
\(793\) −10.0000 −0.355110
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 5.00000 0.176777
\(801\) 42.0000 1.48400
\(802\) −34.0000 −1.20058
\(803\) 14.0000 0.494049
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 30.0000 1.05540
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 18.0000 0.632456
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −48.0000 −1.67931
\(818\) 2.00000 0.0699284
\(819\) −3.00000 −0.104828
\(820\) 12.0000 0.419058
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) −7.00000 −0.242681
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) −32.0000 −1.10542
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) −28.0000 −0.963800
\(845\) −2.00000 −0.0688021
\(846\) 12.0000 0.412568
\(847\) −1.00000 −0.0343604
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 10.0000 0.342193
\(855\) −24.0000 −0.820783
\(856\) −36.0000 −1.23045
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) −6.00000 −0.203186
\(873\) 42.0000 1.42148
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 3.00000 0.101015
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) −28.0000 −0.938562
\(891\) 9.00000 0.301511
\(892\) −4.00000 −0.133930
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −24.0000 −0.800445
\(900\) −3.00000 −0.100000
\(901\) 20.0000 0.666297
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) 54.0000 1.79601
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 28.0000 0.929213
\(909\) −30.0000 −0.995037
\(910\) 2.00000 0.0662994
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −4.00000 −0.132381
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 22.0000 0.724531
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −24.0000 −0.788689
\(927\) −12.0000 −0.394132
\(928\) −30.0000 −0.984798
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) 4.00000 0.130814
\(936\) 9.00000 0.294174
\(937\) 46.0000 1.50275 0.751377 0.659873i \(-0.229390\pi\)
0.751377 + 0.659873i \(0.229390\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) −14.0000 −0.454459
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) −30.0000 −0.971286
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) −10.0000 −0.322917
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 6.00000 0.193448
\(963\) 36.0000 1.16008
\(964\) 18.0000 0.579741
\(965\) 44.0000 1.41641
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 3.00000 0.0964237
\(969\) 0 0
\(970\) −28.0000 −0.899026
\(971\) 32.0000 1.02693 0.513464 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) −14.0000 −0.447442
\(980\) 2.00000 0.0638877
\(981\) 6.00000 0.191565
\(982\) −28.0000 −0.893516
\(983\) 28.0000 0.893061 0.446531 0.894768i \(-0.352659\pi\)
0.446531 + 0.894768i \(0.352659\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) −6.00000 −0.190693
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 20.0000 0.635001
\(993\) 0 0
\(994\) 0 0
\(995\) −40.0000 −1.26809
\(996\) 0 0
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 4.00000 0.126618
\(999\) 0 0
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 1001.2.a.b.1.1 1
3.2 odd 2 9009.2.a.k.1.1 1
7.6 odd 2 7007.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1001.2.a.b.1.1 1 1.1 even 1 trivial
7007.2.a.b.1.1 1 7.6 odd 2
9009.2.a.k.1.1 1 3.2 odd 2