Properties

Label 3120.2.a.bf.1.1
Level $3120$
Weight $2$
Character 3120.1
Self dual yes
Analytic conductor $24.913$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3120,2,Mod(1,3120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3120.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.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: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 3120.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.70156 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.70156 q^{7} +1.00000 q^{9} -0.701562 q^{11} -1.00000 q^{13} -1.00000 q^{15} -0.701562 q^{17} +2.00000 q^{19} -2.70156 q^{21} -6.70156 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.40312 q^{29} +9.40312 q^{31} -0.701562 q^{33} +2.70156 q^{35} -6.70156 q^{37} -1.00000 q^{39} +10.7016 q^{41} -4.00000 q^{43} -1.00000 q^{45} +1.40312 q^{47} +0.298438 q^{49} -0.701562 q^{51} +10.7016 q^{53} +0.701562 q^{55} +2.00000 q^{57} +14.8062 q^{59} +2.70156 q^{61} -2.70156 q^{63} +1.00000 q^{65} +4.00000 q^{67} -6.70156 q^{69} -15.5078 q^{71} +13.4031 q^{73} +1.00000 q^{75} +1.89531 q^{77} -4.70156 q^{79} +1.00000 q^{81} -4.00000 q^{83} +0.701562 q^{85} +9.40312 q^{87} +8.10469 q^{89} +2.70156 q^{91} +9.40312 q^{93} -2.00000 q^{95} +18.1047 q^{97} -0.701562 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} + 5 q^{11} - 2 q^{13} - 2 q^{15} + 5 q^{17} + 4 q^{19} + q^{21} - 7 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 6 q^{31} + 5 q^{33} - q^{35} - 7 q^{37} - 2 q^{39} + 15 q^{41} - 8 q^{43} - 2 q^{45} - 10 q^{47} + 7 q^{49} + 5 q^{51} + 15 q^{53} - 5 q^{55} + 4 q^{57} + 4 q^{59} - q^{61} + q^{63} + 2 q^{65} + 8 q^{67} - 7 q^{69} + q^{71} + 14 q^{73} + 2 q^{75} + 23 q^{77} - 3 q^{79} + 2 q^{81} - 8 q^{83} - 5 q^{85} + 6 q^{87} - 3 q^{89} - q^{91} + 6 q^{93} - 4 q^{95} + 17 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.70156 −1.02109 −0.510547 0.859850i \(-0.670557\pi\)
−0.510547 + 0.859850i \(0.670557\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.701562 −0.211529 −0.105764 0.994391i \(-0.533729\pi\)
−0.105764 + 0.994391i \(0.533729\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −0.701562 −0.170154 −0.0850769 0.996374i \(-0.527114\pi\)
−0.0850769 + 0.996374i \(0.527114\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −2.70156 −0.589529
\(22\) 0 0
\(23\) −6.70156 −1.39737 −0.698686 0.715428i \(-0.746232\pi\)
−0.698686 + 0.715428i \(0.746232\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.40312 1.74612 0.873058 0.487616i \(-0.162133\pi\)
0.873058 + 0.487616i \(0.162133\pi\)
\(30\) 0 0
\(31\) 9.40312 1.68885 0.844425 0.535673i \(-0.179942\pi\)
0.844425 + 0.535673i \(0.179942\pi\)
\(32\) 0 0
\(33\) −0.701562 −0.122126
\(34\) 0 0
\(35\) 2.70156 0.456647
\(36\) 0 0
\(37\) −6.70156 −1.10173 −0.550865 0.834594i \(-0.685702\pi\)
−0.550865 + 0.834594i \(0.685702\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 10.7016 1.67130 0.835652 0.549260i \(-0.185090\pi\)
0.835652 + 0.549260i \(0.185090\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 1.40312 0.204667 0.102333 0.994750i \(-0.467369\pi\)
0.102333 + 0.994750i \(0.467369\pi\)
\(48\) 0 0
\(49\) 0.298438 0.0426340
\(50\) 0 0
\(51\) −0.701562 −0.0982383
\(52\) 0 0
\(53\) 10.7016 1.46997 0.734986 0.678082i \(-0.237189\pi\)
0.734986 + 0.678082i \(0.237189\pi\)
\(54\) 0 0
\(55\) 0.701562 0.0945986
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 14.8062 1.92761 0.963805 0.266609i \(-0.0859033\pi\)
0.963805 + 0.266609i \(0.0859033\pi\)
\(60\) 0 0
\(61\) 2.70156 0.345900 0.172950 0.984931i \(-0.444670\pi\)
0.172950 + 0.984931i \(0.444670\pi\)
\(62\) 0 0
\(63\) −2.70156 −0.340365
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −6.70156 −0.806773
\(70\) 0 0
\(71\) −15.5078 −1.84044 −0.920219 0.391403i \(-0.871990\pi\)
−0.920219 + 0.391403i \(0.871990\pi\)
\(72\) 0 0
\(73\) 13.4031 1.56872 0.784359 0.620308i \(-0.212992\pi\)
0.784359 + 0.620308i \(0.212992\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 1.89531 0.215991
\(78\) 0 0
\(79\) −4.70156 −0.528967 −0.264484 0.964390i \(-0.585201\pi\)
−0.264484 + 0.964390i \(0.585201\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0.701562 0.0760951
\(86\) 0 0
\(87\) 9.40312 1.00812
\(88\) 0 0
\(89\) 8.10469 0.859095 0.429548 0.903044i \(-0.358673\pi\)
0.429548 + 0.903044i \(0.358673\pi\)
\(90\) 0 0
\(91\) 2.70156 0.283201
\(92\) 0 0
\(93\) 9.40312 0.975059
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 18.1047 1.83825 0.919126 0.393963i \(-0.128896\pi\)
0.919126 + 0.393963i \(0.128896\pi\)
\(98\) 0 0
\(99\) −0.701562 −0.0705096
\(100\) 0 0
\(101\) −6.80625 −0.677247 −0.338624 0.940922i \(-0.609961\pi\)
−0.338624 + 0.940922i \(0.609961\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 2.70156 0.263645
\(106\) 0 0
\(107\) −18.1047 −1.75025 −0.875123 0.483900i \(-0.839220\pi\)
−0.875123 + 0.483900i \(0.839220\pi\)
\(108\) 0 0
\(109\) 10.8062 1.03505 0.517525 0.855668i \(-0.326853\pi\)
0.517525 + 0.855668i \(0.326853\pi\)
\(110\) 0 0
\(111\) −6.70156 −0.636084
\(112\) 0 0
\(113\) 6.59688 0.620582 0.310291 0.950642i \(-0.399574\pi\)
0.310291 + 0.950642i \(0.399574\pi\)
\(114\) 0 0
\(115\) 6.70156 0.624924
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 1.89531 0.173743
\(120\) 0 0
\(121\) −10.5078 −0.955256
\(122\) 0 0
\(123\) 10.7016 0.964927
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 13.4031 1.18933 0.594667 0.803972i \(-0.297284\pi\)
0.594667 + 0.803972i \(0.297284\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.8062 1.11889 0.559444 0.828868i \(-0.311015\pi\)
0.559444 + 0.828868i \(0.311015\pi\)
\(132\) 0 0
\(133\) −5.40312 −0.468510
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 4.70156 0.398781 0.199391 0.979920i \(-0.436104\pi\)
0.199391 + 0.979920i \(0.436104\pi\)
\(140\) 0 0
\(141\) 1.40312 0.118164
\(142\) 0 0
\(143\) 0.701562 0.0586676
\(144\) 0 0
\(145\) −9.40312 −0.780887
\(146\) 0 0
\(147\) 0.298438 0.0246147
\(148\) 0 0
\(149\) 1.29844 0.106372 0.0531861 0.998585i \(-0.483062\pi\)
0.0531861 + 0.998585i \(0.483062\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −0.701562 −0.0567179
\(154\) 0 0
\(155\) −9.40312 −0.755277
\(156\) 0 0
\(157\) 8.80625 0.702815 0.351408 0.936223i \(-0.385703\pi\)
0.351408 + 0.936223i \(0.385703\pi\)
\(158\) 0 0
\(159\) 10.7016 0.848689
\(160\) 0 0
\(161\) 18.1047 1.42685
\(162\) 0 0
\(163\) −11.2984 −0.884962 −0.442481 0.896778i \(-0.645902\pi\)
−0.442481 + 0.896778i \(0.645902\pi\)
\(164\) 0 0
\(165\) 0.701562 0.0546165
\(166\) 0 0
\(167\) 22.8062 1.76480 0.882400 0.470500i \(-0.155926\pi\)
0.882400 + 0.470500i \(0.155926\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) −2.70156 −0.204219
\(176\) 0 0
\(177\) 14.8062 1.11291
\(178\) 0 0
\(179\) 22.0000 1.64436 0.822179 0.569230i \(-0.192758\pi\)
0.822179 + 0.569230i \(0.192758\pi\)
\(180\) 0 0
\(181\) −1.29844 −0.0965121 −0.0482561 0.998835i \(-0.515366\pi\)
−0.0482561 + 0.998835i \(0.515366\pi\)
\(182\) 0 0
\(183\) 2.70156 0.199705
\(184\) 0 0
\(185\) 6.70156 0.492709
\(186\) 0 0
\(187\) 0.492189 0.0359925
\(188\) 0 0
\(189\) −2.70156 −0.196510
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −22.1047 −1.59113 −0.795565 0.605868i \(-0.792826\pi\)
−0.795565 + 0.605868i \(0.792826\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −14.2094 −1.01238 −0.506188 0.862423i \(-0.668946\pi\)
−0.506188 + 0.862423i \(0.668946\pi\)
\(198\) 0 0
\(199\) −2.80625 −0.198930 −0.0994648 0.995041i \(-0.531713\pi\)
−0.0994648 + 0.995041i \(0.531713\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −25.4031 −1.78295
\(204\) 0 0
\(205\) −10.7016 −0.747430
\(206\) 0 0
\(207\) −6.70156 −0.465791
\(208\) 0 0
\(209\) −1.40312 −0.0970561
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) −15.5078 −1.06258
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −25.4031 −1.72448
\(218\) 0 0
\(219\) 13.4031 0.905699
\(220\) 0 0
\(221\) 0.701562 0.0471922
\(222\) 0 0
\(223\) −3.40312 −0.227890 −0.113945 0.993487i \(-0.536349\pi\)
−0.113945 + 0.993487i \(0.536349\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 10.5969 0.703339 0.351670 0.936124i \(-0.385614\pi\)
0.351670 + 0.936124i \(0.385614\pi\)
\(228\) 0 0
\(229\) −25.4031 −1.67869 −0.839343 0.543602i \(-0.817060\pi\)
−0.839343 + 0.543602i \(0.817060\pi\)
\(230\) 0 0
\(231\) 1.89531 0.124702
\(232\) 0 0
\(233\) −10.1047 −0.661980 −0.330990 0.943634i \(-0.607383\pi\)
−0.330990 + 0.943634i \(0.607383\pi\)
\(234\) 0 0
\(235\) −1.40312 −0.0915297
\(236\) 0 0
\(237\) −4.70156 −0.305399
\(238\) 0 0
\(239\) 6.10469 0.394879 0.197440 0.980315i \(-0.436737\pi\)
0.197440 + 0.980315i \(0.436737\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.298438 −0.0190665
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 18.2094 1.14937 0.574683 0.818376i \(-0.305125\pi\)
0.574683 + 0.818376i \(0.305125\pi\)
\(252\) 0 0
\(253\) 4.70156 0.295585
\(254\) 0 0
\(255\) 0.701562 0.0439335
\(256\) 0 0
\(257\) 20.2094 1.26063 0.630313 0.776341i \(-0.282927\pi\)
0.630313 + 0.776341i \(0.282927\pi\)
\(258\) 0 0
\(259\) 18.1047 1.12497
\(260\) 0 0
\(261\) 9.40312 0.582039
\(262\) 0 0
\(263\) −4.59688 −0.283456 −0.141728 0.989906i \(-0.545266\pi\)
−0.141728 + 0.989906i \(0.545266\pi\)
\(264\) 0 0
\(265\) −10.7016 −0.657392
\(266\) 0 0
\(267\) 8.10469 0.495999
\(268\) 0 0
\(269\) −14.8062 −0.902753 −0.451376 0.892334i \(-0.649067\pi\)
−0.451376 + 0.892334i \(0.649067\pi\)
\(270\) 0 0
\(271\) 6.59688 0.400732 0.200366 0.979721i \(-0.435787\pi\)
0.200366 + 0.979721i \(0.435787\pi\)
\(272\) 0 0
\(273\) 2.70156 0.163506
\(274\) 0 0
\(275\) −0.701562 −0.0423058
\(276\) 0 0
\(277\) 8.80625 0.529116 0.264558 0.964370i \(-0.414774\pi\)
0.264558 + 0.964370i \(0.414774\pi\)
\(278\) 0 0
\(279\) 9.40312 0.562950
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −28.9109 −1.70656
\(288\) 0 0
\(289\) −16.5078 −0.971048
\(290\) 0 0
\(291\) 18.1047 1.06132
\(292\) 0 0
\(293\) 15.4031 0.899860 0.449930 0.893064i \(-0.351449\pi\)
0.449930 + 0.893064i \(0.351449\pi\)
\(294\) 0 0
\(295\) −14.8062 −0.862053
\(296\) 0 0
\(297\) −0.701562 −0.0407088
\(298\) 0 0
\(299\) 6.70156 0.387561
\(300\) 0 0
\(301\) 10.8062 0.622862
\(302\) 0 0
\(303\) −6.80625 −0.391009
\(304\) 0 0
\(305\) −2.70156 −0.154691
\(306\) 0 0
\(307\) 13.8953 0.793047 0.396524 0.918024i \(-0.370216\pi\)
0.396524 + 0.918024i \(0.370216\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 16.8062 0.949945 0.474973 0.880001i \(-0.342458\pi\)
0.474973 + 0.880001i \(0.342458\pi\)
\(314\) 0 0
\(315\) 2.70156 0.152216
\(316\) 0 0
\(317\) 14.2094 0.798078 0.399039 0.916934i \(-0.369344\pi\)
0.399039 + 0.916934i \(0.369344\pi\)
\(318\) 0 0
\(319\) −6.59688 −0.369354
\(320\) 0 0
\(321\) −18.1047 −1.01051
\(322\) 0 0
\(323\) −1.40312 −0.0780719
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 10.8062 0.597587
\(328\) 0 0
\(329\) −3.79063 −0.208984
\(330\) 0 0
\(331\) 14.2094 0.781018 0.390509 0.920599i \(-0.372299\pi\)
0.390509 + 0.920599i \(0.372299\pi\)
\(332\) 0 0
\(333\) −6.70156 −0.367243
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −6.20937 −0.338246 −0.169123 0.985595i \(-0.554094\pi\)
−0.169123 + 0.985595i \(0.554094\pi\)
\(338\) 0 0
\(339\) 6.59688 0.358293
\(340\) 0 0
\(341\) −6.59688 −0.357241
\(342\) 0 0
\(343\) 18.1047 0.977561
\(344\) 0 0
\(345\) 6.70156 0.360800
\(346\) 0 0
\(347\) −6.10469 −0.327717 −0.163858 0.986484i \(-0.552394\pi\)
−0.163858 + 0.986484i \(0.552394\pi\)
\(348\) 0 0
\(349\) −34.8062 −1.86314 −0.931568 0.363567i \(-0.881559\pi\)
−0.931568 + 0.363567i \(0.881559\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 12.8062 0.681608 0.340804 0.940134i \(-0.389301\pi\)
0.340804 + 0.940134i \(0.389301\pi\)
\(354\) 0 0
\(355\) 15.5078 0.823069
\(356\) 0 0
\(357\) 1.89531 0.100311
\(358\) 0 0
\(359\) 21.6125 1.14066 0.570332 0.821414i \(-0.306815\pi\)
0.570332 + 0.821414i \(0.306815\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −10.5078 −0.551517
\(364\) 0 0
\(365\) −13.4031 −0.701552
\(366\) 0 0
\(367\) −1.19375 −0.0623133 −0.0311567 0.999515i \(-0.509919\pi\)
−0.0311567 + 0.999515i \(0.509919\pi\)
\(368\) 0 0
\(369\) 10.7016 0.557101
\(370\) 0 0
\(371\) −28.9109 −1.50098
\(372\) 0 0
\(373\) −8.59688 −0.445129 −0.222565 0.974918i \(-0.571443\pi\)
−0.222565 + 0.974918i \(0.571443\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −9.40312 −0.484286
\(378\) 0 0
\(379\) −15.6125 −0.801960 −0.400980 0.916087i \(-0.631330\pi\)
−0.400980 + 0.916087i \(0.631330\pi\)
\(380\) 0 0
\(381\) 13.4031 0.686663
\(382\) 0 0
\(383\) 5.40312 0.276087 0.138043 0.990426i \(-0.455919\pi\)
0.138043 + 0.990426i \(0.455919\pi\)
\(384\) 0 0
\(385\) −1.89531 −0.0965941
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 25.6125 1.29861 0.649303 0.760530i \(-0.275061\pi\)
0.649303 + 0.760530i \(0.275061\pi\)
\(390\) 0 0
\(391\) 4.70156 0.237768
\(392\) 0 0
\(393\) 12.8062 0.645990
\(394\) 0 0
\(395\) 4.70156 0.236561
\(396\) 0 0
\(397\) −22.7016 −1.13936 −0.569679 0.821867i \(-0.692933\pi\)
−0.569679 + 0.821867i \(0.692933\pi\)
\(398\) 0 0
\(399\) −5.40312 −0.270495
\(400\) 0 0
\(401\) 0.806248 0.0402621 0.0201311 0.999797i \(-0.493592\pi\)
0.0201311 + 0.999797i \(0.493592\pi\)
\(402\) 0 0
\(403\) −9.40312 −0.468403
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 4.70156 0.233048
\(408\) 0 0
\(409\) −4.80625 −0.237654 −0.118827 0.992915i \(-0.537913\pi\)
−0.118827 + 0.992915i \(0.537913\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) −40.0000 −1.96827
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 4.70156 0.230236
\(418\) 0 0
\(419\) −27.6125 −1.34896 −0.674479 0.738294i \(-0.735632\pi\)
−0.674479 + 0.738294i \(0.735632\pi\)
\(420\) 0 0
\(421\) −25.6125 −1.24828 −0.624138 0.781314i \(-0.714550\pi\)
−0.624138 + 0.781314i \(0.714550\pi\)
\(422\) 0 0
\(423\) 1.40312 0.0682222
\(424\) 0 0
\(425\) −0.701562 −0.0340308
\(426\) 0 0
\(427\) −7.29844 −0.353196
\(428\) 0 0
\(429\) 0.701562 0.0338717
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 12.5969 0.605367 0.302684 0.953091i \(-0.402117\pi\)
0.302684 + 0.953091i \(0.402117\pi\)
\(434\) 0 0
\(435\) −9.40312 −0.450845
\(436\) 0 0
\(437\) −13.4031 −0.641158
\(438\) 0 0
\(439\) 1.89531 0.0904584 0.0452292 0.998977i \(-0.485598\pi\)
0.0452292 + 0.998977i \(0.485598\pi\)
\(440\) 0 0
\(441\) 0.298438 0.0142113
\(442\) 0 0
\(443\) 26.3141 1.25022 0.625109 0.780537i \(-0.285054\pi\)
0.625109 + 0.780537i \(0.285054\pi\)
\(444\) 0 0
\(445\) −8.10469 −0.384199
\(446\) 0 0
\(447\) 1.29844 0.0614140
\(448\) 0 0
\(449\) 3.89531 0.183831 0.0919156 0.995767i \(-0.470701\pi\)
0.0919156 + 0.995767i \(0.470701\pi\)
\(450\) 0 0
\(451\) −7.50781 −0.353529
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.70156 −0.126651
\(456\) 0 0
\(457\) 7.29844 0.341407 0.170703 0.985322i \(-0.445396\pi\)
0.170703 + 0.985322i \(0.445396\pi\)
\(458\) 0 0
\(459\) −0.701562 −0.0327461
\(460\) 0 0
\(461\) 19.8953 0.926617 0.463309 0.886197i \(-0.346662\pi\)
0.463309 + 0.886197i \(0.346662\pi\)
\(462\) 0 0
\(463\) 4.10469 0.190761 0.0953805 0.995441i \(-0.469593\pi\)
0.0953805 + 0.995441i \(0.469593\pi\)
\(464\) 0 0
\(465\) −9.40312 −0.436059
\(466\) 0 0
\(467\) −18.1047 −0.837785 −0.418892 0.908036i \(-0.637582\pi\)
−0.418892 + 0.908036i \(0.637582\pi\)
\(468\) 0 0
\(469\) −10.8062 −0.498986
\(470\) 0 0
\(471\) 8.80625 0.405771
\(472\) 0 0
\(473\) 2.80625 0.129031
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 10.7016 0.489991
\(478\) 0 0
\(479\) 28.7016 1.31141 0.655704 0.755018i \(-0.272372\pi\)
0.655704 + 0.755018i \(0.272372\pi\)
\(480\) 0 0
\(481\) 6.70156 0.305565
\(482\) 0 0
\(483\) 18.1047 0.823792
\(484\) 0 0
\(485\) −18.1047 −0.822091
\(486\) 0 0
\(487\) 36.3141 1.64555 0.822774 0.568369i \(-0.192426\pi\)
0.822774 + 0.568369i \(0.192426\pi\)
\(488\) 0 0
\(489\) −11.2984 −0.510933
\(490\) 0 0
\(491\) 6.20937 0.280225 0.140113 0.990136i \(-0.455254\pi\)
0.140113 + 0.990136i \(0.455254\pi\)
\(492\) 0 0
\(493\) −6.59688 −0.297108
\(494\) 0 0
\(495\) 0.701562 0.0315329
\(496\) 0 0
\(497\) 41.8953 1.87926
\(498\) 0 0
\(499\) −15.1938 −0.680166 −0.340083 0.940395i \(-0.610455\pi\)
−0.340083 + 0.940395i \(0.610455\pi\)
\(500\) 0 0
\(501\) 22.8062 1.01891
\(502\) 0 0
\(503\) −31.4031 −1.40020 −0.700098 0.714047i \(-0.746860\pi\)
−0.700098 + 0.714047i \(0.746860\pi\)
\(504\) 0 0
\(505\) 6.80625 0.302874
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −32.3141 −1.43230 −0.716148 0.697949i \(-0.754096\pi\)
−0.716148 + 0.697949i \(0.754096\pi\)
\(510\) 0 0
\(511\) −36.2094 −1.60181
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) 12.0000 0.528783
\(516\) 0 0
\(517\) −0.984379 −0.0432929
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) −2.70156 −0.117906
\(526\) 0 0
\(527\) −6.59688 −0.287364
\(528\) 0 0
\(529\) 21.9109 0.952649
\(530\) 0 0
\(531\) 14.8062 0.642536
\(532\) 0 0
\(533\) −10.7016 −0.463536
\(534\) 0 0
\(535\) 18.1047 0.782734
\(536\) 0 0
\(537\) 22.0000 0.949370
\(538\) 0 0
\(539\) −0.209373 −0.00901832
\(540\) 0 0
\(541\) 40.2094 1.72874 0.864368 0.502860i \(-0.167719\pi\)
0.864368 + 0.502860i \(0.167719\pi\)
\(542\) 0 0
\(543\) −1.29844 −0.0557213
\(544\) 0 0
\(545\) −10.8062 −0.462889
\(546\) 0 0
\(547\) 25.6125 1.09511 0.547556 0.836769i \(-0.315558\pi\)
0.547556 + 0.836769i \(0.315558\pi\)
\(548\) 0 0
\(549\) 2.70156 0.115300
\(550\) 0 0
\(551\) 18.8062 0.801173
\(552\) 0 0
\(553\) 12.7016 0.540125
\(554\) 0 0
\(555\) 6.70156 0.284465
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0.492189 0.0207803
\(562\) 0 0
\(563\) −22.1047 −0.931601 −0.465801 0.884890i \(-0.654234\pi\)
−0.465801 + 0.884890i \(0.654234\pi\)
\(564\) 0 0
\(565\) −6.59688 −0.277533
\(566\) 0 0
\(567\) −2.70156 −0.113455
\(568\) 0 0
\(569\) −27.4031 −1.14880 −0.574399 0.818575i \(-0.694764\pi\)
−0.574399 + 0.818575i \(0.694764\pi\)
\(570\) 0 0
\(571\) −26.3141 −1.10121 −0.550605 0.834766i \(-0.685603\pi\)
−0.550605 + 0.834766i \(0.685603\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.70156 −0.279474
\(576\) 0 0
\(577\) −31.5078 −1.31169 −0.655844 0.754897i \(-0.727687\pi\)
−0.655844 + 0.754897i \(0.727687\pi\)
\(578\) 0 0
\(579\) −22.1047 −0.918639
\(580\) 0 0
\(581\) 10.8062 0.448319
\(582\) 0 0
\(583\) −7.50781 −0.310942
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) 24.2094 0.999228 0.499614 0.866248i \(-0.333475\pi\)
0.499614 + 0.866248i \(0.333475\pi\)
\(588\) 0 0
\(589\) 18.8062 0.774898
\(590\) 0 0
\(591\) −14.2094 −0.584495
\(592\) 0 0
\(593\) −3.40312 −0.139750 −0.0698748 0.997556i \(-0.522260\pi\)
−0.0698748 + 0.997556i \(0.522260\pi\)
\(594\) 0 0
\(595\) −1.89531 −0.0777003
\(596\) 0 0
\(597\) −2.80625 −0.114852
\(598\) 0 0
\(599\) 42.8062 1.74902 0.874508 0.485011i \(-0.161184\pi\)
0.874508 + 0.485011i \(0.161184\pi\)
\(600\) 0 0
\(601\) 9.29844 0.379291 0.189646 0.981853i \(-0.439266\pi\)
0.189646 + 0.981853i \(0.439266\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 10.5078 0.427203
\(606\) 0 0
\(607\) 41.6125 1.68900 0.844500 0.535556i \(-0.179898\pi\)
0.844500 + 0.535556i \(0.179898\pi\)
\(608\) 0 0
\(609\) −25.4031 −1.02939
\(610\) 0 0
\(611\) −1.40312 −0.0567643
\(612\) 0 0
\(613\) −45.7172 −1.84650 −0.923250 0.384200i \(-0.874477\pi\)
−0.923250 + 0.384200i \(0.874477\pi\)
\(614\) 0 0
\(615\) −10.7016 −0.431529
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) −39.6125 −1.59216 −0.796080 0.605191i \(-0.793097\pi\)
−0.796080 + 0.605191i \(0.793097\pi\)
\(620\) 0 0
\(621\) −6.70156 −0.268924
\(622\) 0 0
\(623\) −21.8953 −0.877217
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.40312 −0.0560354
\(628\) 0 0
\(629\) 4.70156 0.187464
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) −13.4031 −0.531887
\(636\) 0 0
\(637\) −0.298438 −0.0118245
\(638\) 0 0
\(639\) −15.5078 −0.613480
\(640\) 0 0
\(641\) −10.2094 −0.403246 −0.201623 0.979463i \(-0.564622\pi\)
−0.201623 + 0.979463i \(0.564622\pi\)
\(642\) 0 0
\(643\) −31.2984 −1.23429 −0.617145 0.786849i \(-0.711711\pi\)
−0.617145 + 0.786849i \(0.711711\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) −20.1047 −0.790397 −0.395198 0.918596i \(-0.629324\pi\)
−0.395198 + 0.918596i \(0.629324\pi\)
\(648\) 0 0
\(649\) −10.3875 −0.407745
\(650\) 0 0
\(651\) −25.4031 −0.995627
\(652\) 0 0
\(653\) 15.6125 0.610964 0.305482 0.952198i \(-0.401182\pi\)
0.305482 + 0.952198i \(0.401182\pi\)
\(654\) 0 0
\(655\) −12.8062 −0.500382
\(656\) 0 0
\(657\) 13.4031 0.522906
\(658\) 0 0
\(659\) 27.4031 1.06747 0.533737 0.845650i \(-0.320787\pi\)
0.533737 + 0.845650i \(0.320787\pi\)
\(660\) 0 0
\(661\) −12.0000 −0.466746 −0.233373 0.972387i \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 0 0
\(663\) 0.701562 0.0272464
\(664\) 0 0
\(665\) 5.40312 0.209524
\(666\) 0 0
\(667\) −63.0156 −2.43997
\(668\) 0 0
\(669\) −3.40312 −0.131572
\(670\) 0 0
\(671\) −1.89531 −0.0731678
\(672\) 0 0
\(673\) −3.19375 −0.123110 −0.0615550 0.998104i \(-0.519606\pi\)
−0.0615550 + 0.998104i \(0.519606\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −4.10469 −0.157756 −0.0788780 0.996884i \(-0.525134\pi\)
−0.0788780 + 0.996884i \(0.525134\pi\)
\(678\) 0 0
\(679\) −48.9109 −1.87703
\(680\) 0 0
\(681\) 10.5969 0.406073
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −25.4031 −0.969190
\(688\) 0 0
\(689\) −10.7016 −0.407697
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 0 0
\(693\) 1.89531 0.0719970
\(694\) 0 0
\(695\) −4.70156 −0.178340
\(696\) 0 0
\(697\) −7.50781 −0.284379
\(698\) 0 0
\(699\) −10.1047 −0.382194
\(700\) 0 0
\(701\) −25.6125 −0.967371 −0.483685 0.875242i \(-0.660702\pi\)
−0.483685 + 0.875242i \(0.660702\pi\)
\(702\) 0 0
\(703\) −13.4031 −0.505508
\(704\) 0 0
\(705\) −1.40312 −0.0528447
\(706\) 0 0
\(707\) 18.3875 0.691533
\(708\) 0 0
\(709\) 42.8062 1.60762 0.803811 0.594884i \(-0.202802\pi\)
0.803811 + 0.594884i \(0.202802\pi\)
\(710\) 0 0
\(711\) −4.70156 −0.176322
\(712\) 0 0
\(713\) −63.0156 −2.35995
\(714\) 0 0
\(715\) −0.701562 −0.0262369
\(716\) 0 0
\(717\) 6.10469 0.227984
\(718\) 0 0
\(719\) −9.19375 −0.342869 −0.171435 0.985196i \(-0.554840\pi\)
−0.171435 + 0.985196i \(0.554840\pi\)
\(720\) 0 0
\(721\) 32.4187 1.20734
\(722\) 0 0
\(723\) 2.00000 0.0743808
\(724\) 0 0
\(725\) 9.40312 0.349223
\(726\) 0 0
\(727\) 9.40312 0.348743 0.174371 0.984680i \(-0.444211\pi\)
0.174371 + 0.984680i \(0.444211\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.80625 0.103793
\(732\) 0 0
\(733\) 41.7172 1.54086 0.770430 0.637525i \(-0.220042\pi\)
0.770430 + 0.637525i \(0.220042\pi\)
\(734\) 0 0
\(735\) −0.298438 −0.0110080
\(736\) 0 0
\(737\) −2.80625 −0.103369
\(738\) 0 0
\(739\) 8.80625 0.323943 0.161972 0.986795i \(-0.448215\pi\)
0.161972 + 0.986795i \(0.448215\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 10.5969 0.388762 0.194381 0.980926i \(-0.437730\pi\)
0.194381 + 0.980926i \(0.437730\pi\)
\(744\) 0 0
\(745\) −1.29844 −0.0475711
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 48.9109 1.78717
\(750\) 0 0
\(751\) 42.3141 1.54406 0.772031 0.635585i \(-0.219241\pi\)
0.772031 + 0.635585i \(0.219241\pi\)
\(752\) 0 0
\(753\) 18.2094 0.663586
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −47.4031 −1.72290 −0.861448 0.507846i \(-0.830442\pi\)
−0.861448 + 0.507846i \(0.830442\pi\)
\(758\) 0 0
\(759\) 4.70156 0.170656
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) −29.1938 −1.05688
\(764\) 0 0
\(765\) 0.701562 0.0253650
\(766\) 0 0
\(767\) −14.8062 −0.534623
\(768\) 0 0
\(769\) 51.4031 1.85364 0.926822 0.375501i \(-0.122529\pi\)
0.926822 + 0.375501i \(0.122529\pi\)
\(770\) 0 0
\(771\) 20.2094 0.727823
\(772\) 0 0
\(773\) −32.5969 −1.17243 −0.586214 0.810156i \(-0.699382\pi\)
−0.586214 + 0.810156i \(0.699382\pi\)
\(774\) 0 0
\(775\) 9.40312 0.337770
\(776\) 0 0
\(777\) 18.1047 0.649502
\(778\) 0 0
\(779\) 21.4031 0.766847
\(780\) 0 0
\(781\) 10.8797 0.389306
\(782\) 0 0
\(783\) 9.40312 0.336040
\(784\) 0 0
\(785\) −8.80625 −0.314308
\(786\) 0 0
\(787\) −5.19375 −0.185137 −0.0925686 0.995706i \(-0.529508\pi\)
−0.0925686 + 0.995706i \(0.529508\pi\)
\(788\) 0 0
\(789\) −4.59688 −0.163653
\(790\) 0 0
\(791\) −17.8219 −0.633673
\(792\) 0 0
\(793\) −2.70156 −0.0959353
\(794\) 0 0
\(795\) −10.7016 −0.379545
\(796\) 0 0
\(797\) −14.7016 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(798\) 0 0
\(799\) −0.984379 −0.0348248
\(800\) 0 0
\(801\) 8.10469 0.286365
\(802\) 0 0
\(803\) −9.40312 −0.331829
\(804\) 0 0
\(805\) −18.1047 −0.638106
\(806\) 0 0
\(807\) −14.8062 −0.521205
\(808\) 0 0
\(809\) −0.806248 −0.0283462 −0.0141731 0.999900i \(-0.504512\pi\)
−0.0141731 + 0.999900i \(0.504512\pi\)
\(810\) 0 0
\(811\) 14.2094 0.498959 0.249479 0.968380i \(-0.419741\pi\)
0.249479 + 0.968380i \(0.419741\pi\)
\(812\) 0 0
\(813\) 6.59688 0.231363
\(814\) 0 0
\(815\) 11.2984 0.395767
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 2.70156 0.0944002
\(820\) 0 0
\(821\) −2.49219 −0.0869780 −0.0434890 0.999054i \(-0.513847\pi\)
−0.0434890 + 0.999054i \(0.513847\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) −0.701562 −0.0244253
\(826\) 0 0
\(827\) −34.5969 −1.20305 −0.601526 0.798854i \(-0.705440\pi\)
−0.601526 + 0.798854i \(0.705440\pi\)
\(828\) 0 0
\(829\) −47.6125 −1.65365 −0.826825 0.562459i \(-0.809855\pi\)
−0.826825 + 0.562459i \(0.809855\pi\)
\(830\) 0 0
\(831\) 8.80625 0.305485
\(832\) 0 0
\(833\) −0.209373 −0.00725433
\(834\) 0 0
\(835\) −22.8062 −0.789243
\(836\) 0 0
\(837\) 9.40312 0.325020
\(838\) 0 0
\(839\) −45.1203 −1.55773 −0.778863 0.627194i \(-0.784203\pi\)
−0.778863 + 0.627194i \(0.784203\pi\)
\(840\) 0 0
\(841\) 59.4187 2.04892
\(842\) 0 0
\(843\) 10.0000 0.344418
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 28.3875 0.975406
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 44.9109 1.53953
\(852\) 0 0
\(853\) −21.7172 −0.743582 −0.371791 0.928316i \(-0.621256\pi\)
−0.371791 + 0.928316i \(0.621256\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) 12.7016 0.433877 0.216939 0.976185i \(-0.430393\pi\)
0.216939 + 0.976185i \(0.430393\pi\)
\(858\) 0 0
\(859\) −31.2984 −1.06789 −0.533944 0.845520i \(-0.679291\pi\)
−0.533944 + 0.845520i \(0.679291\pi\)
\(860\) 0 0
\(861\) −28.9109 −0.985282
\(862\) 0 0
\(863\) −13.4031 −0.456248 −0.228124 0.973632i \(-0.573259\pi\)
−0.228124 + 0.973632i \(0.573259\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) −16.5078 −0.560635
\(868\) 0 0
\(869\) 3.29844 0.111892
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) 18.1047 0.612751
\(874\) 0 0
\(875\) 2.70156 0.0913295
\(876\) 0 0
\(877\) −3.61250 −0.121985 −0.0609927 0.998138i \(-0.519427\pi\)
−0.0609927 + 0.998138i \(0.519427\pi\)
\(878\) 0 0
\(879\) 15.4031 0.519534
\(880\) 0 0
\(881\) −12.8062 −0.431453 −0.215727 0.976454i \(-0.569212\pi\)
−0.215727 + 0.976454i \(0.569212\pi\)
\(882\) 0 0
\(883\) 17.6125 0.592708 0.296354 0.955078i \(-0.404229\pi\)
0.296354 + 0.955078i \(0.404229\pi\)
\(884\) 0 0
\(885\) −14.8062 −0.497707
\(886\) 0 0
\(887\) −18.4922 −0.620907 −0.310453 0.950589i \(-0.600481\pi\)
−0.310453 + 0.950589i \(0.600481\pi\)
\(888\) 0 0
\(889\) −36.2094 −1.21442
\(890\) 0 0
\(891\) −0.701562 −0.0235032
\(892\) 0 0
\(893\) 2.80625 0.0939075
\(894\) 0 0
\(895\) −22.0000 −0.735379
\(896\) 0 0
\(897\) 6.70156 0.223759
\(898\) 0 0
\(899\) 88.4187 2.94893
\(900\) 0 0
\(901\) −7.50781 −0.250121
\(902\) 0 0
\(903\) 10.8062 0.359609
\(904\) 0 0
\(905\) 1.29844 0.0431615
\(906\) 0 0
\(907\) 16.2094 0.538223 0.269112 0.963109i \(-0.413270\pi\)
0.269112 + 0.963109i \(0.413270\pi\)
\(908\) 0 0
\(909\) −6.80625 −0.225749
\(910\) 0 0
\(911\) −6.80625 −0.225501 −0.112751 0.993623i \(-0.535966\pi\)
−0.112751 + 0.993623i \(0.535966\pi\)
\(912\) 0 0
\(913\) 2.80625 0.0928733
\(914\) 0 0
\(915\) −2.70156 −0.0893109
\(916\) 0 0
\(917\) −34.5969 −1.14249
\(918\) 0 0
\(919\) −7.08907 −0.233847 −0.116923 0.993141i \(-0.537303\pi\)
−0.116923 + 0.993141i \(0.537303\pi\)
\(920\) 0 0
\(921\) 13.8953 0.457866
\(922\) 0 0
\(923\) 15.5078 0.510446
\(924\) 0 0
\(925\) −6.70156 −0.220346
\(926\) 0 0
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) 8.10469 0.265906 0.132953 0.991122i \(-0.457554\pi\)
0.132953 + 0.991122i \(0.457554\pi\)
\(930\) 0 0
\(931\) 0.596876 0.0195618
\(932\) 0 0
\(933\) −8.00000 −0.261908
\(934\) 0 0
\(935\) −0.492189 −0.0160963
\(936\) 0 0
\(937\) 52.8062 1.72510 0.862552 0.505968i \(-0.168864\pi\)
0.862552 + 0.505968i \(0.168864\pi\)
\(938\) 0 0
\(939\) 16.8062 0.548451
\(940\) 0 0
\(941\) −54.7016 −1.78322 −0.891610 0.452804i \(-0.850424\pi\)
−0.891610 + 0.452804i \(0.850424\pi\)
\(942\) 0 0
\(943\) −71.7172 −2.33543
\(944\) 0 0
\(945\) 2.70156 0.0878818
\(946\) 0 0
\(947\) 9.61250 0.312364 0.156182 0.987728i \(-0.450081\pi\)
0.156182 + 0.987728i \(0.450081\pi\)
\(948\) 0 0
\(949\) −13.4031 −0.435084
\(950\) 0 0
\(951\) 14.2094 0.460770
\(952\) 0 0
\(953\) 4.91093 0.159081 0.0795404 0.996832i \(-0.474655\pi\)
0.0795404 + 0.996832i \(0.474655\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.59688 −0.213247
\(958\) 0 0
\(959\) −48.6281 −1.57028
\(960\) 0 0
\(961\) 57.4187 1.85222
\(962\) 0 0
\(963\) −18.1047 −0.583415
\(964\) 0 0
\(965\) 22.1047 0.711575
\(966\) 0 0
\(967\) −39.8219 −1.28058 −0.640292 0.768131i \(-0.721187\pi\)
−0.640292 + 0.768131i \(0.721187\pi\)
\(968\) 0 0
\(969\) −1.40312 −0.0450748
\(970\) 0 0
\(971\) −57.0156 −1.82972 −0.914859 0.403773i \(-0.867699\pi\)
−0.914859 + 0.403773i \(0.867699\pi\)
\(972\) 0 0
\(973\) −12.7016 −0.407193
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) 26.2094 0.838512 0.419256 0.907868i \(-0.362291\pi\)
0.419256 + 0.907868i \(0.362291\pi\)
\(978\) 0 0
\(979\) −5.68594 −0.181723
\(980\) 0 0
\(981\) 10.8062 0.345017
\(982\) 0 0
\(983\) 20.0000 0.637901 0.318950 0.947771i \(-0.396670\pi\)
0.318950 + 0.947771i \(0.396670\pi\)
\(984\) 0 0
\(985\) 14.2094 0.452748
\(986\) 0 0
\(987\) −3.79063 −0.120657
\(988\) 0 0
\(989\) 26.8062 0.852389
\(990\) 0 0
\(991\) −27.7172 −0.880465 −0.440233 0.897884i \(-0.645104\pi\)
−0.440233 + 0.897884i \(0.645104\pi\)
\(992\) 0 0
\(993\) 14.2094 0.450921
\(994\) 0 0
\(995\) 2.80625 0.0889641
\(996\) 0 0
\(997\) −51.4031 −1.62795 −0.813977 0.580898i \(-0.802702\pi\)
−0.813977 + 0.580898i \(0.802702\pi\)
\(998\) 0 0
\(999\) −6.70156 −0.212028
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 3120.2.a.bf.1.1 2
3.2 odd 2 9360.2.a.ct.1.1 2
4.3 odd 2 1560.2.a.m.1.2 2
12.11 even 2 4680.2.a.bb.1.2 2
20.19 odd 2 7800.2.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.m.1.2 2 4.3 odd 2
3120.2.a.bf.1.1 2 1.1 even 1 trivial
4680.2.a.bb.1.2 2 12.11 even 2
7800.2.a.be.1.1 2 20.19 odd 2
9360.2.a.ct.1.1 2 3.2 odd 2