Properties

Label 4840.2.a.k.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} +1.00000 q^{5} -3.73205 q^{7} +O(q^{10})\) \(q+1.73205 q^{3} +1.00000 q^{5} -3.73205 q^{7} +3.46410 q^{13} +1.73205 q^{15} -2.00000 q^{19} -6.46410 q^{21} +0.535898 q^{23} +1.00000 q^{25} -5.19615 q^{27} -8.92820 q^{29} +5.46410 q^{31} -3.73205 q^{35} -4.00000 q^{37} +6.00000 q^{39} -11.9282 q^{41} +1.73205 q^{43} -10.6603 q^{47} +6.92820 q^{49} -0.535898 q^{53} -3.46410 q^{57} +5.46410 q^{59} -8.46410 q^{61} +3.46410 q^{65} +2.26795 q^{67} +0.928203 q^{69} +9.46410 q^{71} -14.9282 q^{73} +1.73205 q^{75} +6.39230 q^{79} -9.00000 q^{81} +0.535898 q^{83} -15.4641 q^{87} +2.07180 q^{89} -12.9282 q^{91} +9.46410 q^{93} -2.00000 q^{95} +6.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 4 q^{7} - 4 q^{19} - 6 q^{21} + 8 q^{23} + 2 q^{25} - 4 q^{29} + 4 q^{31} - 4 q^{35} - 8 q^{37} + 12 q^{39} - 10 q^{41} - 4 q^{47} - 8 q^{53} + 4 q^{59} - 10 q^{61} + 8 q^{67} - 12 q^{69} + 12 q^{71} - 16 q^{73} - 8 q^{79} - 18 q^{81} + 8 q^{83} - 24 q^{87} + 18 q^{89} - 12 q^{91} + 12 q^{93} - 4 q^{95} - 8 q^{97}+O(q^{100}) \) 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
\(3\) 1.73205 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.73205 −1.41058 −0.705291 0.708918i \(-0.749184\pi\)
−0.705291 + 0.708918i \(0.749184\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) 1.73205 0.447214
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −6.46410 −1.41058
\(22\) 0 0
\(23\) 0.535898 0.111743 0.0558713 0.998438i \(-0.482206\pi\)
0.0558713 + 0.998438i \(0.482206\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) −8.92820 −1.65793 −0.828963 0.559304i \(-0.811069\pi\)
−0.828963 + 0.559304i \(0.811069\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.73205 −0.630832
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −11.9282 −1.86287 −0.931436 0.363905i \(-0.881443\pi\)
−0.931436 + 0.363905i \(0.881443\pi\)
\(42\) 0 0
\(43\) 1.73205 0.264135 0.132068 0.991241i \(-0.457838\pi\)
0.132068 + 0.991241i \(0.457838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.6603 −1.55496 −0.777479 0.628909i \(-0.783502\pi\)
−0.777479 + 0.628909i \(0.783502\pi\)
\(48\) 0 0
\(49\) 6.92820 0.989743
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.535898 −0.0736113 −0.0368057 0.999322i \(-0.511718\pi\)
−0.0368057 + 0.999322i \(0.511718\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.46410 −0.458831
\(58\) 0 0
\(59\) 5.46410 0.711365 0.355683 0.934607i \(-0.384248\pi\)
0.355683 + 0.934607i \(0.384248\pi\)
\(60\) 0 0
\(61\) −8.46410 −1.08372 −0.541859 0.840470i \(-0.682279\pi\)
−0.541859 + 0.840470i \(0.682279\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.46410 0.429669
\(66\) 0 0
\(67\) 2.26795 0.277074 0.138537 0.990357i \(-0.455760\pi\)
0.138537 + 0.990357i \(0.455760\pi\)
\(68\) 0 0
\(69\) 0.928203 0.111743
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) 0 0
\(73\) −14.9282 −1.74721 −0.873607 0.486632i \(-0.838225\pi\)
−0.873607 + 0.486632i \(0.838225\pi\)
\(74\) 0 0
\(75\) 1.73205 0.200000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.39230 0.719190 0.359595 0.933108i \(-0.382915\pi\)
0.359595 + 0.933108i \(0.382915\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 0.535898 0.0588225 0.0294112 0.999567i \(-0.490637\pi\)
0.0294112 + 0.999567i \(0.490637\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −15.4641 −1.65793
\(88\) 0 0
\(89\) 2.07180 0.219610 0.109805 0.993953i \(-0.464977\pi\)
0.109805 + 0.993953i \(0.464977\pi\)
\(90\) 0 0
\(91\) −12.9282 −1.35524
\(92\) 0 0
\(93\) 9.46410 0.981382
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 6.39230 0.649040 0.324520 0.945879i \(-0.394797\pi\)
0.324520 + 0.945879i \(0.394797\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.39230 0.934569 0.467285 0.884107i \(-0.345232\pi\)
0.467285 + 0.884107i \(0.345232\pi\)
\(102\) 0 0
\(103\) 18.3923 1.81225 0.906124 0.423013i \(-0.139027\pi\)
0.906124 + 0.423013i \(0.139027\pi\)
\(104\) 0 0
\(105\) −6.46410 −0.630832
\(106\) 0 0
\(107\) −5.73205 −0.554138 −0.277069 0.960850i \(-0.589363\pi\)
−0.277069 + 0.960850i \(0.589363\pi\)
\(108\) 0 0
\(109\) −14.4641 −1.38541 −0.692705 0.721221i \(-0.743581\pi\)
−0.692705 + 0.721221i \(0.743581\pi\)
\(110\) 0 0
\(111\) −6.92820 −0.657596
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0.535898 0.0499728
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −20.6603 −1.86287
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.33975 0.118883 0.0594416 0.998232i \(-0.481068\pi\)
0.0594416 + 0.998232i \(0.481068\pi\)
\(128\) 0 0
\(129\) 3.00000 0.264135
\(130\) 0 0
\(131\) −14.0000 −1.22319 −0.611593 0.791173i \(-0.709471\pi\)
−0.611593 + 0.791173i \(0.709471\pi\)
\(132\) 0 0
\(133\) 7.46410 0.647220
\(134\) 0 0
\(135\) −5.19615 −0.447214
\(136\) 0 0
\(137\) −10.3923 −0.887875 −0.443937 0.896058i \(-0.646419\pi\)
−0.443937 + 0.896058i \(0.646419\pi\)
\(138\) 0 0
\(139\) 12.3923 1.05110 0.525551 0.850762i \(-0.323859\pi\)
0.525551 + 0.850762i \(0.323859\pi\)
\(140\) 0 0
\(141\) −18.4641 −1.55496
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −8.92820 −0.741447
\(146\) 0 0
\(147\) 12.0000 0.989743
\(148\) 0 0
\(149\) 0.464102 0.0380207 0.0190103 0.999819i \(-0.493948\pi\)
0.0190103 + 0.999819i \(0.493948\pi\)
\(150\) 0 0
\(151\) 16.3923 1.33399 0.666993 0.745064i \(-0.267581\pi\)
0.666993 + 0.745064i \(0.267581\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.46410 0.438887
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) −0.928203 −0.0736113
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −16.1244 −1.26296 −0.631479 0.775393i \(-0.717552\pi\)
−0.631479 + 0.775393i \(0.717552\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −24.5167 −1.89716 −0.948578 0.316543i \(-0.897478\pi\)
−0.948578 + 0.316543i \(0.897478\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.46410 0.263371 0.131685 0.991292i \(-0.457961\pi\)
0.131685 + 0.991292i \(0.457961\pi\)
\(174\) 0 0
\(175\) −3.73205 −0.282117
\(176\) 0 0
\(177\) 9.46410 0.711365
\(178\) 0 0
\(179\) −18.3923 −1.37471 −0.687353 0.726324i \(-0.741227\pi\)
−0.687353 + 0.726324i \(0.741227\pi\)
\(180\) 0 0
\(181\) −3.39230 −0.252148 −0.126074 0.992021i \(-0.540238\pi\)
−0.126074 + 0.992021i \(0.540238\pi\)
\(182\) 0 0
\(183\) −14.6603 −1.08372
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 19.3923 1.41058
\(190\) 0 0
\(191\) −24.9282 −1.80374 −0.901871 0.432006i \(-0.857806\pi\)
−0.901871 + 0.432006i \(0.857806\pi\)
\(192\) 0 0
\(193\) 11.8564 0.853443 0.426721 0.904383i \(-0.359668\pi\)
0.426721 + 0.904383i \(0.359668\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) 12.9282 0.921096 0.460548 0.887635i \(-0.347653\pi\)
0.460548 + 0.887635i \(0.347653\pi\)
\(198\) 0 0
\(199\) −14.3923 −1.02024 −0.510122 0.860102i \(-0.670400\pi\)
−0.510122 + 0.860102i \(0.670400\pi\)
\(200\) 0 0
\(201\) 3.92820 0.277074
\(202\) 0 0
\(203\) 33.3205 2.33864
\(204\) 0 0
\(205\) −11.9282 −0.833102
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −13.4641 −0.926907 −0.463453 0.886121i \(-0.653390\pi\)
−0.463453 + 0.886121i \(0.653390\pi\)
\(212\) 0 0
\(213\) 16.3923 1.12318
\(214\) 0 0
\(215\) 1.73205 0.118125
\(216\) 0 0
\(217\) −20.3923 −1.38432
\(218\) 0 0
\(219\) −25.8564 −1.74721
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.73205 0.249917 0.124958 0.992162i \(-0.460120\pi\)
0.124958 + 0.992162i \(0.460120\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.1962 1.14135 0.570674 0.821176i \(-0.306682\pi\)
0.570674 + 0.821176i \(0.306682\pi\)
\(228\) 0 0
\(229\) −25.3923 −1.67797 −0.838985 0.544154i \(-0.816851\pi\)
−0.838985 + 0.544154i \(0.816851\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.39230 0.418774 0.209387 0.977833i \(-0.432853\pi\)
0.209387 + 0.977833i \(0.432853\pi\)
\(234\) 0 0
\(235\) −10.6603 −0.695398
\(236\) 0 0
\(237\) 11.0718 0.719190
\(238\) 0 0
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 0 0
\(241\) −21.0000 −1.35273 −0.676364 0.736567i \(-0.736446\pi\)
−0.676364 + 0.736567i \(0.736446\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.92820 0.442627
\(246\) 0 0
\(247\) −6.92820 −0.440831
\(248\) 0 0
\(249\) 0.928203 0.0588225
\(250\) 0 0
\(251\) 17.4641 1.10232 0.551162 0.834398i \(-0.314185\pi\)
0.551162 + 0.834398i \(0.314185\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −22.2487 −1.38784 −0.693918 0.720054i \(-0.744117\pi\)
−0.693918 + 0.720054i \(0.744117\pi\)
\(258\) 0 0
\(259\) 14.9282 0.927593
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5.60770 −0.345785 −0.172893 0.984941i \(-0.555311\pi\)
−0.172893 + 0.984941i \(0.555311\pi\)
\(264\) 0 0
\(265\) −0.535898 −0.0329200
\(266\) 0 0
\(267\) 3.58846 0.219610
\(268\) 0 0
\(269\) 29.2487 1.78333 0.891663 0.452700i \(-0.149539\pi\)
0.891663 + 0.452700i \(0.149539\pi\)
\(270\) 0 0
\(271\) −5.07180 −0.308090 −0.154045 0.988064i \(-0.549230\pi\)
−0.154045 + 0.988064i \(0.549230\pi\)
\(272\) 0 0
\(273\) −22.3923 −1.35524
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.14359 −0.247186 −0.123593 0.992333i \(-0.539442\pi\)
−0.123593 + 0.992333i \(0.539442\pi\)
\(282\) 0 0
\(283\) 26.5167 1.57625 0.788126 0.615514i \(-0.211052\pi\)
0.788126 + 0.615514i \(0.211052\pi\)
\(284\) 0 0
\(285\) −3.46410 −0.205196
\(286\) 0 0
\(287\) 44.5167 2.62774
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 11.0718 0.649040
\(292\) 0 0
\(293\) 18.2487 1.06610 0.533051 0.846083i \(-0.321046\pi\)
0.533051 + 0.846083i \(0.321046\pi\)
\(294\) 0 0
\(295\) 5.46410 0.318132
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.85641 0.107359
\(300\) 0 0
\(301\) −6.46410 −0.372585
\(302\) 0 0
\(303\) 16.2679 0.934569
\(304\) 0 0
\(305\) −8.46410 −0.484653
\(306\) 0 0
\(307\) −5.60770 −0.320048 −0.160024 0.987113i \(-0.551157\pi\)
−0.160024 + 0.987113i \(0.551157\pi\)
\(308\) 0 0
\(309\) 31.8564 1.81225
\(310\) 0 0
\(311\) 24.9282 1.41355 0.706774 0.707439i \(-0.250150\pi\)
0.706774 + 0.707439i \(0.250150\pi\)
\(312\) 0 0
\(313\) −3.46410 −0.195803 −0.0979013 0.995196i \(-0.531213\pi\)
−0.0979013 + 0.995196i \(0.531213\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.85641 0.441260 0.220630 0.975358i \(-0.429189\pi\)
0.220630 + 0.975358i \(0.429189\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −9.92820 −0.554138
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.46410 0.192154
\(326\) 0 0
\(327\) −25.0526 −1.38541
\(328\) 0 0
\(329\) 39.7846 2.19340
\(330\) 0 0
\(331\) −7.07180 −0.388701 −0.194351 0.980932i \(-0.562260\pi\)
−0.194351 + 0.980932i \(0.562260\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.26795 0.123911
\(336\) 0 0
\(337\) −33.3205 −1.81508 −0.907542 0.419962i \(-0.862044\pi\)
−0.907542 + 0.419962i \(0.862044\pi\)
\(338\) 0 0
\(339\) −17.3205 −0.940721
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.267949 0.0144679
\(344\) 0 0
\(345\) 0.928203 0.0499728
\(346\) 0 0
\(347\) 2.26795 0.121750 0.0608749 0.998145i \(-0.480611\pi\)
0.0608749 + 0.998145i \(0.480611\pi\)
\(348\) 0 0
\(349\) 25.7128 1.37638 0.688188 0.725533i \(-0.258407\pi\)
0.688188 + 0.725533i \(0.258407\pi\)
\(350\) 0 0
\(351\) −18.0000 −0.960769
\(352\) 0 0
\(353\) 26.5359 1.41236 0.706182 0.708031i \(-0.250416\pi\)
0.706182 + 0.708031i \(0.250416\pi\)
\(354\) 0 0
\(355\) 9.46410 0.502302
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.7846 0.780302 0.390151 0.920751i \(-0.372423\pi\)
0.390151 + 0.920751i \(0.372423\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −14.9282 −0.781378
\(366\) 0 0
\(367\) 2.41154 0.125882 0.0629408 0.998017i \(-0.479952\pi\)
0.0629408 + 0.998017i \(0.479952\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −8.39230 −0.434537 −0.217269 0.976112i \(-0.569715\pi\)
−0.217269 + 0.976112i \(0.569715\pi\)
\(374\) 0 0
\(375\) 1.73205 0.0894427
\(376\) 0 0
\(377\) −30.9282 −1.59288
\(378\) 0 0
\(379\) 14.3923 0.739283 0.369642 0.929174i \(-0.379480\pi\)
0.369642 + 0.929174i \(0.379480\pi\)
\(380\) 0 0
\(381\) 2.32051 0.118883
\(382\) 0 0
\(383\) 19.4641 0.994569 0.497285 0.867587i \(-0.334330\pi\)
0.497285 + 0.867587i \(0.334330\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 25.2487 1.28016 0.640080 0.768308i \(-0.278901\pi\)
0.640080 + 0.768308i \(0.278901\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −24.2487 −1.22319
\(394\) 0 0
\(395\) 6.39230 0.321632
\(396\) 0 0
\(397\) 38.6410 1.93934 0.969669 0.244424i \(-0.0785988\pi\)
0.969669 + 0.244424i \(0.0785988\pi\)
\(398\) 0 0
\(399\) 12.9282 0.647220
\(400\) 0 0
\(401\) 16.8564 0.841769 0.420884 0.907114i \(-0.361720\pi\)
0.420884 + 0.907114i \(0.361720\pi\)
\(402\) 0 0
\(403\) 18.9282 0.942881
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.856406 −0.0423466 −0.0211733 0.999776i \(-0.506740\pi\)
−0.0211733 + 0.999776i \(0.506740\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) −20.3923 −1.00344
\(414\) 0 0
\(415\) 0.535898 0.0263062
\(416\) 0 0
\(417\) 21.4641 1.05110
\(418\) 0 0
\(419\) −11.8564 −0.579223 −0.289612 0.957144i \(-0.593526\pi\)
−0.289612 + 0.957144i \(0.593526\pi\)
\(420\) 0 0
\(421\) 11.5359 0.562225 0.281113 0.959675i \(-0.409297\pi\)
0.281113 + 0.959675i \(0.409297\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 31.5885 1.52867
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.60770 −0.462786 −0.231393 0.972860i \(-0.574328\pi\)
−0.231393 + 0.972860i \(0.574328\pi\)
\(432\) 0 0
\(433\) −33.1769 −1.59438 −0.797190 0.603728i \(-0.793681\pi\)
−0.797190 + 0.603728i \(0.793681\pi\)
\(434\) 0 0
\(435\) −15.4641 −0.741447
\(436\) 0 0
\(437\) −1.07180 −0.0512710
\(438\) 0 0
\(439\) −14.9282 −0.712484 −0.356242 0.934394i \(-0.615942\pi\)
−0.356242 + 0.934394i \(0.615942\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.33975 −0.158676 −0.0793381 0.996848i \(-0.525281\pi\)
−0.0793381 + 0.996848i \(0.525281\pi\)
\(444\) 0 0
\(445\) 2.07180 0.0982126
\(446\) 0 0
\(447\) 0.803848 0.0380207
\(448\) 0 0
\(449\) −4.85641 −0.229188 −0.114594 0.993412i \(-0.536557\pi\)
−0.114594 + 0.993412i \(0.536557\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 28.3923 1.33399
\(454\) 0 0
\(455\) −12.9282 −0.606084
\(456\) 0 0
\(457\) 42.2487 1.97631 0.988156 0.153455i \(-0.0490399\pi\)
0.988156 + 0.153455i \(0.0490399\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −13.2487 −0.617054 −0.308527 0.951216i \(-0.599836\pi\)
−0.308527 + 0.951216i \(0.599836\pi\)
\(462\) 0 0
\(463\) −16.2679 −0.756036 −0.378018 0.925798i \(-0.623394\pi\)
−0.378018 + 0.925798i \(0.623394\pi\)
\(464\) 0 0
\(465\) 9.46410 0.438887
\(466\) 0 0
\(467\) −32.6603 −1.51134 −0.755668 0.654955i \(-0.772688\pi\)
−0.755668 + 0.654955i \(0.772688\pi\)
\(468\) 0 0
\(469\) −8.46410 −0.390836
\(470\) 0 0
\(471\) −34.6410 −1.59617
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 35.4641 1.62040 0.810198 0.586156i \(-0.199360\pi\)
0.810198 + 0.586156i \(0.199360\pi\)
\(480\) 0 0
\(481\) −13.8564 −0.631798
\(482\) 0 0
\(483\) −3.46410 −0.157622
\(484\) 0 0
\(485\) 6.39230 0.290260
\(486\) 0 0
\(487\) 41.3205 1.87241 0.936205 0.351453i \(-0.114312\pi\)
0.936205 + 0.351453i \(0.114312\pi\)
\(488\) 0 0
\(489\) −27.9282 −1.26296
\(490\) 0 0
\(491\) −16.2487 −0.733294 −0.366647 0.930360i \(-0.619494\pi\)
−0.366647 + 0.930360i \(0.619494\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −35.3205 −1.58434
\(498\) 0 0
\(499\) −1.32051 −0.0591141 −0.0295570 0.999563i \(-0.509410\pi\)
−0.0295570 + 0.999563i \(0.509410\pi\)
\(500\) 0 0
\(501\) −42.4641 −1.89716
\(502\) 0 0
\(503\) −21.3397 −0.951492 −0.475746 0.879583i \(-0.657822\pi\)
−0.475746 + 0.879583i \(0.657822\pi\)
\(504\) 0 0
\(505\) 9.39230 0.417952
\(506\) 0 0
\(507\) −1.73205 −0.0769231
\(508\) 0 0
\(509\) −3.53590 −0.156726 −0.0783630 0.996925i \(-0.524969\pi\)
−0.0783630 + 0.996925i \(0.524969\pi\)
\(510\) 0 0
\(511\) 55.7128 2.46459
\(512\) 0 0
\(513\) 10.3923 0.458831
\(514\) 0 0
\(515\) 18.3923 0.810462
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −17.0000 −0.744784 −0.372392 0.928076i \(-0.621462\pi\)
−0.372392 + 0.928076i \(0.621462\pi\)
\(522\) 0 0
\(523\) 25.3205 1.10719 0.553594 0.832787i \(-0.313256\pi\)
0.553594 + 0.832787i \(0.313256\pi\)
\(524\) 0 0
\(525\) −6.46410 −0.282117
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.7128 −0.987514
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −41.3205 −1.78979
\(534\) 0 0
\(535\) −5.73205 −0.247818
\(536\) 0 0
\(537\) −31.8564 −1.37471
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.53590 −0.238007 −0.119003 0.992894i \(-0.537970\pi\)
−0.119003 + 0.992894i \(0.537970\pi\)
\(542\) 0 0
\(543\) −5.87564 −0.252148
\(544\) 0 0
\(545\) −14.4641 −0.619574
\(546\) 0 0
\(547\) 1.60770 0.0687401 0.0343700 0.999409i \(-0.489058\pi\)
0.0343700 + 0.999409i \(0.489058\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 17.8564 0.760708
\(552\) 0 0
\(553\) −23.8564 −1.01448
\(554\) 0 0
\(555\) −6.92820 −0.294086
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.3397 −0.646493 −0.323247 0.946315i \(-0.604774\pi\)
−0.323247 + 0.946315i \(0.604774\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) 0 0
\(567\) 33.5885 1.41058
\(568\) 0 0
\(569\) 36.7128 1.53908 0.769541 0.638598i \(-0.220485\pi\)
0.769541 + 0.638598i \(0.220485\pi\)
\(570\) 0 0
\(571\) 32.2487 1.34957 0.674783 0.738016i \(-0.264237\pi\)
0.674783 + 0.738016i \(0.264237\pi\)
\(572\) 0 0
\(573\) −43.1769 −1.80374
\(574\) 0 0
\(575\) 0.535898 0.0223485
\(576\) 0 0
\(577\) −12.3923 −0.515898 −0.257949 0.966158i \(-0.583047\pi\)
−0.257949 + 0.966158i \(0.583047\pi\)
\(578\) 0 0
\(579\) 20.5359 0.853443
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 43.0526 1.77697 0.888485 0.458906i \(-0.151759\pi\)
0.888485 + 0.458906i \(0.151759\pi\)
\(588\) 0 0
\(589\) −10.9282 −0.450289
\(590\) 0 0
\(591\) 22.3923 0.921096
\(592\) 0 0
\(593\) −14.3923 −0.591021 −0.295511 0.955339i \(-0.595490\pi\)
−0.295511 + 0.955339i \(0.595490\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −24.9282 −1.02024
\(598\) 0 0
\(599\) −11.0718 −0.452381 −0.226191 0.974083i \(-0.572627\pi\)
−0.226191 + 0.974083i \(0.572627\pi\)
\(600\) 0 0
\(601\) 3.85641 0.157306 0.0786531 0.996902i \(-0.474938\pi\)
0.0786531 + 0.996902i \(0.474938\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.5359 0.833526 0.416763 0.909015i \(-0.363164\pi\)
0.416763 + 0.909015i \(0.363164\pi\)
\(608\) 0 0
\(609\) 57.7128 2.33864
\(610\) 0 0
\(611\) −36.9282 −1.49396
\(612\) 0 0
\(613\) 41.7128 1.68476 0.842382 0.538880i \(-0.181153\pi\)
0.842382 + 0.538880i \(0.181153\pi\)
\(614\) 0 0
\(615\) −20.6603 −0.833102
\(616\) 0 0
\(617\) −23.3205 −0.938848 −0.469424 0.882973i \(-0.655538\pi\)
−0.469424 + 0.882973i \(0.655538\pi\)
\(618\) 0 0
\(619\) −30.9282 −1.24311 −0.621555 0.783371i \(-0.713499\pi\)
−0.621555 + 0.783371i \(0.713499\pi\)
\(620\) 0 0
\(621\) −2.78461 −0.111743
\(622\) 0 0
\(623\) −7.73205 −0.309778
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 27.7128 1.10323 0.551615 0.834099i \(-0.314012\pi\)
0.551615 + 0.834099i \(0.314012\pi\)
\(632\) 0 0
\(633\) −23.3205 −0.926907
\(634\) 0 0
\(635\) 1.33975 0.0531662
\(636\) 0 0
\(637\) 24.0000 0.950915
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.78461 −0.109985 −0.0549927 0.998487i \(-0.517514\pi\)
−0.0549927 + 0.998487i \(0.517514\pi\)
\(642\) 0 0
\(643\) −25.1962 −0.993639 −0.496820 0.867854i \(-0.665499\pi\)
−0.496820 + 0.867854i \(0.665499\pi\)
\(644\) 0 0
\(645\) 3.00000 0.118125
\(646\) 0 0
\(647\) 30.1244 1.18431 0.592155 0.805824i \(-0.298277\pi\)
0.592155 + 0.805824i \(0.298277\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −35.3205 −1.38432
\(652\) 0 0
\(653\) −23.8564 −0.933573 −0.466787 0.884370i \(-0.654588\pi\)
−0.466787 + 0.884370i \(0.654588\pi\)
\(654\) 0 0
\(655\) −14.0000 −0.547025
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −15.1769 −0.591209 −0.295604 0.955310i \(-0.595521\pi\)
−0.295604 + 0.955310i \(0.595521\pi\)
\(660\) 0 0
\(661\) −30.3205 −1.17933 −0.589666 0.807648i \(-0.700740\pi\)
−0.589666 + 0.807648i \(0.700740\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.46410 0.289445
\(666\) 0 0
\(667\) −4.78461 −0.185261
\(668\) 0 0
\(669\) 6.46410 0.249917
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 22.5359 0.868695 0.434348 0.900745i \(-0.356979\pi\)
0.434348 + 0.900745i \(0.356979\pi\)
\(674\) 0 0
\(675\) −5.19615 −0.200000
\(676\) 0 0
\(677\) −15.7128 −0.603892 −0.301946 0.953325i \(-0.597636\pi\)
−0.301946 + 0.953325i \(0.597636\pi\)
\(678\) 0 0
\(679\) −23.8564 −0.915525
\(680\) 0 0
\(681\) 29.7846 1.14135
\(682\) 0 0
\(683\) 11.5885 0.443420 0.221710 0.975113i \(-0.428836\pi\)
0.221710 + 0.975113i \(0.428836\pi\)
\(684\) 0 0
\(685\) −10.3923 −0.397070
\(686\) 0 0
\(687\) −43.9808 −1.67797
\(688\) 0 0
\(689\) −1.85641 −0.0707235
\(690\) 0 0
\(691\) 39.5692 1.50528 0.752642 0.658430i \(-0.228779\pi\)
0.752642 + 0.658430i \(0.228779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.3923 0.470067
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 11.0718 0.418774
\(700\) 0 0
\(701\) 41.7128 1.57547 0.787736 0.616013i \(-0.211253\pi\)
0.787736 + 0.616013i \(0.211253\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −18.4641 −0.695398
\(706\) 0 0
\(707\) −35.0526 −1.31829
\(708\) 0 0
\(709\) −0.607695 −0.0228225 −0.0114112 0.999935i \(-0.503632\pi\)
−0.0114112 + 0.999935i \(0.503632\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.92820 0.109662
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −24.2487 −0.905585
\(718\) 0 0
\(719\) −36.6410 −1.36648 −0.683240 0.730194i \(-0.739430\pi\)
−0.683240 + 0.730194i \(0.739430\pi\)
\(720\) 0 0
\(721\) −68.6410 −2.55633
\(722\) 0 0
\(723\) −36.3731 −1.35273
\(724\) 0 0
\(725\) −8.92820 −0.331585
\(726\) 0 0
\(727\) 27.1962 1.00865 0.504325 0.863514i \(-0.331741\pi\)
0.504325 + 0.863514i \(0.331741\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 28.0000 1.03420 0.517102 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(734\) 0 0
\(735\) 12.0000 0.442627
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 11.1769 0.411149 0.205575 0.978641i \(-0.434094\pi\)
0.205575 + 0.978641i \(0.434094\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 7.73205 0.283661 0.141831 0.989891i \(-0.454701\pi\)
0.141831 + 0.989891i \(0.454701\pi\)
\(744\) 0 0
\(745\) 0.464102 0.0170034
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 21.3923 0.781658
\(750\) 0 0
\(751\) 40.5359 1.47918 0.739588 0.673060i \(-0.235020\pi\)
0.739588 + 0.673060i \(0.235020\pi\)
\(752\) 0 0
\(753\) 30.2487 1.10232
\(754\) 0 0
\(755\) 16.3923 0.596577
\(756\) 0 0
\(757\) 9.32051 0.338760 0.169380 0.985551i \(-0.445824\pi\)
0.169380 + 0.985551i \(0.445824\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −31.8564 −1.15479 −0.577397 0.816464i \(-0.695931\pi\)
−0.577397 + 0.816464i \(0.695931\pi\)
\(762\) 0 0
\(763\) 53.9808 1.95423
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.9282 0.683458
\(768\) 0 0
\(769\) 11.8564 0.427553 0.213776 0.976883i \(-0.431424\pi\)
0.213776 + 0.976883i \(0.431424\pi\)
\(770\) 0 0
\(771\) −38.5359 −1.38784
\(772\) 0 0
\(773\) 2.53590 0.0912099 0.0456050 0.998960i \(-0.485478\pi\)
0.0456050 + 0.998960i \(0.485478\pi\)
\(774\) 0 0
\(775\) 5.46410 0.196276
\(776\) 0 0
\(777\) 25.8564 0.927593
\(778\) 0 0
\(779\) 23.8564 0.854744
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 46.3923 1.65793
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) −3.58846 −0.127915 −0.0639573 0.997953i \(-0.520372\pi\)
−0.0639573 + 0.997953i \(0.520372\pi\)
\(788\) 0 0
\(789\) −9.71281 −0.345785
\(790\) 0 0
\(791\) 37.3205 1.32696
\(792\) 0 0
\(793\) −29.3205 −1.04120
\(794\) 0 0
\(795\) −0.928203 −0.0329200
\(796\) 0 0
\(797\) −41.4641 −1.46873 −0.734367 0.678753i \(-0.762521\pi\)
−0.734367 + 0.678753i \(0.762521\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) 50.6603 1.78333
\(808\) 0 0
\(809\) 2.78461 0.0979017 0.0489508 0.998801i \(-0.484412\pi\)
0.0489508 + 0.998801i \(0.484412\pi\)
\(810\) 0 0
\(811\) 23.6077 0.828978 0.414489 0.910054i \(-0.363960\pi\)
0.414489 + 0.910054i \(0.363960\pi\)
\(812\) 0 0
\(813\) −8.78461 −0.308090
\(814\) 0 0
\(815\) −16.1244 −0.564812
\(816\) 0 0
\(817\) −3.46410 −0.121194
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −51.1051 −1.78358 −0.891790 0.452449i \(-0.850551\pi\)
−0.891790 + 0.452449i \(0.850551\pi\)
\(822\) 0 0
\(823\) 1.87564 0.0653809 0.0326904 0.999466i \(-0.489592\pi\)
0.0326904 + 0.999466i \(0.489592\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.1962 0.458875 0.229438 0.973323i \(-0.426311\pi\)
0.229438 + 0.973323i \(0.426311\pi\)
\(828\) 0 0
\(829\) 25.2487 0.876924 0.438462 0.898750i \(-0.355523\pi\)
0.438462 + 0.898750i \(0.355523\pi\)
\(830\) 0 0
\(831\) 38.1051 1.32185
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −24.5167 −0.848434
\(836\) 0 0
\(837\) −28.3923 −0.981382
\(838\) 0 0
\(839\) 20.5359 0.708978 0.354489 0.935060i \(-0.384655\pi\)
0.354489 + 0.935060i \(0.384655\pi\)
\(840\) 0 0
\(841\) 50.7128 1.74872
\(842\) 0 0
\(843\) −7.17691 −0.247186
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 45.9282 1.57625
\(850\) 0 0
\(851\) −2.14359 −0.0734814
\(852\) 0 0
\(853\) −26.9282 −0.922004 −0.461002 0.887399i \(-0.652510\pi\)
−0.461002 + 0.887399i \(0.652510\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35.0718 1.19803 0.599015 0.800738i \(-0.295559\pi\)
0.599015 + 0.800738i \(0.295559\pi\)
\(858\) 0 0
\(859\) −52.9282 −1.80589 −0.902943 0.429759i \(-0.858598\pi\)
−0.902943 + 0.429759i \(0.858598\pi\)
\(860\) 0 0
\(861\) 77.1051 2.62774
\(862\) 0 0
\(863\) 12.5167 0.426072 0.213036 0.977044i \(-0.431665\pi\)
0.213036 + 0.977044i \(0.431665\pi\)
\(864\) 0 0
\(865\) 3.46410 0.117783
\(866\) 0 0
\(867\) −29.4449 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 7.85641 0.266204
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.73205 −0.126166
\(876\) 0 0
\(877\) −56.3923 −1.90423 −0.952116 0.305736i \(-0.901098\pi\)
−0.952116 + 0.305736i \(0.901098\pi\)
\(878\) 0 0
\(879\) 31.6077 1.06610
\(880\) 0 0
\(881\) −26.8564 −0.904815 −0.452408 0.891811i \(-0.649435\pi\)
−0.452408 + 0.891811i \(0.649435\pi\)
\(882\) 0 0
\(883\) 4.53590 0.152645 0.0763226 0.997083i \(-0.475682\pi\)
0.0763226 + 0.997083i \(0.475682\pi\)
\(884\) 0 0
\(885\) 9.46410 0.318132
\(886\) 0 0
\(887\) −32.5167 −1.09180 −0.545901 0.837849i \(-0.683813\pi\)
−0.545901 + 0.837849i \(0.683813\pi\)
\(888\) 0 0
\(889\) −5.00000 −0.167695
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 21.3205 0.713464
\(894\) 0 0
\(895\) −18.3923 −0.614787
\(896\) 0 0
\(897\) 3.21539 0.107359
\(898\) 0 0
\(899\) −48.7846 −1.62706
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −11.1962 −0.372585
\(904\) 0 0
\(905\) −3.39230 −0.112764
\(906\) 0 0
\(907\) 12.9090 0.428635 0.214318 0.976764i \(-0.431247\pi\)
0.214318 + 0.976764i \(0.431247\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.2487 1.20097 0.600487 0.799635i \(-0.294974\pi\)
0.600487 + 0.799635i \(0.294974\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −14.6603 −0.484653
\(916\) 0 0
\(917\) 52.2487 1.72540
\(918\) 0 0
\(919\) 1.21539 0.0400920 0.0200460 0.999799i \(-0.493619\pi\)
0.0200460 + 0.999799i \(0.493619\pi\)
\(920\) 0 0
\(921\) −9.71281 −0.320048
\(922\) 0 0
\(923\) 32.7846 1.07912
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −54.7846 −1.79742 −0.898712 0.438539i \(-0.855496\pi\)
−0.898712 + 0.438539i \(0.855496\pi\)
\(930\) 0 0
\(931\) −13.8564 −0.454125
\(932\) 0 0
\(933\) 43.1769 1.41355
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 52.9282 1.72909 0.864545 0.502556i \(-0.167607\pi\)
0.864545 + 0.502556i \(0.167607\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −13.5359 −0.441258 −0.220629 0.975358i \(-0.570811\pi\)
−0.220629 + 0.975358i \(0.570811\pi\)
\(942\) 0 0
\(943\) −6.39230 −0.208162
\(944\) 0 0
\(945\) 19.3923 0.630832
\(946\) 0 0
\(947\) 14.6795 0.477019 0.238510 0.971140i \(-0.423341\pi\)
0.238510 + 0.971140i \(0.423341\pi\)
\(948\) 0 0
\(949\) −51.7128 −1.67867
\(950\) 0 0
\(951\) 13.6077 0.441260
\(952\) 0 0
\(953\) −4.00000 −0.129573 −0.0647864 0.997899i \(-0.520637\pi\)
−0.0647864 + 0.997899i \(0.520637\pi\)
\(954\) 0 0
\(955\) −24.9282 −0.806658
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 38.7846 1.25242
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.8564 0.381671
\(966\) 0 0
\(967\) 59.1769 1.90300 0.951501 0.307647i \(-0.0995415\pi\)
0.951501 + 0.307647i \(0.0995415\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.7846 1.30884 0.654420 0.756131i \(-0.272913\pi\)
0.654420 + 0.756131i \(0.272913\pi\)
\(972\) 0 0
\(973\) −46.2487 −1.48267
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) 9.07180 0.290232 0.145116 0.989415i \(-0.453644\pi\)
0.145116 + 0.989415i \(0.453644\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −36.2679 −1.15677 −0.578384 0.815765i \(-0.696316\pi\)
−0.578384 + 0.815765i \(0.696316\pi\)
\(984\) 0 0
\(985\) 12.9282 0.411927
\(986\) 0 0
\(987\) 68.9090 2.19340
\(988\) 0 0
\(989\) 0.928203 0.0295151
\(990\) 0 0
\(991\) −34.7846 −1.10497 −0.552485 0.833523i \(-0.686320\pi\)
−0.552485 + 0.833523i \(0.686320\pi\)
\(992\) 0 0
\(993\) −12.2487 −0.388701
\(994\) 0 0
\(995\) −14.3923 −0.456267
\(996\) 0 0
\(997\) 37.3205 1.18195 0.590976 0.806689i \(-0.298743\pi\)
0.590976 + 0.806689i \(0.298743\pi\)
\(998\) 0 0
\(999\) 20.7846 0.657596
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 4840.2.a.k.1.2 2
4.3 odd 2 9680.2.a.br.1.1 2
11.10 odd 2 4840.2.a.l.1.2 yes 2
44.43 even 2 9680.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.k.1.2 2 1.1 even 1 trivial
4840.2.a.l.1.2 yes 2 11.10 odd 2
9680.2.a.bq.1.1 2 44.43 even 2
9680.2.a.br.1.1 2 4.3 odd 2