Properties

Label 4864.2.a.x.1.1
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
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] = [4864,2,Mod(1,4864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4864, 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("4864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.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: \(38.8392355432\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.00000 q^{5} -2.46410 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.00000 q^{5} -2.46410 q^{7} -2.00000 q^{9} -4.00000 q^{11} +2.46410 q^{13} +2.00000 q^{15} +5.92820 q^{17} +1.00000 q^{19} -2.46410 q^{21} +0.464102 q^{23} -1.00000 q^{25} -5.00000 q^{27} -4.46410 q^{29} -8.92820 q^{31} -4.00000 q^{33} -4.92820 q^{35} +2.00000 q^{37} +2.46410 q^{39} +6.92820 q^{41} -8.92820 q^{43} -4.00000 q^{45} +6.92820 q^{47} -0.928203 q^{49} +5.92820 q^{51} -5.53590 q^{53} -8.00000 q^{55} +1.00000 q^{57} +3.92820 q^{59} -4.92820 q^{61} +4.92820 q^{63} +4.92820 q^{65} +7.92820 q^{67} +0.464102 q^{69} -14.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} +9.85641 q^{77} +2.00000 q^{79} +1.00000 q^{81} -10.9282 q^{83} +11.8564 q^{85} -4.46410 q^{87} +12.9282 q^{89} -6.07180 q^{91} -8.92820 q^{93} +2.00000 q^{95} -0.928203 q^{97} +8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{7} - 4 q^{9} - 8 q^{11} - 2 q^{13} + 4 q^{15} - 2 q^{17} + 2 q^{19} + 2 q^{21} - 6 q^{23} - 2 q^{25} - 10 q^{27} - 2 q^{29} - 4 q^{31} - 8 q^{33} + 4 q^{35} + 4 q^{37} - 2 q^{39} - 4 q^{43} - 8 q^{45} + 12 q^{49} - 2 q^{51} - 18 q^{53} - 16 q^{55} + 2 q^{57} - 6 q^{59} + 4 q^{61} - 4 q^{63} - 4 q^{65} + 2 q^{67} - 6 q^{69} - 28 q^{71} - 14 q^{73} - 2 q^{75} - 8 q^{77} + 4 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} - 2 q^{87} + 12 q^{89} - 26 q^{91} - 4 q^{93} + 4 q^{95} + 12 q^{97} + 16 q^{99}+O(q^{100}) \) 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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −2.46410 −0.931343 −0.465671 0.884958i \(-0.654187\pi\)
−0.465671 + 0.884958i \(0.654187\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.46410 0.683419 0.341709 0.939806i \(-0.388994\pi\)
0.341709 + 0.939806i \(0.388994\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) 5.92820 1.43780 0.718900 0.695113i \(-0.244646\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.46410 −0.537711
\(22\) 0 0
\(23\) 0.464102 0.0967719 0.0483859 0.998829i \(-0.484592\pi\)
0.0483859 + 0.998829i \(0.484592\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −4.46410 −0.828963 −0.414481 0.910058i \(-0.636037\pi\)
−0.414481 + 0.910058i \(0.636037\pi\)
\(30\) 0 0
\(31\) −8.92820 −1.60355 −0.801776 0.597624i \(-0.796111\pi\)
−0.801776 + 0.597624i \(0.796111\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −4.92820 −0.833018
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 2.46410 0.394572
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) −8.92820 −1.36154 −0.680769 0.732498i \(-0.738354\pi\)
−0.680769 + 0.732498i \(0.738354\pi\)
\(44\) 0 0
\(45\) −4.00000 −0.596285
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −0.928203 −0.132600
\(50\) 0 0
\(51\) 5.92820 0.830114
\(52\) 0 0
\(53\) −5.53590 −0.760414 −0.380207 0.924901i \(-0.624147\pi\)
−0.380207 + 0.924901i \(0.624147\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 3.92820 0.511409 0.255704 0.966755i \(-0.417693\pi\)
0.255704 + 0.966755i \(0.417693\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 0 0
\(63\) 4.92820 0.620895
\(64\) 0 0
\(65\) 4.92820 0.611268
\(66\) 0 0
\(67\) 7.92820 0.968584 0.484292 0.874906i \(-0.339077\pi\)
0.484292 + 0.874906i \(0.339077\pi\)
\(68\) 0 0
\(69\) 0.464102 0.0558713
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 9.85641 1.12324
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.9282 −1.19953 −0.599763 0.800178i \(-0.704739\pi\)
−0.599763 + 0.800178i \(0.704739\pi\)
\(84\) 0 0
\(85\) 11.8564 1.28601
\(86\) 0 0
\(87\) −4.46410 −0.478602
\(88\) 0 0
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) 0 0
\(91\) −6.07180 −0.636497
\(92\) 0 0
\(93\) −8.92820 −0.925812
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −0.928203 −0.0942448 −0.0471224 0.998889i \(-0.515005\pi\)
−0.0471224 + 0.998889i \(0.515005\pi\)
\(98\) 0 0
\(99\) 8.00000 0.804030
\(100\) 0 0
\(101\) −17.8564 −1.77678 −0.888389 0.459091i \(-0.848175\pi\)
−0.888389 + 0.459091i \(0.848175\pi\)
\(102\) 0 0
\(103\) −10.9282 −1.07679 −0.538394 0.842693i \(-0.680969\pi\)
−0.538394 + 0.842693i \(0.680969\pi\)
\(104\) 0 0
\(105\) −4.92820 −0.480943
\(106\) 0 0
\(107\) 0.0717968 0.00694086 0.00347043 0.999994i \(-0.498895\pi\)
0.00347043 + 0.999994i \(0.498895\pi\)
\(108\) 0 0
\(109\) −13.3923 −1.28275 −0.641375 0.767227i \(-0.721636\pi\)
−0.641375 + 0.767227i \(0.721636\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −0.928203 −0.0873180 −0.0436590 0.999046i \(-0.513902\pi\)
−0.0436590 + 0.999046i \(0.513902\pi\)
\(114\) 0 0
\(115\) 0.928203 0.0865554
\(116\) 0 0
\(117\) −4.92820 −0.455613
\(118\) 0 0
\(119\) −14.6077 −1.33909
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 6.92820 0.624695
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −12.9282 −1.14719 −0.573596 0.819138i \(-0.694452\pi\)
−0.573596 + 0.819138i \(0.694452\pi\)
\(128\) 0 0
\(129\) −8.92820 −0.786084
\(130\) 0 0
\(131\) 20.9282 1.82851 0.914253 0.405144i \(-0.132779\pi\)
0.914253 + 0.405144i \(0.132779\pi\)
\(132\) 0 0
\(133\) −2.46410 −0.213665
\(134\) 0 0
\(135\) −10.0000 −0.860663
\(136\) 0 0
\(137\) −18.8564 −1.61101 −0.805506 0.592588i \(-0.798106\pi\)
−0.805506 + 0.592588i \(0.798106\pi\)
\(138\) 0 0
\(139\) 11.8564 1.00565 0.502824 0.864389i \(-0.332295\pi\)
0.502824 + 0.864389i \(0.332295\pi\)
\(140\) 0 0
\(141\) 6.92820 0.583460
\(142\) 0 0
\(143\) −9.85641 −0.824234
\(144\) 0 0
\(145\) −8.92820 −0.741447
\(146\) 0 0
\(147\) −0.928203 −0.0765569
\(148\) 0 0
\(149\) 2.92820 0.239888 0.119944 0.992781i \(-0.461729\pi\)
0.119944 + 0.992781i \(0.461729\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −11.8564 −0.958534
\(154\) 0 0
\(155\) −17.8564 −1.43426
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) −5.53590 −0.439025
\(160\) 0 0
\(161\) −1.14359 −0.0901278
\(162\) 0 0
\(163\) −3.85641 −0.302057 −0.151029 0.988529i \(-0.548259\pi\)
−0.151029 + 0.988529i \(0.548259\pi\)
\(164\) 0 0
\(165\) −8.00000 −0.622799
\(166\) 0 0
\(167\) −18.9282 −1.46471 −0.732354 0.680924i \(-0.761578\pi\)
−0.732354 + 0.680924i \(0.761578\pi\)
\(168\) 0 0
\(169\) −6.92820 −0.532939
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 0 0
\(175\) 2.46410 0.186269
\(176\) 0 0
\(177\) 3.92820 0.295262
\(178\) 0 0
\(179\) −1.85641 −0.138754 −0.0693772 0.997591i \(-0.522101\pi\)
−0.0693772 + 0.997591i \(0.522101\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −4.92820 −0.364303
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −23.7128 −1.73405
\(188\) 0 0
\(189\) 12.3205 0.896185
\(190\) 0 0
\(191\) −10.4641 −0.757156 −0.378578 0.925569i \(-0.623587\pi\)
−0.378578 + 0.925569i \(0.623587\pi\)
\(192\) 0 0
\(193\) −17.8564 −1.28533 −0.642666 0.766146i \(-0.722172\pi\)
−0.642666 + 0.766146i \(0.722172\pi\)
\(194\) 0 0
\(195\) 4.92820 0.352916
\(196\) 0 0
\(197\) −6.92820 −0.493614 −0.246807 0.969065i \(-0.579381\pi\)
−0.246807 + 0.969065i \(0.579381\pi\)
\(198\) 0 0
\(199\) 20.4641 1.45066 0.725331 0.688400i \(-0.241687\pi\)
0.725331 + 0.688400i \(0.241687\pi\)
\(200\) 0 0
\(201\) 7.92820 0.559212
\(202\) 0 0
\(203\) 11.0000 0.772049
\(204\) 0 0
\(205\) 13.8564 0.967773
\(206\) 0 0
\(207\) −0.928203 −0.0645146
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 2.85641 0.196643 0.0983216 0.995155i \(-0.468653\pi\)
0.0983216 + 0.995155i \(0.468653\pi\)
\(212\) 0 0
\(213\) −14.0000 −0.959264
\(214\) 0 0
\(215\) −17.8564 −1.21780
\(216\) 0 0
\(217\) 22.0000 1.49346
\(218\) 0 0
\(219\) −7.00000 −0.473016
\(220\) 0 0
\(221\) 14.6077 0.982620
\(222\) 0 0
\(223\) −13.0718 −0.875352 −0.437676 0.899133i \(-0.644198\pi\)
−0.437676 + 0.899133i \(0.644198\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) 11.9282 0.791703 0.395851 0.918315i \(-0.370450\pi\)
0.395851 + 0.918315i \(0.370450\pi\)
\(228\) 0 0
\(229\) 16.9282 1.11865 0.559324 0.828949i \(-0.311061\pi\)
0.559324 + 0.828949i \(0.311061\pi\)
\(230\) 0 0
\(231\) 9.85641 0.648504
\(232\) 0 0
\(233\) 11.8564 0.776739 0.388370 0.921504i \(-0.373038\pi\)
0.388370 + 0.921504i \(0.373038\pi\)
\(234\) 0 0
\(235\) 13.8564 0.903892
\(236\) 0 0
\(237\) 2.00000 0.129914
\(238\) 0 0
\(239\) 18.3205 1.18506 0.592528 0.805550i \(-0.298130\pi\)
0.592528 + 0.805550i \(0.298130\pi\)
\(240\) 0 0
\(241\) 16.7846 1.08119 0.540596 0.841282i \(-0.318199\pi\)
0.540596 + 0.841282i \(0.318199\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) −1.85641 −0.118601
\(246\) 0 0
\(247\) 2.46410 0.156787
\(248\) 0 0
\(249\) −10.9282 −0.692547
\(250\) 0 0
\(251\) −0.928203 −0.0585877 −0.0292938 0.999571i \(-0.509326\pi\)
−0.0292938 + 0.999571i \(0.509326\pi\)
\(252\) 0 0
\(253\) −1.85641 −0.116711
\(254\) 0 0
\(255\) 11.8564 0.742477
\(256\) 0 0
\(257\) 12.9282 0.806439 0.403220 0.915103i \(-0.367891\pi\)
0.403220 + 0.915103i \(0.367891\pi\)
\(258\) 0 0
\(259\) −4.92820 −0.306224
\(260\) 0 0
\(261\) 8.92820 0.552642
\(262\) 0 0
\(263\) 10.9282 0.673862 0.336931 0.941529i \(-0.390611\pi\)
0.336931 + 0.941529i \(0.390611\pi\)
\(264\) 0 0
\(265\) −11.0718 −0.680135
\(266\) 0 0
\(267\) 12.9282 0.791193
\(268\) 0 0
\(269\) −19.8564 −1.21067 −0.605333 0.795972i \(-0.706960\pi\)
−0.605333 + 0.795972i \(0.706960\pi\)
\(270\) 0 0
\(271\) −25.3923 −1.54247 −0.771236 0.636549i \(-0.780361\pi\)
−0.771236 + 0.636549i \(0.780361\pi\)
\(272\) 0 0
\(273\) −6.07180 −0.367482
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 11.0718 0.665240 0.332620 0.943061i \(-0.392067\pi\)
0.332620 + 0.943061i \(0.392067\pi\)
\(278\) 0 0
\(279\) 17.8564 1.06904
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) −5.07180 −0.301487 −0.150744 0.988573i \(-0.548167\pi\)
−0.150744 + 0.988573i \(0.548167\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) −17.0718 −1.00772
\(288\) 0 0
\(289\) 18.1436 1.06727
\(290\) 0 0
\(291\) −0.928203 −0.0544122
\(292\) 0 0
\(293\) 26.3205 1.53766 0.768830 0.639453i \(-0.220839\pi\)
0.768830 + 0.639453i \(0.220839\pi\)
\(294\) 0 0
\(295\) 7.85641 0.457418
\(296\) 0 0
\(297\) 20.0000 1.16052
\(298\) 0 0
\(299\) 1.14359 0.0661357
\(300\) 0 0
\(301\) 22.0000 1.26806
\(302\) 0 0
\(303\) −17.8564 −1.02582
\(304\) 0 0
\(305\) −9.85641 −0.564376
\(306\) 0 0
\(307\) −25.8564 −1.47570 −0.737852 0.674963i \(-0.764160\pi\)
−0.737852 + 0.674963i \(0.764160\pi\)
\(308\) 0 0
\(309\) −10.9282 −0.621684
\(310\) 0 0
\(311\) −13.3923 −0.759408 −0.379704 0.925108i \(-0.623974\pi\)
−0.379704 + 0.925108i \(0.623974\pi\)
\(312\) 0 0
\(313\) 10.8564 0.613640 0.306820 0.951767i \(-0.400735\pi\)
0.306820 + 0.951767i \(0.400735\pi\)
\(314\) 0 0
\(315\) 9.85641 0.555346
\(316\) 0 0
\(317\) −7.53590 −0.423258 −0.211629 0.977350i \(-0.567877\pi\)
−0.211629 + 0.977350i \(0.567877\pi\)
\(318\) 0 0
\(319\) 17.8564 0.999767
\(320\) 0 0
\(321\) 0.0717968 0.00400730
\(322\) 0 0
\(323\) 5.92820 0.329854
\(324\) 0 0
\(325\) −2.46410 −0.136684
\(326\) 0 0
\(327\) −13.3923 −0.740596
\(328\) 0 0
\(329\) −17.0718 −0.941199
\(330\) 0 0
\(331\) 31.7846 1.74704 0.873520 0.486788i \(-0.161832\pi\)
0.873520 + 0.486788i \(0.161832\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 15.8564 0.866328
\(336\) 0 0
\(337\) −20.9282 −1.14003 −0.570016 0.821634i \(-0.693063\pi\)
−0.570016 + 0.821634i \(0.693063\pi\)
\(338\) 0 0
\(339\) −0.928203 −0.0504131
\(340\) 0 0
\(341\) 35.7128 1.93396
\(342\) 0 0
\(343\) 19.5359 1.05484
\(344\) 0 0
\(345\) 0.928203 0.0499728
\(346\) 0 0
\(347\) −0.143594 −0.00770851 −0.00385425 0.999993i \(-0.501227\pi\)
−0.00385425 + 0.999993i \(0.501227\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −12.3205 −0.657620
\(352\) 0 0
\(353\) 23.9282 1.27357 0.636785 0.771042i \(-0.280264\pi\)
0.636785 + 0.771042i \(0.280264\pi\)
\(354\) 0 0
\(355\) −28.0000 −1.48609
\(356\) 0 0
\(357\) −14.6077 −0.773121
\(358\) 0 0
\(359\) −17.5359 −0.925509 −0.462755 0.886486i \(-0.653139\pi\)
−0.462755 + 0.886486i \(0.653139\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 0 0
\(367\) −26.9282 −1.40564 −0.702820 0.711367i \(-0.748076\pi\)
−0.702820 + 0.711367i \(0.748076\pi\)
\(368\) 0 0
\(369\) −13.8564 −0.721336
\(370\) 0 0
\(371\) 13.6410 0.708206
\(372\) 0 0
\(373\) −9.39230 −0.486315 −0.243158 0.969987i \(-0.578183\pi\)
−0.243158 + 0.969987i \(0.578183\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) −11.0000 −0.566529
\(378\) 0 0
\(379\) −20.8564 −1.07132 −0.535661 0.844433i \(-0.679937\pi\)
−0.535661 + 0.844433i \(0.679937\pi\)
\(380\) 0 0
\(381\) −12.9282 −0.662332
\(382\) 0 0
\(383\) −16.7846 −0.857653 −0.428827 0.903387i \(-0.641073\pi\)
−0.428827 + 0.903387i \(0.641073\pi\)
\(384\) 0 0
\(385\) 19.7128 1.00466
\(386\) 0 0
\(387\) 17.8564 0.907692
\(388\) 0 0
\(389\) −1.85641 −0.0941235 −0.0470618 0.998892i \(-0.514986\pi\)
−0.0470618 + 0.998892i \(0.514986\pi\)
\(390\) 0 0
\(391\) 2.75129 0.139139
\(392\) 0 0
\(393\) 20.9282 1.05569
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) −4.92820 −0.247339 −0.123670 0.992323i \(-0.539466\pi\)
−0.123670 + 0.992323i \(0.539466\pi\)
\(398\) 0 0
\(399\) −2.46410 −0.123359
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) −22.0000 −1.09590
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −18.8564 −0.930118
\(412\) 0 0
\(413\) −9.67949 −0.476297
\(414\) 0 0
\(415\) −21.8564 −1.07289
\(416\) 0 0
\(417\) 11.8564 0.580611
\(418\) 0 0
\(419\) 10.7846 0.526863 0.263431 0.964678i \(-0.415146\pi\)
0.263431 + 0.964678i \(0.415146\pi\)
\(420\) 0 0
\(421\) 18.4641 0.899885 0.449943 0.893057i \(-0.351444\pi\)
0.449943 + 0.893057i \(0.351444\pi\)
\(422\) 0 0
\(423\) −13.8564 −0.673722
\(424\) 0 0
\(425\) −5.92820 −0.287560
\(426\) 0 0
\(427\) 12.1436 0.587670
\(428\) 0 0
\(429\) −9.85641 −0.475872
\(430\) 0 0
\(431\) 0.928203 0.0447100 0.0223550 0.999750i \(-0.492884\pi\)
0.0223550 + 0.999750i \(0.492884\pi\)
\(432\) 0 0
\(433\) 16.7846 0.806617 0.403308 0.915064i \(-0.367860\pi\)
0.403308 + 0.915064i \(0.367860\pi\)
\(434\) 0 0
\(435\) −8.92820 −0.428075
\(436\) 0 0
\(437\) 0.464102 0.0222010
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 1.85641 0.0884003
\(442\) 0 0
\(443\) 9.85641 0.468292 0.234146 0.972201i \(-0.424771\pi\)
0.234146 + 0.972201i \(0.424771\pi\)
\(444\) 0 0
\(445\) 25.8564 1.22571
\(446\) 0 0
\(447\) 2.92820 0.138499
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) −27.7128 −1.30495
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) −12.1436 −0.569300
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) −29.6410 −1.38352
\(460\) 0 0
\(461\) 13.8564 0.645357 0.322679 0.946509i \(-0.395417\pi\)
0.322679 + 0.946509i \(0.395417\pi\)
\(462\) 0 0
\(463\) 24.7846 1.15184 0.575919 0.817507i \(-0.304644\pi\)
0.575919 + 0.817507i \(0.304644\pi\)
\(464\) 0 0
\(465\) −17.8564 −0.828071
\(466\) 0 0
\(467\) −38.7846 −1.79474 −0.897369 0.441281i \(-0.854524\pi\)
−0.897369 + 0.441281i \(0.854524\pi\)
\(468\) 0 0
\(469\) −19.5359 −0.902084
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 0 0
\(473\) 35.7128 1.64208
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 11.0718 0.506943
\(478\) 0 0
\(479\) −24.7846 −1.13244 −0.566219 0.824255i \(-0.691594\pi\)
−0.566219 + 0.824255i \(0.691594\pi\)
\(480\) 0 0
\(481\) 4.92820 0.224707
\(482\) 0 0
\(483\) −1.14359 −0.0520353
\(484\) 0 0
\(485\) −1.85641 −0.0842951
\(486\) 0 0
\(487\) 34.6410 1.56973 0.784867 0.619664i \(-0.212731\pi\)
0.784867 + 0.619664i \(0.212731\pi\)
\(488\) 0 0
\(489\) −3.85641 −0.174393
\(490\) 0 0
\(491\) 41.8564 1.88895 0.944477 0.328579i \(-0.106570\pi\)
0.944477 + 0.328579i \(0.106570\pi\)
\(492\) 0 0
\(493\) −26.4641 −1.19188
\(494\) 0 0
\(495\) 16.0000 0.719147
\(496\) 0 0
\(497\) 34.4974 1.54742
\(498\) 0 0
\(499\) 18.7846 0.840915 0.420457 0.907312i \(-0.361870\pi\)
0.420457 + 0.907312i \(0.361870\pi\)
\(500\) 0 0
\(501\) −18.9282 −0.845650
\(502\) 0 0
\(503\) −12.3205 −0.549344 −0.274672 0.961538i \(-0.588569\pi\)
−0.274672 + 0.961538i \(0.588569\pi\)
\(504\) 0 0
\(505\) −35.7128 −1.58920
\(506\) 0 0
\(507\) −6.92820 −0.307692
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 17.2487 0.763038
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) −21.8564 −0.963108
\(516\) 0 0
\(517\) −27.7128 −1.21881
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) 26.9282 1.17975 0.589873 0.807496i \(-0.299178\pi\)
0.589873 + 0.807496i \(0.299178\pi\)
\(522\) 0 0
\(523\) 31.0000 1.35554 0.677768 0.735276i \(-0.262948\pi\)
0.677768 + 0.735276i \(0.262948\pi\)
\(524\) 0 0
\(525\) 2.46410 0.107542
\(526\) 0 0
\(527\) −52.9282 −2.30559
\(528\) 0 0
\(529\) −22.7846 −0.990635
\(530\) 0 0
\(531\) −7.85641 −0.340939
\(532\) 0 0
\(533\) 17.0718 0.739462
\(534\) 0 0
\(535\) 0.143594 0.00620809
\(536\) 0 0
\(537\) −1.85641 −0.0801099
\(538\) 0 0
\(539\) 3.71281 0.159922
\(540\) 0 0
\(541\) 28.7846 1.23755 0.618774 0.785569i \(-0.287630\pi\)
0.618774 + 0.785569i \(0.287630\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) −26.7846 −1.14733
\(546\) 0 0
\(547\) −33.8564 −1.44760 −0.723798 0.690012i \(-0.757605\pi\)
−0.723798 + 0.690012i \(0.757605\pi\)
\(548\) 0 0
\(549\) 9.85641 0.420661
\(550\) 0 0
\(551\) −4.46410 −0.190177
\(552\) 0 0
\(553\) −4.92820 −0.209569
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) −22.0000 −0.930501
\(560\) 0 0
\(561\) −23.7128 −1.00116
\(562\) 0 0
\(563\) −23.7128 −0.999376 −0.499688 0.866205i \(-0.666552\pi\)
−0.499688 + 0.866205i \(0.666552\pi\)
\(564\) 0 0
\(565\) −1.85641 −0.0780996
\(566\) 0 0
\(567\) −2.46410 −0.103483
\(568\) 0 0
\(569\) −32.7846 −1.37440 −0.687201 0.726467i \(-0.741161\pi\)
−0.687201 + 0.726467i \(0.741161\pi\)
\(570\) 0 0
\(571\) 12.7846 0.535019 0.267510 0.963555i \(-0.413799\pi\)
0.267510 + 0.963555i \(0.413799\pi\)
\(572\) 0 0
\(573\) −10.4641 −0.437144
\(574\) 0 0
\(575\) −0.464102 −0.0193544
\(576\) 0 0
\(577\) −29.7846 −1.23995 −0.619975 0.784622i \(-0.712857\pi\)
−0.619975 + 0.784622i \(0.712857\pi\)
\(578\) 0 0
\(579\) −17.8564 −0.742087
\(580\) 0 0
\(581\) 26.9282 1.11717
\(582\) 0 0
\(583\) 22.1436 0.917094
\(584\) 0 0
\(585\) −9.85641 −0.407512
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −8.92820 −0.367880
\(590\) 0 0
\(591\) −6.92820 −0.284988
\(592\) 0 0
\(593\) 43.8564 1.80097 0.900483 0.434890i \(-0.143213\pi\)
0.900483 + 0.434890i \(0.143213\pi\)
\(594\) 0 0
\(595\) −29.2154 −1.19771
\(596\) 0 0
\(597\) 20.4641 0.837540
\(598\) 0 0
\(599\) 17.8564 0.729593 0.364796 0.931087i \(-0.381139\pi\)
0.364796 + 0.931087i \(0.381139\pi\)
\(600\) 0 0
\(601\) 0.143594 0.00585730 0.00292865 0.999996i \(-0.499068\pi\)
0.00292865 + 0.999996i \(0.499068\pi\)
\(602\) 0 0
\(603\) −15.8564 −0.645723
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −12.1436 −0.492893 −0.246447 0.969156i \(-0.579263\pi\)
−0.246447 + 0.969156i \(0.579263\pi\)
\(608\) 0 0
\(609\) 11.0000 0.445742
\(610\) 0 0
\(611\) 17.0718 0.690651
\(612\) 0 0
\(613\) 34.6410 1.39914 0.699569 0.714565i \(-0.253375\pi\)
0.699569 + 0.714565i \(0.253375\pi\)
\(614\) 0 0
\(615\) 13.8564 0.558744
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) 0.143594 0.00577151 0.00288576 0.999996i \(-0.499081\pi\)
0.00288576 + 0.999996i \(0.499081\pi\)
\(620\) 0 0
\(621\) −2.32051 −0.0931188
\(622\) 0 0
\(623\) −31.8564 −1.27630
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −4.00000 −0.159745
\(628\) 0 0
\(629\) 11.8564 0.472746
\(630\) 0 0
\(631\) 30.6410 1.21980 0.609900 0.792479i \(-0.291210\pi\)
0.609900 + 0.792479i \(0.291210\pi\)
\(632\) 0 0
\(633\) 2.85641 0.113532
\(634\) 0 0
\(635\) −25.8564 −1.02608
\(636\) 0 0
\(637\) −2.28719 −0.0906217
\(638\) 0 0
\(639\) 28.0000 1.10766
\(640\) 0 0
\(641\) −32.7846 −1.29491 −0.647457 0.762102i \(-0.724168\pi\)
−0.647457 + 0.762102i \(0.724168\pi\)
\(642\) 0 0
\(643\) −19.0718 −0.752118 −0.376059 0.926596i \(-0.622721\pi\)
−0.376059 + 0.926596i \(0.622721\pi\)
\(644\) 0 0
\(645\) −17.8564 −0.703095
\(646\) 0 0
\(647\) 20.4641 0.804527 0.402263 0.915524i \(-0.368224\pi\)
0.402263 + 0.915524i \(0.368224\pi\)
\(648\) 0 0
\(649\) −15.7128 −0.616782
\(650\) 0 0
\(651\) 22.0000 0.862248
\(652\) 0 0
\(653\) −34.7846 −1.36123 −0.680613 0.732643i \(-0.738287\pi\)
−0.680613 + 0.732643i \(0.738287\pi\)
\(654\) 0 0
\(655\) 41.8564 1.63547
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 24.8564 0.968268 0.484134 0.874994i \(-0.339135\pi\)
0.484134 + 0.874994i \(0.339135\pi\)
\(660\) 0 0
\(661\) −23.5359 −0.915440 −0.457720 0.889096i \(-0.651334\pi\)
−0.457720 + 0.889096i \(0.651334\pi\)
\(662\) 0 0
\(663\) 14.6077 0.567316
\(664\) 0 0
\(665\) −4.92820 −0.191108
\(666\) 0 0
\(667\) −2.07180 −0.0802203
\(668\) 0 0
\(669\) −13.0718 −0.505385
\(670\) 0 0
\(671\) 19.7128 0.761005
\(672\) 0 0
\(673\) 22.9282 0.883817 0.441909 0.897060i \(-0.354302\pi\)
0.441909 + 0.897060i \(0.354302\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) −13.3923 −0.514708 −0.257354 0.966317i \(-0.582851\pi\)
−0.257354 + 0.966317i \(0.582851\pi\)
\(678\) 0 0
\(679\) 2.28719 0.0877742
\(680\) 0 0
\(681\) 11.9282 0.457090
\(682\) 0 0
\(683\) −9.85641 −0.377145 −0.188572 0.982059i \(-0.560386\pi\)
−0.188572 + 0.982059i \(0.560386\pi\)
\(684\) 0 0
\(685\) −37.7128 −1.44093
\(686\) 0 0
\(687\) 16.9282 0.645851
\(688\) 0 0
\(689\) −13.6410 −0.519681
\(690\) 0 0
\(691\) 19.8564 0.755373 0.377687 0.925933i \(-0.376720\pi\)
0.377687 + 0.925933i \(0.376720\pi\)
\(692\) 0 0
\(693\) −19.7128 −0.748828
\(694\) 0 0
\(695\) 23.7128 0.899478
\(696\) 0 0
\(697\) 41.0718 1.55571
\(698\) 0 0
\(699\) 11.8564 0.448450
\(700\) 0 0
\(701\) 47.7128 1.80209 0.901044 0.433728i \(-0.142802\pi\)
0.901044 + 0.433728i \(0.142802\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 13.8564 0.521862
\(706\) 0 0
\(707\) 44.0000 1.65479
\(708\) 0 0
\(709\) 30.7846 1.15614 0.578070 0.815987i \(-0.303806\pi\)
0.578070 + 0.815987i \(0.303806\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) −4.14359 −0.155179
\(714\) 0 0
\(715\) −19.7128 −0.737217
\(716\) 0 0
\(717\) 18.3205 0.684192
\(718\) 0 0
\(719\) 7.39230 0.275686 0.137843 0.990454i \(-0.455983\pi\)
0.137843 + 0.990454i \(0.455983\pi\)
\(720\) 0 0
\(721\) 26.9282 1.00286
\(722\) 0 0
\(723\) 16.7846 0.624226
\(724\) 0 0
\(725\) 4.46410 0.165793
\(726\) 0 0
\(727\) 34.3205 1.27288 0.636439 0.771327i \(-0.280407\pi\)
0.636439 + 0.771327i \(0.280407\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −52.9282 −1.95762
\(732\) 0 0
\(733\) −16.1436 −0.596277 −0.298139 0.954523i \(-0.596366\pi\)
−0.298139 + 0.954523i \(0.596366\pi\)
\(734\) 0 0
\(735\) −1.85641 −0.0684746
\(736\) 0 0
\(737\) −31.7128 −1.16816
\(738\) 0 0
\(739\) 17.8564 0.656859 0.328429 0.944529i \(-0.393481\pi\)
0.328429 + 0.944529i \(0.393481\pi\)
\(740\) 0 0
\(741\) 2.46410 0.0905210
\(742\) 0 0
\(743\) 3.07180 0.112693 0.0563466 0.998411i \(-0.482055\pi\)
0.0563466 + 0.998411i \(0.482055\pi\)
\(744\) 0 0
\(745\) 5.85641 0.214562
\(746\) 0 0
\(747\) 21.8564 0.799684
\(748\) 0 0
\(749\) −0.176915 −0.00646432
\(750\) 0 0
\(751\) −46.9282 −1.71243 −0.856217 0.516616i \(-0.827192\pi\)
−0.856217 + 0.516616i \(0.827192\pi\)
\(752\) 0 0
\(753\) −0.928203 −0.0338256
\(754\) 0 0
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) −42.7846 −1.55503 −0.777517 0.628862i \(-0.783521\pi\)
−0.777517 + 0.628862i \(0.783521\pi\)
\(758\) 0 0
\(759\) −1.85641 −0.0673833
\(760\) 0 0
\(761\) 6.71281 0.243339 0.121670 0.992571i \(-0.461175\pi\)
0.121670 + 0.992571i \(0.461175\pi\)
\(762\) 0 0
\(763\) 33.0000 1.19468
\(764\) 0 0
\(765\) −23.7128 −0.857339
\(766\) 0 0
\(767\) 9.67949 0.349506
\(768\) 0 0
\(769\) −47.0000 −1.69486 −0.847432 0.530904i \(-0.821852\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 0 0
\(771\) 12.9282 0.465598
\(772\) 0 0
\(773\) −47.5359 −1.70975 −0.854874 0.518836i \(-0.826365\pi\)
−0.854874 + 0.518836i \(0.826365\pi\)
\(774\) 0 0
\(775\) 8.92820 0.320711
\(776\) 0 0
\(777\) −4.92820 −0.176798
\(778\) 0 0
\(779\) 6.92820 0.248229
\(780\) 0 0
\(781\) 56.0000 2.00384
\(782\) 0 0
\(783\) 22.3205 0.797670
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) 21.7846 0.776537 0.388269 0.921546i \(-0.373073\pi\)
0.388269 + 0.921546i \(0.373073\pi\)
\(788\) 0 0
\(789\) 10.9282 0.389054
\(790\) 0 0
\(791\) 2.28719 0.0813230
\(792\) 0 0
\(793\) −12.1436 −0.431232
\(794\) 0 0
\(795\) −11.0718 −0.392676
\(796\) 0 0
\(797\) −32.1769 −1.13976 −0.569882 0.821726i \(-0.693011\pi\)
−0.569882 + 0.821726i \(0.693011\pi\)
\(798\) 0 0
\(799\) 41.0718 1.45302
\(800\) 0 0
\(801\) −25.8564 −0.913591
\(802\) 0 0
\(803\) 28.0000 0.988099
\(804\) 0 0
\(805\) −2.28719 −0.0806128
\(806\) 0 0
\(807\) −19.8564 −0.698979
\(808\) 0 0
\(809\) 31.9282 1.12254 0.561268 0.827634i \(-0.310314\pi\)
0.561268 + 0.827634i \(0.310314\pi\)
\(810\) 0 0
\(811\) 28.8564 1.01329 0.506643 0.862156i \(-0.330886\pi\)
0.506643 + 0.862156i \(0.330886\pi\)
\(812\) 0 0
\(813\) −25.3923 −0.890547
\(814\) 0 0
\(815\) −7.71281 −0.270168
\(816\) 0 0
\(817\) −8.92820 −0.312358
\(818\) 0 0
\(819\) 12.1436 0.424331
\(820\) 0 0
\(821\) 24.6410 0.859977 0.429989 0.902834i \(-0.358518\pi\)
0.429989 + 0.902834i \(0.358518\pi\)
\(822\) 0 0
\(823\) 30.4641 1.06191 0.530956 0.847399i \(-0.321833\pi\)
0.530956 + 0.847399i \(0.321833\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 40.7128 1.41572 0.707862 0.706351i \(-0.249660\pi\)
0.707862 + 0.706351i \(0.249660\pi\)
\(828\) 0 0
\(829\) 22.3205 0.775223 0.387612 0.921823i \(-0.373300\pi\)
0.387612 + 0.921823i \(0.373300\pi\)
\(830\) 0 0
\(831\) 11.0718 0.384076
\(832\) 0 0
\(833\) −5.50258 −0.190653
\(834\) 0 0
\(835\) −37.8564 −1.31007
\(836\) 0 0
\(837\) 44.6410 1.54302
\(838\) 0 0
\(839\) −22.0000 −0.759524 −0.379762 0.925084i \(-0.623994\pi\)
−0.379762 + 0.925084i \(0.623994\pi\)
\(840\) 0 0
\(841\) −9.07180 −0.312821
\(842\) 0 0
\(843\) −4.00000 −0.137767
\(844\) 0 0
\(845\) −13.8564 −0.476675
\(846\) 0 0
\(847\) −12.3205 −0.423338
\(848\) 0 0
\(849\) −5.07180 −0.174064
\(850\) 0 0
\(851\) 0.928203 0.0318184
\(852\) 0 0
\(853\) −26.9282 −0.922004 −0.461002 0.887399i \(-0.652510\pi\)
−0.461002 + 0.887399i \(0.652510\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 22.1436 0.756411 0.378205 0.925722i \(-0.376541\pi\)
0.378205 + 0.925722i \(0.376541\pi\)
\(858\) 0 0
\(859\) −34.9282 −1.19173 −0.595867 0.803083i \(-0.703192\pi\)
−0.595867 + 0.803083i \(0.703192\pi\)
\(860\) 0 0
\(861\) −17.0718 −0.581805
\(862\) 0 0
\(863\) −26.7846 −0.911759 −0.455879 0.890042i \(-0.650675\pi\)
−0.455879 + 0.890042i \(0.650675\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) 0 0
\(867\) 18.1436 0.616189
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 19.5359 0.661949
\(872\) 0 0
\(873\) 1.85641 0.0628298
\(874\) 0 0
\(875\) 29.5692 0.999622
\(876\) 0 0
\(877\) 17.2487 0.582448 0.291224 0.956655i \(-0.405938\pi\)
0.291224 + 0.956655i \(0.405938\pi\)
\(878\) 0 0
\(879\) 26.3205 0.887769
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) −5.71281 −0.192251 −0.0961257 0.995369i \(-0.530645\pi\)
−0.0961257 + 0.995369i \(0.530645\pi\)
\(884\) 0 0
\(885\) 7.85641 0.264090
\(886\) 0 0
\(887\) 0.928203 0.0311660 0.0155830 0.999879i \(-0.495040\pi\)
0.0155830 + 0.999879i \(0.495040\pi\)
\(888\) 0 0
\(889\) 31.8564 1.06843
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 6.92820 0.231843
\(894\) 0 0
\(895\) −3.71281 −0.124106
\(896\) 0 0
\(897\) 1.14359 0.0381835
\(898\) 0 0
\(899\) 39.8564 1.32929
\(900\) 0 0
\(901\) −32.8179 −1.09332
\(902\) 0 0
\(903\) 22.0000 0.732114
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) 40.8564 1.35661 0.678307 0.734778i \(-0.262714\pi\)
0.678307 + 0.734778i \(0.262714\pi\)
\(908\) 0 0
\(909\) 35.7128 1.18452
\(910\) 0 0
\(911\) 19.8564 0.657872 0.328936 0.944352i \(-0.393310\pi\)
0.328936 + 0.944352i \(0.393310\pi\)
\(912\) 0 0
\(913\) 43.7128 1.44668
\(914\) 0 0
\(915\) −9.85641 −0.325843
\(916\) 0 0
\(917\) −51.5692 −1.70297
\(918\) 0 0
\(919\) −15.3923 −0.507745 −0.253873 0.967238i \(-0.581704\pi\)
−0.253873 + 0.967238i \(0.581704\pi\)
\(920\) 0 0
\(921\) −25.8564 −0.851998
\(922\) 0 0
\(923\) −34.4974 −1.13550
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 21.8564 0.717859
\(928\) 0 0
\(929\) −46.7128 −1.53260 −0.766299 0.642484i \(-0.777904\pi\)
−0.766299 + 0.642484i \(0.777904\pi\)
\(930\) 0 0
\(931\) −0.928203 −0.0304206
\(932\) 0 0
\(933\) −13.3923 −0.438444
\(934\) 0 0
\(935\) −47.4256 −1.55098
\(936\) 0 0
\(937\) −16.0718 −0.525043 −0.262521 0.964926i \(-0.584554\pi\)
−0.262521 + 0.964926i \(0.584554\pi\)
\(938\) 0 0
\(939\) 10.8564 0.354285
\(940\) 0 0
\(941\) −35.1051 −1.14439 −0.572197 0.820116i \(-0.693909\pi\)
−0.572197 + 0.820116i \(0.693909\pi\)
\(942\) 0 0
\(943\) 3.21539 0.104708
\(944\) 0 0
\(945\) 24.6410 0.801572
\(946\) 0 0
\(947\) 3.07180 0.0998200 0.0499100 0.998754i \(-0.484107\pi\)
0.0499100 + 0.998754i \(0.484107\pi\)
\(948\) 0 0
\(949\) −17.2487 −0.559917
\(950\) 0 0
\(951\) −7.53590 −0.244368
\(952\) 0 0
\(953\) −48.7846 −1.58029 −0.790144 0.612921i \(-0.789994\pi\)
−0.790144 + 0.612921i \(0.789994\pi\)
\(954\) 0 0
\(955\) −20.9282 −0.677221
\(956\) 0 0
\(957\) 17.8564 0.577216
\(958\) 0 0
\(959\) 46.4641 1.50040
\(960\) 0 0
\(961\) 48.7128 1.57138
\(962\) 0 0
\(963\) −0.143594 −0.00462724
\(964\) 0 0
\(965\) −35.7128 −1.14964
\(966\) 0 0
\(967\) 16.7846 0.539757 0.269878 0.962894i \(-0.413017\pi\)
0.269878 + 0.962894i \(0.413017\pi\)
\(968\) 0 0
\(969\) 5.92820 0.190441
\(970\) 0 0
\(971\) −1.85641 −0.0595749 −0.0297875 0.999556i \(-0.509483\pi\)
−0.0297875 + 0.999556i \(0.509483\pi\)
\(972\) 0 0
\(973\) −29.2154 −0.936602
\(974\) 0 0
\(975\) −2.46410 −0.0789144
\(976\) 0 0
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) −51.7128 −1.65275
\(980\) 0 0
\(981\) 26.7846 0.855167
\(982\) 0 0
\(983\) 48.6410 1.55141 0.775704 0.631097i \(-0.217395\pi\)
0.775704 + 0.631097i \(0.217395\pi\)
\(984\) 0 0
\(985\) −13.8564 −0.441502
\(986\) 0 0
\(987\) −17.0718 −0.543401
\(988\) 0 0
\(989\) −4.14359 −0.131759
\(990\) 0 0
\(991\) 39.5692 1.25696 0.628479 0.777827i \(-0.283678\pi\)
0.628479 + 0.777827i \(0.283678\pi\)
\(992\) 0 0
\(993\) 31.7846 1.00865
\(994\) 0 0
\(995\) 40.9282 1.29751
\(996\) 0 0
\(997\) −1.21539 −0.0384918 −0.0192459 0.999815i \(-0.506127\pi\)
−0.0192459 + 0.999815i \(0.506127\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 4864.2.a.x.1.1 2
4.3 odd 2 4864.2.a.u.1.2 2
8.3 odd 2 4864.2.a.v.1.2 2
8.5 even 2 4864.2.a.s.1.1 2
16.3 odd 4 2432.2.c.d.1217.1 yes 4
16.5 even 4 2432.2.c.c.1217.2 4
16.11 odd 4 2432.2.c.d.1217.3 yes 4
16.13 even 4 2432.2.c.c.1217.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.c.1217.2 4 16.5 even 4
2432.2.c.c.1217.4 yes 4 16.13 even 4
2432.2.c.d.1217.1 yes 4 16.3 odd 4
2432.2.c.d.1217.3 yes 4 16.11 odd 4
4864.2.a.s.1.1 2 8.5 even 2
4864.2.a.u.1.2 2 4.3 odd 2
4864.2.a.v.1.2 2 8.3 odd 2
4864.2.a.x.1.1 2 1.1 even 1 trivial