Properties

Label 1840.2.a.q.1.1
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
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] = [1840,2,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, 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("1840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.47283 q^{3} +1.00000 q^{5} -0.527166 q^{7} +3.11491 q^{9} +O(q^{10})\) \(q-2.47283 q^{3} +1.00000 q^{5} -0.527166 q^{7} +3.11491 q^{9} -3.11491 q^{11} +4.11491 q^{13} -2.47283 q^{15} -4.39905 q^{17} +3.70265 q^{19} +1.30359 q^{21} +1.00000 q^{23} +1.00000 q^{25} -0.284147 q^{27} -9.10170 q^{29} -4.83076 q^{31} +7.70265 q^{33} -0.527166 q^{35} +9.74378 q^{37} -10.1755 q^{39} +6.93246 q^{41} +4.45963 q^{43} +3.11491 q^{45} -0.642074 q^{47} -6.72210 q^{49} +10.8781 q^{51} -3.89134 q^{53} -3.11491 q^{55} -9.15604 q^{57} -8.79811 q^{59} -3.45339 q^{61} -1.64207 q^{63} +4.11491 q^{65} +8.60719 q^{67} -2.47283 q^{69} -12.3642 q^{71} -5.81756 q^{73} -2.47283 q^{75} +1.64207 q^{77} +3.17548 q^{79} -8.64207 q^{81} -4.71585 q^{83} -4.39905 q^{85} +22.5070 q^{87} -5.43171 q^{89} -2.16924 q^{91} +11.9457 q^{93} +3.70265 q^{95} -4.06058 q^{97} -9.70265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9} - 3 q^{11} + 6 q^{13} - 2 q^{15} - 5 q^{17} - 7 q^{19} - 6 q^{21} + 3 q^{23} + 3 q^{25} + q^{27} - q^{29} - 10 q^{31} + 5 q^{33} - 7 q^{35} + 2 q^{37} - 7 q^{39} - 10 q^{41} - 12 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} - 9 q^{51} + 10 q^{53} - 3 q^{55} - 12 q^{57} - 10 q^{59} - 13 q^{61} - 4 q^{63} + 6 q^{65} + 6 q^{67} - 2 q^{69} - 10 q^{71} + 7 q^{73} - 2 q^{75} + 4 q^{77} - 14 q^{79} - 25 q^{81} - 16 q^{83} - 5 q^{85} + 5 q^{87} - 20 q^{89} - 11 q^{91} + 25 q^{93} - 7 q^{95} + 5 q^{97} - 11 q^{99}+O(q^{100}) \) 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
\(3\) −2.47283 −1.42769 −0.713846 0.700303i \(-0.753048\pi\)
−0.713846 + 0.700303i \(0.753048\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.527166 −0.199250 −0.0996250 0.995025i \(-0.531764\pi\)
−0.0996250 + 0.995025i \(0.531764\pi\)
\(8\) 0 0
\(9\) 3.11491 1.03830
\(10\) 0 0
\(11\) −3.11491 −0.939180 −0.469590 0.882885i \(-0.655598\pi\)
−0.469590 + 0.882885i \(0.655598\pi\)
\(12\) 0 0
\(13\) 4.11491 1.14127 0.570635 0.821204i \(-0.306697\pi\)
0.570635 + 0.821204i \(0.306697\pi\)
\(14\) 0 0
\(15\) −2.47283 −0.638483
\(16\) 0 0
\(17\) −4.39905 −1.06693 −0.533464 0.845823i \(-0.679110\pi\)
−0.533464 + 0.845823i \(0.679110\pi\)
\(18\) 0 0
\(19\) 3.70265 0.849446 0.424723 0.905323i \(-0.360372\pi\)
0.424723 + 0.905323i \(0.360372\pi\)
\(20\) 0 0
\(21\) 1.30359 0.284468
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −0.284147 −0.0546842
\(28\) 0 0
\(29\) −9.10170 −1.69014 −0.845072 0.534653i \(-0.820442\pi\)
−0.845072 + 0.534653i \(0.820442\pi\)
\(30\) 0 0
\(31\) −4.83076 −0.867630 −0.433815 0.901002i \(-0.642833\pi\)
−0.433815 + 0.901002i \(0.642833\pi\)
\(32\) 0 0
\(33\) 7.70265 1.34086
\(34\) 0 0
\(35\) −0.527166 −0.0891073
\(36\) 0 0
\(37\) 9.74378 1.60187 0.800934 0.598753i \(-0.204337\pi\)
0.800934 + 0.598753i \(0.204337\pi\)
\(38\) 0 0
\(39\) −10.1755 −1.62938
\(40\) 0 0
\(41\) 6.93246 1.08267 0.541334 0.840807i \(-0.317919\pi\)
0.541334 + 0.840807i \(0.317919\pi\)
\(42\) 0 0
\(43\) 4.45963 0.680087 0.340044 0.940410i \(-0.389558\pi\)
0.340044 + 0.940410i \(0.389558\pi\)
\(44\) 0 0
\(45\) 3.11491 0.464343
\(46\) 0 0
\(47\) −0.642074 −0.0936561 −0.0468280 0.998903i \(-0.514911\pi\)
−0.0468280 + 0.998903i \(0.514911\pi\)
\(48\) 0 0
\(49\) −6.72210 −0.960299
\(50\) 0 0
\(51\) 10.8781 1.52324
\(52\) 0 0
\(53\) −3.89134 −0.534516 −0.267258 0.963625i \(-0.586118\pi\)
−0.267258 + 0.963625i \(0.586118\pi\)
\(54\) 0 0
\(55\) −3.11491 −0.420014
\(56\) 0 0
\(57\) −9.15604 −1.21275
\(58\) 0 0
\(59\) −8.79811 −1.14542 −0.572708 0.819759i \(-0.694107\pi\)
−0.572708 + 0.819759i \(0.694107\pi\)
\(60\) 0 0
\(61\) −3.45339 −0.442161 −0.221080 0.975256i \(-0.570958\pi\)
−0.221080 + 0.975256i \(0.570958\pi\)
\(62\) 0 0
\(63\) −1.64207 −0.206882
\(64\) 0 0
\(65\) 4.11491 0.510391
\(66\) 0 0
\(67\) 8.60719 1.05154 0.525768 0.850628i \(-0.323778\pi\)
0.525768 + 0.850628i \(0.323778\pi\)
\(68\) 0 0
\(69\) −2.47283 −0.297694
\(70\) 0 0
\(71\) −12.3642 −1.46736 −0.733678 0.679497i \(-0.762198\pi\)
−0.733678 + 0.679497i \(0.762198\pi\)
\(72\) 0 0
\(73\) −5.81756 −0.680893 −0.340447 0.940264i \(-0.610578\pi\)
−0.340447 + 0.940264i \(0.610578\pi\)
\(74\) 0 0
\(75\) −2.47283 −0.285538
\(76\) 0 0
\(77\) 1.64207 0.187132
\(78\) 0 0
\(79\) 3.17548 0.357270 0.178635 0.983915i \(-0.442832\pi\)
0.178635 + 0.983915i \(0.442832\pi\)
\(80\) 0 0
\(81\) −8.64207 −0.960230
\(82\) 0 0
\(83\) −4.71585 −0.517632 −0.258816 0.965927i \(-0.583332\pi\)
−0.258816 + 0.965927i \(0.583332\pi\)
\(84\) 0 0
\(85\) −4.39905 −0.477144
\(86\) 0 0
\(87\) 22.5070 2.41300
\(88\) 0 0
\(89\) −5.43171 −0.575760 −0.287880 0.957667i \(-0.592950\pi\)
−0.287880 + 0.957667i \(0.592950\pi\)
\(90\) 0 0
\(91\) −2.16924 −0.227398
\(92\) 0 0
\(93\) 11.9457 1.23871
\(94\) 0 0
\(95\) 3.70265 0.379884
\(96\) 0 0
\(97\) −4.06058 −0.412289 −0.206144 0.978522i \(-0.566092\pi\)
−0.206144 + 0.978522i \(0.566092\pi\)
\(98\) 0 0
\(99\) −9.70265 −0.975153
\(100\) 0 0
\(101\) −15.5529 −1.54757 −0.773784 0.633450i \(-0.781638\pi\)
−0.773784 + 0.633450i \(0.781638\pi\)
\(102\) 0 0
\(103\) −17.9519 −1.76885 −0.884427 0.466678i \(-0.845451\pi\)
−0.884427 + 0.466678i \(0.845451\pi\)
\(104\) 0 0
\(105\) 1.30359 0.127218
\(106\) 0 0
\(107\) 12.2298 1.18230 0.591150 0.806561i \(-0.298674\pi\)
0.591150 + 0.806561i \(0.298674\pi\)
\(108\) 0 0
\(109\) 9.10795 0.872383 0.436192 0.899854i \(-0.356327\pi\)
0.436192 + 0.899854i \(0.356327\pi\)
\(110\) 0 0
\(111\) −24.0947 −2.28697
\(112\) 0 0
\(113\) −13.1755 −1.23945 −0.619723 0.784821i \(-0.712755\pi\)
−0.619723 + 0.784821i \(0.712755\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 12.8176 1.18498
\(118\) 0 0
\(119\) 2.31903 0.212585
\(120\) 0 0
\(121\) −1.29735 −0.117941
\(122\) 0 0
\(123\) −17.1428 −1.54572
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −19.5613 −1.73579 −0.867894 0.496750i \(-0.834527\pi\)
−0.867894 + 0.496750i \(0.834527\pi\)
\(128\) 0 0
\(129\) −11.0279 −0.970955
\(130\) 0 0
\(131\) 4.15604 0.363115 0.181557 0.983380i \(-0.441886\pi\)
0.181557 + 0.983380i \(0.441886\pi\)
\(132\) 0 0
\(133\) −1.95191 −0.169252
\(134\) 0 0
\(135\) −0.284147 −0.0244555
\(136\) 0 0
\(137\) −11.1344 −0.951272 −0.475636 0.879642i \(-0.657782\pi\)
−0.475636 + 0.879642i \(0.657782\pi\)
\(138\) 0 0
\(139\) 6.49452 0.550858 0.275429 0.961321i \(-0.411180\pi\)
0.275429 + 0.961321i \(0.411180\pi\)
\(140\) 0 0
\(141\) 1.58774 0.133712
\(142\) 0 0
\(143\) −12.8176 −1.07186
\(144\) 0 0
\(145\) −9.10170 −0.755855
\(146\) 0 0
\(147\) 16.6226 1.37101
\(148\) 0 0
\(149\) −5.47283 −0.448352 −0.224176 0.974549i \(-0.571969\pi\)
−0.224176 + 0.974549i \(0.571969\pi\)
\(150\) 0 0
\(151\) 0.0954606 0.00776848 0.00388424 0.999992i \(-0.498764\pi\)
0.00388424 + 0.999992i \(0.498764\pi\)
\(152\) 0 0
\(153\) −13.7026 −1.10779
\(154\) 0 0
\(155\) −4.83076 −0.388016
\(156\) 0 0
\(157\) −3.40530 −0.271772 −0.135886 0.990724i \(-0.543388\pi\)
−0.135886 + 0.990724i \(0.543388\pi\)
\(158\) 0 0
\(159\) 9.62263 0.763124
\(160\) 0 0
\(161\) −0.527166 −0.0415465
\(162\) 0 0
\(163\) −18.5745 −1.45487 −0.727435 0.686177i \(-0.759288\pi\)
−0.727435 + 0.686177i \(0.759288\pi\)
\(164\) 0 0
\(165\) 7.70265 0.599650
\(166\) 0 0
\(167\) 12.4596 0.964155 0.482078 0.876129i \(-0.339882\pi\)
0.482078 + 0.876129i \(0.339882\pi\)
\(168\) 0 0
\(169\) 3.93246 0.302497
\(170\) 0 0
\(171\) 11.5334 0.881982
\(172\) 0 0
\(173\) 3.49228 0.265513 0.132757 0.991149i \(-0.457617\pi\)
0.132757 + 0.991149i \(0.457617\pi\)
\(174\) 0 0
\(175\) −0.527166 −0.0398500
\(176\) 0 0
\(177\) 21.7563 1.63530
\(178\) 0 0
\(179\) −6.38585 −0.477301 −0.238650 0.971106i \(-0.576705\pi\)
−0.238650 + 0.971106i \(0.576705\pi\)
\(180\) 0 0
\(181\) −20.8781 −1.55186 −0.775930 0.630819i \(-0.782719\pi\)
−0.775930 + 0.630819i \(0.782719\pi\)
\(182\) 0 0
\(183\) 8.53965 0.631269
\(184\) 0 0
\(185\) 9.74378 0.716377
\(186\) 0 0
\(187\) 13.7026 1.00204
\(188\) 0 0
\(189\) 0.149793 0.0108958
\(190\) 0 0
\(191\) 6.83700 0.494708 0.247354 0.968925i \(-0.420439\pi\)
0.247354 + 0.968925i \(0.420439\pi\)
\(192\) 0 0
\(193\) 7.10170 0.511192 0.255596 0.966784i \(-0.417728\pi\)
0.255596 + 0.966784i \(0.417728\pi\)
\(194\) 0 0
\(195\) −10.1755 −0.728681
\(196\) 0 0
\(197\) 0.850207 0.0605748 0.0302874 0.999541i \(-0.490358\pi\)
0.0302874 + 0.999541i \(0.490358\pi\)
\(198\) 0 0
\(199\) −8.56829 −0.607390 −0.303695 0.952769i \(-0.598220\pi\)
−0.303695 + 0.952769i \(0.598220\pi\)
\(200\) 0 0
\(201\) −21.2841 −1.50127
\(202\) 0 0
\(203\) 4.79811 0.336761
\(204\) 0 0
\(205\) 6.93246 0.484184
\(206\) 0 0
\(207\) 3.11491 0.216501
\(208\) 0 0
\(209\) −11.5334 −0.797783
\(210\) 0 0
\(211\) −9.63511 −0.663309 −0.331654 0.943401i \(-0.607607\pi\)
−0.331654 + 0.943401i \(0.607607\pi\)
\(212\) 0 0
\(213\) 30.5745 2.09493
\(214\) 0 0
\(215\) 4.45963 0.304144
\(216\) 0 0
\(217\) 2.54661 0.172875
\(218\) 0 0
\(219\) 14.3859 0.972106
\(220\) 0 0
\(221\) −18.1017 −1.21765
\(222\) 0 0
\(223\) −9.43171 −0.631594 −0.315797 0.948827i \(-0.602272\pi\)
−0.315797 + 0.948827i \(0.602272\pi\)
\(224\) 0 0
\(225\) 3.11491 0.207661
\(226\) 0 0
\(227\) 3.91774 0.260030 0.130015 0.991512i \(-0.458497\pi\)
0.130015 + 0.991512i \(0.458497\pi\)
\(228\) 0 0
\(229\) −22.0947 −1.46006 −0.730031 0.683414i \(-0.760494\pi\)
−0.730031 + 0.683414i \(0.760494\pi\)
\(230\) 0 0
\(231\) −4.06058 −0.267166
\(232\) 0 0
\(233\) −12.5334 −0.821091 −0.410545 0.911840i \(-0.634662\pi\)
−0.410545 + 0.911840i \(0.634662\pi\)
\(234\) 0 0
\(235\) −0.642074 −0.0418843
\(236\) 0 0
\(237\) −7.85244 −0.510071
\(238\) 0 0
\(239\) 12.4123 0.802882 0.401441 0.915885i \(-0.368509\pi\)
0.401441 + 0.915885i \(0.368509\pi\)
\(240\) 0 0
\(241\) 5.74378 0.369989 0.184995 0.982740i \(-0.440773\pi\)
0.184995 + 0.982740i \(0.440773\pi\)
\(242\) 0 0
\(243\) 22.2229 1.42560
\(244\) 0 0
\(245\) −6.72210 −0.429459
\(246\) 0 0
\(247\) 15.2361 0.969447
\(248\) 0 0
\(249\) 11.6615 0.739019
\(250\) 0 0
\(251\) 6.33624 0.399940 0.199970 0.979802i \(-0.435915\pi\)
0.199970 + 0.979802i \(0.435915\pi\)
\(252\) 0 0
\(253\) −3.11491 −0.195833
\(254\) 0 0
\(255\) 10.8781 0.681215
\(256\) 0 0
\(257\) 22.2772 1.38961 0.694806 0.719197i \(-0.255490\pi\)
0.694806 + 0.719197i \(0.255490\pi\)
\(258\) 0 0
\(259\) −5.13659 −0.319172
\(260\) 0 0
\(261\) −28.3510 −1.75488
\(262\) 0 0
\(263\) 4.66776 0.287827 0.143913 0.989590i \(-0.454031\pi\)
0.143913 + 0.989590i \(0.454031\pi\)
\(264\) 0 0
\(265\) −3.89134 −0.239043
\(266\) 0 0
\(267\) 13.4317 0.822007
\(268\) 0 0
\(269\) 9.70889 0.591962 0.295981 0.955194i \(-0.404354\pi\)
0.295981 + 0.955194i \(0.404354\pi\)
\(270\) 0 0
\(271\) 6.75698 0.410457 0.205229 0.978714i \(-0.434206\pi\)
0.205229 + 0.978714i \(0.434206\pi\)
\(272\) 0 0
\(273\) 5.36417 0.324654
\(274\) 0 0
\(275\) −3.11491 −0.187836
\(276\) 0 0
\(277\) 19.0194 1.14277 0.571384 0.820683i \(-0.306407\pi\)
0.571384 + 0.820683i \(0.306407\pi\)
\(278\) 0 0
\(279\) −15.0474 −0.900863
\(280\) 0 0
\(281\) −32.8370 −1.95889 −0.979446 0.201708i \(-0.935351\pi\)
−0.979446 + 0.201708i \(0.935351\pi\)
\(282\) 0 0
\(283\) −13.7827 −0.819295 −0.409647 0.912244i \(-0.634348\pi\)
−0.409647 + 0.912244i \(0.634348\pi\)
\(284\) 0 0
\(285\) −9.15604 −0.542357
\(286\) 0 0
\(287\) −3.65456 −0.215722
\(288\) 0 0
\(289\) 2.35168 0.138334
\(290\) 0 0
\(291\) 10.0411 0.588621
\(292\) 0 0
\(293\) 15.0668 0.880213 0.440106 0.897946i \(-0.354941\pi\)
0.440106 + 0.897946i \(0.354941\pi\)
\(294\) 0 0
\(295\) −8.79811 −0.512246
\(296\) 0 0
\(297\) 0.885092 0.0513583
\(298\) 0 0
\(299\) 4.11491 0.237971
\(300\) 0 0
\(301\) −2.35097 −0.135507
\(302\) 0 0
\(303\) 38.4596 2.20945
\(304\) 0 0
\(305\) −3.45339 −0.197740
\(306\) 0 0
\(307\) 5.91302 0.337474 0.168737 0.985661i \(-0.446031\pi\)
0.168737 + 0.985661i \(0.446031\pi\)
\(308\) 0 0
\(309\) 44.3921 2.52538
\(310\) 0 0
\(311\) 25.7911 1.46248 0.731241 0.682119i \(-0.238942\pi\)
0.731241 + 0.682119i \(0.238942\pi\)
\(312\) 0 0
\(313\) 16.7306 0.945668 0.472834 0.881152i \(-0.343231\pi\)
0.472834 + 0.881152i \(0.343231\pi\)
\(314\) 0 0
\(315\) −1.64207 −0.0925204
\(316\) 0 0
\(317\) 28.1623 1.58175 0.790876 0.611977i \(-0.209625\pi\)
0.790876 + 0.611977i \(0.209625\pi\)
\(318\) 0 0
\(319\) 28.3510 1.58735
\(320\) 0 0
\(321\) −30.2423 −1.68796
\(322\) 0 0
\(323\) −16.2882 −0.906297
\(324\) 0 0
\(325\) 4.11491 0.228254
\(326\) 0 0
\(327\) −22.5224 −1.24549
\(328\) 0 0
\(329\) 0.338479 0.0186610
\(330\) 0 0
\(331\) 25.8176 1.41906 0.709531 0.704675i \(-0.248907\pi\)
0.709531 + 0.704675i \(0.248907\pi\)
\(332\) 0 0
\(333\) 30.3510 1.66322
\(334\) 0 0
\(335\) 8.60719 0.470261
\(336\) 0 0
\(337\) 33.5676 1.82854 0.914271 0.405103i \(-0.132764\pi\)
0.914271 + 0.405103i \(0.132764\pi\)
\(338\) 0 0
\(339\) 32.5808 1.76955
\(340\) 0 0
\(341\) 15.0474 0.814861
\(342\) 0 0
\(343\) 7.23382 0.390590
\(344\) 0 0
\(345\) −2.47283 −0.133133
\(346\) 0 0
\(347\) −7.63984 −0.410128 −0.205064 0.978749i \(-0.565740\pi\)
−0.205064 + 0.978749i \(0.565740\pi\)
\(348\) 0 0
\(349\) 3.01945 0.161627 0.0808136 0.996729i \(-0.474248\pi\)
0.0808136 + 0.996729i \(0.474248\pi\)
\(350\) 0 0
\(351\) −1.16924 −0.0624094
\(352\) 0 0
\(353\) −25.1840 −1.34041 −0.670203 0.742177i \(-0.733793\pi\)
−0.670203 + 0.742177i \(0.733793\pi\)
\(354\) 0 0
\(355\) −12.3642 −0.656222
\(356\) 0 0
\(357\) −5.73458 −0.303506
\(358\) 0 0
\(359\) 5.32304 0.280939 0.140470 0.990085i \(-0.455139\pi\)
0.140470 + 0.990085i \(0.455139\pi\)
\(360\) 0 0
\(361\) −5.29039 −0.278442
\(362\) 0 0
\(363\) 3.20813 0.168383
\(364\) 0 0
\(365\) −5.81756 −0.304505
\(366\) 0 0
\(367\) −8.24230 −0.430245 −0.215122 0.976587i \(-0.569015\pi\)
−0.215122 + 0.976587i \(0.569015\pi\)
\(368\) 0 0
\(369\) 21.5940 1.12414
\(370\) 0 0
\(371\) 2.05138 0.106502
\(372\) 0 0
\(373\) 1.91774 0.0992970 0.0496485 0.998767i \(-0.484190\pi\)
0.0496485 + 0.998767i \(0.484190\pi\)
\(374\) 0 0
\(375\) −2.47283 −0.127697
\(376\) 0 0
\(377\) −37.4527 −1.92891
\(378\) 0 0
\(379\) −30.1692 −1.54969 −0.774845 0.632151i \(-0.782172\pi\)
−0.774845 + 0.632151i \(0.782172\pi\)
\(380\) 0 0
\(381\) 48.3719 2.47817
\(382\) 0 0
\(383\) 1.64903 0.0842617 0.0421309 0.999112i \(-0.486585\pi\)
0.0421309 + 0.999112i \(0.486585\pi\)
\(384\) 0 0
\(385\) 1.64207 0.0836878
\(386\) 0 0
\(387\) 13.8913 0.706136
\(388\) 0 0
\(389\) −16.6414 −0.843750 −0.421875 0.906654i \(-0.638628\pi\)
−0.421875 + 0.906654i \(0.638628\pi\)
\(390\) 0 0
\(391\) −4.39905 −0.222470
\(392\) 0 0
\(393\) −10.2772 −0.518415
\(394\) 0 0
\(395\) 3.17548 0.159776
\(396\) 0 0
\(397\) 36.0257 1.80808 0.904039 0.427450i \(-0.140588\pi\)
0.904039 + 0.427450i \(0.140588\pi\)
\(398\) 0 0
\(399\) 4.82675 0.241640
\(400\) 0 0
\(401\) −5.62263 −0.280781 −0.140390 0.990096i \(-0.544836\pi\)
−0.140390 + 0.990096i \(0.544836\pi\)
\(402\) 0 0
\(403\) −19.8781 −0.990200
\(404\) 0 0
\(405\) −8.64207 −0.429428
\(406\) 0 0
\(407\) −30.3510 −1.50444
\(408\) 0 0
\(409\) −2.76546 −0.136743 −0.0683716 0.997660i \(-0.521780\pi\)
−0.0683716 + 0.997660i \(0.521780\pi\)
\(410\) 0 0
\(411\) 27.5334 1.35812
\(412\) 0 0
\(413\) 4.63807 0.228224
\(414\) 0 0
\(415\) −4.71585 −0.231492
\(416\) 0 0
\(417\) −16.0599 −0.786455
\(418\) 0 0
\(419\) 17.5962 0.859632 0.429816 0.902917i \(-0.358579\pi\)
0.429816 + 0.902917i \(0.358579\pi\)
\(420\) 0 0
\(421\) 25.5676 1.24609 0.623044 0.782187i \(-0.285896\pi\)
0.623044 + 0.782187i \(0.285896\pi\)
\(422\) 0 0
\(423\) −2.00000 −0.0972433
\(424\) 0 0
\(425\) −4.39905 −0.213386
\(426\) 0 0
\(427\) 1.82051 0.0881006
\(428\) 0 0
\(429\) 31.6957 1.53028
\(430\) 0 0
\(431\) −17.9736 −0.865757 −0.432879 0.901452i \(-0.642502\pi\)
−0.432879 + 0.901452i \(0.642502\pi\)
\(432\) 0 0
\(433\) 37.0382 1.77994 0.889971 0.456018i \(-0.150725\pi\)
0.889971 + 0.456018i \(0.150725\pi\)
\(434\) 0 0
\(435\) 22.5070 1.07913
\(436\) 0 0
\(437\) 3.70265 0.177122
\(438\) 0 0
\(439\) −38.8168 −1.85263 −0.926313 0.376754i \(-0.877040\pi\)
−0.926313 + 0.376754i \(0.877040\pi\)
\(440\) 0 0
\(441\) −20.9387 −0.997081
\(442\) 0 0
\(443\) −25.3726 −1.20549 −0.602745 0.797934i \(-0.705927\pi\)
−0.602745 + 0.797934i \(0.705927\pi\)
\(444\) 0 0
\(445\) −5.43171 −0.257488
\(446\) 0 0
\(447\) 13.5334 0.640108
\(448\) 0 0
\(449\) 33.4589 1.57902 0.789512 0.613735i \(-0.210334\pi\)
0.789512 + 0.613735i \(0.210334\pi\)
\(450\) 0 0
\(451\) −21.5940 −1.01682
\(452\) 0 0
\(453\) −0.236058 −0.0110910
\(454\) 0 0
\(455\) −2.16924 −0.101696
\(456\) 0 0
\(457\) −27.5264 −1.28763 −0.643816 0.765180i \(-0.722650\pi\)
−0.643816 + 0.765180i \(0.722650\pi\)
\(458\) 0 0
\(459\) 1.24998 0.0583440
\(460\) 0 0
\(461\) 36.3161 1.69141 0.845704 0.533652i \(-0.179181\pi\)
0.845704 + 0.533652i \(0.179181\pi\)
\(462\) 0 0
\(463\) 22.5544 1.04819 0.524095 0.851660i \(-0.324404\pi\)
0.524095 + 0.851660i \(0.324404\pi\)
\(464\) 0 0
\(465\) 11.9457 0.553967
\(466\) 0 0
\(467\) −22.8370 −1.05677 −0.528385 0.849005i \(-0.677202\pi\)
−0.528385 + 0.849005i \(0.677202\pi\)
\(468\) 0 0
\(469\) −4.53742 −0.209518
\(470\) 0 0
\(471\) 8.42074 0.388007
\(472\) 0 0
\(473\) −13.8913 −0.638724
\(474\) 0 0
\(475\) 3.70265 0.169889
\(476\) 0 0
\(477\) −12.1212 −0.554989
\(478\) 0 0
\(479\) 35.5264 1.62324 0.811622 0.584182i \(-0.198585\pi\)
0.811622 + 0.584182i \(0.198585\pi\)
\(480\) 0 0
\(481\) 40.0947 1.82816
\(482\) 0 0
\(483\) 1.30359 0.0593156
\(484\) 0 0
\(485\) −4.06058 −0.184381
\(486\) 0 0
\(487\) 10.4945 0.475552 0.237776 0.971320i \(-0.423582\pi\)
0.237776 + 0.971320i \(0.423582\pi\)
\(488\) 0 0
\(489\) 45.9317 2.07711
\(490\) 0 0
\(491\) −42.1685 −1.90304 −0.951519 0.307589i \(-0.900478\pi\)
−0.951519 + 0.307589i \(0.900478\pi\)
\(492\) 0 0
\(493\) 40.0389 1.80326
\(494\) 0 0
\(495\) −9.70265 −0.436102
\(496\) 0 0
\(497\) 6.51797 0.292371
\(498\) 0 0
\(499\) −30.1560 −1.34997 −0.674985 0.737832i \(-0.735850\pi\)
−0.674985 + 0.737832i \(0.735850\pi\)
\(500\) 0 0
\(501\) −30.8106 −1.37652
\(502\) 0 0
\(503\) −22.9729 −1.02431 −0.512155 0.858893i \(-0.671153\pi\)
−0.512155 + 0.858893i \(0.671153\pi\)
\(504\) 0 0
\(505\) −15.5529 −0.692093
\(506\) 0 0
\(507\) −9.72433 −0.431873
\(508\) 0 0
\(509\) 11.8176 0.523804 0.261902 0.965094i \(-0.415650\pi\)
0.261902 + 0.965094i \(0.415650\pi\)
\(510\) 0 0
\(511\) 3.06682 0.135668
\(512\) 0 0
\(513\) −1.05210 −0.0464512
\(514\) 0 0
\(515\) −17.9519 −0.791056
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) −8.63583 −0.379071
\(520\) 0 0
\(521\) −35.7438 −1.56596 −0.782982 0.622045i \(-0.786302\pi\)
−0.782982 + 0.622045i \(0.786302\pi\)
\(522\) 0 0
\(523\) −14.3899 −0.629225 −0.314612 0.949220i \(-0.601875\pi\)
−0.314612 + 0.949220i \(0.601875\pi\)
\(524\) 0 0
\(525\) 1.30359 0.0568935
\(526\) 0 0
\(527\) 21.2508 0.925698
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −27.4053 −1.18929
\(532\) 0 0
\(533\) 28.5264 1.23562
\(534\) 0 0
\(535\) 12.2298 0.528741
\(536\) 0 0
\(537\) 15.7911 0.681438
\(538\) 0 0
\(539\) 20.9387 0.901894
\(540\) 0 0
\(541\) 14.5209 0.624303 0.312152 0.950032i \(-0.398950\pi\)
0.312152 + 0.950032i \(0.398950\pi\)
\(542\) 0 0
\(543\) 51.6282 2.21558
\(544\) 0 0
\(545\) 9.10795 0.390142
\(546\) 0 0
\(547\) −19.6289 −0.839270 −0.419635 0.907693i \(-0.637842\pi\)
−0.419635 + 0.907693i \(0.637842\pi\)
\(548\) 0 0
\(549\) −10.7570 −0.459097
\(550\) 0 0
\(551\) −33.7004 −1.43569
\(552\) 0 0
\(553\) −1.67401 −0.0711860
\(554\) 0 0
\(555\) −24.0947 −1.02276
\(556\) 0 0
\(557\) 1.13659 0.0481588 0.0240794 0.999710i \(-0.492335\pi\)
0.0240794 + 0.999710i \(0.492335\pi\)
\(558\) 0 0
\(559\) 18.3510 0.776163
\(560\) 0 0
\(561\) −33.8844 −1.43060
\(562\) 0 0
\(563\) −25.8913 −1.09119 −0.545595 0.838049i \(-0.683696\pi\)
−0.545595 + 0.838049i \(0.683696\pi\)
\(564\) 0 0
\(565\) −13.1755 −0.554297
\(566\) 0 0
\(567\) 4.55581 0.191326
\(568\) 0 0
\(569\) −44.0558 −1.84692 −0.923459 0.383698i \(-0.874650\pi\)
−0.923459 + 0.383698i \(0.874650\pi\)
\(570\) 0 0
\(571\) 41.8991 1.75342 0.876711 0.481017i \(-0.159732\pi\)
0.876711 + 0.481017i \(0.159732\pi\)
\(572\) 0 0
\(573\) −16.9068 −0.706291
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 1.34544 0.0560114 0.0280057 0.999608i \(-0.491084\pi\)
0.0280057 + 0.999608i \(0.491084\pi\)
\(578\) 0 0
\(579\) −17.5613 −0.729824
\(580\) 0 0
\(581\) 2.48604 0.103138
\(582\) 0 0
\(583\) 12.1212 0.502007
\(584\) 0 0
\(585\) 12.8176 0.529941
\(586\) 0 0
\(587\) −25.0411 −1.03356 −0.516779 0.856119i \(-0.672869\pi\)
−0.516779 + 0.856119i \(0.672869\pi\)
\(588\) 0 0
\(589\) −17.8866 −0.737005
\(590\) 0 0
\(591\) −2.10242 −0.0864821
\(592\) 0 0
\(593\) −22.5947 −0.927853 −0.463927 0.885874i \(-0.653560\pi\)
−0.463927 + 0.885874i \(0.653560\pi\)
\(594\) 0 0
\(595\) 2.31903 0.0950711
\(596\) 0 0
\(597\) 21.1880 0.867166
\(598\) 0 0
\(599\) −14.7111 −0.601080 −0.300540 0.953769i \(-0.597167\pi\)
−0.300540 + 0.953769i \(0.597167\pi\)
\(600\) 0 0
\(601\) 19.2904 0.786871 0.393436 0.919352i \(-0.371286\pi\)
0.393436 + 0.919352i \(0.371286\pi\)
\(602\) 0 0
\(603\) 26.8106 1.09181
\(604\) 0 0
\(605\) −1.29735 −0.0527448
\(606\) 0 0
\(607\) 20.7547 0.842409 0.421205 0.906966i \(-0.361607\pi\)
0.421205 + 0.906966i \(0.361607\pi\)
\(608\) 0 0
\(609\) −11.8649 −0.480791
\(610\) 0 0
\(611\) −2.64207 −0.106887
\(612\) 0 0
\(613\) −43.3091 −1.74924 −0.874619 0.484810i \(-0.838889\pi\)
−0.874619 + 0.484810i \(0.838889\pi\)
\(614\) 0 0
\(615\) −17.1428 −0.691266
\(616\) 0 0
\(617\) 13.5621 0.545988 0.272994 0.962016i \(-0.411986\pi\)
0.272994 + 0.962016i \(0.411986\pi\)
\(618\) 0 0
\(619\) 12.9045 0.518677 0.259339 0.965786i \(-0.416495\pi\)
0.259339 + 0.965786i \(0.416495\pi\)
\(620\) 0 0
\(621\) −0.284147 −0.0114024
\(622\) 0 0
\(623\) 2.86341 0.114720
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 28.5202 1.13899
\(628\) 0 0
\(629\) −42.8634 −1.70908
\(630\) 0 0
\(631\) −45.0404 −1.79303 −0.896515 0.443013i \(-0.853910\pi\)
−0.896515 + 0.443013i \(0.853910\pi\)
\(632\) 0 0
\(633\) 23.8260 0.947000
\(634\) 0 0
\(635\) −19.5613 −0.776268
\(636\) 0 0
\(637\) −27.6608 −1.09596
\(638\) 0 0
\(639\) −38.5132 −1.52356
\(640\) 0 0
\(641\) −3.08074 −0.121682 −0.0608410 0.998147i \(-0.519378\pi\)
−0.0608410 + 0.998147i \(0.519378\pi\)
\(642\) 0 0
\(643\) −4.52493 −0.178446 −0.0892229 0.996012i \(-0.528438\pi\)
−0.0892229 + 0.996012i \(0.528438\pi\)
\(644\) 0 0
\(645\) −11.0279 −0.434224
\(646\) 0 0
\(647\) −34.8719 −1.37096 −0.685478 0.728094i \(-0.740407\pi\)
−0.685478 + 0.728094i \(0.740407\pi\)
\(648\) 0 0
\(649\) 27.4053 1.07575
\(650\) 0 0
\(651\) −6.29735 −0.246813
\(652\) 0 0
\(653\) −35.3726 −1.38424 −0.692119 0.721783i \(-0.743323\pi\)
−0.692119 + 0.721783i \(0.743323\pi\)
\(654\) 0 0
\(655\) 4.15604 0.162390
\(656\) 0 0
\(657\) −18.1212 −0.706973
\(658\) 0 0
\(659\) 2.10866 0.0821419 0.0410710 0.999156i \(-0.486923\pi\)
0.0410710 + 0.999156i \(0.486923\pi\)
\(660\) 0 0
\(661\) 20.0342 0.779239 0.389619 0.920976i \(-0.372607\pi\)
0.389619 + 0.920976i \(0.372607\pi\)
\(662\) 0 0
\(663\) 44.7625 1.73843
\(664\) 0 0
\(665\) −1.95191 −0.0756919
\(666\) 0 0
\(667\) −9.10170 −0.352419
\(668\) 0 0
\(669\) 23.3230 0.901721
\(670\) 0 0
\(671\) 10.7570 0.415269
\(672\) 0 0
\(673\) −5.07530 −0.195638 −0.0978191 0.995204i \(-0.531187\pi\)
−0.0978191 + 0.995204i \(0.531187\pi\)
\(674\) 0 0
\(675\) −0.284147 −0.0109368
\(676\) 0 0
\(677\) −13.4876 −0.518369 −0.259184 0.965828i \(-0.583454\pi\)
−0.259184 + 0.965828i \(0.583454\pi\)
\(678\) 0 0
\(679\) 2.14060 0.0821486
\(680\) 0 0
\(681\) −9.68793 −0.371242
\(682\) 0 0
\(683\) −25.4379 −0.973356 −0.486678 0.873581i \(-0.661792\pi\)
−0.486678 + 0.873581i \(0.661792\pi\)
\(684\) 0 0
\(685\) −11.1344 −0.425422
\(686\) 0 0
\(687\) 54.6366 2.08452
\(688\) 0 0
\(689\) −16.0125 −0.610027
\(690\) 0 0
\(691\) −22.6506 −0.861668 −0.430834 0.902431i \(-0.641780\pi\)
−0.430834 + 0.902431i \(0.641780\pi\)
\(692\) 0 0
\(693\) 5.11491 0.194299
\(694\) 0 0
\(695\) 6.49452 0.246351
\(696\) 0 0
\(697\) −30.4963 −1.15513
\(698\) 0 0
\(699\) 30.9930 1.17226
\(700\) 0 0
\(701\) −41.8099 −1.57914 −0.789569 0.613662i \(-0.789696\pi\)
−0.789569 + 0.613662i \(0.789696\pi\)
\(702\) 0 0
\(703\) 36.0778 1.36070
\(704\) 0 0
\(705\) 1.58774 0.0597978
\(706\) 0 0
\(707\) 8.19894 0.308353
\(708\) 0 0
\(709\) 2.87117 0.107829 0.0539146 0.998546i \(-0.482830\pi\)
0.0539146 + 0.998546i \(0.482830\pi\)
\(710\) 0 0
\(711\) 9.89134 0.370954
\(712\) 0 0
\(713\) −4.83076 −0.180913
\(714\) 0 0
\(715\) −12.8176 −0.479349
\(716\) 0 0
\(717\) −30.6935 −1.14627
\(718\) 0 0
\(719\) 50.1623 1.87074 0.935369 0.353674i \(-0.115068\pi\)
0.935369 + 0.353674i \(0.115068\pi\)
\(720\) 0 0
\(721\) 9.46364 0.352444
\(722\) 0 0
\(723\) −14.2034 −0.528230
\(724\) 0 0
\(725\) −9.10170 −0.338029
\(726\) 0 0
\(727\) 26.8198 0.994691 0.497345 0.867553i \(-0.334308\pi\)
0.497345 + 0.867553i \(0.334308\pi\)
\(728\) 0 0
\(729\) −29.0272 −1.07508
\(730\) 0 0
\(731\) −19.6182 −0.725604
\(732\) 0 0
\(733\) 35.9861 1.32918 0.664588 0.747210i \(-0.268607\pi\)
0.664588 + 0.747210i \(0.268607\pi\)
\(734\) 0 0
\(735\) 16.6226 0.613135
\(736\) 0 0
\(737\) −26.8106 −0.987581
\(738\) 0 0
\(739\) −5.84396 −0.214974 −0.107487 0.994207i \(-0.534280\pi\)
−0.107487 + 0.994207i \(0.534280\pi\)
\(740\) 0 0
\(741\) −37.6762 −1.38407
\(742\) 0 0
\(743\) −14.2012 −0.520991 −0.260495 0.965475i \(-0.583886\pi\)
−0.260495 + 0.965475i \(0.583886\pi\)
\(744\) 0 0
\(745\) −5.47283 −0.200509
\(746\) 0 0
\(747\) −14.6894 −0.537459
\(748\) 0 0
\(749\) −6.44714 −0.235574
\(750\) 0 0
\(751\) −33.0404 −1.20566 −0.602831 0.797869i \(-0.705961\pi\)
−0.602831 + 0.797869i \(0.705961\pi\)
\(752\) 0 0
\(753\) −15.6685 −0.570991
\(754\) 0 0
\(755\) 0.0954606 0.00347417
\(756\) 0 0
\(757\) 18.2034 0.661614 0.330807 0.943698i \(-0.392679\pi\)
0.330807 + 0.943698i \(0.392679\pi\)
\(758\) 0 0
\(759\) 7.70265 0.279588
\(760\) 0 0
\(761\) 34.1010 1.23616 0.618080 0.786115i \(-0.287911\pi\)
0.618080 + 0.786115i \(0.287911\pi\)
\(762\) 0 0
\(763\) −4.80140 −0.173822
\(764\) 0 0
\(765\) −13.7026 −0.495420
\(766\) 0 0
\(767\) −36.2034 −1.30723
\(768\) 0 0
\(769\) 35.1366 1.26706 0.633529 0.773719i \(-0.281606\pi\)
0.633529 + 0.773719i \(0.281606\pi\)
\(770\) 0 0
\(771\) −55.0878 −1.98394
\(772\) 0 0
\(773\) 1.13659 0.0408803 0.0204401 0.999791i \(-0.493493\pi\)
0.0204401 + 0.999791i \(0.493493\pi\)
\(774\) 0 0
\(775\) −4.83076 −0.173526
\(776\) 0 0
\(777\) 12.7019 0.455679
\(778\) 0 0
\(779\) 25.6685 0.919669
\(780\) 0 0
\(781\) 38.5132 1.37811
\(782\) 0 0
\(783\) 2.58622 0.0924241
\(784\) 0 0
\(785\) −3.40530 −0.121540
\(786\) 0 0
\(787\) 8.33848 0.297235 0.148617 0.988895i \(-0.452518\pi\)
0.148617 + 0.988895i \(0.452518\pi\)
\(788\) 0 0
\(789\) −11.5426 −0.410928
\(790\) 0 0
\(791\) 6.94567 0.246960
\(792\) 0 0
\(793\) −14.2104 −0.504625
\(794\) 0 0
\(795\) 9.62263 0.341279
\(796\) 0 0
\(797\) −0.190921 −0.00676278 −0.00338139 0.999994i \(-0.501076\pi\)
−0.00338139 + 0.999994i \(0.501076\pi\)
\(798\) 0 0
\(799\) 2.82452 0.0999242
\(800\) 0 0
\(801\) −16.9193 −0.597813
\(802\) 0 0
\(803\) 18.1212 0.639482
\(804\) 0 0
\(805\) −0.527166 −0.0185802
\(806\) 0 0
\(807\) −24.0085 −0.845138
\(808\) 0 0
\(809\) −14.5202 −0.510503 −0.255252 0.966875i \(-0.582158\pi\)
−0.255252 + 0.966875i \(0.582158\pi\)
\(810\) 0 0
\(811\) 51.5474 1.81007 0.905037 0.425332i \(-0.139843\pi\)
0.905037 + 0.425332i \(0.139843\pi\)
\(812\) 0 0
\(813\) −16.7089 −0.586006
\(814\) 0 0
\(815\) −18.5745 −0.650638
\(816\) 0 0
\(817\) 16.5124 0.577697
\(818\) 0 0
\(819\) −6.75698 −0.236108
\(820\) 0 0
\(821\) 13.7827 0.481019 0.240509 0.970647i \(-0.422686\pi\)
0.240509 + 0.970647i \(0.422686\pi\)
\(822\) 0 0
\(823\) 37.1531 1.29508 0.647538 0.762034i \(-0.275799\pi\)
0.647538 + 0.762034i \(0.275799\pi\)
\(824\) 0 0
\(825\) 7.70265 0.268172
\(826\) 0 0
\(827\) 39.8510 1.38576 0.692878 0.721055i \(-0.256343\pi\)
0.692878 + 0.721055i \(0.256343\pi\)
\(828\) 0 0
\(829\) 20.5933 0.715234 0.357617 0.933868i \(-0.383589\pi\)
0.357617 + 0.933868i \(0.383589\pi\)
\(830\) 0 0
\(831\) −47.0319 −1.63152
\(832\) 0 0
\(833\) 29.5709 1.02457
\(834\) 0 0
\(835\) 12.4596 0.431183
\(836\) 0 0
\(837\) 1.37265 0.0474456
\(838\) 0 0
\(839\) 36.5669 1.26243 0.631214 0.775609i \(-0.282557\pi\)
0.631214 + 0.775609i \(0.282557\pi\)
\(840\) 0 0
\(841\) 53.8410 1.85659
\(842\) 0 0
\(843\) 81.2005 2.79669
\(844\) 0 0
\(845\) 3.93246 0.135281
\(846\) 0 0
\(847\) 0.683919 0.0234998
\(848\) 0 0
\(849\) 34.0823 1.16970
\(850\) 0 0
\(851\) 9.74378 0.334012
\(852\) 0 0
\(853\) −19.3208 −0.661532 −0.330766 0.943713i \(-0.607307\pi\)
−0.330766 + 0.943713i \(0.607307\pi\)
\(854\) 0 0
\(855\) 11.5334 0.394434
\(856\) 0 0
\(857\) 54.6017 1.86516 0.932580 0.360963i \(-0.117552\pi\)
0.932580 + 0.360963i \(0.117552\pi\)
\(858\) 0 0
\(859\) 32.9541 1.12438 0.562190 0.827008i \(-0.309959\pi\)
0.562190 + 0.827008i \(0.309959\pi\)
\(860\) 0 0
\(861\) 9.03712 0.307984
\(862\) 0 0
\(863\) 54.6840 1.86147 0.930733 0.365701i \(-0.119171\pi\)
0.930733 + 0.365701i \(0.119171\pi\)
\(864\) 0 0
\(865\) 3.49228 0.118741
\(866\) 0 0
\(867\) −5.81532 −0.197499
\(868\) 0 0
\(869\) −9.89134 −0.335541
\(870\) 0 0
\(871\) 35.4178 1.20009
\(872\) 0 0
\(873\) −12.6483 −0.428081
\(874\) 0 0
\(875\) −0.527166 −0.0178215
\(876\) 0 0
\(877\) 16.7500 0.565608 0.282804 0.959178i \(-0.408735\pi\)
0.282804 + 0.959178i \(0.408735\pi\)
\(878\) 0 0
\(879\) −37.2577 −1.25667
\(880\) 0 0
\(881\) −31.7049 −1.06816 −0.534082 0.845432i \(-0.679343\pi\)
−0.534082 + 0.845432i \(0.679343\pi\)
\(882\) 0 0
\(883\) −51.9185 −1.74720 −0.873599 0.486646i \(-0.838220\pi\)
−0.873599 + 0.486646i \(0.838220\pi\)
\(884\) 0 0
\(885\) 21.7563 0.731329
\(886\) 0 0
\(887\) −36.2897 −1.21849 −0.609244 0.792983i \(-0.708527\pi\)
−0.609244 + 0.792983i \(0.708527\pi\)
\(888\) 0 0
\(889\) 10.3121 0.345856
\(890\) 0 0
\(891\) 26.9193 0.901829
\(892\) 0 0
\(893\) −2.37737 −0.0795558
\(894\) 0 0
\(895\) −6.38585 −0.213455
\(896\) 0 0
\(897\) −10.1755 −0.339749
\(898\) 0 0
\(899\) 43.9681 1.46642
\(900\) 0 0
\(901\) 17.1182 0.570290
\(902\) 0 0
\(903\) 5.81355 0.193463
\(904\) 0 0
\(905\) −20.8781 −0.694013
\(906\) 0 0
\(907\) 25.4442 0.844861 0.422430 0.906395i \(-0.361177\pi\)
0.422430 + 0.906395i \(0.361177\pi\)
\(908\) 0 0
\(909\) −48.4457 −1.60684
\(910\) 0 0
\(911\) −47.9597 −1.58897 −0.794487 0.607281i \(-0.792260\pi\)
−0.794487 + 0.607281i \(0.792260\pi\)
\(912\) 0 0
\(913\) 14.6894 0.486150
\(914\) 0 0
\(915\) 8.53965 0.282312
\(916\) 0 0
\(917\) −2.19092 −0.0723506
\(918\) 0 0
\(919\) −2.78562 −0.0918892 −0.0459446 0.998944i \(-0.514630\pi\)
−0.0459446 + 0.998944i \(0.514630\pi\)
\(920\) 0 0
\(921\) −14.6219 −0.481808
\(922\) 0 0
\(923\) −50.8774 −1.67465
\(924\) 0 0
\(925\) 9.74378 0.320373
\(926\) 0 0
\(927\) −55.9185 −1.83661
\(928\) 0 0
\(929\) 40.2508 1.32059 0.660293 0.751008i \(-0.270432\pi\)
0.660293 + 0.751008i \(0.270432\pi\)
\(930\) 0 0
\(931\) −24.8896 −0.815722
\(932\) 0 0
\(933\) −63.7772 −2.08797
\(934\) 0 0
\(935\) 13.7026 0.448125
\(936\) 0 0
\(937\) −38.9868 −1.27364 −0.636822 0.771011i \(-0.719751\pi\)
−0.636822 + 0.771011i \(0.719751\pi\)
\(938\) 0 0
\(939\) −41.3719 −1.35012
\(940\) 0 0
\(941\) 38.3682 1.25077 0.625383 0.780318i \(-0.284943\pi\)
0.625383 + 0.780318i \(0.284943\pi\)
\(942\) 0 0
\(943\) 6.93246 0.225752
\(944\) 0 0
\(945\) 0.149793 0.00487276
\(946\) 0 0
\(947\) −15.0217 −0.488139 −0.244070 0.969758i \(-0.578483\pi\)
−0.244070 + 0.969758i \(0.578483\pi\)
\(948\) 0 0
\(949\) −23.9387 −0.777083
\(950\) 0 0
\(951\) −69.6406 −2.25825
\(952\) 0 0
\(953\) −12.1303 −0.392940 −0.196470 0.980510i \(-0.562948\pi\)
−0.196470 + 0.980510i \(0.562948\pi\)
\(954\) 0 0
\(955\) 6.83700 0.221240
\(956\) 0 0
\(957\) −70.1072 −2.26624
\(958\) 0 0
\(959\) 5.86965 0.189541
\(960\) 0 0
\(961\) −7.66376 −0.247218
\(962\) 0 0
\(963\) 38.0947 1.22759
\(964\) 0 0
\(965\) 7.10170 0.228612
\(966\) 0 0
\(967\) −21.2882 −0.684581 −0.342290 0.939594i \(-0.611203\pi\)
−0.342290 + 0.939594i \(0.611203\pi\)
\(968\) 0 0
\(969\) 40.2779 1.29391
\(970\) 0 0
\(971\) −11.4006 −0.365862 −0.182931 0.983126i \(-0.558558\pi\)
−0.182931 + 0.983126i \(0.558558\pi\)
\(972\) 0 0
\(973\) −3.42369 −0.109758
\(974\) 0 0
\(975\) −10.1755 −0.325876
\(976\) 0 0
\(977\) 41.2849 1.32082 0.660411 0.750904i \(-0.270382\pi\)
0.660411 + 0.750904i \(0.270382\pi\)
\(978\) 0 0
\(979\) 16.9193 0.540742
\(980\) 0 0
\(981\) 28.3704 0.905798
\(982\) 0 0
\(983\) 42.8448 1.36654 0.683268 0.730168i \(-0.260558\pi\)
0.683268 + 0.730168i \(0.260558\pi\)
\(984\) 0 0
\(985\) 0.850207 0.0270899
\(986\) 0 0
\(987\) −0.837003 −0.0266421
\(988\) 0 0
\(989\) 4.45963 0.141808
\(990\) 0 0
\(991\) −8.73307 −0.277415 −0.138707 0.990333i \(-0.544295\pi\)
−0.138707 + 0.990333i \(0.544295\pi\)
\(992\) 0 0
\(993\) −63.8425 −2.02598
\(994\) 0 0
\(995\) −8.56829 −0.271633
\(996\) 0 0
\(997\) −44.0808 −1.39605 −0.698027 0.716072i \(-0.745938\pi\)
−0.698027 + 0.716072i \(0.745938\pi\)
\(998\) 0 0
\(999\) −2.76867 −0.0875968
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 1840.2.a.q.1.1 3
4.3 odd 2 920.2.a.i.1.3 3
5.4 even 2 9200.2.a.ci.1.3 3
8.3 odd 2 7360.2.a.bw.1.1 3
8.5 even 2 7360.2.a.cf.1.3 3
12.11 even 2 8280.2.a.bl.1.1 3
20.3 even 4 4600.2.e.q.4049.6 6
20.7 even 4 4600.2.e.q.4049.1 6
20.19 odd 2 4600.2.a.v.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.3 3 4.3 odd 2
1840.2.a.q.1.1 3 1.1 even 1 trivial
4600.2.a.v.1.1 3 20.19 odd 2
4600.2.e.q.4049.1 6 20.7 even 4
4600.2.e.q.4049.6 6 20.3 even 4
7360.2.a.bw.1.1 3 8.3 odd 2
7360.2.a.cf.1.3 3 8.5 even 2
8280.2.a.bl.1.1 3 12.11 even 2
9200.2.a.ci.1.3 3 5.4 even 2