Properties

Label 1911.2.a.h.1.1
Level $1911$
Weight $2$
Character 1911.1
Self dual yes
Analytic conductor $15.259$
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] = [1911,2,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, 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("1911.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.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: \(15.2594118263\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1911.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.82843 q^{5} +2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.82843 q^{5} +2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +6.82843 q^{10} -2.00000 q^{11} -3.82843 q^{12} +1.00000 q^{13} +2.82843 q^{15} +3.00000 q^{16} +3.65685 q^{17} -2.41421 q^{18} -2.82843 q^{19} -10.8284 q^{20} +4.82843 q^{22} -4.00000 q^{23} +4.41421 q^{24} +3.00000 q^{25} -2.41421 q^{26} -1.00000 q^{27} +2.00000 q^{29} -6.82843 q^{30} +6.82843 q^{31} +1.58579 q^{32} +2.00000 q^{33} -8.82843 q^{34} +3.82843 q^{36} +3.65685 q^{37} +6.82843 q^{38} -1.00000 q^{39} +12.4853 q^{40} -10.8284 q^{41} +9.65685 q^{43} -7.65685 q^{44} -2.82843 q^{45} +9.65685 q^{46} +0.343146 q^{47} -3.00000 q^{48} -7.24264 q^{50} -3.65685 q^{51} +3.82843 q^{52} -2.00000 q^{53} +2.41421 q^{54} +5.65685 q^{55} +2.82843 q^{57} -4.82843 q^{58} +3.65685 q^{59} +10.8284 q^{60} +9.31371 q^{61} -16.4853 q^{62} -9.82843 q^{64} -2.82843 q^{65} -4.82843 q^{66} +1.17157 q^{67} +14.0000 q^{68} +4.00000 q^{69} +2.00000 q^{71} -4.41421 q^{72} -11.6569 q^{73} -8.82843 q^{74} -3.00000 q^{75} -10.8284 q^{76} +2.41421 q^{78} +11.3137 q^{79} -8.48528 q^{80} +1.00000 q^{81} +26.1421 q^{82} +7.65685 q^{83} -10.3431 q^{85} -23.3137 q^{86} -2.00000 q^{87} +8.82843 q^{88} -9.17157 q^{89} +6.82843 q^{90} -15.3137 q^{92} -6.82843 q^{93} -0.828427 q^{94} +8.00000 q^{95} -1.58579 q^{96} +7.65685 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 8 q^{10} - 4 q^{11} - 2 q^{12} + 2 q^{13} + 6 q^{16} - 4 q^{17} - 2 q^{18} - 16 q^{20} + 4 q^{22} - 8 q^{23} + 6 q^{24} + 6 q^{25} - 2 q^{26} - 2 q^{27} + 4 q^{29} - 8 q^{30} + 8 q^{31} + 6 q^{32} + 4 q^{33} - 12 q^{34} + 2 q^{36} - 4 q^{37} + 8 q^{38} - 2 q^{39} + 8 q^{40} - 16 q^{41} + 8 q^{43} - 4 q^{44} + 8 q^{46} + 12 q^{47} - 6 q^{48} - 6 q^{50} + 4 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} - 4 q^{58} - 4 q^{59} + 16 q^{60} - 4 q^{61} - 16 q^{62} - 14 q^{64} - 4 q^{66} + 8 q^{67} + 28 q^{68} + 8 q^{69} + 4 q^{71} - 6 q^{72} - 12 q^{73} - 12 q^{74} - 6 q^{75} - 16 q^{76} + 2 q^{78} + 2 q^{81} + 24 q^{82} + 4 q^{83} - 32 q^{85} - 24 q^{86} - 4 q^{87} + 12 q^{88} - 24 q^{89} + 8 q^{90} - 8 q^{92} - 8 q^{93} + 4 q^{94} + 16 q^{95} - 6 q^{96} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 2.41421 0.985599
\(7\) 0 0
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) 6.82843 2.15934
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −3.82843 −1.10517
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 3.00000 0.750000
\(17\) 3.65685 0.886917 0.443459 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) −2.41421 −0.569036
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) −10.8284 −2.42131
\(21\) 0 0
\(22\) 4.82843 1.02942
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 4.41421 0.901048
\(25\) 3.00000 0.600000
\(26\) −2.41421 −0.473466
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −6.82843 −1.24669
\(31\) 6.82843 1.22642 0.613211 0.789919i \(-0.289878\pi\)
0.613211 + 0.789919i \(0.289878\pi\)
\(32\) 1.58579 0.280330
\(33\) 2.00000 0.348155
\(34\) −8.82843 −1.51406
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 6.82843 1.10772
\(39\) −1.00000 −0.160128
\(40\) 12.4853 1.97410
\(41\) −10.8284 −1.69112 −0.845558 0.533883i \(-0.820732\pi\)
−0.845558 + 0.533883i \(0.820732\pi\)
\(42\) 0 0
\(43\) 9.65685 1.47266 0.736328 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) −7.65685 −1.15431
\(45\) −2.82843 −0.421637
\(46\) 9.65685 1.42383
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) −3.00000 −0.433013
\(49\) 0 0
\(50\) −7.24264 −1.02426
\(51\) −3.65685 −0.512062
\(52\) 3.82843 0.530907
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 2.41421 0.328533
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) −4.82843 −0.634004
\(59\) 3.65685 0.476082 0.238041 0.971255i \(-0.423495\pi\)
0.238041 + 0.971255i \(0.423495\pi\)
\(60\) 10.8284 1.39794
\(61\) 9.31371 1.19250 0.596249 0.802799i \(-0.296657\pi\)
0.596249 + 0.802799i \(0.296657\pi\)
\(62\) −16.4853 −2.09363
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) −2.82843 −0.350823
\(66\) −4.82843 −0.594338
\(67\) 1.17157 0.143130 0.0715652 0.997436i \(-0.477201\pi\)
0.0715652 + 0.997436i \(0.477201\pi\)
\(68\) 14.0000 1.69775
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −4.41421 −0.520220
\(73\) −11.6569 −1.36433 −0.682166 0.731198i \(-0.738962\pi\)
−0.682166 + 0.731198i \(0.738962\pi\)
\(74\) −8.82843 −1.02628
\(75\) −3.00000 −0.346410
\(76\) −10.8284 −1.24211
\(77\) 0 0
\(78\) 2.41421 0.273356
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) −8.48528 −0.948683
\(81\) 1.00000 0.111111
\(82\) 26.1421 2.88692
\(83\) 7.65685 0.840449 0.420224 0.907420i \(-0.361951\pi\)
0.420224 + 0.907420i \(0.361951\pi\)
\(84\) 0 0
\(85\) −10.3431 −1.12187
\(86\) −23.3137 −2.51398
\(87\) −2.00000 −0.214423
\(88\) 8.82843 0.941113
\(89\) −9.17157 −0.972185 −0.486092 0.873907i \(-0.661578\pi\)
−0.486092 + 0.873907i \(0.661578\pi\)
\(90\) 6.82843 0.719779
\(91\) 0 0
\(92\) −15.3137 −1.59656
\(93\) −6.82843 −0.708075
\(94\) −0.828427 −0.0854457
\(95\) 8.00000 0.820783
\(96\) −1.58579 −0.161849
\(97\) 7.65685 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 11.4853 1.14853
\(101\) 3.65685 0.363871 0.181935 0.983311i \(-0.441764\pi\)
0.181935 + 0.983311i \(0.441764\pi\)
\(102\) 8.82843 0.874145
\(103\) −13.6569 −1.34565 −0.672825 0.739802i \(-0.734919\pi\)
−0.672825 + 0.739802i \(0.734919\pi\)
\(104\) −4.41421 −0.432849
\(105\) 0 0
\(106\) 4.82843 0.468978
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) −3.82843 −0.368391
\(109\) −17.3137 −1.65835 −0.829176 0.558987i \(-0.811190\pi\)
−0.829176 + 0.558987i \(0.811190\pi\)
\(110\) −13.6569 −1.30213
\(111\) −3.65685 −0.347093
\(112\) 0 0
\(113\) 17.3137 1.62874 0.814368 0.580348i \(-0.197084\pi\)
0.814368 + 0.580348i \(0.197084\pi\)
\(114\) −6.82843 −0.639541
\(115\) 11.3137 1.05501
\(116\) 7.65685 0.710921
\(117\) 1.00000 0.0924500
\(118\) −8.82843 −0.812723
\(119\) 0 0
\(120\) −12.4853 −1.13975
\(121\) −7.00000 −0.636364
\(122\) −22.4853 −2.03572
\(123\) 10.8284 0.976366
\(124\) 26.1421 2.34763
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) 20.5563 1.81694
\(129\) −9.65685 −0.850239
\(130\) 6.82843 0.598893
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 7.65685 0.666444
\(133\) 0 0
\(134\) −2.82843 −0.244339
\(135\) 2.82843 0.243432
\(136\) −16.1421 −1.38418
\(137\) −5.17157 −0.441837 −0.220919 0.975292i \(-0.570906\pi\)
−0.220919 + 0.975292i \(0.570906\pi\)
\(138\) −9.65685 −0.822046
\(139\) −15.3137 −1.29889 −0.649446 0.760408i \(-0.724999\pi\)
−0.649446 + 0.760408i \(0.724999\pi\)
\(140\) 0 0
\(141\) −0.343146 −0.0288981
\(142\) −4.82843 −0.405193
\(143\) −2.00000 −0.167248
\(144\) 3.00000 0.250000
\(145\) −5.65685 −0.469776
\(146\) 28.1421 2.32906
\(147\) 0 0
\(148\) 14.0000 1.15079
\(149\) −14.8284 −1.21479 −0.607396 0.794399i \(-0.707786\pi\)
−0.607396 + 0.794399i \(0.707786\pi\)
\(150\) 7.24264 0.591359
\(151\) −20.4853 −1.66707 −0.833534 0.552468i \(-0.813686\pi\)
−0.833534 + 0.552468i \(0.813686\pi\)
\(152\) 12.4853 1.01269
\(153\) 3.65685 0.295639
\(154\) 0 0
\(155\) −19.3137 −1.55131
\(156\) −3.82843 −0.306519
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −27.3137 −2.17296
\(159\) 2.00000 0.158610
\(160\) −4.48528 −0.354593
\(161\) 0 0
\(162\) −2.41421 −0.189679
\(163\) 13.1716 1.03168 0.515839 0.856686i \(-0.327480\pi\)
0.515839 + 0.856686i \(0.327480\pi\)
\(164\) −41.4558 −3.23716
\(165\) −5.65685 −0.440386
\(166\) −18.4853 −1.43474
\(167\) −7.65685 −0.592505 −0.296253 0.955110i \(-0.595737\pi\)
−0.296253 + 0.955110i \(0.595737\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 24.9706 1.91515
\(171\) −2.82843 −0.216295
\(172\) 36.9706 2.81898
\(173\) 0.343146 0.0260889 0.0130444 0.999915i \(-0.495848\pi\)
0.0130444 + 0.999915i \(0.495848\pi\)
\(174\) 4.82843 0.366042
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) −3.65685 −0.274866
\(178\) 22.1421 1.65962
\(179\) −0.686292 −0.0512958 −0.0256479 0.999671i \(-0.508165\pi\)
−0.0256479 + 0.999671i \(0.508165\pi\)
\(180\) −10.8284 −0.807103
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −9.31371 −0.688489
\(184\) 17.6569 1.30168
\(185\) −10.3431 −0.760443
\(186\) 16.4853 1.20876
\(187\) −7.31371 −0.534831
\(188\) 1.31371 0.0958120
\(189\) 0 0
\(190\) −19.3137 −1.40116
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) 9.82843 0.709306
\(193\) −17.3137 −1.24627 −0.623134 0.782115i \(-0.714141\pi\)
−0.623134 + 0.782115i \(0.714141\pi\)
\(194\) −18.4853 −1.32717
\(195\) 2.82843 0.202548
\(196\) 0 0
\(197\) −16.4853 −1.17453 −0.587264 0.809396i \(-0.699795\pi\)
−0.587264 + 0.809396i \(0.699795\pi\)
\(198\) 4.82843 0.343141
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) −13.2426 −0.936396
\(201\) −1.17157 −0.0826364
\(202\) −8.82843 −0.621166
\(203\) 0 0
\(204\) −14.0000 −0.980196
\(205\) 30.6274 2.13911
\(206\) 32.9706 2.29717
\(207\) −4.00000 −0.278019
\(208\) 3.00000 0.208013
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −7.65685 −0.525875
\(213\) −2.00000 −0.137038
\(214\) −27.3137 −1.86713
\(215\) −27.3137 −1.86278
\(216\) 4.41421 0.300349
\(217\) 0 0
\(218\) 41.7990 2.83098
\(219\) 11.6569 0.787697
\(220\) 21.6569 1.46010
\(221\) 3.65685 0.245987
\(222\) 8.82843 0.592525
\(223\) −4.48528 −0.300357 −0.150178 0.988659i \(-0.547985\pi\)
−0.150178 + 0.988659i \(0.547985\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) −41.7990 −2.78043
\(227\) −5.31371 −0.352683 −0.176342 0.984329i \(-0.556426\pi\)
−0.176342 + 0.984329i \(0.556426\pi\)
\(228\) 10.8284 0.717130
\(229\) −21.3137 −1.40845 −0.704225 0.709977i \(-0.748705\pi\)
−0.704225 + 0.709977i \(0.748705\pi\)
\(230\) −27.3137 −1.80101
\(231\) 0 0
\(232\) −8.82843 −0.579615
\(233\) −26.9706 −1.76690 −0.883450 0.468525i \(-0.844786\pi\)
−0.883450 + 0.468525i \(0.844786\pi\)
\(234\) −2.41421 −0.157822
\(235\) −0.970563 −0.0633125
\(236\) 14.0000 0.911322
\(237\) −11.3137 −0.734904
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 8.48528 0.547723
\(241\) −11.6569 −0.750884 −0.375442 0.926846i \(-0.622509\pi\)
−0.375442 + 0.926846i \(0.622509\pi\)
\(242\) 16.8995 1.08634
\(243\) −1.00000 −0.0641500
\(244\) 35.6569 2.28270
\(245\) 0 0
\(246\) −26.1421 −1.66676
\(247\) −2.82843 −0.179969
\(248\) −30.1421 −1.91403
\(249\) −7.65685 −0.485233
\(250\) −13.6569 −0.863735
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 13.6569 0.856907
\(255\) 10.3431 0.647713
\(256\) −29.9706 −1.87316
\(257\) 15.6569 0.976648 0.488324 0.872662i \(-0.337608\pi\)
0.488324 + 0.872662i \(0.337608\pi\)
\(258\) 23.3137 1.45145
\(259\) 0 0
\(260\) −10.8284 −0.671551
\(261\) 2.00000 0.123797
\(262\) −19.3137 −1.19320
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −8.82843 −0.543352
\(265\) 5.65685 0.347498
\(266\) 0 0
\(267\) 9.17157 0.561291
\(268\) 4.48528 0.273982
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −6.82843 −0.415565
\(271\) −11.7990 −0.716738 −0.358369 0.933580i \(-0.616667\pi\)
−0.358369 + 0.933580i \(0.616667\pi\)
\(272\) 10.9706 0.665188
\(273\) 0 0
\(274\) 12.4853 0.754263
\(275\) −6.00000 −0.361814
\(276\) 15.3137 0.921777
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 36.9706 2.21735
\(279\) 6.82843 0.408807
\(280\) 0 0
\(281\) 26.8284 1.60045 0.800225 0.599700i \(-0.204713\pi\)
0.800225 + 0.599700i \(0.204713\pi\)
\(282\) 0.828427 0.0493321
\(283\) 4.97056 0.295469 0.147735 0.989027i \(-0.452802\pi\)
0.147735 + 0.989027i \(0.452802\pi\)
\(284\) 7.65685 0.454351
\(285\) −8.00000 −0.473879
\(286\) 4.82843 0.285511
\(287\) 0 0
\(288\) 1.58579 0.0934434
\(289\) −3.62742 −0.213377
\(290\) 13.6569 0.801958
\(291\) −7.65685 −0.448853
\(292\) −44.6274 −2.61162
\(293\) 26.1421 1.52724 0.763620 0.645666i \(-0.223420\pi\)
0.763620 + 0.645666i \(0.223420\pi\)
\(294\) 0 0
\(295\) −10.3431 −0.602201
\(296\) −16.1421 −0.938243
\(297\) 2.00000 0.116052
\(298\) 35.7990 2.07378
\(299\) −4.00000 −0.231326
\(300\) −11.4853 −0.663103
\(301\) 0 0
\(302\) 49.4558 2.84586
\(303\) −3.65685 −0.210081
\(304\) −8.48528 −0.486664
\(305\) −26.3431 −1.50840
\(306\) −8.82843 −0.504688
\(307\) 17.1716 0.980033 0.490017 0.871713i \(-0.336991\pi\)
0.490017 + 0.871713i \(0.336991\pi\)
\(308\) 0 0
\(309\) 13.6569 0.776911
\(310\) 46.6274 2.64826
\(311\) −34.6274 −1.96354 −0.981770 0.190071i \(-0.939128\pi\)
−0.981770 + 0.190071i \(0.939128\pi\)
\(312\) 4.41421 0.249906
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −24.1421 −1.36242
\(315\) 0 0
\(316\) 43.3137 2.43659
\(317\) −8.48528 −0.476581 −0.238290 0.971194i \(-0.576587\pi\)
−0.238290 + 0.971194i \(0.576587\pi\)
\(318\) −4.82843 −0.270765
\(319\) −4.00000 −0.223957
\(320\) 27.7990 1.55401
\(321\) −11.3137 −0.631470
\(322\) 0 0
\(323\) −10.3431 −0.575508
\(324\) 3.82843 0.212690
\(325\) 3.00000 0.166410
\(326\) −31.7990 −1.76118
\(327\) 17.3137 0.957450
\(328\) 47.7990 2.63926
\(329\) 0 0
\(330\) 13.6569 0.751785
\(331\) −2.14214 −0.117742 −0.0588712 0.998266i \(-0.518750\pi\)
−0.0588712 + 0.998266i \(0.518750\pi\)
\(332\) 29.3137 1.60880
\(333\) 3.65685 0.200394
\(334\) 18.4853 1.01147
\(335\) −3.31371 −0.181047
\(336\) 0 0
\(337\) −13.3137 −0.725244 −0.362622 0.931936i \(-0.618118\pi\)
−0.362622 + 0.931936i \(0.618118\pi\)
\(338\) −2.41421 −0.131316
\(339\) −17.3137 −0.940352
\(340\) −39.5980 −2.14750
\(341\) −13.6569 −0.739560
\(342\) 6.82843 0.369239
\(343\) 0 0
\(344\) −42.6274 −2.29832
\(345\) −11.3137 −0.609110
\(346\) −0.828427 −0.0445365
\(347\) −31.3137 −1.68101 −0.840504 0.541805i \(-0.817741\pi\)
−0.840504 + 0.541805i \(0.817741\pi\)
\(348\) −7.65685 −0.410450
\(349\) 7.65685 0.409862 0.204931 0.978776i \(-0.434303\pi\)
0.204931 + 0.978776i \(0.434303\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −3.17157 −0.169045
\(353\) −17.4558 −0.929081 −0.464540 0.885552i \(-0.653780\pi\)
−0.464540 + 0.885552i \(0.653780\pi\)
\(354\) 8.82843 0.469226
\(355\) −5.65685 −0.300235
\(356\) −35.1127 −1.86097
\(357\) 0 0
\(358\) 1.65685 0.0875675
\(359\) 1.02944 0.0543316 0.0271658 0.999631i \(-0.491352\pi\)
0.0271658 + 0.999631i \(0.491352\pi\)
\(360\) 12.4853 0.658032
\(361\) −11.0000 −0.578947
\(362\) 33.7990 1.77644
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 32.9706 1.72576
\(366\) 22.4853 1.17532
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) −12.0000 −0.625543
\(369\) −10.8284 −0.563705
\(370\) 24.9706 1.29816
\(371\) 0 0
\(372\) −26.1421 −1.35541
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 17.6569 0.913014
\(375\) −5.65685 −0.292119
\(376\) −1.51472 −0.0781156
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) −16.4853 −0.846792 −0.423396 0.905945i \(-0.639162\pi\)
−0.423396 + 0.905945i \(0.639162\pi\)
\(380\) 30.6274 1.57115
\(381\) 5.65685 0.289809
\(382\) 46.6274 2.38567
\(383\) 2.97056 0.151789 0.0758943 0.997116i \(-0.475819\pi\)
0.0758943 + 0.997116i \(0.475819\pi\)
\(384\) −20.5563 −1.04901
\(385\) 0 0
\(386\) 41.7990 2.12751
\(387\) 9.65685 0.490885
\(388\) 29.3137 1.48818
\(389\) 6.97056 0.353422 0.176711 0.984263i \(-0.443454\pi\)
0.176711 + 0.984263i \(0.443454\pi\)
\(390\) −6.82843 −0.345771
\(391\) −14.6274 −0.739740
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 39.7990 2.00504
\(395\) −32.0000 −1.61009
\(396\) −7.65685 −0.384771
\(397\) 2.97056 0.149088 0.0745441 0.997218i \(-0.476250\pi\)
0.0745441 + 0.997218i \(0.476250\pi\)
\(398\) 24.9706 1.25166
\(399\) 0 0
\(400\) 9.00000 0.450000
\(401\) −2.14214 −0.106973 −0.0534866 0.998569i \(-0.517033\pi\)
−0.0534866 + 0.998569i \(0.517033\pi\)
\(402\) 2.82843 0.141069
\(403\) 6.82843 0.340148
\(404\) 14.0000 0.696526
\(405\) −2.82843 −0.140546
\(406\) 0 0
\(407\) −7.31371 −0.362527
\(408\) 16.1421 0.799155
\(409\) 1.02944 0.0509024 0.0254512 0.999676i \(-0.491898\pi\)
0.0254512 + 0.999676i \(0.491898\pi\)
\(410\) −73.9411 −3.65169
\(411\) 5.17157 0.255095
\(412\) −52.2843 −2.57586
\(413\) 0 0
\(414\) 9.65685 0.474608
\(415\) −21.6569 −1.06309
\(416\) 1.58579 0.0777496
\(417\) 15.3137 0.749916
\(418\) −13.6569 −0.667979
\(419\) 30.6274 1.49625 0.748124 0.663559i \(-0.230955\pi\)
0.748124 + 0.663559i \(0.230955\pi\)
\(420\) 0 0
\(421\) 14.6863 0.715766 0.357883 0.933766i \(-0.383499\pi\)
0.357883 + 0.933766i \(0.383499\pi\)
\(422\) 28.9706 1.41026
\(423\) 0.343146 0.0166843
\(424\) 8.82843 0.428746
\(425\) 10.9706 0.532150
\(426\) 4.82843 0.233938
\(427\) 0 0
\(428\) 43.3137 2.09365
\(429\) 2.00000 0.0965609
\(430\) 65.9411 3.17996
\(431\) 19.6569 0.946837 0.473419 0.880838i \(-0.343020\pi\)
0.473419 + 0.880838i \(0.343020\pi\)
\(432\) −3.00000 −0.144338
\(433\) −1.31371 −0.0631328 −0.0315664 0.999502i \(-0.510050\pi\)
−0.0315664 + 0.999502i \(0.510050\pi\)
\(434\) 0 0
\(435\) 5.65685 0.271225
\(436\) −66.2843 −3.17444
\(437\) 11.3137 0.541208
\(438\) −28.1421 −1.34468
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) −24.9706 −1.19042
\(441\) 0 0
\(442\) −8.82843 −0.419925
\(443\) 41.9411 1.99268 0.996342 0.0854611i \(-0.0272364\pi\)
0.996342 + 0.0854611i \(0.0272364\pi\)
\(444\) −14.0000 −0.664411
\(445\) 25.9411 1.22973
\(446\) 10.8284 0.512741
\(447\) 14.8284 0.701361
\(448\) 0 0
\(449\) 7.79899 0.368057 0.184029 0.982921i \(-0.441086\pi\)
0.184029 + 0.982921i \(0.441086\pi\)
\(450\) −7.24264 −0.341421
\(451\) 21.6569 1.01978
\(452\) 66.2843 3.11775
\(453\) 20.4853 0.962482
\(454\) 12.8284 0.602068
\(455\) 0 0
\(456\) −12.4853 −0.584677
\(457\) 3.65685 0.171060 0.0855302 0.996336i \(-0.472742\pi\)
0.0855302 + 0.996336i \(0.472742\pi\)
\(458\) 51.4558 2.40437
\(459\) −3.65685 −0.170687
\(460\) 43.3137 2.01951
\(461\) −10.8284 −0.504330 −0.252165 0.967684i \(-0.581143\pi\)
−0.252165 + 0.967684i \(0.581143\pi\)
\(462\) 0 0
\(463\) −7.51472 −0.349239 −0.174619 0.984636i \(-0.555869\pi\)
−0.174619 + 0.984636i \(0.555869\pi\)
\(464\) 6.00000 0.278543
\(465\) 19.3137 0.895652
\(466\) 65.1127 3.01629
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 3.82843 0.176969
\(469\) 0 0
\(470\) 2.34315 0.108081
\(471\) −10.0000 −0.460776
\(472\) −16.1421 −0.743002
\(473\) −19.3137 −0.888045
\(474\) 27.3137 1.25456
\(475\) −8.48528 −0.389331
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) −4.82843 −0.220847
\(479\) −2.68629 −0.122740 −0.0613699 0.998115i \(-0.519547\pi\)
−0.0613699 + 0.998115i \(0.519547\pi\)
\(480\) 4.48528 0.204724
\(481\) 3.65685 0.166738
\(482\) 28.1421 1.28184
\(483\) 0 0
\(484\) −26.7990 −1.21814
\(485\) −21.6569 −0.983387
\(486\) 2.41421 0.109511
\(487\) 31.7990 1.44095 0.720475 0.693481i \(-0.243924\pi\)
0.720475 + 0.693481i \(0.243924\pi\)
\(488\) −41.1127 −1.86108
\(489\) −13.1716 −0.595639
\(490\) 0 0
\(491\) −14.6274 −0.660126 −0.330063 0.943959i \(-0.607070\pi\)
−0.330063 + 0.943959i \(0.607070\pi\)
\(492\) 41.4558 1.86897
\(493\) 7.31371 0.329393
\(494\) 6.82843 0.307225
\(495\) 5.65685 0.254257
\(496\) 20.4853 0.919816
\(497\) 0 0
\(498\) 18.4853 0.828345
\(499\) −2.14214 −0.0958952 −0.0479476 0.998850i \(-0.515268\pi\)
−0.0479476 + 0.998850i \(0.515268\pi\)
\(500\) 21.6569 0.968524
\(501\) 7.65685 0.342083
\(502\) 0 0
\(503\) −15.3137 −0.682805 −0.341402 0.939917i \(-0.610902\pi\)
−0.341402 + 0.939917i \(0.610902\pi\)
\(504\) 0 0
\(505\) −10.3431 −0.460264
\(506\) −19.3137 −0.858599
\(507\) −1.00000 −0.0444116
\(508\) −21.6569 −0.960868
\(509\) 27.7990 1.23217 0.616084 0.787680i \(-0.288718\pi\)
0.616084 + 0.787680i \(0.288718\pi\)
\(510\) −24.9706 −1.10572
\(511\) 0 0
\(512\) 31.2426 1.38074
\(513\) 2.82843 0.124878
\(514\) −37.7990 −1.66724
\(515\) 38.6274 1.70213
\(516\) −36.9706 −1.62754
\(517\) −0.686292 −0.0301831
\(518\) 0 0
\(519\) −0.343146 −0.0150624
\(520\) 12.4853 0.547516
\(521\) −2.68629 −0.117689 −0.0588443 0.998267i \(-0.518742\pi\)
−0.0588443 + 0.998267i \(0.518742\pi\)
\(522\) −4.82843 −0.211335
\(523\) −7.31371 −0.319806 −0.159903 0.987133i \(-0.551118\pi\)
−0.159903 + 0.987133i \(0.551118\pi\)
\(524\) 30.6274 1.33796
\(525\) 0 0
\(526\) −28.9706 −1.26318
\(527\) 24.9706 1.08773
\(528\) 6.00000 0.261116
\(529\) −7.00000 −0.304348
\(530\) −13.6569 −0.593216
\(531\) 3.65685 0.158694
\(532\) 0 0
\(533\) −10.8284 −0.469031
\(534\) −22.1421 −0.958184
\(535\) −32.0000 −1.38348
\(536\) −5.17157 −0.223378
\(537\) 0.686292 0.0296157
\(538\) 43.4558 1.87351
\(539\) 0 0
\(540\) 10.8284 0.465981
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 28.4853 1.22355
\(543\) 14.0000 0.600798
\(544\) 5.79899 0.248630
\(545\) 48.9706 2.09767
\(546\) 0 0
\(547\) 0.686292 0.0293437 0.0146719 0.999892i \(-0.495330\pi\)
0.0146719 + 0.999892i \(0.495330\pi\)
\(548\) −19.7990 −0.845771
\(549\) 9.31371 0.397499
\(550\) 14.4853 0.617654
\(551\) −5.65685 −0.240990
\(552\) −17.6569 −0.751526
\(553\) 0 0
\(554\) 4.82843 0.205140
\(555\) 10.3431 0.439042
\(556\) −58.6274 −2.48636
\(557\) 31.7990 1.34737 0.673683 0.739020i \(-0.264711\pi\)
0.673683 + 0.739020i \(0.264711\pi\)
\(558\) −16.4853 −0.697878
\(559\) 9.65685 0.408441
\(560\) 0 0
\(561\) 7.31371 0.308785
\(562\) −64.7696 −2.73214
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −1.31371 −0.0553171
\(565\) −48.9706 −2.06021
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −8.82843 −0.370433
\(569\) −9.02944 −0.378534 −0.189267 0.981926i \(-0.560611\pi\)
−0.189267 + 0.981926i \(0.560611\pi\)
\(570\) 19.3137 0.808962
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) −7.65685 −0.320149
\(573\) 19.3137 0.806842
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) −9.82843 −0.409518
\(577\) −35.9411 −1.49625 −0.748124 0.663559i \(-0.769045\pi\)
−0.748124 + 0.663559i \(0.769045\pi\)
\(578\) 8.75736 0.364258
\(579\) 17.3137 0.719533
\(580\) −21.6569 −0.899252
\(581\) 0 0
\(582\) 18.4853 0.766240
\(583\) 4.00000 0.165663
\(584\) 51.4558 2.12926
\(585\) −2.82843 −0.116941
\(586\) −63.1127 −2.60716
\(587\) −22.9706 −0.948097 −0.474048 0.880499i \(-0.657208\pi\)
−0.474048 + 0.880499i \(0.657208\pi\)
\(588\) 0 0
\(589\) −19.3137 −0.795807
\(590\) 24.9706 1.02802
\(591\) 16.4853 0.678114
\(592\) 10.9706 0.450887
\(593\) 3.51472 0.144332 0.0721661 0.997393i \(-0.477009\pi\)
0.0721661 + 0.997393i \(0.477009\pi\)
\(594\) −4.82843 −0.198113
\(595\) 0 0
\(596\) −56.7696 −2.32537
\(597\) 10.3431 0.423317
\(598\) 9.65685 0.394898
\(599\) −0.686292 −0.0280411 −0.0140206 0.999902i \(-0.504463\pi\)
−0.0140206 + 0.999902i \(0.504463\pi\)
\(600\) 13.2426 0.540629
\(601\) −44.6274 −1.82039 −0.910195 0.414180i \(-0.864069\pi\)
−0.910195 + 0.414180i \(0.864069\pi\)
\(602\) 0 0
\(603\) 1.17157 0.0477101
\(604\) −78.4264 −3.19113
\(605\) 19.7990 0.804943
\(606\) 8.82843 0.358630
\(607\) 25.9411 1.05292 0.526459 0.850201i \(-0.323519\pi\)
0.526459 + 0.850201i \(0.323519\pi\)
\(608\) −4.48528 −0.181902
\(609\) 0 0
\(610\) 63.5980 2.57501
\(611\) 0.343146 0.0138822
\(612\) 14.0000 0.565916
\(613\) −36.3431 −1.46789 −0.733943 0.679211i \(-0.762322\pi\)
−0.733943 + 0.679211i \(0.762322\pi\)
\(614\) −41.4558 −1.67302
\(615\) −30.6274 −1.23502
\(616\) 0 0
\(617\) −29.1716 −1.17440 −0.587202 0.809441i \(-0.699770\pi\)
−0.587202 + 0.809441i \(0.699770\pi\)
\(618\) −32.9706 −1.32627
\(619\) 15.7990 0.635015 0.317508 0.948256i \(-0.397154\pi\)
0.317508 + 0.948256i \(0.397154\pi\)
\(620\) −73.9411 −2.96955
\(621\) 4.00000 0.160514
\(622\) 83.5980 3.35197
\(623\) 0 0
\(624\) −3.00000 −0.120096
\(625\) −31.0000 −1.24000
\(626\) 14.4853 0.578948
\(627\) −5.65685 −0.225913
\(628\) 38.2843 1.52771
\(629\) 13.3726 0.533200
\(630\) 0 0
\(631\) −19.1127 −0.760865 −0.380432 0.924809i \(-0.624225\pi\)
−0.380432 + 0.924809i \(0.624225\pi\)
\(632\) −49.9411 −1.98655
\(633\) 12.0000 0.476957
\(634\) 20.4853 0.813574
\(635\) 16.0000 0.634941
\(636\) 7.65685 0.303614
\(637\) 0 0
\(638\) 9.65685 0.382319
\(639\) 2.00000 0.0791188
\(640\) −58.1421 −2.29827
\(641\) −26.2843 −1.03817 −0.519083 0.854724i \(-0.673727\pi\)
−0.519083 + 0.854724i \(0.673727\pi\)
\(642\) 27.3137 1.07799
\(643\) −17.1716 −0.677181 −0.338590 0.940934i \(-0.609950\pi\)
−0.338590 + 0.940934i \(0.609950\pi\)
\(644\) 0 0
\(645\) 27.3137 1.07548
\(646\) 24.9706 0.982454
\(647\) 11.3137 0.444788 0.222394 0.974957i \(-0.428613\pi\)
0.222394 + 0.974957i \(0.428613\pi\)
\(648\) −4.41421 −0.173407
\(649\) −7.31371 −0.287088
\(650\) −7.24264 −0.284080
\(651\) 0 0
\(652\) 50.4264 1.97485
\(653\) −2.68629 −0.105123 −0.0525614 0.998618i \(-0.516739\pi\)
−0.0525614 + 0.998618i \(0.516739\pi\)
\(654\) −41.7990 −1.63447
\(655\) −22.6274 −0.884126
\(656\) −32.4853 −1.26834
\(657\) −11.6569 −0.454777
\(658\) 0 0
\(659\) −24.6863 −0.961641 −0.480821 0.876819i \(-0.659661\pi\)
−0.480821 + 0.876819i \(0.659661\pi\)
\(660\) −21.6569 −0.842992
\(661\) 1.02944 0.0400405 0.0200202 0.999800i \(-0.493627\pi\)
0.0200202 + 0.999800i \(0.493627\pi\)
\(662\) 5.17157 0.200999
\(663\) −3.65685 −0.142020
\(664\) −33.7990 −1.31166
\(665\) 0 0
\(666\) −8.82843 −0.342095
\(667\) −8.00000 −0.309761
\(668\) −29.3137 −1.13418
\(669\) 4.48528 0.173411
\(670\) 8.00000 0.309067
\(671\) −18.6274 −0.719103
\(672\) 0 0
\(673\) −28.6274 −1.10351 −0.551753 0.834008i \(-0.686041\pi\)
−0.551753 + 0.834008i \(0.686041\pi\)
\(674\) 32.1421 1.23807
\(675\) −3.00000 −0.115470
\(676\) 3.82843 0.147247
\(677\) −49.3137 −1.89528 −0.947640 0.319341i \(-0.896538\pi\)
−0.947640 + 0.319341i \(0.896538\pi\)
\(678\) 41.7990 1.60528
\(679\) 0 0
\(680\) 45.6569 1.75086
\(681\) 5.31371 0.203622
\(682\) 32.9706 1.26251
\(683\) −19.9411 −0.763026 −0.381513 0.924363i \(-0.624597\pi\)
−0.381513 + 0.924363i \(0.624597\pi\)
\(684\) −10.8284 −0.414035
\(685\) 14.6274 0.558885
\(686\) 0 0
\(687\) 21.3137 0.813169
\(688\) 28.9706 1.10449
\(689\) −2.00000 −0.0761939
\(690\) 27.3137 1.03982
\(691\) 34.1421 1.29883 0.649414 0.760435i \(-0.275014\pi\)
0.649414 + 0.760435i \(0.275014\pi\)
\(692\) 1.31371 0.0499397
\(693\) 0 0
\(694\) 75.5980 2.86966
\(695\) 43.3137 1.64298
\(696\) 8.82843 0.334641
\(697\) −39.5980 −1.49988
\(698\) −18.4853 −0.699678
\(699\) 26.9706 1.02012
\(700\) 0 0
\(701\) 38.9706 1.47190 0.735949 0.677037i \(-0.236736\pi\)
0.735949 + 0.677037i \(0.236736\pi\)
\(702\) 2.41421 0.0911186
\(703\) −10.3431 −0.390099
\(704\) 19.6569 0.740846
\(705\) 0.970563 0.0365535
\(706\) 42.1421 1.58604
\(707\) 0 0
\(708\) −14.0000 −0.526152
\(709\) 40.6274 1.52579 0.762897 0.646520i \(-0.223776\pi\)
0.762897 + 0.646520i \(0.223776\pi\)
\(710\) 13.6569 0.512533
\(711\) 11.3137 0.424297
\(712\) 40.4853 1.51725
\(713\) −27.3137 −1.02291
\(714\) 0 0
\(715\) 5.65685 0.211554
\(716\) −2.62742 −0.0981912
\(717\) −2.00000 −0.0746914
\(718\) −2.48528 −0.0927499
\(719\) −37.9411 −1.41497 −0.707483 0.706731i \(-0.750169\pi\)
−0.707483 + 0.706731i \(0.750169\pi\)
\(720\) −8.48528 −0.316228
\(721\) 0 0
\(722\) 26.5563 0.988325
\(723\) 11.6569 0.433523
\(724\) −53.5980 −1.99195
\(725\) 6.00000 0.222834
\(726\) −16.8995 −0.627199
\(727\) 21.6569 0.803208 0.401604 0.915813i \(-0.368453\pi\)
0.401604 + 0.915813i \(0.368453\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −79.5980 −2.94605
\(731\) 35.3137 1.30612
\(732\) −35.6569 −1.31792
\(733\) −8.62742 −0.318661 −0.159330 0.987225i \(-0.550934\pi\)
−0.159330 + 0.987225i \(0.550934\pi\)
\(734\) −57.9411 −2.13865
\(735\) 0 0
\(736\) −6.34315 −0.233811
\(737\) −2.34315 −0.0863109
\(738\) 26.1421 0.962305
\(739\) 10.1421 0.373084 0.186542 0.982447i \(-0.440272\pi\)
0.186542 + 0.982447i \(0.440272\pi\)
\(740\) −39.5980 −1.45565
\(741\) 2.82843 0.103905
\(742\) 0 0
\(743\) 2.00000 0.0733729 0.0366864 0.999327i \(-0.488320\pi\)
0.0366864 + 0.999327i \(0.488320\pi\)
\(744\) 30.1421 1.10506
\(745\) 41.9411 1.53660
\(746\) −24.1421 −0.883906
\(747\) 7.65685 0.280150
\(748\) −28.0000 −1.02378
\(749\) 0 0
\(750\) 13.6569 0.498678
\(751\) 32.9706 1.20311 0.601556 0.798830i \(-0.294547\pi\)
0.601556 + 0.798830i \(0.294547\pi\)
\(752\) 1.02944 0.0375397
\(753\) 0 0
\(754\) −4.82843 −0.175841
\(755\) 57.9411 2.10869
\(756\) 0 0
\(757\) −15.9411 −0.579390 −0.289695 0.957119i \(-0.593554\pi\)
−0.289695 + 0.957119i \(0.593554\pi\)
\(758\) 39.7990 1.44556
\(759\) −8.00000 −0.290382
\(760\) −35.3137 −1.28096
\(761\) −15.5147 −0.562408 −0.281204 0.959648i \(-0.590734\pi\)
−0.281204 + 0.959648i \(0.590734\pi\)
\(762\) −13.6569 −0.494736
\(763\) 0 0
\(764\) −73.9411 −2.67510
\(765\) −10.3431 −0.373957
\(766\) −7.17157 −0.259119
\(767\) 3.65685 0.132041
\(768\) 29.9706 1.08147
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) −15.6569 −0.563868
\(772\) −66.2843 −2.38562
\(773\) −5.85786 −0.210693 −0.105346 0.994436i \(-0.533595\pi\)
−0.105346 + 0.994436i \(0.533595\pi\)
\(774\) −23.3137 −0.837994
\(775\) 20.4853 0.735853
\(776\) −33.7990 −1.21331
\(777\) 0 0
\(778\) −16.8284 −0.603328
\(779\) 30.6274 1.09734
\(780\) 10.8284 0.387720
\(781\) −4.00000 −0.143131
\(782\) 35.3137 1.26282
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) −28.2843 −1.00951
\(786\) 19.3137 0.688897
\(787\) −32.7696 −1.16811 −0.584054 0.811715i \(-0.698534\pi\)
−0.584054 + 0.811715i \(0.698534\pi\)
\(788\) −63.1127 −2.24830
\(789\) −12.0000 −0.427211
\(790\) 77.2548 2.74860
\(791\) 0 0
\(792\) 8.82843 0.313704
\(793\) 9.31371 0.330739
\(794\) −7.17157 −0.254510
\(795\) −5.65685 −0.200628
\(796\) −39.5980 −1.40351
\(797\) −35.6569 −1.26303 −0.631515 0.775363i \(-0.717567\pi\)
−0.631515 + 0.775363i \(0.717567\pi\)
\(798\) 0 0
\(799\) 1.25483 0.0443928
\(800\) 4.75736 0.168198
\(801\) −9.17157 −0.324062
\(802\) 5.17157 0.182615
\(803\) 23.3137 0.822723
\(804\) −4.48528 −0.158184
\(805\) 0 0
\(806\) −16.4853 −0.580669
\(807\) 18.0000 0.633630
\(808\) −16.1421 −0.567878
\(809\) 41.3137 1.45251 0.726256 0.687424i \(-0.241259\pi\)
0.726256 + 0.687424i \(0.241259\pi\)
\(810\) 6.82843 0.239926
\(811\) 1.85786 0.0652384 0.0326192 0.999468i \(-0.489615\pi\)
0.0326192 + 0.999468i \(0.489615\pi\)
\(812\) 0 0
\(813\) 11.7990 0.413809
\(814\) 17.6569 0.618872
\(815\) −37.2548 −1.30498
\(816\) −10.9706 −0.384047
\(817\) −27.3137 −0.955586
\(818\) −2.48528 −0.0868958
\(819\) 0 0
\(820\) 117.255 4.09472
\(821\) 15.7990 0.551389 0.275694 0.961245i \(-0.411092\pi\)
0.275694 + 0.961245i \(0.411092\pi\)
\(822\) −12.4853 −0.435474
\(823\) 48.9706 1.70701 0.853503 0.521088i \(-0.174473\pi\)
0.853503 + 0.521088i \(0.174473\pi\)
\(824\) 60.2843 2.10010
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −26.0000 −0.904109 −0.452054 0.891990i \(-0.649309\pi\)
−0.452054 + 0.891990i \(0.649309\pi\)
\(828\) −15.3137 −0.532188
\(829\) −5.31371 −0.184553 −0.0922764 0.995733i \(-0.529414\pi\)
−0.0922764 + 0.995733i \(0.529414\pi\)
\(830\) 52.2843 1.81481
\(831\) 2.00000 0.0693792
\(832\) −9.82843 −0.340739
\(833\) 0 0
\(834\) −36.9706 −1.28019
\(835\) 21.6569 0.749466
\(836\) 21.6569 0.749018
\(837\) −6.82843 −0.236025
\(838\) −73.9411 −2.55425
\(839\) 47.2548 1.63142 0.815709 0.578462i \(-0.196347\pi\)
0.815709 + 0.578462i \(0.196347\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −35.4558 −1.22189
\(843\) −26.8284 −0.924020
\(844\) −45.9411 −1.58136
\(845\) −2.82843 −0.0973009
\(846\) −0.828427 −0.0284819
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −4.97056 −0.170589
\(850\) −26.4853 −0.908438
\(851\) −14.6274 −0.501421
\(852\) −7.65685 −0.262320
\(853\) 7.65685 0.262166 0.131083 0.991371i \(-0.458155\pi\)
0.131083 + 0.991371i \(0.458155\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) −49.9411 −1.70695
\(857\) −29.5980 −1.01105 −0.505524 0.862813i \(-0.668701\pi\)
−0.505524 + 0.862813i \(0.668701\pi\)
\(858\) −4.82843 −0.164840
\(859\) 23.3137 0.795453 0.397727 0.917504i \(-0.369799\pi\)
0.397727 + 0.917504i \(0.369799\pi\)
\(860\) −104.569 −3.56576
\(861\) 0 0
\(862\) −47.4558 −1.61635
\(863\) −39.6569 −1.34994 −0.674968 0.737847i \(-0.735842\pi\)
−0.674968 + 0.737847i \(0.735842\pi\)
\(864\) −1.58579 −0.0539496
\(865\) −0.970563 −0.0330001
\(866\) 3.17157 0.107774
\(867\) 3.62742 0.123194
\(868\) 0 0
\(869\) −22.6274 −0.767583
\(870\) −13.6569 −0.463011
\(871\) 1.17157 0.0396972
\(872\) 76.4264 2.58812
\(873\) 7.65685 0.259145
\(874\) −27.3137 −0.923900
\(875\) 0 0
\(876\) 44.6274 1.50782
\(877\) −14.2843 −0.482346 −0.241173 0.970482i \(-0.577532\pi\)
−0.241173 + 0.970482i \(0.577532\pi\)
\(878\) −40.9706 −1.38269
\(879\) −26.1421 −0.881752
\(880\) 16.9706 0.572078
\(881\) −53.5980 −1.80576 −0.902881 0.429891i \(-0.858552\pi\)
−0.902881 + 0.429891i \(0.858552\pi\)
\(882\) 0 0
\(883\) −51.5980 −1.73641 −0.868205 0.496205i \(-0.834726\pi\)
−0.868205 + 0.496205i \(0.834726\pi\)
\(884\) 14.0000 0.470871
\(885\) 10.3431 0.347681
\(886\) −101.255 −3.40172
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 16.1421 0.541695
\(889\) 0 0
\(890\) −62.6274 −2.09928
\(891\) −2.00000 −0.0670025
\(892\) −17.1716 −0.574947
\(893\) −0.970563 −0.0324786
\(894\) −35.7990 −1.19730
\(895\) 1.94113 0.0648847
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) −18.8284 −0.628313
\(899\) 13.6569 0.455482
\(900\) 11.4853 0.382843
\(901\) −7.31371 −0.243655
\(902\) −52.2843 −1.74088
\(903\) 0 0
\(904\) −76.4264 −2.54190
\(905\) 39.5980 1.31628
\(906\) −49.4558 −1.64306
\(907\) 20.9706 0.696316 0.348158 0.937436i \(-0.386807\pi\)
0.348158 + 0.937436i \(0.386807\pi\)
\(908\) −20.3431 −0.675111
\(909\) 3.65685 0.121290
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 8.48528 0.280976
\(913\) −15.3137 −0.506810
\(914\) −8.82843 −0.292018
\(915\) 26.3431 0.870878
\(916\) −81.5980 −2.69607
\(917\) 0 0
\(918\) 8.82843 0.291382
\(919\) −19.3137 −0.637100 −0.318550 0.947906i \(-0.603196\pi\)
−0.318550 + 0.947906i \(0.603196\pi\)
\(920\) −49.9411 −1.64651
\(921\) −17.1716 −0.565823
\(922\) 26.1421 0.860945
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) 10.9706 0.360710
\(926\) 18.1421 0.596188
\(927\) −13.6569 −0.448550
\(928\) 3.17157 0.104112
\(929\) 27.7990 0.912055 0.456028 0.889966i \(-0.349272\pi\)
0.456028 + 0.889966i \(0.349272\pi\)
\(930\) −46.6274 −1.52897
\(931\) 0 0
\(932\) −103.255 −3.38222
\(933\) 34.6274 1.13365
\(934\) −19.3137 −0.631964
\(935\) 20.6863 0.676514
\(936\) −4.41421 −0.144283
\(937\) −1.31371 −0.0429170 −0.0214585 0.999770i \(-0.506831\pi\)
−0.0214585 + 0.999770i \(0.506831\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) −3.71573 −0.121194
\(941\) −5.85786 −0.190961 −0.0954805 0.995431i \(-0.530439\pi\)
−0.0954805 + 0.995431i \(0.530439\pi\)
\(942\) 24.1421 0.786593
\(943\) 43.3137 1.41049
\(944\) 10.9706 0.357061
\(945\) 0 0
\(946\) 46.6274 1.51599
\(947\) −54.9706 −1.78630 −0.893152 0.449756i \(-0.851511\pi\)
−0.893152 + 0.449756i \(0.851511\pi\)
\(948\) −43.3137 −1.40676
\(949\) −11.6569 −0.378398
\(950\) 20.4853 0.664630
\(951\) 8.48528 0.275154
\(952\) 0 0
\(953\) 51.6569 1.67333 0.836665 0.547715i \(-0.184502\pi\)
0.836665 + 0.547715i \(0.184502\pi\)
\(954\) 4.82843 0.156326
\(955\) 54.6274 1.76770
\(956\) 7.65685 0.247640
\(957\) 4.00000 0.129302
\(958\) 6.48528 0.209530
\(959\) 0 0
\(960\) −27.7990 −0.897209
\(961\) 15.6274 0.504110
\(962\) −8.82843 −0.284640
\(963\) 11.3137 0.364579
\(964\) −44.6274 −1.43735
\(965\) 48.9706 1.57642
\(966\) 0 0
\(967\) −10.1421 −0.326149 −0.163075 0.986614i \(-0.552141\pi\)
−0.163075 + 0.986614i \(0.552141\pi\)
\(968\) 30.8995 0.993147
\(969\) 10.3431 0.332270
\(970\) 52.2843 1.67875
\(971\) −7.31371 −0.234708 −0.117354 0.993090i \(-0.537441\pi\)
−0.117354 + 0.993090i \(0.537441\pi\)
\(972\) −3.82843 −0.122797
\(973\) 0 0
\(974\) −76.7696 −2.45986
\(975\) −3.00000 −0.0960769
\(976\) 27.9411 0.894374
\(977\) 13.8579 0.443352 0.221676 0.975120i \(-0.428847\pi\)
0.221676 + 0.975120i \(0.428847\pi\)
\(978\) 31.7990 1.01682
\(979\) 18.3431 0.586249
\(980\) 0 0
\(981\) −17.3137 −0.552784
\(982\) 35.3137 1.12691
\(983\) −2.68629 −0.0856794 −0.0428397 0.999082i \(-0.513640\pi\)
−0.0428397 + 0.999082i \(0.513640\pi\)
\(984\) −47.7990 −1.52378
\(985\) 46.6274 1.48567
\(986\) −17.6569 −0.562309
\(987\) 0 0
\(988\) −10.8284 −0.344498
\(989\) −38.6274 −1.22828
\(990\) −13.6569 −0.434043
\(991\) 27.3137 0.867649 0.433824 0.900998i \(-0.357164\pi\)
0.433824 + 0.900998i \(0.357164\pi\)
\(992\) 10.8284 0.343803
\(993\) 2.14214 0.0679786
\(994\) 0 0
\(995\) 29.2548 0.927441
\(996\) −29.3137 −0.928840
\(997\) −51.2548 −1.62326 −0.811628 0.584174i \(-0.801419\pi\)
−0.811628 + 0.584174i \(0.801419\pi\)
\(998\) 5.17157 0.163703
\(999\) −3.65685 −0.115698
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 1911.2.a.h.1.1 2
3.2 odd 2 5733.2.a.u.1.2 2
7.6 odd 2 39.2.a.b.1.1 2
21.20 even 2 117.2.a.c.1.2 2
28.27 even 2 624.2.a.k.1.2 2
35.13 even 4 975.2.c.h.274.4 4
35.27 even 4 975.2.c.h.274.1 4
35.34 odd 2 975.2.a.l.1.2 2
56.13 odd 2 2496.2.a.bf.1.1 2
56.27 even 2 2496.2.a.bi.1.1 2
63.13 odd 6 1053.2.e.m.703.2 4
63.20 even 6 1053.2.e.e.352.1 4
63.34 odd 6 1053.2.e.m.352.2 4
63.41 even 6 1053.2.e.e.703.1 4
77.76 even 2 4719.2.a.p.1.2 2
84.83 odd 2 1872.2.a.w.1.1 2
91.6 even 12 507.2.j.f.361.4 8
91.20 even 12 507.2.j.f.361.1 8
91.34 even 4 507.2.b.e.337.1 4
91.41 even 12 507.2.j.f.316.1 8
91.48 odd 6 507.2.e.h.484.2 4
91.55 odd 6 507.2.e.h.22.2 4
91.62 odd 6 507.2.e.d.22.1 4
91.69 odd 6 507.2.e.d.484.1 4
91.76 even 12 507.2.j.f.316.4 8
91.83 even 4 507.2.b.e.337.4 4
91.90 odd 2 507.2.a.h.1.2 2
105.62 odd 4 2925.2.c.u.2224.4 4
105.83 odd 4 2925.2.c.u.2224.1 4
105.104 even 2 2925.2.a.v.1.1 2
168.83 odd 2 7488.2.a.co.1.2 2
168.125 even 2 7488.2.a.cl.1.2 2
273.83 odd 4 1521.2.b.j.1351.1 4
273.125 odd 4 1521.2.b.j.1351.4 4
273.272 even 2 1521.2.a.f.1.1 2
364.363 even 2 8112.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.1 2 7.6 odd 2
117.2.a.c.1.2 2 21.20 even 2
507.2.a.h.1.2 2 91.90 odd 2
507.2.b.e.337.1 4 91.34 even 4
507.2.b.e.337.4 4 91.83 even 4
507.2.e.d.22.1 4 91.62 odd 6
507.2.e.d.484.1 4 91.69 odd 6
507.2.e.h.22.2 4 91.55 odd 6
507.2.e.h.484.2 4 91.48 odd 6
507.2.j.f.316.1 8 91.41 even 12
507.2.j.f.316.4 8 91.76 even 12
507.2.j.f.361.1 8 91.20 even 12
507.2.j.f.361.4 8 91.6 even 12
624.2.a.k.1.2 2 28.27 even 2
975.2.a.l.1.2 2 35.34 odd 2
975.2.c.h.274.1 4 35.27 even 4
975.2.c.h.274.4 4 35.13 even 4
1053.2.e.e.352.1 4 63.20 even 6
1053.2.e.e.703.1 4 63.41 even 6
1053.2.e.m.352.2 4 63.34 odd 6
1053.2.e.m.703.2 4 63.13 odd 6
1521.2.a.f.1.1 2 273.272 even 2
1521.2.b.j.1351.1 4 273.83 odd 4
1521.2.b.j.1351.4 4 273.125 odd 4
1872.2.a.w.1.1 2 84.83 odd 2
1911.2.a.h.1.1 2 1.1 even 1 trivial
2496.2.a.bf.1.1 2 56.13 odd 2
2496.2.a.bi.1.1 2 56.27 even 2
2925.2.a.v.1.1 2 105.104 even 2
2925.2.c.u.2224.1 4 105.83 odd 4
2925.2.c.u.2224.4 4 105.62 odd 4
4719.2.a.p.1.2 2 77.76 even 2
5733.2.a.u.1.2 2 3.2 odd 2
7488.2.a.cl.1.2 2 168.125 even 2
7488.2.a.co.1.2 2 168.83 odd 2
8112.2.a.bm.1.1 2 364.363 even 2