Properties

Label 8043.2.a.t.1.25
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $52$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, 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("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(0\)
Dimension: \(52\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.25
Character \(\chi\) \(=\) 8043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.120027 q^{2} -1.00000 q^{3} -1.98559 q^{4} +1.73267 q^{5} +0.120027 q^{6} +1.00000 q^{7} +0.478379 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.120027 q^{2} -1.00000 q^{3} -1.98559 q^{4} +1.73267 q^{5} +0.120027 q^{6} +1.00000 q^{7} +0.478379 q^{8} +1.00000 q^{9} -0.207967 q^{10} -4.99112 q^{11} +1.98559 q^{12} +1.07949 q^{13} -0.120027 q^{14} -1.73267 q^{15} +3.91377 q^{16} -1.52581 q^{17} -0.120027 q^{18} +0.00579525 q^{19} -3.44038 q^{20} -1.00000 q^{21} +0.599069 q^{22} +7.05455 q^{23} -0.478379 q^{24} -1.99785 q^{25} -0.129568 q^{26} -1.00000 q^{27} -1.98559 q^{28} +1.18505 q^{29} +0.207967 q^{30} -6.26094 q^{31} -1.42652 q^{32} +4.99112 q^{33} +0.183138 q^{34} +1.73267 q^{35} -1.98559 q^{36} -0.336663 q^{37} -0.000695587 q^{38} -1.07949 q^{39} +0.828873 q^{40} +2.38034 q^{41} +0.120027 q^{42} +9.11735 q^{43} +9.91033 q^{44} +1.73267 q^{45} -0.846737 q^{46} -7.60839 q^{47} -3.91377 q^{48} +1.00000 q^{49} +0.239796 q^{50} +1.52581 q^{51} -2.14342 q^{52} -9.21286 q^{53} +0.120027 q^{54} -8.64796 q^{55} +0.478379 q^{56} -0.00579525 q^{57} -0.142238 q^{58} +10.3896 q^{59} +3.44038 q^{60} -1.68451 q^{61} +0.751482 q^{62} +1.00000 q^{63} -7.65632 q^{64} +1.87039 q^{65} -0.599069 q^{66} +3.51374 q^{67} +3.02964 q^{68} -7.05455 q^{69} -0.207967 q^{70} -2.92691 q^{71} +0.478379 q^{72} +1.51241 q^{73} +0.0404087 q^{74} +1.99785 q^{75} -0.0115070 q^{76} -4.99112 q^{77} +0.129568 q^{78} +3.99546 q^{79} +6.78127 q^{80} +1.00000 q^{81} -0.285705 q^{82} -9.24882 q^{83} +1.98559 q^{84} -2.64372 q^{85} -1.09433 q^{86} -1.18505 q^{87} -2.38765 q^{88} -7.16310 q^{89} -0.207967 q^{90} +1.07949 q^{91} -14.0075 q^{92} +6.26094 q^{93} +0.913213 q^{94} +0.0100413 q^{95} +1.42652 q^{96} +4.19603 q^{97} -0.120027 q^{98} -4.99112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.120027 −0.0848719 −0.0424360 0.999099i \(-0.513512\pi\)
−0.0424360 + 0.999099i \(0.513512\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.98559 −0.992797
\(5\) 1.73267 0.774874 0.387437 0.921896i \(-0.373360\pi\)
0.387437 + 0.921896i \(0.373360\pi\)
\(6\) 0.120027 0.0490008
\(7\) 1.00000 0.377964
\(8\) 0.478379 0.169132
\(9\) 1.00000 0.333333
\(10\) −0.207967 −0.0657650
\(11\) −4.99112 −1.50488 −0.752439 0.658662i \(-0.771123\pi\)
−0.752439 + 0.658662i \(0.771123\pi\)
\(12\) 1.98559 0.573191
\(13\) 1.07949 0.299396 0.149698 0.988732i \(-0.452170\pi\)
0.149698 + 0.988732i \(0.452170\pi\)
\(14\) −0.120027 −0.0320786
\(15\) −1.73267 −0.447374
\(16\) 3.91377 0.978442
\(17\) −1.52581 −0.370063 −0.185031 0.982733i \(-0.559239\pi\)
−0.185031 + 0.982733i \(0.559239\pi\)
\(18\) −0.120027 −0.0282906
\(19\) 0.00579525 0.00132952 0.000664761 1.00000i \(-0.499788\pi\)
0.000664761 1.00000i \(0.499788\pi\)
\(20\) −3.44038 −0.769292
\(21\) −1.00000 −0.218218
\(22\) 0.599069 0.127722
\(23\) 7.05455 1.47098 0.735488 0.677538i \(-0.236953\pi\)
0.735488 + 0.677538i \(0.236953\pi\)
\(24\) −0.478379 −0.0976487
\(25\) −1.99785 −0.399571
\(26\) −0.129568 −0.0254103
\(27\) −1.00000 −0.192450
\(28\) −1.98559 −0.375242
\(29\) 1.18505 0.220059 0.110029 0.993928i \(-0.464906\pi\)
0.110029 + 0.993928i \(0.464906\pi\)
\(30\) 0.207967 0.0379695
\(31\) −6.26094 −1.12450 −0.562249 0.826968i \(-0.690064\pi\)
−0.562249 + 0.826968i \(0.690064\pi\)
\(32\) −1.42652 −0.252175
\(33\) 4.99112 0.868842
\(34\) 0.183138 0.0314080
\(35\) 1.73267 0.292875
\(36\) −1.98559 −0.330932
\(37\) −0.336663 −0.0553471 −0.0276735 0.999617i \(-0.508810\pi\)
−0.0276735 + 0.999617i \(0.508810\pi\)
\(38\) −0.000695587 0 −0.000112839 0
\(39\) −1.07949 −0.172856
\(40\) 0.828873 0.131056
\(41\) 2.38034 0.371747 0.185873 0.982574i \(-0.440489\pi\)
0.185873 + 0.982574i \(0.440489\pi\)
\(42\) 0.120027 0.0185206
\(43\) 9.11735 1.39038 0.695191 0.718825i \(-0.255320\pi\)
0.695191 + 0.718825i \(0.255320\pi\)
\(44\) 9.91033 1.49404
\(45\) 1.73267 0.258291
\(46\) −0.846737 −0.124844
\(47\) −7.60839 −1.10980 −0.554899 0.831918i \(-0.687243\pi\)
−0.554899 + 0.831918i \(0.687243\pi\)
\(48\) −3.91377 −0.564904
\(49\) 1.00000 0.142857
\(50\) 0.239796 0.0339123
\(51\) 1.52581 0.213656
\(52\) −2.14342 −0.297239
\(53\) −9.21286 −1.26548 −0.632742 0.774363i \(-0.718070\pi\)
−0.632742 + 0.774363i \(0.718070\pi\)
\(54\) 0.120027 0.0163336
\(55\) −8.64796 −1.16609
\(56\) 0.478379 0.0639261
\(57\) −0.00579525 −0.000767600 0
\(58\) −0.142238 −0.0186768
\(59\) 10.3896 1.35261 0.676307 0.736620i \(-0.263579\pi\)
0.676307 + 0.736620i \(0.263579\pi\)
\(60\) 3.44038 0.444151
\(61\) −1.68451 −0.215679 −0.107840 0.994168i \(-0.534393\pi\)
−0.107840 + 0.994168i \(0.534393\pi\)
\(62\) 0.751482 0.0954384
\(63\) 1.00000 0.125988
\(64\) −7.65632 −0.957040
\(65\) 1.87039 0.231994
\(66\) −0.599069 −0.0737403
\(67\) 3.51374 0.429272 0.214636 0.976694i \(-0.431144\pi\)
0.214636 + 0.976694i \(0.431144\pi\)
\(68\) 3.02964 0.367397
\(69\) −7.05455 −0.849268
\(70\) −0.207967 −0.0248568
\(71\) −2.92691 −0.347360 −0.173680 0.984802i \(-0.555566\pi\)
−0.173680 + 0.984802i \(0.555566\pi\)
\(72\) 0.478379 0.0563775
\(73\) 1.51241 0.177014 0.0885069 0.996076i \(-0.471790\pi\)
0.0885069 + 0.996076i \(0.471790\pi\)
\(74\) 0.0404087 0.00469741
\(75\) 1.99785 0.230692
\(76\) −0.0115070 −0.00131995
\(77\) −4.99112 −0.568791
\(78\) 0.129568 0.0146706
\(79\) 3.99546 0.449524 0.224762 0.974414i \(-0.427839\pi\)
0.224762 + 0.974414i \(0.427839\pi\)
\(80\) 6.78127 0.758169
\(81\) 1.00000 0.111111
\(82\) −0.285705 −0.0315509
\(83\) −9.24882 −1.01519 −0.507595 0.861596i \(-0.669465\pi\)
−0.507595 + 0.861596i \(0.669465\pi\)
\(84\) 1.98559 0.216646
\(85\) −2.64372 −0.286752
\(86\) −1.09433 −0.118004
\(87\) −1.18505 −0.127051
\(88\) −2.38765 −0.254524
\(89\) −7.16310 −0.759287 −0.379644 0.925133i \(-0.623953\pi\)
−0.379644 + 0.925133i \(0.623953\pi\)
\(90\) −0.207967 −0.0219217
\(91\) 1.07949 0.113161
\(92\) −14.0075 −1.46038
\(93\) 6.26094 0.649230
\(94\) 0.913213 0.0941907
\(95\) 0.0100413 0.00103021
\(96\) 1.42652 0.145593
\(97\) 4.19603 0.426043 0.213021 0.977048i \(-0.431670\pi\)
0.213021 + 0.977048i \(0.431670\pi\)
\(98\) −0.120027 −0.0121246
\(99\) −4.99112 −0.501626
\(100\) 3.96692 0.396692
\(101\) 7.52131 0.748398 0.374199 0.927348i \(-0.377918\pi\)
0.374199 + 0.927348i \(0.377918\pi\)
\(102\) −0.183138 −0.0181334
\(103\) 16.5673 1.63242 0.816212 0.577753i \(-0.196070\pi\)
0.816212 + 0.577753i \(0.196070\pi\)
\(104\) 0.516403 0.0506375
\(105\) −1.73267 −0.169091
\(106\) 1.10579 0.107404
\(107\) −0.431970 −0.0417601 −0.0208800 0.999782i \(-0.506647\pi\)
−0.0208800 + 0.999782i \(0.506647\pi\)
\(108\) 1.98559 0.191064
\(109\) 14.3480 1.37429 0.687144 0.726521i \(-0.258864\pi\)
0.687144 + 0.726521i \(0.258864\pi\)
\(110\) 1.03799 0.0989684
\(111\) 0.336663 0.0319546
\(112\) 3.91377 0.369816
\(113\) 8.78452 0.826378 0.413189 0.910645i \(-0.364415\pi\)
0.413189 + 0.910645i \(0.364415\pi\)
\(114\) 0.000695587 0 6.51477e−5 0
\(115\) 12.2232 1.13982
\(116\) −2.35303 −0.218473
\(117\) 1.07949 0.0997985
\(118\) −1.24704 −0.114799
\(119\) −1.52581 −0.139871
\(120\) −0.828873 −0.0756654
\(121\) 13.9112 1.26466
\(122\) 0.202187 0.0183051
\(123\) −2.38034 −0.214628
\(124\) 12.4317 1.11640
\(125\) −12.1250 −1.08449
\(126\) −0.120027 −0.0106929
\(127\) −11.9872 −1.06369 −0.531847 0.846840i \(-0.678502\pi\)
−0.531847 + 0.846840i \(0.678502\pi\)
\(128\) 3.77200 0.333401
\(129\) −9.11735 −0.802738
\(130\) −0.224498 −0.0196898
\(131\) −7.27404 −0.635536 −0.317768 0.948169i \(-0.602933\pi\)
−0.317768 + 0.948169i \(0.602933\pi\)
\(132\) −9.91033 −0.862583
\(133\) 0.00579525 0.000502512 0
\(134\) −0.421744 −0.0364331
\(135\) −1.73267 −0.149125
\(136\) −0.729915 −0.0625897
\(137\) 10.8734 0.928981 0.464491 0.885578i \(-0.346237\pi\)
0.464491 + 0.885578i \(0.346237\pi\)
\(138\) 0.846737 0.0720790
\(139\) −13.9991 −1.18739 −0.593693 0.804692i \(-0.702331\pi\)
−0.593693 + 0.804692i \(0.702331\pi\)
\(140\) −3.44038 −0.290765
\(141\) 7.60839 0.640742
\(142\) 0.351308 0.0294811
\(143\) −5.38784 −0.450554
\(144\) 3.91377 0.326147
\(145\) 2.05330 0.170518
\(146\) −0.181530 −0.0150235
\(147\) −1.00000 −0.0824786
\(148\) 0.668476 0.0549484
\(149\) −1.87263 −0.153412 −0.0767059 0.997054i \(-0.524440\pi\)
−0.0767059 + 0.997054i \(0.524440\pi\)
\(150\) −0.239796 −0.0195793
\(151\) −10.9305 −0.889514 −0.444757 0.895651i \(-0.646710\pi\)
−0.444757 + 0.895651i \(0.646710\pi\)
\(152\) 0.00277233 0.000224865 0
\(153\) −1.52581 −0.123354
\(154\) 0.599069 0.0482743
\(155\) −10.8482 −0.871344
\(156\) 2.14342 0.171611
\(157\) 8.84714 0.706079 0.353039 0.935608i \(-0.385148\pi\)
0.353039 + 0.935608i \(0.385148\pi\)
\(158\) −0.479563 −0.0381520
\(159\) 9.21286 0.730627
\(160\) −2.47168 −0.195404
\(161\) 7.05455 0.555976
\(162\) −0.120027 −0.00943021
\(163\) −3.83641 −0.300491 −0.150245 0.988649i \(-0.548006\pi\)
−0.150245 + 0.988649i \(0.548006\pi\)
\(164\) −4.72639 −0.369069
\(165\) 8.64796 0.673243
\(166\) 1.11011 0.0861611
\(167\) −10.0104 −0.774625 −0.387312 0.921949i \(-0.626596\pi\)
−0.387312 + 0.921949i \(0.626596\pi\)
\(168\) −0.478379 −0.0369077
\(169\) −11.8347 −0.910362
\(170\) 0.317318 0.0243372
\(171\) 0.00579525 0.000443174 0
\(172\) −18.1033 −1.38037
\(173\) 6.57865 0.500166 0.250083 0.968224i \(-0.419542\pi\)
0.250083 + 0.968224i \(0.419542\pi\)
\(174\) 0.142238 0.0107830
\(175\) −1.99785 −0.151023
\(176\) −19.5341 −1.47244
\(177\) −10.3896 −0.780932
\(178\) 0.859766 0.0644422
\(179\) 5.55411 0.415134 0.207567 0.978221i \(-0.433446\pi\)
0.207567 + 0.978221i \(0.433446\pi\)
\(180\) −3.44038 −0.256431
\(181\) −0.787899 −0.0585641 −0.0292820 0.999571i \(-0.509322\pi\)
−0.0292820 + 0.999571i \(0.509322\pi\)
\(182\) −0.129568 −0.00960418
\(183\) 1.68451 0.124522
\(184\) 3.37475 0.248790
\(185\) −0.583326 −0.0428870
\(186\) −0.751482 −0.0551014
\(187\) 7.61549 0.556900
\(188\) 15.1072 1.10180
\(189\) −1.00000 −0.0727393
\(190\) −0.00120522 −8.74361e−5 0
\(191\) 26.8943 1.94601 0.973003 0.230791i \(-0.0741313\pi\)
0.973003 + 0.230791i \(0.0741313\pi\)
\(192\) 7.65632 0.552547
\(193\) 4.32077 0.311016 0.155508 0.987835i \(-0.450299\pi\)
0.155508 + 0.987835i \(0.450299\pi\)
\(194\) −0.503637 −0.0361591
\(195\) −1.87039 −0.133942
\(196\) −1.98559 −0.141828
\(197\) −1.36194 −0.0970339 −0.0485169 0.998822i \(-0.515449\pi\)
−0.0485169 + 0.998822i \(0.515449\pi\)
\(198\) 0.599069 0.0425740
\(199\) −13.8600 −0.982507 −0.491254 0.871017i \(-0.663461\pi\)
−0.491254 + 0.871017i \(0.663461\pi\)
\(200\) −0.955731 −0.0675804
\(201\) −3.51374 −0.247840
\(202\) −0.902760 −0.0635180
\(203\) 1.18505 0.0831743
\(204\) −3.02964 −0.212117
\(205\) 4.12435 0.288057
\(206\) −1.98852 −0.138547
\(207\) 7.05455 0.490325
\(208\) 4.22486 0.292941
\(209\) −0.0289248 −0.00200077
\(210\) 0.207967 0.0143511
\(211\) −2.02882 −0.139670 −0.0698349 0.997559i \(-0.522247\pi\)
−0.0698349 + 0.997559i \(0.522247\pi\)
\(212\) 18.2930 1.25637
\(213\) 2.92691 0.200548
\(214\) 0.0518481 0.00354426
\(215\) 15.7974 1.07737
\(216\) −0.478379 −0.0325496
\(217\) −6.26094 −0.425020
\(218\) −1.72215 −0.116638
\(219\) −1.51241 −0.102199
\(220\) 17.1713 1.15769
\(221\) −1.64709 −0.110795
\(222\) −0.0404087 −0.00271205
\(223\) 15.2233 1.01943 0.509714 0.860344i \(-0.329751\pi\)
0.509714 + 0.860344i \(0.329751\pi\)
\(224\) −1.42652 −0.0953131
\(225\) −1.99785 −0.133190
\(226\) −1.05438 −0.0701363
\(227\) −7.35715 −0.488311 −0.244156 0.969736i \(-0.578511\pi\)
−0.244156 + 0.969736i \(0.578511\pi\)
\(228\) 0.0115070 0.000762071 0
\(229\) −0.521614 −0.0344693 −0.0172346 0.999851i \(-0.505486\pi\)
−0.0172346 + 0.999851i \(0.505486\pi\)
\(230\) −1.46712 −0.0967387
\(231\) 4.99112 0.328391
\(232\) 0.566904 0.0372190
\(233\) 7.30939 0.478854 0.239427 0.970914i \(-0.423040\pi\)
0.239427 + 0.970914i \(0.423040\pi\)
\(234\) −0.129568 −0.00847009
\(235\) −13.1828 −0.859953
\(236\) −20.6296 −1.34287
\(237\) −3.99546 −0.259533
\(238\) 0.183138 0.0118711
\(239\) 18.2355 1.17956 0.589779 0.807564i \(-0.299215\pi\)
0.589779 + 0.807564i \(0.299215\pi\)
\(240\) −6.78127 −0.437729
\(241\) −12.0047 −0.773292 −0.386646 0.922228i \(-0.626366\pi\)
−0.386646 + 0.922228i \(0.626366\pi\)
\(242\) −1.66973 −0.107334
\(243\) −1.00000 −0.0641500
\(244\) 3.34475 0.214126
\(245\) 1.73267 0.110696
\(246\) 0.285705 0.0182159
\(247\) 0.00625590 0.000398053 0
\(248\) −2.99510 −0.190189
\(249\) 9.24882 0.586120
\(250\) 1.45532 0.0920428
\(251\) 19.1802 1.21064 0.605322 0.795981i \(-0.293044\pi\)
0.605322 + 0.795981i \(0.293044\pi\)
\(252\) −1.98559 −0.125081
\(253\) −35.2101 −2.21364
\(254\) 1.43879 0.0902778
\(255\) 2.64372 0.165556
\(256\) 14.8599 0.928743
\(257\) −1.76135 −0.109870 −0.0549349 0.998490i \(-0.517495\pi\)
−0.0549349 + 0.998490i \(0.517495\pi\)
\(258\) 1.09433 0.0681299
\(259\) −0.336663 −0.0209192
\(260\) −3.71384 −0.230323
\(261\) 1.18505 0.0733528
\(262\) 0.873081 0.0539391
\(263\) −24.3387 −1.50079 −0.750395 0.660990i \(-0.770137\pi\)
−0.750395 + 0.660990i \(0.770137\pi\)
\(264\) 2.38765 0.146949
\(265\) −15.9628 −0.980590
\(266\) −0.000695587 0 −4.26492e−5 0
\(267\) 7.16310 0.438375
\(268\) −6.97686 −0.426179
\(269\) 11.8401 0.721904 0.360952 0.932584i \(-0.382452\pi\)
0.360952 + 0.932584i \(0.382452\pi\)
\(270\) 0.207967 0.0126565
\(271\) 23.8145 1.44663 0.723313 0.690521i \(-0.242619\pi\)
0.723313 + 0.690521i \(0.242619\pi\)
\(272\) −5.97166 −0.362085
\(273\) −1.07949 −0.0653335
\(274\) −1.30511 −0.0788444
\(275\) 9.97152 0.601305
\(276\) 14.0075 0.843150
\(277\) 5.40217 0.324585 0.162293 0.986743i \(-0.448111\pi\)
0.162293 + 0.986743i \(0.448111\pi\)
\(278\) 1.68027 0.100776
\(279\) −6.26094 −0.374833
\(280\) 0.828873 0.0495346
\(281\) −1.04357 −0.0622540 −0.0311270 0.999515i \(-0.509910\pi\)
−0.0311270 + 0.999515i \(0.509910\pi\)
\(282\) −0.913213 −0.0543810
\(283\) −10.3751 −0.616737 −0.308369 0.951267i \(-0.599783\pi\)
−0.308369 + 0.951267i \(0.599783\pi\)
\(284\) 5.81165 0.344858
\(285\) −0.0100413 −0.000594793 0
\(286\) 0.646687 0.0382394
\(287\) 2.38034 0.140507
\(288\) −1.42652 −0.0840583
\(289\) −14.6719 −0.863053
\(290\) −0.246452 −0.0144722
\(291\) −4.19603 −0.245976
\(292\) −3.00302 −0.175739
\(293\) 22.2229 1.29827 0.649137 0.760672i \(-0.275130\pi\)
0.649137 + 0.760672i \(0.275130\pi\)
\(294\) 0.120027 0.00700012
\(295\) 18.0018 1.04810
\(296\) −0.161053 −0.00936099
\(297\) 4.99112 0.289614
\(298\) 0.224766 0.0130204
\(299\) 7.61529 0.440403
\(300\) −3.96692 −0.229030
\(301\) 9.11735 0.525515
\(302\) 1.31196 0.0754947
\(303\) −7.52131 −0.432088
\(304\) 0.0226813 0.00130086
\(305\) −2.91870 −0.167124
\(306\) 0.183138 0.0104693
\(307\) 3.21540 0.183513 0.0917563 0.995781i \(-0.470752\pi\)
0.0917563 + 0.995781i \(0.470752\pi\)
\(308\) 9.91033 0.564693
\(309\) −16.5673 −0.942480
\(310\) 1.30207 0.0739527
\(311\) 5.36542 0.304245 0.152123 0.988362i \(-0.451389\pi\)
0.152123 + 0.988362i \(0.451389\pi\)
\(312\) −0.516403 −0.0292356
\(313\) 22.7716 1.28713 0.643563 0.765393i \(-0.277456\pi\)
0.643563 + 0.765393i \(0.277456\pi\)
\(314\) −1.06190 −0.0599263
\(315\) 1.73267 0.0976249
\(316\) −7.93336 −0.446286
\(317\) −10.8058 −0.606916 −0.303458 0.952845i \(-0.598141\pi\)
−0.303458 + 0.952845i \(0.598141\pi\)
\(318\) −1.10579 −0.0620097
\(319\) −5.91473 −0.331161
\(320\) −13.2659 −0.741585
\(321\) 0.431970 0.0241102
\(322\) −0.846737 −0.0471868
\(323\) −0.00884245 −0.000492007 0
\(324\) −1.98559 −0.110311
\(325\) −2.15665 −0.119630
\(326\) 0.460473 0.0255032
\(327\) −14.3480 −0.793446
\(328\) 1.13871 0.0628745
\(329\) −7.60839 −0.419464
\(330\) −1.03799 −0.0571394
\(331\) 25.7589 1.41584 0.707918 0.706295i \(-0.249635\pi\)
0.707918 + 0.706295i \(0.249635\pi\)
\(332\) 18.3644 1.00788
\(333\) −0.336663 −0.0184490
\(334\) 1.20151 0.0657439
\(335\) 6.08815 0.332631
\(336\) −3.91377 −0.213514
\(337\) 3.94021 0.214637 0.107318 0.994225i \(-0.465774\pi\)
0.107318 + 0.994225i \(0.465774\pi\)
\(338\) 1.42049 0.0772642
\(339\) −8.78452 −0.477110
\(340\) 5.24936 0.284687
\(341\) 31.2491 1.69223
\(342\) −0.000695587 0 −3.76130e−5 0
\(343\) 1.00000 0.0539949
\(344\) 4.36155 0.235159
\(345\) −12.2232 −0.658075
\(346\) −0.789616 −0.0424500
\(347\) −29.1493 −1.56482 −0.782408 0.622766i \(-0.786009\pi\)
−0.782408 + 0.622766i \(0.786009\pi\)
\(348\) 2.35303 0.126136
\(349\) 28.0215 1.49996 0.749979 0.661462i \(-0.230064\pi\)
0.749979 + 0.661462i \(0.230064\pi\)
\(350\) 0.239796 0.0128177
\(351\) −1.07949 −0.0576187
\(352\) 7.11991 0.379492
\(353\) 5.79656 0.308520 0.154260 0.988030i \(-0.450701\pi\)
0.154260 + 0.988030i \(0.450701\pi\)
\(354\) 1.24704 0.0662792
\(355\) −5.07136 −0.269160
\(356\) 14.2230 0.753818
\(357\) 1.52581 0.0807544
\(358\) −0.666644 −0.0352332
\(359\) −11.2440 −0.593434 −0.296717 0.954965i \(-0.595892\pi\)
−0.296717 + 0.954965i \(0.595892\pi\)
\(360\) 0.828873 0.0436854
\(361\) −19.0000 −0.999998
\(362\) 0.0945692 0.00497045
\(363\) −13.9112 −0.730151
\(364\) −2.14342 −0.112346
\(365\) 2.62050 0.137163
\(366\) −0.202187 −0.0105685
\(367\) 32.0013 1.67045 0.835226 0.549906i \(-0.185337\pi\)
0.835226 + 0.549906i \(0.185337\pi\)
\(368\) 27.6099 1.43926
\(369\) 2.38034 0.123916
\(370\) 0.0700149 0.00363990
\(371\) −9.21286 −0.478308
\(372\) −12.4317 −0.644553
\(373\) 29.7796 1.54193 0.770964 0.636878i \(-0.219775\pi\)
0.770964 + 0.636878i \(0.219775\pi\)
\(374\) −0.914065 −0.0472652
\(375\) 12.1250 0.626131
\(376\) −3.63969 −0.187703
\(377\) 1.27925 0.0658845
\(378\) 0.120027 0.00617352
\(379\) −9.67410 −0.496925 −0.248463 0.968641i \(-0.579925\pi\)
−0.248463 + 0.968641i \(0.579925\pi\)
\(380\) −0.0199379 −0.00102279
\(381\) 11.9872 0.614124
\(382\) −3.22805 −0.165161
\(383\) −1.00000 −0.0510976
\(384\) −3.77200 −0.192489
\(385\) −8.64796 −0.440741
\(386\) −0.518610 −0.0263965
\(387\) 9.11735 0.463461
\(388\) −8.33162 −0.422974
\(389\) 30.3305 1.53782 0.768910 0.639357i \(-0.220800\pi\)
0.768910 + 0.639357i \(0.220800\pi\)
\(390\) 0.224498 0.0113679
\(391\) −10.7639 −0.544353
\(392\) 0.478379 0.0241618
\(393\) 7.27404 0.366927
\(394\) 0.163469 0.00823545
\(395\) 6.92282 0.348325
\(396\) 9.91033 0.498013
\(397\) 3.03220 0.152182 0.0760909 0.997101i \(-0.475756\pi\)
0.0760909 + 0.997101i \(0.475756\pi\)
\(398\) 1.66357 0.0833873
\(399\) −0.00579525 −0.000290126 0
\(400\) −7.81913 −0.390957
\(401\) 5.36053 0.267692 0.133846 0.991002i \(-0.457267\pi\)
0.133846 + 0.991002i \(0.457267\pi\)
\(402\) 0.421744 0.0210347
\(403\) −6.75860 −0.336670
\(404\) −14.9343 −0.743007
\(405\) 1.73267 0.0860971
\(406\) −0.142238 −0.00705916
\(407\) 1.68032 0.0832906
\(408\) 0.729915 0.0361362
\(409\) 19.6850 0.973363 0.486681 0.873580i \(-0.338207\pi\)
0.486681 + 0.873580i \(0.338207\pi\)
\(410\) −0.495033 −0.0244480
\(411\) −10.8734 −0.536347
\(412\) −32.8959 −1.62066
\(413\) 10.3896 0.511240
\(414\) −0.846737 −0.0416148
\(415\) −16.0252 −0.786644
\(416\) −1.53990 −0.0755000
\(417\) 13.9991 0.685537
\(418\) 0.00347176 0.000169809 0
\(419\) 7.02090 0.342993 0.171497 0.985185i \(-0.445140\pi\)
0.171497 + 0.985185i \(0.445140\pi\)
\(420\) 3.44038 0.167873
\(421\) 16.8381 0.820641 0.410320 0.911941i \(-0.365417\pi\)
0.410320 + 0.911941i \(0.365417\pi\)
\(422\) 0.243513 0.0118540
\(423\) −7.60839 −0.369933
\(424\) −4.40724 −0.214034
\(425\) 3.04834 0.147866
\(426\) −0.351308 −0.0170209
\(427\) −1.68451 −0.0815191
\(428\) 0.857716 0.0414593
\(429\) 5.38784 0.260127
\(430\) −1.89611 −0.0914386
\(431\) −14.7959 −0.712691 −0.356346 0.934354i \(-0.615977\pi\)
−0.356346 + 0.934354i \(0.615977\pi\)
\(432\) −3.91377 −0.188301
\(433\) 8.66002 0.416174 0.208087 0.978110i \(-0.433276\pi\)
0.208087 + 0.978110i \(0.433276\pi\)
\(434\) 0.751482 0.0360723
\(435\) −2.05330 −0.0984484
\(436\) −28.4893 −1.36439
\(437\) 0.0408829 0.00195569
\(438\) 0.181530 0.00867382
\(439\) 10.9857 0.524317 0.262159 0.965025i \(-0.415566\pi\)
0.262159 + 0.965025i \(0.415566\pi\)
\(440\) −4.13700 −0.197224
\(441\) 1.00000 0.0476190
\(442\) 0.197695 0.00940340
\(443\) −23.6676 −1.12448 −0.562241 0.826974i \(-0.690060\pi\)
−0.562241 + 0.826974i \(0.690060\pi\)
\(444\) −0.668476 −0.0317245
\(445\) −12.4113 −0.588352
\(446\) −1.82721 −0.0865208
\(447\) 1.87263 0.0885723
\(448\) −7.65632 −0.361727
\(449\) 28.8511 1.36157 0.680784 0.732484i \(-0.261639\pi\)
0.680784 + 0.732484i \(0.261639\pi\)
\(450\) 0.239796 0.0113041
\(451\) −11.8806 −0.559434
\(452\) −17.4425 −0.820425
\(453\) 10.9305 0.513561
\(454\) 0.883057 0.0414439
\(455\) 1.87039 0.0876854
\(456\) −0.00277233 −0.000129826 0
\(457\) 28.2758 1.32269 0.661344 0.750083i \(-0.269987\pi\)
0.661344 + 0.750083i \(0.269987\pi\)
\(458\) 0.0626078 0.00292547
\(459\) 1.52581 0.0712186
\(460\) −24.2703 −1.13161
\(461\) 2.02729 0.0944205 0.0472102 0.998885i \(-0.484967\pi\)
0.0472102 + 0.998885i \(0.484967\pi\)
\(462\) −0.599069 −0.0278712
\(463\) −16.2149 −0.753568 −0.376784 0.926301i \(-0.622970\pi\)
−0.376784 + 0.926301i \(0.622970\pi\)
\(464\) 4.63802 0.215315
\(465\) 10.8482 0.503071
\(466\) −0.877324 −0.0406412
\(467\) 7.84438 0.362995 0.181497 0.983391i \(-0.441906\pi\)
0.181497 + 0.983391i \(0.441906\pi\)
\(468\) −2.14342 −0.0990797
\(469\) 3.51374 0.162249
\(470\) 1.58230 0.0729859
\(471\) −8.84714 −0.407655
\(472\) 4.97018 0.228771
\(473\) −45.5058 −2.09236
\(474\) 0.479563 0.0220271
\(475\) −0.0115781 −0.000531238 0
\(476\) 3.02964 0.138863
\(477\) −9.21286 −0.421828
\(478\) −2.18876 −0.100111
\(479\) −8.98025 −0.410318 −0.205159 0.978729i \(-0.565771\pi\)
−0.205159 + 0.978729i \(0.565771\pi\)
\(480\) 2.47168 0.112816
\(481\) −0.363423 −0.0165707
\(482\) 1.44089 0.0656308
\(483\) −7.05455 −0.320993
\(484\) −27.6221 −1.25555
\(485\) 7.27034 0.330129
\(486\) 0.120027 0.00544454
\(487\) −5.69157 −0.257909 −0.128955 0.991650i \(-0.541162\pi\)
−0.128955 + 0.991650i \(0.541162\pi\)
\(488\) −0.805833 −0.0364784
\(489\) 3.83641 0.173488
\(490\) −0.207967 −0.00939500
\(491\) 18.5466 0.836996 0.418498 0.908218i \(-0.362557\pi\)
0.418498 + 0.908218i \(0.362557\pi\)
\(492\) 4.72639 0.213082
\(493\) −1.80816 −0.0814355
\(494\) −0.000750877 0 −3.37835e−5 0
\(495\) −8.64796 −0.388697
\(496\) −24.5039 −1.10026
\(497\) −2.92691 −0.131290
\(498\) −1.11011 −0.0497452
\(499\) 6.95469 0.311335 0.155667 0.987810i \(-0.450247\pi\)
0.155667 + 0.987810i \(0.450247\pi\)
\(500\) 24.0753 1.07668
\(501\) 10.0104 0.447230
\(502\) −2.30214 −0.102750
\(503\) 31.4094 1.40048 0.700238 0.713910i \(-0.253077\pi\)
0.700238 + 0.713910i \(0.253077\pi\)
\(504\) 0.478379 0.0213087
\(505\) 13.0319 0.579914
\(506\) 4.22616 0.187876
\(507\) 11.8347 0.525598
\(508\) 23.8018 1.05603
\(509\) 20.3220 0.900755 0.450378 0.892838i \(-0.351289\pi\)
0.450378 + 0.892838i \(0.351289\pi\)
\(510\) −0.317318 −0.0140511
\(511\) 1.51241 0.0669049
\(512\) −9.32758 −0.412225
\(513\) −0.00579525 −0.000255867 0
\(514\) 0.211409 0.00932486
\(515\) 28.7056 1.26492
\(516\) 18.1033 0.796955
\(517\) 37.9744 1.67011
\(518\) 0.0404087 0.00177546
\(519\) −6.57865 −0.288771
\(520\) 0.894757 0.0392377
\(521\) 13.6375 0.597471 0.298736 0.954336i \(-0.403435\pi\)
0.298736 + 0.954336i \(0.403435\pi\)
\(522\) −0.142238 −0.00622560
\(523\) −16.4548 −0.719519 −0.359760 0.933045i \(-0.617141\pi\)
−0.359760 + 0.933045i \(0.617141\pi\)
\(524\) 14.4433 0.630958
\(525\) 1.99785 0.0871934
\(526\) 2.92130 0.127375
\(527\) 9.55300 0.416135
\(528\) 19.5341 0.850112
\(529\) 26.7667 1.16377
\(530\) 1.91597 0.0832245
\(531\) 10.3896 0.450871
\(532\) −0.0115070 −0.000498892 0
\(533\) 2.56955 0.111299
\(534\) −0.859766 −0.0372057
\(535\) −0.748461 −0.0323588
\(536\) 1.68090 0.0726038
\(537\) −5.55411 −0.239678
\(538\) −1.42113 −0.0612693
\(539\) −4.99112 −0.214983
\(540\) 3.44038 0.148050
\(541\) 29.7250 1.27798 0.638990 0.769215i \(-0.279353\pi\)
0.638990 + 0.769215i \(0.279353\pi\)
\(542\) −2.85838 −0.122778
\(543\) 0.787899 0.0338120
\(544\) 2.17659 0.0933205
\(545\) 24.8603 1.06490
\(546\) 0.129568 0.00554498
\(547\) −27.5504 −1.17797 −0.588984 0.808144i \(-0.700472\pi\)
−0.588984 + 0.808144i \(0.700472\pi\)
\(548\) −21.5902 −0.922289
\(549\) −1.68451 −0.0718931
\(550\) −1.19685 −0.0510339
\(551\) 0.00686767 0.000292573 0
\(552\) −3.37475 −0.143639
\(553\) 3.99546 0.169904
\(554\) −0.648407 −0.0275482
\(555\) 0.583326 0.0247608
\(556\) 27.7965 1.17883
\(557\) 26.0151 1.10229 0.551147 0.834408i \(-0.314190\pi\)
0.551147 + 0.834408i \(0.314190\pi\)
\(558\) 0.751482 0.0318128
\(559\) 9.84205 0.416274
\(560\) 6.78127 0.286561
\(561\) −7.61549 −0.321526
\(562\) 0.125256 0.00528361
\(563\) −25.3919 −1.07014 −0.535071 0.844807i \(-0.679715\pi\)
−0.535071 + 0.844807i \(0.679715\pi\)
\(564\) −15.1072 −0.636127
\(565\) 15.2207 0.640339
\(566\) 1.24530 0.0523437
\(567\) 1.00000 0.0419961
\(568\) −1.40017 −0.0587498
\(569\) 37.0776 1.55437 0.777187 0.629269i \(-0.216646\pi\)
0.777187 + 0.629269i \(0.216646\pi\)
\(570\) 0.00120522 5.04812e−5 0
\(571\) 20.0764 0.840169 0.420085 0.907485i \(-0.362000\pi\)
0.420085 + 0.907485i \(0.362000\pi\)
\(572\) 10.6981 0.447309
\(573\) −26.8943 −1.12353
\(574\) −0.285705 −0.0119251
\(575\) −14.0940 −0.587758
\(576\) −7.65632 −0.319013
\(577\) 37.6452 1.56719 0.783595 0.621272i \(-0.213384\pi\)
0.783595 + 0.621272i \(0.213384\pi\)
\(578\) 1.76103 0.0732490
\(579\) −4.32077 −0.179565
\(580\) −4.07703 −0.169289
\(581\) −9.24882 −0.383706
\(582\) 0.503637 0.0208764
\(583\) 45.9824 1.90440
\(584\) 0.723503 0.0299388
\(585\) 1.87039 0.0773313
\(586\) −2.66734 −0.110187
\(587\) −1.07341 −0.0443044 −0.0221522 0.999755i \(-0.507052\pi\)
−0.0221522 + 0.999755i \(0.507052\pi\)
\(588\) 1.98559 0.0818845
\(589\) −0.0362837 −0.00149505
\(590\) −2.16070 −0.0889547
\(591\) 1.36194 0.0560225
\(592\) −1.31762 −0.0541539
\(593\) 22.5860 0.927497 0.463749 0.885967i \(-0.346504\pi\)
0.463749 + 0.885967i \(0.346504\pi\)
\(594\) −0.599069 −0.0245801
\(595\) −2.64372 −0.108382
\(596\) 3.71828 0.152307
\(597\) 13.8600 0.567251
\(598\) −0.914040 −0.0373779
\(599\) 35.0987 1.43409 0.717047 0.697025i \(-0.245493\pi\)
0.717047 + 0.697025i \(0.245493\pi\)
\(600\) 0.955731 0.0390175
\(601\) 37.6565 1.53604 0.768020 0.640426i \(-0.221242\pi\)
0.768020 + 0.640426i \(0.221242\pi\)
\(602\) −1.09433 −0.0446015
\(603\) 3.51374 0.143091
\(604\) 21.7036 0.883106
\(605\) 24.1036 0.979951
\(606\) 0.902760 0.0366721
\(607\) −31.0322 −1.25956 −0.629779 0.776774i \(-0.716855\pi\)
−0.629779 + 0.776774i \(0.716855\pi\)
\(608\) −0.00826702 −0.000335272 0
\(609\) −1.18505 −0.0480207
\(610\) 0.350323 0.0141841
\(611\) −8.21315 −0.332269
\(612\) 3.02964 0.122466
\(613\) −2.88519 −0.116532 −0.0582658 0.998301i \(-0.518557\pi\)
−0.0582658 + 0.998301i \(0.518557\pi\)
\(614\) −0.385935 −0.0155751
\(615\) −4.12435 −0.166310
\(616\) −2.38765 −0.0962010
\(617\) −26.7012 −1.07495 −0.537474 0.843280i \(-0.680621\pi\)
−0.537474 + 0.843280i \(0.680621\pi\)
\(618\) 1.98852 0.0799901
\(619\) 5.45259 0.219158 0.109579 0.993978i \(-0.465050\pi\)
0.109579 + 0.993978i \(0.465050\pi\)
\(620\) 21.5400 0.865068
\(621\) −7.05455 −0.283089
\(622\) −0.643996 −0.0258219
\(623\) −7.16310 −0.286984
\(624\) −4.22486 −0.169130
\(625\) −11.0193 −0.440773
\(626\) −2.73320 −0.109241
\(627\) 0.0289248 0.00115514
\(628\) −17.5668 −0.700993
\(629\) 0.513683 0.0204819
\(630\) −0.207967 −0.00828562
\(631\) 37.3560 1.48712 0.743559 0.668670i \(-0.233136\pi\)
0.743559 + 0.668670i \(0.233136\pi\)
\(632\) 1.91134 0.0760292
\(633\) 2.02882 0.0806384
\(634\) 1.29699 0.0515101
\(635\) −20.7699 −0.824229
\(636\) −18.2930 −0.725364
\(637\) 1.07949 0.0427708
\(638\) 0.709927 0.0281063
\(639\) −2.92691 −0.115787
\(640\) 6.53563 0.258343
\(641\) 25.3736 1.00220 0.501098 0.865391i \(-0.332930\pi\)
0.501098 + 0.865391i \(0.332930\pi\)
\(642\) −0.0518481 −0.00204628
\(643\) 6.54646 0.258167 0.129084 0.991634i \(-0.458796\pi\)
0.129084 + 0.991634i \(0.458796\pi\)
\(644\) −14.0075 −0.551972
\(645\) −15.7974 −0.622020
\(646\) 0.00106133 4.17576e−5 0
\(647\) 5.54073 0.217829 0.108914 0.994051i \(-0.465263\pi\)
0.108914 + 0.994051i \(0.465263\pi\)
\(648\) 0.478379 0.0187925
\(649\) −51.8558 −2.03552
\(650\) 0.258857 0.0101532
\(651\) 6.26094 0.245386
\(652\) 7.61755 0.298326
\(653\) 6.66414 0.260788 0.130394 0.991462i \(-0.458376\pi\)
0.130394 + 0.991462i \(0.458376\pi\)
\(654\) 1.72215 0.0673413
\(655\) −12.6035 −0.492460
\(656\) 9.31611 0.363733
\(657\) 1.51241 0.0590046
\(658\) 0.913213 0.0356007
\(659\) −34.9347 −1.36086 −0.680432 0.732811i \(-0.738208\pi\)
−0.680432 + 0.732811i \(0.738208\pi\)
\(660\) −17.1713 −0.668393
\(661\) −9.75107 −0.379273 −0.189636 0.981854i \(-0.560731\pi\)
−0.189636 + 0.981854i \(0.560731\pi\)
\(662\) −3.09176 −0.120165
\(663\) 1.64709 0.0639676
\(664\) −4.42444 −0.171702
\(665\) 0.0100413 0.000389384 0
\(666\) 0.0404087 0.00156580
\(667\) 8.36000 0.323701
\(668\) 19.8765 0.769045
\(669\) −15.2233 −0.588567
\(670\) −0.730743 −0.0282311
\(671\) 8.40758 0.324571
\(672\) 1.42652 0.0550290
\(673\) −7.59691 −0.292839 −0.146420 0.989223i \(-0.546775\pi\)
−0.146420 + 0.989223i \(0.546775\pi\)
\(674\) −0.472932 −0.0182167
\(675\) 1.99785 0.0768974
\(676\) 23.4989 0.903805
\(677\) −40.3451 −1.55059 −0.775294 0.631600i \(-0.782399\pi\)
−0.775294 + 0.631600i \(0.782399\pi\)
\(678\) 1.05438 0.0404932
\(679\) 4.19603 0.161029
\(680\) −1.26470 −0.0484991
\(681\) 7.35715 0.281927
\(682\) −3.75074 −0.143623
\(683\) −0.308757 −0.0118143 −0.00590713 0.999983i \(-0.501880\pi\)
−0.00590713 + 0.999983i \(0.501880\pi\)
\(684\) −0.0115070 −0.000439982 0
\(685\) 18.8401 0.719843
\(686\) −0.120027 −0.00458265
\(687\) 0.521614 0.0199008
\(688\) 35.6832 1.36041
\(689\) −9.94515 −0.378880
\(690\) 1.46712 0.0558521
\(691\) 0.137321 0.00522394 0.00261197 0.999997i \(-0.499169\pi\)
0.00261197 + 0.999997i \(0.499169\pi\)
\(692\) −13.0625 −0.496563
\(693\) −4.99112 −0.189597
\(694\) 3.49870 0.132809
\(695\) −24.2558 −0.920074
\(696\) −0.566904 −0.0214884
\(697\) −3.63195 −0.137570
\(698\) −3.36334 −0.127304
\(699\) −7.30939 −0.276466
\(700\) 3.96692 0.149936
\(701\) 3.69096 0.139406 0.0697029 0.997568i \(-0.477795\pi\)
0.0697029 + 0.997568i \(0.477795\pi\)
\(702\) 0.129568 0.00489021
\(703\) −0.00195105 −7.35852e−5 0
\(704\) 38.2136 1.44023
\(705\) 13.1828 0.496494
\(706\) −0.695744 −0.0261847
\(707\) 7.52131 0.282868
\(708\) 20.6296 0.775307
\(709\) 46.7741 1.75664 0.878320 0.478073i \(-0.158665\pi\)
0.878320 + 0.478073i \(0.158665\pi\)
\(710\) 0.608701 0.0228441
\(711\) 3.99546 0.149841
\(712\) −3.42668 −0.128420
\(713\) −44.1681 −1.65411
\(714\) −0.183138 −0.00685378
\(715\) −9.33535 −0.349122
\(716\) −11.0282 −0.412144
\(717\) −18.2355 −0.681019
\(718\) 1.34958 0.0503659
\(719\) −2.28764 −0.0853147 −0.0426574 0.999090i \(-0.513582\pi\)
−0.0426574 + 0.999090i \(0.513582\pi\)
\(720\) 6.78127 0.252723
\(721\) 16.5673 0.616998
\(722\) 2.28051 0.0848718
\(723\) 12.0047 0.446460
\(724\) 1.56445 0.0581422
\(725\) −2.36756 −0.0879289
\(726\) 1.66973 0.0619693
\(727\) 19.6653 0.729344 0.364672 0.931136i \(-0.381181\pi\)
0.364672 + 0.931136i \(0.381181\pi\)
\(728\) 0.516403 0.0191392
\(729\) 1.00000 0.0370370
\(730\) −0.314531 −0.0116413
\(731\) −13.9113 −0.514529
\(732\) −3.34475 −0.123625
\(733\) 29.5659 1.09204 0.546021 0.837772i \(-0.316142\pi\)
0.546021 + 0.837772i \(0.316142\pi\)
\(734\) −3.84102 −0.141775
\(735\) −1.73267 −0.0639105
\(736\) −10.0634 −0.370943
\(737\) −17.5375 −0.646002
\(738\) −0.285705 −0.0105170
\(739\) 10.7447 0.395251 0.197625 0.980278i \(-0.436677\pi\)
0.197625 + 0.980278i \(0.436677\pi\)
\(740\) 1.15825 0.0425781
\(741\) −0.00625590 −0.000229816 0
\(742\) 1.10579 0.0405949
\(743\) 5.65769 0.207561 0.103780 0.994600i \(-0.466906\pi\)
0.103780 + 0.994600i \(0.466906\pi\)
\(744\) 2.99510 0.109806
\(745\) −3.24465 −0.118875
\(746\) −3.57436 −0.130866
\(747\) −9.24882 −0.338397
\(748\) −15.1213 −0.552888
\(749\) −0.431970 −0.0157838
\(750\) −1.45532 −0.0531409
\(751\) −11.6510 −0.425152 −0.212576 0.977144i \(-0.568185\pi\)
−0.212576 + 0.977144i \(0.568185\pi\)
\(752\) −29.7775 −1.08587
\(753\) −19.1802 −0.698966
\(754\) −0.153544 −0.00559175
\(755\) −18.9390 −0.689261
\(756\) 1.98559 0.0722153
\(757\) 37.2997 1.35568 0.677840 0.735209i \(-0.262916\pi\)
0.677840 + 0.735209i \(0.262916\pi\)
\(758\) 1.16115 0.0421750
\(759\) 35.2101 1.27804
\(760\) 0.00480353 0.000174242 0
\(761\) −8.42854 −0.305534 −0.152767 0.988262i \(-0.548818\pi\)
−0.152767 + 0.988262i \(0.548818\pi\)
\(762\) −1.43879 −0.0521219
\(763\) 14.3480 0.519432
\(764\) −53.4012 −1.93199
\(765\) −2.64372 −0.0955840
\(766\) 0.120027 0.00433675
\(767\) 11.2155 0.404967
\(768\) −14.8599 −0.536210
\(769\) 18.8486 0.679699 0.339850 0.940480i \(-0.389624\pi\)
0.339850 + 0.940480i \(0.389624\pi\)
\(770\) 1.03799 0.0374065
\(771\) 1.76135 0.0634334
\(772\) −8.57930 −0.308776
\(773\) −34.0120 −1.22333 −0.611664 0.791118i \(-0.709499\pi\)
−0.611664 + 0.791118i \(0.709499\pi\)
\(774\) −1.09433 −0.0393348
\(775\) 12.5084 0.449316
\(776\) 2.00729 0.0720577
\(777\) 0.336663 0.0120777
\(778\) −3.64048 −0.130518
\(779\) 0.0137947 0.000494246 0
\(780\) 3.71384 0.132977
\(781\) 14.6085 0.522734
\(782\) 1.29196 0.0462003
\(783\) −1.18505 −0.0423503
\(784\) 3.91377 0.139777
\(785\) 15.3292 0.547122
\(786\) −0.873081 −0.0311418
\(787\) 45.9741 1.63880 0.819400 0.573222i \(-0.194307\pi\)
0.819400 + 0.573222i \(0.194307\pi\)
\(788\) 2.70425 0.0963349
\(789\) 24.3387 0.866481
\(790\) −0.830925 −0.0295630
\(791\) 8.78452 0.312342
\(792\) −2.38765 −0.0848413
\(793\) −1.81840 −0.0645734
\(794\) −0.363946 −0.0129160
\(795\) 15.9628 0.566144
\(796\) 27.5203 0.975430
\(797\) −28.9758 −1.02638 −0.513188 0.858276i \(-0.671536\pi\)
−0.513188 + 0.858276i \(0.671536\pi\)
\(798\) 0.000695587 0 2.46235e−5 0
\(799\) 11.6089 0.410695
\(800\) 2.84997 0.100762
\(801\) −7.16310 −0.253096
\(802\) −0.643408 −0.0227195
\(803\) −7.54860 −0.266384
\(804\) 6.97686 0.246055
\(805\) 12.2232 0.430812
\(806\) 0.811215 0.0285738
\(807\) −11.8401 −0.416791
\(808\) 3.59803 0.126578
\(809\) 27.6999 0.973876 0.486938 0.873437i \(-0.338114\pi\)
0.486938 + 0.873437i \(0.338114\pi\)
\(810\) −0.207967 −0.00730723
\(811\) −18.5133 −0.650088 −0.325044 0.945699i \(-0.605379\pi\)
−0.325044 + 0.945699i \(0.605379\pi\)
\(812\) −2.35303 −0.0825752
\(813\) −23.8145 −0.835209
\(814\) −0.201684 −0.00706903
\(815\) −6.64723 −0.232842
\(816\) 5.97166 0.209050
\(817\) 0.0528373 0.00184854
\(818\) −2.36274 −0.0826112
\(819\) 1.07949 0.0377203
\(820\) −8.18928 −0.285982
\(821\) −33.0738 −1.15428 −0.577141 0.816644i \(-0.695832\pi\)
−0.577141 + 0.816644i \(0.695832\pi\)
\(822\) 1.30511 0.0455208
\(823\) −23.5507 −0.820927 −0.410464 0.911877i \(-0.634633\pi\)
−0.410464 + 0.911877i \(0.634633\pi\)
\(824\) 7.92544 0.276096
\(825\) −9.97152 −0.347164
\(826\) −1.24704 −0.0433899
\(827\) −31.5501 −1.09711 −0.548553 0.836116i \(-0.684821\pi\)
−0.548553 + 0.836116i \(0.684821\pi\)
\(828\) −14.0075 −0.486793
\(829\) 32.6150 1.13277 0.566384 0.824142i \(-0.308342\pi\)
0.566384 + 0.824142i \(0.308342\pi\)
\(830\) 1.92345 0.0667640
\(831\) −5.40217 −0.187399
\(832\) −8.26489 −0.286533
\(833\) −1.52581 −0.0528661
\(834\) −1.68027 −0.0581829
\(835\) −17.3446 −0.600236
\(836\) 0.0574329 0.00198636
\(837\) 6.26094 0.216410
\(838\) −0.842698 −0.0291105
\(839\) 27.8475 0.961403 0.480701 0.876884i \(-0.340382\pi\)
0.480701 + 0.876884i \(0.340382\pi\)
\(840\) −0.828873 −0.0285988
\(841\) −27.5957 −0.951574
\(842\) −2.02103 −0.0696494
\(843\) 1.04357 0.0359423
\(844\) 4.02841 0.138664
\(845\) −20.5057 −0.705416
\(846\) 0.913213 0.0313969
\(847\) 13.9112 0.477996
\(848\) −36.0570 −1.23820
\(849\) 10.3751 0.356073
\(850\) −0.365883 −0.0125497
\(851\) −2.37501 −0.0814142
\(852\) −5.81165 −0.199104
\(853\) −27.9335 −0.956424 −0.478212 0.878245i \(-0.658715\pi\)
−0.478212 + 0.878245i \(0.658715\pi\)
\(854\) 0.202187 0.00691868
\(855\) 0.0100413 0.000343404 0
\(856\) −0.206645 −0.00706299
\(857\) 10.0055 0.341781 0.170891 0.985290i \(-0.445336\pi\)
0.170891 + 0.985290i \(0.445336\pi\)
\(858\) −0.646687 −0.0220775
\(859\) 15.2413 0.520027 0.260014 0.965605i \(-0.416273\pi\)
0.260014 + 0.965605i \(0.416273\pi\)
\(860\) −31.3671 −1.06961
\(861\) −2.38034 −0.0811218
\(862\) 1.77590 0.0604875
\(863\) 29.0605 0.989231 0.494615 0.869112i \(-0.335309\pi\)
0.494615 + 0.869112i \(0.335309\pi\)
\(864\) 1.42652 0.0485311
\(865\) 11.3986 0.387565
\(866\) −1.03944 −0.0353215
\(867\) 14.6719 0.498284
\(868\) 12.4317 0.421959
\(869\) −19.9418 −0.676480
\(870\) 0.246452 0.00835550
\(871\) 3.79303 0.128522
\(872\) 6.86378 0.232437
\(873\) 4.19603 0.142014
\(874\) −0.00490705 −0.000165984 0
\(875\) −12.1250 −0.409899
\(876\) 3.00302 0.101463
\(877\) 30.6895 1.03631 0.518155 0.855287i \(-0.326619\pi\)
0.518155 + 0.855287i \(0.326619\pi\)
\(878\) −1.31858 −0.0444998
\(879\) −22.2229 −0.749558
\(880\) −33.8461 −1.14095
\(881\) −39.6196 −1.33482 −0.667409 0.744692i \(-0.732597\pi\)
−0.667409 + 0.744692i \(0.732597\pi\)
\(882\) −0.120027 −0.00404152
\(883\) 28.7131 0.966273 0.483136 0.875545i \(-0.339498\pi\)
0.483136 + 0.875545i \(0.339498\pi\)
\(884\) 3.27045 0.109997
\(885\) −18.0018 −0.605124
\(886\) 2.84075 0.0954369
\(887\) 1.54036 0.0517202 0.0258601 0.999666i \(-0.491768\pi\)
0.0258601 + 0.999666i \(0.491768\pi\)
\(888\) 0.161053 0.00540457
\(889\) −11.9872 −0.402039
\(890\) 1.48969 0.0499345
\(891\) −4.99112 −0.167209
\(892\) −30.2273 −1.01209
\(893\) −0.0440926 −0.00147550
\(894\) −0.224766 −0.00751731
\(895\) 9.62345 0.321676
\(896\) 3.77200 0.126014
\(897\) −7.61529 −0.254267
\(898\) −3.46291 −0.115559
\(899\) −7.41954 −0.247455
\(900\) 3.96692 0.132231
\(901\) 14.0571 0.468308
\(902\) 1.42599 0.0474802
\(903\) −9.11735 −0.303406
\(904\) 4.20233 0.139767
\(905\) −1.36517 −0.0453798
\(906\) −1.31196 −0.0435869
\(907\) 38.0481 1.26337 0.631684 0.775226i \(-0.282364\pi\)
0.631684 + 0.775226i \(0.282364\pi\)
\(908\) 14.6083 0.484794
\(909\) 7.52131 0.249466
\(910\) −0.224498 −0.00744203
\(911\) −9.62453 −0.318875 −0.159437 0.987208i \(-0.550968\pi\)
−0.159437 + 0.987208i \(0.550968\pi\)
\(912\) −0.0226813 −0.000751052 0
\(913\) 46.1619 1.52774
\(914\) −3.39386 −0.112259
\(915\) 2.91870 0.0964892
\(916\) 1.03571 0.0342210
\(917\) −7.27404 −0.240210
\(918\) −0.183138 −0.00604446
\(919\) −29.0365 −0.957824 −0.478912 0.877863i \(-0.658969\pi\)
−0.478912 + 0.877863i \(0.658969\pi\)
\(920\) 5.84733 0.192781
\(921\) −3.21540 −0.105951
\(922\) −0.243330 −0.00801365
\(923\) −3.15955 −0.103998
\(924\) −9.91033 −0.326026
\(925\) 0.672603 0.0221151
\(926\) 1.94622 0.0639568
\(927\) 16.5673 0.544141
\(928\) −1.69049 −0.0554932
\(929\) 3.74697 0.122934 0.0614670 0.998109i \(-0.480422\pi\)
0.0614670 + 0.998109i \(0.480422\pi\)
\(930\) −1.30207 −0.0426966
\(931\) 0.00579525 0.000189932 0
\(932\) −14.5135 −0.475405
\(933\) −5.36542 −0.175656
\(934\) −0.941538 −0.0308081
\(935\) 13.1951 0.431527
\(936\) 0.516403 0.0168792
\(937\) 11.4736 0.374827 0.187414 0.982281i \(-0.439990\pi\)
0.187414 + 0.982281i \(0.439990\pi\)
\(938\) −0.421744 −0.0137704
\(939\) −22.7716 −0.743122
\(940\) 26.1758 0.853759
\(941\) −25.3351 −0.825902 −0.412951 0.910753i \(-0.635502\pi\)
−0.412951 + 0.910753i \(0.635502\pi\)
\(942\) 1.06190 0.0345985
\(943\) 16.7922 0.546831
\(944\) 40.6626 1.32345
\(945\) −1.73267 −0.0563638
\(946\) 5.46192 0.177582
\(947\) −34.6245 −1.12514 −0.562572 0.826748i \(-0.690188\pi\)
−0.562572 + 0.826748i \(0.690188\pi\)
\(948\) 7.93336 0.257664
\(949\) 1.63262 0.0529971
\(950\) 0.00138968 4.50872e−5 0
\(951\) 10.8058 0.350403
\(952\) −0.729915 −0.0236567
\(953\) 32.1189 1.04043 0.520217 0.854034i \(-0.325851\pi\)
0.520217 + 0.854034i \(0.325851\pi\)
\(954\) 1.10579 0.0358013
\(955\) 46.5990 1.50791
\(956\) −36.2084 −1.17106
\(957\) 5.91473 0.191196
\(958\) 1.07787 0.0348245
\(959\) 10.8734 0.351122
\(960\) 13.2659 0.428154
\(961\) 8.19940 0.264497
\(962\) 0.0436206 0.00140638
\(963\) −0.431970 −0.0139200
\(964\) 23.8365 0.767721
\(965\) 7.48648 0.240998
\(966\) 0.846737 0.0272433
\(967\) −8.80911 −0.283282 −0.141641 0.989918i \(-0.545238\pi\)
−0.141641 + 0.989918i \(0.545238\pi\)
\(968\) 6.65485 0.213895
\(969\) 0.00884245 0.000284060 0
\(970\) −0.872638 −0.0280187
\(971\) −42.7141 −1.37076 −0.685380 0.728185i \(-0.740364\pi\)
−0.685380 + 0.728185i \(0.740364\pi\)
\(972\) 1.98559 0.0636879
\(973\) −13.9991 −0.448790
\(974\) 0.683142 0.0218893
\(975\) 2.15665 0.0690682
\(976\) −6.59278 −0.211030
\(977\) 0.547556 0.0175179 0.00875894 0.999962i \(-0.497212\pi\)
0.00875894 + 0.999962i \(0.497212\pi\)
\(978\) −0.460473 −0.0147243
\(979\) 35.7519 1.14263
\(980\) −3.44038 −0.109899
\(981\) 14.3480 0.458096
\(982\) −2.22609 −0.0710374
\(983\) −44.1091 −1.40686 −0.703431 0.710764i \(-0.748350\pi\)
−0.703431 + 0.710764i \(0.748350\pi\)
\(984\) −1.13871 −0.0363006
\(985\) −2.35979 −0.0751890
\(986\) 0.217028 0.00691159
\(987\) 7.60839 0.242178
\(988\) −0.0124217 −0.000395186 0
\(989\) 64.3188 2.04522
\(990\) 1.03799 0.0329895
\(991\) 1.20554 0.0382954 0.0191477 0.999817i \(-0.493905\pi\)
0.0191477 + 0.999817i \(0.493905\pi\)
\(992\) 8.93133 0.283570
\(993\) −25.7589 −0.817433
\(994\) 0.351308 0.0111428
\(995\) −24.0148 −0.761319
\(996\) −18.3644 −0.581898
\(997\) −5.91391 −0.187295 −0.0936477 0.995605i \(-0.529853\pi\)
−0.0936477 + 0.995605i \(0.529853\pi\)
\(998\) −0.834751 −0.0264236
\(999\) 0.336663 0.0106515
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 8043.2.a.t.1.25 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.t.1.25 52 1.1 even 1 trivial