Properties

Label 2001.2.a.g.1.4
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $1$
Dimension $4$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, 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("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.291367\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} +2.14073 q^{5} +2.72347 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} +2.14073 q^{5} +2.72347 q^{7} +1.00000 q^{9} -0.451253 q^{11} +2.00000 q^{12} -5.41294 q^{13} -2.14073 q^{15} +4.00000 q^{16} -3.24046 q^{17} -3.89595 q^{19} -4.28146 q^{20} -2.72347 q^{21} +1.00000 q^{23} -0.417266 q^{25} -1.00000 q^{27} -5.44693 q^{28} +1.00000 q^{29} -1.55800 q^{31} +0.451253 q^{33} +5.83021 q^{35} -2.00000 q^{36} -4.21572 q^{37} +5.41294 q^{39} +3.10674 q^{41} +10.6287 q^{43} +0.902507 q^{44} +2.14073 q^{45} -2.14073 q^{47} -4.00000 q^{48} +0.417266 q^{49} +3.24046 q^{51} +10.8259 q^{52} -0.834532 q^{53} -0.966013 q^{55} +3.89595 q^{57} -14.0049 q^{59} +4.28146 q^{60} +6.69441 q^{61} +2.72347 q^{63} -8.00000 q^{64} -11.5877 q^{65} -5.44693 q^{67} +6.48092 q^{68} -1.00000 q^{69} -1.82528 q^{71} -8.72347 q^{73} +0.417266 q^{75} +7.79190 q^{76} -1.22897 q^{77} -8.73048 q^{79} +8.56293 q^{80} +1.00000 q^{81} +0.403693 q^{83} +5.44693 q^{84} -6.93696 q^{85} -1.00000 q^{87} +4.22498 q^{89} -14.7420 q^{91} -2.00000 q^{92} +1.55800 q^{93} -8.34019 q^{95} -1.86852 q^{97} -0.451253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{9} + 8 q^{12} + 2 q^{15} + 16 q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} + 2 q^{21} + 4 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{28} + 4 q^{29} + 2 q^{31} + 4 q^{35} - 8 q^{36} + 4 q^{37} + 6 q^{41} - 2 q^{45} + 2 q^{47} - 16 q^{48} + 4 q^{49} - 2 q^{51} - 8 q^{53} - 8 q^{55} - 4 q^{57} - 22 q^{59} - 4 q^{60} - 16 q^{61} - 2 q^{63} - 32 q^{64} - 10 q^{65} + 4 q^{67} - 4 q^{68} - 4 q^{69} - 22 q^{71} - 22 q^{73} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 20 q^{79} - 8 q^{80} + 4 q^{81} - 10 q^{83} - 4 q^{84} - 2 q^{85} - 4 q^{87} - 14 q^{89} - 26 q^{91} - 8 q^{92} - 2 q^{93} - 14 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350
\(4\) −2.00000 −1.00000
\(5\) 2.14073 0.957364 0.478682 0.877988i \(-0.341115\pi\)
0.478682 + 0.877988i \(0.341115\pi\)
\(6\) 0 0
\(7\) 2.72347 1.02937 0.514687 0.857378i \(-0.327908\pi\)
0.514687 + 0.857378i \(0.327908\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.451253 −0.136058 −0.0680290 0.997683i \(-0.521671\pi\)
−0.0680290 + 0.997683i \(0.521671\pi\)
\(12\) 2.00000 0.577350
\(13\) −5.41294 −1.50128 −0.750640 0.660711i \(-0.770255\pi\)
−0.750640 + 0.660711i \(0.770255\pi\)
\(14\) 0 0
\(15\) −2.14073 −0.552735
\(16\) 4.00000 1.00000
\(17\) −3.24046 −0.785927 −0.392963 0.919554i \(-0.628550\pi\)
−0.392963 + 0.919554i \(0.628550\pi\)
\(18\) 0 0
\(19\) −3.89595 −0.893792 −0.446896 0.894586i \(-0.647471\pi\)
−0.446896 + 0.894586i \(0.647471\pi\)
\(20\) −4.28146 −0.957364
\(21\) −2.72347 −0.594309
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −0.417266 −0.0834532
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −5.44693 −1.02937
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −1.55800 −0.279825 −0.139912 0.990164i \(-0.544682\pi\)
−0.139912 + 0.990164i \(0.544682\pi\)
\(32\) 0 0
\(33\) 0.451253 0.0785531
\(34\) 0 0
\(35\) 5.83021 0.985485
\(36\) −2.00000 −0.333333
\(37\) −4.21572 −0.693061 −0.346530 0.938039i \(-0.612640\pi\)
−0.346530 + 0.938039i \(0.612640\pi\)
\(38\) 0 0
\(39\) 5.41294 0.866765
\(40\) 0 0
\(41\) 3.10674 0.485192 0.242596 0.970127i \(-0.422001\pi\)
0.242596 + 0.970127i \(0.422001\pi\)
\(42\) 0 0
\(43\) 10.6287 1.62086 0.810428 0.585838i \(-0.199234\pi\)
0.810428 + 0.585838i \(0.199234\pi\)
\(44\) 0.902507 0.136058
\(45\) 2.14073 0.319121
\(46\) 0 0
\(47\) −2.14073 −0.312258 −0.156129 0.987737i \(-0.549902\pi\)
−0.156129 + 0.987737i \(0.549902\pi\)
\(48\) −4.00000 −0.577350
\(49\) 0.417266 0.0596095
\(50\) 0 0
\(51\) 3.24046 0.453755
\(52\) 10.8259 1.50128
\(53\) −0.834532 −0.114632 −0.0573159 0.998356i \(-0.518254\pi\)
−0.0573159 + 0.998356i \(0.518254\pi\)
\(54\) 0 0
\(55\) −0.966013 −0.130257
\(56\) 0 0
\(57\) 3.89595 0.516031
\(58\) 0 0
\(59\) −14.0049 −1.82329 −0.911643 0.410982i \(-0.865186\pi\)
−0.911643 + 0.410982i \(0.865186\pi\)
\(60\) 4.28146 0.552735
\(61\) 6.69441 0.857131 0.428566 0.903511i \(-0.359019\pi\)
0.428566 + 0.903511i \(0.359019\pi\)
\(62\) 0 0
\(63\) 2.72347 0.343124
\(64\) −8.00000 −1.00000
\(65\) −11.5877 −1.43727
\(66\) 0 0
\(67\) −5.44693 −0.665449 −0.332724 0.943024i \(-0.607968\pi\)
−0.332724 + 0.943024i \(0.607968\pi\)
\(68\) 6.48092 0.785927
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −1.82528 −0.216621 −0.108310 0.994117i \(-0.534544\pi\)
−0.108310 + 0.994117i \(0.534544\pi\)
\(72\) 0 0
\(73\) −8.72347 −1.02100 −0.510502 0.859876i \(-0.670540\pi\)
−0.510502 + 0.859876i \(0.670540\pi\)
\(74\) 0 0
\(75\) 0.417266 0.0481818
\(76\) 7.79190 0.893792
\(77\) −1.22897 −0.140055
\(78\) 0 0
\(79\) −8.73048 −0.982256 −0.491128 0.871087i \(-0.663415\pi\)
−0.491128 + 0.871087i \(0.663415\pi\)
\(80\) 8.56293 0.957364
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.403693 0.0443110 0.0221555 0.999755i \(-0.492947\pi\)
0.0221555 + 0.999755i \(0.492947\pi\)
\(84\) 5.44693 0.594309
\(85\) −6.93696 −0.752418
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 4.22498 0.447847 0.223923 0.974607i \(-0.428113\pi\)
0.223923 + 0.974607i \(0.428113\pi\)
\(90\) 0 0
\(91\) −14.7420 −1.54538
\(92\) −2.00000 −0.208514
\(93\) 1.55800 0.161557
\(94\) 0 0
\(95\) −8.34019 −0.855685
\(96\) 0 0
\(97\) −1.86852 −0.189719 −0.0948597 0.995491i \(-0.530240\pi\)
−0.0948597 + 0.995491i \(0.530240\pi\)
\(98\) 0 0
\(99\) −0.451253 −0.0453527
\(100\) 0.834532 0.0834532
\(101\) 3.17472 0.315896 0.157948 0.987447i \(-0.449512\pi\)
0.157948 + 0.987447i \(0.449512\pi\)
\(102\) 0 0
\(103\) −7.66042 −0.754804 −0.377402 0.926050i \(-0.623182\pi\)
−0.377402 + 0.926050i \(0.623182\pi\)
\(104\) 0 0
\(105\) −5.83021 −0.568970
\(106\) 0 0
\(107\) 4.15924 0.402088 0.201044 0.979582i \(-0.435567\pi\)
0.201044 + 0.979582i \(0.435567\pi\)
\(108\) 2.00000 0.192450
\(109\) −14.4334 −1.38246 −0.691232 0.722632i \(-0.742932\pi\)
−0.691232 + 0.722632i \(0.742932\pi\)
\(110\) 0 0
\(111\) 4.21572 0.400139
\(112\) 10.8939 1.02937
\(113\) −3.03399 −0.285414 −0.142707 0.989765i \(-0.545581\pi\)
−0.142707 + 0.989765i \(0.545581\pi\)
\(114\) 0 0
\(115\) 2.14073 0.199624
\(116\) −2.00000 −0.185695
\(117\) −5.41294 −0.500427
\(118\) 0 0
\(119\) −8.82528 −0.809012
\(120\) 0 0
\(121\) −10.7964 −0.981488
\(122\) 0 0
\(123\) −3.10674 −0.280126
\(124\) 3.11600 0.279825
\(125\) −11.5969 −1.03726
\(126\) 0 0
\(127\) −7.47599 −0.663387 −0.331693 0.943387i \(-0.607620\pi\)
−0.331693 + 0.943387i \(0.607620\pi\)
\(128\) 0 0
\(129\) −10.6287 −0.935802
\(130\) 0 0
\(131\) −0.0185035 −0.00161666 −0.000808329 1.00000i \(-0.500257\pi\)
−0.000808329 1.00000i \(0.500257\pi\)
\(132\) −0.902507 −0.0785531
\(133\) −10.6105 −0.920046
\(134\) 0 0
\(135\) −2.14073 −0.184245
\(136\) 0 0
\(137\) −4.61942 −0.394663 −0.197332 0.980337i \(-0.563228\pi\)
−0.197332 + 0.980337i \(0.563228\pi\)
\(138\) 0 0
\(139\) 2.51476 0.213299 0.106650 0.994297i \(-0.465988\pi\)
0.106650 + 0.994297i \(0.465988\pi\)
\(140\) −11.6604 −0.985485
\(141\) 2.14073 0.180282
\(142\) 0 0
\(143\) 2.44261 0.204261
\(144\) 4.00000 0.333333
\(145\) 2.14073 0.177778
\(146\) 0 0
\(147\) −0.417266 −0.0344155
\(148\) 8.43145 0.693061
\(149\) −13.2475 −1.08528 −0.542638 0.839967i \(-0.682574\pi\)
−0.542638 + 0.839967i \(0.682574\pi\)
\(150\) 0 0
\(151\) −0.267282 −0.0217511 −0.0108756 0.999941i \(-0.503462\pi\)
−0.0108756 + 0.999941i \(0.503462\pi\)
\(152\) 0 0
\(153\) −3.24046 −0.261976
\(154\) 0 0
\(155\) −3.33526 −0.267894
\(156\) −10.8259 −0.866765
\(157\) 16.4152 1.31007 0.655037 0.755597i \(-0.272653\pi\)
0.655037 + 0.755597i \(0.272653\pi\)
\(158\) 0 0
\(159\) 0.834532 0.0661827
\(160\) 0 0
\(161\) 2.72347 0.214639
\(162\) 0 0
\(163\) 13.0438 1.02167 0.510836 0.859678i \(-0.329336\pi\)
0.510836 + 0.859678i \(0.329336\pi\)
\(164\) −6.21349 −0.485192
\(165\) 0.966013 0.0752040
\(166\) 0 0
\(167\) −24.0531 −1.86128 −0.930642 0.365930i \(-0.880751\pi\)
−0.930642 + 0.365930i \(0.880751\pi\)
\(168\) 0 0
\(169\) 16.3000 1.25384
\(170\) 0 0
\(171\) −3.89595 −0.297931
\(172\) −21.2573 −1.62086
\(173\) 15.7968 1.20101 0.600505 0.799621i \(-0.294966\pi\)
0.600505 + 0.799621i \(0.294966\pi\)
\(174\) 0 0
\(175\) −1.13641 −0.0859045
\(176\) −1.80501 −0.136058
\(177\) 14.0049 1.05267
\(178\) 0 0
\(179\) −25.3204 −1.89253 −0.946267 0.323386i \(-0.895179\pi\)
−0.946267 + 0.323386i \(0.895179\pi\)
\(180\) −4.28146 −0.319121
\(181\) 15.7284 1.16908 0.584541 0.811364i \(-0.301275\pi\)
0.584541 + 0.811364i \(0.301275\pi\)
\(182\) 0 0
\(183\) −6.69441 −0.494865
\(184\) 0 0
\(185\) −9.02474 −0.663512
\(186\) 0 0
\(187\) 1.46227 0.106932
\(188\) 4.28146 0.312258
\(189\) −2.72347 −0.198103
\(190\) 0 0
\(191\) −8.87122 −0.641899 −0.320949 0.947096i \(-0.604002\pi\)
−0.320949 + 0.947096i \(0.604002\pi\)
\(192\) 8.00000 0.577350
\(193\) −0.330935 −0.0238212 −0.0119106 0.999929i \(-0.503791\pi\)
−0.0119106 + 0.999929i \(0.503791\pi\)
\(194\) 0 0
\(195\) 11.5877 0.829810
\(196\) −0.834532 −0.0596095
\(197\) −15.9414 −1.13578 −0.567890 0.823105i \(-0.692240\pi\)
−0.567890 + 0.823105i \(0.692240\pi\)
\(198\) 0 0
\(199\) 2.51537 0.178310 0.0891548 0.996018i \(-0.471583\pi\)
0.0891548 + 0.996018i \(0.471583\pi\)
\(200\) 0 0
\(201\) 5.44693 0.384197
\(202\) 0 0
\(203\) 2.72347 0.191150
\(204\) −6.48092 −0.453755
\(205\) 6.65071 0.464506
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) −21.6518 −1.50128
\(209\) 1.75806 0.121608
\(210\) 0 0
\(211\) 5.33587 0.367336 0.183668 0.982988i \(-0.441203\pi\)
0.183668 + 0.982988i \(0.441203\pi\)
\(212\) 1.66906 0.114632
\(213\) 1.82528 0.125066
\(214\) 0 0
\(215\) 22.7531 1.55175
\(216\) 0 0
\(217\) −4.24316 −0.288044
\(218\) 0 0
\(219\) 8.72347 0.589477
\(220\) 1.93203 0.130257
\(221\) 17.5404 1.17990
\(222\) 0 0
\(223\) 10.1272 0.678165 0.339082 0.940757i \(-0.389883\pi\)
0.339082 + 0.940757i \(0.389883\pi\)
\(224\) 0 0
\(225\) −0.417266 −0.0278177
\(226\) 0 0
\(227\) −8.16849 −0.542162 −0.271081 0.962557i \(-0.587381\pi\)
−0.271081 + 0.962557i \(0.587381\pi\)
\(228\) −7.79190 −0.516031
\(229\) 28.5084 1.88388 0.941942 0.335774i \(-0.108998\pi\)
0.941942 + 0.335774i \(0.108998\pi\)
\(230\) 0 0
\(231\) 1.22897 0.0808605
\(232\) 0 0
\(233\) 1.07723 0.0705714 0.0352857 0.999377i \(-0.488766\pi\)
0.0352857 + 0.999377i \(0.488766\pi\)
\(234\) 0 0
\(235\) −4.58273 −0.298945
\(236\) 28.0099 1.82329
\(237\) 8.73048 0.567106
\(238\) 0 0
\(239\) −28.8197 −1.86419 −0.932094 0.362216i \(-0.882020\pi\)
−0.932094 + 0.362216i \(0.882020\pi\)
\(240\) −8.56293 −0.552735
\(241\) −17.2388 −1.11045 −0.555225 0.831700i \(-0.687368\pi\)
−0.555225 + 0.831700i \(0.687368\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −13.3888 −0.857131
\(245\) 0.893255 0.0570680
\(246\) 0 0
\(247\) 21.0886 1.34183
\(248\) 0 0
\(249\) −0.403693 −0.0255830
\(250\) 0 0
\(251\) −11.7552 −0.741983 −0.370991 0.928636i \(-0.620982\pi\)
−0.370991 + 0.928636i \(0.620982\pi\)
\(252\) −5.44693 −0.343124
\(253\) −0.451253 −0.0283701
\(254\) 0 0
\(255\) 6.93696 0.434409
\(256\) 16.0000 1.00000
\(257\) 9.34558 0.582961 0.291481 0.956577i \(-0.405852\pi\)
0.291481 + 0.956577i \(0.405852\pi\)
\(258\) 0 0
\(259\) −11.4814 −0.713418
\(260\) 23.1753 1.43727
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −21.2956 −1.31315 −0.656573 0.754263i \(-0.727995\pi\)
−0.656573 + 0.754263i \(0.727995\pi\)
\(264\) 0 0
\(265\) −1.78651 −0.109744
\(266\) 0 0
\(267\) −4.22498 −0.258564
\(268\) 10.8939 0.665449
\(269\) 4.00000 0.243884 0.121942 0.992537i \(-0.461088\pi\)
0.121942 + 0.992537i \(0.461088\pi\)
\(270\) 0 0
\(271\) 24.1783 1.46873 0.734365 0.678755i \(-0.237480\pi\)
0.734365 + 0.678755i \(0.237480\pi\)
\(272\) −12.9618 −0.785927
\(273\) 14.7420 0.892225
\(274\) 0 0
\(275\) 0.188293 0.0113545
\(276\) 2.00000 0.120386
\(277\) 17.6507 1.06053 0.530264 0.847832i \(-0.322093\pi\)
0.530264 + 0.847832i \(0.322093\pi\)
\(278\) 0 0
\(279\) −1.55800 −0.0932749
\(280\) 0 0
\(281\) 15.8784 0.947225 0.473612 0.880733i \(-0.342950\pi\)
0.473612 + 0.880733i \(0.342950\pi\)
\(282\) 0 0
\(283\) −9.88893 −0.587836 −0.293918 0.955831i \(-0.594959\pi\)
−0.293918 + 0.955831i \(0.594959\pi\)
\(284\) 3.65056 0.216621
\(285\) 8.34019 0.494030
\(286\) 0 0
\(287\) 8.46111 0.499444
\(288\) 0 0
\(289\) −6.49942 −0.382319
\(290\) 0 0
\(291\) 1.86852 0.109535
\(292\) 17.4469 1.02100
\(293\) 11.5899 0.677089 0.338545 0.940950i \(-0.390065\pi\)
0.338545 + 0.940950i \(0.390065\pi\)
\(294\) 0 0
\(295\) −29.9808 −1.74555
\(296\) 0 0
\(297\) 0.451253 0.0261844
\(298\) 0 0
\(299\) −5.41294 −0.313039
\(300\) −0.834532 −0.0481818
\(301\) 28.9468 1.66847
\(302\) 0 0
\(303\) −3.17472 −0.182383
\(304\) −15.5838 −0.893792
\(305\) 14.3309 0.820587
\(306\) 0 0
\(307\) 30.4617 1.73854 0.869271 0.494336i \(-0.164589\pi\)
0.869271 + 0.494336i \(0.164589\pi\)
\(308\) 2.45795 0.140055
\(309\) 7.66042 0.435786
\(310\) 0 0
\(311\) 14.4314 0.818332 0.409166 0.912460i \(-0.365820\pi\)
0.409166 + 0.912460i \(0.365820\pi\)
\(312\) 0 0
\(313\) 24.1844 1.36698 0.683492 0.729958i \(-0.260460\pi\)
0.683492 + 0.729958i \(0.260460\pi\)
\(314\) 0 0
\(315\) 5.83021 0.328495
\(316\) 17.4610 0.982256
\(317\) −20.1452 −1.13147 −0.565734 0.824588i \(-0.691407\pi\)
−0.565734 + 0.824588i \(0.691407\pi\)
\(318\) 0 0
\(319\) −0.451253 −0.0252653
\(320\) −17.1259 −0.957364
\(321\) −4.15924 −0.232146
\(322\) 0 0
\(323\) 12.6247 0.702455
\(324\) −2.00000 −0.111111
\(325\) 2.25864 0.125287
\(326\) 0 0
\(327\) 14.4334 0.798167
\(328\) 0 0
\(329\) −5.83021 −0.321430
\(330\) 0 0
\(331\) 22.9618 1.26210 0.631048 0.775743i \(-0.282625\pi\)
0.631048 + 0.775743i \(0.282625\pi\)
\(332\) −0.807385 −0.0443110
\(333\) −4.21572 −0.231020
\(334\) 0 0
\(335\) −11.6604 −0.637077
\(336\) −10.8939 −0.594309
\(337\) −8.62420 −0.469790 −0.234895 0.972021i \(-0.575475\pi\)
−0.234895 + 0.972021i \(0.575475\pi\)
\(338\) 0 0
\(339\) 3.03399 0.164784
\(340\) 13.8739 0.752418
\(341\) 0.703052 0.0380724
\(342\) 0 0
\(343\) −17.9279 −0.968013
\(344\) 0 0
\(345\) −2.14073 −0.115253
\(346\) 0 0
\(347\) −0.276534 −0.0148451 −0.00742257 0.999972i \(-0.502363\pi\)
−0.00742257 + 0.999972i \(0.502363\pi\)
\(348\) 2.00000 0.107211
\(349\) 10.1753 0.544673 0.272336 0.962202i \(-0.412204\pi\)
0.272336 + 0.962202i \(0.412204\pi\)
\(350\) 0 0
\(351\) 5.41294 0.288922
\(352\) 0 0
\(353\) 11.8895 0.632816 0.316408 0.948623i \(-0.397523\pi\)
0.316408 + 0.948623i \(0.397523\pi\)
\(354\) 0 0
\(355\) −3.90744 −0.207385
\(356\) −8.44995 −0.447847
\(357\) 8.82528 0.467083
\(358\) 0 0
\(359\) −11.1923 −0.590706 −0.295353 0.955388i \(-0.595437\pi\)
−0.295353 + 0.955388i \(0.595437\pi\)
\(360\) 0 0
\(361\) −3.82157 −0.201135
\(362\) 0 0
\(363\) 10.7964 0.566662
\(364\) 29.4839 1.54538
\(365\) −18.6746 −0.977473
\(366\) 0 0
\(367\) 1.37896 0.0719810 0.0359905 0.999352i \(-0.488541\pi\)
0.0359905 + 0.999352i \(0.488541\pi\)
\(368\) 4.00000 0.208514
\(369\) 3.10674 0.161731
\(370\) 0 0
\(371\) −2.27282 −0.117999
\(372\) −3.11600 −0.161557
\(373\) −5.90744 −0.305875 −0.152938 0.988236i \(-0.548873\pi\)
−0.152938 + 0.988236i \(0.548873\pi\)
\(374\) 0 0
\(375\) 11.5969 0.598862
\(376\) 0 0
\(377\) −5.41294 −0.278781
\(378\) 0 0
\(379\) −27.2114 −1.39776 −0.698878 0.715241i \(-0.746317\pi\)
−0.698878 + 0.715241i \(0.746317\pi\)
\(380\) 16.6804 0.855685
\(381\) 7.47599 0.383007
\(382\) 0 0
\(383\) −12.5939 −0.643518 −0.321759 0.946822i \(-0.604274\pi\)
−0.321759 + 0.946822i \(0.604274\pi\)
\(384\) 0 0
\(385\) −2.63090 −0.134083
\(386\) 0 0
\(387\) 10.6287 0.540286
\(388\) 3.73704 0.189719
\(389\) −12.6491 −0.641334 −0.320667 0.947192i \(-0.603907\pi\)
−0.320667 + 0.947192i \(0.603907\pi\)
\(390\) 0 0
\(391\) −3.24046 −0.163877
\(392\) 0 0
\(393\) 0.0185035 0.000933378 0
\(394\) 0 0
\(395\) −18.6896 −0.940377
\(396\) 0.902507 0.0453527
\(397\) 17.4358 0.875076 0.437538 0.899200i \(-0.355851\pi\)
0.437538 + 0.899200i \(0.355851\pi\)
\(398\) 0 0
\(399\) 10.6105 0.531189
\(400\) −1.66906 −0.0834532
\(401\) −8.47228 −0.423085 −0.211543 0.977369i \(-0.567849\pi\)
−0.211543 + 0.977369i \(0.567849\pi\)
\(402\) 0 0
\(403\) 8.43336 0.420095
\(404\) −6.34944 −0.315896
\(405\) 2.14073 0.106374
\(406\) 0 0
\(407\) 1.90236 0.0942965
\(408\) 0 0
\(409\) 0.768468 0.0379983 0.0189992 0.999819i \(-0.493952\pi\)
0.0189992 + 0.999819i \(0.493952\pi\)
\(410\) 0 0
\(411\) 4.61942 0.227859
\(412\) 15.3208 0.754804
\(413\) −38.1419 −1.87684
\(414\) 0 0
\(415\) 0.864198 0.0424218
\(416\) 0 0
\(417\) −2.51476 −0.123148
\(418\) 0 0
\(419\) 15.2573 0.745370 0.372685 0.927958i \(-0.378437\pi\)
0.372685 + 0.927958i \(0.378437\pi\)
\(420\) 11.6604 0.568970
\(421\) 1.23855 0.0603632 0.0301816 0.999544i \(-0.490391\pi\)
0.0301816 + 0.999544i \(0.490391\pi\)
\(422\) 0 0
\(423\) −2.14073 −0.104086
\(424\) 0 0
\(425\) 1.35213 0.0655881
\(426\) 0 0
\(427\) 18.2320 0.882308
\(428\) −8.31847 −0.402088
\(429\) −2.44261 −0.117930
\(430\) 0 0
\(431\) 11.8458 0.570594 0.285297 0.958439i \(-0.407908\pi\)
0.285297 + 0.958439i \(0.407908\pi\)
\(432\) −4.00000 −0.192450
\(433\) −27.4131 −1.31739 −0.658695 0.752410i \(-0.728891\pi\)
−0.658695 + 0.752410i \(0.728891\pi\)
\(434\) 0 0
\(435\) −2.14073 −0.102640
\(436\) 28.8667 1.38246
\(437\) −3.89595 −0.186369
\(438\) 0 0
\(439\) 21.5246 1.02731 0.513657 0.857996i \(-0.328290\pi\)
0.513657 + 0.857996i \(0.328290\pi\)
\(440\) 0 0
\(441\) 0.417266 0.0198698
\(442\) 0 0
\(443\) −0.671122 −0.0318860 −0.0159430 0.999873i \(-0.505075\pi\)
−0.0159430 + 0.999873i \(0.505075\pi\)
\(444\) −8.43145 −0.400139
\(445\) 9.04454 0.428752
\(446\) 0 0
\(447\) 13.2475 0.626584
\(448\) −21.7877 −1.02937
\(449\) 21.9517 1.03597 0.517983 0.855391i \(-0.326683\pi\)
0.517983 + 0.855391i \(0.326683\pi\)
\(450\) 0 0
\(451\) −1.40193 −0.0660143
\(452\) 6.06797 0.285414
\(453\) 0.267282 0.0125580
\(454\) 0 0
\(455\) −31.5586 −1.47949
\(456\) 0 0
\(457\) 0.310521 0.0145256 0.00726279 0.999974i \(-0.497688\pi\)
0.00726279 + 0.999974i \(0.497688\pi\)
\(458\) 0 0
\(459\) 3.24046 0.151252
\(460\) −4.28146 −0.199624
\(461\) 15.9604 0.743349 0.371675 0.928363i \(-0.378784\pi\)
0.371675 + 0.928363i \(0.378784\pi\)
\(462\) 0 0
\(463\) 16.6451 0.773563 0.386781 0.922171i \(-0.373587\pi\)
0.386781 + 0.922171i \(0.373587\pi\)
\(464\) 4.00000 0.185695
\(465\) 3.33526 0.154669
\(466\) 0 0
\(467\) −12.7126 −0.588268 −0.294134 0.955764i \(-0.595031\pi\)
−0.294134 + 0.955764i \(0.595031\pi\)
\(468\) 10.8259 0.500427
\(469\) −14.8345 −0.684995
\(470\) 0 0
\(471\) −16.4152 −0.756372
\(472\) 0 0
\(473\) −4.79622 −0.220531
\(474\) 0 0
\(475\) 1.62565 0.0745899
\(476\) 17.6506 0.809012
\(477\) −0.834532 −0.0382106
\(478\) 0 0
\(479\) −33.1686 −1.51551 −0.757757 0.652537i \(-0.773705\pi\)
−0.757757 + 0.652537i \(0.773705\pi\)
\(480\) 0 0
\(481\) 22.8195 1.04048
\(482\) 0 0
\(483\) −2.72347 −0.123922
\(484\) 21.5927 0.981488
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −23.8851 −1.08234 −0.541168 0.840915i \(-0.682018\pi\)
−0.541168 + 0.840915i \(0.682018\pi\)
\(488\) 0 0
\(489\) −13.0438 −0.589863
\(490\) 0 0
\(491\) −0.290716 −0.0131198 −0.00655991 0.999978i \(-0.502088\pi\)
−0.00655991 + 0.999978i \(0.502088\pi\)
\(492\) 6.21349 0.280126
\(493\) −3.24046 −0.145943
\(494\) 0 0
\(495\) −0.966013 −0.0434190
\(496\) −6.23199 −0.279825
\(497\) −4.97109 −0.222984
\(498\) 0 0
\(499\) 16.0735 0.719549 0.359775 0.933039i \(-0.382854\pi\)
0.359775 + 0.933039i \(0.382854\pi\)
\(500\) 23.1938 1.03726
\(501\) 24.0531 1.07461
\(502\) 0 0
\(503\) −6.77517 −0.302090 −0.151045 0.988527i \(-0.548264\pi\)
−0.151045 + 0.988527i \(0.548264\pi\)
\(504\) 0 0
\(505\) 6.79622 0.302428
\(506\) 0 0
\(507\) −16.3000 −0.723907
\(508\) 14.9520 0.663387
\(509\) 38.8024 1.71988 0.859942 0.510391i \(-0.170499\pi\)
0.859942 + 0.510391i \(0.170499\pi\)
\(510\) 0 0
\(511\) −23.7581 −1.05099
\(512\) 0 0
\(513\) 3.89595 0.172010
\(514\) 0 0
\(515\) −16.3989 −0.722622
\(516\) 21.2573 0.935802
\(517\) 0.966013 0.0424852
\(518\) 0 0
\(519\) −15.7968 −0.693404
\(520\) 0 0
\(521\) −24.2961 −1.06443 −0.532216 0.846609i \(-0.678641\pi\)
−0.532216 + 0.846609i \(0.678641\pi\)
\(522\) 0 0
\(523\) −6.94682 −0.303763 −0.151882 0.988399i \(-0.548533\pi\)
−0.151882 + 0.988399i \(0.548533\pi\)
\(524\) 0.0370070 0.00161666
\(525\) 1.13641 0.0495970
\(526\) 0 0
\(527\) 5.04863 0.219922
\(528\) 1.80501 0.0785531
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −14.0049 −0.607762
\(532\) 21.2210 0.920046
\(533\) −16.8166 −0.728409
\(534\) 0 0
\(535\) 8.90381 0.384945
\(536\) 0 0
\(537\) 25.3204 1.09266
\(538\) 0 0
\(539\) −0.188293 −0.00811035
\(540\) 4.28146 0.184245
\(541\) 10.0692 0.432908 0.216454 0.976293i \(-0.430551\pi\)
0.216454 + 0.976293i \(0.430551\pi\)
\(542\) 0 0
\(543\) −15.7284 −0.674970
\(544\) 0 0
\(545\) −30.8980 −1.32352
\(546\) 0 0
\(547\) −3.63220 −0.155302 −0.0776509 0.996981i \(-0.524742\pi\)
−0.0776509 + 0.996981i \(0.524742\pi\)
\(548\) 9.23883 0.394663
\(549\) 6.69441 0.285710
\(550\) 0 0
\(551\) −3.89595 −0.165973
\(552\) 0 0
\(553\) −23.7772 −1.01111
\(554\) 0 0
\(555\) 9.02474 0.383079
\(556\) −5.02952 −0.213299
\(557\) 17.0260 0.721413 0.360706 0.932679i \(-0.382536\pi\)
0.360706 + 0.932679i \(0.382536\pi\)
\(558\) 0 0
\(559\) −57.5324 −2.43336
\(560\) 23.3208 0.985485
\(561\) −1.46227 −0.0617370
\(562\) 0 0
\(563\) 5.80051 0.244463 0.122231 0.992502i \(-0.460995\pi\)
0.122231 + 0.992502i \(0.460995\pi\)
\(564\) −4.28146 −0.180282
\(565\) −6.49495 −0.273245
\(566\) 0 0
\(567\) 2.72347 0.114375
\(568\) 0 0
\(569\) 16.6197 0.696736 0.348368 0.937358i \(-0.386736\pi\)
0.348368 + 0.937358i \(0.386736\pi\)
\(570\) 0 0
\(571\) 9.26789 0.387849 0.193925 0.981016i \(-0.437878\pi\)
0.193925 + 0.981016i \(0.437878\pi\)
\(572\) −4.88522 −0.204261
\(573\) 8.87122 0.370600
\(574\) 0 0
\(575\) −0.417266 −0.0174012
\(576\) −8.00000 −0.333333
\(577\) 43.2098 1.79885 0.899423 0.437079i \(-0.143987\pi\)
0.899423 + 0.437079i \(0.143987\pi\)
\(578\) 0 0
\(579\) 0.330935 0.0137532
\(580\) −4.28146 −0.177778
\(581\) 1.09944 0.0456126
\(582\) 0 0
\(583\) 0.376586 0.0155966
\(584\) 0 0
\(585\) −11.5877 −0.479091
\(586\) 0 0
\(587\) −35.2690 −1.45571 −0.727853 0.685733i \(-0.759482\pi\)
−0.727853 + 0.685733i \(0.759482\pi\)
\(588\) 0.834532 0.0344155
\(589\) 6.06988 0.250105
\(590\) 0 0
\(591\) 15.9414 0.655743
\(592\) −16.8629 −0.693061
\(593\) 15.8508 0.650913 0.325457 0.945557i \(-0.394482\pi\)
0.325457 + 0.945557i \(0.394482\pi\)
\(594\) 0 0
\(595\) −18.8926 −0.774520
\(596\) 26.4950 1.08528
\(597\) −2.51537 −0.102947
\(598\) 0 0
\(599\) −36.6357 −1.49689 −0.748447 0.663195i \(-0.769200\pi\)
−0.748447 + 0.663195i \(0.769200\pi\)
\(600\) 0 0
\(601\) −9.28448 −0.378722 −0.189361 0.981908i \(-0.560642\pi\)
−0.189361 + 0.981908i \(0.560642\pi\)
\(602\) 0 0
\(603\) −5.44693 −0.221816
\(604\) 0.534565 0.0217511
\(605\) −23.1121 −0.939642
\(606\) 0 0
\(607\) −9.75298 −0.395861 −0.197931 0.980216i \(-0.563422\pi\)
−0.197931 + 0.980216i \(0.563422\pi\)
\(608\) 0 0
\(609\) −2.72347 −0.110360
\(610\) 0 0
\(611\) 11.5877 0.468787
\(612\) 6.48092 0.261976
\(613\) −23.0624 −0.931480 −0.465740 0.884922i \(-0.654212\pi\)
−0.465740 + 0.884922i \(0.654212\pi\)
\(614\) 0 0
\(615\) −6.65071 −0.268182
\(616\) 0 0
\(617\) 31.1963 1.25592 0.627959 0.778247i \(-0.283891\pi\)
0.627959 + 0.778247i \(0.283891\pi\)
\(618\) 0 0
\(619\) −20.8368 −0.837500 −0.418750 0.908101i \(-0.637532\pi\)
−0.418750 + 0.908101i \(0.637532\pi\)
\(620\) 6.67051 0.267894
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 11.5066 0.461001
\(624\) 21.6518 0.866765
\(625\) −22.7396 −0.909582
\(626\) 0 0
\(627\) −1.75806 −0.0702102
\(628\) −32.8304 −1.31007
\(629\) 13.6609 0.544695
\(630\) 0 0
\(631\) −4.18959 −0.166785 −0.0833926 0.996517i \(-0.526576\pi\)
−0.0833926 + 0.996517i \(0.526576\pi\)
\(632\) 0 0
\(633\) −5.33587 −0.212082
\(634\) 0 0
\(635\) −16.0041 −0.635103
\(636\) −1.66906 −0.0661827
\(637\) −2.25864 −0.0894905
\(638\) 0 0
\(639\) −1.82528 −0.0722070
\(640\) 0 0
\(641\) 24.9755 0.986475 0.493237 0.869895i \(-0.335813\pi\)
0.493237 + 0.869895i \(0.335813\pi\)
\(642\) 0 0
\(643\) 12.4455 0.490802 0.245401 0.969422i \(-0.421080\pi\)
0.245401 + 0.969422i \(0.421080\pi\)
\(644\) −5.44693 −0.214639
\(645\) −22.7531 −0.895904
\(646\) 0 0
\(647\) −38.3894 −1.50924 −0.754622 0.656160i \(-0.772180\pi\)
−0.754622 + 0.656160i \(0.772180\pi\)
\(648\) 0 0
\(649\) 6.31977 0.248073
\(650\) 0 0
\(651\) 4.24316 0.166302
\(652\) −26.0877 −1.02167
\(653\) 38.3414 1.50042 0.750208 0.661202i \(-0.229954\pi\)
0.750208 + 0.661202i \(0.229954\pi\)
\(654\) 0 0
\(655\) −0.0396110 −0.00154773
\(656\) 12.4270 0.485192
\(657\) −8.72347 −0.340335
\(658\) 0 0
\(659\) −31.3052 −1.21948 −0.609738 0.792603i \(-0.708725\pi\)
−0.609738 + 0.792603i \(0.708725\pi\)
\(660\) −1.93203 −0.0752040
\(661\) −0.243763 −0.00948129 −0.00474065 0.999989i \(-0.501509\pi\)
−0.00474065 + 0.999989i \(0.501509\pi\)
\(662\) 0 0
\(663\) −17.5404 −0.681214
\(664\) 0 0
\(665\) −22.7142 −0.880819
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 48.1062 1.86128
\(669\) −10.1272 −0.391539
\(670\) 0 0
\(671\) −3.02088 −0.116620
\(672\) 0 0
\(673\) −16.6506 −0.641832 −0.320916 0.947108i \(-0.603991\pi\)
−0.320916 + 0.947108i \(0.603991\pi\)
\(674\) 0 0
\(675\) 0.417266 0.0160606
\(676\) −32.5999 −1.25384
\(677\) 11.8162 0.454133 0.227066 0.973879i \(-0.427087\pi\)
0.227066 + 0.973879i \(0.427087\pi\)
\(678\) 0 0
\(679\) −5.08885 −0.195292
\(680\) 0 0
\(681\) 8.16849 0.313017
\(682\) 0 0
\(683\) 37.5343 1.43621 0.718106 0.695934i \(-0.245009\pi\)
0.718106 + 0.695934i \(0.245009\pi\)
\(684\) 7.79190 0.297931
\(685\) −9.88893 −0.377837
\(686\) 0 0
\(687\) −28.5084 −1.08766
\(688\) 42.5147 1.62086
\(689\) 4.51728 0.172095
\(690\) 0 0
\(691\) −13.1048 −0.498531 −0.249266 0.968435i \(-0.580189\pi\)
−0.249266 + 0.968435i \(0.580189\pi\)
\(692\) −31.5937 −1.20101
\(693\) −1.22897 −0.0466848
\(694\) 0 0
\(695\) 5.38343 0.204205
\(696\) 0 0
\(697\) −10.0673 −0.381325
\(698\) 0 0
\(699\) −1.07723 −0.0407444
\(700\) 2.27282 0.0859045
\(701\) 4.19113 0.158297 0.0791483 0.996863i \(-0.474780\pi\)
0.0791483 + 0.996863i \(0.474780\pi\)
\(702\) 0 0
\(703\) 16.4243 0.619452
\(704\) 3.61003 0.136058
\(705\) 4.58273 0.172596
\(706\) 0 0
\(707\) 8.64624 0.325175
\(708\) −28.0099 −1.05267
\(709\) 47.8449 1.79685 0.898426 0.439126i \(-0.144712\pi\)
0.898426 + 0.439126i \(0.144712\pi\)
\(710\) 0 0
\(711\) −8.73048 −0.327419
\(712\) 0 0
\(713\) −1.55800 −0.0583475
\(714\) 0 0
\(715\) 5.22897 0.195553
\(716\) 50.6408 1.89253
\(717\) 28.8197 1.07629
\(718\) 0 0
\(719\) 49.5524 1.84799 0.923996 0.382402i \(-0.124903\pi\)
0.923996 + 0.382402i \(0.124903\pi\)
\(720\) 8.56293 0.319121
\(721\) −20.8629 −0.776975
\(722\) 0 0
\(723\) 17.2388 0.641119
\(724\) −31.4568 −1.16908
\(725\) −0.417266 −0.0154969
\(726\) 0 0
\(727\) 49.3100 1.82881 0.914403 0.404805i \(-0.132661\pi\)
0.914403 + 0.404805i \(0.132661\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −34.4418 −1.27387
\(732\) 13.3888 0.494865
\(733\) −5.79190 −0.213929 −0.106964 0.994263i \(-0.534113\pi\)
−0.106964 + 0.994263i \(0.534113\pi\)
\(734\) 0 0
\(735\) −0.893255 −0.0329482
\(736\) 0 0
\(737\) 2.45795 0.0905396
\(738\) 0 0
\(739\) 31.3623 1.15368 0.576841 0.816857i \(-0.304285\pi\)
0.576841 + 0.816857i \(0.304285\pi\)
\(740\) 18.0495 0.663512
\(741\) −21.0886 −0.774708
\(742\) 0 0
\(743\) 32.2217 1.18210 0.591050 0.806635i \(-0.298714\pi\)
0.591050 + 0.806635i \(0.298714\pi\)
\(744\) 0 0
\(745\) −28.3593 −1.03900
\(746\) 0 0
\(747\) 0.403693 0.0147703
\(748\) −2.92454 −0.106932
\(749\) 11.3275 0.413899
\(750\) 0 0
\(751\) 10.0644 0.367257 0.183628 0.982996i \(-0.441216\pi\)
0.183628 + 0.982996i \(0.441216\pi\)
\(752\) −8.56293 −0.312258
\(753\) 11.7552 0.428384
\(754\) 0 0
\(755\) −0.572180 −0.0208238
\(756\) 5.44693 0.198103
\(757\) 12.9551 0.470862 0.235431 0.971891i \(-0.424350\pi\)
0.235431 + 0.971891i \(0.424350\pi\)
\(758\) 0 0
\(759\) 0.451253 0.0163795
\(760\) 0 0
\(761\) −26.0304 −0.943602 −0.471801 0.881705i \(-0.656396\pi\)
−0.471801 + 0.881705i \(0.656396\pi\)
\(762\) 0 0
\(763\) −39.3088 −1.42307
\(764\) 17.7424 0.641899
\(765\) −6.93696 −0.250806
\(766\) 0 0
\(767\) 75.8079 2.73726
\(768\) −16.0000 −0.577350
\(769\) 5.79831 0.209092 0.104546 0.994520i \(-0.466661\pi\)
0.104546 + 0.994520i \(0.466661\pi\)
\(770\) 0 0
\(771\) −9.34558 −0.336573
\(772\) 0.661870 0.0238212
\(773\) 44.6714 1.60672 0.803358 0.595496i \(-0.203044\pi\)
0.803358 + 0.595496i \(0.203044\pi\)
\(774\) 0 0
\(775\) 0.650100 0.0233523
\(776\) 0 0
\(777\) 11.4814 0.411892
\(778\) 0 0
\(779\) −12.1037 −0.433661
\(780\) −23.1753 −0.829810
\(781\) 0.823664 0.0294730
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 1.66906 0.0596095
\(785\) 35.1405 1.25422
\(786\) 0 0
\(787\) 16.7171 0.595900 0.297950 0.954582i \(-0.403697\pi\)
0.297950 + 0.954582i \(0.403697\pi\)
\(788\) 31.8828 1.13578
\(789\) 21.2956 0.758145
\(790\) 0 0
\(791\) −8.26296 −0.293797
\(792\) 0 0
\(793\) −36.2365 −1.28679
\(794\) 0 0
\(795\) 1.78651 0.0633610
\(796\) −5.03074 −0.178310
\(797\) −31.3789 −1.11150 −0.555748 0.831351i \(-0.687568\pi\)
−0.555748 + 0.831351i \(0.687568\pi\)
\(798\) 0 0
\(799\) 6.93696 0.245412
\(800\) 0 0
\(801\) 4.22498 0.149282
\(802\) 0 0
\(803\) 3.93649 0.138916
\(804\) −10.8939 −0.384197
\(805\) 5.83021 0.205488
\(806\) 0 0
\(807\) −4.00000 −0.140807
\(808\) 0 0
\(809\) −10.4174 −0.366257 −0.183128 0.983089i \(-0.558622\pi\)
−0.183128 + 0.983089i \(0.558622\pi\)
\(810\) 0 0
\(811\) −40.6886 −1.42877 −0.714386 0.699752i \(-0.753294\pi\)
−0.714386 + 0.699752i \(0.753294\pi\)
\(812\) −5.44693 −0.191150
\(813\) −24.1783 −0.847972
\(814\) 0 0
\(815\) 27.9234 0.978113
\(816\) 12.9618 0.453755
\(817\) −41.4088 −1.44871
\(818\) 0 0
\(819\) −14.7420 −0.515126
\(820\) −13.3014 −0.464506
\(821\) 9.34688 0.326208 0.163104 0.986609i \(-0.447849\pi\)
0.163104 + 0.986609i \(0.447849\pi\)
\(822\) 0 0
\(823\) −43.0610 −1.50101 −0.750506 0.660863i \(-0.770190\pi\)
−0.750506 + 0.660863i \(0.770190\pi\)
\(824\) 0 0
\(825\) −0.188293 −0.00655551
\(826\) 0 0
\(827\) 52.3168 1.81923 0.909617 0.415447i \(-0.136375\pi\)
0.909617 + 0.415447i \(0.136375\pi\)
\(828\) −2.00000 −0.0695048
\(829\) 26.5597 0.922456 0.461228 0.887282i \(-0.347409\pi\)
0.461228 + 0.887282i \(0.347409\pi\)
\(830\) 0 0
\(831\) −17.6507 −0.612296
\(832\) 43.3036 1.50128
\(833\) −1.35213 −0.0468487
\(834\) 0 0
\(835\) −51.4912 −1.78193
\(836\) −3.51612 −0.121608
\(837\) 1.55800 0.0538523
\(838\) 0 0
\(839\) −38.9318 −1.34407 −0.672037 0.740517i \(-0.734581\pi\)
−0.672037 + 0.740517i \(0.734581\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −15.8784 −0.546880
\(844\) −10.6717 −0.367336
\(845\) 34.8939 1.20039
\(846\) 0 0
\(847\) −29.4035 −1.01032
\(848\) −3.33813 −0.114632
\(849\) 9.88893 0.339387
\(850\) 0 0
\(851\) −4.21572 −0.144513
\(852\) −3.65056 −0.125066
\(853\) 33.8334 1.15843 0.579216 0.815174i \(-0.303359\pi\)
0.579216 + 0.815174i \(0.303359\pi\)
\(854\) 0 0
\(855\) −8.34019 −0.285228
\(856\) 0 0
\(857\) −31.9332 −1.09082 −0.545409 0.838170i \(-0.683626\pi\)
−0.545409 + 0.838170i \(0.683626\pi\)
\(858\) 0 0
\(859\) 3.13194 0.106860 0.0534302 0.998572i \(-0.482985\pi\)
0.0534302 + 0.998572i \(0.482985\pi\)
\(860\) −45.5063 −1.55175
\(861\) −8.46111 −0.288354
\(862\) 0 0
\(863\) −34.4962 −1.17426 −0.587132 0.809491i \(-0.699743\pi\)
−0.587132 + 0.809491i \(0.699743\pi\)
\(864\) 0 0
\(865\) 33.8168 1.14981
\(866\) 0 0
\(867\) 6.49942 0.220732
\(868\) 8.48631 0.288044
\(869\) 3.93966 0.133644
\(870\) 0 0
\(871\) 29.4839 0.999025
\(872\) 0 0
\(873\) −1.86852 −0.0632398
\(874\) 0 0
\(875\) −31.5838 −1.06773
\(876\) −17.4469 −0.589477
\(877\) −10.6747 −0.360461 −0.180230 0.983624i \(-0.557684\pi\)
−0.180230 + 0.983624i \(0.557684\pi\)
\(878\) 0 0
\(879\) −11.5899 −0.390918
\(880\) −3.86405 −0.130257
\(881\) −18.4669 −0.622166 −0.311083 0.950383i \(-0.600692\pi\)
−0.311083 + 0.950383i \(0.600692\pi\)
\(882\) 0 0
\(883\) 26.3253 0.885917 0.442959 0.896542i \(-0.353929\pi\)
0.442959 + 0.896542i \(0.353929\pi\)
\(884\) −35.0809 −1.17990
\(885\) 29.9808 1.00779
\(886\) 0 0
\(887\) 25.0617 0.841491 0.420745 0.907179i \(-0.361769\pi\)
0.420745 + 0.907179i \(0.361769\pi\)
\(888\) 0 0
\(889\) −20.3606 −0.682873
\(890\) 0 0
\(891\) −0.451253 −0.0151176
\(892\) −20.2543 −0.678165
\(893\) 8.34019 0.279094
\(894\) 0 0
\(895\) −54.2042 −1.81184
\(896\) 0 0
\(897\) 5.41294 0.180733
\(898\) 0 0
\(899\) −1.55800 −0.0519622
\(900\) 0.834532 0.0278177
\(901\) 2.70427 0.0900922
\(902\) 0 0
\(903\) −28.9468 −0.963290
\(904\) 0 0
\(905\) 33.6703 1.11924
\(906\) 0 0
\(907\) −34.3691 −1.14121 −0.570604 0.821225i \(-0.693291\pi\)
−0.570604 + 0.821225i \(0.693291\pi\)
\(908\) 16.3370 0.542162
\(909\) 3.17472 0.105299
\(910\) 0 0
\(911\) 22.5447 0.746941 0.373470 0.927642i \(-0.378168\pi\)
0.373470 + 0.927642i \(0.378168\pi\)
\(912\) 15.5838 0.516031
\(913\) −0.182168 −0.00602887
\(914\) 0 0
\(915\) −14.3309 −0.473766
\(916\) −57.0167 −1.88388
\(917\) −0.0503936 −0.00166414
\(918\) 0 0
\(919\) −17.3790 −0.573279 −0.286639 0.958039i \(-0.592538\pi\)
−0.286639 + 0.958039i \(0.592538\pi\)
\(920\) 0 0
\(921\) −30.4617 −1.00375
\(922\) 0 0
\(923\) 9.88014 0.325209
\(924\) −2.45795 −0.0808605
\(925\) 1.75908 0.0578382
\(926\) 0 0
\(927\) −7.66042 −0.251601
\(928\) 0 0
\(929\) 44.4583 1.45863 0.729315 0.684178i \(-0.239839\pi\)
0.729315 + 0.684178i \(0.239839\pi\)
\(930\) 0 0
\(931\) −1.62565 −0.0532785
\(932\) −2.15445 −0.0705714
\(933\) −14.4314 −0.472464
\(934\) 0 0
\(935\) 3.13033 0.102373
\(936\) 0 0
\(937\) 44.0111 1.43778 0.718890 0.695124i \(-0.244651\pi\)
0.718890 + 0.695124i \(0.244651\pi\)
\(938\) 0 0
\(939\) −24.1844 −0.789229
\(940\) 9.16547 0.298945
\(941\) 43.8742 1.43026 0.715129 0.698992i \(-0.246368\pi\)
0.715129 + 0.698992i \(0.246368\pi\)
\(942\) 0 0
\(943\) 3.10674 0.101170
\(944\) −56.0197 −1.82329
\(945\) −5.83021 −0.189657
\(946\) 0 0
\(947\) −53.9967 −1.75466 −0.877329 0.479889i \(-0.840677\pi\)
−0.877329 + 0.479889i \(0.840677\pi\)
\(948\) −17.4610 −0.567106
\(949\) 47.2196 1.53281
\(950\) 0 0
\(951\) 20.1452 0.653253
\(952\) 0 0
\(953\) 1.62727 0.0527126 0.0263563 0.999653i \(-0.491610\pi\)
0.0263563 + 0.999653i \(0.491610\pi\)
\(954\) 0 0
\(955\) −18.9909 −0.614531
\(956\) 57.6393 1.86419
\(957\) 0.451253 0.0145870
\(958\) 0 0
\(959\) −12.5808 −0.406256
\(960\) 17.1259 0.552735
\(961\) −28.5726 −0.921698
\(962\) 0 0
\(963\) 4.15924 0.134029
\(964\) 34.4777 1.11045
\(965\) −0.708443 −0.0228056
\(966\) 0 0
\(967\) 22.3204 0.717775 0.358888 0.933381i \(-0.383156\pi\)
0.358888 + 0.933381i \(0.383156\pi\)
\(968\) 0 0
\(969\) −12.6247 −0.405563
\(970\) 0 0
\(971\) −36.5661 −1.17346 −0.586731 0.809782i \(-0.699585\pi\)
−0.586731 + 0.809782i \(0.699585\pi\)
\(972\) 2.00000 0.0641500
\(973\) 6.84886 0.219564
\(974\) 0 0
\(975\) −2.25864 −0.0723343
\(976\) 26.7776 0.857131
\(977\) −14.7024 −0.470373 −0.235186 0.971950i \(-0.575570\pi\)
−0.235186 + 0.971950i \(0.575570\pi\)
\(978\) 0 0
\(979\) −1.90653 −0.0609331
\(980\) −1.78651 −0.0570680
\(981\) −14.4334 −0.460822
\(982\) 0 0
\(983\) −9.76221 −0.311366 −0.155683 0.987807i \(-0.549758\pi\)
−0.155683 + 0.987807i \(0.549758\pi\)
\(984\) 0 0
\(985\) −34.1263 −1.08735
\(986\) 0 0
\(987\) 5.83021 0.185578
\(988\) −42.1771 −1.34183
\(989\) 10.6287 0.337972
\(990\) 0 0
\(991\) 12.6899 0.403109 0.201555 0.979477i \(-0.435401\pi\)
0.201555 + 0.979477i \(0.435401\pi\)
\(992\) 0 0
\(993\) −22.9618 −0.728672
\(994\) 0 0
\(995\) 5.38473 0.170707
\(996\) 0.807385 0.0255830
\(997\) 18.8584 0.597252 0.298626 0.954370i \(-0.403472\pi\)
0.298626 + 0.954370i \(0.403472\pi\)
\(998\) 0 0
\(999\) 4.21572 0.133380
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 2001.2.a.g.1.4 4
3.2 odd 2 6003.2.a.g.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.g.1.4 4 1.1 even 1 trivial
6003.2.a.g.1.1 4 3.2 odd 2