Properties

Label 9072.2.a.bp.1.1
Level $9072$
Weight $2$
Character 9072.1
Self dual yes
Analytic conductor $72.440$
Analytic rank $0$
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] = [9072,2,Mod(1,9072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9072, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9072.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.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: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1134)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.267949 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+0.267949 q^{5} +1.00000 q^{7} -6.19615 q^{11} -6.46410 q^{13} +7.00000 q^{17} -0.732051 q^{19} +4.19615 q^{23} -4.92820 q^{25} +1.53590 q^{29} -8.19615 q^{31} +0.267949 q^{35} +10.6603 q^{37} +2.53590 q^{41} +1.46410 q^{43} -4.73205 q^{47} +1.00000 q^{49} -9.46410 q^{53} -1.66025 q^{55} -4.19615 q^{59} +3.92820 q^{61} -1.73205 q^{65} +6.73205 q^{67} -6.53590 q^{71} +8.26795 q^{73} -6.19615 q^{77} +9.12436 q^{79} -16.5885 q^{83} +1.87564 q^{85} +9.92820 q^{89} -6.46410 q^{91} -0.196152 q^{95} +10.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 2 q^{7} - 2 q^{11} - 6 q^{13} + 14 q^{17} + 2 q^{19} - 2 q^{23} + 4 q^{25} + 10 q^{29} - 6 q^{31} + 4 q^{35} + 4 q^{37} + 12 q^{41} - 4 q^{43} - 6 q^{47} + 2 q^{49} - 12 q^{53} + 14 q^{55} + 2 q^{59} - 6 q^{61} + 10 q^{67} - 20 q^{71} + 20 q^{73} - 2 q^{77} - 6 q^{79} - 2 q^{83} + 28 q^{85} + 6 q^{89} - 6 q^{91} + 10 q^{95} + 8 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.267949 0.119831 0.0599153 0.998203i \(-0.480917\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.19615 −1.86821 −0.934105 0.356998i \(-0.883800\pi\)
−0.934105 + 0.356998i \(0.883800\pi\)
\(12\) 0 0
\(13\) −6.46410 −1.79282 −0.896410 0.443227i \(-0.853834\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) −0.732051 −0.167944 −0.0839720 0.996468i \(-0.526761\pi\)
−0.0839720 + 0.996468i \(0.526761\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.19615 0.874958 0.437479 0.899229i \(-0.355871\pi\)
0.437479 + 0.899229i \(0.355871\pi\)
\(24\) 0 0
\(25\) −4.92820 −0.985641
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.53590 0.285209 0.142605 0.989780i \(-0.454452\pi\)
0.142605 + 0.989780i \(0.454452\pi\)
\(30\) 0 0
\(31\) −8.19615 −1.47207 −0.736036 0.676942i \(-0.763305\pi\)
−0.736036 + 0.676942i \(0.763305\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.267949 0.0452917
\(36\) 0 0
\(37\) 10.6603 1.75253 0.876267 0.481825i \(-0.160026\pi\)
0.876267 + 0.481825i \(0.160026\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.53590 0.396041 0.198020 0.980198i \(-0.436549\pi\)
0.198020 + 0.980198i \(0.436549\pi\)
\(42\) 0 0
\(43\) 1.46410 0.223273 0.111637 0.993749i \(-0.464391\pi\)
0.111637 + 0.993749i \(0.464391\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.73205 −0.690241 −0.345120 0.938558i \(-0.612162\pi\)
−0.345120 + 0.938558i \(0.612162\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.46410 −1.29999 −0.649997 0.759937i \(-0.725230\pi\)
−0.649997 + 0.759937i \(0.725230\pi\)
\(54\) 0 0
\(55\) −1.66025 −0.223869
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.19615 −0.546293 −0.273146 0.961973i \(-0.588064\pi\)
−0.273146 + 0.961973i \(0.588064\pi\)
\(60\) 0 0
\(61\) 3.92820 0.502955 0.251477 0.967863i \(-0.419084\pi\)
0.251477 + 0.967863i \(0.419084\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.73205 −0.214834
\(66\) 0 0
\(67\) 6.73205 0.822451 0.411225 0.911534i \(-0.365101\pi\)
0.411225 + 0.911534i \(0.365101\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.53590 −0.775668 −0.387834 0.921729i \(-0.626777\pi\)
−0.387834 + 0.921729i \(0.626777\pi\)
\(72\) 0 0
\(73\) 8.26795 0.967690 0.483845 0.875154i \(-0.339240\pi\)
0.483845 + 0.875154i \(0.339240\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.19615 −0.706117
\(78\) 0 0
\(79\) 9.12436 1.02657 0.513285 0.858218i \(-0.328428\pi\)
0.513285 + 0.858218i \(0.328428\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −16.5885 −1.82082 −0.910410 0.413707i \(-0.864234\pi\)
−0.910410 + 0.413707i \(0.864234\pi\)
\(84\) 0 0
\(85\) 1.87564 0.203442
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.92820 1.05239 0.526194 0.850365i \(-0.323619\pi\)
0.526194 + 0.850365i \(0.323619\pi\)
\(90\) 0 0
\(91\) −6.46410 −0.677622
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.196152 −0.0201248
\(96\) 0 0
\(97\) 10.9282 1.10959 0.554795 0.831987i \(-0.312797\pi\)
0.554795 + 0.831987i \(0.312797\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.92820 0.888389 0.444195 0.895930i \(-0.353490\pi\)
0.444195 + 0.895930i \(0.353490\pi\)
\(102\) 0 0
\(103\) 8.39230 0.826918 0.413459 0.910523i \(-0.364320\pi\)
0.413459 + 0.910523i \(0.364320\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 3.19615 0.306136 0.153068 0.988216i \(-0.451085\pi\)
0.153068 + 0.988216i \(0.451085\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.73205 −0.539226 −0.269613 0.962969i \(-0.586896\pi\)
−0.269613 + 0.962969i \(0.586896\pi\)
\(114\) 0 0
\(115\) 1.12436 0.104847
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.00000 0.641689
\(120\) 0 0
\(121\) 27.3923 2.49021
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.66025 −0.237940
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.5359 −0.920526 −0.460263 0.887783i \(-0.652245\pi\)
−0.460263 + 0.887783i \(0.652245\pi\)
\(132\) 0 0
\(133\) −0.732051 −0.0634769
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.26795 −0.706379 −0.353189 0.935552i \(-0.614903\pi\)
−0.353189 + 0.935552i \(0.614903\pi\)
\(138\) 0 0
\(139\) 3.26795 0.277184 0.138592 0.990350i \(-0.455742\pi\)
0.138592 + 0.990350i \(0.455742\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 40.0526 3.34936
\(144\) 0 0
\(145\) 0.411543 0.0341768
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) 5.80385 0.472310 0.236155 0.971715i \(-0.424113\pi\)
0.236155 + 0.971715i \(0.424113\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.19615 −0.176399
\(156\) 0 0
\(157\) −1.00000 −0.0798087 −0.0399043 0.999204i \(-0.512705\pi\)
−0.0399043 + 0.999204i \(0.512705\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.19615 0.330703
\(162\) 0 0
\(163\) 13.4641 1.05459 0.527295 0.849682i \(-0.323206\pi\)
0.527295 + 0.849682i \(0.323206\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.80385 0.139586 0.0697930 0.997561i \(-0.477766\pi\)
0.0697930 + 0.997561i \(0.477766\pi\)
\(168\) 0 0
\(169\) 28.7846 2.21420
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.26795 0.476543 0.238272 0.971199i \(-0.423419\pi\)
0.238272 + 0.971199i \(0.423419\pi\)
\(174\) 0 0
\(175\) −4.92820 −0.372537
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.19615 0.164148 0.0820741 0.996626i \(-0.473846\pi\)
0.0820741 + 0.996626i \(0.473846\pi\)
\(180\) 0 0
\(181\) −16.3923 −1.21843 −0.609215 0.793005i \(-0.708515\pi\)
−0.609215 + 0.793005i \(0.708515\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.85641 0.210007
\(186\) 0 0
\(187\) −43.3731 −3.17175
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.66025 0.409562 0.204781 0.978808i \(-0.434352\pi\)
0.204781 + 0.978808i \(0.434352\pi\)
\(192\) 0 0
\(193\) 18.8564 1.35731 0.678657 0.734455i \(-0.262562\pi\)
0.678657 + 0.734455i \(0.262562\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.7846 1.12461 0.562303 0.826931i \(-0.309915\pi\)
0.562303 + 0.826931i \(0.309915\pi\)
\(198\) 0 0
\(199\) 19.1244 1.35569 0.677845 0.735205i \(-0.262914\pi\)
0.677845 + 0.735205i \(0.262914\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.53590 0.107799
\(204\) 0 0
\(205\) 0.679492 0.0474578
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.53590 0.313755
\(210\) 0 0
\(211\) 17.2679 1.18877 0.594387 0.804179i \(-0.297395\pi\)
0.594387 + 0.804179i \(0.297395\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.392305 0.0267550
\(216\) 0 0
\(217\) −8.19615 −0.556391
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −45.2487 −3.04376
\(222\) 0 0
\(223\) 25.4641 1.70520 0.852601 0.522562i \(-0.175024\pi\)
0.852601 + 0.522562i \(0.175024\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.9282 1.25631 0.628154 0.778089i \(-0.283811\pi\)
0.628154 + 0.778089i \(0.283811\pi\)
\(228\) 0 0
\(229\) −2.46410 −0.162832 −0.0814162 0.996680i \(-0.525944\pi\)
−0.0814162 + 0.996680i \(0.525944\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.80385 −0.183686 −0.0918431 0.995773i \(-0.529276\pi\)
−0.0918431 + 0.995773i \(0.529276\pi\)
\(234\) 0 0
\(235\) −1.26795 −0.0827119
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.0526 0.650246 0.325123 0.945672i \(-0.394594\pi\)
0.325123 + 0.945672i \(0.394594\pi\)
\(240\) 0 0
\(241\) −14.2679 −0.919079 −0.459540 0.888157i \(-0.651986\pi\)
−0.459540 + 0.888157i \(0.651986\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.267949 0.0171186
\(246\) 0 0
\(247\) 4.73205 0.301093
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.0526 1.39195 0.695973 0.718068i \(-0.254973\pi\)
0.695973 + 0.718068i \(0.254973\pi\)
\(252\) 0 0
\(253\) −26.0000 −1.63461
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.46410 0.403220 0.201610 0.979466i \(-0.435383\pi\)
0.201610 + 0.979466i \(0.435383\pi\)
\(258\) 0 0
\(259\) 10.6603 0.662396
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.33975 0.390925 0.195463 0.980711i \(-0.437379\pi\)
0.195463 + 0.980711i \(0.437379\pi\)
\(264\) 0 0
\(265\) −2.53590 −0.155779
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.58846 0.340734 0.170367 0.985381i \(-0.445505\pi\)
0.170367 + 0.985381i \(0.445505\pi\)
\(270\) 0 0
\(271\) 19.5167 1.18555 0.592776 0.805367i \(-0.298032\pi\)
0.592776 + 0.805367i \(0.298032\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 30.5359 1.84138
\(276\) 0 0
\(277\) −18.7846 −1.12866 −0.564329 0.825550i \(-0.690865\pi\)
−0.564329 + 0.825550i \(0.690865\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −13.1962 −0.787216 −0.393608 0.919278i \(-0.628773\pi\)
−0.393608 + 0.919278i \(0.628773\pi\)
\(282\) 0 0
\(283\) −15.3205 −0.910710 −0.455355 0.890310i \(-0.650488\pi\)
−0.455355 + 0.890310i \(0.650488\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.53590 0.149689
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.33975 0.195110 0.0975550 0.995230i \(-0.468898\pi\)
0.0975550 + 0.995230i \(0.468898\pi\)
\(294\) 0 0
\(295\) −1.12436 −0.0654625
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −27.1244 −1.56864
\(300\) 0 0
\(301\) 1.46410 0.0843894
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.05256 0.0602693
\(306\) 0 0
\(307\) 21.8564 1.24741 0.623706 0.781659i \(-0.285626\pi\)
0.623706 + 0.781659i \(0.285626\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.1962 −0.578171 −0.289085 0.957303i \(-0.593351\pi\)
−0.289085 + 0.957303i \(0.593351\pi\)
\(312\) 0 0
\(313\) 25.5885 1.44635 0.723173 0.690667i \(-0.242683\pi\)
0.723173 + 0.690667i \(0.242683\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −31.3923 −1.76317 −0.881584 0.472028i \(-0.843522\pi\)
−0.881584 + 0.472028i \(0.843522\pi\)
\(318\) 0 0
\(319\) −9.51666 −0.532831
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.12436 −0.285127
\(324\) 0 0
\(325\) 31.8564 1.76708
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.73205 −0.260886
\(330\) 0 0
\(331\) −12.3923 −0.681143 −0.340571 0.940219i \(-0.610620\pi\)
−0.340571 + 0.940219i \(0.610620\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.80385 0.0985547
\(336\) 0 0
\(337\) −16.3923 −0.892946 −0.446473 0.894797i \(-0.647320\pi\)
−0.446473 + 0.894797i \(0.647320\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 50.7846 2.75014
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.4641 −1.15225 −0.576127 0.817360i \(-0.695437\pi\)
−0.576127 + 0.817360i \(0.695437\pi\)
\(348\) 0 0
\(349\) 1.46410 0.0783716 0.0391858 0.999232i \(-0.487524\pi\)
0.0391858 + 0.999232i \(0.487524\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −1.75129 −0.0929488
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.9282 0.576769 0.288384 0.957515i \(-0.406882\pi\)
0.288384 + 0.957515i \(0.406882\pi\)
\(360\) 0 0
\(361\) −18.4641 −0.971795
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.21539 0.115959
\(366\) 0 0
\(367\) −11.1244 −0.580687 −0.290343 0.956923i \(-0.593770\pi\)
−0.290343 + 0.956923i \(0.593770\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −9.46410 −0.491352
\(372\) 0 0
\(373\) −6.14359 −0.318103 −0.159052 0.987270i \(-0.550844\pi\)
−0.159052 + 0.987270i \(0.550844\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.92820 −0.511328
\(378\) 0 0
\(379\) 27.5167 1.41344 0.706718 0.707495i \(-0.250175\pi\)
0.706718 + 0.707495i \(0.250175\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −19.7128 −1.00728 −0.503639 0.863914i \(-0.668006\pi\)
−0.503639 + 0.863914i \(0.668006\pi\)
\(384\) 0 0
\(385\) −1.66025 −0.0846144
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13.4641 −0.682657 −0.341329 0.939944i \(-0.610877\pi\)
−0.341329 + 0.939944i \(0.610877\pi\)
\(390\) 0 0
\(391\) 29.3731 1.48546
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.44486 0.123014
\(396\) 0 0
\(397\) −21.0000 −1.05396 −0.526980 0.849878i \(-0.676676\pi\)
−0.526980 + 0.849878i \(0.676676\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.5167 0.525177 0.262588 0.964908i \(-0.415424\pi\)
0.262588 + 0.964908i \(0.415424\pi\)
\(402\) 0 0
\(403\) 52.9808 2.63916
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −66.0526 −3.27410
\(408\) 0 0
\(409\) 17.3397 0.857395 0.428698 0.903448i \(-0.358973\pi\)
0.428698 + 0.903448i \(0.358973\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.19615 −0.206479
\(414\) 0 0
\(415\) −4.44486 −0.218190
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.46410 0.462352 0.231176 0.972912i \(-0.425743\pi\)
0.231176 + 0.972912i \(0.425743\pi\)
\(420\) 0 0
\(421\) 0.124356 0.00606072 0.00303036 0.999995i \(-0.499035\pi\)
0.00303036 + 0.999995i \(0.499035\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −34.4974 −1.67337
\(426\) 0 0
\(427\) 3.92820 0.190099
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.5359 0.700170 0.350085 0.936718i \(-0.386153\pi\)
0.350085 + 0.936718i \(0.386153\pi\)
\(432\) 0 0
\(433\) 15.7321 0.756034 0.378017 0.925799i \(-0.376606\pi\)
0.378017 + 0.925799i \(0.376606\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.07180 −0.146944
\(438\) 0 0
\(439\) −23.3205 −1.11303 −0.556514 0.830839i \(-0.687861\pi\)
−0.556514 + 0.830839i \(0.687861\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.2679 0.725402 0.362701 0.931906i \(-0.381855\pi\)
0.362701 + 0.931906i \(0.381855\pi\)
\(444\) 0 0
\(445\) 2.66025 0.126108
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.8564 0.748310 0.374155 0.927366i \(-0.377933\pi\)
0.374155 + 0.927366i \(0.377933\pi\)
\(450\) 0 0
\(451\) −15.7128 −0.739887
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.73205 −0.0811998
\(456\) 0 0
\(457\) −6.85641 −0.320729 −0.160365 0.987058i \(-0.551267\pi\)
−0.160365 + 0.987058i \(0.551267\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.78461 −0.315991 −0.157995 0.987440i \(-0.550503\pi\)
−0.157995 + 0.987440i \(0.550503\pi\)
\(462\) 0 0
\(463\) 1.41154 0.0656000 0.0328000 0.999462i \(-0.489558\pi\)
0.0328000 + 0.999462i \(0.489558\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.5885 0.767622 0.383811 0.923412i \(-0.374611\pi\)
0.383811 + 0.923412i \(0.374611\pi\)
\(468\) 0 0
\(469\) 6.73205 0.310857
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −9.07180 −0.417122
\(474\) 0 0
\(475\) 3.60770 0.165532
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.5167 0.983121 0.491561 0.870843i \(-0.336427\pi\)
0.491561 + 0.870843i \(0.336427\pi\)
\(480\) 0 0
\(481\) −68.9090 −3.14198
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.92820 0.132963
\(486\) 0 0
\(487\) −2.58846 −0.117294 −0.0586471 0.998279i \(-0.518679\pi\)
−0.0586471 + 0.998279i \(0.518679\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −26.5359 −1.19755 −0.598774 0.800918i \(-0.704345\pi\)
−0.598774 + 0.800918i \(0.704345\pi\)
\(492\) 0 0
\(493\) 10.7513 0.484214
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.53590 −0.293175
\(498\) 0 0
\(499\) −19.8038 −0.886542 −0.443271 0.896388i \(-0.646182\pi\)
−0.443271 + 0.896388i \(0.646182\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −40.0526 −1.78586 −0.892928 0.450200i \(-0.851353\pi\)
−0.892928 + 0.450200i \(0.851353\pi\)
\(504\) 0 0
\(505\) 2.39230 0.106456
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 31.8564 1.41201 0.706005 0.708207i \(-0.250496\pi\)
0.706005 + 0.708207i \(0.250496\pi\)
\(510\) 0 0
\(511\) 8.26795 0.365753
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.24871 0.0990901
\(516\) 0 0
\(517\) 29.3205 1.28951
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −33.1769 −1.45073 −0.725363 0.688367i \(-0.758328\pi\)
−0.725363 + 0.688367i \(0.758328\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −57.3731 −2.49921
\(528\) 0 0
\(529\) −5.39230 −0.234448
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.3923 −0.710030
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.19615 −0.266887
\(540\) 0 0
\(541\) 3.33975 0.143587 0.0717934 0.997420i \(-0.477128\pi\)
0.0717934 + 0.997420i \(0.477128\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.856406 0.0366844
\(546\) 0 0
\(547\) 22.7321 0.971952 0.485976 0.873972i \(-0.338464\pi\)
0.485976 + 0.873972i \(0.338464\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.12436 −0.0478992
\(552\) 0 0
\(553\) 9.12436 0.388007
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.9282 1.01387 0.506935 0.861984i \(-0.330778\pi\)
0.506935 + 0.861984i \(0.330778\pi\)
\(558\) 0 0
\(559\) −9.46410 −0.400289
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.7128 −0.830796 −0.415398 0.909640i \(-0.636358\pi\)
−0.415398 + 0.909640i \(0.636358\pi\)
\(564\) 0 0
\(565\) −1.53590 −0.0646157
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.1962 0.972433 0.486217 0.873838i \(-0.338377\pi\)
0.486217 + 0.873838i \(0.338377\pi\)
\(570\) 0 0
\(571\) 22.7321 0.951307 0.475653 0.879633i \(-0.342212\pi\)
0.475653 + 0.879633i \(0.342212\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −20.6795 −0.862394
\(576\) 0 0
\(577\) 24.6603 1.02662 0.513310 0.858203i \(-0.328419\pi\)
0.513310 + 0.858203i \(0.328419\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16.5885 −0.688205
\(582\) 0 0
\(583\) 58.6410 2.42866
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.7321 −0.608057 −0.304028 0.952663i \(-0.598332\pi\)
−0.304028 + 0.952663i \(0.598332\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22.1769 −0.910697 −0.455348 0.890313i \(-0.650485\pi\)
−0.455348 + 0.890313i \(0.650485\pi\)
\(594\) 0 0
\(595\) 1.87564 0.0768939
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.1244 0.617964 0.308982 0.951068i \(-0.400012\pi\)
0.308982 + 0.951068i \(0.400012\pi\)
\(600\) 0 0
\(601\) 19.1962 0.783027 0.391514 0.920172i \(-0.371952\pi\)
0.391514 + 0.920172i \(0.371952\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.33975 0.298403
\(606\) 0 0
\(607\) 24.5885 0.998015 0.499007 0.866598i \(-0.333698\pi\)
0.499007 + 0.866598i \(0.333698\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 30.5885 1.23748
\(612\) 0 0
\(613\) 26.7846 1.08182 0.540910 0.841080i \(-0.318080\pi\)
0.540910 + 0.841080i \(0.318080\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11.9808 0.482327 0.241164 0.970484i \(-0.422471\pi\)
0.241164 + 0.970484i \(0.422471\pi\)
\(618\) 0 0
\(619\) 23.7128 0.953098 0.476549 0.879148i \(-0.341887\pi\)
0.476549 + 0.879148i \(0.341887\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.92820 0.397765
\(624\) 0 0
\(625\) 23.9282 0.957128
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 74.6218 2.97537
\(630\) 0 0
\(631\) 3.66025 0.145712 0.0728562 0.997342i \(-0.476789\pi\)
0.0728562 + 0.997342i \(0.476789\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.21539 −0.127599
\(636\) 0 0
\(637\) −6.46410 −0.256117
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 39.4449 1.55798 0.778989 0.627037i \(-0.215733\pi\)
0.778989 + 0.627037i \(0.215733\pi\)
\(642\) 0 0
\(643\) 9.41154 0.371155 0.185578 0.982630i \(-0.440584\pi\)
0.185578 + 0.982630i \(0.440584\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.39230 0.172679 0.0863397 0.996266i \(-0.472483\pi\)
0.0863397 + 0.996266i \(0.472483\pi\)
\(648\) 0 0
\(649\) 26.0000 1.02059
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.2487 −1.18372 −0.591862 0.806039i \(-0.701607\pi\)
−0.591862 + 0.806039i \(0.701607\pi\)
\(654\) 0 0
\(655\) −2.82309 −0.110307
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.3923 1.41764 0.708821 0.705388i \(-0.249227\pi\)
0.708821 + 0.705388i \(0.249227\pi\)
\(660\) 0 0
\(661\) −12.8564 −0.500056 −0.250028 0.968239i \(-0.580440\pi\)
−0.250028 + 0.968239i \(0.580440\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.196152 −0.00760646
\(666\) 0 0
\(667\) 6.44486 0.249546
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −24.3397 −0.939625
\(672\) 0 0
\(673\) −18.3205 −0.706204 −0.353102 0.935585i \(-0.614873\pi\)
−0.353102 + 0.935585i \(0.614873\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.0000 1.38359 0.691796 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 0 0
\(679\) 10.9282 0.419386
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.85641 0.0710334 0.0355167 0.999369i \(-0.488692\pi\)
0.0355167 + 0.999369i \(0.488692\pi\)
\(684\) 0 0
\(685\) −2.21539 −0.0846457
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 61.1769 2.33065
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.875644 0.0332151
\(696\) 0 0
\(697\) 17.7513 0.672378
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 27.3923 1.03459 0.517297 0.855806i \(-0.326938\pi\)
0.517297 + 0.855806i \(0.326938\pi\)
\(702\) 0 0
\(703\) −7.80385 −0.294328
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.92820 0.335780
\(708\) 0 0
\(709\) −3.87564 −0.145553 −0.0727764 0.997348i \(-0.523186\pi\)
−0.0727764 + 0.997348i \(0.523186\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −34.3923 −1.28800
\(714\) 0 0
\(715\) 10.7321 0.401356
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.46410 0.352951 0.176476 0.984305i \(-0.443530\pi\)
0.176476 + 0.984305i \(0.443530\pi\)
\(720\) 0 0
\(721\) 8.39230 0.312546
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.56922 −0.281114
\(726\) 0 0
\(727\) −51.3205 −1.90337 −0.951686 0.307072i \(-0.900651\pi\)
−0.951686 + 0.307072i \(0.900651\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.2487 0.379062
\(732\) 0 0
\(733\) 47.3205 1.74782 0.873911 0.486085i \(-0.161576\pi\)
0.873911 + 0.486085i \(0.161576\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −41.7128 −1.53651
\(738\) 0 0
\(739\) 13.2679 0.488069 0.244035 0.969767i \(-0.421529\pi\)
0.244035 + 0.969767i \(0.421529\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 40.3923 1.48185 0.740925 0.671588i \(-0.234387\pi\)
0.740925 + 0.671588i \(0.234387\pi\)
\(744\) 0 0
\(745\) 2.41154 0.0883521
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 22.1436 0.808031 0.404016 0.914752i \(-0.367614\pi\)
0.404016 + 0.914752i \(0.367614\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.55514 0.0565972
\(756\) 0 0
\(757\) 20.7846 0.755429 0.377715 0.925922i \(-0.376710\pi\)
0.377715 + 0.925922i \(0.376710\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 37.0000 1.34125 0.670624 0.741797i \(-0.266026\pi\)
0.670624 + 0.741797i \(0.266026\pi\)
\(762\) 0 0
\(763\) 3.19615 0.115708
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27.1244 0.979404
\(768\) 0 0
\(769\) −4.41154 −0.159084 −0.0795421 0.996832i \(-0.525346\pi\)
−0.0795421 + 0.996832i \(0.525346\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −4.12436 −0.148343 −0.0741714 0.997246i \(-0.523631\pi\)
−0.0741714 + 0.997246i \(0.523631\pi\)
\(774\) 0 0
\(775\) 40.3923 1.45093
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.85641 −0.0665127
\(780\) 0 0
\(781\) 40.4974 1.44911
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.267949 −0.00956352
\(786\) 0 0
\(787\) 28.3923 1.01208 0.506038 0.862511i \(-0.331109\pi\)
0.506038 + 0.862511i \(0.331109\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.73205 −0.203808
\(792\) 0 0
\(793\) −25.3923 −0.901707
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.4449 −1.04299 −0.521495 0.853254i \(-0.674626\pi\)
−0.521495 + 0.853254i \(0.674626\pi\)
\(798\) 0 0
\(799\) −33.1244 −1.17186
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −51.2295 −1.80785
\(804\) 0 0
\(805\) 1.12436 0.0396283
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 32.1244 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(810\) 0 0
\(811\) −18.1962 −0.638953 −0.319477 0.947594i \(-0.603507\pi\)
−0.319477 + 0.947594i \(0.603507\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.60770 0.126372
\(816\) 0 0
\(817\) −1.07180 −0.0374974
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.9282 0.904901 0.452450 0.891790i \(-0.350550\pi\)
0.452450 + 0.891790i \(0.350550\pi\)
\(822\) 0 0
\(823\) −0.784610 −0.0273498 −0.0136749 0.999906i \(-0.504353\pi\)
−0.0136749 + 0.999906i \(0.504353\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23.3205 0.810934 0.405467 0.914110i \(-0.367109\pi\)
0.405467 + 0.914110i \(0.367109\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.00000 0.242536
\(834\) 0 0
\(835\) 0.483340 0.0167267
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.46410 −0.0505464 −0.0252732 0.999681i \(-0.508046\pi\)
−0.0252732 + 0.999681i \(0.508046\pi\)
\(840\) 0 0
\(841\) −26.6410 −0.918656
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.71281 0.265329
\(846\) 0 0
\(847\) 27.3923 0.941211
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 44.7321 1.53339
\(852\) 0 0
\(853\) −49.7128 −1.70213 −0.851067 0.525057i \(-0.824044\pi\)
−0.851067 + 0.525057i \(0.824044\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −16.8564 −0.575804 −0.287902 0.957660i \(-0.592958\pi\)
−0.287902 + 0.957660i \(0.592958\pi\)
\(858\) 0 0
\(859\) −24.3923 −0.832255 −0.416127 0.909306i \(-0.636613\pi\)
−0.416127 + 0.909306i \(0.636613\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7.12436 0.242516 0.121258 0.992621i \(-0.461307\pi\)
0.121258 + 0.992621i \(0.461307\pi\)
\(864\) 0 0
\(865\) 1.67949 0.0571044
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −56.5359 −1.91785
\(870\) 0 0
\(871\) −43.5167 −1.47451
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.66025 −0.0899330
\(876\) 0 0
\(877\) −36.5167 −1.23308 −0.616540 0.787324i \(-0.711466\pi\)
−0.616540 + 0.787324i \(0.711466\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.2487 1.01910 0.509552 0.860440i \(-0.329811\pi\)
0.509552 + 0.860440i \(0.329811\pi\)
\(882\) 0 0
\(883\) 9.66025 0.325093 0.162547 0.986701i \(-0.448029\pi\)
0.162547 + 0.986701i \(0.448029\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 56.4449 1.89523 0.947617 0.319410i \(-0.103485\pi\)
0.947617 + 0.319410i \(0.103485\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.46410 0.115922
\(894\) 0 0
\(895\) 0.588457 0.0196700
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.5885 −0.419849
\(900\) 0 0
\(901\) −66.2487 −2.20706
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.39230 −0.146005
\(906\) 0 0
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −42.2487 −1.39976 −0.699881 0.714259i \(-0.746764\pi\)
−0.699881 + 0.714259i \(0.746764\pi\)
\(912\) 0 0
\(913\) 102.785 3.40167
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.5359 −0.347926
\(918\) 0 0
\(919\) −26.9808 −0.890013 −0.445007 0.895527i \(-0.646799\pi\)
−0.445007 + 0.895527i \(0.646799\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 42.2487 1.39063
\(924\) 0 0
\(925\) −52.5359 −1.72737
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −51.4974 −1.68958 −0.844788 0.535101i \(-0.820273\pi\)
−0.844788 + 0.535101i \(0.820273\pi\)
\(930\) 0 0
\(931\) −0.732051 −0.0239920
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −11.6218 −0.380073
\(936\) 0 0
\(937\) −25.8372 −0.844064 −0.422032 0.906581i \(-0.638683\pi\)
−0.422032 + 0.906581i \(0.638683\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −34.1244 −1.11242 −0.556211 0.831041i \(-0.687745\pi\)
−0.556211 + 0.831041i \(0.687745\pi\)
\(942\) 0 0
\(943\) 10.6410 0.346519
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.2487 1.50288 0.751441 0.659801i \(-0.229359\pi\)
0.751441 + 0.659801i \(0.229359\pi\)
\(948\) 0 0
\(949\) −53.4449 −1.73489
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −41.5885 −1.34718 −0.673591 0.739104i \(-0.735249\pi\)
−0.673591 + 0.739104i \(0.735249\pi\)
\(954\) 0 0
\(955\) 1.51666 0.0490780
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.26795 −0.266986
\(960\) 0 0
\(961\) 36.1769 1.16700
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.05256 0.162648
\(966\) 0 0
\(967\) 3.66025 0.117706 0.0588529 0.998267i \(-0.481256\pi\)
0.0588529 + 0.998267i \(0.481256\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −8.87564 −0.284833 −0.142416 0.989807i \(-0.545487\pi\)
−0.142416 + 0.989807i \(0.545487\pi\)
\(972\) 0 0
\(973\) 3.26795 0.104766
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 57.7128 1.84640 0.923198 0.384324i \(-0.125565\pi\)
0.923198 + 0.384324i \(0.125565\pi\)
\(978\) 0 0
\(979\) −61.5167 −1.96608
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −58.6410 −1.87036 −0.935179 0.354176i \(-0.884762\pi\)
−0.935179 + 0.354176i \(0.884762\pi\)
\(984\) 0 0
\(985\) 4.22947 0.134762
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.14359 0.195355
\(990\) 0 0
\(991\) 27.6603 0.878657 0.439328 0.898327i \(-0.355216\pi\)
0.439328 + 0.898327i \(0.355216\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.12436 0.162453
\(996\) 0 0
\(997\) 41.2487 1.30636 0.653180 0.757203i \(-0.273435\pi\)
0.653180 + 0.757203i \(0.273435\pi\)
\(998\) 0 0
\(999\) 0 0
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 9072.2.a.bp.1.1 2
3.2 odd 2 9072.2.a.y.1.2 2
4.3 odd 2 1134.2.a.l.1.1 2
12.11 even 2 1134.2.a.m.1.2 yes 2
28.27 even 2 7938.2.a.bg.1.2 2
36.7 odd 6 1134.2.f.s.757.2 4
36.11 even 6 1134.2.f.r.757.1 4
36.23 even 6 1134.2.f.r.379.1 4
36.31 odd 6 1134.2.f.s.379.2 4
84.83 odd 2 7938.2.a.bt.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.1 2 4.3 odd 2
1134.2.a.m.1.2 yes 2 12.11 even 2
1134.2.f.r.379.1 4 36.23 even 6
1134.2.f.r.757.1 4 36.11 even 6
1134.2.f.s.379.2 4 36.31 odd 6
1134.2.f.s.757.2 4 36.7 odd 6
7938.2.a.bg.1.2 2 28.27 even 2
7938.2.a.bt.1.1 2 84.83 odd 2
9072.2.a.y.1.2 2 3.2 odd 2
9072.2.a.bp.1.1 2 1.1 even 1 trivial