Properties

Label 3744.2.a.z.1.3
Level $3744$
Weight $2$
Character 3744.1
Self dual yes
Analytic conductor $29.896$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3744,2,Mod(1,3744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, 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("3744.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.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: \(29.8959905168\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1248)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 3744.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.96239 q^{5} -3.35026 q^{7} +O(q^{10})\) \(q+2.96239 q^{5} -3.35026 q^{7} -1.61213 q^{11} +1.00000 q^{13} -2.00000 q^{17} +3.35026 q^{19} -6.70052 q^{23} +3.77575 q^{25} -2.00000 q^{29} -6.57452 q^{31} -9.92478 q^{35} +7.92478 q^{37} -6.96239 q^{41} +0.775746 q^{43} +2.38787 q^{47} +4.22425 q^{49} -11.9248 q^{53} -4.77575 q^{55} +0.312650 q^{59} +14.6253 q^{61} +2.96239 q^{65} -8.12601 q^{67} -4.31265 q^{71} +0.0752228 q^{73} +5.40105 q^{77} -12.0000 q^{79} -8.31265 q^{83} -5.92478 q^{85} -8.88717 q^{89} -3.35026 q^{91} +9.92478 q^{95} -7.92478 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{5} - 4 q^{11} + 3 q^{13} - 6 q^{17} + 13 q^{25} - 6 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{37} - 10 q^{41} + 4 q^{43} + 8 q^{47} + 11 q^{49} - 14 q^{53} - 16 q^{55} - 20 q^{59} + 2 q^{61} - 2 q^{65} - 16 q^{67} + 8 q^{71} + 22 q^{73} - 24 q^{77} - 36 q^{79} - 4 q^{83} + 4 q^{85} + 6 q^{89} + 8 q^{95} - 2 q^{97}+O(q^{100}) \) 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
\(5\) 2.96239 1.32482 0.662410 0.749141i \(-0.269534\pi\)
0.662410 + 0.749141i \(0.269534\pi\)
\(6\) 0 0
\(7\) −3.35026 −1.26628 −0.633140 0.774037i \(-0.718234\pi\)
−0.633140 + 0.774037i \(0.718234\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.61213 −0.486075 −0.243037 0.970017i \(-0.578144\pi\)
−0.243037 + 0.970017i \(0.578144\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 3.35026 0.768603 0.384301 0.923208i \(-0.374442\pi\)
0.384301 + 0.923208i \(0.374442\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.70052 −1.39716 −0.698578 0.715534i \(-0.746183\pi\)
−0.698578 + 0.715534i \(0.746183\pi\)
\(24\) 0 0
\(25\) 3.77575 0.755149
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −6.57452 −1.18082 −0.590409 0.807104i \(-0.701034\pi\)
−0.590409 + 0.807104i \(0.701034\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.92478 −1.67759
\(36\) 0 0
\(37\) 7.92478 1.30283 0.651413 0.758724i \(-0.274177\pi\)
0.651413 + 0.758724i \(0.274177\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.96239 −1.08734 −0.543671 0.839298i \(-0.682966\pi\)
−0.543671 + 0.839298i \(0.682966\pi\)
\(42\) 0 0
\(43\) 0.775746 0.118300 0.0591501 0.998249i \(-0.481161\pi\)
0.0591501 + 0.998249i \(0.481161\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.38787 0.348307 0.174154 0.984719i \(-0.444281\pi\)
0.174154 + 0.984719i \(0.444281\pi\)
\(48\) 0 0
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −11.9248 −1.63799 −0.818997 0.573798i \(-0.805470\pi\)
−0.818997 + 0.573798i \(0.805470\pi\)
\(54\) 0 0
\(55\) −4.77575 −0.643961
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.312650 0.0407036 0.0203518 0.999793i \(-0.493521\pi\)
0.0203518 + 0.999793i \(0.493521\pi\)
\(60\) 0 0
\(61\) 14.6253 1.87258 0.936289 0.351231i \(-0.114237\pi\)
0.936289 + 0.351231i \(0.114237\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.96239 0.367439
\(66\) 0 0
\(67\) −8.12601 −0.992750 −0.496375 0.868108i \(-0.665336\pi\)
−0.496375 + 0.868108i \(0.665336\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.31265 −0.511817 −0.255909 0.966701i \(-0.582375\pi\)
−0.255909 + 0.966701i \(0.582375\pi\)
\(72\) 0 0
\(73\) 0.0752228 0.00880416 0.00440208 0.999990i \(-0.498599\pi\)
0.00440208 + 0.999990i \(0.498599\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.40105 0.615506
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.31265 −0.912432 −0.456216 0.889869i \(-0.650796\pi\)
−0.456216 + 0.889869i \(0.650796\pi\)
\(84\) 0 0
\(85\) −5.92478 −0.642632
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.88717 −0.942038 −0.471019 0.882123i \(-0.656114\pi\)
−0.471019 + 0.882123i \(0.656114\pi\)
\(90\) 0 0
\(91\) −3.35026 −0.351203
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.92478 1.01826
\(96\) 0 0
\(97\) −7.92478 −0.804639 −0.402320 0.915499i \(-0.631796\pi\)
−0.402320 + 0.915499i \(0.631796\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.4010 −1.53246 −0.766231 0.642566i \(-0.777870\pi\)
−0.766231 + 0.642566i \(0.777870\pi\)
\(102\) 0 0
\(103\) 4.62530 0.455744 0.227872 0.973691i \(-0.426823\pi\)
0.227872 + 0.973691i \(0.426823\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.5501 1.40661 0.703305 0.710889i \(-0.251707\pi\)
0.703305 + 0.710889i \(0.251707\pi\)
\(108\) 0 0
\(109\) −15.4010 −1.47515 −0.737576 0.675264i \(-0.764030\pi\)
−0.737576 + 0.675264i \(0.764030\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.47627 0.891452 0.445726 0.895169i \(-0.352945\pi\)
0.445726 + 0.895169i \(0.352945\pi\)
\(114\) 0 0
\(115\) −19.8496 −1.85098
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.70052 0.614236
\(120\) 0 0
\(121\) −8.40105 −0.763732
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.62672 −0.324383
\(126\) 0 0
\(127\) −5.29948 −0.470252 −0.235126 0.971965i \(-0.575550\pi\)
−0.235126 + 0.971965i \(0.575550\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.14903 0.799355 0.399677 0.916656i \(-0.369122\pi\)
0.399677 + 0.916656i \(0.369122\pi\)
\(132\) 0 0
\(133\) −11.2243 −0.973266
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.513881 −0.0439038 −0.0219519 0.999759i \(-0.506988\pi\)
−0.0219519 + 0.999759i \(0.506988\pi\)
\(138\) 0 0
\(139\) 13.9248 1.18108 0.590542 0.807007i \(-0.298914\pi\)
0.590542 + 0.807007i \(0.298914\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.61213 −0.134813
\(144\) 0 0
\(145\) −5.92478 −0.492026
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.8872 1.05576 0.527879 0.849320i \(-0.322987\pi\)
0.527879 + 0.849320i \(0.322987\pi\)
\(150\) 0 0
\(151\) −13.2750 −1.08031 −0.540154 0.841566i \(-0.681634\pi\)
−0.540154 + 0.841566i \(0.681634\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −19.4763 −1.56437
\(156\) 0 0
\(157\) 21.0738 1.68187 0.840936 0.541134i \(-0.182005\pi\)
0.840936 + 0.541134i \(0.182005\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 22.4485 1.76919
\(162\) 0 0
\(163\) 21.9003 1.71537 0.857683 0.514178i \(-0.171903\pi\)
0.857683 + 0.514178i \(0.171903\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.4617 −1.35123 −0.675613 0.737257i \(-0.736121\pi\)
−0.675613 + 0.737257i \(0.736121\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −21.8496 −1.66119 −0.830595 0.556876i \(-0.812000\pi\)
−0.830595 + 0.556876i \(0.812000\pi\)
\(174\) 0 0
\(175\) −12.6497 −0.956230
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −21.9248 −1.63873 −0.819367 0.573269i \(-0.805675\pi\)
−0.819367 + 0.573269i \(0.805675\pi\)
\(180\) 0 0
\(181\) −18.6253 −1.38441 −0.692204 0.721702i \(-0.743360\pi\)
−0.692204 + 0.721702i \(0.743360\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 23.4763 1.72601
\(186\) 0 0
\(187\) 3.22425 0.235781
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.7743 1.57554 0.787768 0.615972i \(-0.211237\pi\)
0.787768 + 0.615972i \(0.211237\pi\)
\(192\) 0 0
\(193\) 19.2506 1.38569 0.692844 0.721087i \(-0.256357\pi\)
0.692844 + 0.721087i \(0.256357\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.51388 0.321601 0.160800 0.986987i \(-0.448592\pi\)
0.160800 + 0.986987i \(0.448592\pi\)
\(198\) 0 0
\(199\) 9.14903 0.648558 0.324279 0.945962i \(-0.394878\pi\)
0.324279 + 0.945962i \(0.394878\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.70052 0.470285
\(204\) 0 0
\(205\) −20.6253 −1.44053
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.40105 −0.373598
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.29806 0.156727
\(216\) 0 0
\(217\) 22.0263 1.49525
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −18.4241 −1.23377 −0.616883 0.787054i \(-0.711605\pi\)
−0.616883 + 0.787054i \(0.711605\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.61213 0.107001 0.0535003 0.998568i \(-0.482962\pi\)
0.0535003 + 0.998568i \(0.482962\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.0752228 −0.00492801 −0.00246400 0.999997i \(-0.500784\pi\)
−0.00246400 + 0.999997i \(0.500784\pi\)
\(234\) 0 0
\(235\) 7.07381 0.461444
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.08840 0.587880 0.293940 0.955824i \(-0.405033\pi\)
0.293940 + 0.955824i \(0.405033\pi\)
\(240\) 0 0
\(241\) 15.7743 1.01611 0.508057 0.861323i \(-0.330364\pi\)
0.508057 + 0.861323i \(0.330364\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.5139 0.799483
\(246\) 0 0
\(247\) 3.35026 0.213172
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.2243 −0.960946 −0.480473 0.877009i \(-0.659535\pi\)
−0.480473 + 0.877009i \(0.659535\pi\)
\(252\) 0 0
\(253\) 10.8021 0.679122
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.07522 0.254205 0.127103 0.991890i \(-0.459432\pi\)
0.127103 + 0.991890i \(0.459432\pi\)
\(258\) 0 0
\(259\) −26.5501 −1.64974
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.7757 −0.787786 −0.393893 0.919156i \(-0.628872\pi\)
−0.393893 + 0.919156i \(0.628872\pi\)
\(264\) 0 0
\(265\) −35.3258 −2.17005
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −21.8496 −1.33219 −0.666095 0.745867i \(-0.732036\pi\)
−0.666095 + 0.745867i \(0.732036\pi\)
\(270\) 0 0
\(271\) 0.499293 0.0303299 0.0151649 0.999885i \(-0.495173\pi\)
0.0151649 + 0.999885i \(0.495173\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.08698 −0.367059
\(276\) 0 0
\(277\) −13.8496 −0.832139 −0.416070 0.909333i \(-0.636593\pi\)
−0.416070 + 0.909333i \(0.636593\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.4387 0.861338 0.430669 0.902510i \(-0.358278\pi\)
0.430669 + 0.902510i \(0.358278\pi\)
\(282\) 0 0
\(283\) −6.55008 −0.389362 −0.194681 0.980867i \(-0.562367\pi\)
−0.194681 + 0.980867i \(0.562367\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 23.3258 1.37688
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.4387 0.843515 0.421758 0.906709i \(-0.361413\pi\)
0.421758 + 0.906709i \(0.361413\pi\)
\(294\) 0 0
\(295\) 0.926192 0.0539250
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.70052 −0.387501
\(300\) 0 0
\(301\) −2.59895 −0.149801
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 43.3258 2.48083
\(306\) 0 0
\(307\) −15.8740 −0.905977 −0.452988 0.891516i \(-0.649642\pi\)
−0.452988 + 0.891516i \(0.649642\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.25202 0.467929 0.233964 0.972245i \(-0.424830\pi\)
0.233964 + 0.972245i \(0.424830\pi\)
\(312\) 0 0
\(313\) −3.40105 −0.192239 −0.0961193 0.995370i \(-0.530643\pi\)
−0.0961193 + 0.995370i \(0.530643\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.4387 −0.586293 −0.293147 0.956067i \(-0.594702\pi\)
−0.293147 + 0.956067i \(0.594702\pi\)
\(318\) 0 0
\(319\) 3.22425 0.180524
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.70052 −0.372827
\(324\) 0 0
\(325\) 3.77575 0.209441
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −6.57452 −0.361368 −0.180684 0.983541i \(-0.557831\pi\)
−0.180684 + 0.983541i \(0.557831\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −24.0724 −1.31522
\(336\) 0 0
\(337\) 31.4010 1.71052 0.855262 0.518196i \(-0.173396\pi\)
0.855262 + 0.518196i \(0.173396\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.5990 0.573965
\(342\) 0 0
\(343\) 9.29948 0.502125
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.2506 −1.14079 −0.570396 0.821370i \(-0.693210\pi\)
−0.570396 + 0.821370i \(0.693210\pi\)
\(348\) 0 0
\(349\) −19.9248 −1.06655 −0.533274 0.845942i \(-0.679039\pi\)
−0.533274 + 0.845942i \(0.679039\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.43866 0.342695 0.171348 0.985211i \(-0.445188\pi\)
0.171348 + 0.985211i \(0.445188\pi\)
\(354\) 0 0
\(355\) −12.7757 −0.678066
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.5647 −1.50759 −0.753793 0.657112i \(-0.771778\pi\)
−0.753793 + 0.657112i \(0.771778\pi\)
\(360\) 0 0
\(361\) −7.77575 −0.409250
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.222839 0.0116639
\(366\) 0 0
\(367\) 19.3258 1.00880 0.504400 0.863470i \(-0.331714\pi\)
0.504400 + 0.863470i \(0.331714\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 39.9511 2.07416
\(372\) 0 0
\(373\) −25.6991 −1.33065 −0.665325 0.746554i \(-0.731707\pi\)
−0.665325 + 0.746554i \(0.731707\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 24.7513 1.27139 0.635695 0.771941i \(-0.280714\pi\)
0.635695 + 0.771941i \(0.280714\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.6121 1.10433 0.552164 0.833735i \(-0.313802\pi\)
0.552164 + 0.833735i \(0.313802\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −25.3258 −1.28407 −0.642035 0.766675i \(-0.721910\pi\)
−0.642035 + 0.766675i \(0.721910\pi\)
\(390\) 0 0
\(391\) 13.4010 0.677720
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −35.5487 −1.78865
\(396\) 0 0
\(397\) −3.92478 −0.196979 −0.0984895 0.995138i \(-0.531401\pi\)
−0.0984895 + 0.995138i \(0.531401\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.4109 0.869459 0.434729 0.900561i \(-0.356844\pi\)
0.434729 + 0.900561i \(0.356844\pi\)
\(402\) 0 0
\(403\) −6.57452 −0.327500
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12.7757 −0.633270
\(408\) 0 0
\(409\) −15.9248 −0.787430 −0.393715 0.919233i \(-0.628810\pi\)
−0.393715 + 0.919233i \(0.628810\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.04746 −0.0515422
\(414\) 0 0
\(415\) −24.6253 −1.20881
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 36.9986 1.80750 0.903750 0.428062i \(-0.140803\pi\)
0.903750 + 0.428062i \(0.140803\pi\)
\(420\) 0 0
\(421\) −39.4010 −1.92029 −0.960145 0.279503i \(-0.909830\pi\)
−0.960145 + 0.279503i \(0.909830\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.55149 −0.366301
\(426\) 0 0
\(427\) −48.9986 −2.37121
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −35.6385 −1.71664 −0.858322 0.513111i \(-0.828493\pi\)
−0.858322 + 0.513111i \(0.828493\pi\)
\(432\) 0 0
\(433\) 33.0738 1.58943 0.794713 0.606986i \(-0.207621\pi\)
0.794713 + 0.606986i \(0.207621\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −22.4485 −1.07386
\(438\) 0 0
\(439\) −32.4749 −1.54994 −0.774970 0.631998i \(-0.782235\pi\)
−0.774970 + 0.631998i \(0.782235\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.14903 0.434684 0.217342 0.976096i \(-0.430261\pi\)
0.217342 + 0.976096i \(0.430261\pi\)
\(444\) 0 0
\(445\) −26.3272 −1.24803
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.3634 1.14978 0.574891 0.818230i \(-0.305045\pi\)
0.574891 + 0.818230i \(0.305045\pi\)
\(450\) 0 0
\(451\) 11.2243 0.528529
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9.92478 −0.465281
\(456\) 0 0
\(457\) 29.4763 1.37884 0.689421 0.724361i \(-0.257865\pi\)
0.689421 + 0.724361i \(0.257865\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.11283 0.331278 0.165639 0.986186i \(-0.447031\pi\)
0.165639 + 0.986186i \(0.447031\pi\)
\(462\) 0 0
\(463\) 42.7974 1.98896 0.994481 0.104918i \(-0.0334580\pi\)
0.994481 + 0.104918i \(0.0334580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.6253 0.584229 0.292115 0.956383i \(-0.405641\pi\)
0.292115 + 0.956383i \(0.405641\pi\)
\(468\) 0 0
\(469\) 27.2243 1.25710
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.25060 −0.0575027
\(474\) 0 0
\(475\) 12.6497 0.580410
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −34.3390 −1.56899 −0.784494 0.620136i \(-0.787077\pi\)
−0.784494 + 0.620136i \(0.787077\pi\)
\(480\) 0 0
\(481\) 7.92478 0.361339
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −23.4763 −1.06600
\(486\) 0 0
\(487\) −21.5271 −0.975484 −0.487742 0.872988i \(-0.662179\pi\)
−0.487742 + 0.872988i \(0.662179\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.0263 1.17455 0.587276 0.809387i \(-0.300200\pi\)
0.587276 + 0.809387i \(0.300200\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.4485 0.648104
\(498\) 0 0
\(499\) 1.42548 0.0638135 0.0319067 0.999491i \(-0.489842\pi\)
0.0319067 + 0.999491i \(0.489842\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.67276 −0.0745847 −0.0372924 0.999304i \(-0.511873\pi\)
−0.0372924 + 0.999304i \(0.511873\pi\)
\(504\) 0 0
\(505\) −45.6239 −2.03024
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.58910 −0.292057 −0.146028 0.989280i \(-0.546649\pi\)
−0.146028 + 0.989280i \(0.546649\pi\)
\(510\) 0 0
\(511\) −0.252016 −0.0111485
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13.7019 0.603780
\(516\) 0 0
\(517\) −3.84955 −0.169303
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −26.3733 −1.15543 −0.577717 0.816237i \(-0.696056\pi\)
−0.577717 + 0.816237i \(0.696056\pi\)
\(522\) 0 0
\(523\) 3.32582 0.145428 0.0727141 0.997353i \(-0.476834\pi\)
0.0727141 + 0.997353i \(0.476834\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13.1490 0.572781
\(528\) 0 0
\(529\) 21.8970 0.952044
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.96239 −0.301575
\(534\) 0 0
\(535\) 43.1030 1.86350
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.81003 −0.293329
\(540\) 0 0
\(541\) 33.8496 1.45531 0.727653 0.685945i \(-0.240611\pi\)
0.727653 + 0.685945i \(0.240611\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −45.6239 −1.95431
\(546\) 0 0
\(547\) −25.7743 −1.10203 −0.551015 0.834495i \(-0.685759\pi\)
−0.551015 + 0.834495i \(0.685759\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.70052 −0.285452
\(552\) 0 0
\(553\) 40.2031 1.70961
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.88717 0.207076 0.103538 0.994626i \(-0.466984\pi\)
0.103538 + 0.994626i \(0.466984\pi\)
\(558\) 0 0
\(559\) 0.775746 0.0328106
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.2228 1.35803 0.679015 0.734124i \(-0.262407\pi\)
0.679015 + 0.734124i \(0.262407\pi\)
\(564\) 0 0
\(565\) 28.0724 1.18101
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.2228 0.763941 0.381971 0.924174i \(-0.375246\pi\)
0.381971 + 0.924174i \(0.375246\pi\)
\(570\) 0 0
\(571\) 36.9986 1.54834 0.774171 0.632976i \(-0.218167\pi\)
0.774171 + 0.632976i \(0.218167\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −25.2995 −1.05506
\(576\) 0 0
\(577\) −1.10299 −0.0459179 −0.0229589 0.999736i \(-0.507309\pi\)
−0.0229589 + 0.999736i \(0.507309\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 27.8496 1.15539
\(582\) 0 0
\(583\) 19.2243 0.796187
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 37.0884 1.53080 0.765401 0.643554i \(-0.222541\pi\)
0.765401 + 0.643554i \(0.222541\pi\)
\(588\) 0 0
\(589\) −22.0263 −0.907580
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −39.8397 −1.63602 −0.818010 0.575204i \(-0.804923\pi\)
−0.818010 + 0.575204i \(0.804923\pi\)
\(594\) 0 0
\(595\) 19.8496 0.813752
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.0263 0.899972 0.449986 0.893036i \(-0.351429\pi\)
0.449986 + 0.893036i \(0.351429\pi\)
\(600\) 0 0
\(601\) −35.4010 −1.44404 −0.722019 0.691873i \(-0.756786\pi\)
−0.722019 + 0.691873i \(0.756786\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −24.8872 −1.01181
\(606\) 0 0
\(607\) 10.0752 0.408941 0.204470 0.978873i \(-0.434453\pi\)
0.204470 + 0.978873i \(0.434453\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.38787 0.0966030
\(612\) 0 0
\(613\) 14.3733 0.580532 0.290266 0.956946i \(-0.406256\pi\)
0.290266 + 0.956946i \(0.406256\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −38.6615 −1.55645 −0.778227 0.627984i \(-0.783881\pi\)
−0.778227 + 0.627984i \(0.783881\pi\)
\(618\) 0 0
\(619\) 8.12601 0.326612 0.163306 0.986575i \(-0.447784\pi\)
0.163306 + 0.986575i \(0.447784\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 29.7743 1.19288
\(624\) 0 0
\(625\) −29.6225 −1.18490
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −15.8496 −0.631963
\(630\) 0 0
\(631\) 22.5745 0.898677 0.449339 0.893362i \(-0.351660\pi\)
0.449339 + 0.893362i \(0.351660\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −15.6991 −0.623000
\(636\) 0 0
\(637\) 4.22425 0.167371
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.8496 0.705015 0.352508 0.935809i \(-0.385329\pi\)
0.352508 + 0.935809i \(0.385329\pi\)
\(642\) 0 0
\(643\) 2.47295 0.0975234 0.0487617 0.998810i \(-0.484473\pi\)
0.0487617 + 0.998810i \(0.484473\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22.9525 −0.902357 −0.451179 0.892434i \(-0.648996\pi\)
−0.451179 + 0.892434i \(0.648996\pi\)
\(648\) 0 0
\(649\) −0.504032 −0.0197850
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.4010 0.759222 0.379611 0.925146i \(-0.376058\pi\)
0.379611 + 0.925146i \(0.376058\pi\)
\(654\) 0 0
\(655\) 27.1030 1.05900
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 15.8496 0.617411 0.308705 0.951158i \(-0.400104\pi\)
0.308705 + 0.951158i \(0.400104\pi\)
\(660\) 0 0
\(661\) 30.8773 1.20099 0.600494 0.799629i \(-0.294971\pi\)
0.600494 + 0.799629i \(0.294971\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −33.2506 −1.28940
\(666\) 0 0
\(667\) 13.4010 0.518891
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −23.5778 −0.910212
\(672\) 0 0
\(673\) 16.3272 0.629369 0.314684 0.949196i \(-0.398101\pi\)
0.314684 + 0.949196i \(0.398101\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −32.0752 −1.23275 −0.616375 0.787452i \(-0.711400\pi\)
−0.616375 + 0.787452i \(0.711400\pi\)
\(678\) 0 0
\(679\) 26.5501 1.01890
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.1344 0.579103 0.289552 0.957162i \(-0.406494\pi\)
0.289552 + 0.957162i \(0.406494\pi\)
\(684\) 0 0
\(685\) −1.52232 −0.0581647
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −11.9248 −0.454298
\(690\) 0 0
\(691\) 19.7235 0.750319 0.375160 0.926960i \(-0.377588\pi\)
0.375160 + 0.926960i \(0.377588\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 41.2506 1.56472
\(696\) 0 0
\(697\) 13.9248 0.527439
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 7.92478 0.299315 0.149657 0.988738i \(-0.452183\pi\)
0.149657 + 0.988738i \(0.452183\pi\)
\(702\) 0 0
\(703\) 26.5501 1.00136
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 51.5975 1.94053
\(708\) 0 0
\(709\) 15.5515 0.584049 0.292024 0.956411i \(-0.405671\pi\)
0.292024 + 0.956411i \(0.405671\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 44.0527 1.64979
\(714\) 0 0
\(715\) −4.77575 −0.178603
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.5501 0.691801 0.345901 0.938271i \(-0.387573\pi\)
0.345901 + 0.938271i \(0.387573\pi\)
\(720\) 0 0
\(721\) −15.4960 −0.577100
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.55149 −0.280455
\(726\) 0 0
\(727\) 11.9511 0.443243 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.55149 −0.0573840
\(732\) 0 0
\(733\) 8.80209 0.325113 0.162556 0.986699i \(-0.448026\pi\)
0.162556 + 0.986699i \(0.448026\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.1002 0.482550
\(738\) 0 0
\(739\) 23.8251 0.876421 0.438211 0.898872i \(-0.355612\pi\)
0.438211 + 0.898872i \(0.355612\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.53690 0.276502 0.138251 0.990397i \(-0.455852\pi\)
0.138251 + 0.990397i \(0.455852\pi\)
\(744\) 0 0
\(745\) 38.1768 1.39869
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −48.7466 −1.78116
\(750\) 0 0
\(751\) −44.6253 −1.62840 −0.814200 0.580584i \(-0.802824\pi\)
−0.814200 + 0.580584i \(0.802824\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −39.3258 −1.43121
\(756\) 0 0
\(757\) −11.6728 −0.424254 −0.212127 0.977242i \(-0.568039\pi\)
−0.212127 + 0.977242i \(0.568039\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.8397 0.719189 0.359594 0.933109i \(-0.382915\pi\)
0.359594 + 0.933109i \(0.382915\pi\)
\(762\) 0 0
\(763\) 51.5975 1.86796
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.312650 0.0112891
\(768\) 0 0
\(769\) −35.1002 −1.26574 −0.632872 0.774256i \(-0.718124\pi\)
−0.632872 + 0.774256i \(0.718124\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.2882 1.23326 0.616631 0.787253i \(-0.288497\pi\)
0.616631 + 0.787253i \(0.288497\pi\)
\(774\) 0 0
\(775\) −24.8237 −0.891694
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −23.3258 −0.835734
\(780\) 0 0
\(781\) 6.95254 0.248781
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 62.4288 2.22818
\(786\) 0 0
\(787\) 31.8251 1.13444 0.567221 0.823565i \(-0.308018\pi\)
0.567221 + 0.823565i \(0.308018\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −31.7480 −1.12883
\(792\) 0 0
\(793\) 14.6253 0.519360
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.0752 −0.569414 −0.284707 0.958615i \(-0.591896\pi\)
−0.284707 + 0.958615i \(0.591896\pi\)
\(798\) 0 0
\(799\) −4.77575 −0.168954
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.121269 −0.00427948
\(804\) 0 0
\(805\) 66.5012 2.34386
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.32582 −0.0466135 −0.0233067 0.999728i \(-0.507419\pi\)
−0.0233067 + 0.999728i \(0.507419\pi\)
\(810\) 0 0
\(811\) −12.3488 −0.433627 −0.216813 0.976213i \(-0.569566\pi\)
−0.216813 + 0.976213i \(0.569566\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 64.8773 2.27255
\(816\) 0 0
\(817\) 2.59895 0.0909259
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 54.1378 1.88942 0.944711 0.327905i \(-0.106343\pi\)
0.944711 + 0.327905i \(0.106343\pi\)
\(822\) 0 0
\(823\) 15.7283 0.548254 0.274127 0.961694i \(-0.411611\pi\)
0.274127 + 0.961694i \(0.411611\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.4880 0.955852 0.477926 0.878400i \(-0.341389\pi\)
0.477926 + 0.878400i \(0.341389\pi\)
\(828\) 0 0
\(829\) 15.6728 0.544337 0.272169 0.962250i \(-0.412259\pi\)
0.272169 + 0.962250i \(0.412259\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −8.44851 −0.292723
\(834\) 0 0
\(835\) −51.7283 −1.79013
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 29.8641 1.03102 0.515512 0.856882i \(-0.327602\pi\)
0.515512 + 0.856882i \(0.327602\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.96239 0.101909
\(846\) 0 0
\(847\) 28.1457 0.967098
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −53.1002 −1.82025
\(852\) 0 0
\(853\) 34.2228 1.17177 0.585884 0.810395i \(-0.300747\pi\)
0.585884 + 0.810395i \(0.300747\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −17.6267 −0.602117 −0.301059 0.953606i \(-0.597340\pi\)
−0.301059 + 0.953606i \(0.597340\pi\)
\(858\) 0 0
\(859\) −6.80209 −0.232084 −0.116042 0.993244i \(-0.537021\pi\)
−0.116042 + 0.993244i \(0.537021\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 32.4142 1.10339 0.551696 0.834045i \(-0.313981\pi\)
0.551696 + 0.834045i \(0.313981\pi\)
\(864\) 0 0
\(865\) −64.7269 −2.20078
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 19.3455 0.656252
\(870\) 0 0
\(871\) −8.12601 −0.275339
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.1504 0.410760
\(876\) 0 0
\(877\) −26.8773 −0.907582 −0.453791 0.891108i \(-0.649929\pi\)
−0.453791 + 0.891108i \(0.649929\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 33.4763 1.12784 0.563922 0.825828i \(-0.309292\pi\)
0.563922 + 0.825828i \(0.309292\pi\)
\(882\) 0 0
\(883\) −26.3996 −0.888418 −0.444209 0.895923i \(-0.646515\pi\)
−0.444209 + 0.895923i \(0.646515\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 13.7743 0.462497 0.231248 0.972895i \(-0.425719\pi\)
0.231248 + 0.972895i \(0.425719\pi\)
\(888\) 0 0
\(889\) 17.7546 0.595471
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) −64.9497 −2.17103
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.1490 0.438545
\(900\) 0 0
\(901\) 23.8496 0.794544
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −55.1754 −1.83409
\(906\) 0 0
\(907\) −3.74798 −0.124450 −0.0622249 0.998062i \(-0.519820\pi\)
−0.0622249 + 0.998062i \(0.519820\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.42075 0.0470714 0.0235357 0.999723i \(-0.492508\pi\)
0.0235357 + 0.999723i \(0.492508\pi\)
\(912\) 0 0
\(913\) 13.4010 0.443510
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −30.6516 −1.01221
\(918\) 0 0
\(919\) 55.5487 1.83238 0.916191 0.400743i \(-0.131248\pi\)
0.916191 + 0.400743i \(0.131248\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.31265 −0.141953
\(924\) 0 0
\(925\) 29.9219 0.983828
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −49.4636 −1.62285 −0.811424 0.584458i \(-0.801307\pi\)
−0.811424 + 0.584458i \(0.801307\pi\)
\(930\) 0 0
\(931\) 14.1524 0.463825
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9.55149 0.312367
\(936\) 0 0
\(937\) −5.07381 −0.165754 −0.0828770 0.996560i \(-0.526411\pi\)
−0.0828770 + 0.996560i \(0.526411\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −18.8119 −0.613252 −0.306626 0.951830i \(-0.599200\pi\)
−0.306626 + 0.951830i \(0.599200\pi\)
\(942\) 0 0
\(943\) 46.6516 1.51919
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.2652 0.496052 0.248026 0.968753i \(-0.420218\pi\)
0.248026 + 0.968753i \(0.420218\pi\)
\(948\) 0 0
\(949\) 0.0752228 0.00244183
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.47627 −0.177394 −0.0886969 0.996059i \(-0.528270\pi\)
−0.0886969 + 0.996059i \(0.528270\pi\)
\(954\) 0 0
\(955\) 64.5040 2.08730
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.72164 0.0555945
\(960\) 0 0
\(961\) 12.2243 0.394331
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 57.0278 1.83579
\(966\) 0 0
\(967\) 46.0216 1.47996 0.739978 0.672632i \(-0.234836\pi\)
0.739978 + 0.672632i \(0.234836\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 18.4485 0.592041 0.296020 0.955182i \(-0.404340\pi\)
0.296020 + 0.955182i \(0.404340\pi\)
\(972\) 0 0
\(973\) −46.6516 −1.49558
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.160295 0.00512828 0.00256414 0.999997i \(-0.499184\pi\)
0.00256414 + 0.999997i \(0.499184\pi\)
\(978\) 0 0
\(979\) 14.3272 0.457901
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 28.8167 0.919109 0.459555 0.888149i \(-0.348009\pi\)
0.459555 + 0.888149i \(0.348009\pi\)
\(984\) 0 0
\(985\) 13.3719 0.426063
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.19791 −0.165284
\(990\) 0 0
\(991\) −40.2228 −1.27772 −0.638860 0.769323i \(-0.720594\pi\)
−0.638860 + 0.769323i \(0.720594\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 27.1030 0.859222
\(996\) 0 0
\(997\) −12.8021 −0.405446 −0.202723 0.979236i \(-0.564979\pi\)
−0.202723 + 0.979236i \(0.564979\pi\)
\(998\) 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 3744.2.a.z.1.3 3
3.2 odd 2 1248.2.a.p.1.1 yes 3
4.3 odd 2 3744.2.a.ba.1.3 3
8.3 odd 2 7488.2.a.cx.1.1 3
8.5 even 2 7488.2.a.cy.1.1 3
12.11 even 2 1248.2.a.o.1.1 3
24.5 odd 2 2496.2.a.bk.1.3 3
24.11 even 2 2496.2.a.bl.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.o.1.1 3 12.11 even 2
1248.2.a.p.1.1 yes 3 3.2 odd 2
2496.2.a.bk.1.3 3 24.5 odd 2
2496.2.a.bl.1.3 3 24.11 even 2
3744.2.a.z.1.3 3 1.1 even 1 trivial
3744.2.a.ba.1.3 3 4.3 odd 2
7488.2.a.cx.1.1 3 8.3 odd 2
7488.2.a.cy.1.1 3 8.5 even 2